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Active delay output feedback control for high performance flexible servo systems
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Active delay output feedback control for high performance flexible servo systems
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ACTIVE DELAY OUTPUT FEEDBACK CONTROL FOR HIGH PERFORMANCE FLEXIBLE SERVO SYSTEMS by Zheng Liu A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (MECHANICAL ENGINEERING) May 2003 Copyright 2003 Zheng Liu UNIVERSITY OF SOUTHERN CALIFORNIA THE GRADUATE SCHOOL UNIVERSITY PARK LOS ANGELES, CALIFORNIA 90089-1695 This dissertation, written by under the direction of hl..?_ dissertation committee, and approved by all its members, has been presented to and accepted by the Director of Graduate and Professional Programs, in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Director Date August 12, 2003 Chair ii Dedication To my family iii Acknowledgements I was lucky to have Dr. Bingen Yang as my Ph.D. adviser. He is one of the most creative and smartest people that I have ever met. I have learned so much from him. Whenever I was on a wrong track, he helped me find the right approach with his insightful comments. I sincerely thank him for his guidance, encouragement, patience, support, and friendship over the past five years. Next I would like to gratefully acknowledge the support of Dr. Henryk Flashner for his extensive knowledge, which greatly ease my literature survey, and for his helpful comments and discussion, which makes me get through a lot of difficulties. Special thanks to my colleagues in Dynamic Systems Lab for their help, support and friendship. iv Appreciation Great thanks are to be given to Nation Science Foundation for the support of research assistantship for flexible systems dynamics and control. And deep appreciation has been given to Aerospace and Mechanical Engineering Department of USC for the support of teaching assistantship in dynamical systems control. v Table of Contents Dedication 11 Acknowledgements 111 Appreciation 1v List of Tables v11 List of Figures vm Abstract xvi 1. Introduction 1 1.1 Motivations and Purposes 1 1.2 Problem Definitions 6 1.3 Overview ofExisting Design Approach 12 1.4 Outline and Scope 18 2. Review of Stability and Performance Criterion for Delayed Feedback Systems 23 2.1 Delay-induced Infinite Dimensional Systems 23 2.2 Review of Stability and Performance Criterion for Feedback Systems 27 2.3 Phase Design Method 35 2.4 Design Examples of Phase Design Method 53 2.5 Weakness of Existing Stability and Performance Criteria 62 3. Delay Transform Technique and Finite Design Region Algorithms 65 3 .1 Finite Design Region Principle 65 3 .2 Delay Transform Technique 78 3 .A Proof of Lemma 3 .1.1 85 3.B Proof of Lemma 3.1.2 87 3 .C Proof of Delay Transforms 87 3 .D. Characteristics of some delay and delay-free control elements 89 vi 4. DTT & FDR Control Algorithm 93 4 .1 Full State Feedback Realization: Analysis and Design 93 4.2 Advantages ofDTT & FDR Method 100 4.3 Systematic Design ofDTT & FDR Controller 114 4.4 Discrete-time DTT & FDR design 118 5. Delay-induced filtering and transient response compensation 120 5 .1 Introduction 120 5 .2 Delay-induced Filtering 121 5.3 Initial Function Compensation 124 5 .4 Illustrative examples 127 6. Delay-based internal model modification (DIMJ\1) technique 140 6.1 Internal Model Modification 141 6.2 Delay-based LTR Method 152 6.3 Illustrative Example 153 7. Simulations and results 160 7 .1 Non-collocated Control of Multi-DOF Flexible Systems using DTT & FDR Method 160 7.2 2-DOF flexible robot arm control by DIMM technique 172 8. Non-collocated Control of Single-link Robot Arm 178 8.1 Model Description 178 8.2 Controller Design 181 9. Experiment and results 189 9 .1 Experiment Descriptions 189 9 .2 Controller Design: Continuous and Discrete 194 9.3 Experimental Results 198 10. Conclusion 200 10 .1 Summary 200 10.2 Open research problems for High Performance Flexible Servo Control 201 Reference List 205 List of Tables Table 1. 4 .1 Disadvantages of the existing control algorithms Table 2.5.1. Weaknesses of existing delay-induced stability-performance Criteria Vll 19 63 viii List of Figures Figure 1.1.1 Block diagram of servo system of a hard disk drive. 2 Figure 1.1.2. Operation modes of the hard disk drive servomechanism 2 Figure 1.2.1. Controller-plant configuration for output and full state feedback 8 Figure 1.2.2. Flexible structure models 11 Figure 2.1.1 Block diagram of the above feedback system 24 Figure 2.1.2 Root-locus plot for the system shown in figure 2.1.1 25 Figure 2.2.1. Feedback configuration of control system 27 Figure 2.3.1. Feedback control of a 2-DOF spring-mass flexible system 38 Figure 2.3 .2. Polar plot of l-be-fwT 43 Figure 2.3.3. Phase plot of l-be-fwT element 45 Figure 2.3 .4. Frequency response of 1- be-sr element 49 Figure 2.3.5. Frequency response of 1 +be-sT element 50 Figure 2.3.6 Frequency response of e- 2 sT +ce-sT +d element (I) 51 Figure 2.3.7. Frequency response of e- 2 sT +ce-sT +d element (II) 52 Figure 2.3.8. Frequency response of e- 2 sT +ce-sT +d element (III) 52 Figure 2.4.1. Phase plot of the open-loop plant with phase crossover 54 Figure 2.4.2. Phase design boundaries and some desirable phase plots 54 Figure 2.4.3. Phase plot of full state feedback controller and phase design boundaries. 56 ix Figure 2.4.4. Frequency response ofloop transfer function for full state feedback system 56 Figure 2.4.5. Closed-loop frequency response for full state feedback system. 57 Figure 2.4.6. Phase plot of the delay-induced and lead-lag compensator based feedback with phase design boundaries. 5 8 Figure 2.4.7. Frequency response of the loop transfer function for the delay-induced and lead-lag compensator based feedback 59 Figure 2.4.8. Comparisons of closed-loop frequency responses of full state feedback and delay-induced lead-lag compensator based feedback 59 Figure 2.4.9. Phase plot of the delay-induced feedback with phase design boundaries. 60 Figure 2.4.10. Frequency response of the loop transfer function for the delay-induced and lead-lag compensator based feedback 61 Figure 2.4.11. Comparisons of closed-loop frequency responses of full state feedback and delay-induced lead-lag compensator based feedback 61 Figure 2.4.12. Frequency Response for full state feedback controller and delay- induced feedback controller. 62 Figure 3 .1.1. Controller-plant configuration 66 Figure 3 .1.2. Root-loci of delay-induced infinite dimensional system 68 Figure 3.1.3. Example of monotonically decreasing magnitude envelope 74 Figure 3.1.4. Diagram of finite design region principle 77 x Figure 3.1.5. Diagram of finite design region principle with inside isolation region 77 Figure 3.2.1 Finite design region diagram toDdt or PDdt -type controllers 82 Figure 3.2.2 Finite design region diagram to PD~tI and PD~n type controllers 83 Figure 3.2.3. Frequency response of PD (s+lOO) and PDdt-type µ(I-be-sT) controller DRC: rod=2nx100 rad/sec, DRI: ro 1 =2nx10000 rad/sec, T=O.OOOls. 83 Figure 3 .2.4. Frequency response of both ( s 2 +s+100) and PD~ti -type µ 52 + s + lOO (1-b 1 e-sT )(1-b 2 e-sT) controller DRC: rod =2nx 10 rad/sec, DRI: ro1 (s + lO)(s + 20) =2nx1000 rad/sec, T=0.001 sec. 84 Figure 4 .1.1. Controller-plant configuration for output feedback and full state feedback 94 Figure 4.1.2. Loop transfer function realization of the full state feedback in terms of output feedback 95 Figure 4.1.3. Non-causal realization ofloop transfer function realization of full state feedback in terms of output feedback 95 Figure 4.1.4. Non-collocated control of2-DOF flexible mass-spring systems 96 Figure 4.2.1 Frequency Response of PD controller and DTT controller 105 Figure 4.2.2. Control input of PD control 105 Figure 4.2.3. Control input ofDTT control 106 Figure 4.2.4. Frequency Response of PD, PD at and Ddt (PMD) controller with T= 1 xi 109 Figure 4.2.5. Frequency Response of PD, PD at and Ddt (PMD) controller with T=0.1 110 Figure 4.2.6. Frequency Response of PD+D 2 , PD;n and Ddt (PMD) controller with T=0.05 Figure 5.3 .1. Delay-induced output feedback configuration Figure 5.3.2. Delay-based output feedback configuration with Initial function 111 125 compensation 126 Figure 5.4.1. Frequency Response of (a) LTR and (b) delay-based controller 128 Figure 5.4.2. Loop Frequency Response of (a) LTR and (b) Delay-based output feedback systems 128 Figure 5.4.3. Closed-loop Frequency Response of (a) LTR and (b) Delay-based output feedback systems with (a) BW=900rad/sec and (b) BW=1405rad/sec. 129 Figure 5.4.4. Step Response of (a) LTR and (b) IFC delay-based output feedback systems 130 Figure 5.4.5. Control efforts of(a) LTR and (b) IFC delay-based output feedback systems 131 Figure 5.4.6. Steady-state error for (a) LTR and (b) delay-based output feedback with noises 131 xii Figure 5.4.7. Steady-state control power for (a) LTR and (b) delay-induced output feedback with noises 132 Figure 5.4.8. Step Responses of (a) full state Feedback and (b) delay-based output feedback 134 Figure 5.4.9. Non-collocated control of 2-DOF flexible mass-spring systems. 135 Figure 5.4.10. Closed-loop frequency responses of (a) full state feedback and (b) delay-induced output feedback 136 Figure 5 .4. 11. Open-loop frequency responses of (a) full state feedback and (b) delay-induced output feedback 13 7 Figure 5.4.12. Step response for (a) full state feedback and (b) delay-induced output feedback 137 Figure 5.4.13. Control efforts for (a) full state feedback and (b) delay-induced output feedback 138 Figure 6.1.1. Augmented plant-based internal model modification 141 Figure 6.1.2. Augmented controller-based internal model modification 143 Figure 6.3.1. Frequency Response ofLTR and delay-induced controller 156 Figure 6.3.2. Loop Frequency Response of (a) LTR and (b) delay-induced feedback systems 156 Figure 6.3.3. Closed-loop Frequency Response of (a) LTR and (b) DTT induced feedback systems with (a) BW=900rad/sec and (b) BW=l405rad/sec. 157 Figure 6.3.4. Square of steady-state error for (a) LTR and (b) delay-induced output feedback with noises 157 xiii Figure 6.3.5. Steady-state control power for (a) LTR and (b) delay-induced output feedback with noises 158 Figure 6.3.6. Step Response of (a) LTR and (b) IFC based delay-induced output feedback systems 158 Figure 7.1.1.1 The Non-collocated control ofN-DOF spring-mass flexible System 161 Figure 7 .1.1.2. Controller-plant configuration for output feedback and full state feedback 162 Figure 7 .1.2.1. Frequency response of full state feedback, LTR output feedback and DTT &FDR output feedback controller 165 Figure 7.1.2.2. Frequency response of the loop transfer function of full state feedback, LTR output feedback and DTT &FDR output feedback systems 166 Figure 7.1.2.3. Frequency response of the closed-loop transfer function of full state feedback, LTR output feedback and DTT &FDR output feedback 167 Figure 7.1.2.4 Frequency Response ofloop transfer function with LQR full state feedback and zero placement via delay feedback. 171 Figure 7.1.2.5 Frequency Response of Closed-loop system with LQR full state feedback and zero placement via delay feedback. 171 Figure 7 .2.1.1. 2-DOF non-minimum phase flexible robot arm system 172 Figure 7.2.2.1. Open-loop frequency responses for full state feedback and IM:M based delay-induced output feedback systems 176 xiv Figure 7 .2.2.2. Closed-loop frequency responses for full state feedback and IMM based delay-induced output feedback systems 176 Figure 7 .2.2.3 Step Response of full state feedback, IMM-LTR and delay-based IMM-LTR feedback 177 Figure 8.1.1. A single-link flexible robot arm system 179 Figure 8.2.1. Bode plot of the truncated beam transfer function 185 Figure 8.2.2. Bode plot for the delay-based loop transfer function (GM=20dB and PM=75 degrees) 185 Figure 8.2.3. Closed-loop gain frequency response (BW>>1500 rad/sec) 186 Figure 8.2.4. Matlab Simulink Diagram for delay-based loop transfer recovery with internal model modification. 187 Figure 8.2.5. Step response of the delay-based loop transfer recovery with internal model modification output feedback system from Matlab Simulink. 187 Figure 8.2.6. Step responses for delay-based control system and theoretical full state feedback system. 188 Figure 9 .1.1 The Torsional Control System 190 Figure 9 .1.2 Closed-loop configuration for full state feedback (continuous) 192 Figure 9.1.3 Closed-loop configuration for delay transformed control 192 Figure 9.2.1 Desirable root-locus for the full state feedback system. 195 Figure 9.1.2 Closed-loop configuration for full state feedback (discrete) 196 Figure 9.3.1 Step responses for both full state feedback and IVC based delay transformed output feedback. 198 Figure 9.3.2 Control efforts for both full state feedback and IVC based delay transformed output feedback xv 199 XVI Abstract In this dissertation, the active delay output feedback control techniques are presented for high-performance flexible servo systems. Among these delay related control techniques, the delay transform with finite design region principle play as the fundamentals of the generalized delay-based infinite dimensional control algorithms. The former is a way to realize any improper polynomial transfer function, which is physically unrealizable in conventional finite dimensional domain. And the latter serves as the systematic and general closed loop stability and performance criteria. Since the improper controller can be realized through delay transform, the full state feedback control is physically realizable by delay-based output feedback with equivalent performance. And this technique is proved to have much better performance than the conventional observer-based or compensator-based output feedback. However, due to the introduction of delay, although the steady-state response is as perfect as full state feedback, the transient response deterioration is inevitable. This calls for a remedy called initial function compensation technique for delay-based controller and it has been proved that the transient response deterioration can be eliminated. Furthermore, to achieve high-performance flexible servo systems, control of non minimum phase system is not uncommon, which is very complicated in practice. The so-called internal model modification technique solves this problem perfectly and combined with the delay-based output feedback, this new technique becomes a prom1smg output feedback technique for high-performance flexible servo systems. 1 Introduction 1.1. Motivation and Purpose With the increasing demand for structural control systems with high performance, such as fast response, disturbance rejection, robust stability and performance, insensitive to high frequency noises and low power consumption, high signal-to-noise ratio in control channel, no steady-state chattering, etc., the flexibility of the structure must be taken into consideration when controller is designed. However, during the past decades, in most of the structural control systems, the rigid body model was assumed for the purpose of easy implementation. Nowadays, in many high-tech engineering applications, such as computer disk drives servomechanisms, industrial/medical robot arms, aerospace structures and high speed high-precision automated weapon systems, this rigid body assumption becomes the bottleneck for further improving performance. That is why the control of flexible systems becomes more and more popular and important. A good illustration that can show the importance of designing controller with consideration of the structure flexibility is the servomechanism for hard disk drive. The block diagram for a typical head-positioning servo system of a hard disk drive is shown in figure 1. 1. 1 and the operation modes of the servo systems, track seeking and track following are shown in the figure 1.1.2. Specifically, the track following servo is used to position the Read/Write head over a desired track on the disk with a minimum deviation from the track center with presence of all kinds of disturbances 2 like RRO (repeatable runout) or NRRO (non-repeatable runout), etc. Since the servo must have the disturbance rejection ability, the feedback control has to be employed. One factor that affects the performance of a disk drive is the recording density, which is a combination of linear bit density BPI (bits per inch) and track density TPI (tracks per inch) [23]. Rotary Actuator with Voice-Coil Motor Spindle Suspension Desi red head location Controller PES Channel Rotating disk Magnetic track Figure 1. 1. 1 Block diagram of a typical head-positioning servo system of a hard disk drive. Suspension arm Slider with R/Whead Figure 1.1.2. Operation modes of the hard disk drive servomechanism: (a) Track seeking, (b) Settling, (c) Track Following 3 Due to the physical limitation, further increase the linear bit density by another order of magnitude doesn't seem possible as predicted by experts in the disk drive industry. Increasing the track density, which still has much room, becomes a must for the future growth of disk storage capacity. However, the track density of a disk drive is mainly determined by the performance of the disk drive servo system in the track following. The actuator suspension-assembly is a flexible structure with multiple modes (resonances) of vibration. At a low track density, the servo system can be regarded as a rigid body manipulator with good fidelity. However, at a high track density, the track misregistration budget (allowable head position deviation from the track center) is very small, thus high frequency vibrations, even with small amplitudes, can significantly affect the performance of the servo system in the following ways: First, an increasing number of high frequency modes of the actuator-suspension assembly become influential in head positioning [24]. Because the damping in a disk drive is apt to be slight, vibration in these modes may lead to significant track misregistration during read and write processes, or even loss of data. Second, the in-plane and out-plane vibrations of the disk spindle assembly are imposed onto the servo system as disturbances [32]. These disturbances, whose frequency spectra may be well beyond the bandwidth of the servo system (around 1. 0-1. 5kHz with current drives), are difficult to compensate. The way to overcome these problems is to further increase the closed-loop bandwidth of the servo system, which is proved to be very difficult with the currently 4 used techniques, such as rigid body based control or employing some certain notch filter to cancel some flexible modes or even using the low pass or band-stop filter to attenuate the high frequency modes of vibrations. Due to the large parameter variations of resonance frequencies under different working conditions or to different drives, the notch filter cannot well fit and sometimes the stability may not be guaranteed. As the order of notch filter, low pass filter or band-stop filter increases, the more phase lags will be brought to the system, which will eventually reduce the phase margin of the system and bring more delay to the systems for large computation time. Therefore, in order to further increase the closed-loop bandwidth of the servo system, more powerful control algorithm with consideration of system's flexibility should be employed. On the other hand, besides being a flexible structure with many modes of vibration, the servo system for high TPI drive has the following special properties: 1. The sensor (R/W head) and actuator (voice coil motor) are non-collocated, which renders the servo system of non-minimum phase, and limits the bandwidth of the track following servo. 2. Due to the aerodynamic forces generated by the spinning disk, the slider disk dynamic coupling and friction of the actuator pivot bearing, accurate parameters (e.g., damping ratios and resonance frequencies) of the high frequency modes of the servo system are difficult to estimate. As a rule of thumb in the disk drive industry, the closed-loop bandwidth of the servo cannot exceed half of the first flexible mode frequency. And the frequencies of 5 non-minimum phase zeros are above the first flexible mode frequency and below the third flexible mode frequency, thus currently the disk drive industry didn't touch the non-minimum phase problem yet. However, in the near future, as long as the bandwidth requirement exceeds the first flexible mode frequency, the non-minimum phase problem will become very stringent and even dominant, so more advan~ed algorithms that can overcome the non-minimum phase problems should be developed. Another way for current disk drive industry to further increase the closed-loop bandwidth is to redesign the actuator suspension-assembly using novel materials that can shift the first flexible mode to much higher frequency range such that the closed loop bandwidth can be further extended by using conventional control method according to the above rule of thumb. However, this hardware redesign will definitely increase the cost and it always has a physical limit, when that limit is hit, advanced control algorithms for flexible system still have to be developed and implemented. The third way of further increasing the closed-loop bandwidth of hard disk drive track following servo for disk drive industry is the other scheme of hardware redesign called multi-stages actuator suspension assembly, which will consist of one coarse tuning servo and one fine tuning servo. And theoretically this will greatly increase the closed-loop bandwidth. However, this scheme will definitely increase the complexity of controller design and render the multivariable control system. And the down side of this method lies in the cost of hardware redesign and still modes of 6 vibration and non-minimum phase limitations. This will also need the advanced control algorithms that can take the flexible dynamics into account to achieve better performance. In general, most of the high performance flexible control systems have several common features. First, due to the operational constraints, a flexible system often consists of sensors and actuator, which are usually non-collocated. Second, the structure of a flexible system is usually a flexible elastic continuum, whose dynamic response is characterized by multiple modes of vibration. Third, due to the non- collocated sensors and actuators, the non-minimum phase behavior is always another feature of the flexible systems. Hence, this study aims at finding a general and systematic way to design the flexible system controller such that higher performance can be achieved, e.g., wider closed-loop bandwidth, higher robustness to system's uncertainties, lower gain and lower bandwidth controller and so on. 1.2 Problem Definitions To mathematically investigate the control algorithms for flexible systems, let us consider the continuous-time SISO flexible system with the following state space model L and transfer function matrices G, { x=Ax+Bu :L: y=Cx (1.2.1) y ( s) = G(s) = C (sf - Af1 B U(s) 7 (1.2.2) And within the entire book, we are going to employ the controller-plant closed- loop configuration shown in figure 1.2.1 for both state feedback and output feedback and use break point at input of the system to define the loop transfer function. This configuration is widely used in the flexible positioning systems [9]. And assume the flexible system is state controllable, which is usually true for most of the flexible structural system. Let K be a full state feedback gain vector such that (a) The closed-loop system is asymptotically stable, i.e., all the eigenvalues of A- BK lie in the left half s-plane. (b) The loop transfer function, when the loop is broken at the input point of the plant, meets some frequency domain specifications such as stability margin, etc. ( c) The closed-loop transfer function meets some frequency domain specifications such as closed-loop bandwidth, high frequency magnitude cutoff rate, etc. The state feedback control is of the following form, u=-Kx (1.2.3) and the loop transfer function evaluated, when the loop is broken at the input point of the plant, is, L(s)=K(sl-Ar 1 B (1.2.4) 8 The corresponding sensitivity and complementary sensitivity functions are, S(s)- 1 T(s)- L(s) -l+L(s)' -l+L(s) (1.2.5) y e Figure 1.2.1. Controller-plant configuration for output feedback and full state feedback Obtaining a proper feedback gain vector K is concerned with the issue of loop shaping, which often involves the use of arbitrary pole-placement or linear quadratic regulator design. For the output feedback configuration, a general delay and delay-free infinite dimensional controller, which is shown in (1.2.6), is employed. Ca(s) = Na(s)+N(s) Da(s)+D(s) (1.2.6) where, Nd and Dd are psuedo-polynomials defined in (1.2.7), while N, D and P are polynomials of s. n Pa(s) = LgkPk(s)e-kTs (1.2.7) k=O 9 The aim of using the above delay-based infinite dimensional controller is to realize the physically unrealizable full state feedback control and guarantee the following performance requirements, t. To asymptotically match the loop of the full state feedback within low frequency range, such that the gain margin and phase margin are acceptable. 11. To increase the closed-loop bandwidth to fast the system's response and have better disturbances rejection ability. 111. To increase the gain of both loop and controller in high frequency region such that the overall system is insensitive to high frequency noises, especially in the control channel. 1v. To reduce the order of the controller such that the computation time for control input can be greatly decreased. v. For minimum phase systems transfer function with greater than 1 relative order (the order difference between numerator and denominator), asymptotic loop transfer functions can be realized without introducing physically unrealizable improper controller. vt. For non-minimum phase systems, the asymptotic loop transfer recovery can be realized without introducing unstable controller. In here, for performance requirements (i-iv and v), the technique called delay transform and finite design region (DTT & FDR) is employed, while for performance requirements (i-iv and vi), the delay-based internal model modification (DIMM) technique is utilized. 10 To clearly understand the principle of the two methodologies of designing the above delay based infinite dimensional control such that the above performance requirements are satisfied, we need to briefly introduce some definitions. Definition 1.2.1 (Loop transfer function): is defined as the open-loop transfer function evaluated at certain break point of the feedback loop, where here, the break · point locates at the input point of the given plant. Definition 1.2.2 (Crossing point) is defined as the point that the root-locus crosses the imaginary axis. Definition 1.2.3 (Crossing root-locus) is defined as the root-locus that crosses the imaginary axis. Definition 1.2.4 (Destabilizing loop gain) is defined as the gain of the crossing point of the crossing root-locus. Definition 1.2.5 (Phase design method) is a method that makes use of the phase information of open-loop system to design the delay-based infinite dimensional controller by phase matching and stabilize the flexible system. Definition 1.2.6 (Finite design region) is a principle of designing delay-based infinite dimensional controller such that the overall infinite dimensional system can behave like a finite dimensional system with arbitrarily small error and simplify the infinite dimensional phase design method by finite dimensional method, e.g., root locus method. This principle is trying to locate the zeros and poles in such a fashion that the frequency response in the low frequency range of both ideal loop transfer function of full state feedback and delay-induced loop transfer function nearly match 11 to guarantee the robustness requirements and the high frequency characteristics mismatch in such fashion that the closed-loop performance of the delay-induced system is better that of the full state feedback system. ,_ ~x. ···~ T (a)~(} n . . ... (b) T ···~ kn-1 ""-. 8 (c) ~ n Figure 1.2.2. Flexible structure models: (a) Multi-DOF flexible longitude mass spring systems; (b) Multi-DOF flexible torsion mass-spring systems; (c) Multi-DOF flexible bending mass-spring systems. Definition 1.2. 7 (Delay transform technique) is a method that can realize the physically unrealizable zeros and place them to the desired locations by meanings of delay-based infinite dimensional controller. Definition 1.2.8 (Internal Model Modification) is a method that employs some certain compensator in the feedback loop, such that the input output relation of the original system can be modified to satisfy certain desired requirements. 12 Definition 1.2.9 (Zero placement) is to place the zeros of the controller to the desired locations. For investigating high performance flexible control systems, the above three types of flexible plants (shown in Figure 1.2.2) may be concerned in this dissertation. These plants are: (1) Multi-DOF flexible longitude mass-spring systems; (2) Multi DOF flexible torsion mass-spring systems; (3) Multi-DOF flexible bending mass spring systems. 1.3 Overview of Existing Design Approach The high performance flexible feedback control systems have been extensively studied and are of increased research interest today. All the existing feedback control algorithms dealing with the regulating problem can be classified into the either full state feedback or output feedback control. 1. 3. 1 Full state feedback control It is well known that the full state feedback control can realize arbitrarily closed loop pole-placement, if the open-loop system is state controllable. The LQR (linear quadratic regulator) method [12-14] is a systematic way to design a full state feedback controller and gives the system very good performance e.g., robustness [ 15]. However, the problem of full state feedback is the unavailability of full states, i.e., we cannot always measure all the states of the system due to limited number of sensors and immeasurable states. Hence, usually the full state feedback is an ideal and theoretical control algorithm and it is physically unrealizable. 13 1.3.2 Output feedback control Another kind of feedback control is the output feedback. There are two research branches in this kind. One is the observer based output feedback (e.g., LQG/LTR (Linear Quadratic Gaussian/Loop Transfer Recovery [6]), another is the compensator based output feedback (e.g., CSS/LTR (Chen, Saberi, Sannuti/LTR)[29]), H-infinity, delay-based output feedback, etc.). 1.3 .2.1 Observer based output feedback One conventional output feedback is the observer based output feedback, which is to construct an observer to recover all the states from the output measurement together with control input and then employ the full state feedback algorithm to achieve good performance. Kalman [16-18] shows that as long as the open-loop system is state observable, we can always construct an observer to estimate all the states from the measurement and he also proposed the well-known LQG (Linear Quadratic Gaussian) algorithm. Kalman and Bucy [19] introduced a state variable version of the Wiener filter. Gunckel and Franklin [ 11] used the Kalman-Bucy filter to design state-estimate feedback systems that minimized the expected value of a quadratic performance measure for sampled-data systems, which is the so-called linear quadratic Gaussian problem (LQG). But the problem of observer based output feedback or LQG is that it has no guarantee of the robustness [7], although it may have exactly the same closed-loop input/output characteristics as that of the full state feedback or LQR [3]. And also the overall cascaded compensator which is equivalent to the observer based controller might be unstable, although both the controller and 14 the observer are stable respectively, which will sometimes leads to internal instability [ 5]. Doyle showed that the LQG feedback system could have arbitrarily small stability margin [7]. In general, the LQG or observer based output feedback cannot guarantee the robustness that the LQR or full state feedback provides. This is the reason that the so-called LQG/L TR came into being [8], and the basic idea of the LQG/L TR is to recover the robust characteristics of the LQR control system to that of the LQG induced feedback system by recovering the loop transfer function of full state feedback.. This is done by choosing suitable parameters for the observer so that the loop transfer function of the LQG system is exactly the same or close to that of the LQR system. This will eventually guarantee the 60-degree phase margin and infinity or very large increasing gain margin that LQR control gives [ 6]. But the problem of conventional LQR or LQG/L TR is that since high gain and high bandwidth controller must be used, the high frequency noise will be amplified greatly, which leads to bad signal-to-noise ratio in the control channel [29]. And another problem of LQG/L TR is that as long as the order of the open-loop plant increases, the order of. the controller is always the same as that of the plant, which means more and more computation delay should be produced when it is implemented by digital controller. This will essentially become one limitation of the overall system performance. Hence, in terms of high frequency noise amplifications due to high gain and high bandwidth controller and the large computation delay due to high order controller, the LQG/LTR is not a suitable method in achieving high performance flexible servo system. 