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Bumpless transfer and fading memory for adaptive switching control

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Bumpless transfer and fading memory for adaptive switching control

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Content BUMPLESS TRANSFER AND FADING MEMORY FOR ADAPTIVE SWITCHING CONTROL by Shin-Young Cheong A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Ful¯llment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) December 2009 Copyright 2009 Shin-Young Cheong Dedication This dissertation is dedicated to my parents and my beloved wife, Hanna. ii Table of Contents Dedication ii List Of Figures v Abstract vii Chapter 1: Introduction 1 1.1 Bumpless Transfer with Slow-fast Controller Decomposition . . . . 2 1.2 Unfalsi¯ed Control Using Limited Memory . . . . . . . . . . . . . . 5 1.3 Organization of the Dissertation . . . . . . . . . . . . . . . . . . . . 7 Chapter 2: Adaptive Switching Control and Bumpless Transfer 8 2.1 Switching Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Switching Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2.1 Hysteresis Switching Algorithm . . . . . . . . . . . . . . . . 10 2.2.2 Dwell Time Switching . . . . . . . . . . . . . . . . . . . . . 13 2.3 Adaptive Switching Control . . . . . . . . . . . . . . . . . . . . . . 14 2.3.1 Supervisory Control . . . . . . . . . . . . . . . . . . . . . . 15 2.3.1.1 Estimator-based Supervisor . . . . . . . . . . . . . 16 2.3.1.2 Performance-based Supervisor . . . . . . . . . . . . 16 2.3.2 Unfalsi¯ed Control . . . . . . . . . . . . . . . . . . . . . . . 17 2.3.2.1 Theory: Validation and Unfalsi¯cation . . . . . . . 17 2.3.2.2 Data-Driven Learning and Adaptive Control . . . . 23 2.3.2.3 Stability and Convergence . . . . . . . . . . . . . . 25 2.4 Bumpless Transfer Techniques . . . . . . . . . . . . . . . . . . . . . 30 2.4.1 Case Study: ACC benchmark problem with 2-cart mass- spring-damper system . . . . . . . . . . . . . . . . . . . . . 31 2.4.2 Bumpless Transfer Using Add-on Type Controller . . . . . . 35 2.4.2.1 L 2 bumpless transfer . . . . . . . . . . . . . . . . . 35 2.4.2.2 Linear quadratic bumpless transfer . . . . . . . . . 38 2.4.3 Controller Modi¯cation Based Bumpless Transfer . . . . . . 40 2.4.3.1 Conditioning Technique . . . . . . . . . . . . . . . 40 iii 2.4.3.2 Continuity Ensuring State Space Realization . . . . 41 Chapter 3: Bumpless Transfer for Adaptive Control 44 3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.1.1 Switching control system . . . . . . . . . . . . . . . . . . . . 44 3.1.2 Slow-fast decomposition . . . . . . . . . . . . . . . . . . . . 46 3.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.3 Bumpless transfer implementation . . . . . . . . . . . . . . . . . . . 48 3.3.1 Bumpless transfer for a PID controller . . . . . . . . . . . . 48 3.3.1.1 Location of switching gains . . . . . . . . . . . . . 50 3.3.1.2 Controller state reset . . . . . . . . . . . . . . . . . 50 3.3.2 Bumpless transfer with slow-fast decomposition . . . . . . . 51 3.3.3 Slow modes controller with observable canonical form . . . . 54 3.3.4 Controller State Augmentation with Uncontrollable Modes . 55 3.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.4.1 Adaptive PID controller . . . . . . . . . . . . . . . . . . . . 57 3.4.2 Applying slow-fast controller decomposition . . . . . . . . . 61 3.4.3 Advantages over continuity assuring method . . . . . . . . . 62 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Chapter 4: Fading Memory and Time-Window 70 4.1 Review of Unfalsi¯ed Control . . . . . . . . . . . . . . . . . . . . . 70 4.1.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.1.2 Fictitious Reference signals in PID controller . . . . . . . . . 71 4.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.3 Computer Simulations . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 Chapter 5: Conclusion 81 Bibliography 83 iv List Of Figures 1.1 Switching control system with 2-degree-of-freedom controllers . . . 4 2.1 Switching control system . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Concept of Hysteresis Switching . . . . . . . . . . . . . . . . . . . . 11 2.3 Hysteresis switching algorithm . . . . . . . . . . . . . . . . . . . . . 11 2.4 Hysteresis Switching algorithm with one ¯xed unfalsi¯ed cost level . 13 2.5 Dwell time switching algorithm . . . . . . . . . . . . . . . . . . . . 14 2.6 Supervisors in adaptive supervisory control . . . . . . . . . . . . . . 15 2.7 Unfalsi¯edadaptivecontrol: unfalsi¯edcontrollerhastwo-tierstruc- ture, consisting of the supervisor and a conventional controller-plant pair. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.8 Unfalsi¯cation concept. Using performance goal and measured data as a sieve, candidate controllers are decided to be unfalsi¯ed or fal- si¯ed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.9 Without cost-detectability, model mismatch can cause an adaptive controller to switch a destabilizing controller C 2 into the loop and keep it, even when the original stabilizing controller C 1 was working well [41] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.10 L 2e -gain related cost functions have the cost-detectability property: They are monotone in t, uniformly bounded for stabilizing K and unbounded for destabilizing K. Cost-minimizing adaptive laws ro- bustlyconvergeirrespectiveofthesizeofplantmodel-mismatch[29], [38] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.11 2-cart mass-spring-damper system . . . . . . . . . . . . . . . . . . . 31 2.12 Bumpy transient with no bumpless transfer method . . . . . . . . . 33 v 2.13 In the switching control system, the actual control signal u follows the output u i when the current online controller is Controller i . . . . 35 2.14 Switching control system withL 2 bumpless transfer . . . . . . . . . 36 2.15 Switching control system with linear quadratic bumpless transfer. . 39 2.16 Feedback control systems in conditioning technique. (a) without conditioning (b) with conditioning . . . . . . . . . . . . . . . . . . . 41 2.17 Theideaofbumplesstransferrealizationforthecontroller K i . Com- monC andDmatricesforallcontrollersensurecontinuityregardless controller switching. . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.1 Slow-fast controller decomposition with state reset . . . . . . . . . . 46 3.2 Adaptive switching PID controller . . . . . . . . . . . . . . . . . . . 49 3.3 Controlleroutputu(t)withoutbumplesstransferandwithbumpless transfer (upper ¯gure); Plant output y(t) without bumpless transfer and with bumpless transfer (lower ¯gure). Controller is switched at t=2.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.4 Magni¯ed u(t) around the switching instant (t=2). . . . . . . . . . 60 3.5 Controlleroutputu(t)withoutbumplesstransferandwithbumpless transfer (upper ¯gure); Plant output y(t) without bumpless transfer and with bumpless transfer (lower ¯gure). Controller is switched at t=5.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.6 Controller output u(t) (upper ¯gure); Plant output y(t) (lower ¯g- ure). Controller is switched at t=2. . . . . . . . . . . . . . . . . . 66 3.7 Magni¯ed u(t) around the switching instant (t=2). . . . . . . . . . 67 4.1 Con¯guration of adaptive PID controller . . . . . . . . . . . . . . . 71 4.2 PID controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.3 Time-windowing on T spec . . . . . . . . . . . . . . . . . . . . . . . . 74 4.4 Fading-memory for T spec . . . . . . . . . . . . . . . . . . . . . . . . 74 4.5 Simulation result of Case 1 and Case 2 in large scale . . . . . . . . 79 4.6 SameasFigure4.5,butwithverticalaxismagni¯edbyfactor0:5£10 20 80 vi Abstract Thisdissertationmainlyfocusesonimplementationtechniquesforadaptiveswitch- ing control. Adaptive switching control can generate bad transients in controller output. These bad transients can be reduced by various bumpless transfer tech- niques. A new bumpless transfer technique is developed based on slow-fast con- trollerdecomposition. Thistechniqueisespeciallywell-suitedtosituationsinwhich the plant model is poor or yet to be identi¯ed, as may be the case in adaptive switching control. A new cost function with fading memory and a ¯nite-duration time-window is introducedinordertoreducethee®ectofolddatainunfalsi¯edadaptivecontrolap- plicationswheretheplantvariesslowlyorinfrequentlywithtime. Thee®ectiveness of the approach is demonstrated via a simple simulation. The result demonstrates that a time-windowed/fading-memory cost function for unfalsi¯ed control is useful for adaptive control system with time-varying plants, even when the plant fails to satisfy the usual 'feasibility' requirement of unfalsi¯ed control that it must be stabilizable by at least one of the candidate controllers. vii Chapter 1 Introduction Adaptive switching control is composed of two main concepts, adaptive control and switching control. Although each area has its own theories and applications developed by numerous researchers, combining two separate control areas raised new problems. Regarding problems in adaptive switching control, two main topics are con- sidered in this dissertation. First, a new bumpless transfer method is presented. Bumpless transfer methods help switching controller to avoid bumpy transients when the online controller is switched. There are existing bumpless transfer meth- ods for switching control, but no method particularly created for `adaptive' switch- ing control exists. A new bumpless transfer method introduced in [6] is designated to be used for adaptive switching control systems. Second, a fading memory and time-window technique for adaptive control cost functions is addressed. Although this technique can be utilized for most adaptive control methods, a data-driven adaptive switching control method called unfalsi¯ed control [35] was used for a particular example. 1 1.1 Bumpless Transfer with Slow-fast Controller Decomposition Controllerswitchinghasbeenfoundtobeusefulinbothadaptiveandnon-adaptive feedbackcontrolsystems. Non-adaptiveapplicationsincludeswitchingfromaman- ual to an automatic control and anti-windup compensation (e.g., [14]). In the adaptivecontrolsetting, switchingamonga¯nitesetofcontrollershasbeeno®ered as an alternative to continuous parameter tuning methods (e.g., [27]). Adaptive switching control has improved existing adaptive control system behavior in many ways, but it has also introduced a new problem not associated with earlier continu- ous adaptive control methods. The problem is that the controller output can have undesired transients, called `bumps', if a current on-line controller and a new con- troller to be switched have di®erent outputs at the switching instant. To attenuate these bumps associated with controller switching, a variety of bumpless transfer methods have been suggested over the years since the 1980's ( [13], [14], [40], [42]), some of which are better suited to adaptive switching problems than others. In adaptive control, the plant is generally not precisely known at the outset, and the goal of adaptive control is to change the controllers to improve perfor- mance as plant data begins to reveal some information about the plant. Thus, in adaptive switching control an exact plant model is generally unavailable at the time of switching. This implies that bumpless transfer methods that may be suit- ablefornon-adaptiveapplicationssuchlikeanti-winduportransferfrommanualto automatic control where the true plant is well-known, may not be ideal for adap- tiveswitchingcontrolapplications. Inparticular,inadaptiveswitchingapplications wherethetrueplantmodelmayonlybepoorlyknownatcontrollerswitchingtimes, 2 it may be preferable to employ a bumpless transfer technique for adaptive control that does not depend on a precise knowledge of the true plant model. While bumpless transfer methods such as [13] and [42] require explicit knowl- edge of the true plant model, other methods do not. For instance, the conditioning methods of [14], the continuous switching method of [2], and linear quadratic opti- mal bumpless transfer method of [40] are examples of methods that do not require a plant model. Likewise, [2] solved the problem of how to ensure control signal continuity without precise plant knowledge, but did not consider transient e®ects that may follow immediately after controller switch. Anothermethodthatwassuggestedparticularlyforadaptiveswitchingcontrols istheslow-fastdecompositionbumplesstransfer[6]. Byappropriatelyre-initializing the states of the slow and fast modes controllers at switching times, this method can ensure that not only will the controller output be continuous, but also that it avoids fast transient bumps after switching. This can be considered as an im- provementoverthemethodin[2], whichisoneofthesimplestmethodsbyassuring continuity [12] but may allow abrupt fast transients after switching. Our slow-fast decomposition bumpless transfer method removes the possibility of abrupt tran- sients. Thenewbumplesstransferwithslow-fastcontrollerdecompositionisinspiredby anadaptivePIDcontrollerin[18]. APIDcontrollerhasapoleandazeroatorigin. It is a special case of the controller which has fast modes (the di®erentiator) and slow modes (the integrator). Generalizing the PID controller case, the bumpless transfer suggested in this report decomposes the original controllers into the fast modes controllers and the slow modes controllers. By appropriately re-initializing the states of the slow and fast modes at switching times, our methods can ensure 3 G K i K j y u r . . . K Switching Signal State Reset Figure 1.1: Switching control system with 2-degree-of-freedom controllers that not only will the controller output be continuous, but also that it avoid fast transient bumps after switching. One recently published paper [7] presents improvements over [6]. The previous results are extended to include 2-degree-of-freedom controllers. For this extension a switching control structure with 2-DOF candidate controllers in Fig. 1.1 is intro- ducedandaproofisalsomodi¯ed. Afterwards, astraightforwardwaybasedonthe standard observable canonical form is suggested. This makes manipulating slow- fastdecomposedcon¯gurationeasyinpractice. Furthermore,specialcasesofwhich acontrollerhasonlyslowmodesoronlyfastmodesareaddressed. Acontrollerwith onlyfastmodesusesstateaugmentationtechniquetocreateaslowmodescontroller. After these descriptions simulation results comparing the method presented in this paper with the other in [2] while the earlier examples in [6] demonstrated di®er- ences between controller switching transients with and without bumpless transfer. The example veri¯es slow-fast decomposition bumpless transfer produces better performance than [2] when controllers have fast parts. 