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Inverse problems, identification and control of distributed parameter systems: Applications to space structures with active materials

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Inverse problems, identification and control of distributed parameter systems: Applications to space structures with active materials

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Content INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality o f th is reproduction is dependent upon the quality o f th e copy subm itted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand corner and continuing from left to right in equal sections with small overlaps. Photographs included in the original manuscript have been reproduced xerographicaily in this copy. Higher quality 6” x 9” black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. ProQuest Information and Learning 300 North Zeeb Road, Ann Arbor, Ml 48106-1346 USA 800-521-0600 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. INVERSE PROBLEMS, IDENTIFICATION AND CONTROL OF DISTRIBUTED PARAMETER SYSTEMS: APPLICATIONS TO SPACE STRUCTURES WITH ACTIVE MATERIALS by Taufiquar Rahm an Khan A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (A pplied M ath em atics) May 2000 Copyright 2000 Taufiquar Rahman Khan Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number 3018009 __ ___ __® UMI UMI Microform 3018009 Copyright 2001 by Bell & Howell Information and Learning Company. Ail rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. Bell & Howell Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UNIVERSITY OP SOUTHERN CALIFORNIA TUB OBADUA- TV SCHOOL u M v m s r r r pa ju c t o s A N O ftn , C A U PO ftN U 90007 This dissertation, written by T a u f iq u a r B a h a a n K h an under the direction of h...±* Dissertation Committee, and approved by adits members, has been presented to and a c c e pted by The Graduate School, in partial fulfillment of re quirements for the degree of DOCTOR OF PHILOSOPHY D m m af Cndmtt S t ud i es D ate ASr i l 2 5 , 2000 DISSERTATION G ITEE Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Dedication ...to my mother and father Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgements In the sum m er of my first year in graduate school, I had the pleasure of meeting Professor Chunming Wang. I am very grateful to Professor Wang for accepting me as his Ph.D. student. I thank him for his support and encouragement. I have yet to encounter an advisor who matches his devotion and support for his students. Thank you for your patience and guidance. I would like to thank Professor Bingen Yang and Professor Gary Rosen for serving on my committee. Their thoughts and criticism have been very helpful. A special thanks to Professor Alan Knoerr for being a constant guide throughout my undergraduate and graduate years. In 1988, after my brother Akik completed his training at the Bangladesh Military Academy, I asked him for some money to apply to various U.S. high schools. W ith his assistance, I was able to correspond with U.S. schools and receive a full scholarship to come to study in the U.S. Thank you for your help. I would like to thank my brothers Khalid and Asif for their insight about higher education and for their encouragement. A final thanks to my mother. You have supported us boys with little or no money as a prim ary school teacher after our father passed away. Your example of hard work and discipline is why I have come this far. It is you th at I dedicate this thesis. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Contents D edication ii A cknow ledgem ents iii List O f Tables vi List O f Figures vii A b stract viii 1 Introduction 1 1.1 Active M aterials................................................................................................ 5 1.2 History of Study ............................................................................................ 6 1.3 Thesis Overview................................................................................................ 10 2 M athem atical M odel 11 2.1 Model O u tlin e ................................................................................................... 11 2.2 Energy Form ulation......................................................................................... 16 2.3 Equation of Undamped M o t i o n .................................................................. 22 2.4 Equation of Damped M o tio n ........................................................................ 30 2.5 Modeling P a ra m e te rs..................................................................................... 32 3 W ell Posedness 36 3.1 Weak Formulation ........................................................................................ 36 3.2 Wellposedness for the ACTEX M odel........................................................ 44 3.3 State Space A p p ro x im a tio n ........................................................................ 48 3.4 The Spline Galerkin M e th o d ........................................................................ 50 4 Param eter E stim ation 54 4.1 Abstract F orm ulation..................................................................................... 55 4.2 Inverse Problem for the ACTEX M odel.................................................... 60 4.3 Convergence of Param eter E s tim a te s ........................................................ 62 iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 C om p u tation al R esults 64 5.1 Model P aram eters............................................................................................. 64 5.2 System A pproxim ation................................................................................... 66 5.3 ACTEX S im ulation......................................................................................... 67 5.4 Model Param eter E stim ation......................................................................... 69 5.5 Param eter C om parison................................................................................... 72 6 C onclusions and Future S tu dy 74 R eferen ce L ist 75 A p p en d ix A Existence and U n iq u en ess...................................................................................... 79 A .l P re lim in a rie s .................................................................................................. 79 A.2 Semigroup Generation T h e o re m .................................................................. 80 A p p en d ix B A p p ro x im a tio n .......................................................................................................... 82 B .l Preliminaries .................................................................................................. 82 B.2 Trotter-K ato T h e o re m .................................................................................. 84 A p p en d ix C Polynomial S pline...................................................................................................... 86 C .l Prelim inaries .................................................................................................. 86 C.2 Approximation T h e o re m ............................................................................... 91 A p p en d ix D Param eter E s tim a tio n ............................................................................................. 92 D .l Prelim inaries .................................................................................................. 92 D.2 Param eter Estimation T h e o re m ..................................................................... 93 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List O f T ables 5.1 The ACTEX Geometric P aram eters.......................................................... 65 5.2 The ACTEX Physical P aram eters.............................................................. 66 5.3 Estim ated Param eters for Simulation D a t a ............................................. 71 5.4 Estim ated Param eters for the A C T E X .................................................... 72 5.5 Param eter Comparison for the A C T E X .................................................... 73 vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List O f Figures 1.1 Observed Body T em perature........................................................................ 3 1.2 Plot of Least Squaxe F u n c tio n a l.................................................................. 3 1.3 PZT as a Sensor and as an A c t u a t o r ........................................................ 6 1.4 The Actual ACTEX trip o d ........................................................................... 7 1.5 The Embedded PZT Sensors and A c tu a to r s ........................................... 7 1.6 Nearly Colocated Transfer F u n c tio n ........................................................... 8 1.7 Therm ostat L o c a tio n ..................................................................................... 9 1.8 Colocated Transfer F u n c tio n ........................................................................ 9 2.1 The ACTEX Tripod (model o n ly ).............................................................. 12 2.2 The ACTEX Leg D eform ation..................................................................... 12 2.3 The ACTEX Leg Cross S e c tio n ................................................................. 13 2.4 The Moment of Inertia for Hollow L e g .................................................... 13 2.5 Translation of the Top P l a t e ........................................................................ 14 2.6 Rotation of the Top P l a t e ........................................................................... 14 2.7 Top Plate Boundary C onstraints................................................................. 15 2.8 The ACTEX Leg and the Rigid Plate C oordinates................................ 18 2.9 Piecewise Constant P a ra m e te r.................................................................... 33 2.10 Characteristic Function for the Embedded P Z T s ......................................... 33 5.1 Comparison of Time R esponse.................................................................... 68 5.2 Comparison of Frequency R e sp o n se .......................................................... 68 5.3 Comparison of Transfer F u n c tio n .............................................................. 69 5.4 Initial Guess for Simulated Id en tificatio n................................................ 70 5.5 Low Temperature Response Using Estim ated P a ra m e te rs.......................71 5.6 High Temperature Response Using Estimated P aram eters....................... 72 5.7 Stiffness P ro f ile ............................................................................................... 73 vii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Abstract In the last decade, active materials have emerged as one of the most promising tools in the development of high performance control devices in many fields of applications, including space vehicles and structural design. The Active Controls Technology Ex perim ent (ACTEX), a guest investigator program funded by the Air Force Office of Scientific Research and operated by TRW, has created the first space structure with embedded active materials. This provides a unique opportunity for structural design and control scientists to evaluate the potential of using active m aterials to control space structures. Our investigation focuses on the modeling of piezoceramic (PZT) as a control device for the ACTEX structure. We also study the inverse problem to identify structural property variations as the space structure moves through different environmental conditions. Our model, derived from first principles, takes the form of a system of partial differential equations. We establish the well-posedness of the governing infinite dimensional or distributed param eter system, the convergence of the Galerkin approximation scheme, and the convergence of the param eter estima tion problem for this model. We also give soluton of the inverse problem using the currently available space experimental data. vm Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 1 Introduction Recent initiatives in interdisciplinary research have stim ulated interest in m athem at ical applications in science and technology. Some applications in ecology, geophysics, and flexible structures involve distributed param eter systems. Distributed param e ter systems can generally be modeled by partial differential equations with spatially varying coefficients. For a further discussion of distributed parameter systems see Om atu [1, p.1-7] and Stavroulakis [2, p.0-2]. D istributed parameter systems arise often in control problems. Refer to Wang [3], Banks and Wang [4], and Banks and Sm ith [5] for control problems in ecology, geophysics, and flexible structures. It is often necessary to estim ate unknown model param eter values from observa tions. For a general introduction to parameter estim ation for distributed param eter systems, see Banks and Kunisch [6, p. 1-31]. We refer the reader to Banks and Wang [7] [8] for ecological applications, to Banks and Rosen [9], Banks and Smith [10] for flexible structure applications, and to Lotosky [11] and Piterbarg and Rosovskii [12] for stochastic applications. Recent advances in m aterial science have led to the development of advanced sensors and actuators for active materials structures using piezoelectric materials, shape memory alloys, and fiber optics. These active m aterials have wide applications in aerospace, mechanical, and civil engineering. For example, fiber optics sensors can be used for damage detection in bridges and highways. Realistic m athem atical models of structures with embedded active m aterials involve distributed param eter systems. For examples of systems involving structures with active materials includ ing piezoelectrics see Banks, Smith, and Wang [13, p. 1-26]. 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The motivation for our research comes from our desire to control a space struc ture with embedded active materials. The space structure is m ounted on a satellite. The host satellite moves through the earth’s atmosphere. The satellite encounters environmental changes as it moves through space. In order to control the mounted structure, a mathematical model is needed that can capture the qualitative fea tures accurately and can translate the observed variations in the system response into variations in the model parameters. In our study, we have constructed such a m athem atical model for the mounted structure. Once the form of a m athem atical model is obtained its param eters must be spec ified so th at the model performance approximates the observed behavior as nearly as possible. This “parameter identification” problem is an inverse problem. To illus trate the main idea behind the techniques for parameter identification, we present an example with an ordinary differential equation (ODE) with one param eter. Ac cording to Newton’s Law of Cooling the rate of change of the tem perature y(t) of a body is proportional to the difference between the body’s tem perature and the surrounding medium’s tem perature Ts. The model for this problem is a first order ODE, y'(t) = —k(y(t) — Ts) (1.1) where k > 0. Suppose the model param eter k cannot be measured directly, but we can measure the temperature of the body accurately for times U, i = 0, • • ■ , N. The problem is to estimate the value of the parameter k from the observations y(U). If k = 2 and Ts = 98°(7, the unique solution satisfying the initial condition y(0) = 50°C can be analytically determined to be z(i) = 98 — 48e-2t. We plot this solution for ti = if 10, i = 0,..., 10, in Figure 1.1. Now suppose we observe the values of z,- but didn’t know th at k — 2. We could simulate our model for various values of k and compare each simulation y(t{, k ) to these observations using the least square “mismatch functional” m = Y:\zi-y (ti,k ) I 2 (1.2) 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In Figure 1.2, we plot this functional as a function of k for values of k from 0 to 5 at intervals 0.4. As one can see the m ism atch functional is minimized for these simulations neax 2.0, the correct value of the parameter. 