15 1.3.2.2 Compensator based output feedback The compensator based output feedback is another version of output feedback which only takes the output measurement in the controller design while the observer based output feedback takes both control input and the output measurement into account when designing the controller [29]. In this branch of output feedback, there exist many different research areas like the robust output feedback control, CSS/L TR, and delay-based output feedback control. • Robust output feedback Among all the robust control algorithms, probably the most famous ones are H infinity, µ-synthesis [31] or H 2 control [30], which are used to control the systems with bounded uncertainties to ensure robust stability and even robust performance. One problem of the above robust control is that they are to some extent conservative due to the small gain theorem [3 1]. And the other one is that it has been shown that none of them is suitable for systems with lightly damped vibration modes because of singularity of the Riccatti equations, which are used to solve the controller. And another one is the complexity of algorithms. Hence, from the points of view in applicability to flexible systems with multiple lightly damped modes of vibration and algorithm complexity, the conventional robust control method is not suitable for achieving high performance flexible servo system. 16 • CSS/LTR CSS/L TR [29] proposed by Saberi, is another way of the loop transfer recovery, which is essentially the compensator based output feedback. As claimed in [29], in order to realize exact loop transfer recovery or asymptotical loop transfer recovery, similar to the LQG/LTR and even LQR, the high gain and high bandwidth controller should be employed, which will lead to great amplification of high frequency noises and bad signal to noise ratio in control channel. And also generally speaking, all the conventional control algorithms can never avoid high order controller especially when the order of the plant is high, which will definitely increase the computation delay in the real implementation. This suggests that the CSS/L TR is still not suitable for high performance flexible servo system. 1.3 .2.3 Delay-based output feedback The third type of the output feedback is of great interest, which is the so-called delay based output feedback. Since it is infinite dimensional compensator, which is different from the previous compensators, we consider it in this separate section. Frequently, time delay is considered as the source of poor performance and even instability in various industrial applications. However, recently, purposely adding time delay actions into the controller is of great interests especially in active control of flexible structural systems. Several reports are available on positive uses of the time delay actions in the controller design. 17 Tallman and Smith [38] and Rubin [28] have shown that a second order lightly damped oscillatory system can be controlled to produce a deadbeat nonoscillatory response by a controller with time delay. Choksy [ 4] briefly explained that an unstable linear time-invariant system could be made stable by introducing a delayed feedback. For the control of process with inherent time delays, Smith [3 3] suggested that a controller with minor-loop feedback, the so-called Smith's linear predictor, be used. This scheme utilizes the action of time delay, the length of which is equal to the plant itself. Abadallah et al., [1] showed that the delayed positive feedback could stabilize the oscillatory systems. Youcel-Toumi [ 44,45] shows that time delay can be used in the input output linearization. Olgac [26,27] used the delayed resonator to construct a tunable active vibration absorber. Yang [40-37,20] showed that the delayed feedback could be used to stabilize infinite number of flexible modes of a series of flexible systems, like string, plate, and beam. Tomizuka [39] proposed the repetitive control, which employs the delayed feedback to cancel the periodic signal in the control channel. Gilchrist [10] has used time-delay actions to construct the state vector from the output. Bien [35-37,28] proposed the so-called proportional minus delay (PMD) method that can be considered as the averaged PD controller in terms of time domain behavior and it is insensitive to the high frequency noise. It can also be used as the state observer. 18 However, the problem of the above delay-based researches is that most of them focus on the stabilizing problem but not focus on the performance related problem. Although t..1e PMD control proposed by Bien has somehow touched some certain performance like IT AE (Integral of time multiplied by absolute error), they did little job on the stability criteria of the delay-based infinite dimensional system and did not give the relation between the stability and performance. This means that more researches should be done regarding both stability criteria of delay feedback system, performance and robustness. 1. 4 Outline and scope In general, a high performance feedback control of flexible structural system usually has a lot of performance requirements, which can be summarized as follows: 1. High closed-loop bandwidth, a measure of disturbance rejection ability. 2. Low control energy, to avoid the actuator saturation. 3. High attenuation of high frequency flexible modes to m1mm1ze residual vibrations. 4. Good signal-to-noise ratio in control force to avoid chattering. 5. Simple and low order controller to minimize the computation delay. 6. High robustness to system's uncertainties. (e. g., phase margin and gam margin) 7. Internal stability guaranteed. 19 As the existing control algorithms shown in the literature, generally speaking, there is no existing control algorithm that can guarantee all the above performance requirements and can be used to achieve high performance :flexible servo systems. The problems of the existing control algorithms are shown in the table 1.4.1. T bl 141 D. d f h "f t l l . h a e .. 1sa vantages o t e ex1s mg con ro a ,gont ms Control Algorithms Disadvantages in high performance servo systems Full state feedback/LQR Unavailability of full states/high gain/unrealizable Observer based/LQG No robustness guarantee/internal stability/high order LQG/LTR and CSS/LTR High gain/high order Robust output feedback Complexity/stability problem Delayed output feedback Infinite dimension/complex stability and performance For clarity, the content of the above table are explained as follows, ( 1). The problems of full state feedback are the unavailability of full states due to the physical limitation (e.g., limited number of sensors and immeasurable states) and high amplification of high frequency noise and unmodeled dynamics, which makes the full state feedback physically unrealizable. (2). The problems of observer based output feedback or LQG are that it has no guarantee of robustness that LQR provides, the overall controller-observer compensator may be unstable due to separate design of controller and observer and the order of controller will become very high as the order of plant increases. (3). The problems of LQG/LTR or CSS/LTR are high controller order as (2) and high amplification of noise and unmodeled dynamics as (1 ). 20 ( 4). The problems of robust output feedback, e.g., H-infinity and µ-synthesis are that they often fail to find solution of controller, because the Ricatti equation is usually singular due to lightly damped or even non-damped flexible modes and also the algorithm is very complicated to use. ( 5). The problems of delay based output feedback are that due the infinite dimension of the controller, the overall stability and performance analysis become very complicated. There is no way to easily handle this kind of systems and most frequently the design of controller is system dependent. Based on the above summary of disadvantages of the existing control algorithms in achieving high performance flexible servo systems, we can conclude that there is no such a way that can guarantee all the above performance simultaneously. Although to find a control algorithm that can guarantee all the above performance requirements simultaneously seems impossible, in this study we still try to find a general way to solve this difficult problem and we do find a systematic design method which is capable of solving this problem. This method is referred to as the delay transform technique and finite design region algorithm and delay-based internal model modification technique. The aim of this study in general is to find a systematic way to design the delay based infinite dimensional controller to achieve high performance SISO flexible servo systems. And the mainly used control algorithm of this research is referred to as the delay transform technique, which is an infinite dimensional control algorithm 21 and can be designed and analyzed by meaning of finite dimensional methods based on the finite design region principle. Hence, the finite design region principle plays a bridge between the finite dimensional domain and the delay-induced infinite dimensional domain, while the delay-based zero placement is a technique that can be used to realize arbitrary physically unrealizable zeros by output feedback with arbitrarily small error by means of the delay-induced infinite dimensional control algorithm. The delay-based internal model modification technique is specific to controls of systems with non-ideal characteristics such as non-minimum phase, vibration modes and even inherent delays. In this dissertation, we study the characteristics of the delay-induced infinite dimensional control algorithms with the emphasis on the delay transform techniques and finite design region principle and employ this type of output feedback control law to control the flexible minimum phase systems with non-collocated sensor and actuator. The non-minimum phase flexible system control is also investigated and the solution, the delay-based internal model modification technique, is proposed. The whole dissertation is organized into two parts: minimum phase system and non-minimum phase system. In part one, a review of stability and performance criterion for linear feedback systems is given in section 2. Section 3 introduces the delay transform technique and finite design region algorithm. The full state feedback realized by delay transform technique and finite design region output feedback is contained in the section 4. To solve the problem of transient response deterioration caused by delay action, the so-called initial function compensation for delay-based 22 control system is proposed in section 5. In part two, the delay-based internal model modification technique is presented in section 6. Simulations of multiple-DOF non collocated flexible system based on the delay transform and finite design region algorithm and 2-DOF flexible non-minimum phase robot arm control through delay based internal model modification technique are given in section 7. In section 8, the non-collocated control a single-link flexible robot arm based on Euler-Bernoulli beam model is investigated. The experimental results for non-collocated control of 2- DOF flexible torsional mass-spring system by using full state feedback and delay transform technique and finite design region output feedback control algorithms are given in section 9. And the conclusion is drawn at the end. 23 2 Review of Stability and Performance Criterion for Delayed Feedback Systems 2.1 Delay-induced infinite dimensional systems In linear control theories, most of the controllers being investigated are of finite dimension, which is apt to have easy analysis, design and implementation. However, when the controllers are incorporated with some certain amount of inherent or purposely adding delays, the overall systems will be of infinite dimension due to the dynamics of delays. It is well known that to the finite dimensional systems, there exist many control algorithms such that full state feedback, observer based output feedback, robust output feedback control, etc, and many existing stability and performance analysis methods such as Nyquist criteria, root-locus analysis, bode based analysis, etc. While to the delay-induced infinite dimensional systems, both controller design and stability-performance analysis will be very complicated and up to now, there is no systematic and effective way to deal with it. To the delay-induced infinite dimensional systems, the usually used stability criteria are the Nyquist criteria and infinite dimensional root-locus method, however, due to the infinite dimension of the systems, the Nyquist criterion becomes very complicated. While the infinite dimensional root-locus is very difficult to generate provided that there is no existing commercialized software, and even if we have this kind of software, the design of controller with certain performance requirements still 24 could be very complicated. A simple first order system with single delay will have infinite branches of root-loci when the feedback loop is closed. And the stability and performance analysis are very complicated and system dependent. This can be seen from the following simple example: Suppose we have a process whose dynamics can be approximated by G(s) = Ke-s s+l (2.1.1) And the characteristic equation of the feedback system (shown in figure 2.1.1) is: R(s) Ke-s s + 1 Y(sl Figure 2.1.1 Block diagram of the above feedback system (2.1.2) The infinite dimensional root-locus plot of the above feedback system is shown in Figure 2.1.2 and from the plot, we can see that the above system has infinite branches of root-loci, which is introduced by the time delay. And the stability of the overall system is determined by all the destabilizing gains of corresponding crossing root-loci. And the performance of the feedback system is determined by the closed- loop bandwidth, which is indirectly related to the locations of infinite number closed- loop poles and zeros, and it is affected by the stability margin, which cannot be found 25 directly from the root-locus plot. This gives us the feeling of how complicated the stability and performance analysis or controller design will be to the delay-induced infinite dimensional systems. J-7E---------- j~-re- - - - - - - - - - - - - - - j~'lr- - - - - - - - - - - - - - - j4-re- - - - - - - - - - - - - - - ~r-------------- Figure 2.1.2 Root-locus plot for the system shown in figure 2.1.1. a On the other hand, the commonly used performance analysis methods are the so- called finite dimensional appro~imation approaches, such as the Pade' s approximation and Taylor's expansion in Equation 2.1.3 and 2.1.4, where the delay terms in infinite dimensional systems can be approximated by the finite dimensional 26 systems and reduce the infinite dimensional system to a finite dimensional systems so that the stability and performance analysis can be approximated by the existing finite dimensional methods. T1s2 T3s3 e-sT =I-Ts+-----+... (Taylor's Expansion) (2.1.3) 2! 3! Ts T 2 s 2 T 3 s 3 I--+-----+ ... e-sT = 2 ~ 2 ;s 3 (Pade' s Approximation) (2.1.4) Ts Ts Ts I+-+--+--+ ... 2 8 48 And if the delay is of very small amount, the following approximation is also used frequently, e-sT =I-Ts -sT I e =-- I+Ts I-Ts -sT 2 e =-- I+ Ts 2 (2.1.5) (2.1.6) (2.1.7) However, the finite dimensional delay approximation methods only fit in the systems with single or very few delay terms without loss of accuracy. To the systems with a lot of delay terms, this kind of approximation will become very inaccurate and the whole analysis will also become very complicated. In general, to the delay- induced infinite dimensional systems with any number of delay terms, there is no effective and precise method to deal with the stability and performance analysis. 27 2.2 Overview of Existing Stability and Performance Criteria for Delay-induced Infinite Dimensional Systems It is well known that to any linear feedback systems, the sufficient and necessary stability conditions are either Nquist's criteria in terms of frequency response or the root-locus method in terms of closed-loop eigen-structures. As to the controller design such that the performance e.g., closed-loop bandwidth, stability margin, etc., can be satisfied, the Nyquist's criteria and bode based method can be directly used while the root-locus method can_be indirectly used. • Stability Criteria (1). Nyquist's Criterion Assume the following feedback configuration shown in Figure 2.2.1 is employed and the open-loop poles of the G(s)H(s) may locate exactly on the jro axis, the generalized Nyquist's criteria based on e-indented Nyquist's plot are given as follows [25], • s ......_ , Y(s) H(s) ~ ~ Figure 2.2.1. Feedback configuration of control system In the system shown in figure 2.2.1, if the open-loop transfer function G(s)H(s) has k poles in the right half s plane, then for stability, the G(s)H (s) locus, when the l 28 representative point s traces on the modified Nyquist path in the clockwise direction, must encircle the -1 +jO point k times in the counterclockwise direction. This criterion can be expressed as Z=N+P where Z = number of closed-loop right half plane poles N = number of clockwise encirclements of the -1 +jO point P = number of open-loop right half plane poles If the open-loop system is unstable and has P right half plane poles, for closed loop stability, Z=O, or N=-P, which means that we must have P counterclockwise encirclements of -1 +jO point. If the system is open-loop stable, i.e., all the open-loop poles locate in the left half plane of complex plane, for closed-loop stability, there must be no encirclement of -1 +jO point by the modified Nyquist locus. Although the Nyquist criterion is the sufficient and necessary stability, it doesn't give us a design methodology of the controller such that the closed-loop system is stable because it is too general. Thus, in many control system design, people add some constraints on the Nyquist's criterion, which makes it a necessary stability condition when designing a controller. Such constraints are usually the following, I. Small gain theorem Small gain theorem is a well-known and simplest stability criterion, which assumes that the system is open-loop stable and if the gains of the open-loop transfer 29 function over all frequencies are less than one, the closed-loop system is stable. This is because all the loci locate inside the unit circle around the origin, i.e., there will be definitely no encirclement of -1 +jO point by the Nyquist locus. This is the most conservative but simplest stability criterion, because no phase but gain is considered in designing a controller. This criterion is actually extensively applied to the conventional robust output feedback theory, such as H-infinity, where either the sensitivity function or complementary sensitivity function for nominal system is designed stable and the generalized plant with bounded uncertainties is consider as a new feedback system, the small gain theorem is employed to ensure the overall stability of the system. Since the uncertainties are usually unstructured and may be very complicated in both gain and phase , the small gain theorem can greatly simplify the whole design procedure without taking the phase conditions into consideration when design the feedback controller, although it is very conservative. II. Non-conditional stability From the Nyquist criterion, when the Nyquist locus has even number of (no less than two) encirclements of -1 +jO point with exactly the same number of clockwise . and counterclockwise encirclements, the closed-loop system is said to be conditionally stable. This means that the stability of this kind of system depends on the overall gain. Different gain may cause stable or unstable closed-loop system. Since the conditional stability is usually undesirable in practice and also in order to simplify the design of controller, people sometimes add some constraints such that 30 the closed-loop system is non-conditionally stable, which are the following phase and gain conditions (2.2.1) This criterion can be interpreted into root-locus method to delay-induced fe(dha.ck control of flexible stru~hi~h will be introduced lat~ III. Relative stability In terms of robustness to system's uncertainties which can be usually interpreted as some phase and gain uncertainties, some certain relative stability must be guaranteed for the closed-loop system. This requires some phase margin and gain margin, or sometimes vector stability margin. Specifically, when the constraints applied to the Nyquist criterion are non-conditional stability and some certain phase margin a'rid gain margin, Equation (2.2.1) can be modified as { IG(m)H(m )I~ 1 m E (O,mcJ \ G(m)H(m)I < 1 m E (mc, oo) </J(m) > -ir m E (0, mp] (2.2.2) Here, the phase margin and gain margin can be considered as the open-loop performance requirements, which need the open-loop frequency shaping to guarantee it. As for the vector stability margin defined as VSM = II + G(m)H(m)I (2.2.3) 31 it is a more accurate relative stability definition, which can be consider as the closed- loop performance requirement and can be guaranteed through the closed-loop frequency shaping techniques, e.g., the sensitivity function shaping. Note that there is the following relation between vector stability margin and magnitude of sensitivity function, 1 1 IS(w)I = ll +G(m)H(w)I = VSM (2.2.4) All the above constraints can be used to simplify the controller design but introduce some certain conservativeness. There is a tradeoff between design complexity and system performance. However, frequently, adding some certain constraints initially is used to check if the desirable closed-loop performance is satisfied, otherwise releasing some constraints and trying again is an effective and simple design methodology, because systems with more constraints will always have lower design complexity. On the other hand, in many systems' design, although initially several constraints are added to the stability and performance criteria for simplifying the design and analysis complexity, the final version of design can usually release some of these constraints, which is one feature of delay-based feedback system. (2). Root-locus Method In the analysis of system's eigen-structure, we know that right half plane poles mean unstable system while the left half plane poles are stable, which intuitively 32 motivates us to investigate the eigen-structure of the closed-loop system to judge the closed-loop stability. If all the closed-loop poles are in the closed left half plane, the system is stable, otherwise it is unstable. The investigation of closed-loop eigen- structure is referred to as the eigen-value problem in linear state space, while in the ~ frequency domain it involves solving zeros of the closed-loop characteristic ~----------------·-------------.--. '--~-- -· · polynomial, which is usually very difficult to obtain analytically especially when the - --------- system is of high order or even infinite dimension. The root-locus method is a way to find the closed-loop poles by the open-loop eigen-structure and zero-structure when the overall gain changes from zero to infinity. These loci origin from the open-loop poles and end at the open-loop zeros of the system according to some rules. Although the root-locus method is suitable for any linear system, and plays a sufficient and necessary stability condition, to the high order or infinite dimensional system, it is very difficult to obtain the root-locus plot to design the controller. This suggests that some constraints should be added to the root-locus design method, such that the whole design procedure can be simplified without needing to obtain the complete root-locus plot. For example, for the open-loop system with all the poles . -------.:._____ imaginary, some constraints on departure angles and minimum destabilizing gain are added to the stability criterion as follows, Re(:k J<O µ µ=0+ (2.2.5) 33 where icx.k are all the intersections of the root loci and the imaginary axis when µ>O and Sk are the open-loop poles of the plant. The first condition is the departure angle requirement, which is essentially the phase condition, and the second one is the minimum destabilizing gain requirement. In this case, only few values should be evaluated and other segments of the root-loci are not necessary to obtain, this greatly simplifies the design procedure when the system is of high order or infinite dimension. Some other constraints like non conditional stability are also frequently used in the root-locus method, but sometimes this constraint can be released at the final stage of the whole design as will be shown in the following chapters. • Performance Criteria (1). Nyquist's Criterion and Bode based Method As shown before, not only the Nyquist criterion can be used as the stability criterion, but also it can be used as the performance criterion in that the stability margin and closed-loop bandwidth. Bode based method is another version ofNyquist criterion, and the only difference between bode based method and Nyquist criterion is that the former is not a sufficient and necessary stability condition while the latter is. While in terms of performance criterion provided that the system is closed-loop stable, the bode based method is direct and easy to use, because it has decoupled both phase and gain, which is helpful in designing lead/lag compensator with guaranteed closed-loop phase margin and gain margin. And since the open-loop gain 34 crossover frequency with OdB is to some extent near to the closed-loop bandwidth frequency in value, people usually use it to approximate the closed-loop bandwidth. Additionally, employing some bode based closed-loop frequency response shaping can easily evaluate the closed-loop bandwidth, a measure of system's response speed and disturbance rejection ability, closed-loop gain peak value, a measure of sensitivity and the high frequency cut-off rate, a measure of noise attenuation ability. (2). Root-locus Method Although root-locus method can give us all the eigen-structure of the closed-loop system, it is indirect in designing the controller such that certain performance is guaranteed. Only to very low order system or system with very few dominant poles and many high frequency poles, the closed-loop performance like bandwidth can be predicted. And in the root-locus analysis, only gain margin can be predicted, while the phase margin is difficult to find. On the other hand, when the closed-loop system is of high order or even infinite dimension and there may be bunch of dominant poles, the closed-loop performance is usually difficult or even impossible to predict. This makes the root-locus method an indirect design methodology for performance improvement. As shown in previous section, the control system incorporated with delay will become an infinite dimensional system and the stability and performance analysis of this kind of infinite dimensional system will be very complicated no matter what criterion we employ. The only way to design this kind of controller such that the stability and performance are satisfied is to simplify the design criterion by adding 35 more constraints and after checking and releasing procedure to find a suitable solution in the final stage of design. 2.3 Phase Design Method As discussed before, for any linear feedback system, the Nyquist criterion is the sufficient and necessary stability condition. And for most flexible mechanical systems which are usually open-loop stable, the Nyquist criterion can be interpreted that the Nyquist plot has no clockwise encir~l~m~~!_~rQun_cJ_Jhe_-1 +jO point. And -----" - ------ ~-----·-- . since in practice the conditional stability is not desirable [25], and if we assume that the closed-loop §ystem has no conditional stability, the Nyquist criteria can be interpreted as the following phase and gain conditions, (2.3.1) where w)s the open-loop gain cross-over frequency [25], ¢(co) and jG(w)j are the phase and gain of the open-loop transfer function. To ensure some robustness to system uncertainties, which can be modeled as some phase and gain uncertainties, some phase and gain margin should be concerned. This gives the following phase and gain conditions, l jG(w )j ~ 1 co E (0, we] IG(w)l<l WE(Wc,oo) ¢(co) >-n OJ E (O,wP] (2.3.2) where we< WP to ensure positive gain and phase margin. J 36 It is well known that the root-locus method is another way to decide the closed- loop stability. Due that the Nyquist criterion deals with frequency response of the overall system and the root-locus method deals with the closed-loop eigen-structures, usually there is no explicit relation between them, although essentially they are in accordance with each other. However, in terms of the phase conditions specifically, <-._____ there is some certain relation betw::::e:en:-:ti:'he::r:o:ot;-_r,lo::c::u:s-:m:e~t:th::;:o':fd-:an~ar-N~y"'qwut"s•t .... c...;ti~te-1~ ----------------------------------i:;?;:;--------------------------->-- Th is can be referred to as the following relation, ~ (~ (2.3.3) where iak are the intersection points (also referred to as crossing points) of the root- loci with the imaginary axis, and a rnin is the minimum frequency of all the crossing points, which is exactly _:qual to the phase cross-over frequency provided that the root-locus plot does ha~e th~-~-;o~ point, othCrwise th( phase cross-over '-- frequency is infinity. --- From the above analysis, in order to guarantee positive stability margin, the gain 1 crossover frequency must be less than the phase crossover frequency. And as we know, usually the gain cross-over frequency can also be used to approximate the closed-loop bandwidth of the feedback system [25], which suggests that in order to guarantee some certain closed-loop bandwidth, a major performance requirement for flexible systems, the phase cross-over frequency must be greater than this bandwidth. This condition tells us that in terms of the crossing point frequency of the root-locus '--- . ____. . ) \fl j . plot, the minimum crossing p'oint frequency should be greater than the desired 37 closed-loop bandwidth frequency. From the eigen-structure point of view, in the ~~--~~~ ·----------- complex s plane, the distance between the closed-loop complex pole and the origin is referred to as the natural frequency Qf this flexible mode, whiGh is related to closed- ~-------- --~ -· loop bandwidth of this mode. This suggests that the minimum crossing point frequency, which is the distance between origin and the nearest crossing point of the root-locus plot, should be greater than the desired closed-loop bandwidth frequency. J This is called the phase design strategy of the root-locus in terms of closed-loop bandwidth, which means that the larger the minimum crossing point frequency the root-locus plot has, the larger closed-loop bandwidth the feedback system may be achieved. This is a necessary but not sufficient condition in terms of bandwidth improvement, but it can be used as one design requirement of the feedback controller, especially for the delay-based infinite dimensional controller, where the root-loc~s plot usually has infinite crossing points. • Stability criteria To investigate the stability conditions for the delay-based infinite dimensional systems, we need to employ the stability criteria proposed by Yang [40-42]. For simplicity of derivation, let's take the following 2-DOF mass-spring flexible system as an example shown in Figure 2.3 .1 and investigate the stability conditions for the following type of delayed controller based output feedback systems. The transfer ~nction of the plant is (2.3.4) 38 In general, the delay-based controller can be of the following form, n C(s)=Ctif(s)+ LCdi(s) (2.3.5) i=l where Ctif(s) is the delay-free controller and Cdi(s) are delayed controllers 41 Controller~ m 1 m 2 t jSensor Figure 2.3 .1. Feedback control of a 2-DOF spring-mass flexible system And each different delay based controller has different characteristics. For simplicity and without loss of generality, we take the following delay based controller to derive the stability conditions in this section and generalize to arbitrary delay-based controller later n µC(s) =µfl (1-bie-sTi ), 0 <bi< l,J; > O,µ > 0 (2.3.6) i=l According to the root locus analysis:> one sufficient condition for stability is the following. (2.3 .7) 39 which is exactly the stability criterion proposed in [ 40-42], where iak are all the intersections of the root loci and the imaginary axis when µ>O and Sk are the open- loop poles of the plant. The first condition can be interpreted as the departure angle requirement and the second one is the minimum destabilizing gain condition. In terms of the closed-loop bandwidth requirement proposed before, we hope the following crossing point condition is satisfied, (2.3.8) In this specific case of non-collocated feedback control of a 2-DOF flexible system with the above controller, the stability and performance criteria are shown as follows, 1. n23 (in equation 2.3 .6) 11. { Jr< ¢(w 0 ) < 27r bk and T,. should be chosen such that where OJ E (0, OJ 0 ) 0 <¢(w) < 2n 111. { 2( 2 2)} - . ak Wo -ak - + - µ < µer - mf 2 , where ¢( ak) - _(2k + l)Jr, k - 0, 1, 2, ... k C(ak)w1 The detailed analytic proofs are shown as follows, Proof: 1. Departure angle at origin Since the open-loop system has double integrators, we use perturbed poles of ±ie, '----- ... where e~O+. Thus, we need to check the departure angle at ie. -------- .. . .. 