4 1.2 Unfalsi¯ed Control Using Limited Memory Whennoone¯xedcontrollerisadequatetocontrolanunknownorhighlyuncertain plant, adaptive control methods can theoretically be used to iteratively identify a suitable controller from a given pool of candidate controllers. Early adaptive con- trol methods were unreliable because they required excessive assumptions such as minimum phase plant, known upper bound of plant order, or no measurement noise [30]. Because of this, most engineers had been reluctant until recently to use adaptive control methods for safety critical applications. To overcome the limita- tions of the earlier adaptive methods, various new paradigms for adaptive control have been proposed [27], [35]. Unfalsi¯ed control, one of the newest paradigms, is introduced by Safonov and Tsao [35]. This formulation built on the idea of Morse, Mayne, and Goodwin [27] shows that adaptive control system can be interpreted as minimizing a cost function. It is attractive because unfalsi¯ed control theory provides a uni¯ed framework which explains the behavior of adaptive controllers in terms of the minimization of a certain data-driven cost function whose value for eachcontrollercanbecomputed ateachtime frommeasuredplantdata. Thebasic idea is that the adaptive control supervisor unit chooses a controller from the pool that either minimizes the cost function or at least maintains the cost at or below a prescribed cost level. The unfalsi¯ed control paradigm has facilitated the discovery of new classes adaptive control laws that reliably stabilize unknown plants under only the very weakest of assumptions; viz., that there exists at least one ¯xed but a priori unknown controller in the candidate controller pool that can stabilize the plant [37]. Attheheartofeveryadaptivecontrolsystemisaunitcalledthe supervisor that selects the currently active controller from a pool of candidate controllers. In the 5 unfalsi¯ed control paradigm, the supervisor is modeled as a device that evaluates and compares the unfalsi¯ed performance levels of candidate controllers using a data-driven cost function. Then, the supervisor tags candidates that achieve a prescribed unfalsi¯ed cost level as unfalsi¯ed controllers. Other controllers that fail the test at a given unfalsi¯ed cost level are not used unless the unfalsi¯ed cost level of the currently active controller increases above another controller by at least some small amount " called the hysteresis constant. The basic elements of unfalsi¯ed control theory are described in [35], and the use of the theory for reliableadaptivePID controllergain tuning was described in [18]. The relationship with Morse-Mayne-Goodwin convergencelemma [27] and importance of using cost- detectable cost-functions whose behavior accurately re°ects stability was addressed in [37]. Moreover, [37] provides the proof of stability. In the unfalsi¯ed control paradigm, the choice of cost function plays the key role in determining the performance and behavior of the adaptive control system. Previous unfalsi¯ed control works have developed several cost functions for achiev- ing performance goals and maximal stability robustness using the concept of cost detectability [38]. Nevertheless, these did not handle the possibility of the falsi¯- cation of all controllers, which can happen when no one controller in the candidate poolcanrobustlystabilizethetime-varyingplant{evenwhentheplantvariesonly slowly or infrequently over a set of plants each of which is stabilizable by one on the controllers in the pool. We address this problem by introducing fading memory andtime-windowingmodi¯cationsintothe unfalsi¯edcontrolcost function. Angeli andMoscasuggestasimilarapproachforhandlingtime-varyingplantsusingfading memory [1]. 6 1.3 Organization of the Dissertation ThefollowingChapter2describesconceptsofadaptiveswitchingcontrolandbump- less transfer. Then, the details of two newly suggested idea will be addressed. Chapter 3 presents a solution to bumpless transfer for adaptive control based on slow-fast controller decomposition. Notation and the switching control sys- tem con¯guration to describe the bumpless transfer method are introduced in Sec- tion 3.1. The bumpless transfer problem formulation is presented in Section 3.2. The slow-fast bumpless transfer theory and implementing method are presented in Section 3.3. Section 3.4 shows simulation results and Section 3.5 summarizes Chapter 3. Next, Chapter 4 shows fading memory and time-window. Basic concepts of the unfalsi¯ed control and ¯ctitious reference signal for the PID controller are re- viewed in Section 4.1. Section 4.2 formally introduces the procedure for adding time-windowed fading memory to unfalsi¯ed control cost functions. Section 4.3 contains an example that demonstrates the advantages of the approach via a sim- ple simulation that shows that the time-windowed fading-memory modi¯cation to the cost-function allows an unfalsi¯ed adaptive system to work without the usual feasibilityassumptionofunfalsi¯edcontrol. Then,theentirechapterissummarized in Section 4.4. Finally, Chapter 5 concludes this dissertation over all. 7 Chapter 2 Adaptive Switching Control and Bumpless Transfer As mentioned in Chapter 1, introducing `switching control' to adaptive control has beenachievedenhancementsoverconventionaladaptivecontrolmethods. However, adaptive switching control has di®erent structure from the conventional adaptive controls and this results in di®erent characteristics. Therefore, one needs to recog- nize what switching control system is and how adaptive laws work in the switching control system. Adaptive switching control system is introduced in this chapter. After describing adaptive switching control, bumpless transfer controller switching is addressed. Bumpless transfer is necessary when a switching control system is built in practice since switching one online controller into another controller gener- ates unexpected `bumpy' control signal even when the switching control system is asymptotically stable. The bumpy signal may seriously a®ect to the performance particularly when candidate controller is open-loop unstable. Details of this situa- tion and examples of bumpless transfer are shown in Section 2.4. 8 1 i n e y u r . . . . . . + - Figure 2.1: Switching control system 2.1 Switching Control A switching control system selects at each time a controller from a set of candidate controllers, rather than using only one ¯xed controller. Fig. 2.1 shows the basic block diagram of the switching control system with a supervisor. One controller is located online, Controller i in this case. This makes the Controller i is called the online controller. A supervisor generates switching signal using data u, y, and r. The switching signal generated by the supervisor decides whether the online controller would be switched to another controller and/or chooses a controller to be switched. Switching control can be used in the cases which traditional continuous control approaches do not work. According to [19], the following cases may bene¯t from switching control: 1) Control problems that cannot be solved by continuous control; 2) Control problems with sensor and/or actuator limitations; 9 3) Control problems for which plant model is highly uncertain or for which one single model cannot be determined. Regarding the third case, one can ¯nd similarity between adaptive switching control and other adaptive control schemes because both endeavor to solve the problems involving highly uncertain models and changes of models. 2.2 Switching Algorithms The switching algorithm plays a central role in switching control. The switching signal in Fig. 2.1 is determined by the switching algorithm. It de¯nes the char- acteristics of the switching control system and a®ects adaptive law. The methods most widely used in adaptive switching control are hysteresis switching proposed for adaptive control in [27] and dwell time switching in [24], [25]. Adaptive control with fading and windowed memory cost function presented in Chapter 4 is one of the examples using the hysteresis switching algorithm while the bumpless transfer in Chapter 3 can be used either way. 2.2.1 Hysteresis Switching Algorithm Hysteresis is a frequently observed natural phenomenon and one of the typical nonlinear operators. Fig. 2.2 shows the basic concept of the hysteresis switching. Although hysteresis phenomena explain how one switch is turned on and o® in general, it can be easily expanded to switching among multiple controllers. Expanding on the basic concept above, the hysteresis switching algorithm for twoormorecandidatecontrollersisestablishedasshowninFig.2.3. Now, suppose 10 Figure 2.2: Concept of Hysteresis Switching min min{ } i i K J J ! " # K min i K K $ Figure 2.3: Hysteresis switching algorithm 11 a controller set K with n controllers K=fK 1 ;¢¢¢ ;K i ;¢¢¢ ;K n g (i=1;2;¢¢¢ ;n): Each controller has its own cost level J i . The controller K min is the currently cost minimizing controller in the setK and the corresponding cost level is J min . By the algorithm in Fig. 2.3, supervisor compares cost levels of each candidate controllers in every time step. If one cost level is less than the currently minimum cost level minus the hysteresis constant ", the supervisor generates the switching signal to replace the current online controller with the other controller. If not, the current online controller keep staying online. This is the basic framework of hysteresis switching algorithm. The hysteresis switching algorithm for adaptive control was introduced in [27]. Since the ¯rst introduction, various re¯nements have been developed. Scale inde- pendent hysteresis switching algorithm [16] made the hysteresis switching possible to use multiplicative hysteresis constant. It can be achieved by replacing the com- parison in Fig. 2.3 with min K i 2K fJ i g·(1¡")J min : A hierarchical hysteresis switching algorithm was suggested in [17] to treat the unknown parameters with continuum. Multiple set of controllers are used for the hierarchical hysteresis switching. K=fK 1 ;K 2 ;¢¢¢ ;K n g Another classi¯cation for hysteresis switching is found in unfalsi¯ed control theory which is one of adaptive switching controls using hysteresis switching. The 12 Unfalsified min i K K min min{ } i i K J J ! " # $ K fixed for each , i i K J J # Falsified Figure 2.4: Hysteresis Switching algorithm with one ¯xed unfalsi¯ed cost level cost level J min is the criterion whether a controller is unfalsi¯ed or not. It is called unfalsi¯ed cost level. In unfalsi¯ed control theory, the unfalsi¯ed cost level can be a ¯xed level or variable level just as the algorithm in Fig. 2.3. While the recent works have used the variable unfalsi¯ed cost level [32], some papers have adopted the ¯xed level for controller unfalsi¯cation [18], [31]. Note that [18], [31] is not a switching control application, so that it is not considered further. If the unfalsi¯ed control system uses the ¯xed unfalsi¯ed cost level, the hysteresis algorithm can be applied as in Fig. 2.4. 