1 2 0 1-------- . 1 1 1------- 0.2 0.4 0.6 0.8 tlin* (hr) Figure 1.1: Observed Body Temperature 0.5 1.5 2.5 3.5 4.5 para mater k Figure 1.2: Plot of Least Square Functional Our example is illustrative, but in real applications one needs to derive a model before estim ating its parameters. In this thesis we concentrate on modeling and param eter estim ation for the space structure. Our model is a set of paxtial differential equations (PDEs). The parameters are the coefficients of the PDEs. Param eter 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. identification for PDE models is more complicated than for the preceeding example because both the solution and param eter spaces are typically infinite-dimensional. Before presenting our model we give an example of another param eter estimation problem involving PDEs. Consider the following one dimensional model of an Euler- Bemoulli beam, . d2w d2 ( — . . d2w — d3w \ fi(x)a F + d ^ [ EI(i:)d ^ + CDld ^ d i ) = / (L3) where w(x, t) is the beam deformation, p(x) the mass density, E I{x ) the beam stiffness, C£>I(x) the Kelvin-Voigt damping coefficient, and /( x , t) the forcing term . Let q(x) = (p{x),EI{x),cDI{x)) (1.4) denote the vector of model param eters, which are unknown a priori. These param e ters are said to be “distributed” because they may vary with the spatial variable x. To solve the param eter identification problem for this distributed param eter systems one could attem pt to estim ate the param eters by comparing numerical simulations of the model with the observed beam deformation. Our goal is to use similar param eter identification techniques to estim ate model param eters for the space structure and to translate the observed variations in the system response into variations in the model parameters. The outline of our study is as follows. In chapter 2 we develop a m athem ati cal model for the space structure with active materials. In chapter 3 we establish the well-posedness of our model and provide a com putational foundation for sim ulation. In chapter 4 we pose the param eter estim ation problem and theoretically establish that our param eter identifiacation scheme converges. Finally in chapter 5 we establish the validity of our model by comparing numerical simulations with the experimental data. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.1 Active Materials Recent advances in material science have led to the development of shape memory alloys, magnetostrictive materials, and piezoelectric elements. Shape memory alloys deform under applied stress but recover their original shape when stress is removed. For examples of shape memory alloys and their applications see B arrett [14] and Jardine [15] [16]. Magnetostrictive materials change shape when subjected to a strong magnetic field. An example is the metal alloy Terfenol-D. If the magnetic field changes rapidly, the Terfenol-D m aterial expands and contracts at that same frequency, creating a powerful movement. Piezo-electric materials are electrically polarized by mechanical stress; this is called the “direct effect”. For a discussion of piezo-electric materials and their applications see Banks, Smith, and Wang [13, p.3-7] We now focus our attention on piezo-electric materials that are embedded in our space structure for vibration control. For example, consider a lead zirconate titanate ceramic (PZT) wafer. If we put mechanical stress on the top and bottom surfaces of the wafer and connect the two surfaces by an electrical wire, the applied stress creates a potential difference between the surfaces. In this way any stress in the wafer can be detected by measuring the voltage across the wire. Conversely piezo-electric materials are strained by an applied electric field. Suppose we apply an electric potential across a PZT wafer, the potential will create stress on the top and bottom of the wafer surface. Thus piezo-electric material can be used both as a sensor to detect stress and as an actuator to create stress. For example, suppose we attach PZT wafers on both the top and bottom surface of a beam. If we vibrate and deform the beam, both the top and bottom PZT wafers will be stressed. The applied stress creates a potential difference, and the deformation of the beam can be detected by measuring the voltage across the wafers. Thus the PZT wafers act as sensors to detect vibration of a beam. Conversely, suppose the beam is vibrating and deforming. If we apply a voltage across the wafers, the potential will create stress on the beam. If we time this properly, this stress will deform the beam in the opposite direction of its original motion and cancel the vibration. Thus PZT wafers can actuate and dampen vibrations in the beam. The use of PZT both as a sensor and as an actuator is shown in Figure 1.3. 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (a) (b) (c) vibration ------------ /■ — - V ------- ( 5T & ■ o —0 anti : C * j A c t J i vibration \ raaldual- «0 (•) (0 Figure 1.3: PZT as a Sensor and as an Actuator Piezo-electric elements are used for sensing and controlling vibrations in heli copter rotors, in the wings of commercial airplanes, in waterskis, and in snowboards. Although wide commercial applications of active materials began in the 1980s, the first use of active materials for space applications was in 1996. For examples of applications of active materials in commercial airplanes see Austin [17] [18] and [19] M artin [20]. 1.2 History of Study The U.S. Air Force Laboratory and TRW launched the Advanced Controls Tech nology Experim ent (ACTEX, see Manning, Quassim [21, p.3-32] [22]) in 1996. The objective was to demonstrate in-orbit the use of embedded PZT sensors and actu ators for active and passive vibration suppression. TRW built a tripod which was mounted in an Air Force satellite. A picture of the tripod is shown in Figure 1.4. The tripod mimics a deployable antenna in space. The top plate is like a antenna dish, supported by the legs of the tripod. If the vibrations of the legs can be controlled using PZTs, the top plate can be stabilized. One could stabilize a dish antenna orbiting the earth in a similar way. A stable dish antenna can transm it and receive signals more accurately. 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 1.4: The A ctual A C TEX Tripod Various remote system identification experiments were performed to detect any system atic changes as the ACTEX structure moved into different space environ ments. A white noise voltage was fed into the embedded PZT actuator in Leg A of the tripod as shown in Figure 1.5. The potential difference in the PZT deformed the leg and produced vibration of the ACTEX tripod system. The produced vibration was sensed at the nearly colocated and colocated PZT sensors as shown in Figure 1.5. Leg A Nearly Colocated Colocated Sensor Sensor Figure 1.5: The Embedded PZT Sensors and Actuators 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 1.6 depicts the Leg A nearly colocated sensor to the actuator transfer function. In the top left panel of Figure 1.6, the transfer function of experiment 1 run 5 is shown. In the top right panel, the corresponding tem perature profile is given. The therm ostat location is shown in Figure 1.7. In the lower left and right panels of Figure 1.6, the transfer function of experiment 1 run 13 and the corresponding tem perature profile are shown. Similarly in Figure 1.8, the Leg A colocated sensor to actuator transfer function for experiment 1 run 4 and run 10 are compared. In each case, the transfer function Z(k) = Y(k)(X(k) is calculated as follows: Y(k) = - L j 2 Vs(ti)e~jK2^ { iyt i=o X(k) = j r V ac(ti)e-jki2*/Nc)i t i=0 (1.5) 3 § 8 20 40 60 80 100 hertz O 60 E -20 2 4 6 8 thermostat number I 1 2 f 8 £ 4 100 % o 60 - v 1 s 1 20 1 I 8. a M s K a A A _ ~ a E -20 © ” I " -60 20 40 60 80 100 hertz 2 4 6 8 thermostat number Figure 1.6: Nearly Colocated Transfer Function k = 0,1, • • •, Nt — 1 where Vs(ti), Vac{ti) are the Leg A sensor and actuator voltages at tim e t,- = iAt. In this experiment the samping interval is A t = 0.001s and 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 1.7: T herm ostat Location. 100 1 o 60 1 20 * E -20 -60 A A / * ' > * V v v 20 40 60 80 100 hertz 2 4 6 8 thermostat number 20 40 60 80 100 hertz 100 O 60 9 ratui 1 0 o & E -20 e -60 thermostat number Figure 1.8: Colocated Transfer Function Nt = 1000 is the number of samples. Thus for each k , |F(/:)| and |^f(A:)| give the magnitudes of the system responses at frequency fk A t- Nt ' ( 1.6) 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and the corresponding m agnitudes axe and |X(fc)j. Figure 1.6 and Figure 1.8 indicate th at observable variations exist in the system response due to tem perature and other environmental effects. Therefore we need to derive a physical model of the ACTEX structure and form ulate an inverse problem to track any systematic changes as it moves into different space environments. This will enable us to translate the observed variations in the system response into variations of structural properties such as density, stiffness and damping factors. This will help us understand these environmental effects and improve the design and control of similar space structures in the future. 1.3 Thesis Overview The m ain objective of this thesis is to provide a com putational foundation for es tim ating parameters in the ACTEX distributed param eter system. This will help us sim ulate the ACTEX response and answer practical questions relating to the system atic changes of the ACTEX structure. In chapter 2 we formulate a PD E model of the ACTEX structure. Hamilton’s principle of least action is used to derive the equation of m otion for this system. In chapter 3 we establish well-posedness of our model for the ACTEX structure. We cast our system of equations in a weak form and provide a computational founda tion for simulation. In chapter 4 we formulate the inverse problem to estim ate the model parameters using a set of observations of the state of the system. Theoretical convergence of our param eter estimation scheme is then established. In chapter 5 we evaluate the validity of our model by comparing numerical simulations to actual experim ental data. We present and discuss the results of numerical studies for the param eter estimation problem involving the ACTEX distributed parameter system. In chapter 6 we discuss the future directions of the work presented in this thesis. In particular, we discuss the practical implications of param eter estimation techniques in controlling space structures with active materials. 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 2 Mathematical Model In this chapter we derive a m athematical model of the ACTEX structure. We seek a model of the ACTEX structure that can capture its main dynamics, yet be simple enough to facilitate further analysis. We use physical principles to derive our model. In section 2.1 we discuss the physical assumptions behind our model. In section 2.2 we discuss the energy formulation for constrained mechanical systems and find expressions for the energies for the ACTEX tripod. In section 2.3 we derive the unforced and undam ped modeling equations for the ACTEX structure. In section 2.4 we incorporate damping and actuator dynamics. In section 2.5 we derive models of the density, stiffness, and damping for the ACTEX structure. 2.1 Model Outline In this section we discuss the physical assumptions made in modeling the ACTEX structure. We model the structure as a tripod, as shown in Figure 2.1. The tripod consists of three legs. Each leg is clamped to the bottom plate and rigidly attached to the top plate. We model each leg as two one-dimensional Euler-Bernoulli beams. Our Euler-Bernoulli model assumes only transverse deformation (see Figure 2.2). For each leg, we model the transverse deformation in the Y{Z{ plane as a superposition of two one-dimensional Euler- Bernoulli beams with deformations U { and u ,- in local coordinates. The deformations are thus functions of the A T , coordinates along the leg, and we assume the deformations are small. We also assume th at there is no torsion present. 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Leg B Lei Top Plate Bottom Plate -5 20 12 -1 0 x Figure 2.1: The ACTEX Tripod (model only) Vi(X,t) T op P late B ottom P late Figure 2.2: The ACTEX Leg Deformation Throughout this thesis, we refer to the deformation of Leg A as Ui and Vi, the deformation of Leg B as u2 and u2, and the deformation of Leg C as uz and vz- Now consider the cross section of each leg. Each leg is identical and has a hollow cross section. The PZTs are symmetrically embedded as shown in Figure 2.3. 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PZT Actuator PZT Sensor Graphite Leg Hollow Inside Figure 2.3: The ACTEX Leg Cross Section The four black rectangles, shown at the bottom left com er of Figure 2.3, represent the embedded PZT actuator where voltages are fed into to control the legs. The white rectangles represent the embedded PZT sensors and are used to sense the vibrations in each leg. We take into account the hollow cross section by modeling the moment of inertia as shown in Figure 2.4. We can think of the inner rectangle of the cross section as having a negative moment of inertia. The moment of inertia determines the stiffness and damping properties of the hollow legs. We will derive expressions for density, stiffness, and damping for the hollow legs in Section 2.5. It * 2 * * » • Figure 2.4: The Moment of Inertia for Hollow Leg 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The top plate is modeled as a rigid body. We consider translation, of the plate due to end deformations u,-(/) and ut '(0 as shown in Figure 2.5. Top Plate / < Y . X . I Figure 2.5: Translation of the Top Plate We assume that the plate translates only in the lfZ t - plane. We also assume that the plate rotates only in the X{Z{ and X{Yi plane as shown in Figure 2.6. Top Plate Y . Figure 2.6: Rotation of the Top Plate The angles of rotation axe denoted by &xz and Ox y - We assume no rotation in the YiZi plane because we do not consider torsion in our model (dyz = 0). 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. As a careful reader will notice, we have modeled the interaction of each leg with the top plate independently of each other. The rigidity of the top plate forces additional geometric constraints into our model, as shown in Figure 2.7. Top Plate □ Leg End Figure 2.7: Top Plate Boundary Constraints In Figure 2.7, the end deformations of Leg A, ui(l) and vx(l) axe depicted. Since the plate is rigid, the end deformations of Leg B, u2(l) and v2(l), and Leg C, u3(/) and vz(l) are forced to be in sine with Leg A deformations u x(l) and vx{l). The rigid plate also forces the angle of orientation of Leg B and Leg C with respect to Leg A to be fixed. The angle of orientation 6 is depicted in Figure 2.7. The modeling equations are derived using the energy method. For this, we will need to find expressions for the potential and kinetic energy of the three legs and the translational and rotational kinetic energy of the rigid plate. We will first use the energy methods to derive the unforced and undamped equation of motion. Then we will incorporate damping and actuator dynamics into our system. 