40 First, the characteristic equation is (2.3.9) Rewrite (2.3 .9) as • 2 s -1& = - mi C(s) µ ( s + i & )( s 2 + m:) Take limit of the above equation as µ~O, s~iE and E~O+ gives ds dµ s=ie (2.3.10) For stability at small gain µ, we need Re( : J < 0 which gives µ S=I& Im{ C(i&)) > o :::> Jm( fi (1-bke-iET,)) > Q =>Im ( (1- b1e-i&1i )(l-b2e-iET2 ) . .. (1-bNe-ieTN)) > 0 ( . T)2 (. T)3 e~o+ -ieT 1 . T 1 £ 1£ 1 . T e = -1£ + - + ... = -1£ 2! 3! => lm{(l-b 1 +ib 1 £1'i)(l-b 2 +ib 2 £T 2 ) ... (1-bN +ibN&TN)) > 0 denote a;= 1-b; > 0 =>Im( (al+ ib1&1'i)(a2 + ib2&T2) ... (aN +ibN&TN)) > 0 41 a; > 0, b; > 0, I; > 0, & > 0 :. Limlm((a 1 +ib 1 &I;)(a 2 +ib 2 &T;_) ... (aN +ibN&TN)) &-->0+ ::::::>Re(ds ]=-Mc =- M <0 dµ s=i& 2&0J~ 20J~ where,M=t[b,T,:Qa 1 )>0 ~" · )~I From the above results, we know that when µ is small, the branches of the double integrators will go into the LHP of the complex ~!ape( which is stable with the above delay-induced controllers. rr . i • . ' ' 2. Departure angle at s=iroo \) Rewrite (2.3.9) as _s_-_im_ 0 = _ w: C(s) (s + im 0 )s 2 µ Take limit of the above equation as µ~O & s~iroo gives ds m 2 = . i 3 C(imo) dµ s=iw 0 21 OJ O (2.3.11) For stability at small gainµ, we need Re(: J < 0 which gives µ s=112> 0 ;r < LC(im 0 ) < 2;r 3. Crossing points at imaginary axis Rewrite the (2.3.9) as with s=j co, the above equation becomes where and we know F(m) >0 F(m) < 0 F(m) = 0 F(w) = C(w) OJ E (0, m 0 ) m E {m 0 ,+oo) Note that C( co) is complex in general, a crossing point ico * exists (i) in L'.\ 0 = (0, W 0 ) (ii) in ~ '° = (mo , +oo) if C( ro *) is real and negative This can be restated as 42 (2 .3.12)~ (2.3.13) (2.3.14) (2.3.16) {2.3.17) 43 Note that m = m 0 is not a crossing point, becauseF(m 0 ) = O,but,C(m 0 ) ts always non-zero. 1 0.8 0.6 0.4 co.2 ~ 0 ·rn rn-0.2 E --0.4 -0.6 -0.8 </J.kmax -1 L--~~-'--~~-'-~~__,_~~---'-~~--'-~~__J -1 -0.5 0 0.5 1 1.5 2 Real Figure 2.3.2. Polar plot of l-be- fmT Let's make use of the phase information of C(ro). N Write C(m) =fl Ck (w),C k (w) = l -bke - imT 1 k=l { l ~Ct< N </J(m) = L<Pk(m), k=l By figure 2.3 .2, we ~--~¢;-ax, wh~~ ·: 0 <bk < 1, :. 0 < sin -l (bk) < Jr, and</J;:ax occurs at 2 44 (2.3.18) 7r A. ( ) _ 1 ( b" sinwT 1 ) 7r --<'Y w =tan <- 2 " l-b 1 coswT 1 2 (2.3.19) N Note that C(O) =fl (l -b 1 ) > 0, :. LC(O) = </J(O) = 0 k=l From (2.3.12), we know in order to guarantee the closed-loop system to be stable for small µ, the controller must give at least 180 degrees phase lead at ro 0 , and the maximum phase lead of single 1-be-imT element is less than 90 degrees. So we can conclude that ~ast three 1 :-~elements should be used in our control~/ . stabilize the system when µ is very smal I. Note that no matter how many (no less than one) 1-be-imT elements will definitely stabilize the rigid body mode when µ is very small. And in order to have no crossing point in A 0 , we need that there is no crossing point with 2k7t for the phase plot of the controller (refer to (2.3.17)) as shown in figure 2.3.3. In figure 2.3.3, we set roo=lOO (rad/sec), and we can see that the phase of the controller initially increases from zero and in (0,roo) there is no crossing point with 0 or 360 degrees line. )And the phase at roo is above 180 degree, so the system is stable for some µ<JJ.cr , which is the aain corresponding to the lowest crossing point -.c:::::::::::: frequency with ±180 degree line above ro 0 (as shown in the above figure). --------~-- 1 45 Let all crossing point with ± 180 degrees line corresponding to the frequency above roo be a~ where k=l,2,3, .... ~~ ft,~{kp~ ~ 250 200 150 t~j ........, I Q) Q , (fl ' m 50 .c - 0.._100 -150 -200 -250 -- -~,~~, ~.~--~~-~~~ i ~' 0 100 .J 200 300 400 500 600 700 800 900 1000 <. , Frequency (Rad/sec.) Figure 2.3 .3. Phase plot oj-be-Jo>1' .Alement CrS» • Stability condition: (i). At least three of the above delay and delay-free elements should be used and ./JW O)W) { 1! < </J(wo) < 27! r. bk,Tk are chosen such that 0 < ¢J(w) < 27!, (I) E (Q, w 0 ) wIAc{i ;'fill guarC\.~\~es A , ~'~~~~~\\~ON'~~' the departure angle conditions and there is ~ro~sinJpoints in (O,ro 0 ) . . (ii). For (roo ,+oo ), there are infinite number of crossing points. This is because for (coo , +oo ), there are infinite Clk at which </J( a k) = +tr, C (a k ) < 0 (refer to equation (2.3.17) for C(m) ). Note F(ak) < Oforak > OJ 0 • (refer to equation (2.3.16) for F(m)) 46 So, where w: is referred to equation (2.3 .9) The upper bound ofµ is (2.3.20) This is the end of analytic proof of the above stability conditions. From this simple example, we can easily see that employing some certain phase desi method, we can s atically deal with the stability of delay-induced infinite dimensional system and achieve the possibility of increasing the closed-loop bandwidth. However, this phase design method still needs to be extended to any type ~ . of delay-induced controller and higher order systems, which will be discussed in ,,. section 4. As discussed before, the phase design method can be used to systematically analyze the stability of the delay-based infinite dimensional system and achieve the potential of increasing the closed-loop bandwidth. In this section, we are going to generalize this phase design method to any high order system with arbitrary type of delay-based infinite dimensional controller. -----~------------- -- Assume we have the flexible system with the following transfer function with one rigid body mode and several flexible modes, 47 (2.3.21) (2.3.22) where Ctf' (s) is the delay-free controller and Cdi (s) are delayed controllers And a general delay-based controller of equation (23) is employed to stabilize the system and satisfy the closed-loop bandwidth potential of mB , which means that the closed-loop bandwidth of the feedback system may be greater than mB . According to the necessary condition of closed-loop bandwidth improvement proposed in section 2, this can be interpreted that the minimum crossing point frequency of the root-loci should be at least greater than mB, and to ensure some phase margin, the minimum crossing point frequency should be greater than the sum of mB and mcm~ch-i.s_ -- ---- ~~ed as the desired p~e cross-over frequency m P . := Assume that the closed-loop system is not conditionally stable, which is desirable in practice. And according to the Nyquist criterion, to the above open-loop stable system, the stability conditions of the above delay-induced infinite dimensional feedback system are, (2.3 .23) 48 where iak are all the intersections of the root loci and the imaginary axis and ak > OJ P and µ 0 s is the guard band gain, which is used to guarantee some gain margin. r/J(OJ) is the phase of the loop transfer function C(s)G(s), which can be written as </>(OJ)= LC(OJ)G(OJ) = LC(OJ) + LG(OJ) (2.3.24) and then the above phase conditions can be written as, LC(OJ) >-n-LG(OJ) OJ E (O,wP] (2.3.25) i.e., in order to guarantee the above stability conditions, the phase of the controller is bounded by a phase plot below the phase crossover frequency. On the other hand, due to the periodic properties of the delay-induced open-loop frequency response, within bandwidth, the phase of the controller should be also bounded by the following phase curve, LC(OJ) > n- LG(OJ) OJ E (0,0JP] (2.3.26) To perform the phase design method in the delay-based controller design, the characteristics of different delay based controller must be investigated. Among all the delay based controllers, some major delay-induced control elements must be concerned, such that employing the combinations of these delay-induced control elements and all other basic delay-free control elements e.g., lead/lag compensator will be able to satisfy all kinds of phase design requirements. Here, three major used delay-induced control elements with three different frequency responses are introduced as follows, Type!. C(s)=1-be-sr,0<b:::;1,T>0 C(jw) = 1-be- 101 r = (1-bcoswT)+ jbsinwT = IC(jw)ILC(jw) IC(Jw)I = ~(1-b coswT) 2 +b 2 sin 2 wT /C(. ) _ 1 ( bsinwT J L ]OJ =tan 1-bcoswT The frequency response is shown in figure 2.3.4. Gain 1+b I I I I I I 1-b ----,----,------------ 0 0 ! I I I I I rtfT -1--- 1 I I I -1------ 1 I 2rtfT Figure 2.3 .4. Frequency response of 1-be-sr element Frequency Frequency 49 Type II. C(s) =I +be-sr, 0<b:-::;;1,T > 0 C(jw) =I+ be-fmT = (1 + b coswT)- jb sin wT = IC(jw )I L.C(jw) IC(jw )I= ~(1 + b cos wT) 2 + b 2 sin 2 wT /C( . ) _ 1 ( -b sinwT J L ]OJ =tan I +bcoswT The frequency response is shown in figure 2.3.5. 1+b 1-b 0 Gain I I I I ----,----r--------- Phase 1 I I I TIIT 2 TI/T Frequency I I _I ______ _ I I I ___ I _____ _ I Frequency Figure 2.3. 5. Frequency response of 1 + be-sT element 50 51 Type III. C(s) = e-zsT + ce-sT + d, d~l, c<O, 4d>c 2 C(jOJ) = e-fZwT +ce-fwT +d = (d +ccosOJT +cos 2mT)- j(c sinOJT +sin 20JT) = IC(jOJ)I L.C(jOJ) IC(JOJ)I = ~(d +ccosmT + cos20JT) 2 +(csinOJT + sin20JT) 2 /C(. ) _ 1(-(csinOJT+sin20JT) J L ]OJ =tan d + c cos OJT +cos 2wT The frequency response is shown in figure 2.3.6-8. and in the figures, ror is the following, [ _ (J4d-c 2 JJ 2 In 2 d + 4 tan 1 c OJ =~~~~~~~~~~ / r 4y2 L-=--- - -- -- - - (2.3.27) 1-c+d ·-----/legion I l+c+d~--.-----~------~._.,,j--,.._... __ -=--=-; Log-Frequency Phase .- - - - - - --, ---r.------=:..-=..;=-=::::::..=_ <I> max Log-Frequency Figure 2.3.6 Frequency response of e-zsr +ce-sT +d element (I) Gain ,-------------------------------- I ~o----------~--------- I I I .. - - - - - - - - - - - - - - - - - - 1- - - - - - - - - - - - _, Log-Frequency Phase ,-----·-----------~egion 1 <f>max ----------------.;.----------'----- Log-Frequency Figure 2.3.7. Frequency response of e- 2 sT +ce-sT +d element (II) in l+c+d I ~o - - - -1: ::-: =-- -:-_-:_-: .:i:: ::-: :-_ -:_-:_-::: ::-: - I I TI!f -Wr Log-Frequency Wr+1t!f Phase ~ - - - - - - - - - - - - - - - - - - - - - - -- - --.~ egion 2 I I <f>m~~ - - - - - - ~ - - - - - - - - - - - -,r-__._: _______ _ 0 - - - - - - -:- - - -.----------1j I I I I -<f>m-""1-----------------~ I 1 !. - - - - • I - - - - - - - - - - - - - - - -1- - - - -' I I Tiff-w Log-Frequency w +nff r r Figure 2.3.8. Frequency response of e- 2 sr +ce-sT +d element (III) 52 53 Based on the frequency characteristics of the above three mainly used delay- induced controllers, we can employ them and other finite dimensional compensators to construct the controller that can guarantee the above phase requirements. The phase design method together with the minimum destabilizing gain condition is a ---------·------- .. ......._____ ·-- -·· ·-··-·---~· . ' sufficie_ nt ~tab _i~~ty condition, it_ will always guarantee closecr=roopsiability, while in terms of the closed-loop bandwidth improvement, since it is an necessary condition, the closed-loop bandwidth improvement is not always guaranteed, which requires ,_____ so~~i~~~~nLb..a.@~!~h requirement. Thus phase design ---...... ·------- .. method is a systematic design procedure of delay-induced infinite dimensional controller that can guarantee stability and it can also increase the possibility of bandwidth improvement. This is not only the advantage but also the disadvantage of of phase design method in that it is not general and system dependent. Different bandwidth requirement with different controller structures may have different design complexity, although the design procedure is systematic. 2.4 Design Example of Phase Design Method Consider the non-collocated control of a 2-DOF mass-spring flexible system with one rigid body mode and one flexible mode. The dominant dynamics can be modeled by the following transfer function, G(s) = 5000 s2{s 2 + 10000) (2.4.1) The desirable closed-loop bandwidth is no less than 600 rad/sec, and at least 45 degrees phase margin and 1 OdB gain margin are required. The open-loop frequency 54 response is shown in figure 2.4.1. According to the phase design method proposed in section 4, we can easily find the following phase design boundary with the guard band frequency of 400 rad/sec shown in figure 2.4.2. 0~~-~~--~~~-~~~~~B~~~··-~~~~ -50 -100 ~ -~:~ -200 in ,-c 250 I 300 -350 -400 -450 I I I I I I I I -500 L---~~_......-~~~ ............... --~- 1 .._____.___. __ ~____. 0 10 10 1 10 3 10 4 Frequency (rad/sec) Figure 2.4.1. Phase plot of the open-loop plant with phase crossover frequency and bandwidth potential frequency 600 500 400 1 10 2 10 Frequency (rad/sec) WB W ~~ ~ (fl\'- l Figure 2.4.2. Phase design boundaries and some desirable phase plots " ,,- 55 Design 1. Full state feedback According to the above phase design boundary and bandwidth requirement, a full state feedback control has been employed. Based on the following state space representation, one static state feedback gain vector has been designed, which can satisfy the above performance requirements. 0 1 0 0 0 0 0 1 0 0 x= x+ u 0 0 0 1 0 0 0 -10000 0 1 y = [5000 0 0 o]x 1 x = 5000[.y y ji y] (2.4.2) u=Kx=[k 1 k2 k3 k ]x = kly + k2y + k3ji + k4}i 4 5000 K=[6.25xl0 7 3.875xl0 6 6xl0 4 6.23xl0 2 ] (2.4.3) Employing the transfer function transformation, we can easily derive the transfer function of the above full state feedback controller as follows, (2.4.4) Therefore the frequency response of the above full state feedback controller is shown in figure 2. 4. 3, from which we can see that it satisfies the phase design boundary conditions. To satisfy the stability conditions, we still need to check the frequency response of the loop transfer function, which is shown in figure 2.4.4 with infinity increasing gain margin and 80 degrees phase margin. 500 400 200 100 102 Frequency (rad/sec) lower phase bound 56 Figure 2.4.3. Phase plot of full state feedback controller and phase design boundaries. 100 iii' 50 ::!:!. c: '(ij 0 (.!) -50 100 50 0 --:- O> -50 41 e. ; -100 ~ a. -150 -200 0 10 : I : / : ~_ .... ,,,. :: : ; ./ p11~i:: •• , ., .•• ~~ ae gr1~es 102 Frequency (rad'sec) Figure 2.4.4. Frequency response ofloop transfer function for full state feedback system 57 And from the closed-loop frequency response shown in figure 2.4.5, we can see that closed-loop bandwidth is 700 rad/sec, which satisfies the performance requirement. al 5~~~~~-~~~..,---~~~-.-.--.--.--~~~~ 3 -------------------------------------- 01--------- -5 Bandwidth is 700rad/sec ~ -10 i:: "iii Cl -15 -20 -25 '---~~..........._~~-~~..._.______~-~L...-...L-~~~.....J 10° 102 Frequency (rad/sec) Figure 2.4.5. Closed-loop frequency response for full state feedback system. Design 2. Delay-induced with lead-lag compensator According to the above phase design boundary requirements, the characteristics of the type-II delay-induced control unit and also the characteristics of conventional lead-lag compensator, the following delay-induced and lead-lag compensator based infinite dimensional controller is designed such that all the stability and performance requirements are satisfied. The infinite dimensional controller with the parameters is shown as follows, C(s) = µ s 2 + 74 .4 7 s+ 4593 · 54 fl (I-b.e_,r ),T = 0.0005 sec,b, = 0.989 . (s+10)(s+30) k=I (2.4.5) b 2 = 0.995,b 3 = 0.99,µ = 9.9xl0 8 58 From the phase frequency response of the above controller shown in figure 2. 4. 6, we could find out the above phase boundary conditions are satisfied and from the frequency response of the loop transfer function in figure 2.4.7, we could see that the system is stable with 5 5 degrees phase margin and I OdB gain margin, which also satisfies the performance requirements. Additionally, from the comparisons of closed-loop frequency responses between full state feedback and the delayed induced lead-lag compensator based feedback systems in figure 2.4.8, we found that the closed-loop bandwidth of the latter is much larger than the former shown in figure 2.4.5, while the loop transfer function frequency responses of them are similar in low frequency region shown in figure. This suggests that the delay-induced infinite ---~--~--------:--:--:-_,,__-..-~---------~- dim a have larger closed-loop bandwidth than the full state ~-·-·· -··--" -- feedback. r- 600 ~IJ) ·-----------· 500 :::: upper phase boWld ~ 400 iii" 300 delay-induced/lead-lag ~ c: "(ii (!) 200 100 0 0 1 10 2 3 4 10 10 10 10 Frequency (rad/sec) Figure 2.4.6. Phase plot of the delay-induced and lead-lag compensator based feedback with phase design boundaries. ill ~ c " (ii C) 0 r--..' . 50 ~ .... ;' .. c-:-.:"""'-' .. :.;.; .......... ; ...... ; .... ; ... ; .. ; .. :::-~ ,... ---~~.....-:" "-..• . -- . . . ' . . . . . . . · AM ktate· -rJedba6k · : · '· .: : ; : :: . : . . . .. . . ··~:.:.: ····· ,~-,::l'!~-...;..;.J -50 .. , ... , .. ,.; ... ....... ,,~--- . .. ---.~N --:- : ' DDel~~JHn,1uc<!<L'l•~ad-~'ag ' 0 10 I 10 2 10 3 10 10 -600 '----'--'--'--'....L....1..-'L...LI...__.;._..___.___._._-'-'-'-...____.___.__.__._.__.._.L..LI..._...J-.-_.__.._._.--'--LI..J 4 0 I 2 3 4 10 10 10 10 10 Frequency (rad/sec) 59 Figure 2.4.7. Frequency response of the loop transfer function for the delay-induced and lead-lag compensator based feedback full state feedback 3 oi...-~~~~~~-----~------=::::::::::::--- -3 ------ -- ------ - ----- ------- - --- - ----------~ --- - -10 -40 -50 -60 Bandwidth=700rad/sec Bandwidth=l200rad/sec Frequency (rad/sec) I I Figure 2.4.8. Comparisons of closed-loop frequency responses of full state feedback and delay-induced lead-lag compensator based feedback 60 Design III. Delay-induced Employing both type I and type III delay-induced control units, we can also construct an infinite dimensional controller of (34) that can guarantee all the performance requirements, such as 700 rad/sec bandwidth, 55 degrees phase margin and I OdB gain margin shown in figure 2.4. 9-11. And the same phenomena are observed that delay-induced controller can to some extent increase the closed-loop l \ bandwidth compared with the full state feedback control. 1 I .£ () • I) V ~~ I '. ! C(s) =µ(I -be-n )(e- 2 sr + ce-sT +d~ V'-1" , T=0.0005sec. ~7~ l ~. ::~·.~::,:1:: . 03 =1.03; ~) »-? ,, (JV/ I 500 400 ~ 300 c 'iii <!l 200 100 10 0 delay-induced ~ , 10 2 10 Frequency (rad/sec) 3 10 4 10 Figure 2.4.9. Phase plot of the delay-induced feedback with phase design boundaries. '°' ~ c "(ij r> 100 50 0 -50 -100 0 10 1 10 2 10 3 10 10 200~~~~~ ,,~~~~~~~~~~~~~ -~ .. ~~~~~ .. 0 : : /: g -200 l- _ _ _ _,__ _ ~'.j- .. 7.t" .-:-;;:: ,.-- .. .... ,~ ..... -:'····<···:···: : ···:-. ,. .,. .,.. ....... ., .. .. .. , .. .. ., .. ., .. ., .. . .. ,.., .. --.....:.... ...... :.;..,,, '~·· ··-: ·: .. •·<·<· · :-i ~ ' ~~ ,,/" ~ full statej feedback ''n:~' -· -· -&. -400 ................................................. .......... ......... ....... .... ,. L ... ..... L -~ 1 -'-Yt. 1 Pfl~,~~<1. .... : ..... : .. .. ; ' .:~ .1''-~.U , , 0 10 10 1 102 Frequency (rad/sec) 3 10 : .\ 61 4 Figure 2.4.10. Frequency response of the loop transfer function for the delay-induced and lead-lag compensator based feedback in full state feedback 3 ol--~~--~~~-----~------==:::::::-- -3 -10 Bandwidth=700rad/ sec. -20 !::.. -30 -~ Bandwidth=l200rad/sec. <!> -40 -50 -60 -70~~~~~~~~~~~~~~~~~~~~~~__, 10° 10 1 10 2 10 3 10 4 Frequency (rad/sec) Figure 2.4.11. Comparisons of closed-loop frequency responses of full state feedback and delay-induced lead-lag compensator based feedback 62 Additionally, from the frequency response of the controllers of the above three different design methods in figure 2.4.12, we can easily see that the high frequency gain of the delay and delay-free control is much lower than that of the full state feedback, which means the attenuation of high frequency noise and unmodeled dynamics of the delay-induced control is much higher than full state feedback with guaranteed closed-loop performance. This suggests another advantage of delay- induced control over full state feedback control. : : : :_.. 250 >-····· ····· ······· Fu i ''" 1•~ ; "'..;;,,ai.-n · __,:.,.,"' :----:-...~.,.___=~ .= ·~· - . : .. -: ~ 200 ~- _,/,_ .;.;;~ ... ~ \ n ~ l'I -~ 1so~ ............... , ....... , .......... , ....... , .... , .... , ............. -... ~ .. ·~~~., ......... , ... ,./.,/~~··.,,, ........ , .. 111--..:·~ (.!) ...,..,,,,,,.. ~- 100r""'': ":···:"·:··::"'"'''":":"·c··":='-"_.,,,,.,,.~ ...... : ..... c.: .. : .. : .. : .. c: ...... : .... : ....... c.:.:;:,,+ ........ : .. F·f. SOL--~~_._._.._.J___.._._.__._._......J___.._.._._._....__...___._.._._.._........~-'-'--'-'"-~ 0 1 2 3 4 5 10 10 10 10 10 10 Frequency (Rad/sec) 400~~~~~~~.~~,~.~~~~~~~.~ .. ~.l~~~~ ~•lai.e f¢edlb~ck:-:.:_;_~ .... -400 l-----L-'-'-L-'-U...J-1-----L-'--'-'-"-LU-L-----'---L.....l-L..W.J..u,__-'---'---'-.l-l...J....L.l..L--'--~..J.....LI..UJ 0 1 2 3 4 5 10 10 10 10 10 10 Frequency (Rad/sec) Figure 2.4.12. Frequency Response for full state feedback controller and delay induced feedback controller. 2.5 Weakness of Existing Stability and Performance Criteria So far we have reviewed all the existing stability and performance criteria to deal with the linear systems including delay-induced infinite dimensional systems. And 63 also the so-called phase design method is introduced as a systematic design methodology for delay-induced infinite dimensional controller design. Specifically, to deal with the stability and performance of delay-induced infinite dimensional systems, all the existing methods including phase design method have some ·weaknesses, which are shown in table 2. 5 .1. Table 2.5.1. Weaknesses of existin dela -induced stabilit - erformance criteria Existing Method Weakness of the Method Nyquist Criterion Too general and complex to use, coupled phase and gain Bode based Method No guarantee stability Root-locus Method Too complicated to get infinite dimensional root-locus Difficult to predict the closed-loop performance Phase Design Method Not general and system dependent In details, the contents of the above table are explained as follows, (1 ). Nyquist Criterion is too general to use and it includes all kinds of situations which maybe very complex to design and analyze sometimes. In designing controller using Nyquist criterion directly, some constraints should be added to simplify the design procedure. And since the Nyquist locus contains both phase and gain information, which are coupled with each other, it is not clear in designing controller that satisfied some certain gain and phase requirements. 64 (2). Bode based Method is the decoupled phase and gain version of Nyquist criterion, and just because of this decoupling, it cannot always guarantee stability, i.e., we cannot say a system is closed-loop stable if Bode based Method shows it is stable, we must double check the stability by either root-locus method or Nyquist criterion. (3). The root-locus method is difficult to use to predict the closed-loop performance like stability margin and closed-loop bandwidth. And also, when the system is delay-induced, or of infinite dimension, the root-locus will be very difficult to generate and the whole design will be very complicated. ( 4). The phase design method is a systematic way to deal with stability and performance analysis of the delay-induced infinite dimensional systems, but it is not general and usually depends on the bandwidth requirement and controller structure. Therefore, more general and simpler method regarding to the stability and performance analysis of delay-induced infinite dimensional systems should be developed, which is referred to as the finite design region method proposed in chapter 3, which can make the delay-induced infinite dimensional system -- - --· ------- ------ - -- asymptotically approach a finite dimensional system and all the finite dimensional stability and performance criteria can be directly applied to the delay-induced infinite dimensional systems based on the finite design region principle. 65 3. Delay Transform Technique and Finite Design Region Principle 3 .1 Finite Design Region Principle In §2.3, we notice that to the 2-DOF mass-spring flexible system with one rigid body mode and one flexible mode, one can always employ some certain delay-based feedback to stabilize the system and in order to guarantee some closed-loop performance like bandwidth, he can always design certain parameters in the controller such that there is no crossing point with imaginary axis for the root-loci in the frequency range of [-m 0 , m 0 ] , where m 0 is the natural frequency of the flexible mode. Although in the stabilizing conditions, the departure angle conditions of all the open-loop poles of the plant have been satisfied, which are essentially the phase conditions, the minimum de · izing gain condition must be also satisfied according to ~ Nyquist crite~ The problem in estimating this minimum destabilizing gain is that due to the infinite root-loci caused by the dynamics of delay actions, the relation between these high frequency root-loci and their destabilizing gain is not obvious. In other word, we cannot guarante~ some certain relations -------- between the destabilizing gain of each high frequency delay-related root-locus. "-.._ is an infinite dimensional system, although the lant is of finite dimension, there are - ----------------- - infinite number of delay-related root-loci whose destabilizing gain could be so small - that the overall performance such as bandwidth cannot be guaranteed. Question is: 66 Can we find some conditions that can guarantee the infinite root-loci caused by high frequency dynamics of delay cross the imaginary axis with very high gain, which will not violate the performance in the low frequency range we care about? The answer is yes, and the condition is referred to as the following two lemmas. In order to introduce the above-mentioned conditions, some assumptions are given as follows, Assumptions: Here, the SISO minimum phase flexible systems with the following proper transfer function are considered, (3.1.1) And the feedback control configuration is shown in figure 3 .1.1. To guarantee the closed-loop stability by using delay-based controller, the following stability conditions should be satisfied, Re(:•J <0 µ µ-)0+ · f( 1 J µ<m - a C(ia)G 0 (ia) (3.1.2) where Skare poles of the open-loop plant and µC(s) is the delay-based controller. yr + Figure 3 .1.1. Controller-plant configuration 67 As discussed in §2.3, we know that the first condition is to guarantee all the departure angle conditions at the open-loop poles of the plant when the loop gain is small, and the second one is to guarantee the loop gain to be less than minimum destabilizing gain µer. However, since the delay induced feedback system is of infinite dimension, which means there may be infinite destabilizing root-loci. This --· makes the evah~~Jion._of the minimum· destabilizing gain v~ry complicated. From the ______.-...------ -~ - ----.. .. __ ____ common sense, we have an intuition that as the frequency of the crossing point with imaginary axis increase, the destabilizing gain should monotonically increases. However, in reality, this is true only when some certain conditions are satisfied. This condition is referred to lemma 3 .1.1. In order to introduce the lemma 3 .1.1, the following definition of magnitude bounded should be mentioned. Definition 3.1.1: Magnitude bounded is defined that the magnitude of a certain transfer function is bounded by a finite number in some certain frequency range. Lemma 3.1.1: Assume we have the above flexible systems and we can arbitrarily choose a frequency OJd, which is larger than all natural frequencies of the flexible poles and zeros of the plant. A delay induced infinite dimensional controller is employed to stabilize the overall system shown in figure 3 .1.1. If the delay-induced infinite dimensional controller is magnitude bounded in the frequency range [OJ d, oo) , then the destabilizing loop gain of each srossing point with frecui~es beyond OJ d --....._____._, 68 be formulated as follows: Assume iaoutk with aoutk+I ~ aoutk ~ OJd are all the frequencies of the crossing point with the imaginary axis for the root-loci of the system outside the frequency range [-m d, m d] . And denote µoutk are corresponding destabilizing loop gains of the kth crossing root-loci shown in figure 3 .1.2. If md is larger than any natural frequencies of the flexible poles and zeros of the open-loop plant and IC(Jro)I <M < oo within the frequency range of [md,oo), then we have µoutk+i ~ µoutk' which implies that the destabilizing gains of the crossing root-loci will monotonically increase as the crossing point frequency increases. 400 300 200 100 Cl) ~ 0 C> ro .E -100 -200 -300 -400 L---'~=:=:::=;~d~___.__ _ __.._ _ _.___J -400 -300 -200 -100 0 100 200 300 40 Real Axis Figure 3 .1.2. Root-loci of delay-induced infinite dimensional system 69 Lemma 3 .1.1 actually gives an easier way to evaluate the minimum destabilizing gain of all the crossing root-loci outside the frequency range [-OJ a, OJ a] . The minimum destabilizing gain outside [-OJ a, OJ a] is exactly the destabilizing gain of the first nearest crossing root-loci outside [-OJ a, OJ a] , which will greatly simplify the who le evaluation procedure of minimum destabilizing gain outside [-OJ a, OJ a] without calculating all the destabilizing loop gain of all high frequency crossing root- loci. However, in order to guarantee the overall closed-loop stability, the overall minimum destabilizing loop gain µer should be found. This only requires evaluating the minimum destabilizing loop gain of the finite crossing root-loci within [-OJ a, OJ a] . And the final stability conditions can be referred to lemma 2 as follows, Lemma 2: Denote the minimum destabilizing loop gain outside [-OJa,OJa] by µout, which is exactly the destabilizing loop gain of the first nearest crossing root-loci outside [-OJ a, OJ a] . Denote the minimum destabilizing loop gain within [-OJ a, OJ a] by µin. Then the final stability conditions are, ! Re(~kJ <0 µ µ~O+ µ<µer= min {µin' µout} (3.1.3) Although the stability can be guaranteed according to the above lemmas, the performance of the closed-loop systems (e.g., bandwidth, robustness) is still very 70 difficult to analyze due to the complexity of infinite dimension. As discussed before, for general delay induced system, since there are usually many delay terms in the controller, it is unsuitable to use so-called open-loop approximation of delay (e.g., Taylor's series or Pade's approximation) to approximate the infinite dimensional system by a finite dimensional system due to high order and inaccuracy. Here, a new kind of design technique called finite design region is developed to solve this problem very easily and accurately, which essentially employs two finite design regions. One is called DRC (design region of closed-loop system), which usually relates to the closed-loop bandwidth requirement and the other is called DRI (design region of isolation), which usually depends on the closed-loop robustness requirement. The definitions of the above two design regions are given below, Definition of Design Region: Assume that we have the above-defined minimum phase SISO system with n stable poles and m zeros in the LHP of complex plane (n>m), and we employ some certain infinite dimensional controller with infinite LHP poles and zeros. If we hope that the overall feedback system behaves as an Nth-order system (N~n) within a large pre-designated design region (DRC), the sufficient and necessary condition from root-locus point of view is that within the design region (DRC) there are only N branches of root-loci, which can be translated as some necessary and sufficient conditions imposed on the infinite dimensional controller. The condition is that the infinite dimensional controller has only K poles and L zeros in the LHP of complex plane within a greatly larger design region DRI (note DRI is 71 greater than DRC), with K=N-n, and L=N-m-1. And these K poles and L zeros should all reside in DRC, which means between DRI and DRC, there is no pole or zero of either plant or controller. This is why we call DRI the design region of ~isolation~---- This concept can be formulated as follows, Assume we have a finite dimensional SISO m1mmum phase plant with the following transfer function, m koil (s-zo) G(s) = /= 1 , n > m, Re(z 01 ) :s; 0, Re(p 0 ;) :s; 0, k 0 > 0 (3.1.4) TI (s- Po;) i=l And the pre-designated design region DRC (a flipped "D" contour in complex plane) is defined as follows, DRC: {Isl~ mdandRes ~ 0, wheres= (j + jm} (3.1.5) where m dis the radius frequency of the design region DRC. The infinite dimensional controller is of the following form, <X> kc fl (s - zCj) C(s) = ..,j=l ,Re(zCj) ~ O,Re(pc;) ~ O,kc > 0 fl (s - Pc;) (3.1.6) i=l And the desired closed-loop transfer function is 72 (3 .1.7) So within the DRI>>DRC, the infinite dimensional control must be of the following form, L kcIJ(s-zc) CDR-1(s) = /= 1 ,Re(zCj) s O,Re(Pc;) s O,kc > o,lzcjl sRDRc,IPcil sRDRC IT (s- PcJ i=1 (3.1.8) where K=N-n, L=N-m-1. And the design region DRI (a flipped "D" contour in complex plane) is defined as follows, DR!: {Isl~ w 1 andRes < 0, wheres= a+ jw} (3.1.9) where OJ 1 ( OJ 1 > > OJ d) is the radius frequency of the design region DRI. Note that we want all these K poles and L zeros must reside in the DRC, which means that there is no pole or zero of either plant or controller in between DRC and DRI. Based on the phase design method proposed in §2.3, some conditions of the design parameters of the controller could be found that can guarantee no crossing points with imaginary axis for the root-loci of the system within the frequency range of [-OJ 1 , OJ 1 ] and according to the lemma 1 and 2, since OJ 1 >> OJd and OJd is chosen 73 larger than all natural frequencies of the poles and zeros of the plant, the minimum destabilizing loop gain µer of the overall system is the destabilizing gain of the first nearest crossing root-locus outside [-OJ 1 , OJ 1 ], which is µ 1 . Then since the design region DRI is much larger than DRC and if we choose the overall loop gain less than the minimum destabilizing gain, we can always neglect those high frequency delay- related dynamics outside [-OJ 1 , OJ 1 ] , because they will definitely be stable and with small gain due to the proper loop transfer function, which can be described as follows, Consider a general SISO minimum phase flexible system with a proper transfer function (3.1.10) And we can always find a magnitude-bounded controller C(s) such that the magnitude envelops of overall loop transfer function C(s)G(s) is monotonically decreasing outside the frequency range [-OJ 1 , OJ 1 ] . Thus assume that according to the phase design method, we find some controller such that the root-loci have no crossing poi.1t with the imaginary axis within [ -OJ 1 , OJ 1 ] and the first nearest crossing point outside the design region [ -OJ 1 , OJ 1 ] is ia 1 , then according to lemma 3 .1.1 and 3 .1.2, the minimum destabilizing loop gain of the overall system is just the destabilizing gain of this root-loci. 74 1 (3.1.11) µer= µl = -C(' )G (' ) za 1 0 za 1 And then with the loop gain µ < µer, all the high frequency dynamics caused by delay outside [-OJI, OJI] are stable. Additionally, due to the monotonically decreasing magnitude envelop of the loop transfer function C(s)G 0 (s) outside [-OJI, OJI], the high frequency dynamics have small magnitude, which can be neglected without significant influence. An example of monotonically decreasing magnitude envelope is shown in the figure 3 .1.3 . Cl) "O 0.09 0.08 0.07 0.06 ~ 0.05 0) <ti ~ 0.04 0.03 0.02 0.01 Monotonically Decreasing Magnitude Envelop OL--~-'--------'-~--'-1-----1-~L--"--l----'!