2.2.2 Dwell Time Switching Dwelltimeswitchingwassuggestedtoremove\chatter"orrepairoscillatingswitch at the beginning of supervisory control in [24], [25]. The supervisory control has 13 ! argmin i i K K J " # K Figure 2.5: Dwell time switching algorithm obtainedimportantproperty, guaranteedcontrollerconvergencewith¯nitenumber of switching, by using dwell time switching algorithm. There is a signi¯cant di®erence, in dwell time switching, from hysteresis switch- ing. As noticed by the name `dwell time' switching, a ¯xed dwell time ¿ D should be de¯ned. After de¯ning ¿ D > 0, dwell time switching algorithm is achieved as shown in Fig. 2.5. The key point of this algorithm is waiting for ¿ D seconds to switch the cost minimizing controller. Once the algorithm ¯nds the controller, the supervisor switches the controller found into the controller loop. Then, it waits for ¿ D seconds again before the next competition among controllers. The dwell time ¿ D mayvarybasedonthespeci¯cationofthecontrolsystem. Itshould be properly chosen because not properly chosen dwell time makes controller adaptation slow. 2.3 Adaptive Switching Control There are various examples that switching control has been merged into adaptive control since Mº artenson introduced the possibility of adaptive switching control 14 Decision Logic Plant Models & Estimators Switching Signal Decision Logic Performance Monitor Switching Signal J 1 J n (a) Estimator-based supervisor (b) Performance-based supervisor u y u y y y P1 y Pn e P1 e Pn Figure 2.6: Supervisors in adaptive supervisory control in [21]. Each control theory can be categorized by the type of its supervisor. Note thatadaptiveswitchingcontrolinthisdissertationislimitedto`supervisorycontrol' due to the structure that uses supervisor or equivalent mechanism to choose an online controller. 2.3.1 Supervisory Control The term \supervisory control" was ¯rstly suggested by Morse in ( [24], [25]) to design ??? of an adaptive control system and it has been adopted by others. Su- pervisory control, or more speci¯cally adaptive supervisory control, has the archi- tecture shown in Fig. 2.1. The supervisor contains the adaptive control law or switching algorithm for the controller adaptation. Adaptive supervisory control can be classi¯ed according to the type of its supervisor [15] as shown in Fig. 2.6. 15 2.3.1.1 Estimator-based Supervisor An estimator-based Supervisor consists of the multiple plant model candidates and the corresponding controllers for each plant model. The supervisor continuously monitors data to determine which plant model is the closest to the real plant. Fig. 2.6 (a) shows the architecture of the estimator-based supervisor. The decision logic compares estimation error e P i for all i 2 f1;2;¢¢¢ ;ng and chooses the best matching plant model which has the smallest error. The best matching model selected by the supervisor is considered as `an estimate of the actual plant' [15]. The best plant model has a pre-matched controller which the supervisor switches thecontrollerintothecontrolloop. Estimator-basedsupervisorispresentedin[24], [25], [28]. 2.3.1.2 Performance-based Supervisor Unlike estimator-based supervisor, performance-based supervisor does not use a plant model estimator to choose the online controller. The supervisor in Fig. 2.6 (b) has the block called performance monitor which calculate the cost of each candidate controllers based on the input and the output of the controllers. Then the cost minimizing controller is selected as the best controller at the current time. The decision logic generates switching signal by the hysteresis switching algorithm described in Section 2.2.1. Because of the property that only the measured signals fromcontrollerareusedforcontrollerdecision,adaptivecontrolusingperformance- basedsupervisorisalsocalleddata-drivenadaptiveswitchingcontrol. Oneexample of this category is unfalsi¯ed control [5], [35]. Since unfalsi¯ed control is a major concern of Chapter 4, details shall be described in the following section. 16 2.3.2 Unfalsi¯ed Control Validation or more precisely unfalsi¯cation of hypotheses against physical data is the central aspect of the process of scienti¯c discovery. This validation process allows scientists to sift the elegant tautologies of pure mathematics in order to discover mathematical descriptions of nature that are not only for logically self- consistent, but also consistent with physically observed data. This data-driven process of validation is also a key part engineering design. Successful engineering design techniques inevitably arrive at a point where pure introspective theory and model-based analysis must be tested against physical data. But, in control engi- neering in particular, the validation process is one that has been much neglected by theoreticians. Here, the theory trying control designs to physical data has for the most part focused on pre-control-design `system identi¯cation.' Otherwise, the mathematization of the processes of post-design validation and re-design has remained relatively unexplored virgin territory. In particular, a satisfactory quan- titative mathematical theory for direct feedback of experimental design-validation data into the control design process has been lacking, though this seems to be changing with the recent introduction of a theory of unfalsi¯ed control [35]. 2.3.2.1 Theory: Validation and Unfalsi¯cation Unfalsi¯ed control is essentially a data-driven adaptive switching control theory that permits learning based on physical data via a process of elimination, much like the candidate elimination algorithm of Mitchell [22]. The theory concerns the feedback control con¯guration in Fig. 2.7. As always in control theory, the goal is to determine a control law K for the plant P such that the closed-loop system response, say T, satis¯es given speci¯cations. Unfalsi¯ed control theory is 17 r u y Figure2.7: Unfalsi¯edadaptivecontrol: unfalsi¯edcontrollerhastwo-tierstructure, consisting of the supervisor and a conventional controller-plant pair. concerned with the case in which the plant is either unknown or is only partially known and one wishes to fully utilize information from measurements in selecting the control law K. In the theory of unfalsi¯ed control, learning takes place when new information in measurement data enables one to eliminate from consideration one or more candidate controllers. AsindicatedinFig.2.8threeelementsthatde¯netheunfalsi¯edcontrolproblem are (1) plant measurement data, (2) a class of candidate controllers, and (3) a performance speci¯cation, say T spec , consisting of a set of admissible 3-tuples of signals. T spec (°;¿)=f(r;y;u)jT spec (r;y;u;¿)·°g (2.1) where T spec (r;y;u;¿) is a scalar-valued cost function, and ° and ¿ are real numbers that represent cost and time respectively. More precisely, we have the following de¯nition. De¯nition 2.3.1 [35] A controller K2K is said to be falsi¯ed at cost level ° at time ¿ by plant data (y;u) if this data is su±cient to deduce that the performance 18 K K K K K K K K K K K K K K K K K K K K K Figure 2.8: Unfalsi¯cation concept. Using performance goal and measured data as a sieve, candidate controllers are decided to be unfalsi¯ed or falsi¯ed. speci¯cation(r;y;u)2 T spec (°;¿)8r wouldbeviolatedifthatcontrollerwereinthe feedback loop. Otherwise, the controller K is said to be unfalsi¯ed. The least value of ° for which a controller K is unfalsi¯ed is called the unfalsi¯ed cost-level of K at time ¿. To put plant models, data and controller models on an equal footing with per- formance speci¯cations, these like T spec are regarded as sets of 3-tuples of signals - that is, they are regarded as relations in R£Y £U. For example, if P : U ! Y and K :R£Y !U then P=f(r;y;u)j y =Pug K= 8 > < > : (r;y;u)j u=K 2 6 4 r y 3 7 5 9 > = > ; . 19 And, the performance speci¯cation would be simply the set as de¯ned in Eq. 2.1. Ontheotherhand,experimentalinformationfromaplantcorrespondstopartial knowledge of the plant P. Loosely, data may be regarded as providing a sort of an \interpolation constraint" on the graph of P { i.e., a `point' or set of `points' through which the in¯nite-dimensional graph of dynamical operator P must pass. Typically, the available measurement information will depend on the current time, say ¿. For example, if we have complete data on (u;y) from time 0 up to time ¿ >0, then the measurement information is characterized by the set [35] P data = 8 > < > : (r;y;u)2R£U£Y ¯ ¯ ¯ ¯ ¯ ¯ ¯ P ¿ 2 6 4 u¡u data y¡y data 3 7 5 =0 9 > = > ; (2.2) where P ¿ is the familiar time-truncation operator of input-output stability theory (cf. [36], [43]), viz., bP ¿ xc(t)= 8 > < > : x(t); if 0·t·¿ 0; otherwise : Themainresultofunfalsi¯edcontroltheoryisthefollowingtheoremwhichgives necessary andsu±cientconditions for past open-loop plantdata P data to falsify the hypothesis that controller K can satisfy the performance speci¯cation T spec . Theorem 2.3.1 [35] A control law K 2K is unfalsi¯ed at time ¿ by plant data (y 0 ;u 0 ) if, and only if, for each triple (r 0 ;y 0 ;u 0 )2 P data (¿)\ K, there exists at least one pair (y 1 ;u 1 ) such that (r 0 ;y 1 ;u 1 )2 P data (¿)\ K\ T spec (°;¿) (2.3) 20 where P data (¿) = f(r;y;u)jy(t) = y 0 (t) and u(t) = u 0 (t) for all t · ¿g and K = f(r;y;u)ju=K 2 6 4 r y 3 7 5 g. Proof: With controller K in the loop, a command signal r 0 2 R could have producedthemeasurementinformationif, andonlyif, (r 0 ;y 0 ;u 0 )2 P data (¿)\Kfor some (u 0 ;y 0 ). The controller K is unfalsi¯ed if and only if for each such r 0 there is at least one (possibly di®erent) pair (u 1 ;y 1 ) which also could have produced the measurement information with K in the loop and which additionally satis¯es the performance speci¯cation (r 0 ;y 1 ;u 1 )2 T spec . That is, K is unfalsi¯ed if and only if for each such r 0 , condition (2.3) holds. Q.E.D. The Unfalsi¯ed Control Theorem constitutes a mathematically precise state- ment of what it means for experimental data and a performance speci¯cation to be inconsistent with a particular controller. It has some interesting implications: ² TheUnfalsi¯ed ControlTheorem is nonconservative; i.e., it gives \if and only if" conditions on K. It uses all the information in the past data { and no more. It provides a mathematically precise sieve which rejects any controller which, based on experimental evidence, is demonstrably incapable of meeting a given performance speci¯cation. ² The Unfalsi¯ed Control Theorem is model free. No plant model is needed to test its conditions. There are no assumptions about the plant. ² Information P data which invalidates a particular controller K need not have beengeneratedwith thatcontrollerin the feedbackloop; it maybeopenloop data or data generated by some other control law (which need not even be in K). 21 ² When the sets P data , K and T spec are each expressible in terms of equations and/or inequalities, then falsi¯cation of a controller reduces to a minimax optimization problem. For some forms of inequalities and equalities (e.g., linear or quadratic), this optimization problem may be solved analytically, leading to procedures for direct identi¯cation of controllers { as the example in [33]. ² Given data (u 0 ;y 0 ) and a candidate controller K, the r 0 's satisfying the con- ditions of the Unfalsi¯ed Control Theorem are called the ¯ctitious reference signals. When K has a causal inverse, the r 0 is uniquely determined by (u 0 ;y 0 ) and a candidate controller K; that is, there exists a causal function ~ r(K;u 0 ;y 0 ) such that r 0 = ~ r(K;u 0 ;y 0 ) . The function ~ r(K;u 0 ;y 0 ) is called the ¯ctitious reference signal [35]; and is closely related to the virtual reference signal of [1]. ² Inadaptivecontrolthefactthatthesupervisorchoosesacontrollermeans, at leastimplicitly,thatthereisareal-valueddata-drivencostfunctionV(K;y 0 ;u 0 ;t) such that at any give time ¿ the active controller is cost-minimizing ^ K(y 0 ;u 0 ;t)=argmin K2K V(K;y 0 ;u 0 ;t) where K is a set of candidate controllers. In the case of unfalsi¯ed control, the cost function V is determined from the cost J(r;y;u;t) and evaluating it with r equal to the ¯ctitious reference signal ~ r(K;u 0 ;y 0 ) : V(K;y 0 ;u 0 ;t)=J(~ r(K;u 0 ;y 0 );y 0 ;u 0 ;t) . (2.4) 22 The cost J can be chosen in various way. One of the cost functions, J for slowly time-varying plant, shall be shown in Chapter 4 later. 2.3.2.2 Data-Driven Learning and Adaptive Control Theunfalsi¯edcontroltheoremsayssimplythatcontrollerfalsi¯cationcanbetested by computing an intersection of certain sets of signals. A noteworthy feature of the unfalsi¯ed control theory is that a controller need not be in the loop to be falsi¯ed. Broad classes of controllers can be falsi¯ed with open-loop plant data or even data acquired while other controllers were in the loop. Adaptive control is achieved within this framework by using the unfalsi¯cation process as the key element of a supervisory controller( [24], [25]). The supervisor switchesan unfalsi¯ed controller into the feedback loop whenever the current controller in the loop is amongst those falsi¯edbyobservedplantdata. Consequently,thesupervisorchoosesasthecurrent control law one that is not falsi¯ed by the past data, resulting in a control law that is adaptive in the sense that it learns in real time and changes based on what it learns. Like the controllers of [11], [25], this approach to adaptive real-time unfalsi¯ed control leads to a sort of \switching control." Controllers which are determined to be incapable of satisfactory performance are switched out of the feedback loop and replaced by others which, based on the information in past data, have not yet been found to be inconsistent with the performance speci¯cation. However, adaptive unfalsi¯ed controllers generally would not be expected to exhibit the poor transient response associated with switching methods such as [11]. The reason is that, unlikethetheoryin[11], unfalsi¯edcontroltheorye±cientlyeliminatesbroad classes of controllers before they are ever inserted in the feedback loop. The salient feature that distinguishes unfalsi¯ed control from other adaptive methods is that 23 in unfalsi¯ed control the adaptive supervisor evaluates candidate controllers objec- tivelybasedonexperimentaldataalone,withoutprejudicialassumptionsaboutthe plant. While, inprinciple, the unfalsi¯edcontroltheoryallowsforthe setK toinclude continuously parameterized sets of controllers, restricting attention to candidate controller setsK with only a ¯nite number of elements can simplify computations. Further simpli¯cations result by restricting attention to candidate controllers that are \causally-left-invertible" in the sense that, given a K 2 K, the current value of r(t) is uniquely determined by past values of u(t) and y(t). If the candidate controller is not causally-left-invertible, modi¯ed version of ¯ctitious signal can be usedasin[20]. WhenEq.2.2holds,theserestrictionson T spec and Karesu±cientto permit the unfalsi¯ed set to be evaluated in real-time via the conceptual algorithm in Fig. 2.3 or Algorithm 4.2.1. At the beginning of the procedure, functions and parameters are initialized. The data-window duration would normally be selected to be somewhat less than the time-scale over which signi¯cant plant variations may occur that cannot be robustly accommodated by any one candidate controller in the controller set K. After initialization, output data u(t) and y(t) are measured and ¯ctitious signals arecomputedbygivenstructureofK. CostJ(r;y;u;t)ofeachcandidatecontroller was obtained based on the measured data and the ¯ctitious signals by Eq. 2.4. If the cost is larger than the current unfalsi¯ed cost level °, the controller is falsi¯ed at this cost level and is not used. It is important to note that while the above algorithm is geared towards the caseofanintegralinequalityperformancecriterion T spec anda¯nitesetofK's, the underlying theory is, in principle, applicable to arbitrary non-¯nite controller sets K [37] and to hybrid systems with both discrete and continuous time elements. 