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.2 Energy Formulation The dynamics of the ACTEX tripod will be derived using Ham ilton’s Principle. Ham ilton’s Principle postulates that of all the possible paths along which a dynami cal system may move from one point to another within a specific tim e interval [f i, t2], the actual path w followed is a stationary point of the functional J = f*2 (T(w) — V(w))dt (2.1) Jt i where T is the kinetic energy, V is the potential energy, and w is in a Hilbert space H such that it possesses the necessary smoothness intrinsic in writing the action integral J and the essential boundary conditions imposed upon the system. Moreover the equation of m otion is obtained by setting the G ateaux differential of J to be zero 8J(w\r\) = 0 (2.2) where t j is a test function in H and 5J is defined as 5J(w; rj) = lim + arf) — J («;)] (2.3) where a is real. Since we model the top plate of the tripod as a rigid body, the functional J for the ACTEX structure is subject to additional constraints (see Figure 2.7). Therefore the problem is to find w which is a stationary point of J = f (T (w )—V(w))dt (2-4) J ti subject to a constraint g{w) — 0. (2.5) For our formulation for the ACTEX structure, we will use penalty function method (see Reddy [23, p.116-119]) which reduces the above constrained problem (2.4)- (2.5) to one without constraints by introducing a penalty term associated with the 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. constraints. As applied to the problem in (2.4)- (2.5), the technique involves seeking the stationary point of a modified functional obtained by adding a quadratic term associated with the constraint in (2 .5 ) such that the constraint is satisfied in the least square sense: I = / V (*») - V(w))dt + i I*’ 7 (A M ) 2 dt (2.6) Z Jti where 7 is a positive real number. Therefore the constrained equation of motion is obtained by setting the Gateux differential of I to be zero 6I(w ;r])= 0 (2.7) We must therefore find expressions for both the kinetic and potential energies of the ACTEX tripod system and also find the quadratic term associated with the constraints. We first find expressions for kinetic and potential energies. T he potential energy of the ACTEX tripod system is made up of the internal strain energy of the deformed legs. In this discussion, we will consider only the internal strain energy. If we assume that the legs undergo small linear deformation, the linear theory of beams and rods allows us to write an expression for the potential energy of the deformed legs (see Gere [24, p.599-611] and Meirovitch [25, p.368-377]): Here EI{pc), the same for all legs, is the bending stiffness or flexural rigidity. This is the product of Young’s modulus of elasticity, E, and the area moment of inertia, I(x) (see Gere [24, p.812]). The total kinetic energy of the system is made up of the kinetic energy of the legs plus the kinetic energy of the top plate, both translational and rotational. For 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. an element dx of the undamped legs, the instantaneous linear kinetic energy dT[eg is given by 'duj dt , + dx (2.9) where p(x), the mass density function per unit length, is the same for all legs. Integrating over the total length of each leg and summing over all legs, we find that the total instantaneous kinetic energy T/eg is ™ = * * ( ( % ) ' + { % ) ) * (2.10) We now concentrate on the kinetic energy of the top plate. This kinetic energy includes both the translational and rotational components. In the spirit of Banks and Smith [26, p .1-19], let X, Y, and Z designate an inertial cartesian coordinate system parallel to the ith leg’s neutral axis and Z ,- in the undeformed state as depicted in Figure 2.8. Figure 2.8: The ACTEX Leg and the Rigid Plate Coordinates 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Let Q denote the mass center of the end plate. Let T, denote the plane passing through Q and perpendicular to the X axis. Let P' denote the tip of the z'th leg and P the orthogonal projection of P' onto T9. Let < z t - denote the distance between P and P '. Let c ,- denote the length of the vertical component (in the Y direction) of PQ and let 6, - denote the length of the horizontal component (in the Z direction) of PQ, as depicted in Figure 2.8. Now let X{, Yi, and Zi designate a second cartesian coordinate system, attached to the top plate, with origin at point P as shown in Figure 2.8. In the zth leg’s undeformed state the axes X{, Yi, and Z, are parallel to the X ,Y , and Z, respectively. The motion of the rigid body can be described in term s of this new coordinate system. Let d . Q (t) denote the position of the center of mass of the end plate with respect to the inertial coordinate system. Let ex,eY, &z and exi,eYi, eZi denote the unit vectors in coordinates of the inertial frame and the zth leg’s local coordinates respectively. Then the position of the center of mass of the plate can be written as dQ(t) = lex + Ui(t, l)eY + Vi{t, l)ez +a«ext + 6,-eyj + CieZi (2 .1 1 ) Now since eZi eY i eZi vary with t and with respect to ex&yez we need to find the relationship between ex, ey;. eZi and exeYez . In general, when a coordinate system X Y Z is rotated through differential angles 0xY ,0YZ, and Oxz, a new coordinate system X,YiZ{ is obtained. The m atrix R = ( 1 —OxY 0XZ ^ Ox y 1 —Oy z —6xz 0YZ 1 ) transforms a vector expressed in the frame X{YiZi into an equivalent vector in X Y Z coordinates. For our model the differential angles 0YZ = 0 because we do not consider torsion. If we further assume the differential angles are small, we can approximate them as _ dui(l) 0xr = S T ' 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. @xz — dvi(l) dx ' (2.12) If we now let = (ex, eYez ) and tE h = (ex.eV.ez.) then we have = R T^ 0 where R is given by, f 1 dui(l) dvdl) \ dx dx R = dui(l) d x 1 0 \ dvi(l) dx 0 1 / Using this coordinate transform ation we can rewrite dp as, dQ(t) = + + ^w,(/) + O f - C i— i e X dx ' J,‘ ) dx du{(l) dx + & ,-) eY + ( Vi(l) + a,i-V ^ —C i)ez and its time derivative d t* * v TtdQ{t) -( dx + + ( 0 + + O -i dtdx d2uj(l)\ dtdx J d2vi(iy dtdx eY ez (2.13) (2.14) (2.15) (2.16) The total translational kinetic energy of the plate is given by squaring the derivative of dQ r r w - + / duj(t, I) d2Uj(t,l)\ 2 \ dt Q l dtdx J 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where M is the mass of the plate. If we assume the plate undergoes rotation through small angles, the standard theory of rigid body mechanics allows us to write the expression for the rotational kinetic energy of the plate as follows (see Goldstein [27, p. 149]): r r ’m = i t {/. ( ^ ) 2+/, ( ^ ) 2 } ( 2 .1 8 ) where Iy and Iz are the moments of inertia of the plate for rotations in and X{Z{ planes respectively. The total kinetic energy of the plate is given by Tp(t) = Tp (< r) + T^rot\ We have found expressions for the combined kinetic and potential energies of the ACTEX tripod legs and the rigid plate: T(t) = T,'S( t) + T p(t) V(t) = (2.19) Because we are only considering internal strain energy and regard the plate as rigid, there is no contribution to potential energy from the plate. We now concentrate on incorporating the constraints for the ACTEX tripod system. T he rigidity of the top plate forces additional geometric constraints into our model, as shown in Figure 2.7. These constraints leads to the following equations dux(l) du2(l) Rdv2(l) = St dt dt P m jl) d2u2(l) d*v2(l) = dtdx dtdx dtdx dvi(l) _dv2(l) , „du2(l) ~ ai— a - § r +(3- § r = 0 y » i ( 0 a V ( /) , t.& uili) = n dtdx dtdx ■ * "p dtdx 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. du2{l) du3{l) dt dt d2u2(l) d2u3(l) dtdx dtdx dv2(l) dv3(l) dt dt d2v2{l) d2v3{l) dtdx dtdx = 0 = 0 = 0 = 0 (2.20) where a = cos{9) and (3 = sin{9), 6 is the orientation of the leg B and C with respect to leg A as shown in Figure 2.7. Now let A = (a,-j) be 8 x 12 m atrix such that aX iX = 1 , a i t5 = —a , aX j = —(3, a 2,2 = lj <22,6 = ~Oi, 0-2,8 = ~ P i <23,3 = 1, <23,5 = P , 03j = —O', a4j4 = 1, a4i6 = P i < 2 4 ,8 = — OC, 0 5 , 5 = 1 , 0 5 ,9 = — l j <26,6 = 1 5 O6 ti 0 = — 1 , 0 7 , 7 = 1 , 0 7 ^ 1 = — 1 , O s,8 = l j < 2 8 ,1 2 = — 1 , then the equation of constraints (2 .2 0 )becomes Aw(l) = 0 (2.21) where w(l) = f t («,(/), ■■■, u3(t), v3(/), *£?■). The modified Ham ilton’s functional can be w ritten as I = /V - V)dl + i P (g (w )f dt (2 .2 2 ) Jti 2. Jti where g(w) = YAw and T is a diagonal 8 x 8 m atrix with parameter 7 , • which are positive real. In this section, we have found expressions for the kinetic energy T, potential energy V, and the quadratic penalty constraint term g. In the next section, we derive the undamped and unforced equations for the ACTEX tripod system using the energy methods discussed in this section. 2.3 Equation of Undamped Motion In this section, we derive the undamped and unforced equation of motion for the ACTEX structure. Since the damping and the actuator dynamics we consider for the 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACTEX structure can not be derived from a quadratic functional, the energy meth ods can not be used to derive the equation of damped and forced motion. However, we will incorporate damping and actuator dynamics from physical consideration in the next section. For an arbitrary tim e interval [to, £1], consider the action integral 1 = [ \ t (w) - V(w))dt + - [ 2 (< 7 ( » ) 2 dt (2.23) J t i I J ti where w = (wx(t, •), w2(t, /)) is given by ^ lfo * ) = (u i(x,t),v l(x ,t),---,u 3(xit),v 3(x,t)) (2.24) r* n _ f n\ & ul{l) dvx(l) , n du3(l) m dv3(l)\ ,n w2(t,l) - 1 ^ ( 0 , dx ,v i(0 , d x , ” - , « 3 ( 0 . d x , w s ( 0 . Qx J(2-25) possesses the necessary smoothness and essential boundary conditions. We assume th at our beams axe truly cantilevered and satisfy the essential or clamped boundary conditions at x = 0 -< o ) = ^ = o ,3 (0 = * g p . = o “ 3 (0 ) - = 0 “3(0) = = 0 (2'26) A class of functions possessing the necessary smoothness and the essential boundary conditions is the Hilbert Space H = f[ #£(°’ 0 x ^1 2 (2 ‘ 2 7 ) 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where #£(0, I) is the usual Sobolev space of £ 2 (0 , 1) functions with first and second derivative in £ 2 (0 , I) satisfying the essential boundary conditions (2.26). We denote the function that minimizes J by to and form a one param eter family of comparison functions, parametrized by a and defined as W (t1 x,a) = w(t, x) + ar}{t, x) (2.28) where 77 = (171 (t, ■),r}2(t, I)) is given by Vi t o ) = (2.29) th(M) = The family W (t, x, a ) is chosen so that W ( t , - , a ) e H (2.31) Now the resultant Hamilton’s action integral takes the form, / = J ‘2{Tlel( W ) - V ,',( W ) + T p( W ) } d t + ± j ‘\ g ( W ) ) 2dt (2.32) where Tieg is the kinetic energy of the deformed legs, Tp is the plate kinetic energy, both translational and rotational, Vieg is the potential energy of the deformed legs, and g(w) is the constrained penalty term . A necessary condition for a stationary trajectory w is that 81(w] 77) = = 0 (2.33) a=0 We now concentrate on calculating 8l(w, rj). Let I(w; 77) = I^w , 77) + I2(w, 77) + I3(w; 77) (2.34) where A t o 77) = [ t2{Tleg( W ) - V leg(W )}dt Jti 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I2(w; rj) = f 2 Tp(W)dt Jti h(w \n) = \ £ ' (s(W )f it. (2.35) Therefore we get 8I(w; 77) = SIi(w, 77) + 8I2{w■ 77) + SI3(w; 77) (2.36) We first calculate 8Ix(w; 77). We recall *U «0 - 5§j£'«*){(|£ ), + ( ^ ) ,}‘ f a (2 - 3 7 > w « ) - { ( § £ ) ’ + (I? )’ }* and if we use W (t, x,a ) = w(t, x) + 0 : 77(2, x) (2.38) we get w = ltJ^l{w+ a f)2 + {^+ a w))d x ^ K . W - 5 g /o 'B 7 w { ( ^ + “0 ) 2 + © + “0 ) 2 } ^ « ) Let Ii(w,T}) = l i l\w;r}) + l[2\w;T 7) where 4 V ; * ? ) = f 2Tle g {w)dt Jti l[2\ w i 77) = - F Vleg(W)dt. (2.41) Jh Therefore 8I(w\ 77) = <£/^(u;; 77) + < L /j2^(u;; 77). We will first calculate 77) 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - ij:£**(% £‘St)" We integrate by parts in t and we get S 4 l>(w;n) = £ p ( x ) ^ t < + ^ ? j dxdt (2.42) + ' dui . dvi r*) (2.43) If we further assume that 0,-, tpi vanish at ti and <2 we get S I ? \w ,V) = - ± J * J j { x ) (^L<t,i + y ± i ,^ d x d t . (2.44) We calculate Sl[2\w ; 77) as follows s i ? \ v w ) = ^ 4 2)L = 0 ■ + M ) and we integrate by parts in x twice, we get +g £ {£ (® C » )f£ ) * + s f ® 7 ^ ) *}[* - §r {( m x )^) § + (^ 0 ) g}[ * • (2 ^ 6 ) Now if we further assume that d > ,- and satisfies essential boundary condition at x = 0 , we get dx 2 \ d x2J si?\«,-,ri) = - E / o' £ { ^ 5 + * } * = * dt + 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - t £ { l 7 m & ! 2 M ! i + H r o « > » g 2 } , M Now if we let, * ( » ) = 7 ( 0 ^ ) (2-48) we can rewrite 77) as si™ = ~ i t f o J , ' { ^ { m ( x ) ^ ) * i + ^ { T i ( x ) ^ ) * ‘} dxdt +rfih{w) (2.49) where * ( * . 0 = ( a w . * ( 0 , M l), ^ ) ( 2 . 5 0 ) Now we calculate 5l2{w\ri) where I2(w,r})= f*2 Tp(W)dt. (2.51) Jt\ We recall ( dvi{t,l) d2u i( t,l) \2 + — 51------- 1 " ar dt 1 dtdx t fd u i(t,i) _ d2vi(t,iy + d i ~ + a i~ m i 1.2. + 2 ' 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We expand each term in (2.52), we get r , ( « ) = + + ( M ! ^ ) ) 2 + a ? ( ^ M ) 2 + ( t )2 -K ^ )2 } ( - ) If we let u n _ (.. f,\ d tii(0 /n dvx(l) m du3(l) /n ^ ( O V o * ^ W2 (^5 0 ( M l(0> ^ JU l(0> ^ 5 * * " 7 ^ 3(^ )1 ^ 5^3 (0 ? ^ I (2.54) Then we can write (2.53) as Tp(w) = vj^A.W2 (2.55) where A is a block diagonal m atrix consisting of A ,-, i = 1,2,3, where ( M Mai 0 0 ^ M n - T. -4- M n 2 . 4 - M b '* 0 M h r - A ,- = V M Mai 0 0 Mai Iz + Ma} + Mb} 0 Mbid 0 0 M Ma{ 0 Mbid Mai Iy + Ma} + Mb} Therefore we get J rt2 j. ' (ii> 2 + cwft) A (u> 2 + a^ 2) u dt and 6I2(w , 77) = d (2.56) (2.57) Or=0 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and if we integrate by parts in t and assume that rj2(t,l) vanishes at ti and t2, we get SI2(w;r}) = 772 Aw2 (2.59) We now calculate 5I3(w , ij). We recall h (w ' rl ) =z\ j t dt (2.60) 1 'H where g{w) = TAw2 (2.61) and we calculate 5I3(w , t ]) similar to SI2(w]r]), we get SI3(w,r}) = r\^Bw2 (2.62) where B = A TFA. Now using equations (2.44), (2.49),(2.59), and (2.62) we get SI(w; 77) = 6Ii(w,rj) + SI2(w;r}) + 5 I 3 ( w , t } ) (2.63) = ir C S l K x ) + d x d t * + j? * ■ } ix i* +772A^ 2 + t j2B w2 (2.64) We derive the equation of undam ped and unforced m otion be setting 8I[w; 77) = 0 for arbitrary 4> i,ipi and r}2. Therefore we arrive at the following equation of undamped and unforced motion for the ACTEX structure: 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. dt2 d x 2 \ The boundary conditions at x = 0 are d2Ui d 2 f ^ r sd2Ui\ P^ ~ d P + f a ? \ ( ^ l h ? J = **>?£+£fa(«>f£) 0 (2.65) dui dvi m = — — = Vi = — = 0 dx dx (2.66) where we assume th at legs are clamped at the bottom plate. The boundary condi tions at x = I is given by (A + B) r 9 2ui(i) dt2 - £ / ( , ) I 93ui(0 dt2d x m > ? g p a2vi(D at2 - EI(l d 3vi{l) dt2dx £ / ( / ) % P- d 2u2{l) at2 - E I ( l ) S 0 2 a 3u2(i) dt2dx + 92V 2{1) at2 - E I ( l ) ^ 0 i d 3v2 (I) dt2dx E I ( l ) ^ 2 d 2u 3(l) at2 d3u3{l) dt2dx d 2v3(i) at2 -E I (l) ^ a 3v3(i) - dt2d x - E I ( l ) ^ 0 1 _ = 0 . (2.67) Thus, the undam ped model for the ACTEX structure is given by Equations (2.65)- (2.67). In the next section, we incorporate damping into our model. 2.4 Equation of Damped Motion In this section we will incorporate damping and actuator dynamics for the ACTEX model. To introduce internal damping into our model we adopt the Kelvin-Voigt hypothesis (see Meirovitch [25, p.484]). This states that stress is proportional to a combination of strain and strain rate. We m ay introduce strain rate into the 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. equations of motion through, the moments. For an Euler-Bemoulli beam the moment is given by d2 u - M (x, t) = E I(x) (2.68) which can be used to write the undamped Euler-Bemoulli beam equation ^ ~ d W ’ + dx2 = F W (2-69) The Kelvin-Voigt hypothesis implies the following form for the moment M (x, t) = T l ( x ) ^ + c T / ( x ) ^ . (2.70) Furthermore we assume the following forcing term for the actuator dynamics (see Banks [13, p.94-96]): = (2.71) where /,-(i) is the voltage fed into the embedded PZTs in the legs, k f is the actuator constant, and Xac determines the location of the actuator. Therefore altered equa tions including damping and actuator dynamics for the ACTEX structure becomes -r , & ( ~ r , , \ p > d t 2 d x 2 V ^d x2) d x2 \ C D ^ d x 2d t) = 2 k f J j k (X^ m ) ,d 2Vi , d 2 , d 2 l ~ , N a 3 ® ; p^ dt2 d x 2 v f o 2 J d x 2 vC D^ d x 2d t) r\ 2 = 2 * f a J I ( W i ( 0 ) ( 2 .7 2 ) along with the essential boimdary conditions at x = 0 dui dv; = ° < 2 - 7 3 > 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and natural boundary conditions at x = I (A + B) r 92 «i(o ■ at2 -E I{1 )3 ^ ± 1 ■ 1 a3 «i (i) d t2d x cDm a i n § d 2vi (I) d t2 - E m 2 ^ 1 a3vt(i) dt2d x E I (l)? g P cD/ ( 0 f e # a2u 2(i) at2 - E m ^ g P - - c o m & g d 3u2(i) d t2d x + E I (l)* W - + cDm ^ § d 2V2(D d t2 - E IW S 0 1 -C D n o l s f d 3v2(l) d t2d x EI(l C D l(l)^ & d 2u3(i) dt2 - E I ( t ) ^ P - c D m ^ P d 3u3U) d t2d x E m ^ g P - cDm ^ § d 2v3{l) d t2 - cdI ( 1 ) ^ P d 3v3{l) . d t2d x - E l ( l) ? g P . cdI(1)tS § J = 0 (2.74) Model equations (2.72)-(2.74) take the form of six one-dimensional Euler-Bernoulli beam equations. Two equations are used for each leg to model the transverse de formations in local coordinates. In the next section we model the density, stiffness, and damping parameters. 