~-'"-_y_-'-'""---~~'--1"""'--~~~ 0 100 200 300 400 500 600 700 800 900 1000 Frequency Figure 3 .1.3. Example of monotonically decreasing magnitude envelope 75 On th~he latger the design region ORI becomes, from the lemma ;> 3 .1.1 and 3 .1.2, we know that the larger the minimum destabilizing loop gain will become. And due to property of monotonically ecreasmg magnitude of the loop transfer function, the smaller magnitude the high frequency dynamics have, the less influence on the performance of the overall system will be. In the limiting case, when DRI becomes infinitely large, the µer will goes to infinity and the high frequency dynamics caused by delay will have zero magnitude, i.e., the infinite dimensional system will become exactly a finite dimensional system. This tells us system approac9 a finite dimensional system asymptotically. To sum up, the finite design region principle is a design method in dealing with the stability and performance of the delay-induced infinite dimensional system by using the finite dimensional design method, and eventually the infinite dimensional system can be equivalent to a finite dimensional system with arbitrarily small error. Moreover, let's check the finite design region in more details regarding the root- loci analysis. As shown above, the finite design region is a general method to design an infinite dimensional controller with guaranteed closed-loop stability and performance. This design methodology employs two design regions DRC (design region of closed-loop system) and DRI (design region of isolation) shown in figure 3 .1.4. While DRC is usually related to the highest natural frequency of all flexible . ~ 76 m~ of the plant or the desired closed-loop bandwidth and th~ DRI is usually .........__ related to the closed-loop robustness requirements like phase margm and gain ------------- margm. From figure 3.1.4, we can see that all the desired low frequency dynamics are within the design region DRC, which is based on the closed-loop performance requirement. And between DRI and DRC, there is no dynamics of either plant or controller. This DRI plays a role of isolating the high frequency dynamics caused by delay from entering the desired · closed-loop dynamics region DRC. And as the design region DRI becomes larger and larger, the delay-related high frequency dynamics will be farther and farther away from the desired closed-loop dynamics, which has less and less influence on the performance of the overall system. In other word, the eventual system's performance will be only dominated by the finite dimensional dynamics within the design region DRC, which makes the overall infinite dimensional system equivalent to a finite dimensional system. However, sometimes it is possible for some high frequency root-loci to go into the design region DRI, this can be seen from figure 3 .1. 5. In this case, since the finite design region has already guaranteed that there is no crossing point of the root-loci with imaginary axis within the frequency range [-m 1 , co 1 ] , eventually this root-locus will definitely go out ofDRI when the loop gain increases. Usually this problem can be compensated by enlarging the design region DRI, so that these dynamics can be more negligible. 77 300 200 100 en ~ 0 C') ctl .§ -100 -200 -300 -400~~--'-~--'~~~~~~~~~~~~~~~ -400 -300 -200 -100 0 1 00 200 300 400 Real Axis Figure 3 .1.4. Diagram of finite design region principle 400 300 200 100 en ~ 0 OJ ro .E -100 -200 -300 -400~~--'-~~'--~--'-~~~~--'-~~'--~----'-~--' -400 -300 -200 -100 0 100 200 300 40 Real Axis Figure 3 .1. 5. Diagram of finite design region principle with inside isolation region 78 In a word, the finite design region principle provides a general methodology to deal with the stability and performance analysis to the controller induced infinite dimensional system. 3 .2 Delay Transform Technique As.discussed in §2.3, for the delay-induced infinite dimensional feedback system, · one way to analyze the stability and performance is the phase design method, which can be generalized to be the finite dimensional root-locus method together with the minimum destabilizing gain conditions. However, since the phase design method is quite system dependent and complex, it cannot be applied to arbitrary system. And ----------~~~~------=::::=-. ------~----------- due to the asymptotic properties of the finite design region principle presented finite dimensional system with arbitrarily small error, the stability and performance - analysis of delay-induced infinite dimensional control system can be done through the use of finite dimensional root-locus method. However, due to the infinite dimension of the system, the stability is also limited by the minimum destabilizing .\A, k gain, thus the final stability and performance analysis method for delay-induced infinite dimensional feedback system is finite dimensional root-locus method -·- together with the minimum destabilizing gain conditio~ based on the finite design region principle. In order to satisfy the finite design region principle proposed before, some certain form of delay-induced controller must be employed such that only finite number of 79 poles and zeros locate within the design region and exclude all the other infinite number of high frequency zeros and poles due to the dynamics of delay outside the design region. And it is well known that the pure poles are physically realizable while the pure zero means differentiation, which is physically unrealizable. Thus, the using the delay-based infinite dimensional controller. In general, there are infinite number of delay-based controllers that can satisfy the finite design region principle and can be used to realize zeros. However, since all the zeros can be classified into either real or complex conjugated zeros and these two types of zeros can be realized by the following four types of delay transform techniques. D dt : Differentiator Delay Transform 1 -R - e T T-.O+ ) S it!j J>t y d· ~ UWI 'ril{v This delay transform technique can be used to realize the pure differentiator s (Laplace operator) or to realize any improper transfer function by transform s into delay space. PD dt : Proportional and Derivative Delay Transform µ(1-be- sT ) T-.o+ > s +a ' (0 < b ~ 1, a ~ 0) This delay transform technique can be used to realize the real zeros or to realize the improper controller with all zeros real. P D ! 1 : Second-order Proportional and Derivative Delay Transform- I f µ s2+cs+d (1-e-(s+a)T)(l-e-(s+b)T) r~o+ s2+cs+d (a,b,c,d>O) (s+a)(s+b) 80 This delay transform technique can be used to realize the complex conjugated zeros or transfer functions with all complex conjugated zeros. PD~t2 : Second-order Proportional and Derivative Delay Transform-2 µ(e-2sT + ce-sT + d) r ~ 0 + > s 2 +as + b (a b c d > 0) ' ' ' This delay transform technique is another way to realize any complex conjugated zeros or improper transfer function with all complex conjugated zeros. And based on the finite design region principle, the above realizations can have arbitrarily small error as long as the design region of isolation becomes large enough. And the detailed proofs for the above delay transforms are shown in §3. C. In the finite design region principle, in order to guarantee the closed-loop system to behave like a finite dimensional system within some frequency range, the design region of closed-loop DRC is defined and to guarantee some stability robustness, the open-loop system should behave like a finite dimensional system within a much larger frequency range, the design region of isolation DRI is also defined (shown in figure 3.1.4). Assume that the radius frequency of the design region of closed-loop ------- .-.. DRC is m d, which is larger than the natural frequencies of any poles and zeros of the - ------ plant, and the natural frequencies of all the poles and zeros of the controller must be also less than m d . This plays the constraints on the location of the zeros and poles of the delay-induced infinite dimensional controller within the design region DRC. For 81 the purpose of guaranteeing stability robustness, the radius frequency of the design region of isolation DRI is OJ 1 >> OJd and all the other high frequency poles and zeros of the delay-induced infinite dimensional controller are excluded outside DRI. Between the design region DRC and DRI, there is no pole or zero of either the open- loop plant or the delay-induced infinite dimensional controller, which is why we call it design region of isolation. And based on the above two radius frequencies of both DRC and DRI, we can always choose some certain amount of delay such that the above four delay transformed controllers can satisfy the finite design region principle and realize the zeros with arbitrarily small error. In detailed, to satisfy the finite design region principle for the above four types of delay transformed controllers, the following conditions correspondin~~ach_ different delay transformed controller should be guaranteed. For Ddt ~ PDdt, andPD! 1 controllers, the condition is, (3.2.1) For PD! 2 , controllers, the condition is, 1 .J4d-c 2 1 [ _ .J4d- c 2 ] OJ 1 = -(tan- 1 +kn) > OJ 1 (:::) O+ < T < - tan 1 +kn (3 .2.2) _ r .. c ~ c._ . ~, All the above four types of delay and delay-free controllers are called delay transformed controllers and from figure 3 .2.1 and 3 .2.2, we can see how these types 82 of delay transformed controllers satisfy the finite design region principle with all the parameters defined above. Although we know that the above three zero placement via delay controller can realize the corresponding zeros with arbitrarily small error, it is still necessary for us to double check whether the frequency responses of the above zero placement via delay controllers are equivalent to their corresponding finite dimensional zeros. From figure 3.2.3, we can see that with some certain delay, the frequency response of a PD controller (negative real zero) is almost the same as that of the corresponding PDdt -type controller, which proves the accuracy of the finite design region principle. And since the D dt -type controller has similar principle to the PD dt - type one, we find that Dd 1 -type controller has the similar frequency response to the pure differentiator, which is in accordance with the finite design region principle. 400 300 200 100 c:- ~ 0 ·~ ..§ -100 -200 -300 -400 '-----'------'----L----L--~-_,._ _ __.______, -400 -300 -200 -100 0 100 200 300 400 Real Figure 3.2.1 Finite design region diagram toDdt or PDdt -type controllers ~ ~ "6> ro .§ 200 100 0 -100 -200 -300 -400~~~~~~~~~~~~~~~~~~~~ -400 -300 -200 -100 0 Real 100 200 300 400 Figure 3.2.2 Finite design region diagram to PD~ 1 and PD~ 2 type controllers 60~~~~~~~~~~~~-,-,r-r-r~ , ~-.--~~~~ : :,Ii ~ss~ .......... , ............. , ...... , .. , .... " ....... .. ..... , ...... ., ... < .. .,, .. , ... , ........... " ... . , ..... ... " ,. 1 ,/ .... ,..~ CXl v ~ ~~- .. 2 / "§, / ~ 45~ ......... , ...... .... ; .... ; .. ; ... .. ;.; .. ......... ; ...... ; ..... ; .... ; ... ; .. ; .. ; ... ; .... ~ ... ; ....... ; .......... ; ... ; ... ; .. ~ - .-~ Ci cu e.. cu <JJ <':! &. 401--~.i...--'-.............. '-'--~~~ ....... ~:::!::::i...._i_i.....i...i.j__~-'---'-_...i_i_j_~ 0 10 100 80 60 40 20 0 0 10 1 10 Frequency (Rad/sec.) Frequency (Rad/sec.) 2 10 3 10 Figure 3.2.3 . Frequency response of PD (s+lOO) and PDdt-type µ(1-be-sr) controller DRC: cod=27txl00 rad/sec, DRI: cor =27txl0000 rad/sec, T=0.0001 sec. 83 -~ 60 ~ ~ ~ 40r---~~-7--~;.......~~~···:···: ~ c: Cl ~ 201-··················'······ ······················ o~~~~~~~~~~~~~~~~~~~~~ 0 1 2 10 10 10 Frequency (Rad/sec.) ~ 1~r······················:······ · ·····::· ·· ······:······, .... , .. , ... , . ....... ~, ............................................................ . e. 100 1-····················•'"·•···· ····-<•·········'.·······'.······'.·•·•·'.····!···>··<f·····················'.············ dl ~ 50 -· . I _/ OL-~~-'--~i..-"""=::::i:=:~·:_i___· ~'L'~~___j_~_j____i__i___j__i_.i_i_J 0 1 2 10 10 10 Frequency (Rad/sec.) Figure 3.2.4. Frequency response ofboth (s 2 +s+lOO) and PD! 1 -type 84 s 2 +s+lOO _ µ (1-b 1 e sT)(l-b 2 e-sT) controllerDRC: rod =27txl0 rad/sec, DRI: ror (s + lO)(s + 20) =2Jtxl000 rad/sec, T=0.001 sec. A general way named the finite design region principle to analyze the stability and performance of delay-induced infinite dimensional system is presented in this chapter. Due to the asymptotic property of the finite design region principle, we can always make some certain delay-induced controller which satisfies the finite design region principle approach a finite dimensional system with arbitrarily small error. Based on the finite design region principle, delay transform technique is introduced to asymptotically realize any improper controller, which can be used to realize the physically unrealizable full state feedback by output feedback, as will be shown in chapter 4. 85 Appendix: 3 .A Proof of Lemma 3 .1.1 · Assume we employ the following magnitude bounded delay-induced controller, µC(s) = µ[ CvFo(s)+ ~CvFJs)e-•T,] (3.A. l) and!C(Jw)I ~M < ro w E [wa,oo) , I where CDFi (s) are delay-free controllers and Ti>O. - _,:· V j< . - e The closed-loop characteristic equation is 1 + µC(s)G 0 (s) = 0 (3 .A.2) As shown above we have chosen a frequency w a such that w a i~ greater than any natural frequency of the poles and zeros of the plant and assume the crossing point frequency of root-loci beyond Wa is a, which definitely satisfies the characteristic equation as follows, Rewrite equation (3 .1.4) and substitute s=ia, yields _!_ = -C(ia)G 0 (ia) > 0 µ The following results can be derived from the above equation, _!_ = -C(ia)G 0 (ia) = !C(ia)G 0 (ia)! µ 1 =>- = jC(ia)!jG 0 (ia)I µ (3.A.3) (3.A.4) Since we already assume that the controller is magnitude bounded such that 86 (3.A.5) and the gain of the plant is also bounded when a ~ OJ d due to the following, since a----)oo, I / (3.A.7) V ' when a <oo, and due to a > OJ d , where OJ d is greater than all the natural frequencies of the open-loop poles and zeros, which means that the denominator of the following fraction is finite and nonzero, and the numerator is also finite. Since a finite number divided by a finite number gives a finite number, we can always find the upper bound of the following fraction, which is denoted by Ni as follows, (3.A.8) a a m m-1 + n-m-1 + ... +-0 S +an-ls + ... +an-m S Sm s=ia Therefore, the magnitude of the plant outside the frequency range [-OJ d, OJ d] is bounded by the following number, 87 (3.A.9) Equation (3. 1. 6) can be reduced to, 1 MN an-m -::;--=>µ~-- n>m µ an-m MN, (3.A.10) This implies that when a> md, all the destabilizing gain of high frequency root- loci will monotonically increase as a increases. 3.B Proof of Lemma 3.1.2 The proof is just to combine the following stability conditions and lemma 1. Re(~•J <0 µ µ~O+ (3.B. l) where iak are all the infinite number of crossing points with the imaginary axis for the root-loci of the system. 3.C Proof of Delay Transforms Proof: for D dt delay transform 1 . (1-e-sTJ 1 . (l-(1-sT+s 2 T 2 /2!- ... )) sT im = im =-=s T~O+ T T~O+ T T 1 -sT -e ~s T 88 ) Proof: for PDdt delay transform .~ ....... , 00 1 lim p(l-be-aT) = lim µ(1-b)(l-~)[J (1-~)(1-~) '- T~+ ( T~o+_ Zq. i.:n z" zk z e ln(b:)\ =-InW') + j = -z 0 v 0 ;,--1-~ T · ,µ (l-b) _.,,.. cr=: ~ - - - t ,be-~"T 70 :.µ(1-be-aT)j ~s-z 0<1'«1 ° Q ~t - I Proof: for P D! 1 delay transform 0 1- e-(s+a)T . · 1- e-(a+b)T ·1 lim = s +a, hm = s + b, µ = - - I T-+0+ T T-+0+ T y2 u ~ - - ( 2 d - :. lim µ _s +cs+ ) (l-e-<s+a)T)(l-e:;(·Hb)T )=s2+cs+d T_,,o+ (s+a)(s +b) :.[µ (s2 +cs+d) (l-e-(s+a)T )(I-e-(s+b)T )] ~s2 +cs+d (s + a)(s + b) 0<7'«l Proor:' for P D! 2 delay transform ' lim µ(e- 2 sT +ce-sT +d) = lim Ii(1-~J(1-~ J T_,,or , . ~ Pl T-+O+ k-0 z z G I''.~· . I ( -t ../4d-c: k )• :oZo , zk ~+J- tan + 7r ,µ= -5_ T c Q_+c+d) :. lim µ(e- 2 sT +ce-sT +d) = (s-z 0 )(s -:Zo) T-+0+ 89 3.D. Characteristics of some delay and delay-free control elements The following are some frequency domain characteristics of several basic delay and delay-free control elements. Consider the controller to be the following form: (3.D.1) Here, we want to find the locations of the open-loop poles pq 00 and open-loop zeros zq 00 of the above delay and delay-free controller. First let's denote the open- loop transfer function by G 0 (s), which has the following form: (3.D.2) The characteristic equation can be written as 1 + µC(s)G 0 (s) = 0 (3 .D.3) and it can also be written by 1 -- = f(s) = C(s)G 0 (s) (3.D.4) µ According to the definition of Zq 00 and pq 00 , (3 .D.5) ) 1 00 j(s) = C(s G 0 (s) = --~ -oo => µ --t O+ => s ~ Pq µ (3.D.6) Find conditions that guarantee system has open-loop poles of -co±)m, i.e. lsl~oo. (3.D.7) 90 Case 1. n-m=O, f (s) = (k 1 + k 21 e-sT )am ~ -oo => k 2 ame-aT e-J(J)T ~ -oo rr ~ -oo,m = ( 2 k; l)K for k 2 > 0 :.=> 2kn CY~ -oo,m = T for k 2 < 0 k=0,±1,±2, ... (3.D.8) Case 2. n-m>O 1 k a e-ar . f( s) = (k +k e-sT)a -- = 2 m (-l)n-me-J())T ~ -oo l 2 m (J"n-m lo-ln-m (2k + l)n CY~ -oo,m = for k 2 > 0,n-m = 21 T :.=> 2kn o-~ -oo,m = T for k 2 < 0,n-m = 21 2kn o-~-oo,m=y for k 2 >0,n-m=2l+l (3.D.9) (2k + l)n o-~ -oo, OJ = for k 2 < 0, n - m = 21 + 1 T Based on the above results, we know that no matter what parameters are in the above delay and delay-free controller, the open-loop poles are all at infinity in LHP, .' \ which are definitely stable and have no effect on the open-loop frequency response, which just behave like introducing pure zeros to the system. Here we choose the two types of delay and delay-free controller as the examples. 91 • b<l 00 µ(1-be-sr) = µ'(s-z 0 )IJ (s-zil)(s-Z; 2 ) i=l where _ _!_ _ _!_ + . 2kn _ z 0 - In b < 0, z . 1 2 - In b _ J , k - 1, 2, ... T I, T T / From the above analysis, we know that all the zeros of this kind of delay and delay-free controller are in the LHP of complex plane. And all the open-loop poles are at infinity in LHP. • b>l 00 µ(1-be-sr) = µ '(s -z 0 ) IJ (s- zil)(s- Z; 2 ) i=l where 1 1 . 2kn z 0 =-lnb >0,z; 12 =-lnb±J--,k =1,2, ... T ' T T From the above aqalysis, we know that all the zeros of this kind of delay and delay-free controller are in the RHP of complex plane. And all the open-loop poles are at infinity in LHP. This type of delay and delay-free controller is undesired due to non-minimum phase zeros. _____ ~sually we do not use this type of controller in the control of flexible system. • b=l 00 µ(l-e-sT) = µ'(s-z 0 )IJ (s-zil)(s-z;J i=l where - O - + · 2 kn k - 1 2 Zo - 'Z;1,2 - - J T ' - ' ' · · · 92 From the above analysis, we know that all the zeros of this kind of delay and delay-free controller are purely imaginary except one pure differentiator. And all the open-loop poles are at infinity in LHP. The same as before, to eliminate the non-minimum phase case, we get the following delay and delay-free controller, µ(1+be-s 7 ),0<b~1 (3.D.11) where µ(1 + be-sr) = µ 'fJ: (s- zi 1 )(s - zii), zi 1 , 2 = _!_ ln b ± j ( 2 k + l)n, k = 0, 1, 2 ... i=l T T and _!_ ln b ~ 0 0 < b ~ 1 T As seen above, the above form of delay and delay-free controller does not satisfy the design region principle, so we cannot use it to realize the improper controller with arbitrarily small error. Hence, it is not a delay-'transformed controller. 93 4. Delay Transform Technique and Finite Design Region Control Algorithms 4.1 Full State Feedback Realization: Analysis and Design It is well known that the full state feedback control can give linear system very \ good performance as long as the system is state controllable. And specifically, in flexible positioning cont.fol system, the following plant-controller feedback configuration is usually employed, which is shown in figure 4.1.1. To the full state feedback, the breaking point is put in between the state feedback block and plant to define the loop transfer function follows, L(s) = V(s) = K(sl -Ari B =NL (s) [](s) IJL(s) (4.1.1) In this section, all the state space equation of the system is preferred to be of the following form, ( 4.1.2) From the zero placement points of view, the full state feedback controller can be regarded as a way to arbitrarily place the zeros of the loop transfer function with different gain, which can be derived by using the transfer function transformation. In 94 this way, the full state feedback can be considered as a physically unrealizable improper output feedback controller. Yr+> O e C(s) v f- +.., e xr Figure 4.1.1 . Controller-plant configuration for output feedback and full state feedback In terms of loop transfer function, we can always change the above full state feedback into the unit output feedback system shown in figure 4.1.2. And after decomposition of the loop transfer function (L TF), we can consider the system as a ·- - new plant ~bich is exactly t_ l}e__ product of the inverse of the denominator of the L TF ............. • ' • ---------=---- -• ----- •• -- • -, - • • •- - • ••' • • • • • • - L - .r • a~ an improper c_gntr~xactly the numerator of t~-~.J~TF shown in ·---·----- figure 4.1.3. This improper controller can be consider-as some pure zeros with some certain gain, which will shift the closed-loop poles to the desired location. This is the so-called the zero placement mechanism of full state feedback. Based on the transfer function transformation, we can see that in common sense, the improper controller is unrealizable unless we have full states available. But in practice, not always can we have the full states be available, and sometimes we only 95 have small number of measurable states, which is why people often use the output feedback in practice. From the above loop transfer function decomposition, we can easily find that in order to realize the full state feedback by output feedback, the improper controller must be used. Since the improper controller is physically unrealizable, which suggests that the output feedback realization of full state feedback is very difficult or even impossible by using conventional controller. However, this is not true if we employ the delay transform technique and finite design region method discussed previously. Figure 4.1.2. Loop transfer function realization of the full state feedback in terms of output feedback Figure 4.1.3. Non-causal realization ofloop transfer function realization of full state feedback in terms of output feedback As shown above, we know that in order to realize the full state feedback by output feedback, we must find a way to realize the improper controller, which is impossible in common sense. However, in the previous section, the delay transform 96 I technique and finite design region method can easily realize the improper controller. This is why we use this method to realize the full state feedback by output feedback. In order to clarify this problem, let's take the non-collocated control of a 2-DOF mass-spring flexible system as an ex~mple shown in Figure 4.1.4 and see how we change the full state feedback into output feedback with improper controller and how ---~- we can realize the im ro er controller by delay transform technique and finite design ---- - - ---- region method, which is physically realizable. ~Controller~ m 2 t~------~jSensor Figure 4.1.4. Non-collocated control o~a 2-DOF flexible mass-spring systems il 0 0 I 0 x1 0 ~,-:. '( ~ .xl -OJl 0)2 0 0 il I (4.1.3) l l + = u il 0 0 0 I x2 0 .xl OJl l -co2 l 0 · O il 0 y = x 2 = [o 0 I o}x 1 il x2 xJ First, equation (4.1.3) is the state space representation of the 2-DOF spring-mass flexible systems. We can easily check that the system is controllable. So from the linear control theory, we know that we can arbitrarily place the closed-loop poles and give system arbitrarily good performance by full state feedback. Assume we have a designed full state feedback gain vector K, which satisfies our control performance requirement. And the control force is the following (shown in Figure 4.1.1.) 97 v = -Kx = -[k 1 x, + k 2 ±, + k 3 x 2 + k 4 ±J K = [k 1 k 2 k 3 kj x = [x, ± 1 x 2 xJ (4.1.4) In Figure 4.1.4, we know that we have only one sensor of displacement x 2 , which makes the full state feedback unrealizable. We must employ the output feedback control algorithms to this system. However, in order to realize the full state feedback by output feedback we must first find the equivalent output feedback controller to the full state feedback controller, which is the so-called generalized controller. This generalized controller is actually improper controller and can be found by transfer function transformation shown below. From the equation (4.1.3), we know that the actual output y= x2, and the transfer function from u to x 2 , which is denoted by The other transfer function from u to x 1 denoted by Gux 1 is and take Laplace transform on both side of ( 4 .1. 4), yields, V ( s) = - KX ( s) = -[ k 1 X 1 ( s) + k 2 sX 1 ( s) + k 3 X 2 ( s) + k 4 sX 2 ( s)] => V(s) =-[ k 1 Gux, (s)+ k 1 sGux 1 (s)+ k 3 Gux 2 (s)+ k 4 sGux 2 (s) ]U(s) => V(s) = (k1 +k2s)(s2 +w12)+ (k3 + k4s)w12 U(s) s 2 (s 2 + w;) Additionally, (4.1.5) ( 4.1.6) (4.1.7) 98 (4.1.8) Thus, V(s) = (k 1 +k 2 s)(s 2 +m 1 2 )+(k 3 +k 4 s)m 1 2 Y(s) m: (4.1.9) \\.// Equation (4.1.9) is the so-called generalized controller, which is improper and equivalent to the full state feedback controller shown before. Then we need to realize the improper controller by delay transform technique and finite design region method. In order to find the zero-placement via delay controller, we need to find the zeros of the generalized controller which are denoted by Zi, shown as follows, (4.1.10) Since the zeros are whether negative real or LHP complex conjugated, according to the delay transform technique in previous section, we can deal with these two kinds of zeros by four types of delay transformed controllers and find the delay transformed controller as follows, Case I. All zeros are negative real. 3 C(s) = µ IJ (1-bie-s1j) (4.1.11) i=l where all the parameters are chosen according the previous section. Case II. One negative real zero and two LHP complex conjugated zeros (4.1.12) 99 where Z(s) denotes the complex conjugated zero realization by PD! 1 or PD! 2 delay transform techniques with all parameters chosen accordingly described in previous section. To satisfy the finite design region principle, we need to define DRC and find DRI, which can be used to determine the amount of delays in the above delay transform controller, which has been presented before. Although the delay transform technique can realize any kind of improper controller, it does not mean that we can realize the full state feedback by output feedback to any system. Actually there are some conditions imposed on the input/output relations of the system to be controlled. Co. nditions on output feedback realization of full state feedback: {1) For the minimum phase SISO system who~e input/output transfer function has no zero or negative real zeros or~ damped complex conjuga~os or any combination ~ we can always use the delay transform technique method to realize the full state feedback by output feedback. On the contrary, if the open-loop transfer function has lightly damped zeros or even non-damped zeros, sometimes it might destabilize the closed-loop system using the generalized controller realized by delay transform technique. (2) For non-minimum phase SISO system, exactly realization of full state feedback by output feedback is impossible, some design tradeoff must be used. 100 The reasons are that there is always pole-zero cancellation in the generalized controller induced loop transfer function, if the open-loop system has non-damped or lightly damped zeros, there must have the same value of poles in the generalized controller to cancel them, which may sometimes cause instability problem to the closed-loop system, especially when the model i~ which is always true in the real world. The problem here is that to most of the process control systems, we can always realize the full state feedback by output feedback using delay transform 7 technique, whose zeros are always over-damped. While to the flexible systems, the lightly damped zeros might or might not show up in the open-loop transfer function, which means that sometimes we can realize full state feedback by output feedback and sometimes we cannot. The way to deal with the flexible system whose input/output transfer function does have light damped zeros is to employ some other techniques, which will not be covered in this dissertation. 4.2 _Advantages ofDTT-FDR In general, there are many advantages of the delay transform technique and finite design region control algorithms. Among all the advantages, probably the most important contribution is the realization of improper controller, which makes the physically unrealizable controller realizable. And some of these advantages over some existing control algorithms will be discussed below. Currently, most control engineering applications involve the output feedback, although it is well known that the full state feedback control can achieve arbitrarily 101 optimal performance to the linear systems theoretically. It is well known that the problem of full state feedback is the unavailability of full states due to the limited sensors or immeasurable states. Thus, in most cases, the full state feedback is physically unrealizable. The only choice is to employ the output feedback control laws. Among all the output feedback algorithms, one main branch is the observer based output feedback, which involves the conventional observer based, Kalman- ~----- Bucy filter based LQG/L TR, PMD-observer based and Gilchrist's observer based output feedback. And another branch is the compensator based output feedback. As we will see, the delay transform technique and finite design region control algorithm have more advantages over any of the above control laws even the full state feedback. 