24 If the plant is slowly time-varying, very old data ought to be discarded before evaluating controller falsi¯cation. This may be e®ected within the context of the hysteresis switching adaptive control algorithm by ¿ 0 and regarding t¡¿ 0 as the deviation from the current time. The result is an algorithm which only considers data from moving time-window of ¯xed duration ¿ 0 time-units prior to the current real-time. In this case the unfalsi¯ed controller set no longer shrinks monotonically as it would if were increasing in lockstep with real-time. Details are addressed in Chapter 4. 2.3.2.3 Stability and Convergence When modeling assumptions about the plant fail to hold, there is the possibility that badly designed adaptive algorithms can fail to stabilize { even when the adap- tive control problem is theoretically feasible in the sense that one of the candidate controllers K 2 K is stabilizing. The problem is that in most cases, `proofs' of stability in adaptive control only hold when there is no mismatch between model assumptions and the true plant. Well-know standard assumptions in adaptive con- trol include upperbounds on plant order, assumption the plant is minimum phase and has no time-delays, or that the plant is `su±ciently close' to one of several presumed prior plant models. There are studies illustrating how adaptive systems can fail in the presence of a mismatch between assumptions and reality [30], [41]. One result of model mismatch instability is shown in Fig. 2.9 which illustrates the consequences of model mismatch instability for a typical multi-model adaptive (MMAC) system. But, though currently popular adaptive algorithms are suscepti- ble to model-mismatch instability, optimal unfalsi¯ed adaptive design designs with a suitable cost-detectability properties as well as some lesser known early adaptive methods robustly avoid model mismatch instability, provided that the unfalsi¯ed 25 C 1 C 2 Figure 2.9: Without cost-detectability, model mismatch can cause an adaptive controller to switch a destabilizing controller C 2 into the loop and keep it, even when the original stabilizing controller C 1 was working well [41] cost function J(r;y;u;t) is chosen to have a property called cost-detectability [37], [38], [41]. For example, the multi-controller adaptive control (MCAC) switching algo- rithms by Mº artensson [21] and by Fu and Barmish [11] are robust against model- mismatch. They require essentially only feasibility to assure convergence. They work robustly in the presence of model mismatch because the cost penalty is assigned to destabilizing controllers that tends to in¯nity for destabilizing can- didate controllers. This is the essential feature of the cost-detectability. Cost- detectability[37],[38]isafeaturenottypicallypresentedformostcurrentlypopular adaptive algorithms, and it explains why they are susceptible to model mismatch instability. Withoutthecost-detectabilityproperty,adaptivecontrolalgorithmsare ingeneralnotabletoreliablydistinguishstableandunstablebehaviorwhenmodel- mismatch exceeds certain thresholds. While the early cost-detectable algorithms 26 of [11], [21] are too slow for applications requiring real-time adaptive stabilization, this is not true of unfalsi¯ed adaptive control. De¯nition 2.3.2 (Cost-Detectability [38]) Consider the adaptive control sys- tem of Fig. 2.7 with input r and output d = (u;y). The pair (V;K) is said to be cost-detectable if, without any assumptions on the plant P and for every switched sequence of controllers K(t i ) 2 K with ¯nitely many switching times and ¯nal controller K, the following statements are equivalent: (a) V(K;y;u;t) is monotone in t, and bounded as t increases to in¯nity; (b) Stability of the closed-loop system is unfalsi¯ed by the input-output data (r;d). So-called `L 2e -gain related' cost functions like the unfalsi¯ed `mixed-sensitivity' cost J(r;y;u;t)=max ¿·t kw 1 ¤(r¡y)k 2 ¿ +kw 2 ¤uk 2 ¿ ¡¾ 2 ¿ krk 2 ¿ +½ (2.5) genericallyassuresthecostdetectabilitypropertyfor costfunction. Theplantdoes not need to be minimum phase, nor does it need to satisfy any other standard assumptions for conventional adaptive control theories. So, with L 2e -gain related cost functions, destabilizing controllers are never retained when a stabilizing can- didate controller is available, irrespective of model mismatch and irrespective of whether standard assumptions or other prior beliefs about the plant fail to hold. Consequently, model-mismatch instability cannot occur and the adaptive system is stable whenever the adaptive control problem is feasible (feasibility assump- tion [37], [38]). Moreover unlike the early cost-detectable adaptive algorithms of Mº artensson [21] and of Fu and Barmish [11], unfalsi¯ed adaptive control systems adapt with optimally rapidity. Simulation studies [18] demonstrate that unfalsi¯ed adaptive systems can adapt fast enough to do real-time stabilization of open-loop 27 unstable plants in cases where the measurement signal to noise ratio is not too great. Usingthehysteresisswitchinglemma[26],Stefanovicetal. provedthefollowing theorem about controller convergence and stability. Theorem 2.3.2 (Convergence and Stability [38]) Aswitchedsequenceofcon- trollers K(t i )(i = 1;2;¢¢¢) that minimize the current unfalsi¯ed cost V(K;y;u;t) at each switching time t i will stabilize the plant P if the pair (V;K) has cost- detectability property. Proof: AsillustratedinFig.2.10,cost-detectabilityassuresthatthecost-minimizing controller tends towards one with ¯nite cost, which implies stability when the cost hasthecost-detectabilityproperty. See[38]and[37]formoredetailedproof. Q.E.D. While Morse et al. [26] were able to demonstrate a property similar to cost- detectabilitycalled'tunability'onlybyintroducingassumptionsontheplant,open- ing the possibility for model mismatch instability. In contrast, cost detectability by de¯nition entails no plant assumptions. Consequently, adaptive control laws de- signed using a cost-detectable pair (V;K) can reliably and robustly sift candidate controllers without risk of model mismatch instability. In design studies including [18], we have demonstrated that the unfalsi¯ed con- trol approach with an L 2e -gain related cost function converges quickly and reliably in real time to a stabilizing controller that robustly achieves speci¯ed performance goals, often converging within a fraction of an unstable plant's fastest unstable time constant. This speed of adaptive response means that \bursting phenomena" that plague conventional slow adaptive systems do not occur, even in the absence 28 K i K(t 1 )=arg min V(K,t 1 ) K(t 2 ) K(t 3 ) V(K,z data ,t 1 ) V(K,z data ,t 2 ) V(K,z data ,t 3 ) V TRUE K RSP Figure 2.10: L 2e -gain related cost functions have the cost-detectability property: They are monotone in t, uniformly bounded for stabilizing K and unbounded for destabilizing K. Cost-minimizing adaptive laws robustly converge irrespective of the size of plant model-mismatch [29], [38] 29 of persistently exciting disturbance signals. Because our unfalsi¯ed adaptive sys- temsperformreliablyirrespectiveofplantmodelmismatch,theyhavethepotential to reliably achieve rapid real-time failure recovery for battle-damaged aircraft and similar systems. 2.4 Bumpless Transfer Techniques As mentioned in Chapter 1, adaptive switching control gave rise to the necessity of bumpless transfer due to its `switching' property. Like other adaptive switching control schemes, the data driven adaptive switching control, unfalsi¯ed control introduced in Section 2.3.2, needs bumpless transfer to avoid sudden changes of control output. When the study on ACC benchmark problem with 2-cart mass- spring-damper system was performed to compare unfalsi¯ed control with robust multiple-model adaptive control (RMMAC) introduce by Fekri and Athans [9], the necessity of bumpless transfer method occurred. While the RMMAC generates control output not by switching but by continuously changing probability-based control signal generator, unfalsi¯ed control uses the switching control structure in Fig. 2.1 and this caused discontinuities in control signal at switching instants. In this section, the case study which indicates bumpy transients at the switching instants is shown. Then, the existing bumpless transfer methods will be brie°y presented. 30 m 2 m 1 k 1 k 2 b 1 b 2 x 1 x 2 u Figure 2.11: 2-cart mass-spring-damper system 2.4.1 Case Study: ACC benchmark problem with 2-cart mass-spring-damper system The purpose of this section is to show the control output without any bumpless transfer scheme in the adaptive switching control system and eventually to claim the necessity of bumpless transfer for adaptive switching control system. Aplantisslightlymodi¯edversionofthewell-knownACCbenchmarkproblem. It is a physical system with two carts and the carts were tied with a spring and a damper. Parameters for the system are assigned as in Fig 2.11. The plant is expressed by the following di®erential equation _ x(t)=Ax(t)+Bu(t)+L»(t) (2.6) y(t)=Cx(t) 31 where x=[x 1 (t) x 2 (t) _ x 1 (t) _ x 2 (t) d(t)] and the system matrices are A= 2 6 6 6 6 6 6 6 6 6 6 4 0 0 1 0 0 0 0 0 1 0 ¡k 1 =m 1 k 1 =m 1 ¡b 1 =m 1 b 1 =m 1 0 k 1 =m 2 ¡(k 1 +k 2 )=m 2 b 1 =m 2 ¡(b 1 +b 2 )=m 2 1=m 2 0 0 0 0 ¡® 3 7 7 7 7 7 7 7 7 7 7 5 ; B = 2 6 6 6 6 6 6 6 6 6 6 4 0 0 1=m 1 0 0 3 7 7 7 7 7 7 7 7 7 7 5 ; L= 2 6 6 6 6 6 6 6 6 6 6 4 0 0 0 0 ® 3 7 7 7 7 7 7 7 7 7 7 5 ; C = · 0 1 0 0 0 ¸ . L»(t) in Eq. 2.6 is a disturbance term. The disturbance is applied to Cart 2. When the disturbance d(t) is applied, it is written as d(s)=W d (s)»(s) and W d (s)= ® s+® where ® is the parameter in the matrix L. Forasetofcandidatecontrollers,fourrobustcontrollersdesignedbythemixed- ¹ synthesis method are used. The controller con¯guration is same to the set of LNARCs (local non-adaptive robust compensators) in [10]. For details, see Section 3.5 of [10]. Using the plant in Eq. 2.6 and applying four LNARCs as the candidate con- trollers of unfalsi¯ed control system, the result is shown in Fig. 2.12. The ¯rst ¯gure indicates the current controller index. Change of the controller index implies that the online controller is switches one to another. According to the ¯rst ¯gure, 32 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 1.5 2 2.5 3 3.5 4 Controller Index Controller 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −15 −10 −5 0 5 x 10 4 Control Output u(t) u(t) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −15000 −10000 −5000 0 5000 Control Output u(t), magnified time t, second u(t) Bump2 Bump3 Bump1 Figure 2.12: Bumpy transient with no bumpless transfer method 33 online controller is LNARC4 at the beginning. The online controller is switched to LNARC1, LNARC2, and LNARC3 in time. Lower two ¯gures indicate the con- troller output u(t). One can observe three bumpy transient when the controller switched. The third ¯gure is the magni¯ed version with y-axis u(t). By reading the third ¯gure, it is recognized that the ¯rst two bumps show discontinuity and the third is continuous, but shows the bump after switching. In most of adaptive switching control system having the structure of Fig. 2.13, the inputs of the controllers are always connected to the current signal and the outputs of the controllers depend on the controller status. The output of the online controller is directly connected with the actual control signal u(t) and the outputs of other o®-line candidate controllers are open. Because the inputs are continuously excited by the current plant output y(t) or related signal, for example e(t)=r(t)¡y(t), each candidate controller has the di®erent output even though it is not located in the control loop. This structural property of switching controller possibly creates discontinuity in u(t) when the online controller is switched. If the candidate controller is open-loop unstable, discontinuity may burst exponentially in time. Therefore, the primary goal of bumpless transfer in switching control is to achieve u i = u in Fig. 2.13 when the controller is switched from one controller to u i . Another possible type of bump is Bump3 in Fig. 2.12. Bump3 does not make discontinuity, but the abrupt change of the control signal is followed after the switching. This phenomenon is produced by the natural property of the switched controller if the switched controller has fast modes exited by the input signal. 