2.5 Modeling Parameters In the last section we derived the modeling equations for the ACTEX structure. In this section we model the coefficients of the PDEs: /5(x), E I(x), C £ > / ( x ) , and k f . We model these coefficients as piecewise constant functions (see Figure 2.9). We use piecewise constant functions because we have PZTs embedded in the legs of the tripod as shown in Figure 2.10. These embedded PZTs passively contribute to the density, the stiffness, and the damping coefficient for the hollow legs. 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Leg Density with PZT Leg Density without PZT (0 c a X 'S Leg Axis Figure 2.9: Piecewise Constant Parameters z Colocated Sensor Nearly Colocated ^(X > Sensor ^ — J!!i Actuator Figure 2.10: Characteristic Function for the Embedded PZTs Lets define the characteristic functions as follows ✓ J 1 ,ncs < X <C S ^ .n c s 1 = 1 0 else / Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. *-<*> = { o = { ; * ~ < ^ * ~ } (2-75> which determine the location of the piezoceramic embedded in the legs of the ACTEX structure (see Figure 2.10). Following Banks [13, p.93-117], we write p { x ) Pbe-A-be “f* (.Ppe P b e) X ( A nC)S^ nc s -(- A Cja^ ClS -f- AacXac) E I ( x ) = E b e lb c "f* ( E p e E b e ) (I n c ,s X n c ,s T E ,s X c ,s Ia.cX .ac) fie I be ”t~ (^-Z),pe CD fie ) X ( In c ,s X n c ,s “f - I c ,s X c ,s "f“ la c X a c ) kf = — ( A t e + Aac) Evedz\ ( 2 .7 6 ) where pbe-, Ebe, and c^fie are the density, the modulus of elasticity and the Kelvin- Voigt damping constant respectively, the same for each leg, and A b e is the beam cross-sectional area and Ib e is the beam area of moment of inertia. Similarly, ppe, Epe, and C £ > iP e are the piezoceramic density, the modulus of elasticity, and the Kelvin- Voigt damping coefficient, respectively, while An C y S and Inc< s, AC tS and IC tS , and Aac and I ac are the cross-section area and the area moment of inertia for the nearly colocated, the colocated sensor, and the actuator embedded in each leg. All the param eters are the same for each leg. In the above equation for k f , is the piezoelectric constant. These param eters in (2.76) along with the governing set of equations (2.72)- (2.74) in the previous section are used to model the vibration of the ACTEX tripod. In this chapter we have formulated a PDE model of the ACTEX structure. Ham ilton’s principle of least action is used to derive the equations of motion for our system. The model takes the form of six one-dimensional Euler-Bemoulli beam equations. Two equations are used for each leg to model the transverse deforma tion in local coordinates. The model param eters are the beam density, the flexural rigidity, and the damping coefficient. We have used piecewise constant functions to describe the beam density, the flexural rigidity, and the damping coefficient to 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. include the active and passive contributions of the embedded PZTs in the internal and external bending moments. A careful reader will notice that the modeling PDEs involve talcing fourth deriva tive of U { and vt and second derivative of piecewise constant functions p(rc), E I(x), cdI(x). In order to address smoothness issues and to accommodate the discontinu ities in our parameters, we cast our model in a weak form in the next chapter and study well-posedness of our model. 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 3 Well Posedness In this chapter, we turn to issues of well-posedness and approxim ation for an abstract second order system m otivated by the model of the ACTEX structure derived in chapter 2. At the end of the last chapter, we noted th a t our modeling equations with piecewise constant param eters are not well-posed because they involve taking derivatives of piecewise constant param eter functions. In order to accommodate the discontinuities in the param eters, we need to cast our set of PDEs in the weak form. The weak form also provides an approximation framework for simulation. In section 3.1 we discuss the weak form and state m ain definitions operator theory. We then state the results for well-posedness of the abstract second order system. In section 3.2 we apply the theoretical framework of section 3.1 to prove well- posedness for the ACTEX modeling PDEs. In section 3.3 we discuss approximation results and use them to develop a computational foundation for simulation. In section 3.4 we discuss the Spline Galerkin method and show how this method can be used to approximate the ACTEX state space. 3.1 Weak Formulation We focus our attention on second order systems of the form d2w dw ~ § F + A * W + A lW = m w{0) = wo dw(0) dx = W 1 (3.1) 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where Ai ajid A i axe generalized stiffness and damping operators, respectively. We consider a variational or weak formulation here. The power of such a framework lies in its well-posedness and its advantages for computational approximations. We shall find that the concept of a sesquilinear from is very im portant to our analysis of the ACTEX distributed param eter system. The theory of sesquilinear forms is analogous to that of linear operators (see Kreyszig [28, p.82-90]). D efin ition 3.1 (Sesquilinear Form ) Given a complex Hilbert space V over the field ?C , a i V x V — tlC is a sesquilinear form if f> a (0, 0 ) is linear and 0 — ► a (0 , 0 ) is conjugate linear. D efin ition 3.2 (V -C ontinuous) The form cr is V —continuous if there exists a k > 0 such that for all 0 , 0 £ V , k ( 0 ,0 ) l < fc |0 lv M v - (3.2) The form cr is symmetric if cr (0, 0 ) = cr(0,0 ) for all 0, 0 £ V. T he most important fact about sesquilinear forms is th at each sesquilinear form <x(-, •) on V corresponds to a unique linear operator A £ L(V, V*) given by, a (0 ,0 ) = (A 0,0)vr.iv 0,0 € V (3.3) Conversely, if A £ T(V, V "*) is given then the above equation defines a sesquilinear form on V. We will see that the potential energy of the ACTEX system will generate a “stiffness” sesquilinear form and a corresponding “stiffness” operator Ai. Given a Hilbert space V, a continuous sesquilinear form cr(-, •) on V, and / £ V*, the weak problem is essentially finding w £ V such that cr(w, 0) = / ( 0 ) for all 0 £ V. Since there is a one-to-one correspondence between continuous sesquilinear forms on V and linear operators from V to V m , there is a unique linear operator A such that a(w, 0) = (A tu ,0 )v ,v , V0 £ V. (3.4) Therefore, to is a solution to the weak problem if and only if w £ V and Aw = /, where / £ V m is given. Now we would like to know if Aw = f implies that w solves a partial differential equation. In general V* cannot always be identified with a space 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of distributions on a domain G in i f 1 . For example, if we take V = H l (G), where we seek a solution w to the Neumann problem i.e. first derivative of w vanishing at the boundary dG. Then the difficulty is that the space of distributions C£°(G) is not dense in V. Therefore to get around this difficulty we define a Gelfand triplet of Hilbert spaces. For a general introduction to Sobolev spaces and the embedding theorem, we refer the reader to Wloka [29, p.11-42] and Showalter [30, p.27-49]. The space V is usually contained in a larger Hilbert space H. We assume th at there exists a Hilbert space H with the property that V is densely and continuously embedded in H. For such an embedding it may be shown that (see Wloka [29, p.165,261]) H m embeds densely and continuously in V*. If we now identify H and H* through the Riesz map, we get the sequence of continuous embeddings V «-+ H = H* V* (3.5) that V,H, and V* form a Gelfand triplet with pivot space H. Although V and V* are not identified through the Riesz map, we will utilize the duality pairing (-, -)v ,v given by the extension of H inner product from H x V to V* x V. T hat is, for v* 6 Vm , hn 6 H, and hn — > ■ v* in V*, we define for all v £ V v*(v) = (v*, v ) v ,v = jig i (^n, v)h (3.6) In the context of a Gelfand triplet, we say that D efinition 3.3 (V -C oercive) A sesquilinear form a on V is V —coercive if there exists c > 0, and \q > 0 such that for all c f> € V Rea((f>,(f) + X0 \(f> \2 H > c\^)\y . (3.7) D efinition 3.4 (V -EU iptic) A sesquilinear form a is V —elliptic if it is V —coercive and A o = 0 . W ith this definition, we can find a requirement which will guarantee th at Aw — f implies a partial differential equation is solved in the sense of a distribution. We state this as a theorem (see Showalter [30, p.55] for a proof): 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T h e o re m 3.1 I f <x(*,-) is V-coercive, then D(A ) = {« 6 V : A u E H } is dense in V , hence dense in H. Note that u E D(A) or A u E H if and only if u E V and there is a K > 0 such that |a (u ,u )| < K\v\h , v E V (3.8) Therefore the operator generated by V-coercive sesquilinear forms corresponds to partial differential equations because D(A) C H is dense in V and the space of distributions C(f(G) is dense in H. The above framework is for the static, ellip tic boundary values problems involving partial differential equations. To move to evolutionary systems we will rely heavily on semigroup theory. We begin with a brief review of some standard results from semigroup theory. For an introduction to analytic semigroups, we refer the reader to Pazy [31, p.60-68]. Let A e denote a sector in the complex plane Ag — {z E C : —0 < arg z < 8} D efin itio n 3.5 (A n a ly tic S e m ig ro u p ) A family of operators \T (z) E C(H) : z € Ae} is called an analytic semigroup of linear operators in Ag, if (i) For w E H, T(z)w -+ w as z 0, z € Ag. (ii) T(z + s) = T(z)T(s) for every z,s E Ag. (Hi) For every w E H, the function w — y T(z)w is analytic on Ag. Associated with every analytic semigroup is an infLntesimal generator. D efin itio n 3.6 (In fin ite sim a l G e n e ra to r) An operator A : D{A) C H — > • H called the infinitesimal generator of the semigroup T (t) D{A) = E H : lim ~ W and . T(t)w — w . n , .. Aw — lim -------------- for w E D(A) *— ► 0 t 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The infinitesimal generator is the connection between semigroups and the abstract evolution initial value problem ^ = **> u>(0) = wo (3.9) Then solution to the initial value problem (3.9) is given by w(t) = T(t)wo. Now we will state the m ain theorem that allows us to establish well-posedness for our model of the ACTEX structure. T h e o re m 3.2 Let V, H be Hilbert spaces with V < — y H = H m e — > ■ V m and suppose that the sesquilinear form a : V x V — > • C is V-continuous and V-coercive. If we define A : D(A) C V H as a{w,tpi) = (—Aw, ip)n w € D(A),ip € V then D(A) is dense in H and A is the infinitesimal generator of an analytic semi group T(t) G C(H), t > 0 such that w(t) = T(t)wo solves = * * ) tu(0 ) = wo furthermore, for every wq € H there is a solution, w e e ([o, o o ) , H) n c ° ° ((o, oo), H) and for each t > 0, w(t) G D(AP) for every integer p > 1 . Proof: By the Analytic Generation Theorem (see Appendix A.3), it suffices to show th at <ta = c r(< f> , ip) + A(0, ip) is V-continuous and V — elliptic. T he V —coercive < r(-, •) satisfies Rea(<p, < p ) > c\(p\\r - \ o\< P \2 h Re<r((p,<p) + \ q \ < P \ 2 h > c\ip\\ (3.10) 40 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Thus a\Q is V — elliptic and V-continuous. The continuity of < T \0 follows from its definition. Therefore, a\0 satisfies conditions of the Generation Theorem A.3 for analytic semigroups in the Appendix A. Thus, for A > A o > 0, A — XI generates an analytic semigroup and hence we have proved the above theorem. □ As we discussed at the beginning of this chapter that we will focus on distributed parameter systems that may be formulated as an abstract second order system d2w dw ~ W + A*~di + AlW ~ m w{0) = w0 dw( 0 ) dt = wi (3-11) where A 2 is the generalized damping operator and is the generalized stiffness operator. Now consider this equation in a weak or variational sense defined via a bounded sym m etric sesquilinear form < X i : V x V — > C and a bounded sesquilinear form a2 : V x V — ) ■ C. As before there exist Ai, A 2 € C(V, V m ) such that 0i(<£>VO = (Ai<f>,if>)v,v (3.12) 1>)v,v- (3.13) We seek solutions w(t) 6 V satisfying + < T 2 +<Tl = w( 0) = wo dw( 0) dt = wi (3.14) where / € L2 ((0, T ) , V*). Now the well-posedness for this system can be established by writing this second order system as an equivalent first order system. Let us define H = V x H and V = V x V in the coordinates (w(t), . Now define a sesquilinear form < r : V x V — > C as a ((^> v )i i>)) = - ( v > < t> )v + <ri(w, tf> ) + cr2(u, if;) (3.15) 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Then the above second order equation can be w ritten as = < /( * ) , * > v . v ( 3 -16) where «?(*) = ( w ( t \ ^ - ) T, q = G V, and / = ( 0 ,/) r E L2((0,T ), V*). We may now use the sesquilinear form a to define an operator A as ( - A r f,d v.,v = cr(T?,C). (3.17) This first order system is formally equivalent to the system dw(t) ~ . - = Aw(t) + f w( 0) = w0 (3.18) provided the operator A is restricted to the domain D{A) = {17 | 77 E V and Aq E and A — 0 I -Ai —A 2 (3.19) The following theorem establishes the well-posedness of our abstract second order distributed param eter system. T h e o re m 3.3 Let V, H be Hilbert space with V '->• H = H * V*. Suppose that o'x and a2 are continuous and V-coercive sesquilinear forms on V and that cri is symmetric. Then the operator A is the infinitesimal generator of an analytic semigroup in'H = V x H . Proof: By the Analytic Generation Theorem (see Appendix A.3), it suffices to show th at cr\ = a (J), fPj +X ( f, tfj is V-continuous and V-elliptic. The K-coercivity of c t\ and cr2 yields Rea ($, (£) > k2\(f> 2 \v — Aol&lh 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. > k2 + |< ^ 2 Iv- ) — ^2 1^ 1 1 v ~ ^o|<^2 |jf > k2\< t> \^ ~ ^\4> \y_ where k2 axL d A axe positive constants. Thus, cr\ is V-elliptic. The continuity of cr\ follows from its definition. Therefore < j\ satisfies conditions of Theorem 3.2. Thus, for A > A > 0, A — A generates an analytic semigroup and hence we have proved the above theorem. □ We can use the above semigroup to define weak solutions for dw — * = A a + f to(0) = w0 (3.20) For to0 £ H and f E T2((0, T), V) the variation of param eters representation w(t) = T(t)w0 + f T(t — s)f(s)ds (3.21) Jo defines a mild solution of (3.20) which for to0 E D(A) with f E C l((0, T), %) is the unique strong solution to the weak problem (3.11). We summarize our results of this section as follows C orollary 3.1 Let V, H be Hilbert space with V ^ H = H* c -> V*. Suppose that c i and (J2 are V-continuous and V-coercive and that cr1 is symmetric. Then there exists a unique solution w of (3.11) with to,to E £ 2((0, T), V ) and to E L2((0, T), V*). Moreover, solutions of (3.11) depend continuously on the data (too,tOi,/) in that the map (wQ ,w u f) -»• (to, to) is continuous from V x H x L 2((0, T), V*) to L2((0 ,T ),V ) x £ 2((0, T), V). The differentiability of the solutions and continuous dependence on initial data follows from (3.21). 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.2 Wellposedness for the ACTEX Model In this section we return to our ACTEX model and show how it can be formulated in the weak form. Lets introduce w(t) in the state space H = ( n ^ ( 0 , Z ) ) x & 1 2 with the standard inner product, (< f> , i>)H = H M i i=l i'= 7 for < j), if) € H , where T < t > = - ' X } ) = (01 and (•, •) denotes the L2 inner product on (0,/). Let = («! (*,.) 