4.2.1 Advantages over LQG/LTR/Observer based output feedback As described above, the observer based output feedback is one main branch of output feedback control laws currently. And within this branch, the c~nventional observer based output feedback and the Kalman-Bucy filter based LQG/LTR belong to the same group, since both of them employ the finite dimensional controller, which are completely different with the PMD-observer based and Gilchrist's observer based infinite dimensional output feedback laws. In modern control theory, we know that the main contribution of the conventional observer based output feedback is that they can achieve exactly the same closed-loop frequency response as the full state feedback does with the assumption of error-free model and noise-free system. The reasons are that there are pole-zero cancellations when the system is 102 closed-loop, and the states of the observer become input/output unobservable, which behaves like no observer exists. But there are three major problems here. The first is that in the real world, there are always uncertainties in the model although these ~an be very small. The second is that the noise is always all over the world. The third is that although the observer based output feedback can have the same closed-loop frequency response as the full state feedback does, it actually cannot guarantee the same loop transfer function as the full state feedback even in some frequency range. It can be shown that sometimes they are completely different. Motivated by the above three problem, the LQG came into being, which take some kind of noise into account, and to some extent solves the second problem, while it still can't solve the first and third problem which are essentially the robustness issues. LTR is a kind of method called loop transfer recovery, which tries to recover the loop transfer function of the full state feedback to the observer based output feedback or even LQG. However, the design procedure of LQG/L TR is complicated. And also, as long as the loop recovery error [29] becomes small, the high gain and high bandwidth controller must be used, which will essentially deteriorate the signal-to-noise ratio in the control channel. The advantages of the proposed delay transform technique and finite design region are the following. First, basically the whole design is to try to realize the same loop transfer function as the full state feedback with arbitrarily small error, so it is more direct than any of the above methods. Second it can realize almost the same 103 loop transfer function within the large design region, which will definitely possess the robustness of the full state feedback. The third, since within a large design region, the loop transfer function is almost the same as that of the full state feedback, eventually the closed-loop transfer functions of both the full state feedback and delay transform technique will be almost the same within the large design region. And the fourth is that there are no assumptions on model error and noise, so it is not conservative. And the fifth advantage, which is the very important, is that the whole design procedure is very simple and easy to implement. The advantages over LTR are that (1) the design procedures are much easier. (2) It has been shown that with the same loop recovery error within some certain frequency range, we can always find a DTT & FDR controller, which is less sensitive to high frequency noise than the L TR method. 4.2.2 Advantages over full state feedback/LQR So ~ar we mention all the time that we try to realize the full state feedback by delay transform technique, which does not mean that the full state feedback is the best. Actually, the main reason that we try to mimic the full state feedback is that it has been well regarded as the best control law from the theoretical point of view. The problem here we want to mention is that in some sense, the delay transform technique is better than the full state feedback. From the control input point of view, for the full state feedback, since the transfer function of generalized controller is actually improper, which means as frequency goes to infinity, the magnitude of the 104. frequency response of the full state feedback controller will monotonically go to infinity. This suggests that the full state feedback controller has very bad high frequency noise rejection ability. While the magnitude frequency response of the delay transformed controller will be finite as the frequency goes to infinity, this suggests that the high frequency noise rejection ability of delay transform technique is much better than the full state feedback controller. And on the other hand, since within the design region, the magnitude frequency responses of both the delay transformed and full state feedback are nearly the same, while outside the design region, the former is always less than the latter, which means the norm of control input of delay transformed controller is much less than the norm of the full state feedback control input. This implies that under the noisy circumstance, the control _ ene~he- filU state feedback is much more than that of the delay transformed controller. This is the main advantage of delay transform technique over full state feedback, which can be seen from the following simple example. Assume we have the following rigid body mass, we want to control the displacement of it. The open-loop transfer function of the plant is 1 G(s)= 2 s we have the following two choices of controller, one is the PD controller which is essentially the full state feedback controller, and the other is the PDdt -type controller with the parameter chosen according! y. And assume there is a high frequency sinusoidal noise w=0.005sin(10000t). And the controllers are as follows, 105 - ---------- C2 ( s) = µ(1-be-sr),b = e-T ,µ =6,T = :0 The frequency responses of both the above controllers are shown in the following figures, al ~ c "iii Cl Ci CD Q. CD Ill Ill .c a. 80 60 40 20 0 -20 10" 1 100 50 0 -50 -100 10" 1 I I t I 1001 I I 0 I I Ill . . . . ..... . .... ,, , , I I I 0 11111 I I I I I 1101 I 0 It 11000 t I I I I 1111 I 0 0 I 11011 t I I 0 0 0101 101 Frequency (Rad/see) I I t I 01 II I I I I ot11 t I I I II II I I I I 0011 I I I I Ottl Figure 4.2.1 Frequency Response of PD controller and DTT controller QI Ul 60 40 20 i 0 QI 0: c. QI I u=; -20 -40 -60 -ao~~---~ ............ ~-'---~~~---~~~---~---~........._~-' 0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 2 Time (sec.) Figure 4.2.2. Control input of PD control 100 80 ~ 60 0 Cl. (I) CJ 0::: Cl. 40 CJ u; 20 -20 L___L___L_____J_---.L_--1_--1_---.L _ ____._ _ ___J_ _ ___j 0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 2 Time (sec.) Figure 4.2.3. Control input ofDTT control 106 For step input, the control input versus time plots of the PD and DTT control system are shown in Figure 4.2.2 and 4.2.3. Both the frequency response of the two types of controller and the time domain response of control input show that the DTT control is better than the full state feedback in terms of noise rejection in control input and control energy consumption. 4.2.3 Advantages over PMD controller Bien (1979, 1980) proposed a delay based feedback control algorithm called proportional minus delay (PMD) method and he claimed that PMD can be regarded ~----------- .. as the averaged PD control, which makes the IT AE (integral of the time multiplied ---~ - by the absolute error) of PMD feedback control less than that of the PD feedback ------ control. And based on this characteristics of PMD, he also claimed for an SISO 107 irreducible plant, i.e., the state vector can be computed from the linear combination of the output and its derivatives, then the state observer can be realized though PMD method, for simplicity, here, we just call it PMD-:observer, which is obviously different from the conventional observer due that it's of infinite dimension. However, actually the PMD method is one special form of our delay transformed controllers, which is Ddt-type controller, and it is time for us to talk about the advantages and disadvantages of all types of delay transformed controller including the Ddt-type (PMD) controller. In order to compare these four methods, let's first see some details of the PMD- method. From the time domain point of view, the pure differentiator control is . D-control: u(t) = e(t) and the PMD controller can be regarded as the averaged derivative control as follows, PMD: 1 1 t • u(t) = -[ e(t)-e(t -T)] = - J e(t)dt T T t-T while in the frequency domain, both the PD and PMD controllers are the following, D-control: U(s) --=s E(s) PMD= Ddt: U(s) 1-e-sr =--- E(s) T 108 ~ To clearly see the advantages and disadvantages of the four types of DTT controllers, let's take two examples of conventional PD control and PD+D 2 control realizations by all the DTT controllers respectively and see what the differences are. (I). Assume we want to realize the following PD controller by both Ddt (P11D), and PDdt, controllers, 0.1. C(s)=10s+1 For each of these two methods, assume the amount of delay is chosen as T=l, CPMD(s) = 1 + lO(l ;e-.r) = 11-loe-', T = 1 CPMD(s) = 1+lO(l-e-sT)=101- lOOe-o.is ,T = 0.1 T which has all its zeros as follows, z 0 =-0.0953,z; 1 , 2 =-0.0953±2ni,i=1,2, ... ,T=1 z 0 =-0.0995,zi 1 , 2 =-0.0995±20ni,i=1,2, ... ,T==0.1 2.) PDdt C (s) = 10 (l-e- 0 · 1 e-s)=10.5-9.5e- 0 · 1 s T = 1 ZPD (l- e-0.l) ' which has all its zeros as follows, 109 z 0 = -0.1,zi 1 , 2 =-0.1±2;ri,i=1,2, ... ,T = 1 z 0 = -0.1,zi 1 , 2 =-0.1±20;ri,i=1,2, ... ,T = 0.1 From the exact loop transfer function realization point of view, one of the zeros of PDdt is exactly the zero of the PD controller. And all the zeros of Ddt (PMD) are different from those zero of PD controller. Here, in terms of exactly zero match, PD dt is better than D d~' which is the advantages of PD dt and it is suitable for inverse dynamics feedfdrward control, where the exact zero match is needed and of course, it can be used in the feedback control of negative real zero realization. However, as we will see, in terms of frequency response in both phase and magnitude, especially with large delay, the Ddt (PMD) is better thanPDdt. This can be seen from the frequency response of the above two PD control realization of both Ddt(PMD), PDdt. 80 60 al % 40 ·n; (.!) 20 0 -2 10 100 50 0, Q) 9. 0 Q) (/) ca .r::: Q._ -50 -100 -2 - 1 0 1 10 10 10 10 Frequency (Rad/sec .) Figure 4.2.4. Frequency Response of PD, PDdt and Ddt (PMD) controller with T=l m ~ c "iii C> C> Q) g, Q) rt> (IS .£: 0... 100 80 60 40 20 100 50 0 -50 -100 10- 1 : 101 Frequency (Rad/sec.) 110 Figure 4.2.5 . Frequency Response of PD, PDdt and Ddt (PMD) controller with T=O. l From the above two frequency responses, we can see that when T is large, D at (PMD) has larger low frequency gain and larger low frequency phase lead than PD and PDdt, which is good in terms of disturbance rejection, while when T is small, both PDdt and Ddt (PMD) are almost the same. This suggests that to realize the PD control, if th~ delay is large, in terms of disturbance rejection within bandwidth and phase margin, PMD is better than PD and Z1-i, but can not guarantee the same loop transfer function as the PD control, which means the closed-loop performance can not be guaranteed. While in order to recover exact the loop transfer function of PD control and guarantee the same closed-loop performance within the design region, the PDdt must be used as Tis large. And when Tis small, both Z1-1 and Zi-2 (PMD) 111 equivalent and both of them can be used to realize the PD control. Question rises that, how about high order PD control? Is Dar (PMD) still better than other delay transformed controllers? The answer is no. When realizing the high order PD control, the Dar (PMD) may have large mismatch, while the PD;n and PD;r 2 can work well, which can be seen from the following frequency responses for PD+D 2 controller, Ddt (P:MD) and PD;n controllers. o~~~·~·~ .. ~ ·~ .. ~~~~~~~~~~~~~ -1 10 200 r-~~~.,.....,--~~--.--.--.......--.--.-,,----,--~.--.--.-....,.---.---,--....-.-,.--,.......,., . : : : : : ~ : + 02 : . " " " " . ~ 100 _______ ; ___ _ ; __ _ L _ _;__;_~-L!i--------L---L--: .. :.:: ·_; - - gi : : : : : l T Zll-~ i l l : i : : e. 0 l l i l i l ill : : : :_; __ ;_;: i--------~- -+-+-:--. f-:+:-- i j j l l jj lll l l l i l llll z:-2 j j l j)\ i ~ - 100 -------r·--rrnn1r·----r--Trrni-rr·-----r-- 1 --·r-rrr-rr: - ----r -200~~~~~~~~~~~~~~~~~~ 10" 1 10° 10, 10 2 10 3 Frequency (Rad/sec.) Figure 4.2.6. Frequency Response of PD+D 2 , PD;n and Dar (PMD) controller with T=0.05 From the above analysis, the advantages of Dar are that when Tis large and it is suitable to realize negative real zeros to increase the disturbance rejection ability and phase margin. And the disadvantages are that it cannot guarantee the equivalent closed-loop performance as the PD control gives. When Tis small, both PD at and 112 D at are nearly equivalent, though there is mismatch in D at, the error can be ignored as T decreases. In realizing the high order PD control, the mismatch of D at with the original PD controller and PD;n controller becomes large, if T is large, so in terms of equivalent closed-loop performance, the DTT controllers except Dat are preferred. But as T goes down, verything will be fine and any DTT controller will work well. In general, when delay can be small enough, all the above DTT controllers can be used to realize any improper controller with guaranteed performance. Only when T is not small, some choice should be made in order to acquire better performance in terms of either disturbance rejection ability or other performance. 4.2.4 Advantages over Gilchrist's observer based output feedback Gilchrist (1966) proposed the n-Observability, another kind of delay based observer, which made use of the delay actions to reconstruct the state vector from the output. However due to the complexity of the algorithm, few people actually use it. But the idea is very good and the observer form is a little bit like PMD-observer, but it is more general. The problems of this algorithm are that it has no guarantee that the zeros of the loop transfer functions are exactly the same as those of the full state feedback loop transfer function, which means exactly loop transfer recovery, and it is very similar to the PMD observer. While the DTT method can recover the loop transfer function of the full state feedback with arbitrarily small error. Another problem is that the closed-loop stability and even performance are difficult to predict, because of infinite dimensional system. This is also the problem of PMD 113 proposed by Bien (1979), which did not give a general method to predict the stability and performance to the delay induced infinite dimensional system. While the delay transform technique and finite design region are more powerful in dealing with this problem as shown before. Thus the finite design region based DTT control algorithm has great advantages over any of the existing delayed based control algorithms. Also, as we will see, the DTT method can be also used in realizing some feedforward control schemes like inverse dynamics and minimum residual vibration control, etc., while the PMD can never be used. This suggests that DTT is more general than the PMD, which is actually a special scenario of the DTT method. 4.2.5 Advantages over compensator based output feedback The other branch of output feedback is the compensator based output feedback, like H-infinity control, conventional compensator, etc. The advantages of the generalized DTT over the compensator based output feedback are that the compensator is usually proper or even bi-proper transfer function for realizability, which will eventually increase the order of the overall system, while as we know, the DTT control can realize the improper controller, which will not increasing the order the whole system. As the order of the system increase, both the stability and performance will possibly deteriorate. And on the other hand, the DTT controller is much easier to design than high order compensator, for example, H-infinity controller is very difficult to design. Above all, the proposed DTT-FDR has many advantages over the existing control algorithms, which makes it very useful in developing high performance flexible servo systems. 114 4.3 Systematic Design of DTT & FDR Controller So far, we have talked about the delay transform technique method and the finite design region principle in details and its advantages over many existing control algorithms. Another important thing is that we need to find a systematic design procedure for the DTT-FDR scheme. The DTT method can be applied to either feedback or feedforward control, when dealing with the feedback control, since the stability and closed-loop performance should be guaranteed, the finite design region should be used. And to deal with inverse dynamics feedforward control, in order to guarantee the performance requirement, the finite design region also must be used. While in the minimum residual vibration control, the finite design region is not necessary to use. However, only the systematic design procedure for the full state feedback and inverse dynamics feedforward control will be discussed in this dissertation. 4. 3 .1 Full state feedback control In section 4 .1, we discussed that under some conditions that impose on the input/output relation of the plant to be controlled, we can always realize the full state feedback by using delay transform technique output feedback. As to those systems that do not satisfy the previous conditions, they should be controlled by other control technique, which will not discussed in this dissertation. Here we just focus on the system that satisfies these conditions, which means the full state feedback can be realized by output feedback. The generalized design procedures of the full state 115 feedback realization to these systems by using the zero placement via delay and "- finite design region algorithms are presented as follows. Assume we have a general SISO minimum phase plant that satisfies the conditions in section 4 .1 and both the input output transfer function and the state space representation are shown as follows, Xn xl = AnxnXnxl + BnxlU y = cl xnx (4.3.1) (4.3.2) Assume we have designed a certain full state feedback gain vector K that satisfies some certain performance requirements by either arbitrary pole placement or LQR (given certain Q, R matrices). The loop transfer function of the full state feedback system is, L(s) = NL (s) = K(sl -At 1 B DL(s) (4.3 .3) Then the generalized output feedback controller that is equivalent to the full state feedback controller is, ( 4.3 .4) - 116 Employing the delay transformed controller including D dt, PD dt, PD~ 1 and PD~t2 to substitute the above real or complex zeros respectively, or even use Ddt to directly substitute each s operators in the numerator of the above generalized controller. According to the closed-loop performance, define the design region DRC and initially choose design region DRI to f times of DRC, where f is an arbitrary chosen positive number. According to the size ofDRI, decide the value of delay in the DTT controller and then decide all other parameters in the controller. And then check the stability by the lemma 3.1.2 and check whether the closed performance like control bandwidth, robustness, etc. are satisfied or not, if not, increase f and iterate the check until we find the smallest design region DRI that can guarantee both stability and performance. Another way is to initially choose f to be a very large positive number like 1000, and decide all the parameters in the DTT controller accordingly. Usually, with very large DRI, both the stability and performance should be satisfied, check both stability and performance, if satisfied, reduce f and iterate the above steps until the minimum DRI is found, whose parameters are the parameters for the final designed controller. The whole systematic design procedures are summarize as follows, 1.) According to performance requirements design full state feedback gam vector. 2.) Calculate the loop transfer function of full state feedback system. 117 3.) Calculate generalized controller. 4.) According to closed-loop performance requirement, usually closed-loop ~Rd·~i~t,h. de~ne design regi C::\ / ~ ~ 5 .)~ .• 6.) Using DTT transform to realize the improper generalized controller and calculate parameters in the DTT controller according to DRI. 7.) Check whether stability and performance are satisfied or not. /~£.and repeat S~7·untilminimum DRI..is.found. If not satisfied, increase f and repeat 5-7 until minimum DRI is found. / --····· 4.3.2 Inverse dynamics feedforward control ----._/// The DTT-FDR can also be applied to the inverse dynamics feedforward control, I which can realize the improper inverse dynamics and cancel all the dynamics of the plant to be controlled so that the eventual input and output become identical theoretically. Actually, here, we only focus on the minimum phase plant, and the DTT-FDR control can only realize the inverse dynamics within the design region DRI, with· all the high frequency mismatch ignored. The design procedure is very simpler than the feedback case, which is presented as follows, 1.) According to the design performance requirements, define the design region DRC. 2.) 118 3.) Using DTT transform to realize the improper inverse dynamics and calculate parameters in the DTT controller according to DRI. 4.) Check whether performances are satisfied or not. If satisfied, decrease f and repeat 2-4 until minimum DR is found. If not satisfied, increase f and repeat 2-4 until minimum DR is found. As to other applications of the delay transform technique and finite design region either in feedback control or in the feedforward control, or even the combined feedback and feedforward 2-DOF control scheme using DTT-FDR algorithms are going to be investigated in the future. 4.4 Discrete-time DTT & FDR design The control of physical systems with a digital computer or microcontroller is becoming more and more common. Examples of elctromechanical servomechanisms exist in aircraft, automobiles, disk drives and robotics. In this dissertation, we only assume that single sampling rate is employed and the sampling frequency is very fast such that the whole digital system can be designed through continuous time, which is referred to as the emulation method. Since the finite design region principle gives us the guidance of choosing· the amount of delay, which will be substituted by the sampling time in the discrete-time system. So we can intuitively employ the continuous-time design methodology to decide the sampling rate such that the closed-loop performance like bandwidth and robustness can be satisfied. And since we assume the sampling rate is fast enough, 119 the error caused by sampling rate and quantization can be ignored. Then the delay transformed controller can be easily transformed into z-domain through z- 1 =e-sT' and other part of finite dimensional transfer function can be transformed into z-domain through frequency response invariance method. In detailed, the four types of delay transformed controller can be transformed into z-domain as follows, D dt : Differentiator Delay Transform which is essentially the backward z-transform PD dt : Proportional and Derivative Delay Transform PD;t 1 : Second-order Proportional and Derivative Delay Transform-I Z (µ S 2 +CS+ d (1- e-(s+a)T)(l-e-(s+b)1») =. z( S 2 +CS+ d )µ(1- e-aT z-1)(1- e-bT z-1) (a,b, C, d > Q) (s + a)(s +b) (s + a)(s+b) Th .c-. f s 2 +cs+ d b b . d h h . h T . e z-trans1orm o can e o tame t roug usmg t e ustm z- ( s +a)( s + b) transform PD;t 2 : Second-order Proportional and Derivative Delay Transform-2 Z (µ(e- 2 sT + ce-sT + d)) = µ(z- 2 + cz- 1 + d) (a, b, c, d > 0) 120 5. Delay-induced Filtering and Transient Response Compensation 5. 1 Introduction In the signal process and control theories, filter (e.g., low-pass, high-pass, band pass or band-stop.) or compensator (e.g., lag, lead, lead-lag, lag-lead.) is widely investigated in finite dimensional domain [9,25]. The infinite dimensional filter and compensator have been rarely studied in the literatures. With the recent reports on the delay-induced infinite dimensional controls, it is worthy of investigating the delay-induced infinite dimensional filtering or compensating techniques and establishing the fundamentals for infinite dimensional signal processing and control systems. Here, the delay-induced filtering or compensation technique is proposed, which can achieve lots of unique advantages over the conventional finite dimensional methods. And since the performance for the filtering or compensating is usually defined in frequency domain, which is in the steady state, the delay-induced filtering and compensating techniques in here are also designed to satisfy these frequency domain performance requirements. However, in terms of time domain behaviors, the major side effect of delay-induced filtering or compensating is the deterioration of transient responses, such as large overshoots/undershoots. The solution is referred to 121 initial function compensation, which can reduce or eliminate the transient response deterioration caused by delay actions without affecting the frequency domain performance. 5.2 Delay-induced Filtering In [21] and [22], the delay-induced controllers are reported having the ability of realizing any zero structure asymptotically, which suggests any finite dimensional filter or compensator can be transform into the delay-induced infinite dimensional domain with the zero-structure realized by delay-induced units and with eigen- structure remaining the same as before. The transformed delay-induced filter or compensator will always be realizable no matter the original one is physically realizable or not. In [21 ], we know that the following two basic types of delay-induced units can be used to realize the real zero and c~mplex C()njugated zeros respectively. (5 .2.1) .----------· - · - --. .. C(s) = µ(e- 21 T +ce- sT +d) ~(~+a ·+ jw)(s .+a- jw) d = e2uT C = - 4d µ = (]'2 + (t)2 ' l+tan 2 (wT)' l+c+d (5 .2.2) And the magnitude frequency responses of the above two types of delay-induced units are always bounded in the overall frequency range, which can be shown as follows. 122 (5.2.3) (5.2.4) This suggests that zero structures realized by the delay-induced units must have bounded magnitude frequency response and hence the magnitude of a general delay- induced filter or compensator is always bounded, which usually has delay-induced zero-structure and finite dimensional eigen-structure. Assume a general finite dimensional filter or compensator is of the following form and has p real zeros and (n-p )/2 pairs of complex conjugated zeros, where n is the total number of zeros of the transfer function of the finite dimensional filter or compensator. (5.2.5) Note that in the above transfer function, n can be less than, equal or even larger than m, which will cause the total transfer function proper, bi-proper or even improper. Specially, when n is greater than m, the improper transfer function is physically unrealizable in the finite dimensional domain. However, by employing the delay-induced units, this transfer function can be made realizable. Employing the two delay-induced control units to realize all real and complex zeros respectively and then multiplying them will give us the following delay- induced controller, 123 n-p p 2 ITµi(l-bie-sT) ITµj(e-2sT + cje-sT +dj) Cn(s) = i=I J=I ,n > m bmsm + bm_,sm-I + ... + b 1 s + b 0 (5.2.6) From Equation (5.2.6), we can see that as the order of denominator increases, the high frequency gain of the overall delay-induced filter or compensator will become smaller and smaller, which means higher signal-to-noise ratio, lower sensitivity to high frequency noises and lower power consumption can be achieved. These are the major performance improvement of the delay-induced filtering or compensating compared with the conventional finite dimensional one. To speak in details, the high frequency magnitude decaying rate of the finite dimensional filter or compensator is proportional to the relative order of filter or compensator, while that of the delay- induced filter or compensator is proportional to the order of the denominator, which can be seen from the following Bode analysis. In Bode analysis, the decaying rate of magnitude for the transfer function of Equation (5 .2.5) in high frequency region is -20x(m-n) dB/decade. While for the transfer function of Equation (5.2.6), it is -20xm dB/decade. This tells us no matter what relation between n and mis, the decaying rate of magnitude of Equation (5.2.6) is always negative except when it is purely improper (i.e., m=O). However, the decaying rate of magnitude of Equation (5.2.5) is much more dependent on the relation between n and m, which can be zero, positive or negative. This tells us, for 124 the same denominator, the finite dimensional compensator could be either realizable or not and always has larger high frequency gain than the corresponding delay induced compensator, which is always realizable. Another unique performance improvement of the delay-induced filtering or compensating is that when it is used in a feedback system, the overall closed-loop bandwidth may be wider than that of the conventional method with appropriate tuned controller parameter, which can be seen in the simulations in section 5.4. 5.3 Initial Function Compensation Recently time delay control is of great interests, however, as discussed before, due to the transient response deterioration, in most systems, this method is not good to use. Elimination of this becomes a must for implementation of delay-based control. This suggests that instead of using zero initial condition, some compensation for the initial values of time delay action should be employed, such that at the beginning of the control, all the negative time initial values are available. This will help make the overall system stable from the very beginning of the control. On the other hand, since the delay-based controller is designed such that certain steady-state performances are satisfied, the initial value compensation should not change the steady-state response of the system. Assume the output feedback control system configuration shown in Figure 5.3.1 is employed, and specifically, the tracking error initial function needs to be predicted to compensate the initial values of the delay-based controller. 125 Figure 5 .3 .1. Delay-induced output feedback configuration The way to derive the initial function here is through use of inverse Laplace transform technique. And since we are trying to predict the negative time initial function for delay-based controller, we can always think that when the time was negative, no feedback control was used, which means only forward loop is concerned. An_d-assume the desired steady state response of the overall feedback .........._.._ ___ _ system is Y*(s), then the desired tracking error initial function E*(s) with desired steady-state output Y(s) will satisfy, E•(s)= y•(s) C(s)G(s) And the time domain function E• ( t) is E• (t) = g-1 { y• (s) } C(s)G(s) (5.3.1) (5.3.2) This gives us a way to find out the tracking error initial function that will not affect the steady-state response of the overall feedback system. The delay-based ---------- - feedback control system with initial function compensation is shown in Figure 5.3.2. 4~ e c~_v_u_,..IG(s)J YI> T- ~ . Figure 5.3 .2. Delay-based output feedback configuration with Initial function compensation 126 As to eliminate the transient response deterioration caused by delay, only some segment of the above initial function is useful. This segment F1(t) with tE(to-T, to) satisfies the time boundary condition of F 1 (to)=-l, where T is the maximum amount of delay in the controller. This initial function will make the control effort at the time Ti, (all delays in the controller) smooth. On the other hand, the real transient response of the system is not sensitive to the prediction error of the boundary time to from numerical simulations, which means that within some bound, all the initial function segments are acceptable. Additionally, to some systems, the initial function segment that satisfies boundary condition is not unique, because E(t) may have some oscillations in some period of time. As a rule of thumb, to avoid some unacceptable phase shifts in systems response, the segment closest to the RHP is the choice. This initial function compensation can be applied to any delay-based control system to eliminate the transient response deterioration caused by delay without affecting the steady-state behavior, which makes the delay-based control system not only satisfy the frequency domain performance but also the time domain performances simultaneously. 127 5.4 Illustrative Examples To clearly see the dynamic behavior of the delay-induced filter or compensator and Initial Function Compensation, several examples are given as follows. Example 1. Delay-based output feedback control versus LTR output feedback control Consider an open-loop plant with the following transfer function, G(s)=s+50 s2 (5.4.1) And assume a reduced-order loop transfer recovery compensator in Equation (5.4.2) is employed such that the overall loop transfer function shown in Equation (5.4.3) can be achieved. C (s) = 800(s + 100) LTR S + 50 (5.4.2) LAs) = 800(s + 100) 52 (5.4.3) Using the delay transform techniques and finite design region principle, the delay and delay-free controller is obtained as follows, 8.4067x10 5 (1-0.9048e-o.oois) c DTT (s) = ------'-------' s + 50 (5.4.4) The difference of the frequency responses between L TR and DTT controllers can be seen from Figure 5 .4.1. And from that, we can easily see the high frequency gain of the DTT controller is much lower than that of the LTR controller, and the control bandwidth ofLTR is infinity while the DTT controller has finite control bandwidth, 128 which implies that the control effort of the DTT should be smaller than that of the L TR and the signal-to-noise ratio in control channel of DTT should be higher that that of the L TR 60 ............... : ........... . . . . f: ::: : : :::::::::::: t.:::::::::r ::~j:[· .. ·:·::····· (.'.) : : : : . . . 0 ......... ..... : ...... .. ...... -~ .............. ·:· . .. .......... ·:· ..... ...... . . . . . . . -20'~--~---~--~---~-----' 1cf 10 1cf 1o' 10 4 HT I - 5$1 9. -1 ~ ~ -1 n... Frequency (Rad/sec) ··············-:···············:···············: .. ~) .. ······· . . . . . . . . . 10 1a2 1o' 10 4 1cT Frequency (Rad/sec) Figure 5.4.1. Frequency Response of (a) LTR and (b) delay-based controller 100 ro=----~---..-----.. ----r-. -----, GM=l6dB: ........................................................ . . . 50 . . . ~ o ··············~·············· :··········· : ............ (a) .......... . j .. 50 -. · · -· · · · · · · · -~- · · · · · · · · · · · · · ~- · · · · · · · · · · · ·:+ ·· · · · :· (o} \· · -100 · · · · · · · · · · · · · · :. · · · · · · · · · · · · · :. · · · · ········I·:.·····. J. • ·· · · :. · · · · · · · · · · · · · : : I'. I : .. 