34 i n i n Figure 2.13: In the switching control system, the actual control signal u follows the output u i when the current online controller is Controller i . There are existing bumpless transfer methods which intend to avoid the bumpy problems mentioned above. Some methods take account only the continuity prob- lem, and others consider both. Now, the existing bumpless transfer methods shall be illustrated. 2.4.2 Bumpless Transfer Using Add-on Type Controller Some of bumpless transfer methods use add-on type controllers. These methods do not change the controllers originally designed for the system and attach the addi- tional bumpless transfer compensator to each candidate controllers. Zaccarian and Teel applied the L 2 anti-windup scheme [39] into bumpless transfer problem and successfully suggested its problem formulation and solution [42]. Another remark- able bumpless transfer method in this category is the linear quadratic bumpless transfer by Turner and Walker [40]. 2.4.2.1 L 2 bumpless transfer TheL 2 bumpless transfer method de¯nes `target response' and solves the problem to guarantee the mismatch between the target response and the actual plant state 35 r u û u c y c v 1 v 2 t t s + + + + + - y Figure 2.14: Switching control system withL 2 bumpless transfer in an L 2 sense. The following description is the summarized version of [42]. Note that for details see Zaccarian and Teel's 2004 paper. The structure of this scheme is shown in Fig. 2.14. As a linear time-invariant plant, the following form is considered: _ x=Ax+Bu y =Cx+Du z =C z x+D z u (2.7) wherexistheplantstateanduistheplantinput,andzistheperformanceoutput. Assume the feedback control system has the paired controller designed to ensure closed loop speci¯cation _ x c =f(x c ;u c ;r) y c =g(x c ;u c ;r) (2.8) 36 Now the bumpless compensator is de¯ned as in Fig. 2.14. _ x e =Ax e +B(u¡y c ) v 2 =¡(Cx e +D(u¡y c )) v 1 =f e (x e ;u¡y c ) (2.9) wherex e isthebumplesscompensatorstate,v 1 andv 2 istheoutputofthebumpless compensator as indicated in Fig. 2.14. The function f e is decided by the LMI constraints. The control signal is de¯ned by the means of switching time t s u(t)= 8 > < > : ^ u(t) for t<t s y c (t)+v 1 (t) for t¸t s (2.10) where ^ u denotes certain control signal before the switching and the control signal afterswitchingisgeneratedbytheoutputoftheoriginalcontrollerandthebumpless compensator. Now, the function f e in Eq. 2.9 can be chosen by the following: f e (x e ;u¡y c ),Kx e (2.11) where the matrix K,XQ ¡1 ; X and Q is the matrices satisfy the LMI constraints 2 6 4 QA T +AQ+BX +X T B T ? C z Q+D z X ¡I 3 7 5 <0 (2.12) 2 6 4 ®I I I Q 3 7 5 >0: (2.13) Bytheframeworkdescribedabove,theswitchingcontrolsystemwithL 2 bump- less transfer in Fig. 2.14 is designed so that z(t)¡z T (t) is small for t ¸ t s and 37 x(t s )¡x T (t s ) isL 2 bounded. This implies that the di®erence between the perfor- mance output and the `target' performance output forced to be small `after' the switching. Furthermore, thedi®erencebetweenactualplantstateanddesiredstate is bounded, likely on the small enough value to ensure no bumpy transient, at the switching time. L 2 bumpless transfer method is remarkable because it is very well-structured in mathematical way and take into account not only continuity at switching instant but also the performance after the switching. However, unfortu- nately, this bumpless transfer scheme rely on the exact plant model and it is not suitable for adaptive control. 2.4.2.2 Linear quadratic bumpless transfer Another recently developed scheme in the add-on type controller category is the linear quadratic bumpless transfer introduced in [40]. In this method, the input of the candidate controller which is not located in the closed loop does not connected into one common controller input e as shown in Fig. 2.15. Adopting the switching either side, input and output of the controller, every single candidate controller forms its own closed loop when it is o®-line. This characteristic makes each o®-line controller loop including bumpless compensator possible to be designed without knowledge of plant mode. Regarding each closed loop of the o®-line controller, the following cost function is de¯ned to solve linear quadratic minimization problem: J(y c ;®;T)= 1 2 Z T 0 z u (t) 0 W u z u (t)+z e (t) 0 W e z e (t)dt+ 1 2 z u (T) 0 Pz u (T); 38 x u û y c t t s + - e r y t t s y Figure 2.15: Switching control system with linear quadratic bumpless transfer where z u (t)=y c (t)¡u(t); z e (t)=®(t)¡e(t): In other words, the bumpless transfer compensator is chosen to minimize the error between (1) the output y c of the controller switched at t s and the actual plant input u; (2) the input ® of the controller switched at t s and the input e of the online controller prior to the switching. Similar to other linear quadratic problems, this bumpless transfer scheme using linear quadratic optimization is limited because it requires a prior data for e and u for the ¯nite horizon problem. Otherwise, in¯nite horizon problem can be easily solvedusingalgebraicRiccatiequationwithminorassumptionofconstantreference signal. 39 2.4.3 Controller Modi¯cation Based Bumpless Transfer Unlike the bumpless transfer methods in the previous section, there is another ap- proach to achieve bumpless transfer. Well-known conditioning technique by Hanus etal.[14]andbumplessswitchingcontrollerrealizationbyArehartandWolovich[2] belong to this category. 2.4.3.1 Conditioning Technique The conditioning technique begins with considering the feedback control system with nonlinearity as in Fig. 2.16 (a). The nonlinearity, called N.L. in Fig. 2.16, between the controller and the plant is frequently observed in the control system with actuator saturation. While Fig. 2.16 (a) is the ordinary structure in anti- windup literature, it can be easily interpreted to switching control. In switching control, there are multiple controllers rather than only one controller K and the nonlinearityisreplacedbytheswitchmechanism. Despitethestructuraldi®erence, the main goal, accomplishing u=u r (or, y c =u in Fig. 2.14 and Fig. 2.15), of both methodologies is the same in general. The term `conditioning' gives an intuition how the scheme work. The condi- tioningtechniquemanipulatestheoriginalcontrollerforthereferencesignal r tobe `conditioned' such that u = u r . Fig. 2.16 (b) shows the modi¯ed feedback control system which the conditioning technique is applied into. Since the conditioning technique does not use the plant model, it seems to par- tially suitable for adaptive control. However, the method in [14] is only applicable to the controller having restriction such as bi-proper and minimum phase proper- ties. Although the improved scheme over the original conditioning technique has been suggested to overcome this limitation, the application to adaptive switching 40 K P r y u u r y K 1 P r y u u r y K 2 v Figure 2.16: Feedback control systems in conditioning technique. (a) without con- ditioning (b) with conditioning controlwithanumberofcandidatecontrollersisstilldi±cultduetoitscomplicated implementation issue [8]. 2.4.3.2 Continuity Ensuring State Space Realization One simple bumpless transfer method was suggested by Arehart and Wolovich [2] in 1996. They introduced a state-space realization scheme for switching control system. Suppose the i th candidate controller K i 2K =fK 1 ;¢¢¢ ;K i ;¢¢¢ ;K n g has state-space realization: _ x=A i x+B i u y i =C i x+D i u (2.14) wherex2R n isthecontrollerstate,uisthecontrollerinput,andy i isthecontroller output. 41 B i 1 s A i D C u y i Figure 2.17: The idea of bumpless transfer realization for the controller K i . Com- mon C and D matrices for all controllers ensure continuity regardless controller switching. Suggested realization of the controller (2.14) is performed to ensure continuous controller output when K i is switched to another controller, say K j . The funda- mental idea of this method is to make a common C matrix for all i under the assumption that a ¯xed matrix D for all i is given. Fig. 2.17 explains the idea with block diagram. Since C i and D i are directly connected to the controller output y i , switching either matrix causes discontinuity, i.e., y i 6= y j . Meanwhile, switching A i and B i do not a®ect to the continuity due to the integral action. Therefore, to achieve y i = y j , the original controller (2.14) is modi¯ed with the new C matrix such as C i =FP i (i=1;2;¢¢¢ ;n): (2.15) where F is a ¯xed matrix for all i and P i de¯nes the following equivalence trans- formation: _ x=P i A i P ¡1 i x+P i B i u y i =Fx+Du (i=1;2;¢¢¢ ;n): (2.16) 42 Finding P i for a given matrix F refer to Section III in [2]. It is very useful that this method can be achieved by the straightforward equivalence transform. However, considering only continuity in control signal does not seem to very ad- vanced comparing with the others previously introduced. Note that this continuity ensuring realization method can be generalized to the bumpless transfer presented in Chapter 3 and the new bumpless transfer method overcomes the limitation of thecontinuityensuredbumplesstransferrealization. Detailswillbefollowedinthe next chapter. 43 Chapter 3 Bumpless Transfer for Adaptive Control 3.1 Preliminaries Before presenting main idea about a new bumpless transfer method for adaptive switching controls, preliminaries about a switching control system is described. Then, slow-fast controller decomposition, a tool for the bumpless transfer method, is introduced. 3.1.1 Switching control system WeconsidertheswitchingcontrolsystemasshowninFig.1.1. Thesystemincludes a plant and a set of controllers K=fK 1 ;¢¢¢ ;K i ;¢¢¢ ;K n g (i=1;2;¢¢¢ ;n): (3.1) Assume that the plant output is continuous when input is continuous; A linear time invariant plant with a proper transfer function is a good example. The input of the plant is u(t) and the output is y(t). Plant input is directly connected to 44 the controller output. Controller inputs are r(t) and y(t) where r(t) is a reference signal. WhenacontrollerK i isinthefeedbackloop, thecontrollerissaidtobe on-line, and the other controllers are said to be o®-line. The i-th controller K i is supposed to have state-space realization _ x i =A i x i +B i z y Ki =C i x i +D i z (3.2) where z =[r T y T ] T is the input and y Ki is the output of K i . Equivalently, we write K i (s) s = 2 6 4 A i B i C i D i 3 7 5 : (3.3) We are interested in the situation in which the on-line controller is switched from K i to K j at time t s , so that u= 8 > < > : y Ki for t<t s y Kj for t¸t s : (3.4) Note that time t s is called switching time or switching instant. Since the controller output y Ki is replaced by y Kj at the switching instant t s , the control signal u can have bumps in the neighborhood of t = t s if y Ki and y Kj have di®erent values. Times immediately before and after t s are denoted as t ¡ s and t + s , respectively. The objective of bumpless transfer is to ensure continuity in the control signal and to smooth `bumpy' transients at, and immediately following, the switching instant. 45 K slow state reset K fast u K r u state reset y r y r y Figure 3.1: Slow-fast controller decomposition with state reset 3.1.2 Slow-fast decomposition We now consider controllers that can be additively decomposed into slow and fast parts as in Fig. 3.1, which is written as the following form; K(s)=K slow (s)+K fast (s) (3.5) with respective minimal realizations K slow (s) s = 2 6 4 A s B s C s D s 3 7 5 and K fast (s) s = 2 6 4 A f B f C f D f 3 7 5 : (3.6) The poles of K slow (s) are of smaller magnitude than the poles of K fast (s), i.e., j¸ i (A s )j·j¸ j (A f )j for all i;j where ¸ i (¢) denotes the i-th eigenvalue. The K slow (s) and K fast (s) of the slow-fast decomposition may be computed by various means, e.g., the Matlab slowfast algorithm, which is based on the stable-antistable decomposition algorithm described in [34]. 46 The slow-fast decomposition of the i-th controller K i in the set K is denoted with the subscript i as (3.6). K islow (s) s = 2 6 4 A is B is C is D is 3 7 5 and K ifast (s) s = 2 6 4 A if B if C if D if 3 7 5 (3.7) Further details on how the controller modes are divided as slow or fast will be described in a later section. 