1 (<,.)>'•* 1 «3 (*>•)> y3 (* » •)) M ^ rx d u i (*» 0 ^ ,x d v 3 (* w2 {t,l) = ^ ----,---,v 3 (t,l), and write w(t) = (wi(t, .),w2(t,l)). Then w G H. Let N be 3 (M )\ 5a: / N = p(x) 0 0 0 0 0 01x12 0 p{x) 0 0 0 0 01X 12 0 0 p(x) 0 0 0 0lxl2 0 0 0 p{x) 0 0 0lxl2 0 0 0 0 p{x) 0 01x12 0 0 0 0 0 p(x) 01x12 0l2xl 0l2xl 0l2xl 0l2xl 0l2xl 0l2xl A + B Then define an inner product on H as { M ) n = {N4>^)h (3.22) (3.23) (3.24) (3.25) (3.26) (3.27) (3.28) 44 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Let H & 0, /) = [ a S ff2(0, /) A(0) = 0, = 0 1 (3.29) where H 2(0,l) is the usual Sobolev space of ^ ( 0 ,/ ) functions with first and second derivatives in L2(0,l). Define the Hilbert space V C H as, V = I ^ € ^ ° ’ 0 > * = * • *. 6 and 1 \ 0: + 6 = ^(O5* = l , 3 , - - - , l l a n d ^ +6 = ^ , i = 2,4,...,12 ' with inner product { M ) v g ( ax 2 , 9a,2 >£2 (3.31) Using Sobolev embedding theorems, one can establish that the associated V norm is equivalent to the standard norm on H. Let H* denote the dual of H and let V m denote the dual of V. We identify H = H m via the Riesz map. Although V and Vm are not identified through the Reisz map, we will utilize the duality pairing (•, - ) v y given by the extension of H inner product from H x V to V* x V (see Wloka[29, p. 165]). Therefore V, H, and Vm defined above is our “Gelfand triplet” with V H = H* V*. We may develop the variational form of our system over V. We form the inner product of our modeling equations (2.72)-(2.74) with test functions to obtain a weak system , d 2u{ , d2 ( ~ f , d2u i\ d2 , d3u i \ , (p(X) dt2 + dx2 ( ^ C * ) dx2 ) + QX2 yCL>I^ d x 2d t) = (2 kf-^(Xacfi(t)),4>i) ,d 2Vi d2 (-pri, ,d 2Vi\ d2 ( — . . d3V { \ \ dt* ' fa* dx*) + fa* i 00 ^ f a * d i ) ’ ^ = (2k? g j (x ~ 9 i(0 ).A ) (3-32) 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. i = 1,2,3. Similarly, we form the inner product of each of the boundary condition with a test function to obtain additional constraints of the form < t > T (A + B) r s v m ■ dt2 a3 «i(n dt2dx cd/ ( 0 & ^ d2v l {l) dt2 - W - c Dm 3 i £ § d3vx(i) dt2dx E m ^ s P - « , / ( ( ) & § * + ¥ \ * d2uz{l) dt2 - T l ( l ) 3 -? 0 1 d3u3(l) dt2dx E m - . W cdI(1)t B § d2vz {Q dt2 - E I ( l ) ^ g ^ - C D/(Z) & £ d3v3(l) - dt2dx - E I ( l ) f y v 3(l) . J = 0. (3.33) where < f > = (07, 4> is)- For < f> , 0 € V we now introduce the “stiffness” sesquilinear form /jl / \ 4^/crfr d2& \ ^ W , « = E ( s / ( x ) - g - r ,-g -r > 1, and the “damping” sesquilinear form t± m — u d^ \ *2 (< £, 0 ) = (3.34) (3.35) Integrating by parts our weak equations (3.32) and incorporating the boundary conditions, we can rewrite the variational form of our system as (3.36) where c f > G V. Recall from the previous chapter that the stiffness and damping coefficients are given by E I(x ) = Ebelbe “ I " (^EpJ Ebe'j X ( f n c , s X n c , s ”1 “ I c ,s X c ,s “ I ” ^ a c X a c ) < -'I?-f(® ) = ^ D fie lb e “I” ([^D ,pe D ,b e) X ( /n e ,s X n c ,s 4 “ I c ,s X c ,s “1 “ Ia.cX.ac) (3.37) 46 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. These coefficients (3.37) are determined by the stiffness and dam ping properties of the leg and the embedded PZTs. The physical parameters Ebe, Epe, co,be and co,p e are bounded above and below by au and ao- From physical considerations EI(x) and ce)I(x) satisfy a i > E I(x), C£i/(x) > a 0 > 0 (3.38) T h eo re m 3.4 a i and a2 are V-continuous and V —coercive. proof: dx2 „ ^ p j W , > 9 V . 2 < ri (9,9) = 2_ E l (i) t=l „ ^ 92* > Oo 2-, t=l = < * 0 \4> \v dx2 Therefore cq is V —coercive. The continuity of cq follows similarly. Moreover cu is symmetric from physical considerations. The damping form cr2 is similarly V- coercive and K-continuous. □ The well-posedness of our ACTEX model is a consequence of the ^-continuity and K-coercivity of the sesquilinear forms cq and a2. We summarize the results in the following theorem and corollary. T h eo re m 3.5 Suppose stiffness and damping coefficeint E I{x) and cr>I(x) satisfy (3.38). Then the operator 0 I —A i —A 2 with domain V = V x V and cq = (Aiyq ijj) and a2 = (A2<p,ip}, generates an analytic semigroup T(t) on H = V x H. C o ro llary 3.2 Suppose stiffness E I ( x ) and C£>I(x) satisfy (3.38). Then there ex ists a unique solution w of the A C TE X modeling PDEs (3.32) -(3.33) and with initial conditions tu(O) = wq ii;(0) = wi. Moreover w ,w 6 L2((0,T ),V ) and A = 47 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. w £ L2((0,T ),V *) and solutions depend continuously on the data (wq, uq, f ) in that the map (w0,w i,f) -» (w,w) is continuous from V x H x L2((0, T), V *) to L2{{0, T), V) x L2((0, T), V). The corollary readily follows from Theorem 3.4 and Corollary 3.1. Theorem 3.5 and Corollary 3.2 guarantee existence, uniqueness, regularity, and continuous dependence of the solution to the ACTEX model. 3.3 State Space Approximation The results of the previous section guarantee th at an appropriate solution to our model exist. However a closed form solution is not tractable. The modeling PDEs are linear, but as function of the parameters p ( x ) , E I ( x ) , ce>I(x ) the solution is non linear. To overcome this difficulty and to obtain a computationally tractable m ethod, we approximate the state space by a sequence of finite-dimensional subspaces. Let us consider a nested sequence {H N} of finite dimensional subspaces H 1 C H 2 C • • • C H satisfying the following condition: A pproxim ation C on dition 3.1 For each r ) £ V , there is a sequence {tj}, r}N £ H N, N = 1 ,2 ,... such that nN - ri\v 0 (3.39) as N oo. We can now define the restriction aw to H N x H N of the sesquilinear form a on V x V in such a way that the boundedness and coercivity properties of < j are preserved. Then there exist bounded linear operators A N on H N th at satisfy C T J V = C A T ) (3.40) 48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We can expect th at each A N is a sectorial operators and generates an analytic semigroup on H N denoted by T N(t) such that T N(t)P N Wo solves the approximate system = A V ( t ) + f N (3.41) uA(0) = P Nw0 (3.42) where, for each N, P N is the projection operator defined as P N : H — > H N. We first give a general approximation theorem for analytic semigroups that is a generalization of the well-known Trotter-Kato Theorem B.5 in the Appendix. We state the theorem and refer the reader to the Appendix for proof. T h e o re m 3.6 Suppose we have Hilbert spaces H and H N, N = 1 ,2 ,..., with H N C H. Let P N : H — > H N denote the orthogonal projection of H onto H N satis fying (S.39) such that P N — > I strongly. Suppose that A N and A are the infinitesimal generators of analytic semigroups T N(t) and T(t) on H N and H, respectively, sat isfying the following: There exists a region E = E* = |A G C : \arg(\ — A 0)| < § + where 8 > 0 such that (E U {Ao}) C p{A) H ^=1 p(AN) and (i) there exists a constant M independent of N such that M IA — A o | for all A 6 E and N = 1 ,2 ,.. (ii) for some A G E and each w G H we have R \(A N)P Nw — > R\w; Then we have (Hi) for each w G H, T N(t)P Nw — > T(t)w uniformly in t on compact subintervals of [ 0, oo) ; (iv) for each w G H and integer k > 1, (A N)kT N (t)P Nw — » A hT(t)w uniformly in t on compact subintervals o f {0, oo) Let us now return to our abstract second order system. We use Theorem 3.3 to establish the following corollary the main convergence result for the second order system. 49 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C o ro lla ry 3.3 Let < X i and ai be V-coercive and V-continuous. Let < J \ be symmetric and H N satisfy condition (3.39). Then (a) The system sesquilinear form a (3.15) is V-coercive and V-continuous in the norms of V and TL and the operator A is the infinitesimal generator of an analytic semigroup T (t) on TL. (b)Let A N denote the operator obtained by the restricting a to *H N x 'HN and let T N(t) denote the corresponding analytic semigroups on TLN . Then we have T N(t)P Nw T(t)w for each w € H- uniformly in t on compact subintervals o/[0, oo )/ For each w E.'H and positive integer k, {A1 *) k T N{t)PNw (A)* T(t)w uniformly in t on compact subintervals of (0, oo). The approximation results of Theorem 3.6 and Corollary 3.3 gives us a compu tational foundation for simulating the ACTEX distributed param eter system. We construct the subspace H N to approximate H and approxim ate the actual solution w by wN. 3.4 The Spline Galerkin Method The finite dimensional approximation theory for our m odel was developed in the previous section. In this section we discuss how polynomial splines can be used to construct our approxim ating subspaces. We define polynomial splines and establish, in particular, that cubic polynomial splines can be used to satisfy the approximation condition (3.39). Following Schumaker [32, p .108-142], let [a, 6] be a finite closed interval and let A = with a — x 0 < Xi ■•• < x^+i = 6, be a partition of [a, 6]. Denote the set of all polynomials p of order m by Pm. 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. D efin ition 3.7 (P olynom ial Splines) Let M = (mi, - •, m*) € 1 < m ,- < m . The space S (P m,A i,A ) = {s : there exist S{ 6 P m such that s(x) = s,(x) dJ d3 for x € [xi, x,-+1), i = 0 ,..., k, and ^ - j s f_1(a;1 -) = ^ y s ( x t), for j = 0 ,1 ,..., m — 1 — nti, i = l , . . . , k } is called a space of polynomial splines of order m and A i is called the multiplicity vector. I f A i= ( 1 ,..., 1) put Sm(A) = S (P m, (1 ,..., 1), A) and call Sm(A) space of polynomial splines with simple knots. The dimension of S (P m,A i, A) can be shown to be m + H t=i m-i = m + K. Denote the space of cubic polynomial basis splines by S^B. The following theo rem is proven in the Appendix C (see Theorem C.6). T h e o re m 3.7 Let P N be the standard orthogonal projection of L2(0, /) onto S$B and let rj 6 H 2(0,l). Then there exists a constant C, independent of N and f , such that Furthermore I P Nr ,~ v \c .,< C N - 1 ^ { P NT ] — J?)| 0 as N -> ■ oo. dt) dx l2 This establishes that cubic polynomial splines satisfies the approximation condi tion (3.39). To simulate the ACTEX space structure, we utilize a class of schemes based upon choosing the H N as subspaces generated by cubic splines. We construct a family of approximating solutions using cubic spline-based Galerkin approximations. For each N = 1 ,2 ,..., let 11^ denote the uniform partition 0 = x0 < xx < x2 < • • - < xjv = I of [0, /]. Let flF(x) denote the standard cubic polynomial on each subinterval [(* — *17]) z = 1, Recall that V is a subset of (lTf=i Hf, (0, /)) x 3 ft12. In order for H N to satisfy the essential boundary conditions at x = 0, the first six coordinates of the approximating elements must also satisfy these essential boundary conditions. A linear combination of the spline elements { /^ (x ) j-. * axe utilized for 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the problem at hand. Six sets of elements are used for the six one-dimensional Euler- Bernoulli beams, namely {r /V (x) } ^ 1- Lets denote the basis elements = ( t f U , 0,0,. # U + 1 = (o, 0 f ( - ) . O , . . . , O , 0 f ( / ) , ^ j ^ , . . . ) $U(W+i) = (o,0,...,rjiv( . ) , 0 , 0 , . . . , T f ( 0 , ^ ^ V (3.43) The Galerkin equations corresponding to wN(t) £ H N are given by ( ^ > / ) + I , ( / ( i ) / ) + I , ( ^ ( ! ! / ) = (P Km , 4 F ) wn {Q) = P NWq ^ (3.44) where c f > N £ H N . If we set wN(t) = 2Dfg.+6 x i* ($)& ■ > the resulting initial value problem is equivalent to the linear, homogenous, second order 6iV + 6-dimensional system given by U n ^ - + Cn ^ - + K n x n = F N{t) at* dt a^(0 ) = PNw o = P NWl. (3.45) ai/ The (67V + 6) x (67V + 6) matrices M N, C N and K N are given by M g = e g = K& = (3-46) 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. For each t > 0 and i = 1, — , 6N 6, This second- system can be solved using any num ber of solvers for second-order ODEs. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 4 Parameter Estimation We established the well-posedness of the ACTEX structure model iu the previous chapter. In this chapter we turn to parameter estim ation or identification techniques for our ACTEX models. The problem is to estim ate our model parameters using a set of observations of the ACTEX tripod response. We formulate this as a minimization problem using a least-square criterion. This problem involves an infinite-dimensional state space and, in general, an infinite-dimensional param eter space. To overcome this difficulty and to obtain a computationally tractable m ethod, we approximate the state space by a sequence of finite-dimensional subspaces, as explained in Chapter 3 and the param eter set by a sequence of finite-dimensional sets. For a discussion of parameter estim ation problems for distributed param eter systems we refer the reader to Banks and Kunisch [6, p. 1-31]. In section 4.1 we discuss the abstract param eter estim ation problem in a Hilbert space. We state m ain results for parameter estim ation and convergence for abstract second order systems. For a survey of the recent results in param eter estim ation for distributed param eter systems, we refer the reader to Banks and Ito [33]. In section 4.2 we pose the param eter estimation problem for the ACTEX modeling PDEs. In section 4.3 we apply the theoretical framework of section 4.1 to prove convergence for the ACTEX param eter estimation problem. 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.1 Abstract Formulation We consider the second order system of the previous chapter, d2w dw + A 2(q)-g£ + Ai(q)w = f ( t ) to(0) = w0 (q) —g f - = v>i(q) (4.1) with solutions in a Hilbert space H , where q is a vector of model param eters in a m etric space (Q,d ) with m etric d. The operators A\(q) and A ^q) are defined via parameter-dependent sesquilinear forms < T i and < j% . We assume th at we are given a Hilbert space V C H that is continuously and densely embedded in H. Suppose the symmetric sesquilinear form cri(q) ' . V x - V — tC satisfies the following conditions. (A) ^-continuity: There exists ci > 0 such that q 6 Q, < fi, if) € V implies k i(« )(& ^ )l ^ ci\<f>\v\i!)\v. (B) V-Coercivity: There exists C 2 > 0, A o > 0 such that q 6 Q 4 > € V implies Recrx(q) ( < < f > , < f> ) + Xo\4$j > c2\4>\v. (C) V-Parameter Continuity: For q,q € Q, we have for all € V K (?)(<£,^) - < d(q,q)\cf)\v \^\v . Then for each q £ Q we may define a continuous linear operator Ai(q) : V — » • Vm given by * /> ) = $)v»,v (4.2) for all € V. The sesquilinear form crl and its associated operator A i will correspond to stiffness in the examples we treat below. 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We assume th at we are also given a sesquilinear form cr 2(q) : V x V — » • C satis fying conditions (A), (B) and (C). Under these conditions we similarly m ay define a continuous linear operator A 2(q) : V — + V m given by (4.3) Now we shall rewrite our second-order weak system in first-order form sim ilar to the development of Chapter 3. Define T -L = V x H and V = V x V . Define a sesquilinear form cr(q) : V x V — > C as, a(q) ( ( w , v = — (v, cfr)v + ip) + < T 2(q)(v,i>). (4.4) Then the above second-order equation can be w ritten as (^ ^ ,V )u + < T (q )(w (t),ff) = (/(*), »7)v\v, where w(t) = (w (t), , if = (<£,^)T E V, and / = (0, f ) T € L2((0, T), V*). Now define an operator A(q) : V — > V' by (~A(q)j?, c>v,v = &(q) (n, C) • (4.5) The first-order system is formally equivalent to the system = A(q)w(t) + f(t) w( 0) = w0, (4.6) where the operator A{q) is defined in the domain D(A(q)) = {77 | if € V and A{q)ff e %]• and A{q) = 0 I ~Ai(q) ~ A 2(q) (4.7) 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. We can argue that under the above conditions (A),(B), and (C) on cr\{q) and cr2(g), -'4(9) generates an analytic semigroup T(t;q) on H. Now suppose we axe given an observation Z { 6 'H . of iZ?