150~--__._ ___ ...._ __ ............ _........_ _ _..__ __ ____. 10° 10 2 10 3 10~ Frequency (Rad/sec) -50 .-----...----,..-----~-~-.,..------. : : • ~ a a r::: ~:--~.~+~~~.~+ .. -~~.;thJ T.T j -200 .. -:-. :-: .-:-. :-.. --: -~- :-:.-:-.:-.. ~.-:-. ~.-.... -: ::-. :-: .-:~~- :: .-:-......... . :.. j PM=60~Deg. j j -250 10° 102 !OJ Frequency (Rad/sec) 10~ Figure 5.4.2. Loop Frequency Response of (a) LTR and (b) Delay-based output feedback systems 3 -----~------~-----~------~----- 0 -----~------~----- -3 . . BW=900 rad/Sec. ~ . -cl . : : ; -40 ............... : ..... .......... : ... .. ..... ..... : ........ . ~ : ~ ~ . . . . . . . -60 .. ... .. ... .. .. · : · . . .. .......... , .. ... .. ..... . .. , . ... . ' . ' ' . . ... <· . . . . . . . ' .. . . . . . . . . . . -80 .. ............. : ....... ....... ~· .............. ; ... .. ' .. . .. ... ·:· . ....... . -100 '----~............., __ ____.. ___ _..__ __ ___._. __ ---..-l;I 100 Frequency (Rad/sec) Figure 5.4.3. Closed-loop Frequency Response of (a) LTR and (b) Delay-based output feedback systems with (a) BW=900rad/sec and (b) BW=1405rad/sec. 129 From Figure 5.4.2, we can see that both LTR and DTT induced feedback systems have acceptable stability margin. And from the closed-loop frequency response of the above two feedback systems shown in Figure 5.4.3, we can easily find out that the closed-loop bandwidth of the DTT induced feedback system is higher than that of the L TR induced feedback system. This suggests that the response of the DTT feedback system should be faster than that of the L TR feedback system. And the high frequency gain of DTT feedback system is still lower than that of the LTR feedback system, which means higher signal-to-noise ratio can be achieved by using DTT. Additionally, in this case, since the LTR compensator is of bi-proper, the loop transfer function of corresponding full state feedback can be exactly recovered, hence the closed-loop frequency response of full state feedback system will be 130 exactly the same as that of the LTR output feedback system. This suggests DTT induced feedback can achieve much wider closed-loop bandwidth and higher signal- to-noise ratio than the full state feedback. By employing the method presented previously, the following Initial Function Compensator (IFC) can be achieved, JFC(t) = 8(t)-_!_ae- 100 t ,t E [-0.0218,-0.0208),a = 760667 8 (5.4.5) and using Matlab Simulink, the time domain responses can be obtained. To clearly see that signal-to-noise ratio, we assume the sensed output might be coupled with white noises. . . . 0.8 .. ·····:·········:········:······ ·-'.········'.·· ······:···· ····'.······ ··: ····· ···:···· ··· ~ l 0.6 . ······:·········~···· ····~······· ·~········~···· ····~····· ··· ~······· ·~ · ·······~·· ····· 7. . . . . . . . ~ ~ ~ ~ ~ ~ ~ ~ r::... . • • . . . • :;) . : : : : : . : z: 0.4 ······:· ··· ··· ··~· ·······:········~·· · · ·· ···~· ··· ····~········~······ · ·~· ·······~······· . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . . . . . . . . . . . . . . . . . . . . . . . . . . . 02 . . . . . . . . . . ...... :· ....... ·:· ....... : .... ... ·: .. ...... ~ ....... ·:· ... .... ~ ........ : ... .. .. ·:· ... .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . " . o---~..__~_..__~...._~_.._~_.__~_._~_._~_..__~___..__~__. 0 0.005 O.Gl O.Gl5 O.G2 0.025 0.03 O.G35 0.04 0.G45 0.05 Time (sec.) Figure 5.4.4. Step Response of (a) LTR and (b) IFC delay-based output feedback systems 131 800~-~-~-~-~~~~~-~~~~~~~ 700 (~) ... -~·.. . . .. . . ..... J....... . ...... -~·.... . . . . . . .. . . . . . . .. . . .. .... J .. ... . 600 . ······:········ ········:··· ····· ... ·····'.········ ............... ········:······· 500 .. _c_L? !,l. . . . . . . . . ...... ·:-....... . ..... -<·.. .. . . . . . . .. . . . . . . . . . . . ....... :- ..... . . . . . . . . . 'h : : : : e 400 ...... : ................ : ................ : ............................... : ...... . :.:.I : : : : -;::: : : : : i:::: • • . • § 300 . . ... -~· .............. -~· .............. ·:· ............................. -~· ..... . ::._) : : : : . . . 200 . . . . ··~· .............. ·:·· ............. ·:··· .......................... ··:· ..... . . . . . . . . . . . . . . . . . 100 ..... ··:··· ...... ·······:······· .... ···· ·'.··· ......................... ···:··· ... . . . . . . . . . . . . . . . . . 0 ....... : ................ ; ................ : ........... .................... : ...... . . . . . . . . . . . . . . . . . . . . . -100 "-----'-----+--.......i.......-............... _ __._ _ _....+ _ ___.._ .............. h-.-..-_...__........_._.._. 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Time (sec.) Figure 5 .4. 5. Control efforts of (a) LTR and (b) IFC delay-based output feedback systems 0.9 ············ · ···:·········· ········:············· ..... 0.8 ····· ······ ···· · ·:···· ···· · · ········:·············. . . 0.7 0.15 0.2 0.25 0.3 0.35 0.4 Time (sec) Figure 5.4.6.Steady-state square of error for (a) LTR and (b) delay-based output feedback with noises 250.------.----~--~----.-------.---~ 200 ................ . .............. . ~ ~ 150 l5 .b 8 ...... 0 ~ ~ 100 ii (a) Time (sec) 132 Figure 5.4.7. Steady-state control power for (a) LTR and (b) delay-induced output feedback with noises From the step responses of both L TR and DTT induced feedback systems without sensory noises shown in Figure 5.4.4, we can easily see that that all the overshoot, rising time and settling time of DTT induced system are smaller that those of the LTR induced system. And from the control efforts comparisons shown in figure 5.4.5, we can see that the control effort of the DTT system is much smaller than that of the L TR system, which is in accordance with the previous claims. From the step responses with sensory noises shown in Figure 5.4.6, the steady- state error of DTT system is smaller than the L TR system, and so does the control effort shown in Figure 5.4.7. Example 2. Delay-based output feedback control versus full state feedback control 133 Theoretically speaking, the full state feedback is always considered as the ideal control algorithm that can arbitrarily place the closed-loop poles as long as the system is state controllable. However, due to some physical limitations, such as limited number of sensors and immeasurable states, the full states might not always be available, which makes the full state feedback control physically unrealizable. Here, the delay-based control algorithm based on IFC compensator can be used to realize the full state feedback control by output feedback and can achieve wider closed-loop bandwidth than full state feedback. This can be seen from the one DOF rigid body system control as follows, Employing the IFC in Equation (5.4.5), the delay-based controller in Equation (5.4.6) and the full state feedback controller (Proportional+velocity feedback controller in this case) in Equation ( 5 .4. 7), through use of Matlab Simulink, we can obtain the step response for both full state feedback and delay-based output feedback systems shown in Figure 5.4.8, from which we can easily see that the overall performance of delay-based system is nearly the same as that of the full state feedback system. CDTT (s) = 8.4067x10 5 (1-0.9048e- 0 · 001 s) cp+v(s) = 800(s+ 100) (5.4.6) (5.4.7) Note that when we employ the IFC in Equation (5.4.5) without changing any other thing, the step responses of both feedback systems will be nearly the same. 134 This approach of IFC based delay and delay-free control, in general, can be applied to any linear non-vibrational minimum phase systems to realize the full state feedback. And with the help of the internal model modification technique, it can also be applied to the vibrational or non-minimum phase system to realize the full state feedback by output feedback, which is not the subject covered in this paper. I ........ " ........ " ........ " .... '___'...' . :.....:..· ":;....:· ·~· . ....,__ . ..__......;....--~----:------:-----:--~-- ~ (a) : . . . . . . . . . 0.8 ............. ···-····················· .............. ··-···· ........... ··-· ............... . (lj) . . . . . . . . o o • • • • o " o o o • • • • •, • • • • • • • o •' o o • • • • • '• • • • • o o • • '• • • • o • o o • '\ • ' • • • • • • I • • • • • • • •" • • • • • • • • ~ o o • o • • • ....... : ......... : ......... : ......... : .... ····<···· .. ···:········=········:·· ....... : ....... . . . . . . . . . .. . . . . . . . . . . . . . . . . . . o 0 o I o 0 0 0 o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.2 ............................................. ··············'···· ................... ·····""'· ..... . 0"'----'-~--'~-_.__~____.._ __ __.__ _ __..~--+--~-'-~-......_ _ _. 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Time (sec.) Figure 5.4.8. Step Responses of (a) full state Feedback and (b) delay-based output feedback Example 3. Delay-based output feedback control of a 2-DOF flexible system In this example, a non-collocated feedback control of a 2-DOF flexible mechanical system in Figure 5.4.9 is concerned. 4 Controller +---'-'--u~ m2 Sensor Figure 5.4.9. Non-collocated control of a 2-DOF flexible mass-spring systems The governing equation of motion is, 135 (5.4.8) Here, assume mi=m2=lkg and k=5000N/m, the dynamics of the plant can be modeled as, Gs _ 5000 ( ) - s 2 (s 2 +10000) (5.4.9) Both the delay induced output feedback and full state feedback controls are employed and compared with each other in this example. The full state feedback here is physically unrealizable, because we have only one displacement sensor. For comparison with delay-induced output feedback control, we use the theoretical results. The equivalent full state feedback controller in terms of the measured displacement is of the following physically unrealizable improper transfer function, ~ CFsF(s) = 0.2(s 3 + 70s 2 + 4100s +87000) (5.4.10) -------~~~~~------~- And the delay-induced output feedback controller is of the following form, C (s)= -9.995xl0 14 (0.997e- 3 sr -2.998e-zsr +3.005e-sr -1.004) D 5000 where T=O. lms. The virtual initial function for this case is, V(t) = t5(t) + 0.001(-125.77e- 301 +(55.77 cos 50t + 171.9sin 50t)e- 201 ) 5000 where t E [-0.5398,-0.5395). Or--~~--.--~___,_,,_.._..,....~~~···· ........... -~············:·········· .. . . . . . . . . . . . -20 ............ ·:· ........... -~ ............ ·:· ............ : ............ : ........... . : : : : (:a) . . . . . -40 ............ ·'.· ........... -~ ............ ·'.· ........... -~ ........... : ........... . . . . . . . . . . . . . . . . . . . . . ~ -60 ·············:·············:··········· ··:·············~············:············ ·- = ~· -80 ·············~·············~··········· ··~· ············~··········· ; ........... . . . . . . -100 ............. j ............ -~ ............. ~ ............ -~ ........... ~ . (~). ... . . . . . . . . . . . . . . . . . . ' . . -120 ............ ·:· ........... ·: ............ ·:· ........... ·: ......... .. : .. . . . . . . . . . . . . . . . . . . . . -140 ............ ·:· ............ ~ ............ ·:· ........... -~ ........... : .. . . . . . . . . . . . . . . . . . . . . -1601-_._ __ _,.1......... __ ---1.. __ ~....l...-.-.---~--~...L.-.Jl....JL..i.-ll.~ 100 101 102 103 104 105 106 Frequency (Rad/sec) 136 (5.4.11) (5.4.12) Figure 5.4.10. Closed-loop frequency responses of (a) full state feedback and (b) delay-induced output feedback The closed-loop frequency responses ofboth full state feedback and delay- induced output feedback are given in Figure 5.4.10 and the open-loop frequency responses are shown in Figure 5.4.11. 137 ~ 0 ········· ·· ···:····· ···· ~ c: . : 8 -100 .... ....... ... j .. ...... ..... . [ ......... ..... ····· ··· · ····~······ ·(bY ··· ~· ······· ··· . . . . . . . . . . . . . . . . . . -200 .._ __ _.__ __ _._ ___ _._ __ __.__ __ _,___..~_,_,_._,_ 10° Frequency (Rad/sec) . . -- 0 · ·· ··········~·· ··· ····· . ; ...... .. .... .. ; ......... ..... ; .... .. .... .... ; ...... ... .. . ~ : : : : ~) -200 . ....... ... . -~ .. ·· ····· ····~· ...... ....... : ... .... .. .... .: ............ : ... . :J . . . . . i:::::: ::.:::::: . :::: ::: :::::~~ :: ·. : .. . . . . . . . . . . -800'--~~......__~--'-~~~-'--~~-'-~~--'~~______. 100 Frequency (Rad/sec) Figure 5.4.11. Open-loop frequency responses of (a) full state feedback and (b) delay-induced output feedback . . . . . . . . ... ........ . . .......... ,,. ..... . . , ....... .... ... . . ..... .. ... .,. ....... , .... . . . :;,) § 0.6 ....... : ..... .. . ;. ....... ; ........ ; ........ . ; . . ...... ; . .... .. . ; ........ ; .. .. ... ; ... . .. . ~ : : : : : : : ~ ~ ~ : ~ ~ ~ ~ a. . . . . . . :.J • • : : : • : : :;/; 0.4 ······ i· ..... .. : .... .... : ........ : .... .. ... : ..... ... : ........ : ....... . ; ···· ···!······· . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.2 .... ··'.· ... .... ; ... ... .. : ..... .. . : ..... . ·'.·· ...... : ........ ; .. .. .... : ....... : ...... . OL-4--....__.....__.....__~ _ _,_ _ __.__ _ __.__ _ _._ _ _._ _ __, 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Time (sec.) Figure 5.4.12. Step response for (a) full state feedback and (b) delay-induced output feedback The step responses and control efforts by Matlab Simulink are shown in Figure 5.4.12 and 5.4.13. x 10 4 1.5~~-~~~-~-~-~-~-~-~-~ (a) I I • I I 1 ....... ; ....... ~- ...... ~- ...... ~- ....... : ........ : ....... ·=· ....... : ....... -~ ..... . f 0.5 tE u c l::s 15 u 0 . . . . . . .................................................................................. . . . . . . . . -0.5 ...... : ...... ·:·· ..... :· ...... :· ...... ·:· ..... ··:· ...... ·:· ...... ·:· ...... ·: ..... . . . . . . . . . . I I I I I I I I I I I I I I I I I I I I I I 0 I I I I . . . . . . . . . I I I I I I I o I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -1"'--~h-..-.....--''----'----1-~____. _ __._~__..~____.___...-.-..._ _ ___. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Time (sec.) 138 Figure 5.4.13. Control efforts for (a) full state feedback and (b) delay-induced output feedback Based on the above results, we can see that combined with suitable initial function compensator, the delay-induced output feedback can be used to realize the physically unrealizable full state feedback control with guarantee the performance. """'..,.~..,.,.1 efforts point of view, the delay-based control system has more oscillations than the theoretical full state feedback, it does make sense because ~ --- instead of using full states information, the output feedback, especially in this case, the non-collocated actuator and sensor, must have some phase difference and inherent delay in terms of wave propagation theory, which will definite cause more oscillations than full state feedback. Therefore, in this scenario, where full state 139 feedback is physically unrealizable which is true in most of real applications, the delay-based output feedback with proper initial function compensation is acceptable to realize the full state feedback with guaranteed performance. 140 6. Delay-based internal model modification (DIMM) technique It has been a long time since the feedback control of non-minimum phase system became of interests, however, up to today the feedback control of this type of system is still a big problem tJ2) any control engineer. Due to the non-minimum-phase zeros, almost all the existed control methods except the full state feedback fail to achieve good performance, e.g., wide closed-loop bandwidth, high stability margin, etc. As discussed before, since the full state feedback control is usually physically unrealizable, the only practical way to control the non-minimum phase systems is through use of output feedback. Among all the existed output feedback control algorithms, the one aiming to achieve the performance of the full state feedback as much as possible is the loop transfer recovery method [8]. However, this method is only effective for the minimum phase systems in general [29]. To the non-minimum phase systems, arbitrary loop transfer recovery becomes unrealizable unless the unstable controller is involved. Additionally, the closed-loop bandwidth of non minimum phase system is usually very limited, which is also undesirable in practice [31]. Motivated by the above problems, here we investigate a new method called delay-based internal model modification, which can be applied to output feedback control of the non-minimum phase systems with known model, and achieve arbitrary loop transfer recovery without introducing unstable controller. 141 6.1 Internal Model Modification It is well-known that to the non-mm1mum phase system, the conventional feedback control can never achieve arbitrary loop transfer recovery unless the RHP pole-zero cancellations are employed, which will definitely cause closed-loop instability in practice. However, if changing the non-minimum phase plant into minimum phase is possible, the delay-based loop transfer recovery method will definitely be capable of realizing asymptotic loop transfer recovery as described in chapter 4. Questions arise: How can we change the plant properties? Will this change affect the steady-state response of the original system? And How does this change affect both time-domain and frequency domain responses of the system? The way that can answer all the above questions is referred to as the delay-based internal model modification technique, which will be explained as follows. r-------------------- 1 Minimwn phase plant I r + ... e Controller lu - Non-minimum y ... ... ,,,- '.I , I , phase plant ... , ' I - I I I + 1 Augmented +_ '' I ______. I 'f- I pant ~-------------------- Figure 6.1.1. Augmented plant-based internal model modification As we know, the non-minimum phase system is defined as a system with RHP zeros in its transfer function. Therefore, it is obvious that multiplying a minimum 142 phase transfer function to the original non-minimum phase system is impossible to change it to minimum phase system because the RHP zeros are still zeros of the overall transfer function. The only way is to use addition, which means that we can ~- - ---- add a minimum phase transfer function to the original non-minimum phase system ... -------- --·- and make the overall system minimum phase. This can be referred to as the .,------- ---- augmented-plant based internal model modification shown in Figure 6.1 .1. To ensure both the original and modified plants have the same steady-state response, some constraints should be imposed on the augmented plant model, which can be formulated as follows, Suppose the original non-minimum phase plant transfer function is, (6.1.1) and the augmented plant transfer function is, (6.1.2) Note that the denominator of the original plant is assumed exactly known here, because in practice the eigen-structure is much easier and more precise to predict than the zero-structure. The overall modified plant transfer function is of the following form, (6.1.3) 143 and for the steady-state response invariance to step input, the constraint of augmented plant transfer function is that at least c 0 is zero, which can be easily proved by final value theorem as follows, limsG 0 (s)R(s) = limsG 0 (s)~ = G 0 (0)r = bor s~O s~O S a 0 T b 0 r (b 0 +c 0 )r l . . . d o ensure - = , at east c 0 1s zero 1s reqmre . ao ao On the other hand, after some certain block diagram transform, the augment- plant based internal model modification can have another form, which is referred to as the augmented-controller based internal model modification shown in Figure 6.1.2, which suggests that by adding some certain inner feedback loop in the system, the overall system can have much better performance than the original non-minimum phase system. r + ... e ..... Controller u - Non-minimum y - ~ ' " I"' ~ phase plant ,... )' , ' - - Augmented +-- Controller Figure 6.1.2. Augmented controller-based internal model modification I ... 144 Assume either the augmented-plant or augmented-controller based internal model modification technique is employed to change the original non-minimum phase plant property to minimum phase, to achieve as close performance as full state feedback through output feedback, the conventional loop transfer recovery method can be employed. However, to different loop transfer recovery requirements of either open-loop or closed-loop loop transfer recovery, different constraints can be achieved in designing the internal model modification compensator, which will be introduced below. 6.1.1 Open-loop Transfer Recovery As a matter of fact, the loop transfer recovery controller was originally designed by using state space method, which is in time domain, e.g., LQG/LTR [8], compensator based LTR [29]. However, in the frequency domain, the loop transfer recovery can be more directly understood, which is a way to ~en lo transfer function of the output feedback system approach that of a full state --~ f~ack exactly or asymptoticallyrovided that the plant is of minimum phase. Therefore, it is obvious that the controller's denominator should have one or two factors: one is exactly the numerator of the plant and the other is some transfer function which can make the overall controller physically realizable, while the numerator of the controller should be exactly the numerator of the desired loop transfer function of full state feedback. This type of controller can be formulated below. 145 Assume the open-loop transfer function of the plant is, (6.1.1.1) and the desired loop transfer function of the full state feedback is, (6.1.1.2) Thus, the transfer function of the loop transfer recovery controller must be, (6.1.1.3) ,/ while to make the above controller physically realizable, the transfer function must be either bi-proper or strictly proper, which means p(s) should be of the following forms, ( ) I' n-m I' n-m-l I' I' Pi S =Jn-ms + Jn-m-ls + ... + J1S+ Jo (6.1.1.4) (6.1.1.5) The controller with pj(s) is called full-order observer or compensator based loop transfer recovery controller, and one with pr(s) is called reduced-order observer or compensator based loop transfer recovery controller. Note that bothpr(s) and pj(s) have no term with negative power of sand specially when m=n-1, pr(s) will become a constant and pj(s) will be a first order polynomial ~:And the zeros ofbothp,(s) and pj(s) are usually chosen to be outside the desired closed-loop bandwidth to minimize the difference between the real loop transfer function and the desired one. 146 In the special case, when m=n-1, using the reduced order L TR controller can realize exact loop transfer function of full state feedback because the denominator of the controller will exactly cancel the numerator of the plant such that the overall loop transfer function of the output feedback system is exactly the same as that of the full state feedback. W~othe_r __ ~J!iat mis less than n-1, no matter reduced order or full order LTR controller will never be able to realize exact loop transfer recovery and it can only approach the loop transfer function of full state feedback asymptotically. However, in practice, exact loop transfer recovery usually causes high bandwidth controller, which will deteriorate the signal-to-noise ratio in control channel and the whole system will be very sensitive to high frequency noises. Therefore, people often employ the full order L TR controller although it can only realize the asymptotic loop transfer recovery. To further improve the performance of the systems, the delay-based loop transfer recovery comes into being, which can be used to realize any improper controller and therefore the denominator of the delay-based LTR controller can be exactly the same as the numerator of the plant regardless of the order issue. And it is shown that such delay-based L TR has much better performance than the conventional loop transfer recovery method [21,22]. Nevertheless, the problem for either delay-based LTR or conventional LTR is that when the plant is of non-minimum phase, the overall feedback system could have very bad performance or even instability. This invokes the use of internal 147 model modification technique to change the non-minimum phase to minimum phase, such that either delay-based or conventional LTR can achieve perfect performance. Moreover, in general, to realize arbitrary loop transfer recovery to a general system, the pole-zero cancellation between controller and plant is inevitable. However, in terms of the system's sensitivity to the precision of pole-zero cancellation, the highest one is when the pole and zero are exactly imaginary, because the gain can change from infinity to zero and the phase can change from 180 degrees to -180 degrees with arbitrarily small miscancellation, while the lowest one, which is the most robust one, exists when the pole and zero are on the negative real axis, because both the gain and P.hE-~~- change could be as small as possible with ...____----------~---···--·---· . .. _________ -··. arbitrarily small miscancellation. Therefore, in terms of the system's se · · · ------------ the precision of pole-zero cancellation _the best one is negative real axis pole-zero cancellation, which gives us a guide on how to put the zeros of the modified plant by internal model modification. Based on this guidance and equation (6.1.1-3), assume the step input is concerned, c 0 should be zero according to the steady-state response invariance constraints. We can always find a set of Ci without changing bi such that the zeros of modified plant in equation (6.1.3) are all on the negative real axis and then we can figure out what kind of augmented plant we should employ by substituting Ci into equation (6.1.2). Note that there is no constraint right now on the locations of these negative real axis zeros, which means they could be arbitrarily chosen up to now. 148 Based on this method, the modified plant is obtained with all zeros on the negative real axis without changing the steady-state response. Then we can employ the delay-based loop transfer recovery method to design controller for this modified minimum phase system according to the design methods discussed before [21,22]. 6.1.2 Closed-loop Transfer Recovery As introduced above, we know that by arbitrarily choosing locations of the negative real axis zeros of the modified plant, we can always realize the desired modified plant by augmented-plant based internal model modification method. And then we can use the delay-based LTR output feedback method to asymptotically realize any loop transfer function of full state feedback to the non-minimum _phase systems. Without the internal model modification, this can never be done. However, this can only guarantee the open-loop transfer function recovery. To realize the closed-loop transfer function recovery, some other constraints should be imposed on the locations of the negative real axis zeros of the modified plant. On the other hand, to the closed-loop frequency response, the phase is not as important as that of the open-loop frequency response while only the gain is important to us. -~-- - -~ p ---- Depending on different gain frequency response requirements, different constraints can be derived on the locations of modified plant zeros. Among all the feasible closed-loop gain requirements such as IL constraints, loop shaping constraints, and so on, one is called all-pass filter constraint. This constraint can be used to cancel 149 the gain effect of the numerator of the plant, which behaves like the numerator of the original plant being constant, and the closed-loop gain frequency response will -------~--.....;._ ___________ __ asymptotically approach the following one, I G ( . )j L(jm) cL ]OJ = 1 + L(jm) (6.1.2.1) where LGco) is the frequency response of desired loop transfer function. This constraint is an easy way to design the control system and all we need to do is to design a full state feedback controller for a system transfer function with exactly the same denominator as the plant and the numerator is constant 1 to satisfy all the performance requirements in frequency domain theoretically. And then design the internal model modification based loop transfer recovery controller according to the all-pass filter constraint, and everything is done! Of course, other type constraints can also be used depending on different requirements. The all-pass constraint will be discussed in detail below. To talk about the all-pass filter constraint, let's first check the definition and elements of all-pass filters. An all-pass filter is filter whose gain over all frequency range is constant 1. This type of filter can be of the following two basic forms, s±b s+b Fa(s) = s2 ±cs+d s 2 +cs+d where b, c and dare all positive real numbers. (6.1.2.2) 150 The all-pass filter constraint for internal model modification based loop transfer recovery controller design can be formulated as follows. Assume a general non-minimum phase plant with the following transfer function, a P r A. bmTI (s-aJTI (s+d 1 )Il (s 2 - J,_s+ gr)Il (s 2 +hts+lt) Go ( s) = i=O J=O :=o t=o , m < n ( 6. 1. 2. 3) TI (s-pk) k=I where a ~ 0, p ~ 0, r ~ 0, 2 ~ 0, m = a + /3 + 2y + 22 and ai, dj, fr, gr, ht, lt, are all positive real numbers. To satisfy the all-pass filter constraint, the modified plant transfer function should be of the following form, ,m<n (6.1.2.4) Then the augmented-plant should be, (6.1.2.5) For simplicity, let's denoted the numerator of Go(s) as N(s) and that of GM(s) as M(s) and note that N(s)IM(s) is all-pass filter, i.e., \N(s)IM(s) \=I. And the desired loop transfer function of the full state feedback is, 151 (6.1.2.6) Thus, the transfer function of the conventional loop transfer recovery controller must be, (6.1.2.7) where 11 can be n-m for full order LTR and n-m-1 for reduced order LTR, and T/ T/ p(s) =IT (s-vi)IIJ vi i=l i=l The loop transfer function of the internal model modification based LTR output feedback system is, Nc(s) =--- D(s)p(s) (6.1.2.8) and the closed-loop transfer function is, GcL (s) = Nc(s)N(s) = N/s) N(s) M(s)[p(s)D(s)+Nc(s)] p(s)D(s)+Nc(s) M(s) (6.1.2.9) and the closed-loop gain frequency response is, (6.1.2.10) 152 where N(s) =1 (all-pass filter property). M(s) From equation (6.1.2.8) and (6.1.2.10), we can easily see that both the open-loop transfer function and closed-loop gain frequency response are asymptotically recovered. Specially, when m=n-1 and the reduced order LTR is employed, then p(s)=l, hence both the open-loop transfer function and closed-loop gain frequency response are exactly recovered as shown in equation (6.1.2.11) and (6.1.2.12). L(s) = Nc(s) D(s) (6.1.2.11) (6.1.2.12) This just behaves like the results of full state feedback control of a plant transfer function with exactly the same denominator as the original plant and the constant 1 numerator. As for other types of constraints, the procedure of controller design is similar to the all-pass filter constraint case. 6.2 Delay-based LTR Method As discussed in chapter 3 and 4, the delay-based controller can be used to realize any polynomial transfer function, which suggests that we can realize improper controller by the delay-based controller with arbitrarily small error. This technique can be directly applied to the loop transfer recovery control system where the relative order of the original plant is greater than 1, i.e., the order difference between 153 numerator and denominator is 2 or more. Because to such systems, the denominator of the conventional loop transfer controller should be a product of two parts, one is exactly the numerator of the plant and the other is an additional polynomial transfer function for realizability. And usually there is no fixed criterion on how to choose this additional part, which usually plays an important role in the closed-loop performance and is determined by experience. This increases not only the order of the controller but also the design difficulty. Instead of using the conventional loop transfer recovery method, employing the -~-based loop transfer recovery control can achieve lower controller order, lower ~---- design complexity and higher closed-loop performance, such as wider closed-loop bandwidth, higher signal-to-noise ratio in control channel, etc, as mentioned in [21,22]. 6.3 Illustrative Example Consider an arbitrary non-minimum phase open-loop plant with the following transfer function, G(s) = -s+50 s2 (6.3.1) The reduced order loop transfer recovery compensator in Equation (6.3.2) and an augmented plant in Equation (6.3.3) are employed such that the desired loop transfer function shown in Equation (6.3.4) can be realized. C (s) = 800(s + 100) LTR s+50 (6.3.2) 154 (6.3.3) L(s) = 800(s + 100) s2 (6.3.4) According to the all-pass filter constraint, the modified plant should be, (6.3.5) and then the augmented plant can be obtained by subtracting G from CiM, GA(s) = GM(s)-G(s) = s+50 - -s+SO = 2 s 2 s 2 s The loop transfer function of the internal model modification based L TR output feedback system is, LJMMLTR(s) = CLrR(s)GM(s) = 800(s+ 100) x s+;o = 800(s; 100) (6.3.6) s+50 s · s which is exactly the same as the desired loop transfer function of the full state feedback. And the closed-loop gain frequency response of the internal model modification based LTR output feedback system is, IGcL(s)!=l 2 800(s+100) 11-s+501=1 2 800(s+l00) I= L(s) ( 6 . 