3.2 Problem Formulation Thebumplesstransferin[40]determinedtheoutputofeachcontrollersbyminimiz- ingthedi®erencesbetweenaplantinputandtheoutputsofeacho®-linecontrollers from time t = 0 to t = t s where t s is the switching instant. One of the possible solution of this problem is to make the controller output continuous. In this point of view, the method in [2] can be a solution. It transforms original controllers represented in state-space into another form which can guarantee the continuity of the controller output. This method replaces A i and B i with ¹ A i and ¹ B i , but it ¯xes ¹ C and ¹ D for all controllers. Replacing only matrices located on before an integra- tor makes the controller output to be continuous, and it solved their problem of bumpless transfer which is de¯ned by the continuity of the controller output. However, as we mentioned in Section 1.1, bumpless transfer should perform not onlycontinuouscontrolsignalbutalsosmoothtransientafterswitching. Toaddress both of problems, we de¯ne bumpless transfer as follows. De¯nition 3.2.1 (Bumpless Transfer) A switching controller with slow-fast de- composition (3.5) is said to perform a bumpless transfer if, whenever controller is 47 switched, the new controller state is reset so as to satisfy both of the following two conditions: (a) The control input signal u(t) is continuous at t s whenever r(t)2C 0 , and (b) the state of fast part of controller K fast (s) is reset to zero at t s . } Condition(a)inDe¯nition3.2.1isfrequentlyobservedinotherbumplesstrans- ferliterature[2],[23]. Condition(b)inDe¯nition3.2.1concernscontrolsignalafter switching. This additional requirement for our bumpless transfer is needed to en- sure that there are no rapid transients immediately following controller switching. How controller state reset be performed to simultaneously satisfy both conditions will be described in the following section. 3.3 Bumpless transfer implementation The idea, using slow-fast decomposition of the controller as the basis for bumpless transfer, generalizes a related idea introduced in [18] for adaptive PID controller switching. To clarify this and to put our result in perspective, we begin by pre- senting a brief explanation of our slow-fast decomposition interpretation of the adaptive PID controller controller switching approach. Then, we will introduce our main result. 3.3.1 Bumpless transfer for a PID controller A PID controller has three gains; K P , K I , and K D . In PID adaptive switching control, each of these gains may be changed by switching three PID control gains 48 K I 1 s K P K D 1 s s ε + e u Figure 3.2: Adaptive switching PID controller as shown in Figure 3.2, with the values of each of the three gains taking values in a discrete set. A practical PID feedback controller implementation takes the form u=K(s)e (3.8) where K(s)=K slow (s)+K fast (s) and K slow (s) = K P + K I s K fast (s) = K D s ²s+1 and ²>0 is a small constant (with ²=0 for an ideal PID controller). 49 3.3.1.1 Location of switching gains LocatingswitchingPIDgains(K P ;K I ;K D )playsanimportantroleindetermining the continuity of the controller output. As shown in Figure 3.2, an integrator gain K I maybelocatedbeforeanintegratorinorderthatthecontrolleroutputdoesnot haveadiscontinuitywhentheintegratorgainK I isswitched. However,aderivative gain K D should be located after (i.e., at the output of) a di®erentiator because the change of K D will respond an extreme overshoot or undershoot if a switching gain K D occurs before (i.e., at the input of) the di®erentiator. 3.3.1.2 Controller state reset Locating K I before an integrator as in Figure 3.2 is su±cient to ensure that the output of the integrator remains both continuous and smooth when K I switches, the problem of ensuring that the control signal response remains both continuous and smooth when switching K D or K P is more complicated. To deal with the later, states of the controller in [18] use a PID controller realization that places the gains K D and K P to be switch directly at the input the plant, then reset the state in the integrator at the switching time to a value precisely calculated so as to ensure control signal continuity and thus to achieve the desired bumpless controller switch. The integrator has only one pole at s = 0, which is an in¯nitely slow mode, whereas the di®erentiator term K D s = lim ²!0 K D s=(²s + 1) has an in¯nitely fast mode associated with a pole at s = 1=² (² ! 0). Smoothness and continuity of the control signal at switching times is ensured by resetting only the stateassociatedwiththeslowintegratormodeofthePIDcontrollerandleavingthe stateofthein¯nitelyfastdi®erentiatormodealone. Asweshallshow,thisapproach to bumpless controller switching can be generalized to other types of controllers by 50 restricting switching-time state resets to the states associated with the slow modes of the controllers. Comment: In the PID controller implementation of [18], a command signal r(t) was also included in the control loop and to prevent step switches in external command signals from producing `bumps' or discontinuities in the control signal the derivative term ( K D s ²s+1 ) of the switched PID controller was positioned in the feedback path ahead of the point where the command signal r enters so that step commands r = 1=s could not produce a `bump' by directly exciting the fast mode of the derivative term. Thus, the issue bumps excited by external step or other similarly `bumpy' command signals, if present, can be always addressed this way, i.e., by putting the fast part of the controller K fast (s) in the feedback path so that command signal bumps cannot directly excite fast modes of the controller. In the presentreport, weshallnotexplicitlyconsidertheissueofcommandsignalinduced bumps, but simply note here that they can be always be handled by appropriately positioning the command signal input so that it does not directly excite the input to the fast part of the candidate controllers K ifast (s). 3.3.2 Bumpless transfer with slow-fast decomposition Our bumpless transfer method which will be stated in this section requires the following assumption hold for each of the candidate controllers. Assumption 3.3.1 For each candidate controller K i , the slow part K islow in (3.7) has at least m=dim(u) states. } The Assumption 3.3.1 is su±cient to allow the state of the slow controller K islow (s) to be reset at switching times to ensure both continuity and smoothness of the control signal u(t), as we shall explain. 51 Ingeneral,evenifallthecontrollershavethesameorderandallshareacommon state vector, when the controller switching occurs, any or all of the slow and fast controllerstate-spacematriceswillbeswitched,whichcanleadtobumpytransients or discontinuity in the control signal u(t) at switching times. However, if only A is or B is are switched and there is common state vector before and after the switch, then the control signal will be continuous and furthermore no `bumpy' fast modes of the controller will be excited. Fast transient `bumps' or discontinuities, when they occur, may arise from switching the D is matrix of the slow controller or from switching any of the state-space matrices (A if ;B if ;C if ;D if ) of the fast controller. In the case of switching the matrices A if or B if switches do actually not result in discontinuous jumps in u(t), but nevertheless can result `bumpy' fast transients in the control signal which, if very fast, may appear to be nearly discontinuous. Our goal in bumpless transfer is to avoid both discontinuity and fast transients inducedbychangingfastmodes. Wewouldlikeourmethodstoworkevenwhenthe order of the controller changes at switching times, and to allow for the possibility thatthetrueplantmaybeimpreciselyknown,wewouldlikeourswitchingalgorithm not to depend on precise knowledge of the true plant. In our method, we can do this by initializing the state of the slow part of the new controller K jslow (s) after each switch to a value computed to ensure continuity, and setting the state of the fast part K jfast (s) to zero. Theorem 3.3.1 (Main Result) Suppose that each of the candidate controllers have slow-fast decomposition (3.7) satisfying Assumption 3.3.1 and suppose that at 52 time t s the online controller is switched from controller K i to controller K j . At t s , let the states of the slow and fast controllers be reset as follows x fast (t + s ) = 0 (3.9) x slow (t + s ) = C y js fu(t ¡ s )¡(D js +D jf )z(t ¡ s )g+» (3.10) where z = [r T y T ] T , C y js is the pseudoinverse matrix of C js , and » is any element of the null space of C js ; C js » =0 . (3.11) Then, bumpless transfer is achieved at the switching time t s . } Proof: The control signal immediately after switching (time t + s ) can be written, based on state space representation model (3.7) of the new controller K j (s), as u(t + s ) = C js x slow (t + s )+C jf x fast (t + s ) +(D js +D jf )z(t + s ) . (3.12) By (3.9) { (3.10), u(t + s ) = C js [C y js fu(t ¡ s )¡(D js +D jf )z(t ¡ s )g+»] +(D js +D jf )z(t + s ) ByAssumption3.3.1,C js C y js =I m£m wheremislargerthanorequaltothenumber of states of K j . This results in u(t + s )=u(t ¡ s )¡(D js +D jf )z(t ¡ s )+(D js +D jf )z(t + s ) . 53 Since z(t ¡ s )=[r T (t ¡ s ) y T (t ¡ s )] T =[r T (t + s ) y T (t + s )] T =z(t + s ) , we ¯nally have u(t + s ) = u(t ¡ s ) . The result follows immediately from the De¯nition 3.2.1. Q.E.D. Comment: Since C js is a full rank matrix which consists of m linearly inde- pendent vectors, C js C T js is invertible and C y js =C T js (C js C T js ) ¡1 . For details, see [4]. } Equations(3.9)and(3.10)nowde¯neourslow-fast bumpless transfer algorithm. An example using this algorithm will be presented in Section 3.4. 3.3.3 Slowmodescontrollerwithobservablecanonicalform Now we introduce a way to build slow modes controller satisfying Theorem 3.3.1. Although there are various ways to make slow modes controller, using observable canonical form of K slow is a good choice since C js in (3.12) has the following form C js = · I 0 ¢¢¢ 0 ¸ : (3.13) in which case, C y js =C T js in (3.10). Additionally, one simple choice for the vector » in (3.11) is » =0. 54 Byhaving(3.13)forallj,oneusestransposematricesratherthanpseudoinverse matrices of C js s. This reduces complexity on state reset procedure. Comment: Note that a bumpless transfer in [2] is a special case of our bump- less transfer method. A method in [2] requires that all controllers have the same numberofstatesandallcontrollersmusthaveastate-spacerealizationwhichshare a common C-matrix and D-matrix. Our slow-fast method does not impose these controller restrictions, and can be used whenever the minimal realization of slow part K islow has order at least equal to the dim(u). However, if the controllers have slow part only (i.e., K ifast = 0 for all i) and each controller has C-matrices with the form of (3.13) and a common D-matrix (e.g. D is = 0 for all i), then, it is expected for both of methods to have the same result. The advantages of the slow- fast bumpless transfer method considered in this paper arise when the controllers have both slow and fast modes, in which case our method is able to exploit the ad- ditional °exibility for state re-initialization provided by the additional fast modes to eliminate the bumpy abrupt transients the might otherwise result. } 3.3.4 Controller State Augmentation with Uncontrollable Modes Situations that some of candidate controllers have only slow modes or only fast modes are addressed in this section as special cases of the slow-fast decomposition bumpless transfer. Ifcandidatecontrollershaveslowmodesonly, usingthemethodinSection3.3.3 and applying (3.10) straightforwardly solve the problem. On the other case, when candidate controllers have only fast modes K =K fast , it is necessary that the controllers are modi¯ed in order to apply Theorem 3.3.1 55 because they do not contain slow parts to be re-initialized. One possible solution is augmenting the controller states with uncontrollable slow modes that was not originally contained in the controller. Then, by slow-fast decomposition, an additive slow mode controller K slow is included in the controller so that ~ K =K slow +K fast where K slow s = 2 6 4 A s 0 C s 0 3 7 5 (3.14) and A s 2 R m£n has only slow modes. Note that the K slow has zero matrices for its B s and D s , so its output is determined solely by initial state. The matrix C s is chosen with the form of (3.13) for the easiest way. Now, (3.9) and (3.10) in Theorem 3.3.1 can be applied, exactly as when there was already a K slow . Since K(s) = ~ K(s), measurements for performance are not e®ected by adding the slow mode controller (3.14) except transients after the switching times, as expected how x slow works. 3.4 Simulation Results One of previously suggested slow-fast decomposition can be found in [3]. Or, con- trollers can be decomposed by inspection if they has poles and zeros which clearly represent fast or slow modes. An example of this type of controller is a PID con- troller. Thus, the simulation shows how the method suggested above works with PID controllers. 