(t,-,q) at times £t -, i = 1 , 2 , . . where w(t; q ) = T(£; q)u;o(q) + f T(t - s; q )/(s, <?)ds. (4.8) ./o is the mild solution of the weak problem in 'hi. We then consider the least-squaxe identification problem of finding q € Q minimizing the functional J (9) = Z 9) - 2 * * 1 * • (4-9) I Note the above problem is, in general, infinite dimensional in both the state w and the param eter q. One must thus consider a sequence of computationally tractable approximations to this problem. We consider next a Galerkin-type approximation as discussed in Chapter 3. We assume we have a sequence {'HN} of approximating subspaces H N satisfying: (H i) : H n C V (H2) : < f > E V ,implies |< £ — P ^ ^ — > 0 as N — > 0 0 where P N is the standard orthogonal projection P A of H onto H N. We note that (H2) also implies that |( f> — — > 0 for each c f> E H because V is continuously and densely embedded in H. Now consider the restriction of <r(q)(-, ■ ) to 7iN x H N . This can be shown to generate analytic semigroups T N(t; q) E H N. The T N(t) are then used to define approximating solutions wN(t; q) = q)PN,»0(q) + f T N(t - s; q)PN f(s , q)ds. (4.10) Jo One thus obtains a sequence of approximating identification problems requiring as to minimizing over Q the functionals j N (q) = Z I^C**; 9 ) - £-|2 (4-11) t 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In problems where the parameter space Q is infinite dimensional one m ust approxi m ate Q by a finite set QN. For this thesis, we consider finite dimensional parameter space Q. The solution qN to the approxim ate identification problem (4.11) leads to a sequence of param eters {g^}. Therefore one needs to address (i) convegence of this sequence {q^}, (ii) suppose qN converges to some parameter q, under which condi tions wN(t; qN) also converges to w(t, q), and (iii) finally is q solution to the original estim ation problem (4.9). For this thesis, we only address (i) and (ii). However our treatm ent of (i) and (ii) provides a foundation for addressing (iii) in the future. To establish convergence and continuous dependence of param eter estimates with respect to observations for the minimizing solutions g^ to (4.11), it suffices (assuming (Q ,d ) is a compact space) to show that for arbitrary |g ^ | C Q, qN — * g implies that wN(t; qN) — > w(t; q) for each t. One can argue wN(t; qN) — > ■ w(t] q) if one shows that T N(t] qN)P Nw — > T (i; q)w for arbitrary qN — > ■ q and w 6 TC . To do this one can use a version of the Trotter-Kato theorem. We state below the resolvent convergence form of this theorem. The proof of this theorem is given in the Appendix (see Theorem D .l). T h e o re m 4.1 Suppose that H N satisfies (Hl)-(H2) and that crfiq), 0 2 (g) are V- parameter continuous, 0 1 (g) is symmetric, &i{q), 0 2 ( g ) are V-continuous and V - coercive. Let g^ — > q in Q. Then for A > 0 we have R \ (JLN (q1 *)} P Nfj — » • R \ ^A(g)) 77 in the V norm for any if € TL. Applying this result and the Trotter-Kato Theorem (see Appendix B.5), we imme diately establish the main result of this section as a corollary. C o ro lla ry 4.1 Suppose that H N satisfies (Hl)-(H2) and that 0'i(g), 0^(g) V-parameter continuous, o'i(g) symmetric, 0 1 (g), 0 2 (g) V-continuous, V-coercive and furthermore q — > f(t,q ) is continuous from Q to L 2 ((0, T), V*). Let qN be arbitrary in Q such that qN — > q in Q, then for all t > 0, as N — ¥ 0 0 , wN (t, qN) — > • w(t, q) in V norm dwN(t,qN) dw(t,q) . — m -----------' - g f - . n V n o r m 58 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. While most commonly encountered least-squares problems involve tim e domain data, it is often advantageous to fit the data in a frequency domain setting. We use the frequency domain setting for our ACTEX structure. If the identification is done in the frequency dom ain then one can construct a least-squaxes functional ■r(«) = E E {W0*T +i-,q)\-\Z(M + j)\Y + i°\k-q\\l (4.12) where W{k\ < ?), Z{k) are the Fourier series coefficients of the solution u)(t,-; q) and the data Z{ respectively and k f and kf are the location of the ith peak of the frequency response W and Z respectively. Here 7 0 is a regularization constant, [| • || is the norm induced by the Q-metric, Np is the number of peaks observed in the real system response function, and [n/, iV } ] is the domain where zth peak is to be matched. Suppose the m easurements are taken with fixed sampling tim e A t = U+i — U for all i, and with a total of N t samples. Then the fourier series coefficients for 0 < i < Nt are given by Nt-1 K tio Nt-l Z (fc) = W E (4.13) .=0 where t,- = iA t and fc = 0,l,...,iVt — 1 . W ith A t as the sampling time, the fc-th value of the frequency corresponding to the &-th coefficient is given by = A t - N t (4‘14) and the corresponding magnitudes are given by \W(k\q)\ and \Z(k)\. Let W N(k; q) be the fourier coefficients for the approximate solution of wN(t;q), W N(k; 9) = 4 E 1 q)e-ik^ IN')i. (4.15) t t= 0 Now we state and prove the m ain result for frequency domain param eter estim ation following Banks, Sm ith, and Wang [13, p .129]. 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T h e o re m 4.2 Suppose {9 ^} C Q with q ^ — * ■ q as N 0 0 . Let W N (k', qN) denote the Fourier series coefficients of the unique solution wN(t; qN) to the initial value problem corresponding to qN, and let W(k",q) denote the Fourier series coefficients of the unique solution w(t] q) to the initial value problem corresponding to q. If wN(t-,qN) — > • w(t;q) in the V-norm and pointwise evaluation is continuous in the V-norm, then E E { \w N(kr +i;«w)| - \W (kr + j;q)\Y - * ■ 0 t = l j = - n ; as N — » ■ 0 0 . Proof: Since pointwise evaluation is continuous in the V topology, we know that converging to w(t;q) in the V- norm implies that W N(k;qN) converges pointwise to W N{k', q) for each k. Then we have k(* — $ ■ ki as N 0 0 , since the significant coefficients of V/N(k\ qN) converge to those of W(k\ q) both in magni tude and phase because the Fourier series coefficients converging for each k. Hence |WrJ V '(fcjv'; <J^)| — > • \W(kr: q)| as N — > • 0 0 and the result follows. □ 4.2 Inverse Problem for the ACTEX Model In this section we pose the param eter estim ation problem for the ACTEX model. The param eter estim ation problem can be form ulated as an inverse problem for the distributed param eter system considered in the previous chapter. Let q = (,p(x), E I(x ), cDI(x ), K s ) (4.16) and consider the space Q = { 9 € [£oo(0, /)]3 x : 91 (x), q2{x), q3(x) > a 0 > o} . In the rem ainder of this thesis we only consider q which belongs to a compact set Q C Q . 60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Now we can write our abstract second order system for the ACTEX structure as < ^ r ^ > + = ( f ( t U ) (4.17) for all < f > 6 V, where V is subspace of test functions in a Hilbert space H with inner product (.,.), t is time, w is the unknown state function in H, Ai(<?) and A 2(q) are the operators describing the damping and stiffness of the system and q is a vector of model param eters in Q. The well-posedness of our model is a consequence of the V —continuity and V —coercivity of the parameter-dependent sesquilinear forms *i(&^) = (A i{q)(f>A) and < r 2{4>^) = (A2(q )< f> , ip). The param eter estimation or inverse problem for our model may be abstractly formulated as follows: find q € Q that minimizes the least-squares output functional, ■/(«.“ ) , € « = E ( { l ^ ” + i ; ? ) l - l ^ ( * f + j ) l } a) + i » ) l l 9 - « l l « . (4-is) 3=-ni where {W^A;)} are Fourier coefficient of the solution w(t{;q) and Z{k) is the actual frequency response function experimentally observed and k f and kf are the location of the ith peak of W and Z respectively. Here w(t; q) is the parameter-dependent weak solution of our modeling equations (2.72)- (2.74). The param eter identification problem (4.18) for the ACTEX structure involves an infinite dimensional state space. To overcome this difficulty and obtain compu tationally tractable minimization problems, we use approximate solutions obtained by and minimize the following least square output functionals: N l JN(q,wN),eQ= £ ( { K " ( i r + i ; « ) | - I ^ W + J ) l } a) + i » l l9 - « l l a , (4-19) J=-»i where {tF^A ;)} are Fourier coefficients of the approximate solution wN(t{; q) and Z{k) is the actual frequency response function experimentally observed. Here wN(t\q ) is the parameter-dependent solution of our modeling equations (2.72)- (2.74). 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.3 Convergence of Parameter Estimates In this section we apply the parameter convergence results of section 4.1 to the ACTEX param eter estim ation problem (4.18)- (4.19). To obtain convergence and continuous dependence of the parameter estim ates q1 * with respect to observations, it suffices to show that 0i(q) and cr2(q) for the ACTEX model are V-continuous, V-coercive and V-param eter continuous. We have already established V-continuity and coercivity in the previous chapter. We now show the <Ti(q) is V-param eter continuous. The argument is similar for a2(q). T h e o re m 4.3 cri(q) is V-parameter continuous sesquilinear form. Proof: Let q, q £ Q. From the definition of ay we have -0-1 (?)(<£, V O I2 = X ){?2-<72}( t=i d24 > j d2tpj d x 2 ’ d x 2 ) i = l \ i = l < 1 \ q - q \ 2 \(f,\v \% ! > \v . d24 > i d x2 d24 > i d x2 This establishes that ay (q) is V-parameter continuous. □ The convergence of solutions to the ACTEX param eter estimation problems (4.18)-(4.19) is a consequence of the V-parameter continuity of the param eter depen dent sesquilinear forms ay(q) and a2(q). We sum m arize our results for this section in the following theorem and corollary. T h e o re m 4.4 Suppose the ACTEX model parameters ( p { x ) , E I ( x ) , c d I ( x ), K s ^ € Q as in (4-17) and q — > ■ f ( t,q ) is continuous from Q to L2 ((0, T), V*). Let qN be arbitrary in Q such that qN — ¥ q in Q. Then for all t > 0,as N — ► oo, wN (^ 9 ^ ) — > vj(t,q) in V-norm dwN(t, ) dw(t, q) . in V-norm dt dt Proof: The theorem follows from an application of Theorem 4.1. □ 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C o ro lla ry 4.2 Suppose the A C T E X model parameters (/5(x), E I (x ), cdI(x), K s^ € Q as in (4-17) with qN q as N — ¥ oo. Let W N{k, q1 *) denote the Fourier se ries coefficients of the unique solution wN(t;qN) to the initial value problem (4-8) corresponding to qN and let W (k ; q) denote the Fourier series coefficient of the unique solution w(t;q) to the initial value problem (4-10) corresponding to q. I f wN{t-qN) — > ■ w(t; q) in V norm, and pointwise evaluation is continuous in V norm, then E E {\wN(kr+j;qN)\-\w(kr+j-,q)i}2 - > o as N — * ■ oo. The corollary follows from an application of Theorem 4.2. These theorems guarantee the convergence of solutions to the approximating ACTEX param eter estimation problem. In this chapter, we have considered the inverse problem to estim ate the model param eters for the ACTEX tripod. We formulate a sequence of approximate mini m ization problems (4.11). We show th at qN converging to q implies that wN{t]qN) converges to w(t; q) for the ACTEX structure. One needs to further address the issue of q being the solution to the original estim ation problem (4.9). We do not consider this here, however our treatm ent provides a foundation to address these issues in the future. 63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 5 Computational Results la this chapter we evaluate the validity of our ACTEX distributed param eter system model by comparing numerical simulations to the actual experimental data. We also discuss the results of numerical studies for the param eter estim ation problem involving the ACTEX structure. In section 5.1 we recall the model parameters described in Chapter 2. We calcu late the geometric and physical param eters for our model. In section 5.2 we discuss numerical techniques to solve our modeling equations. In section 5.3 we simulate the ACTEX response using numerical procedures outlined in section 5.2. We use handbook material specification for the values of the physical parameters. We com pare our simulation with experimental data both in the tim e domain and in the frequency domain. In section 5.4 we set up the parameter estim ation problem to minimize the mismatch between the simulated response and the experimental data. In section 5.5, we list and compare the estim ated model parameters for two different data sets. 5.1 Model Parameters To simulate the ACTEX distributed param eter system we first calculate the model param eters using geometric and structural considerations. Recall the form of the model parameters for the ACTEX structure: p ( s ) = P6eAfee -f- (/Jpe Pbe) X ( A nC)aX n c ,s " H AC v S ^ C ) s A ac X a c ) E I(x) = EbeIbe " " t ~ (Epe Ebe) * ( ln c ,s X n c ,s " b E ,s X c ,s + l a c X a c j 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Param eter Value Units I 17.7 (in) Ab 0.1418 (in2) Beam h 0.0270 (in4) e 60° - C t 1 2 - P & 2 - Xl,7IC S 1.56 (in) NC * ^ 2 ,T I C S 2.56 (in) Sensor A ■ t^T lC S 0.0045 (in2) I-ncs 0.0013 (in4) X1 ,cs 6.76 (in) c x 2 ,cs 7.26 (in) Sensor AC s 0.0045 (m 2) Ic s 0.0013 (in4) TU) i,ac 3.06 (in) T(1) x2,ac 5.56 (in) ~(2) l,oc 5.81 (in) Actuator t(2) x 2 ,ac 8.31 (in) r (3) l,ac 8.56 (in) t(3) x 2,ac 11.06 (in) Aac 0.0090 (in2) la c 0.0025 (in4) Table 5.1: The ACTEX Geometric Parameters Cd I('E) A D ,6eA e "f" (p D ,p e C-D,be) X (-/n c,sX tic,s “t“ A ,s X c ,s “t” Ia.cX .ac) K B = ——{Aije -\-Aac)E pedzi (5.1) specified as piecewise constant functions. The geometric param eters in the modeling equations are listed in Table 5.1. The area of the beam is calculated as Ab = (62 — a2)/ 2 (in2). The area moment of inertia (see Gere and Timoshenko [24]) of the beam is given by (64 — a4) / 24 (m 4). Similarly we have calculated area and area moment of inertia for the actuator and the colocated and nearly colocated sensors. The physical param eters for our ACTEX tripod system are given in Table 5.2 using handbook values. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Param eter Param eter Value Units Pbe 0.07563 (lb fin) Beam Ebe 5.018 xlO6 (lb-force-in2) CD,be 12.9 (I b-force-in2-s) P pe 0.28 {lb fin) PZT Epe 1.0 xlO7 {lb-force-in2) c D ,pe 5.7 {lb-force-in2-s) M 5.598 {lb) Plate Iy 29.5962 {lb-in2) h 29.5962 {lb-in2) Table 5.2: The ACTEX Physical Parameters 5.2 System Approximation In seeking an approximation to the system of partial differential equations for the ACTEX tripod, we consider a Galerkin type approximation. The leg displacements U{, Vi are approximated by *k = ] C n£ iW fc j(x ) i = i ^ = (5-2) 1 in an appropriate N th order finite dimensional space. For each N = 1,2,..., let 11^ denote a uniform partition 0 = x 0 < x x < x2 < • • • < Sjv = / of [0, /]. The basis elements <$!j, are chosen to be in V N, the space of standard cubic B-splines (see Schumaker [32], Prenter [34]) on the interval [0, /] defined with respect to the uniform mesh 11^ with essential boundary conditions at x = 0. The Galerkin equations are then given by M Nc{t) + CNc{t) + K Nc{t) = F N{t) (5.3) where cfc + J -(t), Ck+j+ 3N(t) are u^- and Vk,j respectively. We transform the system of ordinary differential equations in Equation 5.3 into U = A NU + F(t) (5.4) 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where U = A n = c c F = 0 pN K N 0 -1 0 k n ' 0 m n —K N - C N (5.5) The initial conditions Uq for the approximating system are taken to be the orthogonal projection of the initial conditions onto the space V N. Numerical intergration of the Galerkin equations is then carried out via any one of a num ber of standard methods. In particular, we have used second-order Backward Differential Formulas (BDF) in tim e (see Dahlquist [35, p.666]). 5.3 ACTEX Simulation To see if our model could provide a valid description for the actual system response, we simulate the ACTEX response using handbook values of physical parameters (see Table 5.2). We simulate the ACTEX identification experiment 1 run 5. We use the actual PZT actuator voltage Vac for experiment 1 run 5 as the input voltage for the Leg A actuator. The accumulated strain generated due to actuation causes the neaxly colocated sensor in Leg A to generate output voltage Vs ■ We assumed the following form for the sensor output (see Banks, R.C. Sm ith and Y. Wang [13]), K s r ^ dx = K s I * ? (< ■ ?») _ fr°(jl5 l >} (5.6) Jxi ox2 ( dx ox J where K s is the sensor constant (stra in /V ). The tim e domain simulation, using K s = 9.3646 x 107, is compared to the actual response Vs observed. The comparison is shown in Figure 5.1. In the upper panel of Figure 5.1, the actual Vs for experiment 1 run 5 is shown, and in the lower panel of Figure 5.1 response is depicted. We observe that the simulated tim e history is similar to the experimental time response. We compare the frequency response of our simulation to the actual response in Figure 5.2. 