3 .?) s + 800s + 80000 s + 50 s + 800s + 80000 1 + L(s) which is the same as that of the full state feedback introduced in section 6.1.2. Based on the above analysis, since he Il\1M-LTR and the theoretical full state feedback have exactly the same closed-loop gai;-frequencyrespcmse,ln time -------- domain, they should also have the similar behavior, which in this case can be shown 155 by the step response of both methods in Figure 5. In this specific case, since the system's order is low, actually the step response of IMM-LTR and the theoretical full state feedback happen to be almost the same as each other, while in general, the same-frequency response does not have to mean the same time domain response, but usually si~ilar. Additionally, for companson with the delay-based IMM-LTR, both the frequency response and time domain response of conventional IMM-LTR and delay- based IMM-LTR are discussed. And for better transient response in the delay-based IMM-LTR, the so-called initial function compensator is employed. In order to show the better noise attenuation of delay-based IMM-LTR compared non-delayed IMM- L TR, both the frequency response and time domain response with noise issues are discussed. Employing the same augmented plant internal model modification method, the following delay-based LTR controller is employed, 8.4067x10 5 (1- 0.9048e-o.oois) Cvrr(s)=~~~~~~~~~~ s+50 (6.3.8) And for transient response optimization, the following initial function compensation is used to overcome the effects to the transient response of the closed- loop system caused by delay action. ] VIF(t) = 8(t)-_!_ae- 100 t ,t E [-0.0218,-0.0208), a= 760667 8 (6.3.9) 156 60 ········· ·· ···· .......... . .. ........... .. .... .... J~) ........... . m : "tJ 40 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . ... ...... . ·~ (b) : ·~ 20 ... ......... ... ···· ······· ··· ........ ..................... : .... .. . 0 : 0 .. ............. ············ · · .................. ······ ·· ···:· ············ ~0'-----~....._--~~~--...__..-~_...___ __ ____, 1~ 1d 1~ 1~ 1~ 1~ Frequency (Rad/sec) · ;;;; . : . (a) . ~ -50 .............. -~· ........ ...... ~ .............. ~-.. . . . . . . . . . -~· ~ : : : (b) : 9-10 ···············:-·· ··· ·········:········· ·······:········ ··· ··· · . . . . . ~ : : : ~ -15 ............ ... : ............... ; ............... : ............. .; ... . . . . . CL : : : : . . . . . . . . 10 1 10 2 1~ 10 4 1a5 Frequency (Rad/sec) Figure 6.3.1. Frequency Response of (a) LTR and (b) delay-induced controller lOO~--~---~----.----~~--~ . . 50 .. .... ........ : . . .... .... . . ··········· ..... ........ ................ ..... . . . . 0 . . . . ..... .... ·:·· ..... .. ..... : .......... . . . -100 .............. : .... .......... : ...... .... .... :· ............. :· ... ......... . . . . . . . . . -150 ......_ __ ____. ___ ___. ____ .__ __ _____.~------' 10° Frequency (Rad/sec) . . 7.-100 ...... ...... .. ; .... .. ... ..... ; ..... ... . 8 b~ . : : ~. -150 .............. ~ . . .. .. .. . . . ~ .............. "(b) . . . .. . . -~ . . . . . I~ • • • • ] -200 ...... ........ ; .............. ; .. ............. : .... . . ........... . ~ : : : : . . . . . . . . Frequency (Rad/sec) Figure 6.3.2. Loop Frequency Response of (a) LTR and (b) delay-induced feedback systems 157 3 ~-------~-~-__;.:_-_-_-............,..-: .,....,.,-:,.....--r. . .::--:- . . :-:'"-:--':"":" . :-"!'":' .. -;:;::::;::::::;-;-;--- ......... -: - - - - - - :- - - - - - - - - - - -: - - - - - - :- - - - - - - - - - - - - - - - - -3 BW=900 rad/sec. -20 -60 .............. ·:· ........... .. . ·:· .............. ; . . . .. ......... <· .. -80 ............... : .............. :· .......... .... : .............. ·:· ........ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -100 '---~-__..__ ___ __.._ ___ __._ ___ ___....___ __ ______.,. 10 ° Frequency (Rad/sec) Figure 6.3.3. Closed-loop Frequency Response of (a) LTR and (b) DTT induced feedback systems with (a) BW=900rad/sec and (b) BW=l405rad/sec. 0.9 . . . . . 0.8 ··········<·· ······ ···········>············ ·· ····>· ·· ··· ·· ··· ··· ··· i· ····· ············t·· ···· ·· ·· ······· Time (sec) Figure 6.3.4. Square of steady-state error for (a) LTR and (b) delay-induced output feedback with noises 250~--~---~---~--~---~--~ 200 ... . .. ... ..... . ... .......... . . ~ ~ 150 .............. .... .. ... ............. ... ...... . -0 ~ 8 0 ll:! ~ 100 . {1 (a) Time (sec) 158 Figure 6.3 .5. Steady-state control power for (a) LTR and (b) delay-induced output feedback with noises Ilv1M-L TR/fuil state feedback 0.8 0.6 ... .... ..... .. .......... .... ..... .. ... : .. ...... .. .. ...... ; .... ........ ... . . . . . . . . . . . . 0.4 ············ ···· ·· ·· ··· ··· ·· ·· ···· ··· ···· ···· ······· ··· ······ ···· ·· . . . . . (I) . . . . 1!! . . ~ 0.2 ..... ... . ·.· . . . . . . . . .. . . . .. . ~ .. . . ... . .... .... . . i ............ . . .. . a> a:: % 0 .... Delay-b~ed IMM-L'fR Ci) -0.2 . . ····· ·· ....... ; ..... ........ ... . . : ... .... ... .... . .... : . ....... . . . . -0.4 . . ··· ···· ... · ···: ····· ·············:·· · ····· ·· · .. ... ·· · :· · · · · ··· ... . . . . . . . . . . . . -0.6 . . ..... ... ... . . ; .. ... ... .. .. . ... . . : . . ..... ............ : ............ ...... ; .... . .. ....... ... . ; .. . . . -0.8 -1'-------L.----L-----'-----'----'------' 0 0.05 0.1 0.15 Time(sec) 0.2 0.25 0.3 Figure 6.3.6. Step Response of (a) LTR and (b) IFC based delay-induced output feedback systems 159 With the above delay-based LTR compensator, the frequency responses of both delayed and non-delayed Il\1.M-LTR output feedback controller, open-loop transfer function and closed-loop transfer function are given in Figure 6.3 .1-3 . From this we can see that the bandwidth of delayed controller is finite while that of the non delayed controller is infinite because of bi-proper transfer function. Hence the high frequency noise attenuation of delayed feedback system should be much better than that of non-delayed system, which is in accordance with the steady-state error and control effort for step response of both systems with assumed white noises shown in Figure 6.3.4 and 6.3.5. And both the closed-loop bandwidth and peak gain value within bandwidth of delayed feedback system are higher than those of the non delayed feedback system, which suggests that the response speed of non-delayed system maybe faster and at the same time, the transient response such as over/undershoots maybe larger. This transient response deterioration can effectively be eliminated by the proposed initial function compensation method with affecting any steady-state performance, which can be shown by Figure 5. 160 7. Simulations and Results 7.1 Non-collocated Control of Multiple-DOF Flexible Systems using DTT & FDR Method In practice, most of the systems to be controlled are essentially of infinite dimension or very high order. To deal with these control plants, usually the model of the system should be reduced to some extent, which can reduce the complexity of the controller. Although we can use the model reduction method to simplify the system, usually the eventual model is still of high order. To control these kinds of systems, the full state feedback can be regarded as one ideal but physically unrealizable method due to the unavailability of the full states. Here, we are going to deal with the multi-DOF flexible system by using DTT & FDR output feedback and compare it with theoretical full state feedback and LTR output feedback. 7.1.1 Multiple-DOF flexible system description Consider the non-collocated control of the following N-DOF mass-spnng- damper flexible system shown in figure 7 .1.1 and the dominant dynamics of this N- DOF spring-mass flexible system can be modeled as .. . Mx+Dx+Kx=F,M =diag{fni m2 mN} kl -kl 0 0 di -di 0 0 -k I kl +k2 -k2 0 -di di +d2 -d2 0 K= 0 -k 2 k2 +k3 0 D= 0 -d 2 d2 +d3 0 (7 .1.1.1) -kN-1 -dN-1 0 0 0 kN-1 0 0 0 dN-1 F=[l. 0 0 ... or f =bNxJ 161 The input output relation is N-1 X (s) IJ (dis+k;) G(s) = N = ---'--1-1~--- U(s) det{Ms 2 +K} (7.1.1.2) Note that usually the di is negligible. X1 X2 XN-1 x k1 ki kN-2. kN-1 f ID1 ID2 • • • ffiN- ffiN d1 C(s) Figure 7 .1.1 .1 The Non-collocated control ofN-DOF spring-mass flexible system The state space realization is . z=Az+Bu y=Cz A = l ON xN I N xN ] -M- 1 K 0 N xN 2Nx2N (7.1.1.3) B = l~N•I] c = [ ol x(N-1) 1 olxN 1 xN N xl 2Nxl The plant-controller configurations of full state feedback and output feedback are given in figure 7 .1.1.2 and the breaking point is put between controller and plant to define the following loop transfer function, For full state feedback, the loop transfer function is, L(s)=K(sI-Ar 1 B (7.1.1.4) 162 For output feedback, the loop transfer function is, L(s) = C(s)G(s) (7.1.1.5) r~~._____C(.-s)_ "--~ -v~-~~1G_(s-~ 1--1_-r---->- v Figure 7 .1.1.2. Controller-plant configuration for output feedback and full state feedback 7 .1.2 Design Examples Example I. Consider a 4-DOF mass-spring-damper flexible system with the following model, .. . M x+Dx+Kx=F,M =diag{l 1 1 l} 100 -100 0 0 0.1 -0.1 0 0 -100 200 -100 0 -0.1 0.2 -0.1 0 K= D= 0 -100 200 -100 0 -0.1 0.2 -0.1 (7.1.2.1) 0 0 -100 100 0 0 -0.1 0.1 F=(l 0 0 of u 163 and the input-output relation is 3 G (s)- X,(s) _ D (d,s +k,) _ 0.00ls 3 +3s 2 +3000s+ Ix IO' non - U(s) - det{Ms 2 +K}- s 8 +0.6s 7 +600.ls 6 +200s 5 +lxl0 5 s 4 +1.2xl0 4 s 3 +4xl0 6 s 2 (7.1.2.2) And the state space representation is 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 A= B= -100 100 0 0 -0.1 0.1 0 0 1 100 -200 100 0 0.1 -0.2 0.1 0 0 0 100 -200 100 0 0.1 -0.2 0.1 0 0 0 100 -100 0 0 0.1 -0.1 0 C=[O 0 0 1 0 0 0 o] . z=Az+Bu (7.1.2.3) y=Cz Here, we employ the full state feedback, compensator based L TR and DTT & FDR method to control the system. And (1) at least lOdB gain margin, (2) at least 45 degree phase margin, (3) at least 400 rad/sec closed-loop bandwidth (the frequency that the closed-loop magnitude crosses over -3dB from above), (4) less than 3dB magnitude peak, (5) at least -20dB/dec high frequency cutoff rate of closed-loop transfer function should be satisfied. After certain transfer function transformations, one well-designed full state feedback controller can be easily represented by the following improper transfer function in terms of output feedback, 164 C (s) = 400s 1 +10800s 6 + l 91600s 5 +l.967x10 6 s 4 +l.433x10 7 s 3 +6.18lx10 7 s 2 +1.751 xl0 8 s+l.258x10 8 ftfb 0.00ls 3 +3s 2 +3000s+lxl0 6 and the following transfer function is the DTT and FDR controller and a candidate LTR output feedback controller, • DTT & FDR controller I. 3 µ(1-be-sT) fl (e-2sT +c;e-sT + d;) Cnrr(s) = 0.00ls 3 +~; 2 +3000s+lxl0 6 d 1 = 1.00500, c 1 = -2.00497 d 2 =1.00503, C 2 = -2.00501 d 3 = 1.00300, C 3 = -2.00299 b = 0.9995 T = 0.0005s 1.258x10 8 µ= 3 c1-b)IT c1 +c; +d;) i=l C (s) 5.4x10 1 7(400s 7 +10800s 6 +191600s + 1.967 xl0 6 s 4 +1.433x10 7 s3+6.181x10 7 s 2 +l.75lx10 8 s+ l.258x 10 8 ) LTR 0.00ls 8 +2ls 1 +18500s 6 +8.89xl0 8 s5 +2.54xl0 12 s 4 +4.37xl0 15 s 3 +4.42xl0 18 s 2 +2.4xl0 21 s+S.4xl0 23 From the frequency responses of the above three controllers in Figure 7 .1.2.1, we can see that cutoff rate of the magnitude frequency response for the full state feedback is +80dB/dec, and that of LTR is -20dB/dec. They are both much larger than that of the DTT and FDR controller -60dB/dec. This suggests that in terms of high frequency gain or the high frequency noise amplification and bandwidth of the controller, the DTT and FDR controller is much better than the full state feedback and LTR. 165 600 ,--,----.,.---~,,--,-,,,--,--,----,--,--...,-,--,-,..,.----,--,.-.,.---,-,---,-~~-,-----,,---,---,--,--,---,--.-,-~~~~.,........-, ·- 400 co :s. c: ~ 200 Fuli state : feedback (sl~p~ ~qq~f4ec) : . . .. O'--~_._~.........._._,___._~~_._._._,_,____.__..~·~·~ ·~ · ._._._~~~~~=.:...;=--=-~_._._........w Q) (/) (t] .c: 0 10 500 1 10 2 10 3 10 4 10 : : : ; ; ; ; ; ; : : \ F:~l~ ~tate fe~d*~¢~ : ~ ! : : : . . . ..... . . : : ::::: :'-. :: ::::: : : ··:·····:····:··:··:··:·:·:·:········:··· ·:···: .· . . ·······:·~~· ·:··:·:·: : ······· ·:·····:··· . . . . ... . . . . .. . . . . . ... . o ·······~ ····; ... ; .. : .. H.:F·· ··· ···:····Fuitortler-L'fR·::..... . .;._.;._;~;..,..,..,.,.,.~~~~ : : : : : : : :: a. -500 ..... .. ~ .. .. : ... ~ .. ~ . ~ . ~ . : . . : .: .. ....... : ... .. : .. .. : .. : .. : .. :. : . :. : .. 0 10 10 1 2 10 10 Frequency (rad/sec) 3 4 10 5 10 5 10 Figure 7 .1.2.1. Frequency response of full state feedback, LTR output feedback and DTT &FDR output feedback controller From the frequency responses of the loop transfer functions of all the three feedback control systems in Figure 7.1.2.2, we can find out that all of them can guarantee the desirable stability margin. The is shown by that the full state feedback has infinite gain margin and 85 degrees phase margin, and both the L TR and DTT and FDR have lOdB gain margin and 55 degrees phase margin, which all satisfy the performance requirements. And from the frequency responses of the closed-loop transfer functions of the above three controllers based feedback systems in Figure 7.1.2.3, we can also notice 166 that the cutoff rate of closed-loop transfer function of full state feedback in high frequency range is -20dB/dec, which is much larger than that of the DTT and FDR of-160dB/dec. So does the LTR. 0 . cn- .. . . . . . . ..... . :9. -200 c ~ -400 · · : ··: - ~ : .. : · · · ·· • .. : ; ·''Fun 6tdetttR+ ' ' : '.' '.'.. . \(GMflO~a( [\ ~ ... ··": ........... ~ ... ~ ... . :.:. .. .... ··; .. . . . ·~." ... ... ; .... ·~·· -~. -~ .~ . ; -~~-~ .. .. -600 ......_....__.. ......... . _._ .. ....._. .. ........ . _.__._ . ...__. ~· .......... ........___.___...___.__. . ._._ . ......., ... '""--__..__.___.__._.............._......__.__..........._._............, gi -500 ::9.. Ql 8l 6: -1000 foll~statc! feedback: .. '(PM=85deg.)'~ ' ; : : : , : : , , . : : : : : !F.ti11 oided:m.: :: . . ; . ::: .. :· ,(ftM~$54e8Y :f:£'l'T&FDR ::r . -1500 ......__._____,___._ . __._ . ·~ · .....,____.__.....__ . ...._. ;_;,.• ............. ·._..__ .. _____.·~ · ..__. . ._... . ........ :: ....... ' KP_M_.._~_5~5d ......... ¢g_.__.)_...._:: :.__· ___.___.___.__..__._._._..., 0 4 10 10 Frequency (rad/sec) Figure 7.1.2.2. Frequency response of the loop transfer function of full state feedback, L TR output feedback and DTT &FDR output feedback systems In general, to different open-loop plants, in terms of the cutoff rate of magnitude frequency response of controller in high frequency range, the DTT and FDR method could be -nx20dB- -20dB/dec, the full state feedback will be OdB- (n-l)x20dB/dec and the LTR will be always -20dB/dec for full order LTR and OdB for reduced order L TR. This means that in terms of amplification of high frequency noise, the DTT and FDR method is the best and the worst one is the full state feedback. In terms of the cutoff rate of the closed-loop frequency response, the same order is obtained, 167 which means that the best one is the DTT and FDR method and the worst one is the full state feedback. This suggests that in terms of high frequency noise rejection and high frequency dynamics attenuation, the DTT and FDR is the best choice . cc ~ c: 'IQ (!) ... Qt---+-......... .-------............ -------= ........... ...., -50 .. ' . . .. .. . . . . . . . . .... . . ~ i FaiLordCr L1R ~ ~ ~ ~: . [ l : !b~*4width~8oQ r~4{sec : . . .... . - ·· -100 •• •• 1 • ••• ; . , .~ •• ~ . ;.; ,:,,:.; •••• ) • • •• ; •• • ~ •• ~ • • ~ •• ;. ~.;~ • •• ••••• ; •• ••• • • ; , , ; •• :. ; . ~.=-~ ........ ~ . . . . . . . .... . . . '. -150 .. .... · ;· · · ·~···~· · ~· '.· ~ ·:·~·~··· 0 10 . . 1 10 2 10 Frequency (rad/sec) 4 10 Figure 7 .1.2.3. Frequency response of the closed-loop transfer function of full state feedback, LTR output feedback and DTT &FDR output feedback Moreover, from Figure 7.1.2.3, we can see that the closed-loop bandwidth of full state feedback, LTR feedback and the DTT and FDR feedback systems are 410, 800, 900 rad/sec respectively. This tells us that in terms of disturbance rejection ability and fastness of system's response, the DTT and FDR is still the best choice. • Example II. Consider a 4-DOF mass-spring flexible system with the following model, 168 .. Mx+Kx =F,M =diag{l 1 1 1} 5000 -5000 0 0 1 -5000 10000 -5000 0 0 (7.1.2.4) K= F= u 0 -5000 10000 -5000 ' 0 0 0 -5000 5000 0 and the input-output relation is 3 Ilk. 1.25el 1 G (s)- X4(s) - i=l ' = (7.1.2.5) non - U(s) - det{Ms 2 + K} s 2 (s 6 + 3e4s 4 + 2.5e8s 2 + 5el2) And the state space representation is 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 A= B= -5000 5000 0 0 0 0 0 0 1 (7.1.2.6) 5000 -10000 5000 0 0 0 0 0 0 0 5000 -10000 5000 0 0 0 0 0 0 0 5000 -5000 0 0 0 0 0 C=[O 0 0 1 0 0 0 o] . z == Az+Bu y==Cz And the desired performance requirement are (1) the closed-loop bandwidth is no less than 10,000 rad/sec. (2) the overall system should have at least 10 dB gain margin and at least 60 degrees phase margin. 169 1). LQR based full state feedback controller design To guarantee the above desired performance requirements, we employ the LQR control algorithm to place the closed-loop poles. The linear quadratic performance index with some suitable Q, R matrices is shown as follows, CXl J = J ( zr Qz + ur Ru )dt, Q = 1 OOOC z r Cz, R = 1 0 (7.1.2.7) CZ = [600 0 150 150 12 3 0 O] the static state feedback control input is (7.1.2.8) where P satisfies the following ARE, (7.1.2.9) solving the above ARE, we can easily find the state feedback gain vector K and then the loop transfer function as follows, ( ( ) _ 1 Katij{sl -A}B Katij{sl -A}B L s) = K sf - A B = = ------'-----'----- LQR det{s/ -A} s2{s 6 + 3e4s 4 + 2.5e8s 2 + 5e12) (7.1.2.10) 2). DTT & FDR controller design According to the approach in chapter 4, we can easily find the generalized improper output feedback controller, which is equivalent to the full state feedback controller as follows, 2N-l c s = Ka4J'{sI-A}B = 3.so1e4U (s-z,) = iN-1 s-z ,..() l.25ell l.25ell µ,..:gc ,) (?.1. 2 .ll) 170 And we can find that the zeros of the generalized controller are Z; ={ -1.117 ± 128.88j,-1.947±90.92j,-2.839±37.024j,-49.414}={z1,2,Z3,4,Z5,6,z7} Now we can employ the delay transform technique to realize the above non- causal generalized controller. DTT: (both PDdt, andPD~tI-type controllers are employed) where 6 cgen(O) IT Pi b - P;T • - 1 2 J 4 5 6 b - z1T - i=l ;-e ,l-,,,,' '1-e ,f..li- 6 1 IT z;IT (I-bi) i=l i=l Here, according the closed-loop bandwidth requirement 10,000 rad/sec, we can define the radius frequency of design region DRC to be 20,000 rad/sec. And in order to satisfy the robustness, we choose the radius frequency of design region DRI to be 1,000,000 rad/sec so that here, we choose T=le-6 sec (not optimal), and then all other parameters in the DTT controller can be calculated accordingly. 3). Simulations and Comparisons The loop transfer function frequency responses of the LQR full state feedback, and DTT output feedback are shown in Figure 7.1.2.4. and Figure 7.1.2.5 is the closed-loop frequency response of these two methods, from which we can find that 171 both robustness and closed-loop bandwidth are satisfied and DTT control can realize the full state feedback by output feedback. ;; ~. 100 ~.,.,,:."'""'-''"''''''"''''"""""''"'"''"'"''''"'"'··'·'l""'"'''"'"'"''"''"'''''"'""'""'""''""'"''"''J'[:lf'l·"··""""""'"'''""'''"'i ~ ~-.. .. .....__ > . -........_,.;..,, ' ,J : ~ 50~.: .. ::.;:"'·········~J':.;~~·-~~~··'·''···--·········· .. ····· .. ,~ .......... ·········--~ .a . ~~,--;. . --- 'iSi T i;-4~: , ~ 0 ~.. ~+...,;: ::: lJ.tti: : : ; : : ; ; ; -5o~~·-·~·-··~~·-·-·-··~··~~~~~~·-·~··~ .. ~~~~··~· 0 1 1 3 4 5 10 10 10 10 10 10 50.--~~~.--~~,_....,...,~~~~.--~~_......,,-~~..-.-.., :, LQ ~l·un .... ' .... '"-:ii ~ -50 ............ ; ... ; .; .. ;; •. ,., ......................... ; .. ;."'' ::s. ~ -100 - .c c... -150 _ ......................... .. -- ' ..... ./ J I ~r ---- _ ...... ZJ DR . ..... -200 ~-~--~~~--~~~~~~~--~-~--~~ 0 1 1 3 4 10 10 10 10 10 10 Frequency (Rad/sec.) 5 Figure 7.1.2.4 Frequency Response of loop transfer function with LQR full state feedback and zero placement via delay feedback. 5 ~ 0 Q) ] ·c: -5 1:11 m :2 -10 0 10 50 ff 0 LQR F1il Feedb:icl ... ::s. ~ t1I -50 - .c a.. -100 0 10 Frequency (Rad/sec.) Figure 7.1.2.5 Frequency Response of Closed-loop system with LQR full state feedback and zero placement via delay feedback. 7.2 Non-collocated control of a 2-DOF flexible robot arm 7.2.1 System description 172 In the non-collocated control of flexible structures, because of the dislocated sensor and actuator, the plant characteristics usually have the non-minimum phase behavior, which cause great complexity in controller design. And as presented before, only the delay-based internal model modification technique can achieve good closed-loop performance without needing to involve unstable controller. In this section, a 2-DOF bending robot arm shown in Figure 7.2.1.1 is considered, which has non-minimum phase behavior because of non-collocated sensor and actuator. - - - - - - -- - ---------,..,.--- Figure 7.2.1.1. 2-DOF non-minimum phase flexible robot arm system Assume small displacement, the linearized governing equation of motion can be written as, 173 (7.2.1.1) Assume m1=m2=l, and JJ=h=l, 11=/i=l, the dynamics of the non-collocated plant from T to (} 2 can be modeled as, -s 2 +k G(s)---- - 3s 2 (s 2 + 2k) (7.2.1.2) For the purpose of comparison, the physically unrealizable full state feedback, the unstable conventional LTR output feedback and the delay-based IMML TR output feedback proposed in the paper are considered. The state and output feedback configurations are given in Figure 7.1.1.2. Moreover, since here we have only one measurable output for feedback, the full state feedback is always physically unrealizable as long as the plant order is 2 or more. For easy design and comparison, we can employ arbitrary full state feedback configuration because it is theoretical. Here, the full state feedback configuration shown in Figure 7 .1.1.2 is used for purpose of comparison. Note that for this special full state feedback configuration, the loop transfer function is, L(s)=K(s1-Ar 1 B while the closed-loop transfer function is,. G s - K(s1-Ar 1 B = K(s1-Ar 1 B cC )- K(sl -A+BKr 1 B l+K(sl -Ar 1 B (7.2.1.3) (7 .2.1.4) 174 And the loop transfer function for output feedback is, L(s) = C(s)G(s) (7.2.1.5) with the closed-loop transfer function as the following form, G (s) = C(s)G(s) c 1 +C(s)G(s) (7.2.1.6) Note that in the output feedback configuration, the controller C(s) can be either conventional IMM-LTR controller or the delay-based IMM-LTR controller, while the plant model is either of the above two types of non-minimum phase systems. 7.2.2 Simulations In this part of simulation, the 2-DOF flexible robot arm servo system introduced in section 2 is concerned, with k=5000N.m, we have the open-loop plant transfer function shown in equation (37). Since in this case, the plant is of non-minimum phase, in order to achieve complete loop transfer recovery, the internal model modification (IMM) technique must be used, and the theoretical full state feedback, IMM based reduced-order LTR output feedback and IMM based delay-induced output feedback are involved in this example. G(s) = -s 2 + 5000 3s 2 (s 2 + 10000) The desired open-loop transfer function of the feedback system is, L (s) = 1000(s 3 + 70s 2 + 4100s +87000) d s 2 (s 2 +10000) (7.2.2.1) (7.2.2.2) 175 According to the all-pass filter constraint, the modified plant should be, G (s) = s 2 +IOO.J2s+5000 M 3s2{s 2 + 10000) (7.2.2.3) Then the augmented-plant is, G (s) - G (s) - G(s) - 1 oo.J2 A - M - 3s(s 2 +10000) (7.2.2.4) Assume the reduced-order LTR controller is employed and to make the LTR controller physically realizable, one other negative real pole should be added below, C ( ) =9x10 6 (s 3 +70s 2 +4100s+87000) IMMLTR S r,:: (s 2 +1OOv2s + 5000)(s + 3000) (7.2.2.5) Additionally, smce the delay-based controller can be used to realize any improper controller, the additional negative real pole mentioned in equation ( 41) is useless and we can realize the delay-based LTR controller as follows, Cv (s) = -2.9985x10 15 (0.997e- 3 sr - 2.998e- 2 sr + 3.005e-sr -1.004) (s 2 +100J2s+ 5000) (7.2.2.6) To eliminate the transient response deterioration due to delay action, the following initial function compensation is utilized, V(t) = o(t) + 0.001(-125.77e- 30 t + (55.77 cos 50t + 171.9sin 50t)e- 20 t) 5000 where t E [-0.5398, -0.5395). (7.2.2.7) The open-loop and closed-loop frequency responses of theoretical full state feedback, delayed and non-delayed IMM-L TR output feedback systems are given in Figure 7.2.2.1 and 7.2.2.2. And the step response of these three methods with the 176 initial function compensation technique is given in Figure 7.2.2.3, from which we can see that the delayed IMM-LTR can be used to realize loop transfer recovery to non-minimum phase systems with guaranteed robustness, transient and steady-state performance. 100~-~--~--~----~--~ : GM=ZGqB -200 ._____..........._ __ --+--, __ _.___ _ ____,.____~~ 100 I~ I~ I~ Frequency (Rad/sec) 200 . . . . ··~ o ~ ·:_· ~ -~ L · ~ ... ~ . ~ ......... 1·\· ... ...... i ..... .(a)~ .... ... .. . f: ::_~~1~:~~6;:::F:~:~:::(~< ::.:: . . . . . . . . . . -800 .__ _ __.__ __ ........_. __ .___ _ _..... __ _.__ _ ___. 10° 10 1 10 2 Ht 10 4 HT 10 6 Frequency (Rad/sec) Figure 7.2.2.1. Open-loop frequency responses for full state feedback and IMM based delay-induced output feedback systems . . . . . . . . . . 0 - - - - :- - - - :- :-. - - ~ .-:-. "'."". :-:- .'.:-. :-: .-:-. "'."". :-: ."'."". :-: .-:- - - -~---~--- ---------- . . . . . . . . -20 ........ . : ......... ~ ......... ~ .... . ... .. ....... : ....... . . . . . . c: -60 .... ..... : ...... ... ~ . . . . ..... ~ ......... : ..... . ... ; . . ..... . :::=. : : : : : ;~ -80 ...... . .. : .. . ...... ~ ......... : . .. ... ... : .. . ..... ; .. . .. .. . ,,... . . . . : : : : . fh) -100 ......... : ......... ~ ......... ~ ......... : ........ ; ... \~. . . . . . . . . . . . . . . -120 ........ ·:· ... ..... : .. . ...... ~ ....... . ·:·...... . : . . . .. . . . . . . . . . . . -1 40 ......... : ......... ~ . . ... ... . ~ ......... : . . . . . . . . : . . . . . . . . . . . . . . . . . . . . . -160'---~____.._ __ ......_ __ ~-------'--........---'~_,_,_.,~ 1 o 0 id 10 4 Frequm::y (Rad/sec) Figure 7 .2.2.2. Closed-loop frequency responses for full state feedback and IMM based delay-induced output feedback systems Full state feedback :: ·· ••••·· /',:--~---~~ . ..... ......... : IU ~ / · ................ ........... ........ ... Delay-bl!S!;d !MMUR ................ . 0.4 g. 0.2 ......... , ....... .. ; ..... ... .............. :···· ···· ······· ·······:······················: ····················· I . : . : IU a: g- 0 I : . .. ;:r······· · ·····~ · · ···· · · ···· · ···· ·····:··· ···· ... ..... .... : ...... ... ···· · · ······~·· · ··········· · ·· ·· ·· as . . . . -0.2 · · · · · · · · · · · · · · · · · !MM.,LTR · · · · · · · · \ · ·· · · · · · · · · · · · · · · · · · · ·\· · · · · · · · · · · · ·· · · · · · · · · \ · · · · · · · · · · · · · · · · · · · · · . . . . . . . -0.4 ..... . .. .... . .. . .. . : ............. .... ····!··· ................ ···:········· · .... ...... ··: .... ................ . -0.6 -0.8 . ...... .... . .. .... · ......... ...................... .. . .. . . . . ····· ······ ....... ·.········· . . . . . . . -1---~~~----J'--~~~--'~~~~----JL--~~~--'~~~~--' 0 0.05 0.1 0.15 0.2 0.25 Time(sec) 177 Figure 7 .2.2.3 Step Response of full state feedback, IMM-LTR and delay-based IMM-LTR feedback 178 8. Non-collocated Control of Single-link Robot Arm Recently, more and more researches on robotics are focusing on the link with structural flexibility, which can achieve much faster response than the conventional rigid link and reduce the construction cost due to lighter robot. However, link flexibility will usually cause significant technical problems, such as stabilization, non-minimum phase, spillover and bandwidth limitations. Most importantly, the control algorithms are required to compensate for these problems. As described before, due to the non-collocated control of flexible link, the non minimum phase system is not uncommon, which will cause severe difficulties especially in the control design. Through use of the proposed internal model modification technique can help reduce the side effects of the non-minimum phase behaviors and delay-based loop transfer recovery method can help increase the closed-loop bandwidth and noise immunization. In this chapter, a non-collocated control of a single-link flexible robot is investigated using the delay-based loop transfer function and internal model modification technique in controller design. 8.1 Model Description We consider a one-link flexible arm, oflength Land uniform linear mass density p, rotating on a horizontal plane shown in Figure 8.1.1. The arm is driven by an actuator at the base with negligible inertia and torque t(t). The flexible link can be modeled as an Euler-Bernoulli beam with Young Modulus E and inertia of the cross 179 section I, assuming small deformations limited to the plane of motion. Let 8(t) be the angle to the instantaneous center of mass of the link. The transversal bending deformation at a point xE [O, L] along the link is described by w(x, t). y I / / / / / Figure 8.1.1. A single-link flexible robot arm system From Hamilton principle, the motion equations of the flexible arm are ; Elw" "(x, t) + p(w(x, t) + xB(t) )= 0, r(t)-JB = 0 -' (8.1.1) where J = pL 3 I 3 is the total inertia of the arm with respect to the joint axis, with associated dynamic boundary conditions given by w(O,t) = 0 Elw "(O, t) = 0 Elw"(L,t) = 0 Elw "'(L, t) = 0 (8.1.2) 180 in which a prime denotes the spatial derivative with respect to x. By seperation in space and time, assume a finite number n ~fo~~ associated deformation coordinates ch(t), ~-- ---- n w(x,t) = L9Hx)8;(t) (8.1.3) i=l and imposing the boundary conditions (8.1.2), the mode shapes take the form \~(~)~~(~;~:X~~:~2;~nh~~\_ (8.1.4) where /3; 4 = pm; 2 I EI and /3 1 , /3 2 , ... , Pn are the first n roots of the following characteristic equation (8.1.5) and m; are the eigen-frequencies of the flexible arm, for i=l, ... ,n. The resulting Euler-Lagrange equations for the N=n+ I generalized coordinates q=(8, Di) are JO =r (8.1.6) 8; + miz8i =</Ji' (O)r X2 . . . X 2 n r the State space equation of the system is, x=Ax+Bu (8.1.7) U=T 181 where, :0 1 0 0 0 0 0 0 0 0 0 0 0 1/ J 0 ·o 0 1 0 0 0 A= 0 0 -0)2 1 0 0 0 B= <P;(o) 0 0 0 0 0 1 0 0 0 0 0 -OJn 0 2nx2n q)~(O) C=[l 0 ¢1 (L) 0 </Jn(L) o] L L 8.2 Controller Design Assume a link with L=lm, EI=72.2Nm 2 , p=0.8kg/m, and from the boundary conditions, we can obtained both Pi and co i as follows, /31=3.927,/32=7.07,/33=10.21,/34 = 13.35, ... OJ 1 =146.50,0J 2 =474.86,0J 3 =99Q.32,0J 4 =1693.11, ... Although theoretically speaking, the modal expansion of the flexible beam in equation (8.1.3) has infinite number modes, since the amplitudes of the higher modes of flexible manipulator are significantly smaller than the lower modes, the model can be truncated to small number of modes, e.g., one, two, or three. Mathematically this infinite sum holds true, in reality there is always some internal damping of the modes within the beam. These facts, coupled with the impossibility of taking a true infinite sum as a model in designing controller, forces an approximation or truncation to eventually be taken in all realistic applications. 182 In here, we only consider the rigid body mode and the first three flexible modes in controller design. Hence the state space equation of the robot arm is, 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3.75 0 0 0 1 0 0 0 0 0 0 0 -21463 0 0 0 0 0 3.8175 x= x+ "{ 0 0 0 0 0 1 0 0 0 0 0 0 0 -225489 0 0 0 7.0785 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -980731.6 0 10.2095 y= [1 0 -1.4142 0 1.4162 0 -1.41396 o}x And the transfer function from 't to y is, G (s) = (s+855.4)(-s+855.4)(s 2 +387s+6.336xl0 4 )(s 2 -387s+6.336xl0 4 ) ry s 2 (s 2 +2.146x10 4 )(s 2 +2.255xl0 5 )(s 2 +9.807xl0 5 ) According to the all pass filter constraints in designing internal model modification, the desired modified model is, G (s) = (s+855.4)(s+855.4)(s 2 +387s+6.336x10 4 )(s 2 +387s+6.336x10 4 ) rym s 2 (s 2 +2.146xl0 4 )(s 2 +2.255x10 5 )(s 2 +9.807xl0 5 ) and thus the Augmented-plant internal model modification compensator is, G (s)=G (s)-G (s)= (s+855.4)(s 2 +387s+6.336xl0 4 )(2s 2 +7.888x10 5 ) 1MM rym ry S(s 2 +2.146x10 4 )(s 2 +2.255x10 5 )(s 2 + 9.807x10 5 ) To design the controller, first we assume the higher modes of vibration is stable because of its internal damping, and then the desired closed-loop performance requirements of the system are at least 1500 rad/sec closed-loop bandwidth, 45 degree phase margin and 10 dB gain margin can be achieved. In common sense, this 183 requirement is too stringent for usual controller, because system has non-minimum- phase open-loop zeros within the required closed-loop bandwidth. For existing control technology, the only way to realize this performance is through use of full state feedback. However, since the system has only one measurable state, the full state feedback is actually physically unrealizable. To realize the full state feedback by realizable output feedback scheme without introduce the unstable controller, the only method is though use of internal model modification based loop transfer recovery. Additionally, to reduce system's sensitivity to noise and controller's order, the proposed delay-based loop transfer recovery method should be employed. The su1~able internal model modification compensator has been found through use of all-pass filter constraints in the above. The only difficulty in designing the delay-based loop transfer recovery controller is t~ find a reference loop transfer function, which might be physically unrealizable. This reference loop transfer function can be easily found by ful~ stat~ _fe~baclc contioller d~- -~:;,e~, like I pole-placement, root-locus analysis, bode loop shaping, linear quadratic regulator (LQR), etc. A reference loop transfer function for a full state feedback system is achieved, which is shown as.follows, L (s) = lxl0 4 (s +30)(s 2 +60s+l.09xl0 4 )(s 2 +200s +l.325xl0 5 )(s 2 +200s +5.725 xl0 5 ) r.f s 2 (s 2 + 2.146x10 4 )(s 2 + 2.255 x IOS)(s 2 + 9.807x10 5 ) To realize this loop transfer function by loop transfer recovery output feedback scheme, some certain pole-zero cancellation must be employed. However, to the 184 original system, some open-loop zeros are in RHP, direct pole-zero cancellations with this kind of zeros will cause unstable controller. This is why we employ the internal model modification technique, which changes the input output relation of the controlled system from non-minimum phase to minimum phase and since we change the system according to the all pass filter constraints, the eventual gain frequency response of closed-loop transfer function of both modified plant and real plant are exactly the same, which means what performance the modified closed-loop plant has will be those that the real closed-loop system has, e.g., the closed-loop bandwidth, damping ratio, noise rejection, etc. Based on the above description, the following desired loop transfer recovery controller should be realized, ~x~;~~)(s 2 ~ 60s +l.09x10 4 )(s 2 + 200s + 1.325x10')(s 2 + 200s + 5.725x1 O') ~ =~ (s+855_j):_(s_ 2 _ ±3-.8.il±6 336~!