56 3.4.1 Adaptive PID controller A plant in this example is G(s)= s 2 +s+10 s 3 +s 2 +98s¡100 : Two controllers having the structure in (3.4) were used to show the results. Each controllers have three gains to be switched, and the gains are as follows; Controller 1; K P1 =80; K I1 =50; K D1 =0:5 Controller 2; K P2 =5; K I2 =2; K D2 =0:6 A small number ² is 0.01 and the reference input is r = 1. The ² prevents the di®erentiator not to make a in¯nite peak when a discontinuity comes into the controller. A PID controller is naturally decomposed into a slow and a fast part. Since a proportional gain is memoryless component, it can be added to either part. Therefore, the controllers were decomposed into K slow (s) s = 2 6 4 0 K I 1 K P 3 7 5 : (3.15) And, in the same way, K fast can be written by K fast (s) s = 2 6 4 ¡ 1 ² 1 ² ¡ K D ² K D ² 3 7 5 : (3.16) 57 0 2 4 6 8 10 −80 −60 −40 −20 0 controller output u(t) u(t) time t (second) without bumpless transfer with bumpless transfer 0 2 4 6 8 10 0 0.5 1 1.5 2 plant output y(t) y(t) time t (second) without bumpless transfer with bumpless transfer Figure 3.3: Controller output u(t) without bumpless transfer and with bump- less transfer (upper ¯gure); Plant output y(t) without bumpless transfer and with bumpless transfer (lower ¯gure). Controller is switched at t=2. 58 Controller 1 and Controller 2 in this particular case were, respectively, K 1 (s)=K 1slow +K 1fast =80+ 50 s + 0:5s 0:01s+1 K 2 (s)=K 2slow +K 2fast =5+ 2 s + 0:6s 0:01s+1 K 1 (s)wasdesignedtostabilizetheplant,whileK 2 (s)cannotstabilizetheplant. In this experiment, K 2 is the on-line controller at ¯rst. Thus, the plant was not stabilized at early stage. After 2 seconds, the on-line controller was switched into K 1 . The experiments were done twice for a comparison. One included the bumpless transfer method, but the other did not. The upper part of Figure 3.3 shows the controller output of both cases. The output u(t) with bumpless transfer had a smooth transient even at the switching instant. On the other hand, u(t) without bumpless transfer had a large spike after t=2. Figure 3.4 shows the controller output around the switching instant in detail. The solid line is continuous and changes moderately because of using bumpless transfer method. The dotted line is not continuous and has spikes. After the large overshoot, it changes abruptly. The di®erence of controller output resulted in the di®erent plant output as shown in the lower part of Figure 3.3. An abrupt transient in the system without bumplesstransfercausedaundesiredjumpintheplantoutputy(t). Thisistoofast response to occurs in practice. Since generating this kind of response in the plant output needs in¯nitely large energy, this is not desirable and even not feasible in real world. Thus, we can conclude that the plant output in the case with bumpless transfer is more reasonable than the other case. 59 1.5 2 2.5 −30 −25 −20 −15 −10 −5 0 controller output without bumpless transfer (magnified) y(t) time t (second) 1.5 2 2.5 −30 −25 −20 −15 −10 −5 0 controller output with bumpless transfer (magnified) u(t) time t (second) Figure 3.4: Magni¯ed u(t) around the switching instant (t=2). 60 3.4.2 Applying slow-fast controller decomposition Thesecondexampleusedcontrollerswhichhavetwopolesandtwozerosandapplied slow-fast decomposition to the controllers. The plant is G(s)= 1 s 3 +15s 2 +50s : Controller 1 and Controller 2 are, respectively, K 1 (s)= (s+4)(s¡0:1) (s+7:28)(s+10) and K 2 (s)= (s+4)(s¡0:1) (s+7:28)(s+0:01) : Now, slow-fast decomposition is applied to both of controllers. K 1 (s)=K 1slow (s)+K 1fast (s) where K 1slow s = 2 6 4 ¡7:28 ¡51:7 ¡32:02 95:6 3 7 5 ; K 1fast s = 2 6 4 ¡10 39:19 ¡106:5 95:6 3 7 5 : (3.17) In the same way, K 2slow s = 2 6 4 ¡0:01 2:634 3:585 95:6 3 7 5 ; K 2fast s = 2 6 4 ¡7:28 28:98 ¡21:37 95:6 3 7 5 (3.18) Afunctionnamed slowfastin[3]wasusedtohavetheresultsin(3.17)and(3.18). After the slow-fast decomposition, the transform in [2] is used with F = 1 for the continuity of K slow . Then, we have 61 ¹ K 1slow s = 2 6 4 ¡7:28 1655:45 1 95:6 3 7 5 ; ¹ K 2slow s = 2 6 4 ¡0:01 9:44 1 95:6 3 7 5 : In this example, on-line controller is switched from K 1 to K 2 at t=5. The results were shown in Figure 3.5. Similarly to the earlier example, the switching controller with bumpless transfer method shows better transient in controller output and plant output. 3.4.3 Advantages over continuity assuring method This example veri¯es validity of the additional condition De¯nition 3.2.1(b) com- pared with a previously existing method in [2] and demonstrates using 2-DOF controllers. A PID controller has an in¯nitely slow pole and a very fast zero when ² is not zero with the following de¯nition; K(s)=K slow (s)+K fast (s)=K P + K I s + K D s ²s+1 : (3.19) Since a fast zero in its di®erentiator can make a large and fast transient even after controller switching, considering only continuity of controller output as in [2] might not be su±cient to perform bumpless transfer. In this example, we show that our method can suppress undesired transient right after the switching. The results compare our bumpless transfer method properly initializing both fast and slowmodescontrollerswithnon-bumplesstransferswitchingandanotherbumpless transfer in [2]. 62 0 5 10 15 20 25 30 0 20 40 60 80 100 120 controller output time t (second) u(t) Without bumpless transfer With bumpless transfer 0 5 10 15 20 25 30 0 0.5 1 1.5 plant output time t (second) y(t) Without bumpless transfer With bumpless transfer Figure 3.5: Controller output u(t) without bumpless transfer and with bump- less transfer (upper ¯gure); Plant output y(t) without bumpless transfer and with bumpless transfer (lower ¯gure). Controller is switched at t=5. 63 The plant for the comparison is G(s)= s 2 +s+10 s 3 +s 2 +98s¡100 . Two controllers having the structure in (3.4) were used to show the results. Each controllers have three gains to be switched, and the gains are as follows; Controller 1: K P1 =80; K I1 =50; K D1 =0:5 Controller 2: K P2 =5; K I2 =2; K D2 =1:25 A small number ² is 0.01 and the reference input is r = 1. The ² prevents the di®erentiator not to make an in¯nite peak when a discontinuity comes into the controller. A PID controller is naturally decomposed into a slow and a fast part. Since a proportional gain is memoryless component, it can be added to either part. Controllerinputisz =[r T y T ] T for2-DOFcontrollers. Subsequently,thecontrollers were decomposed into K slow (s) s = 2 6 4 0 [K I ¡K I ] 1 [K P ¡K P ] 3 7 5 : (3.20) And, in the same way, K fast can be written by K fast (s) s = 2 6 4 ¡1=² [1=² ¡1=²] ¡K D =² [K D =² ¡K D =²] 3 7 5 : (3.21) 64 Controller 1 and Controller 2 in this particular case were, respectively, K 1 (s) = K 1slow +K 1fast =80+ 50 s + 0:5s 0:01s+1 K 2 (s) = K 2slow +K 2fast =5+ 2 s + 1:25s 0:01s+1 K 1 (s) was designed to stabilize the plant, while K 2 (s) cannot stabilize the plant. In this experiment, K 2 is the on-line controller at ¯rst. Thus, the plant was not stabilized at early stage. After 2 seconds, the on-line controller was switched into K 1 . Comment: The bumpless transfer method in [2] does not include any initializ- ing or state reset procedure at switching instants. Instead, it works allowing only controllers for which there exist state-space realizations such that share common C and D matrices; i.e., C i =C j ,C and D i =D j ,D for all i6=j (3.22) wherethematricesareasin(3.3). Notethatthisisnotpossibleingeneral,unlessall controllershavethesameorderandsame D-matrics. Thoughourslow-fastmethod does not su®er this restriction on controllers, we have chosen for our simulation examplecontrollersthatdoconformtothisrequirementinorderbeabletodirectly compare the two di®erent methods in the same situation. } Threesimulationexperimentsweredone. First,switchingwithoutanybumpless transfer method was performed. Next, for comparison one used the method of [2] and another used our slow-fast method based on Theorem 3.3.1. The upper part of Fig. 3.6 shows the controller output. The solid line (Cheong and Safonov's) of outputu(t)showsasmoothtransientaroundtheswitchinginstantwhilethedashed 65 0 1 2 3 4 5 6 7 8 −80 −70 −60 −50 −40 −30 −20 −10 0 10 Controller Output time t (second) u(t) Without Bumpless Transfer Arehart & Wolovich Cheong & Safonov 0 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Plant Output time t (second) y(t) Without Bumpless Transfer Arehart & Wolovich Cheong & Safonov Figure 3.6: Controller output u(t) (upper ¯gure); Plant output y(t) (lower ¯gure). Controller is switched at t=2. 66 1.96 1.98 2 2.02 2.04 2.06 2.08 2.1 −30 −25 −20 −15 −10 −5 0 Controller Output time t (second) u(t) Without Bumpless Transfer Arehart & Wolovich Cheong & Safonov Figure 3.7: Magni¯ed u(t) around the switching instant (t=2). line (Arehart and Wolovich's) shows a fast transient after switching. The dotted line indicates switching transient without bumpless transfer, which has extremely high peak value generated by derivative controller. If ² approaches to zero, the peak value goes to in¯nity. Fig. 3.7 showsu(t) with the time axis magni¯ed near the switching time. While the output without bumpless transfer has discontinuity at switching time 2 sec- ond, the outputs with bumpless transfer (dashed line and solid line) show contin- uous transient. Note that dashed line (Arehart and Wolovich's) satis¯es De¯ni- tion 3.2.1(a) which coincides with the de¯nition of bumpless transfer used in [2]. However, comparing with the solid line, the dashed line exhibits a fast `bumpy' transient after 2 second. It is excited by changing K fast , which is clearly di®erent result from our method. 67 The resultant plant output y(t) shown in the lower part of Fig. 3.6 likewise exhibits an abrupt transient with the method of [2]. Evidently, both the control signalandtheplantoutputinthecasearesigni¯cantlysmootherwithourslow-fast bumpless transfer method. 3.5 Summary Afterbriefreviewofpreviouslyexistingbumplesstransfermethods,anewde¯nition ofbumplesstransferwhichaddsinadditiontotheusualcontinuityrequirementand additional requirement that there be no transients induced by controller switching. A new bumpless transfer method based on slow-fast decomposition and state re- set has been introduced. Simulation results demonstrate the e®ectiveness of our bumpless transfer method. The bumpless transfer proposed in this dissertation generalized the speci¯c method in [18], and extend the coverage to any 2-degree-of-freedom switching con- trollers which have state-space controller models. The slow-fast controller decom- position bumpless transfer method can be distinguished from existing bumpless transfer methods because the method makes use only of knowledge of the switched controllers' state-space matrices and the value of the control signal just prior to switched. The method is particularly well-suited to adaptive switching control ap- plications where the true plant model is imprecisely known or yet to be identi¯ed. After presenting a new bumpless transfer theory, we explained how to use the standard observable canonical form of the controllers to simplify practical imple- mentation. Special cases that controllers have only slow modes or only fast modes werecommentedandpossiblesolutionswereprovided. Toclarifyadvantagesofour bumpless transfer method, comparison with continuity based bumpless transfer 68 and with non-bumpless transfer were performed. Simulation results demonstrate the e®ectiveness of our slow-fast bumpless transfer method in eliminating abrupt fast switching transients. 69 Chapter 4 Fading Memory and Time-Window 4.1 Review of Unfalsi¯ed Control Elementary concepts and terms of the unfalsi¯ed control were provided in Sec- tion 2.3.2 and further detail may be found in [35], [37]. Section 4.1.2 explains the role and use of ¯ctitious signals in adaptive PID control, the example to which fading memory and time-window cost function applied. 4.1.1 Theory Since the details of unfalsi¯ed theory were already described in Section 2.3.2, a short comment and the feasibility assumption are presented in this section. In the unfalsi¯ed control paradigm the adaptive control supervisor selects the active online controller from among the currently unfalsi¯ed controllers in the candidate pool setK, switching to a new controller when new data falsi¯es the current online controller. It is known that if a few very mild assumptions hold and if additionally theT spec (r;y;u;¿)ismonotoneintimetandsatisfycertainother`costdetectability' conditions, then the following feasibility assumption is su±cient to guarantee that the adaptive system is robustly stable. 70 r u y plant unfalsified algorithm adaptive PID controller y adaptive control supervisor Figure 4.1: Con¯guration of adaptive PID controller Assumption 4.1.1 (Feasibility) [38] The candidate controller set K contains at least one robustly stabilizing and performing controller. InSection4.2,theunfalsi¯edalgorithmisdescribedinfurtherdetail,alongwith a new modi¯cation for handling slowly time-varying plants that may violate this feasibility assumption. 