67 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.2 0.4 0.6 0.8 Figure 5.1: Com parison of Tim e Response 300 ~ 200 2 200 Figure 5.2: Comparison of Frequency Response Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In the top panel of Figure 5.2 the actual frequency response |T(fc)| is shown, and in the lower panel the simulated frequency response is plotted. We also compare the transfer function of our simulation to the actual transfer function in Figure 5.3. 1 n. E m k. « 20 60 80 40 100 hertz i s. E • W I I 20 100 40 60 80 hertz Figure 5.3: Comparison of Transfer Function In the top panel of Figure 5.3, the actual transfer function is shown, and in the lower panel the sim ulated transfer function is plotted. We observe that our simulated response features closely resemble the actual response features except at very low frequencies (below 10 hz). The comparison of Figure 5.3, validates our physical- principle based model for the ACTEX structure. 5.4 Model Parameter Estimation To minimize the difference between our model simulation and the actual data we formulate the following parameter identification problem. Let q = (p(x), E I(x ), ^ I ( x ), K s ) (5.7) 69 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. be the set of param eters in our model. An inverse problem is set up to identify q which achieves the desired system response. We wish to minimize the cost functional = {mf=r+r,9)\-\Z(kt+i)\Y + -lo \W -q f (5.8) i= lj= — T li where W is the transfer function simulated using q and Z is the actual ACTEX transfer function, Np is the number of peaks observed in the real system transfer function, and [ n t, iV } ] is the domain where zth peak is to be matched. The regular ization term 7o||<? — q\\2 facilitates numerical convergence of our scheme. This scheme allows us to translate the observed changes of the ACTEX transfer function into changes of structural properties such as density, stiffness, and damping factors included in q above. To validate our scheme, we first test our identification procedure on sim ulated data. We choose a set of param eters and simulate the transfer function using our model. Then we use an initial guess different from the parameters chosen to generate the data. The value of the given parameter and the initial guess used for identification are given in Table 5.3. We also plot the transfer function for both the given parameter and the initial guess in Figure 5.4. 16 2 0 40 60 hertz 80 100 16 2 0 40 60 hertz 80 100 Figure 5.4: Initial Guess for Simulated Identification In the top panel of Figure 5.4 we plot the transfer function for the given parameter. In the bottom panel of Figure 5.4 we plot the transfer function for the initial guess. 70 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Parameter Given Initial Estim ated Units Pbe 0.09076 0.09529 0.09075 {lb/in) Beam Ebe 8.44 x10s 8.02x10s 8.44 x10s {lb-force-in2) c D,be 10.5 9.975 10.56 {lb-force-in2-s) Ppe 0.2660 0.2527 0.26599 {lb fin) PZT Ep e 0.2 xlO 5 0.19x10s 0.2 x10s {lb-force-in2) P D,pe 12.25 12.86 10.919 {lb-force-in2-s) Table 5.3: Estim ated Parameters for Simulation D ata We find that our scheme converges to the original param eter (see Table 5.3). However one needs to keep in mind that we generated our simulation using a model and used an initial guess close to the given parameter. We next use our identification scheme on the experimental data from the ACTEX experiment. We have optimized the cost functional J using an initial guess of handbook param eters. The estimated param eter is listed in Table 5.4. The plot of the estim ated transfer function is shown in Figure 5.5. In the top panel of Figure 5.5 we plot the transfer function corresponding to the experimental data. In the bottom panel of Figure 5.5 we plot the transfer function corresponding to the estimated param eters. a . i I 20 40 60 hertz 8 0 100 L Figure 5.5: Low Tem perature Response Using Estim ated Parameters We observe th at the estimated param eter minimizes the discrepency at lower fre quencies noted in our simulation based on handbook values only. 71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Param eter Initial Param eter Estim ated Parameter Units Pbe 0.07563 0.28264 (lb/in) Beam Ebe 5.018xl06 0.6785 xlO8 (lb-force-in2) CD,be 12.9 909 (lb-force-in2-s) P pe 0.28 0.43248 (lb/in) PZT E pe l.OxlO7 3.1022 xlO 8 (lb-force-in2) C-D,pe 5.7 1553 (lb-force-in2-s) Table 5.4: Estimated Param eters for the ACTEX 5.5 Parameter Comparison In this section we compare the estim ated param eters for the transfer function with two different tem perature profiles. Recall th a t various remote system identification runs were performed to observe any system atic changes in the ACTEX structure as it moved into different space environments. For the low tem perature profile we found the param eters as shown in Table 5.4. The plot of the transfer function for the high tem perature profile is shown in Figure 5.6. I s. E m W m 2 2 0 60 80 40 100 hertz I1 2 " S . i « < 9 J 4 Figure 5.6: High Temperature Response Using Estimated Parameters We list the comparison of estimated param eters for the high and low tem perature profiles in Table 5.5. We note a 22 percent change in the PZT stiffness due to 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Parameter Low T H ighT Units Percent Change Pbe 0.28264 0.3504 (lb/in) 18 Beam Ebe 0.6785 x10s 0.7218 x10s (lb-force-in2) 6 CD,be 909 384 (lb-force-in2-s) 57 P pe 0.43248 0.4787 (lb/in) 8 PZT Epe 3.1022 x10s 3.7705 x10s (lb-force-in2) 22 c D ,pe 1553 695 (lb-force-in2-s) 55 Table 5.5: Param eter Comparison for the ACTEX tem perature or other environmental effects. We plot the stiffness changes in Figure 5.7. 100 n O 60 H s i r V i 1 20 rJ 1 1 1 8. „ f m i i i E -20 $ -60 • % A / * V * •* Beam Axis 2 4 6 8 thermostat number i r f n 100 0 60 s 1 20 1-20 s -60 beam axis 2 4 6 8 thermostat number Figure 5.7: Stiffness Profile Our parameter estim ation procedure can track such system atic changes of the AC TEX structure as it moves into different space environments and can translate the observed variation in the system response into variations of structural properties such as stiffness. This enables us to better understand these environmental effects and should help improve design and control of similar space structure in the future. 73 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 6 Conclusions and Future Study In this thesis we developed a m athem atical model for a space structure with active m aterial. We established the well-posedness of our model and provided a computa tional foundation for simulation. We posed the param eter estim ation problem and theoretically established param eter convergence. Finally, we validated our model by comparing the numerical simulation with experimental data. This thesis establishes that param eter estimation techniques can be used to model, simulate, track, and diagnose a space structure with active materials. Our param eter estimation procedure can track systematic changes of a space structure moving through different space environments and can translate the observed varia tion in the system response into variations of structural properties such as stiffness. Our understanding of these effects can be used for integrated design and control of space structure with active m aterials in future missions. Our study is a step in that direction. This thesis has generated a num ber of future avenues of research. To conclude we briefly describe a few of them. First, we would like to investigate observability and controllability of the model studied in this thesis. Secondly, we would like to extend our model to a nonlinear setting. Many active materials, including piezoceramic, exhibit a nonlinear relationship between strain and stress (see Smith [36], Banks and Pinter [37], and Banks and Lo [38]). Therefore, realistic models of many future space structures with active m aterials will lead to nonlinear distributed parameter systems. These nonlinearities can lead to interesting param eter estim ation problems. We would like to extend our results to such space structures. 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reference List [1] Sigeru Om atu and John H. Seinfeld. Distributed Parameter Systems: Theory and Applications. Clarendon Press, 1st edition, 1989. [2] Peter Stavroulakis. Distributed Parameter Systems Theory. Hutchinson Ross Publishing, 1st edition, 1983. [3] Chunming Wang. Approximation and num erical experiments for control of tim e periodic parabolic distributed param eter systems. Journal o f Mathematical Systems, Estimation, and Control, 6(4):371-413, 1996. [4] H.T. Banks and C. Wang. Optimal feedback control of infinite-dimensional parabolic evolution equation systems: approximation techniques. S IA M J. Control and Optimization, 27(5):1182— 1219, 1989. [5] H.T. Banks and R.C. Smith. Models for control in smart m aterial structures. Identification and Control in Systems Governed by Partial Differential Equa tions, l(l):26-44, 1993. [6] H.T. Banks and K. Kunisch. Estimation Techniques for Distributed Parameter Systems. Birkhauser, 1st edition, 1989. [7] H.T. Banks, K. Ito, and C. Wang. Exponentially stable approximations of weakly damped wave equations. International Series of Numerical M athemat ics, 100(l):l-33, 1991. [8] H.T. Banks, F. Kappel, and C. Wang. Weak solutions and differentiability for size structured population models. International Series o f Numerical Mathe matics, 100(l):35-50, 1991. 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [9] H.T. Banks, R.K. Powers, and I.G. Rosen. Inverse problems in the modeling of vibrations of flexible beams. Estimation and Control of Distributed Parameter Systems Conference Proceedings, l(l):l-2 2 , 1988. [10] H.T. Banks, R.C. Sm ith, D.E. Brown V.L. Metcalf, and R .J. Silcox. The esti m ation of m aterial and patch parameters in a pde-based circular plate model. Journal o f Sound and Vibration, 199(5) :777— 799, 1997. [11] Sergey V. Lototsky and Boris L. Rosovskii. Spectral asymptotics of some func tional arising in statistical inference for spdes. Stochastic Processes and their Applications, 79(l):69-94, 1999. [12] L. Piterbarg and Boris L. Rosovskii. On asym ptotic problems of parameter estimation in stochastic pdes: discrete tim e sampling. Mathematical Methods o f Statistics, 6(2):200— 223, 1997. [13] H.T. Banks, R.C. Smith, and Y. Wang. Sm art M aterial Structures Modeling, Estimation and Control. John Wiley Sz Sons, 1st edition, 1996. [14] R. Barrett and R.S. Gross. Super-active shape m em ory alloy composites. SPIE Conference Proceeding, 2441(1):110-117, 1995. [15] P. Jardine, J. Flanigan, and C. Martin. Sm art wing shape memory alloy ac tuator design and performance. SPIE Conference Proceeding, 3044(1) :48-55, 1997. [16] P. Jardine, J.N. Kudva, and C. Martin. Shape m em ory alloy tini actuators for twist control of sm art wing designs. SPIE Conference Proceeding, 3044(1):160- 165, 1997. [17] F. Austin, W.V. Nostrand, and P. Aidala. Design and performance predictions of sm art wing for transonic cruise. SPIE Conference Proceeding, 2721(l):17-25, 1996. [18] F. Austin, M.J. Siclari, and W.V. Nostrand. Comparison of smart-wing concepts for transonic cruise drag reduction. S P IE Conference Proceeding, 3044(l):33-40, 1997. 76 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [19] F. Austin, M.J. Rossi, and W.V. Nostrand. Static shape control for adaptive wings. A IA A Journal, 32(9): 1895-1901, 1994. [20] C.A. M artin, L. Jasmin, and J. Flanagan. Smart wing tunnel model design. SPIE Conference Proceeding, 3044(l):41-47, 1997. [21] Ray Manning. A C TE X I Design and Development Technical Report. TRW Space and Electronic Group, 1st edition, 1996. [22] Ray Manning and Ken Qassim. A C T E X Flight Experiment. TRW Space and Electronic Group : www.vs.afrl.af.mil, web page, 1st edition, 1998. [23] J.N. Reddy. Energy and Variational Methods in Applied Mechanics: with an Introduction to the Finite Element Method. John Wiley & Sons, 1st edition, 1984. [24] James M. Gere. Mechanics o f Materials. PWS Publishing Company, 4th edi tion, 1984. [25] Leonaxd Meirovitch. Principles and Techniques of Vibrations. Prentice Hall, 1st edition, 1997. [26] H.T. Banks and C.A. Smith. Modeling and identification of m aterial parameters in coupled torsion and bending. Math. Comput. Modeling, 18(8):1— 19, 1993. [27] Herbert Goldstein. Classical Mechanics. Addison Wesley, 6th edition, 1959. [28] Erwin Kreyszig. Introductory Functional Analysis with Applications. Wiley, 1st edition, 1978. [29] J. Wloka. Partial Differential Equations. Cambridge University Press, 1st edition, 1982. [30] R E Showalter. Hilbert Space Methods fo r Partial Differential Equations. Pit man, 1st edition, 1977. [31] A Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, 1st edition, 1983. 77 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [32] Larry L. Schumaker. Spline Functions: Basic Theory. John Wiley & Sons, 1st edition, 1981. [33] H.T. Banks and K. Ito. A unified framework for approximation in inverse problems for distributed parameter systems. Control Theory and Advanced Technology, 4(l):73-90, 1988. [34] P.M. Prenter. Splines and Variational Methods. Springer-Verlag, 1st edition, 1989. [35] Germund Dahlquist and Ake Bjorck. Numerical Methods. Prentice-Hall, 1st edition, 1974. [36] R.C. Smith. A nonlinear physics-based optim al control method for magne- tostrictive actuators. NASA ICASE Report, 98(4):l-27, 1998. [37] H.T. Banks and Gabriella A. Pinter. Approximation results for param eter esti mation in nonlinear elastomers. International Series o f Numerical Mathematics, 126(1):1— 13, 1998. [38] H.T. Banks, C.K. Lo, Simeon Reich, and I.G. Rosen. Numerical studies of identification in nonlinear distributed param eter systems. International Series o f Numerical Mathematics, 91(l):l-20, 1989. [39] H.T. Banks and D.A. Rebnord. Analytic semigroups: Applications to inverse problems for flexible structures. Differential Equations with Applications in Physics, Biology, and Engineering, 133(l):21-35, 1991. 78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix A Existence and Uniqueness In this Appendix we sum m arize some results from functional analysis and semigroup theory pertinent to our treatm ent of well-posedness of the ACTEX distributed pa ram eter systems. Generally we will not provide proofs for the results that are stated here. Rather we refer the reader to another source. A. 1 Preliminaries We state below the Riesz representation theorem. We refer the reader for a proof to Showalter [30, p. 16]. T h e o re m A .l (R iesz R e p re se n ta tio n T h e o re m ) Let H be a Hilbert space and f £ H*. Then there is an element x £ H (and only one) fo r which f ( y ) = (x,y)} y e H . The function x ^ f x from H to H* is called the Riesz Map and is denoted by R jj. Note that it depends on the scalar product as well as the space. In particular, R jj is an isometry of H onto H m defined by Rn(x)(y) = (x,y)n , x,y € H. We state the following results which are elementary and will be used below for proving semigroup generation theorem. We refer the reader to Showalter [30, p.92] for proofs. L e m m a A .l Let B £ j C(H) with ||B|| < 1. Then ( / — B )-1 £ C(H) and is given by the power series £{H)- 79 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. L e m m a A .2 Let A £ L(D(A), H) where D(A) C H, and assume (p i — A)-1 £ C(H), n £ C. Then (A/— A)-1 £ C(H), A £ C, if and only if [I — (p — A )(pl — A)-1]-1 C(H), and in that case we have (XI - A ) - ‘ = (ftl - A )-1 [ l - ( ! i - X)(fil - /I ) - 1 ] . T h e o re m A .2 Let V and H be Hilbert spaces fo r which the identity V ^ H is continuous. Let a : V x V — > C be continuos, sesquilinear, and V — elliptic. In particular, \cr{u,v)\ < k\u\v\v\v R e a (u ,u ) > c||u||2 where u, v £ V and 0 < c < k. Define D(A) = {u € V : |(j(u,u)| < ku\v\H, v £ V } where ku depends on u, and let A £ L (D(A), H) be given by a { u , v ) = (- A u , v )h u £ D{A) and v £ V . Then D(A) is dense in H and there is a 9q , 0 < 6q < such that fo r each A £ S ( f + #o) = \ z £ C : \arg(z)\ < f + ^o} we have (AI — A)-1 £ C(H). For each 9, 0 < 0 < 9a, there is an Mg such that ||A(A/ —A) 1 ||L(H) < Mg where A £ S (9 + . A.2 Semigroup Generation Theorem For a proof of the following generation theorem see Showalter [30, p.100]. 80 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T heorem A .3 (G eneration o f A n a ly tic Sem igroup) Let A E L(D(A), H) be the operator o f Theorem A.2. Then there is a family o f operators { T (t) : t E S(0q)} C £(H) generates an anlytic semigroup, satisfying the following: (a) T(0) = I; (b) T(t + r) = T(t) ■ T(t), t,r £ S(90) and fo r x,y E H, the function t (T(t)x,y)jf is analytic in S{90); (c) for t E S(0O ), T(t) E L(H, D(A)) and ^ = A- T(t) € L(H); (d) if 0 < e < do, then fo r some constant C(e), \\T(t)\\ < C(e), ||M T (t)|| < C(e), t E S(0O - e), and for x E H , T(t)x — > x as t — > 0. t E S(6q — e) We can use the above semigroup to define mild solutions for dw - * « = A a + f w(0) = tw 0 (A.l) For w0 e n and f E [L2{{Q,T),'H)]2 the variation of param eters representation w(t)=T(t)w0 + f T(t — s)f(s)ds (A.2) Jo defines a mild solution for A .l which for w0 E D(A) and f E [C1((0, T), H)]2 is the unique strong solution to the following weak problem + ^ = ^ ^ ^A '3^ for < f> E D(A). The following corollary readily follows from the Generation Theorem A.3 and (A.2). C orollary A .l (E xistence and U n iq ueness) I f A is an operator o f Theorem A.3, then fo r every wq E H there is a unique w E (7([0,oo ) ,H ) D ^ “ ((Ojoo) ,TT) o f (A .l). For each t > 0, iw(£) E D(AP) fo r every integer p > 1. The infinite differentiabihty of the solutions and the consequential inclusion in the domain of every power of A at each t > 0 follows directly from (A.2). 81 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix B Approximation We summarize in this Appendix some results from approximation theory pertinent to our treatm ent of state space approximation of the ACTEX distributed param eter system. Our goal is to use the Trotter-Kato approxim ation theorem. Trotter-K ato the orem guarantees the continuous dependence of an analytic semigroup T(t) on its infinitesimal generator A and the continuous dependence of A on T(t). Therefore we can deduce that the convergence of a sequence of infinitesimal generators is equivalent to the convergence of the corresponding semigroups. B. 1 Preliminaries Let A be a closed and densely defined operator on H and let R(A : A) = (XI — A)~l be its resolvent. If // and A are in the resolvent set p(A) of A, then we have the resolvent identity R(X : A) - R(p : A) = (p — X)R(X : A)R(p : A) (B .l) The identity motivates our next definition (see Pazy [31, p.36-41]). D efin ition B .l (P seu d o R esolvent) Let A be a subset o f the complex plane. A fam ily J ( A), A £ A , of bounded linear operators on H satisfying J(X) - J(p) = ( p - X)J(X)J(p) 82 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. fo r X,p. E A is called a pseudo resolvent on A . Our m ain objective next is to determ ine conditions under which there exists a densely defined closed linear operator A such th at J ( A) is the resolvent family of A. The following lemma and the two theorems are elementary and will be used below (see Pazy [31, p.36-41]). L em m a B .l Let A be a subset ofC, the complex plane. IfJ(X) is a pseudo resolvent on A , then J(X)J(/j.) = J(p)J( A). The null space N(J(X)) and the range, R(J(X)), are independent o f X 6 A. N(J(X)) is a closed subspace o f H . T h eorem B .l Let A be a subset o f C and let J(X) be a pseudo resolvent on A . Then, J{X) is the resolvent o f a unique densely defined closed linear operator A if and only if N(J(X)) = {0} and R(J(X)) is dense in H. T h eorem B .2 Let A be an unbounded subset ofC and let J(X) be a pseudo resolvent on A . I f /?(«/( A)) is dense in X and there is a sequence Xn € A such that |A n| — y oo and \\XnJ{Xn)\\ < M fo r some M then «/(A ) is the resolvent o f a unique densely defined closed linear operator A. The following lemma and the theorem will be used to prove the Trotter-Kato ap proximation theorem (see Pazy [31, p.85-86]). L em m a B .2 Let A and B be the infinitesimal generators o f Cq semigroups T(t) and S(t) respectively. For every x 6 H and X E p(A) D p(B) we have R(X : B ) [T(t) - S{t)] R(X : A)x = f S(t - s) [R(X : A) - R{X : B)] T(s)xds Jo T h eorem B .3 Let A, A n be infinitesimal generators of Cq semigroups T{t) and Tn{t) such that ||T (t)|| < M e"4 and |[Tn(t)|| < Mew t where M , oj same constants, then the following are equivalent: (a) For every x E H and X with ReX > w. R(X : A n)x — > R(X : A )x as n -* ■ oo. 83 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (b) For every x € H and t > 0, Tn(t)x — F T{t)x as n oo. Moreover the convergence in part (b) is uniform on bounded t intervals. We say that a sequence of operators A n, i— converges to an operator A if for some complex number A, R(X : A n)x — > R (A : A )x for all x 6 H. In the above theorem we assumed the existence of the r — limit A of the sequence An and furthermore assumed th at A generates a Co semigroup T (t) such that ||T(£)|| < Me"*. It turns out that these assumptions axe unnecessary. This will follow from our next theorem (see Pazy [31, p.85-86]). T heorem B .4 Let A n be infinitesimal generators o f a Cq semigroup Tn(t) such that ||Tn(f)|| < Me"*. I f there exists a A 0 with ReA 0 > u > such that (a) fo r every x € H , R (A 0 : A n)x — » ■ R(X0)x as n oo and (b) the range o f R (A 0) is dense in H . then there exists a unique operator A which generates a Co semigroup T (t) such that ||T(£)|| < M ewt and that R(X0) = R (A 0 : A) B.2 Trotter-Kato Theorem A direct consequence of the above Theorem is the following theorem (see Pazy [31, p.87-89]). T heorem B .5 (T rotter-K ato A pproxim ation) Let A n generates Cq semigroup Tn{t) such that ||X'7 l(;£)|| < Me"*. I f for some A o with Re A o > aj we have: (a) A s n -> oo, R(X0 : A n)x — > • R{Xq)x for all x € X and (b) the range of R {A 0) is dense in X . then there exists a unique operator A generating a Co semigroup T{t) such that ||T (t)|| < Me"* and that R(Xq) = R(X0 : A). I f T f t ) is the semigroup generated by A then as n — > oo, Tn{t)x — > T(t)x for all t > 0 and x € X . The limit is uniform in t fo r t bounded intervals. We next give a general approximation theorem for analytic semigroups that is a generalization of the well-known Trotter-Kato theorem, see Banks and Rebnord [39, p.27] for a proof. 84 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T h e o re m B .6 Suppose we have Hilbert spaces H and TCN, N = 1 , 2 , . . with H N C Li. Let P N : Li — > LiN denote the orthogonal projection o f TL onto TiN satisfying P N — > • I strongly. Suppose that A N and A are the infinitesimal generators o f analytic semigroups T N(t) and T(t ) on H N and Li respectively satisfying the following: There exists a region E = Y ,& = |A E C : \arg(\ — A 0)| < f + where S > 0 such that E U {A0} C p{A) fl~ =1 p{AN) and (i) There exists a constant M independent of N such that fo r all A € E and N = 1 ,2 ,___ (ii) fo r some A G S and each w E H. we have R\(AN)PNw R \w . Then we have (iii) fo r each w E H , T N{t)PNw — > T(t)w unformly in t on compact subintervals [0,oo), (iv) fo r each w E H and integer k > 1, (AN)kT N(t)PNw — » • A kT{t)w uniformly in t on compact subintervals (0, oo). 85 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix C Polynomial Spline We summarize in this Appendix som results from polynomial splines approximation that are pertinent to our treatm ent of ACTEX state approximation, see Schumaker [32, p.108-142] and Banks and Kunisch [6, p.299-312]. C.l Preliminaries Let [a, 6] be a finite closed interval and let A = with a = x 0 < x x • • • < £fc+i = 6 be a partition of [a, 6]. Put A = maxo<i<fc(®t+i— xf), and A = minQ<i<k{^i+i— £,). Let A be a positive integer. By P m we denote the set of all polynomials p of order m. D efinition C .l (P o ly n o m ia l Splines) Let M. — (m l5 • • • , m,t) € 1 < m,- < m. The space S(Pm, A4, A) = {s : there exist st - € P m such that s(x) = st -(x) fo r x € [x,-,Xi+i), i = 0,... ,k, and = D 3 's(xi), fo r j = 0,1,..., m — 1 — m i, i = 1,..., fc} is called a space o f polynomial splines o f order m and M. is called the multiplicity vector. I f M = (1,..., 1), then we put Sm(A) = 5 ( P m, (1,..., 1), A) and call 5m(A) space o f polynomial splines with simple knots. The dimension of S(Pm,A4, A) can be shown to be m + 2D £=i m, = m + K. 86 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. D efinition C .2 (E xten ded P artition ) Let A = {xj}|=o+1 be a partition o f [a, b], let M. = (m i,..., m/c), 1 < m,- < m , and K = m,-. Suppose yi < y2 < • • • < y2m+K, is such that yx < - • - ym < a,b < ym+K+i < • < y 2m+K, and mi mfc JAn-t-l ^ ^ y m + K — x h j 5 E k i 5 Then we call A = {yt} ^ +A an extended partition associated with S(P m, M , A). D efinition C .3 (D iv id ed D ifference) Let {ut -}?L! 6e a set o f functions defined on a set I , and let ti, • • • ,tm be points in I such that ti < ■ • • < t m. We put M D ti-, u u h , j tm Ur j tm ( U i ( * i ) U2(ti) U i ( t 2) ^1 (tm) Um(tl) Umifm) j = detM ^ 15 , U ^ tli Ul, 5 77 1 7 U 7 7 1 For f : I -+ R, the rth order divided difference over the points ti, ■ ■ ■, tr+i is given by D ti, •• U i , X, t- „ r — 1 r + 1 / D t \ , ••• U l, X , ? ^r-f-1 „ r . 2 / Lei (a: — = (x — y)% fo r x > y and (x — y)'+ = 0 otherwise. T h e o re m C .l Let A = {yi}]™*K be an extended partition associated with S(Pm,fA, A) and suppose b < y2m+K- For F t -(*) = (— l) m(yl+m — yf) [yf, • - •, yf+m] (a; - i/)+_1, /o r a < x < b, the set \ K form s a basis for S(Pm, A4, A) with B,(x) = 0 for x £ [yu yi+m], Bi{x) > 0 fo r x € (yi, j/l+m) and E jgJ* B{(x) = 1 fo r all x < E [a, 6]. 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. If b = j/2m+iv, then Bm+K(b) is defined to be the l i m ^ B m+K{x). Again is a basis and the remaining properties hold. Let B[a, 6] = { / : [a, 6] — ► R : |/(x )| < oo for x e [a, 6]} Before we can define the quasi-interpolation operator Iq : B[a, 6] — > < S ’ m(A) we need the following lemma, L em m a C .l Let A = {a = xo < x i < — < x^+i = 6} be a partition of [a, 6] and choose rj so that ^ < A . Then there exists a refinement A* = {a = X q < • • • x*+1 = & j- such that — < A* < A* < — V V Corresponding to the partition A* we define the extended partition Ul ' Vm — Q > i J/m+1 = j • , ym+l — U l ? ym+l+l = 2/2m+Z — b Let n = to + I, and for each i = 1,2, • • •, n, put - l Tij = yi + (yi+m ~ Vi){j - l)(m - 1) 0) ,P n = 5 ( - 1) ! ' ( C J ) for j = 1, • • •, m, where (m -1)! 9t'm K 1 m— 1 4> i,m {t') = ( t V i+ r) r=l j —1 = lie * - * * ) = 1 (C.2) 88 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Finally, let * i / = J 2 aijiTiu ■ • Tij]f (C.3) i=i z = 1, • • •, n. T h e o re m C.2 For f € E i?[a,6] /e£ T/zerc /< g zs a linear operator into S m{A*) C < S m(A), and /gp = p, fo r allp € P m - Moreover |/g /|c 7(a ,& ) < ( 2m ) 7 n | / | C (a,6 ) / o r / € C ( a , 6) . Next we introduce some notation. The common notation for Sobolev spaces, Lp, and W fc ,p , fc = 1,2,..., 1 < p < oo will be used; the domain of the functions will be (0,1) or (0,1) x (0,1) and the range will be in R. The notation for the norm will be standard, e.g., | • |^iP for the norm of W k,p. Differentiation of a functon of one variable is denoted by D , whereas a subscript, for instance Dx , is used to denote a partial derivative. A polynomial p is said to be of order m if p(x) = a,-a;1-1, A, E R - i a-jji ^ 0. Lets recall some useful inequalities. T h e o re m C.3 Let g be a polynomial of degree m , — oo < a < b < oo, and 1 < p < q < oc. Then 2(p + l) 2 m Is^lo.p b — a T h e o re m C .4 Let g be a polynomial of degree m and — oo < a < b < oo. Then \Dg\0,o o < r ~ — \g\o,oo b — a I f g is degree m = 1,2 or 3, and 1 < p < oo then \Dg\olP < T ^ — \g\o,P b — a where C is independent of g, b, and a and can be calculated explicitly as a function o f m and p. Finally we state the m ain result of this section, 89 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. T h e o re m C .5 L f l < p < q < o o and 1 < a < m, then for f E \V a'v{a, b) we have iD } < C'1A < T - r+1/9- 1 /Po;m_£ r (Daf-K ) I DrlQfkq J - V >* the first inequality holding for r = 0, • • ■, a — 1, the second fo r r = cr, • • •, m — 1. The constant Ci depends only on m and p. In Theorem C.5, wr(y; £)p, with 1 < p < oo, r a positive integer and 0 < 8 < (b—a)/r, denotes the modulus of smoothness given by Urigifyp = S U p o ^ S \A l9 \LP (Irh) r where Irh = [a, b — rh] and Ar hg(t) = 23;=o(— l) r 1 ^ ) g(t + ® ^)* Many estimates on u > r(g‘ , d)p are known, and we ju st present a few of them. L e m m a C .2 Let r > 1 and 1 < p < oo. Then for g E Lp(a, 6) u r(g\ S)p < 23u r-j(g] S)p, 0 < j ^rigj d)p < 2 | f i r | o , p Moreover ur(g;5)P < Sr \Drg\0 < p if g E W r'p(a, 5), 1 < p < oo. C o ro lla ry C .l For 1 < p < oo, there exists a constant Ci = Ci(m,r), such that fo r all f E Lp(a,b) we have inf{\f ~ 5k P = ^ € <Sm(A)} < Cium (f; A )p 90 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C.2 Approximation Theorem The following theorem shows that polynomial splines form an approximating family satisfying the approximation condition (3.39) in Chapter 3. We refer the reader to Banks and Kunisch [6, p.299-312] for the proof. T h e o re m C .6 Let P N be the orthogonal projection of L2(I) onto and let f € W 1,2(I). Then there exists a constant C independent of N and f such that \ f - P Nf\ax < C N - l \Df\Q 3 Furthermore [D ( f — PNf)\o,2 ~> 0 as Ar — > • oo. 91 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix D Parameter Estimation We summarize in this Appendix some results from param eter estim ation for dis tributed parameters systems pertinent to our treatm ent of the ACTEX param eter identification problem. D .l Preliminaries We assume that we are given an Hilbert space V C H that is continuously and densely embedded in H. Let cri(q) : V x V — » ■ C be a symmetric sesquilinear form which satisfies, (A) V-Continuity: There exists C \ such that, for q E Q, E V , we have k i ( 9 ) ( 0 , 0 ) | < c x\<l>\v\il>\v. (B) V-Coercivity: There exists c2 > 0 such th at for q E Q, 0 E V , we have Reai(q) (0,0) + \o\4>\]j > c2\< t> \v . (C) K-Parameter Continuity: For q,q E Q with m etric d, we have for all 0,0 E V k it e ) ( 0 > 0 ) — ° ‘it e ) ( 0 > 0 ) l ^ < * te > 9 ) l0 M 0 k - We assume approximating subspaces H N satisfying: (HI) : HN C V 92 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (H2) : 4>€V,\ P N<i>-<t\v — » ■ 0 as N — > • oo We note that {H2) also implies th at |< f > — P N< j > | — > • 0 for each 4 > 6 H. D.2 Parameter Estimation Theorem We next give a general estim ation theorem for distributed param eter systems. We refer the reader to Banks and Ito [33, p.86-87] for a proof. T h e o re m D .l Suppose that H N satisfies (H1)-(H2) and that<Ti(q), < r 2(q) V-parameter continuous, &i(q) symmetric, crfiq), cr2(q) V-continuous, V-coercive. Let qN — » • q in Q. Then for A > 0 we have R \ (^AN(qN)> j PNff — > R \ fj in the V norm for any ff € R . 93 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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Creator Khan, Taufiquar Rahman (author)

Core Title Inverse problems, identification and control of distributed parameter systems: Applications to space structures with active materials

Contributor Digitized by ProQuest (provenance)

School Graduate School

Degree Doctor of Philosophy

Degree Program Applied Mathematics

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

Tag mathematics,OAI-PMH Harvest

Language English

Advisor Wang, Chunming (committee chair), Rosen, Gary (committee member), Yang, Bingen (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c16-62500

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Rights Khan, Taufiquar Rahman

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