_0 4 ) 2 ..._______ ___.. -- ------- --------- - - - · -.____ - -- - Since the above loop transfer recovery controller is improper, ~ ---1s not realizable in finite dimensional domain. The easy way to realize it is using the delay- based infinite dimensional control method, which can realize any improper controller with arbitrarily small error. The method to generate the delay-based loop transfer recovery controller is to change the numerator of the above desired loop transfer recovery controller into delayed form while keep the denominator the same. This gives us the following transformed controller, 3 µ(1-be-sT)fJ (e-2sT +c;e-sT +d;) Cv (s) = (s+ 855.4) 2 (s~=~ 387s+6.336x10 4 ) 2 where we choose T=O.OOOOlsec., and b, Ci, di, are shown as follows, 2 u;r b = e-3or ,ci = e d -e2a,r ~1 +tan 2 w,T' j - ' CT 1 ~~- =100,m 2 =3 U 3 =30,m =}00 V / µ-~~ 0)(1.09x10 4 )(1.325 xl0 5 )(5.725 x10 5 ) 1 (1- b) II 1 + s + d;) ' ~ --1,;J - 100 • •. .• • •• • •• .• • •• . •• .• • •• • •. .• • .• • •. .• • ••••.•• ••••..•.•• ••.••••••. .• ; ~ 50 • ••.••.•••. .••.. f ........•....... ~- .. .. . .. .. . .. . .. . ........ .• ..... . : I ~ ·100 •.•••••• ••••..•• ~- ••..•• ••••..• • •• ~ •. .• • .. . . .• - ': ·- • •• ••. .••.• •. l f l . ' -200 ••••..••••••..• • ; •••..••.•••..•••• ' •. •• .•••... ••• • ..••••••..••••• •. . : ·250 •...•..•••. .•..• ~ •..••.•••..• • .• • ; ..• ·300 • •. .• • .• • •. .• • .• ~ •. .• • .• • • . .••..• i· . . .. . .. . . . .. . .. . .. . .. . . . . .... · ·- -350 ••••..• •••••..••••••. .•• ••••. .•••• ~ •. • • •• • •. . • •• • . .••••••.••• ••• ·- 100 10 1 10 2 10 1 10• F~Mlc:y (rad/Ht) Figure 8.2.1. Bode plot of the truncated beam transfer function ·50'---..___--~---'-:-----'------' 10 1 10 1 10 1 10 1 10' 10 1 ·300'----..___ __ _.__ _ _ -'-:-__ _..__ __ __, 10 1 10' 10 2 10 1 10' 10• F19q11onc~ (Rect°l9C) 185 Figure 8.2.2. Bode plot for the delay-based loop transfer function (GM=20dB and PM=75 degrees) 186 The frequency response of the open-loop plant is given in Figure 8.2.1. And the bode plot of delay-based loop transfer function is shown in Figure 8.2.2, from which we can see that both gain margin and phase margin are big enough and the closed- loop bandwidth also satisfies the performance requirement as shown in Figure 8.2.3. -5 Iii -10 :!!. c ~ -15 -20 -25 -30 10" 10 1 - - -......,,r '-.../ v 102 103 Frequency (Rad/sec) ""' " I\ \ \ \ \ Figure 8.2.3. Closed-loop gain frequency response (BW>>1500 rad/sec) To implement this control algorithm, the transient response deterioration due to delay action must be taken into account. Based on the method of initial function compensation described in chapter 5, the following initial function is derived, 1 3 IFC(t) = --8(t)- 0.0412e- 30 t + LeU;t [ ai cos(w/) +bi sin(m/)] 5000 i=l a 1 =-30,a 2 =-100,a 3 =-100,m 1 =100,m 2 =350,m 3 =750 a 1 =-0.03036,b 1 =0.08635,a 2 = -0.0477,b 2 = 0.1411,a 3 = 0.0213,b 3 = 0.096 t E (0.01813- ?T, 0.01813) 187 And using Matlab Simulink shown in Figure 8.2.4, combining internal model modification, delay-based loop transfer recovery and initial function compensation, vey good performance can be achieve with the step response shown in Figure 8.2.5. l~ -D-------- 1 ...---_... :' ____ 0 I Clock~ --·-- . - -· / • _;·7 Ip ' 1 /i ) ,) ! , ' o ... \ r · To V'.\Jrl<sp•c• 1 :-· / 1.., \// V -Vv In itia l Function Non~delay Comsie nsator L TR Partition Subststem t . .... -- Gain _.~/ Figure 8.2.4. Matlab Simulink Diagram for delay-based loop transfer recovery with internal model modification. 1.2 ~ 0.4 I I I 0.8 0.6 0.2 Step .e E ! Ollj!e I v -0.2 -0.4 -0.6 ·0.8 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Time (sec) Figure 8.2.5. Step response of the delay-based loop transfer recovery with internal model modification output feedback system from Matlab Simulink. 188 For comparison, the step response for delay-based control and the physically unrealizable full state feedback control system (mathematically generated) are given in Figure 8.2.6, from which we can see only small difference in transient response can be found. 1 ... .. . .. . , ... .,...... ~· ._.;... _..-..,-:-:-:-;-,.-:-:-:-~-;-.-.---;.---~,......_;..----;:----...;-----1 ii) ~ 0.5 a. ., ~ a. 0 Q) Ci.I . . . . . . . . . . . ~ ........ .. ·:· .... . .. ... : . . ........ ! ...... ... . : .. ... .... ·:· . ..... .. . . ~ ...... . . .. : ... . ... .. . ·:· .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .............. .......... ............ ..... ........ . .......... ........................ .... ... .. ........... . . . . . . . . . . . . . . . . . . . -0.5 ......... ·.· .... .. . . ·:· ... ..... . . ; ... .. .. ... '." ..... . . ·.• . . ..... .. ·.· ...... . .. , .... . . ... .... ... . . . . . 0~2 0~4 0~6 0~8 0.1 0.12 0.14 0.16 0.18 0.2 0 0.005 0.01 0.015 Tine (sec) Figure 8.2.6. Step responses for delay-based control system and theoretical full state feedback system. Upon the applications in this and previous chapters, we can conclude that the delay-based loop transfer recovery method with internal model modification and initial function compensation techniques work fine for either discrete or continuous system. And very good performance can be achieved with general and systematic design methodologies, which may become an practical way to design high performance flexible servo systems. 189 9. Experimental Results-Non co-located Control of Flexible Systems As discussed in the previous chapters, the delay transform technique and finite design region principle are general and systematic ways to design the high performance flexible servo control system, which has been proved theoretically. However, to check the effectiveness of this method, we need to implement it on the real-time control system. Here, in this chapter, we employ the ECP (Education Control Products) torsional control system as the experiment, which implements the delay transform technique and finite design region control algorithm with comparison to the full state feedback. 9 .1 System Description The ECP torsional control system consists of the torsional disks-spring flexible system and a full complement of control hardware and software. This apparatus shown in figure 9 .1.1 represents many such physical plants including rigid bodies; flexibility in drive shafts, gearing and belts; and coupled discrete vibration with actuator at the drive input and sensor collocated or at flexibly coupled output (noncollocated). Thus the plant models may range from a simple double integrator to a sixth order case with four lightly damped poles and either four or no zeros. The experimental control system is comprised of the three subsystems. The first of these is the electromechanical plant, which consists of the torsional mechanism, 190 its actuators and sensors. The design features a brushless DC servo motor, high- resolution encoders, adjustable inertias and reconfigurable plant type. Next is the real-time control unit, which contains the digital signal processor (DSP) based real-time controller, servo/actuator interfaces, servo amplifier and auxiliary power supplies. The DSP is capable of executing control laws at high sampling rates allowing the implementation to be modeled as continuous or discrete time. The controller also interprets trajectory commands and supports such functions as data acquisition, trajectory generation, and system health and safety checks. A logic gate array performs motor commutation and encoder pulse decoding. Personal Computer ------ Real-time Controller Servo Amplifier Figure 9.1.1 The Torsional Control System The third subsystem is the executive program, which runs on a PC in the Windows operating systems. The menu-driven program is the user's interface to the 191 system and support controller specification, trajectory definition, data acquisition, plotting, system execution commands and more. In this experiment, only two disks with masses are considered, which means that the middle disk is removed such that the overall flexible system is of 2-DOF with one rigid body mode and one flexible mode. The overall dynamics of the investigated system can be modeled as the following fourth order transfer functions, B 3 (s) = k U(s) Js 2 (Js 2 +2k) (9.1.1) 0 1 ( s) _ Js 2 + k U(s) - Js 2 (Js 2 + 2k) (9.1.2) Assume we adjust the experiment apparatus such that the moment of inertias of disk 1 and disk 3 are the same and the coefficient of the torsional spring k = k 1 k 2 , kl +k2 where k 1 and k 2 are the coefficients of springs between disk 1 and disk2, disk 3 and disk 2, respectively. After some certain system identifications, we found that k 1 =2.587 Nim and k 2 =2.455N/m and then k=l .262N/m. And J is set to be 0.018kgm 2 . The closed-loop control diagram of the full state feedback experiment is given in figure 9.1.2 and that of the delay transformed control is given in figure 9.1.3. And in both diagrams, the so-called hardware gain khw, of the system is comprised of the product: (9.1.3) 192 where: kc: the DAC gain= 8v/32, 768 DAC counts ka: the Servo Amp. gain= 2 A/v km: the Servo Motor gain= 0.6158 Nm/v ke: the Encoder gain = 16, 000 pulses/27t radians ks: the Controller Software gain= 32 controller counts/encoder counts d k 83 ! Js 1 (Js 2 +2k) ~~~~ Js 2 +k Js2(Js 2 + 2k) F igure 9 .1.2 Closed-loop configuration for full state feedback (continuous) d Figure 9 .1.3 Closed-loop configuration for delay transformed control 193 The control objective is to employ control effort at the disk 1 and control the position of the disk 3, which is essentially the non-collocated control. Both the full state feedback control algorithm and the delay transform technique and finite design region control algorithm are going to be implemented on this real-time experiment to check the equivalence between each other. However, due to the physical limitation of the experiment apparatus in that the overall feedback gain is very limited, the delay transformed controller cannot be directly implemented, since at the initially stage of the control, there are no enough history data available to calculate the control force, unsuitable initial conditions may cause the control effort out of the limit. This will not happens in the following two cases, one is when the limit of the control effort is large enough, which sometimes is true for high performance servo control system, the other is when the initial values are set properly or the so-called initial value compensation (IVC) is employed. In this experiment, since the control energy is limited, the only way to check the effectiveness of the delay transformed control algorithm is to employ the initial function compensation proposed in chapter 5. And it is worth mentioning that either of the above methods, increasing gain level or employing initial function compensation is quite practical to realize high performance servo system. Either configuration can be competent, and to ensure less control effort, the initial value compensation is an effective way, while to decrease the complexity of the control algorithm, increasing the gain level becomes a must. 194 And usually there is a design tradeoff and most frequently both of these methods are employed to further improve the performance. 9.2 Controller Design (I). Full state feedback controller design To design the full state feedback controller, there are many existing methods such as the LQR (linear Quadratic Regulator), pole-placement or zero placement. Here, we employ the so-called zero placement method, which is to design the full state feedback gain vector such that some certain zero structure of the overall loop transfer function for the full state feedback can be guaranteed. Specifically, in this experiment, we hope all the zeros of the loop transfer function is in the stable region (left half complex plane), which is due to the fact of root-locus analysis that as long as all the open-loop zeros are in the stable region, as the gain is large enough, we can always make the system closed-loop stable, since the root-locus will approach the open-loop zeros as gain goes to infinity. Additionally, as discussed before, since the gain level of this experiment is quite limited, the conditional stability is undesirable, which requires the overall system is stable when the overall loop gain is very small. This imposes some phase conditions on the loop transfer function of the full state feedback and according to the phase design methodology and the relation between phase and zero-structure of the loop transfer function proposed before, the following type of root-locus shown in figure 6.2.1 should be employed. 195 Real Axis Figure 9.2.1 Desirable root-locus for the full state feedback system. Based on the above analysis, the zero placement requirement can be formulated that the zeros of the following loop transfer function are in left half complex plane, i.e., where Re( Zi)<O. After some certain experiment, we get the following state feedback gain, vi=0.0516, v2=0.0172, vs=0.0306, v6=0.0136. And the zero structure of the loop transfer function is, 196 Z 1 , 2 = -0.156l±j10.8663,z 3 = -2.6878 To implement the above control law, we must transform the s-domain controller to z-domain first and the discrete-time full state feedback configuration diagram is given in figure 9.2.2 with the sampling period Ts=0.002652s. And the discrete-time full state feedback gain vector is shown as follows, ui=0.0516, u2=0.0172ffs, us=0.0306, u6=0.0136/Ts. d k Js 2 +k Js2(Js 2 + 2k) "'----1u 1 +~{1-z- 1 )/Ts .....---~·------· I I I I -----u 5 +~{1-z- 1 )/Ts .....---~------___, Figure 9.1.2 Closed-loop configuration for full state feedback (discrete) (2) DTT & FDR controller design Based on the above full state feedback controller design, the generalized controller, which is equivalent to the full state feedback in terms of output feedback is such an improper controller shown as follows, C (s) = k(v 1 +v 2 s)+(Js 2 +k)(v 5 +v 6 s) g k (9.2.2) 197 And according to the delay transform technique based on the finite design region principle, the following DTT controller is obtained, (9.2.3) where all the parameters of the above DTT controller can be calculated from the formula presented before according! y. And according to the zero-structure z-transform proposed before, we can transform the above-generalized controller into z-domain as follows, (9.2.4) (3) IFC design As discussed before, since the experiment apparatus we employed here has limited gain, to avoid the nonlinearity caused by the gain saturation or undesirable transients, the sufficient initial values should be compensated, which involves the initial function compensation (IFC) method. The principle of the IFC is to set the _,.....-------~~ · ............. ~·.·.' . initial values properly such that the transients caused by switch can be minimized. Here, the output feedback scheme instead of the state feedback is employed, thus not the initial value of state but several history output values should be compensated. According to the formula discussed in chapter 5, we can easily calculate the following initial values, y(-3)=0.002658, y(-2)=0.001823, y(-1)=0.000887. This initial value compensation will make the delay transformed controller behaves like the full state feedback in our real-time control experiment, and the experiment results are shown in next section. 198 9. 3 Experiment Results In this section, the experiment results of the above designed controllers are illustrated below in time domain response. For easy comparison, the data generated by ECP controller are processed through use of Matlab (because Matlab has great graphical capability). The time-domain closed-loop step responses and control efforts for both discrete-time full state feedback and IVC based delay transformed feedback with 0.002652sec sampling time are shown in figure 9.3.1 and 9.3.2. And from these figures, we can see that both of them are almost equivalent as expected. F•ull state feedback I I I 1000 - I I 800 (i) ---~ ----Dh--rn~ -----:- ----~ ----~ -----r - - - - , - - - - -.- - - - - I I I I I I I I I I I I I I I I c: I I I I I ::3 I I I I CJ I I I I I I ~ 600 c :~ I I I I I I I I I ---,-----r----T----,-----T----,-----~----,-----r---- 1 I I I I I I I I I I I I I I I I I en CJ c.. 400 I I I I I I I I ----1-----~----+----~-----·----~-----~----1-----~---- I I I I I I I I I I I I I I I I I I I I I 1 I I I I I I I I I I 200 ----~-----~----·----~-----·----~-----~----~-----~---- ' I I I I I I I I I I I I I I I I I I 0'--~-'-~_,_~__..__~....._~__._~~~~~~--'-~~'--~~ 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Time (sec) Figure 9.3.1 Step responses for both full state feedback and IVC based delay transformed output feedback. From the closed-loop step response and control effort plot for these two types of control, we can see that the output response of the them are almost equivalent while 199 the control efforts is a little bit different in that the control effort of full state feedback control is a little bit noisy due to high frequency noise amplification and that of the delay transformed control is much more smooth. This is in accordance to the advantages of DTT & FDR method over full state feedback proposed in previous chapters. This suggests that DTT & FDR can be used to realize the usually physically unrealizable full state feedback by output feedback with guaranteed performance and make the overall system insensitive to high frequency noises. ,,......, 2 0 ~ 1:= ~ Q) E! c 0 u 0. 6 - - - - -I - - - - -1- - - - - 1' - - - - -1- - - - - t- - - - - -t - - - - - I- - - - - "1 - - - - -1- - - - - 0.4 0.2 0 -0.2 -0.4 - - - - -~ I I • F.ull state feedback • I I I I I ----~-----r----~----- ' I I I I - - - - -,- -- - - I I I I I - - - - -1- - - - - .. - - - - ~ - - - - - .... - - - - ~ - - - - - ·- - - - - I I I I I I I I I I I I I I I I -' - - - - - L - - - - J - - - - - ._ - - - - J - - - - _ 1_ - - - - I I I I I :on-FQR I I I I I I I ----,-----r----T----~-----r----,-----r----,-----r---- 1 I I I I I I I I I I 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Time (sec) Figure 9.3.2 Control efforts for both full state feedback and IVC based delay transformed output feedback 200 10. Concluding Remark 10. 1 Summary In this dissertation, some delay-based output feedback control technologies for design of high performance flexible servo control system. These technologies are delay transform technique with finite design region principle, initial function compensation for delay-based system and delay-based internal model modification for control of non-minimum phase systems. Several advantages over existing control algorithms such as full state feedback, loop transfer recovery (L TR) output feedback, conventional robust control and existing delay-induced infinite dimensional control are illustrated, which can be summarized as follows, • Lower gain and lower bandwidth controller • Lower amplification of high frequency noise and unmodeled dynamics • Lower order controller • Higher signal-to-noise ratio in control channel, no chattering in steady- state • High robustness (phase margin, gain margin) guaranteed • Higher closed-loop bandwidth guaranteed • Internal stability guaranteed • Systematic and simple design procedure • Easy and general stability and performance analysis for delay-induced feedback system 201 Both the theoretical proof and experiment verification of these delay-based output feedback control technologies are given, which shows that the delay-based output feedback is a general and systematic way to design the high performance feedback system and can be used to realize the full state feedback with much better closed-loop performance that the overall system is insensitive to high frequency noise, and it has good disturbance rejection ability and fast response with guaranteed robustness to system uncertainties like parameter variations and unmodeled dynamics. 10.2 Open research problems for High Performance Flexible Servo Control In this dissertation and related research, we can obtain the following open research problems for design of high performance flexible servo control systems. (1) Arbitrary loop shaping versus open-loop zero location In the loop transfer recovery feedback schemes or DTT & FDR control algorithms, in order to achieve arbitrary loop transfer function that the full state feedback has, the pole-zero cancellation of the compensator's poles and zeros of the open-loop plant must be employed. This will cause severe problem when the open loop zeros have some parameter variations, especially when the zeros are complex ~~~~c:_~~--~9-~l}~j.ro.aginary axis At_that time poor performance and even instability may happens when the system's loop is closed. This tells us in designing the loop transfer recovery output feedback or DTT & FDR controller, the farther the 202 open-loop zeros are away from the imaginary axis in the left half complex plane, the --------- better closed-loop performance will be achieved. Thus to a system with many accessible outputs whose transfer function to system's input have nearly pure imaginary zeros and zeros far away from imaginary axis, the one with all the zeros far away from imaginary axis is preferred to be used to design the loop transfer recovery or DTT & FDR controller. And the closed-loop performance will be much better than those employing output with light damped zeros transfer function. This is because over damped pole-zero cancellation is more robust than light damped one. (2) Non-collocated versus collocated control In the structural control, it is well know that collocated sensor and actuator will make the closed-loop system energy dissipated and the system is robust to system's uncertainties, however, in terms of closed-loop performance such as closed-loop bandwidth, usually the non-collocated control must be employed not only because of physical limitation in some systems such as micro-positioning system, disk drive, but mechanical systems, collocated system usually has pairs of complex conjugated ------ lightly damped zeros, which make the arbitrary loop transfer recovery impractical ----........ ------ du~ that parameters variations will always cause bad pole-zero cancellation and eventually cause poor closed-loop performance. This tells us in terms of closed-loop performance improvement of the flexible control system, usually the non-collocated 203 control is preferred unless the system is near rigid, although the non-collocated ------------- 9 controller is very hard to design. (3) Non-minimum phase versus minimum phase As discussed before, in order to acquire better closed-loop performance, especially arbitrary loop transfer function, the non-collocated control is preferred. However, due to the dislocated sensor and actuator, ~all flexible system is usually non-minimum phase system which has right half plane zeros. This usually places the upper bound on the closed-loop achievable bandwidth according to conventional robust control theory and the closed-loop system is difficult to stabilize, which will increase the complexity of the controller design. Although the delay- based internal model modification technique can be used to change the plant properties from non-minimum phase to minimum phase, which can greatly simplify the whole feedback control design and achieve much better performance, there still has many problems in the robustness issues. For example, if the plant eigen-st~cture has large parameter variations,_ the -~igef!:valu~ difference between the original plant and augmented plant may cause poor closed 7 loP.E._ p~rformance. This encourages us --- - to find more robust way to design the delay-based i~ternal model modification output feedback controller. With one clue from common sense in control theories, the more information we can measure from the system, the better performance we may achieve, which in multivariable system becomes increasing the number of sensors can change the non-minimum phase system to minimum phase one. This 204 suggests some certain sensor locations should be investigated, such that after some manipulation between these measurable outputs, the non-minimum phase plant can be change to minimum phase system. And since on the same flexible structure, the transfer functions from the control input to the measurable outputs have the same eigen-structures, which can make . internal-· model · ·medification_ CQ}ltroller design ... -... -~ .... ·-· ... .. ... ,_ ·-. .. . . . ~ ...... ind~~ef!~ _.Qf th~ . ~tIJJ~iµ.rnl.unc.ertainties.--Since this method is somehow depending on the structure design, we call it integrated structural control technology. Combining the integrated structural control and delay-based output feedback technologies, better performance may be expected in the future research. (4) SISO versus MIMO This dissertation is only focus on the SISO system and the MIMO system is much more complicated and lots of work need to do to establish the theoretical fundamentals for the delay-based control method in many situations such as MIMO minimum phase system, MIMO non-minimum phase system, etc, which has many industrial applications such as multi-stage hard disk servo assembly, multiple-link robot arms and many flexible systems. 205 REFERENCE LIST: [1] Abdallah C., Dorato P., Benitez-Read J. and Byrne R., "Delayed Positive Feedback Can Stabilize Oscillatory Systems,'' Proceeding of the American Control Conference, San Francisco, California, June 1993, pp3 106-3107 [2] Bien Z. and Suh I. H., "Controller with Multiple Time-delays," Proceeding of 1979 ISCAS. [3] Chen C-T., Linear System Theory and Design, Holt, Rinehart and Winston, Inc. New York. 1984. [ 4] Choksy N. H., "Time Lag Systems," in Progress in Control Engineering, vol 1, R.H. MacMillan et al., Eds. New York: Academic, 1962. [5] Dorato P. and Abdallah C., "Linear-Quadratic Control An Introduction,'' 1995, Prentice Hall, Englewood Cliffs, New Jersey. [6] Doyle J. C. and Stein G., "Robustness with Observers,'' IEEE trans. Automatic Control, vol. AC-24, 1979, no. 4, pp 607-611. [7] Doyle J. C., "Guaranteed margins for LQG regulators," IEEE trans. Automatic Control, AC-23, 1978, pp756-757. [8] Doyle J. C. and Stein G., "Robustness with observers,'' IEEE Trans. Automatic Control, AC-24,1979, pp 607-611. [9] Franklin G. F., Powell J. D., and Workman M., Digital Control of Dynamic Systems, Addison-Wesley, Menlo, California, 1997. [ 10] Gilchrist J. D., "n-Ob servability for Linear Systems,'' IEEE trans. Automatic Control, vol. AC-11, No. 3, 1966 pp388-395. [11] Gunckel T. F. and Franklin G. F., "A general solution for linear sampled-data control systems,'' J. Basic Eng. Trans. AS:ME 8 SD ( 1963): 197-203. [12] Kalman R. E. and Koepcke R. W., "Optimal Synthesis of Linear Sampling Control Systems Using Generalized Performance Indices,'' Trans. ASME Ser. D. ( J. Basic Engr.) 80 (1958): 1820-1826. 206 [13] Kalman R. E., "Contribution to the theory of optimal control," Trans. ASME Ser. D. (J. Basic Engr.) 86 (1964):51-60. [14] Kalman R. E. and Bertram J. E., "General synthesis procedures for computer control of single and multiloop linear systems," Trans. AIEE 77, part 2 (1958): 602-609. [ 15] Kalman R. E., "When is a linear control system optimal?," Trans. ASME Ser. D (J. Basic Engr.) 86 (1964): 51-60 [ 16] Kalman R. E., "On the general theory of control systems," presented at the 1961 1st Int. Cong. Automatic Control, Moscow, Russia. [17] Kalman R. E. and Weiss L., "Contributions to linear system theory," Research Inst. for Advanced Studies, April, 1964. [ 18] Kalman R. E., Ho, Y. C., and Narendra K. S., "Controllability of linear dynamical systems," Contributions Differential Equations, vol. 1, pp 189-213. [19] Kalman R. E. and Bucy R. S., "New results in linear filtering and prediction theory," Trans. ASME Ser. D. (J. Basic Engr.) 83 (1961): 95-107. [20] Kang M. S. and Yang B., "Discrete Time Noncollocated Control of Flexible Mechanical Systems Using Time Delay," ASME J. of Dynamic Systems, Measurement, and Control, 1994, vol. 116, pp 216-222. [21] Liu Z. and Yang B., "Phase Design Method in Stability and Performance Analysis of Delay-based Output Feedback Systems", American Control Conference, 2002. [22] Liu Z. and Yang B., "Full State Feedback Realization by Delay-based Output Feedback", American Control Conference, 2002. [23] Mee C. D. and Daniel, E. D., Magnetic Storage Handbook, 1993, McGraw-Hill, New York. [24] Miu D. K. and Karam R. M., "Dynamics and Design of Read/Write Head Suspension for High Performance Small Form Factor Rigid Disk Drives," ASME Advances in Information Storage Systems, 1991, vol. 1, pp 145-153. 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"Closed-Form Force Response of a Damped, Rotating, Multiple Disk/Spindle System," ASME J. of Applied Mechanics, 1997, vol. 64, pp 343- 352. [33] Smith 0. H. M., "Closer Control of Loops with deadtime," Chem. Eng. Prog., vol 53, no. 5, pp217-219,1957 [34] Spector V. A. and Flashner, H., "Modeling and Design Implications of Noncollocated Control in Flexible Systems," AS:ME J. of Dynamic Systems, Measurement, and Control, 1990, vol. 112, pp 186-193. [35] Suh I. H. and Bien Z., "Proportional Minus Delay Controller," IEEE trans. Automatic Control, vol AC-24, No. 2, April, 1979, pp370-371 [36] Suh I. H. and Bien Z., "Use of Time-Delay Actions in the Controller Design," IEEE trans. Automatic Control, vol AC-25, No. 3, 1980, pp600-603. [37] Suh I. H. and Bien Z., "A Root-locus Techniques for Linear Systems with Delay," IEEE trans. Automatic Control, vol AC-27, No. 1, 1982, pp205-208. [38] Tallman G. H. and Smith 0. J., "Analog Study of Deabeat Posicast Control," IRE trans. 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Abstract (if available)
Abstract
In this dissertation, the active delay output feedback control techniques are presented for high-performance flexible servo systems. Among these delay-related control techniques, the delay transform with finite design region principle play as the fundamentals of the generalized delay-based infinite dimensional control algorithms. The former is a way to realize any improper polynomial transfer function, which is physically unrealizable in conventional finite dimensional domain. And the latter serves as the systematic and general closed-loop stability and performance criteria. Since the improper controller can be realized through delay transform, the full state feedback control is physically realizable by delay-based output feedback with equivalent performance. And this technique is proved to have much better performance than the conventional observer-based or compensator-based output feedback. However, due to the introduction of delay, although the steady-state response is as perfect as full state feedback, the transient response deterioration is inevitable. This calls for a remedy called initial function compensation technique for delay-based controller and it has been proved that the transient response deterioration can be eliminated. Furthermore, to achieve high-performance flexible servo systems, control of non-minimum phase system is not uncommon, which is very complicated in practice. The so-called internal model modification technique solves this problem perfectly and combined with the delay-based output feedback, this new technique becomes a promising output feedback technique for high-performance flexible servo systems.
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Liu, Zheng
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Active delay output feedback control for high performance flexible servo systems
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Doctor of Philosophy
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Mechanical Engineering
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08/01/2015
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