4.1.2 Fictitious Reference signals in PID controller An adaptive PID controller in Figure 4.1 is presented as an example of unfalsi¯ed control system. In particular, the PID controller has a structure of Figure 4.2 and this is the same structure used in [18]. AsetofcandidatecontrollersisKandthei-thcontrollerinKisK i =fk P i ;k I i ;k D i g. Fictitiousreferencesignalsanderrorsignalsassociatedwiththe i-thcandidatecon- troller can be computed by the equations ~ r(K i ), ~ r i =y+ s k P i s+k I i µ u+ k D i s "s+1 y ¶ and ~ e(K i ), ~ e i = ~ r i ¡y (4.1) 71 Figure 4.2: PID controller which are obtained by inverting Figure 4.2 to solve for the r and e in terms of (u;y). We assume that for all controllers in the candidate pool the three PID gains (k P i ;k I i ;k D i ) are either all positive or all negative, which ensures that the ¯ctitious reference signal generator systems de¯ned by (4.1) are well-posed and stable. [35] In the implementation of unfalsi¯ed adaptive control, the adaptive control su- pervisor unit uses the ¯ctitious reference signal associated with each candidate controller to iteratively update that controller's current unfalsi¯ed cost level. This is possible because, as shown in [35], if the system de¯ned by (4.1) is well-posed andstablethenthei-thcontrollerK i isunfalsi¯edatcost level ° attime¿ byplant data (y;u) if, and only if, J(K i ;u;y;¿),T spec (~ r(K i );u;y;¿)·° : (4.2) 72 4.2 Problem Formulation Selection of a suitable cost function is a signi¯cant task in building an unfalsi- ¯ed adaptive control law. In this report, the cost-detectable monotone function T spec (~ r(K i );u;y;t)normallyusedtoevaluateunfalsi¯edcostlevelforthecandidate controller K i at each time t is replaced by the following non-monotone modi¯ed cost function J(K i ;u;y;t)= 8 > < > : max l2(t¡¿ 0 ;t) fe ¡¸(t¡l) T spec (~ r i ;y;u;l)g , t¸¿ 0 max l2(0;t) fe ¡¸(t¡l) T spec (~ r i ;y;u;l)g , t<¿ 0 (4.3) where ~ r i = ~ r(K i ). As shown in [35], the i-th controller K i is unfalsi¯ed at cost level ° at time t by plant data (y;u) if, and only if, J(K i ;u;y;t)·° : The modi¯ed cost (4.3) has two di®erences as compared to the original unmodi¯ed cost (4.2). 1) Time-Windowing: The max operator term ignores old data outside the recent-time window (t¡¿ 0 ;t). (Fig. 4.3) 2) Fading-Memory: The term e ¡¸(t¡l) exponentially reduces the e®ect of the older data. (Fig. 4.4) We note that a similar exponential factor arises in [16] in conjunction with scale-independent hysteresis switching algorithms. These di®erences make it possible for older data to be de-weighted and very old datatobeignored. Thiscanbeusefulwhentheplantchangesslowlyorinfrequently. 73 Figure 4.3: Time-windowing on T spec T spec e - t t t T spec t Figure 4.4: Fading-memory for T spec In practice, slow changes of the environment (temperature, air pressure, etc.) or suddenbutinfrequentcomponentfailuresarepossiblecausesofsuchchangesinthe plant. In case of a cost function with monotone non-decreasing property as in [38] without fading memory or windowing of data, a controller falsi¯ed once would not be recycled even though it would be the best candidate controller at sometime after being falsi¯ed. But, with the modi¯ed cost function (4.3) having the fading memory and time-windowing features of 1) and 2) above, Assumption 4.1.1 can be relaxed, as we will demonstrate via an example. Algorithm 4.2.1 (Hysteresis algorithm for unfalsi¯ed control) [18] INITIAL SETTING: ² a ¯nite set K of m candidate controllers K i ; i2I=f1;2;¢¢¢;mg ² initial cost °(0)=0 and J(i;0)=0; 8i2I 74 ² sampling time ¢t ² e®ective time of the max operator ¿ 0 ² the values of ", ¸ ² initial online controller K =K m at t=0 PROCEDURE at each time t = k¢t (k; time step) and for each candidate controller K i ; i2I=f1;2;¢¢¢;mg: 1. Measure u(k¢t), y(k¢t) and set °(k¢t)=J(K(k¢t);u(k¢t);y(k¢t);k¢t). 2. For each i2I, calculate ~ r i (k¢t); ~ e i (k¢t); and J(i;k)=J(K i ;u(k¢t);y(k¢t);k¢t). 3. If min K i 2K fJ(i;k¢t)g·°(k¢t)¡", then K((k+1)¢t)=arg min K i 2K fJ(i;k¢t)g. Otherwise,K((k+1)¢t)=K(k¢t)whereK(k¢t)isonlinecontrollerattime t. 4. Set k =k+1 and repeat. } Comment: The addition of non-zero fading memory and ¯nite time window pa- rameters, ¸ and ¿ 0 , in (4.3) is the di®erence as compared to the cost form (4.2) used in [38]. When these parameters are positive numbers, the modi¯ed cost need not necessarily increase monotonically as it does in [38], which means that after some time has elapsed a controller can be recycled and re-added to the unfalsi¯ed set without increasing the cost level °. Therefore, when either or both of the pa- rameters ¸ and ¿ 0 is non-zero, the set of unfalsi¯ed controllers at each cost level ° computed by Algorithm 4.2.1 no longer necessarily shrinks monotonically with 75 time as in the previous works [18], [35], [38] where there was no fading memory or windowing in the cost function. } At the beginning of the hysteresis switching Algorithm 4.2.1 [27], functions and parametersareinitialized. Thetime-windowduration¿ 0 wouldnormallybeselected tobesomewhatlessthanthetime-scaleoverwhichsigni¯cantplantvariationsmay occur that cannot be robustly accommodated by any one candidate controller in the controller pool K. After initialization, output data u(t) and y(t) are measured and ¯ctitious signals are computed by (4.1). The unfalsi¯ed cost level J(i;k) of each candidate controller K i ; i2I =f1;2;¢¢¢;mg is iteratively updated in step 2 of Algorithm 4.2.1, based on the measured data and the ¯ctitious signals by (4.3) where T spec will be presented in Section ??. If the unfalsi¯ed cost level J(i;k) is larger than the current unfalsi¯ed cost level °, the controller K i is falsi¯ed at this cost level and is not used. The PID controller K i with the currently minimum cost among the unfalsi¯ed controllers is examined, and its cost is compared to the previouscost-minimizingcontroller'scost° minusasmallpositivenumber", called the hysteresis constant. If unfalsi¯ed cost level J(i;k) of this cost-minimizing K i is less than °¡", Algorithm 4.2.1 updates the currently active controller, replacing it with K i . 4.3 Computer Simulations We now describe the results of simulations that were performed to verify the e®ec- tiveness of the modi¯ed cost function. Details of the simulation are as follows. We consider an unfalsi¯ed adaptive PID controller as shown in Figure 4.2. Its three gains fK P ;K I ;K D gare selected from a ¯nite set by the hysteresis algorithm [27]. We shall compare the result of using the hysteresis algorithm with, and without, 76 the fading memory and time-windowing modi¯cation of the cost J(K i ;u;y;t). In both of the two cases considered the hysteresis algorithm is the same. Only the cost function J(K i ;u;y;t) di®ers. The simulations were set up as follows. ² The plant switches from G 1 (s) to G 2 (s) at time t 1 = 100, and from G 2 (s) back to G 1 (s) at time t 2 =300. ² G 1 (s)= 0:1 s+0:1 and G 2 (s)=¡G 1 (s)=¡ 0:1 s+0:1 ² K=fK 1 ;K 2 g=ff3;0:1;0:1g;f¡3;¡0:1;¡0:1gg ² The reference input r(t) is a step function with amplitude 1. - Case 1 (perfect memory): Cost function (4.3) with ¿ 0 !1; ¸=0; - Case 2 (fading and limited memory): Cost function (4.3) with ¿ 0 =5; ¸=0:1. To select a performance speci¯cation T spec , following performance goal is con- sidered kw 1 ¤~ e i k 2 L 2 [t¡¿ 0 ;t] +kw 2 ¤uk 2 L 2 [t¡¿ 0 ;t] ·k~ r i k 2 L 2 [t¡¿ 0 ;t] ; 8t¸¿ 0 (4.4) wherek¢k is a L 2 norm and¤ denotes convolution. Inequality (4.4) means that the error and control output should be small compared to the reference signal. The ¯lters w 1 and w 2 are inspired by controller design method of H 1 mixed-sensitivity loop shaping. Frequency responses of w 1 and w 2 are W 1 and W 2 , which satisfy the following H 1 performance criterion. ° ° ° ° ° ° ° 2 6 4 W 1 S W 2 KS 3 7 5 ° ° ° ° ° ° ° 1 ·1 (4.5) 77 The transfer function S in (4.5) is a sensitivity function of the system. Based on (4.4), the performance speci¯cation set T spec of i-th controller at time t is selected to be T spec (r i (t);y(t);u(t);t)= kw 1 (t)¤(y(t)¡~ r i (t))k 2 L 2 [t¡¿ 0 ;t] +kw 2 (t)¤u(t)k 2 L 2 [t¡¿ 0 ;t] k~ r i (t)k 2 L 2 [t¡¿ 0 ;t] : (4.6) Comment: When the duration of time-window ¿ 0 is in¯nity and fading memory parameter ¸ is zero as in Case 1 above, the performance speci¯cation (4.6) is cost- detectable. As we proved in [38], cost-detectability is su±cient to ensure that the hysteresis switching algorithm robustly converges to a stabilizing controller when- ever a stabilizing controller exists in the candidate controller set. Thus, we may reasonably expect for all intents and purposes that the fading memory and time- windowingalgorithmwillbesimilarlyrobustif¿ 0 and¸arechosentobesu±ciently large. } There are only two elements in the set of candidate controllers. Controller K 1 stabilizes a plant G 1 (s) and controller K 2 stabilizes a plant G 2 (s). On the other hand,K 1 cannotstabilizeG 2 (s)andK 2 cannotstabilizeG 1 (s). Forthissimulation, the two plants are the same except for their sign. Likewise, the controllers K 1 and K 2 the same, except for opposite signs of their gains. The cost level of controller K 2 increases abruptly in the early stage of the simu- lation because the plant is G 1 (s), which is destabilized by K 2 . Later, at simulation timet 1 , the plant changes into G 2 (s). Then, the cost for controller K 1 begins to in- creasequickly,becauseK 1 destabilizestheplantG 2 (s). Inthisstage,thetraditional Case1costfunctionofunfalsi¯edcontrolcannotchooseapropercontrollerbecause the cost level of controller K 1 is still higher than K 2 . Therefore, the output y(t) of 78 Figure 4.5: Simulation result of Case 1 and Case 2 in large scale the Case 1 grows negatively exponentially as shown in Figure 4.5. At time t 2 , the plant comes back to G 1 (s) and the output needs long time to become stabilized. With the Case 2 fading-memory time-windowed cost function, controller K 2 , which was initially falsi¯ed, is recycled and is available to be selected for stabiliza- tion of G 2 (s). The stable simulation result in Figure 4.6 re°ects the fact that the currently active controller was successfully switched from K 1 to K 2 when the plant switched from G 1 (s) to G 2 (s), although several seconds were needed to remove the e®ect of old data after t 1 . Next, after t 2 when the plant switched from G 2 (s) back to G 1 (s), the Case 2 fading memory cost function allowed the controller to cor- rectly switchback to K 1 again. The results in Figure 4.5 and Figure 4.6 show that the fading memory Case 2 cost function controller maintains stability, in contrast to traditional in¯nite-memory Case 1 cost function, which does not. Case 1 has 79 Figure4.6: SameasFigure4.5, butwithverticalaxismagni¯edbyfactor0:5£10 20 exponentially increased peak around time t 2 , but Case 2 shows good performance at all times. 4.4 Summary A new cost function with time-windowing and the fading memory, proposed in [5], was described in Chapter 4. Simulations with slowly time-varying plant demon- strate the e®ectiveness of the modi¯cation for stabilizing a plant that violates the usual feasibility assumption of unfalsi¯ed control and cannot be robustly stabilized byanysinglecontrollerinthecandidateset. Thetime-windowingandfadingmem- ory of the new cost function works because it allows re-use of controllers that were falsi¯ed by very old data that may no longer be relevant in the case of a slowly or infrequently varying plant. 80 Chapter 5 Conclusion Two separate implementation techniques for adaptive switching control were sug- gested to improve performances in practical sense. A new bumpless transfer using slow-fast controller decomposition is expected to enhance the controller output transient when the online controller is switched. This bumpless transfer technique does not require exact information about the plant to be controlled. The suggested bumpless transfer technique is more suit- able for adaptive controls than other bumpless transfer techniques that require an exact plant model or a prior data. The slow-fast controller decomposition and re-initialization of the decomposed controllers provide a tool to manipulate the switching transient after the switching instant. This has not been observed in pre- viouslyexistingbumplesstransfermethodsthatproducethecontrolleroutputonly with the consideration of control signal continuity. Controller state augmentation with uncontrollable modes is another tool giving °exibility to add and manipulate slow mode controller even though there is no slow mode in the original controller. 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Abstract (if available)

Abstract This dissertation mainly focuses on implementation techniques for adaptive switching control. Adaptive switching control can generate bad transients in controller output. These bad transients can be reduced by various bumpless transfer techniques. A new bumpless transfer technique is developed based on slow-fast controller decomposition. This technique is especially well-suited to situations in which the plant model is poor or yet to be identified, as may be the case in adaptiveswitching control.

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Asset Metadata

Creator Cheong, Shin-Young (author)

Core Title Bumpless transfer and fading memory for adaptive switching control

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 09/16/2009

Defense Date 08/25/2009

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag adaptive control,bumpless transfer,feedback control,OAI-PMH Harvest,switching control

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Safonov, Michael G. (committee chair), Flashner, Henryk (committee member), Jonckheere, Edmond A. (committee member)

Creator Email kkoong@gmail.com,sycheong@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m2605

Unique identifier UC1366273

Identifier etd-Cheong-3256 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-254488 (legacy record id),usctheses-m2605 (legacy record id)

Legacy Identifier etd-Cheong-3256.pdf

Dmrecord 254488

Document Type Dissertation

Rights Cheong, Shin-Young

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu