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"' ""' '1""“"‘ 1111“ 111‘“ 11111‘1'1 II . I" 111111 “1‘1‘1‘ "'1" 1 . . I '11 "1'1""1"‘ 1“'1' 1111 “1111111 1“‘ 1‘111 11111111'1'111'111I 1 1 1111“ 1 "'11.“ '11‘11‘11“111'1‘1“1“111‘1‘ ““‘““‘“‘““‘ '. '1'1'1' "' ' “1'1/‘11 11“‘ """1 'f" 1‘1.’ "1‘ "‘11 11 . . 1‘“ -._-—-—-\:lc.— gfif’ __._-—————:'~—dv MM ‘ '_., -—~—_—.—.—;: __..._.—-———-..3 WWW INHIIIWHIHWHUIIHTHWW 31293 01093 0059 111E515 ' LIBRARY man—3am 3““ ' amenity v: ' '—" w v This is to certify that the dissertation entitled PARAMETER ESTIMATION FOR THE FAST AND SLOW SUBSYSTEMS OF A PROCESS OPERATING IN COUPLED SINGULARLY PERTURBED FORM presented by Michael Joel Cook has been accepted towards fulfillment of the requirements for Ph . D . degree in S3; stems Sc ience flaw. fiwgfix Major professor DMWOZ’) /7faZ MS U is an Affirmative Action/Equal Opportunity Institution 0- 12771 MSU LlBRARlES m RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. PARAMETER ESTIMATION FOR THE FAST AND SLOW SUBSYSTEMS OF A PROCESS OPERATING IN COUPLED SINGULARLY PERTURBED FORM BY Michael Joel Cook A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering and Systems Science 1982 © Copyright by MICHAEL JOEL COOK 1982 ABSTRACT PARAMETER ESTIMATION FOR THE FAST AND SLOW SUBSYSTEMS OF A PROCESS OPERATING IN COUPLED SINGULARLY PERTURBED FORM BY Michael Joel Cook The input and output of a deterministic singularly perturbed system, operating in coupled form, are observed over a finite time-interval. The problem under considera- tion is to determine the system parameters of the decoupled subsystems from these measurements. The nature and for- mulation of the singularly perturbed system are examined along with the fundamentals of systems identification. A finite time-interval identification method is inves- tigated which utilizes a filter to annihilate the initial condition response, and models disturbances as solutions to a homogeneous differential equation. The adaptation of this method is applied to the singularly perturbed problem, and a unique procedure for its implementation is presented via a heuristic study of linear time invar- iant systems. The experimental results indicate success of the methodology for reasonable separation of the Michael Joel Cook subsystems, as characterized through the inherent time scale parameter. To Leonard and Rheva, my dad and mom, for their unconditional love and sacrifices, and for their wisdom in sending me to a tutor in seventh grade when I was ready to give up on mathematics. And to kindred air pirates Steve and Sandy, my brother and sister, for the greatest times in my life. They always want the best for me. Who could ask for a better family! iii ACKNOWLEDGMENTS I express my sincere gratitude to Dr. Robert O. Barr, my advisor, whose supervision and guidance and encouragement throughout my program were heightened not only by his friendly nature, but also by his genuine concern and understanding--especia11y through the ebbs and flows of my research. My affirmations also extend to the other members of my committee--Dr. Robert A. Schlueter, Dr. Hassan K. Khalil, and Dr. James H. Stapleton--for all their valued inputs, directions, assistance, and kindness, and for their creative instructions throughout my coursework interactions with them. I also thank Dr. Erik Goodman and the consultants at the Case Center for their readiness and availability in.answering my questions concerning the modus operandi <3f the Prime 750 Computer. And thanks are extended to Ali Saberi--a colleague and a friend--for his feedback and insights, and for the enjoyable discussions we share paralleling systems theory with everyday life. iv My special thanks to Carol Cole, whose fast and excellent typing of this dissertation is greatly appreciated. ‘ I would also like to express my absolute validations to all my friends, whose constant support and love I will cherish always. Three of these folk are very special to me: Karin Montgomery, whose love and confidence in me and my abilities have always given me strength; Kitty Buffington, who has been there each and every day with a smile, giving me her special attention and friendship; and Dr. Peter Brobeil, my white-water sternman and best buddy, whose altruistic beliefs and attitudes towards life and nature enhance the quality friendship that we share. These friends are a treasure to me. I might not have reached my goals without them. Finally, a heartéfelt thanks to a fellow joey, Al "Whistles" Fast, my clown instructor, who kept me (a.k.a. ”Noodles") laughing through all the experiences of the last two years. TABLE OF CONTENTS LIST OF TMLES O O O O O O O O O O O O O O C O O O I LIST OF FIGURES O O O I. O O O O O O O O O O O O O 0 Chapter I. II. III. IV. INTRODUCT ION O O O O C O O O O O O O O O O O O 1.1. Singular Perturbation . . . . . . . . . 1.2. Estimation and Identification . . . . . 1.3. Identification Schemes and Applications 1.4. Object of the Thesis . . . . . . . . . . FORMULATION OF THE PROBLEM . . . . . . . . . . 2.1. The Singularly Perturbed System' . . . . 2.2. Exact Decomposition of the System . . . 2.3. Approximate Decomposition of the System 2.4. Properties of the System . . . . . . . . 2.5. The Identification Problem . . . . . . . PROBLEM SOLUT ION . O O O O O O O I O O O O O O 3.1. Considerations for a Solution to the Problem . . . . . . . . . . . . . . . . 3.2. Model of a System to b Identified . . . 3.3. The Algorithms of H-identification . . . 3.4. Identifying Decoupled Subsystems via H-identification . . . . . . . . . . . . COMPUTATIONAL CONSIDERATIONS AND RESULTS . . . 4.1. Modification of H-identidication 4.2. Computational Aspects . . . . . 4.3. Direct Application and Results . 4.4. Discussion of Results . . . . . SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS . . vi Page viii ix 11 14 15 15 21 30 35 38 38 43 49 57 61 61 69 74 84 87 APPENDIX A . . . . APPENDIX B . . . . APPENDIX C . . . . LIST OF REFERENCES vii Page 90 93 94 124 LIST OF TABLES Table Page 4.1. Examples (n = 1, m = 1) . . . . . . . . . . . 80 4.2. Results--Slow Subsystem . . . . . . . . . . . 81 4.3. Results--Fast Subsystem . . . . . . . . . . . 82 4.4. Example and Result (n = 1, m = 2) . . . . . . 83 viii LIST OF FIGURES Figure Page 3.1. Singularly Perturbed Process . . . . . . . . 39 3.2. Least-Squares Equation Error Model . . . . . 43 3.3. Basic Model . . . . . . . . . . . . . . . . . 49 3.4. Basic Identification Model for Singularly Perturbed Systems . . . . . . . . . . . . . . 57 ix CHAPTER I INTRODUCTION The purpose of this chapter is to serve as a founda- tion for the results of this thesis. Section 1.1 will commence the discussion with the concept of singular ‘perturbation and its significance in systems theory. {The role of estimation in the identification of systems vvill be the theme of Section 1.2. A survey of general identification schemes and their application will be examined in Section 1.3. The aggregated problem of iden- tification and singular perturbation as presented in Section 1.4 will complete this preview. 1.11. Singular Perturbation A system can be defined as a function whose domain is a: set of inputs and whose range is a set of outputs. The: fumctional relationship and behavior between the inputs and outputs are based on the inherent characteris- tics of the physical system under consideration. In studying this behavior it often becomes necessary to construct a mathematical model, in which the relationships between the physical variables in the system are mapped onto the mathematical structures via equations. To acquire a full representation of the system often requires many variables and equations, which tend to increase the com- plexity of the model. This largeness of the model can be due to the inclusion of all factors which affect the system, even those contributions which have little effect on the behavior. They may be nearly negligible because their cumulative effect is small during the operation of the system, or they might be relatively "short-lived" in comparison to the other variables and thus do not dominate the mid-term and long-term phenomena. Since these small contributors must be included for a complete representation in the original model description of the system, they are classified as "parasitic." Examples of such parasitics are small time constants, masses, moments of inertia, capacitances, inductances, and any other relatively unimportant parameters. Besides increasing the dynamic order of the system, these para- sitics introduce "fast modes" making the model "stiff": that is, hard to handle on a digital computer because the equations require small integration intervals. Solu- tion of the system equations becomes overly-complicated, although numerical methods have been developed to increase the efficiency of the solution procedures [CLA], [GE]. A set of dynamic equations containing such parasitics is called a singularly perturbed system, since the solutiontx>the equations can be constructed as a power series in terms of a small perturbation parameter 6 [WA], [GA]. In a singularly perturbed system there are generally many time-scales needed in describing the system behavior. For instance, there can be very fast and very slow phenom- ena requiring three or more separate time-scales [DE], [HO] . In this thesis, the discussion will be limited to the case of two time scales, for very fast and for normal-speed phenomena. The separation between the two time scales is directly related to e in that the smaller 6 is, the wider the separation. The extension of this thesis to multi- time-scaled systems is an area for further research. The small parasitics of the system are considered as proportional to the perturbation parameter c [KO-1]. The effect that the parasitics have on the system behavior occurs immediately upon initiation of the system. There-' fore the states of the system associated with these para- sitics are called "fast" states, and their swift effect dies out rapidly allowing for these states to reach their quasi-steady-states very quickly. The other states not associated with the parasitics take longer to affect a change in the system behavior and are therefore dubbed "slow" states. The underlying assumption in singular perturbation theory is that the slow variables remain constant at the onset of the system and the fast variables are dominant during this short time, and by the time the changes in the slow variables become noticeable, the fast variables have reached their quasi-steady-state. A wealth of studies has arisen in reference to linear singularly perturbed systems (to be discussed below non- chronologically). Kokotovic et a1. [KO-2,3] provide an overview on the use of singular perturbations in reduc- ing the model order by first neglecting the parasitics and then reintroducing them as boundary layer corrections in separate time scales. Kokotovic et a1. [KO-1] develop an iterative procedure to more accurately separate the full-ordered system into two subsystems of slow and fast states which avoids inconsistencies associated with the approach of first neglecting parasitics. Javid [JA] constructs a reduced-order state observer for the slow reduced system wherein parasitics are neglected, and derives two types of observer errors. And Saksena and Cruz [SA] design a robust low-order observer estimating the slow states using only the reduced model. O'Reilly [OR] formulates a full-order observer for the singularly perturbed system as a composition of two observers, one for each of the slow and fast subsystems. Kokotovic and Haddad [KO-4] find criteria for controllability of the slow and fast states of the system by separately analyzing the two subsystems defined by these states, and apply the separation procedure to the time-optimal control problem [KO-5]. The ever-popular linear state regulator problem is treated by O'Malley [OM—1,2] via Hamiltonian methodology and asymptotic expansions. He gives conditions under which the optimal regulator-control problem has a unique asymptotic solution for sufficiently small 6. Basic theorems providing for the uniqueness and uniformness of the Riccati solution to the linear regulator problem are furnished by Kokotovic and Yackel [KO-6]. Stochastic control of the linear singularly perturbed system with additive noise is discussed by Haddad and Kokotovic [HA-1] wherein the optimal control is approximated by a near-optimal control obtained as a combination of a slow controller and a fast controller computed in separate time scales. Conditions on asymp- totically stable feedback controllers by using Hurwitz criteria is dealt with by Porter [PO-1]. By applying frequency-domain techniques, Porter and Shenton [PO-2] use the special structures of the transfer function matrices for singularly perturbed systems to construct controllers. They find that in the frequency-domain the full system with slow and fast states is asymptotically equivalent to parallel connections of the reduced slow subsystem with the fast subsystem. Khalil and Kokotovic [KH] examine stability test criteria for the implementation of effective control laws for linear singularly perturbed systems with multiparameters all of the same order, and for systems with multitime scales. They also design a near-optimal control law which does not depend on the values of the small parameters. State estimation is explored by Haddad [HA-2] for the case of input distur- bances by developing two lower order filters in separate time scales. For unknown 8, Sebald and Haddad [SE] examine the problem of estimation of the slow and fast states of the singularly perturbed system. And Chow et a1. [CHO-l] rely on singular perturba- tion techniques to take a system with slightly damped high frequency oscillations and decompose it into two separate subsystems, one containing the slowly varying dynamics and the other containing only the fast oscilla- tory modes. Then the subsystems are analyzed in different time scales. This decomposition is also shown to work for systems whose slightly damped large eigenvalues result in sustained high frequency oscillations. These situa- tions can occur in mechanical and electromechanical systems such as the spring-mass suspension system and the multi- machine power system. Chow and Kokotovic [CEO-2] also design a near-optimal state regulator by decomposing the singularly perturbed system into two subsystems with separate fast and 81 w modes and then developing a com- posite controller based on the inputs of each subsystem. 1.2. Estimation and Identification The modelling of a system and its analysis has sig- nificance in many fields, e.g. economics, biology, medi- cine, ecology, and certainly in the field of process control. The model building is an enormously important condition for making use of control theory. In order to better understand the dynamic behavior of a system, the system must be properly designed. Treating the system through mathematical representations allows for the model- ling to be accomplished. Most certainly, an ample model of the system to be controlled is necessary or else the construction of a control law is not feasible. A mathematical model can be considered as a function between the physical variables of the system under con- sideration and the mathematical equations presuming the system structure. These equations may be simple alge- braic, differential, or difference types of equations. The plant system is then said to be described or modelled by the set of mathematical equations involving these physical variables. The model is constructed theoretically and/or empiri- cally. By theoretically analyzing the system through the usage of balance equations and the physical laws of conservation, the simple subprocesses of the plant can be described by mathematical equations. Adjoining these equations with the appropriate boundary conditions yields the mathematical model that is desired. The theoretic-construction approach is utilized if experiments in the plant cannot be accomplished, or if the plant is not yet in existence. However, if an experimental analysis of a plant with arbitrary structure is executed, the input and output signals are measured. Evaluation of these measurements through an identification procedure produces the mathematical model of the plant process. This estimated model is then a description of the input-output behavior of the process. A precise definition of identification is now stated [ZA]: Given a class of systems S where each member of the class is completely specified, the identification of a system A consists in finding a system s c S that is input-output equivalent to A. It is important to note that the definition requires input-output equivalence and does not require 3 e S to be identical to A. Cer- tainly, for a given input-output relation, there is gen- erally no unique system representation [KA-l]. In this thesis the systems under consideration are to be completely specified within a parameter set and the purpose of iden- tification is to determine, that is, estimate these parameters. There are three major complications which appear into any real identification problem. The first deals with the absence of knowledge concerning initial condi- tions of the system; the second is the presence of random noises shadowing the input and output observations; and the last is the difficulty in establishing a meaningful and convenient method for estimating the system parameters as a function of the observations. When disturbances are present they act on the process and thus affect the output signals, making more difficult the determination of a mathematical model from correctly measured input and output signals. Hence, the method of identification must separate the piece of output into the information component and the disturbance component. For linear systems, the disturbance is a single additive component of the output superimposed upon the information- carrying part. The identification scheme should overcome the influence of these disturbance signals. The model's validity rests upon the connection between the variables of the mathematical structure and their physical counterparts. Hopefully, the relation between these entities is isomorphic [AH]. That is, the values assumed by the variables in the mathematical model are in a one-to-one property correspondence with the values that are measured. 10 Since one goal of identification is to determine the system model for a process under investigation, it is relevant to discuss the various model classifications that are closely associated with the identification prob- lem. A model described by sets of differential or alge- braic equations is called parametric, and the identifica- tion procedure is to determine the parameters in this structure. The number of these parameters is finite, and their true values uniquely determine the system model [BL]. These parameters may be constant or vary with time. The response description obtained from an experimental analysis of the physical process is a Egg: parametric model, for no a priori structure of the model can be assumed, and no finite number of parameters deter- mines the model. If the dynamics of the system are described by partial differential equations (e.g., parabolic, elliptical), then a distributed parameter model is being used, whereas a lumped parameter model is one using ordinary differen- tial equations for its structure [FA]. A lumped parameter model lends itself to being discretized in time from an original continuous time model. Models may also involve statistical values for some of its variables (stochastic-type), or there may be no probability structures at all (deterministic-type). In 11 the former case, the stochastic phenomena can present themselves in the form of random input and output dis- turbances, or, perhaps, the initial states of the system may be random variables with known or unknown means and covariances. Improper measurements of the inputs and outputs in conjunction with uncertainty in the process are a cause for difficulties in the effective identifi- cation of systems. 1.3. Identification Schemes and Applications There exists quite a variety of strategies dealing with the problem of system identification [AS-1] . Step response and frequency response techniques [RA], [CHE] can accommodate both parametric and nonparametric models [IS], whereas Fourier and spectral analysis as well as correlation techniques [GO], [RA] apply only to nonparam— etric models. It is the tactics of parameter estimation—- applicable solely to parametric models--that will be under consideration in this thesis. The first of the parameter estimation methods is that of Least Squares [ST], [GR], [LEE]. This method is based on the thought that the most probable value of the param- eters is the one "that minimizes the sum of the squares of the differences between the actually observed and computed values multiplied by weighting factors measuring the degree of precision" [GAU]. Within the least squares 12 methodology are the specialized schemes of Generalized Least Squares [CL], Instrumental Variables [W0], [YO], Levin's method [LEV], and the Tally principle [PE-2]. The Maximum-Likelihood method [AS-2] estimates the parameters by selecting the value of the parameters which "makes the observed data most probable in the sense that the likelihood function is maximized" [GOO]. The likeli- hood function is a function of the conditional probability density of the data given the parameters. Thus, the method chooses the parameters' values that makes as proba- ble as is possible the data which is in fact observed [BL]. Another scheme for parameter estimation is through a Bayesian approach [DO], [PE-1]. In this method the estimates are taken from the a posteriori conditional density of the parameters given the input-output data. This is done by the use of Bayes' Theorem [LEE] on the conditional probability density of the data given the parameters--the function which is the argument of the likelihood functional. In both the Maximum Likelihood and Bayesian estima- tion methods, it is necessary to make assumptions on the probability distributions of the data and parameters. The difficulties involved with expressing the a priori information in terms of a probability distribution can be circumvented by using prediction error methods [AS-2]. 13 In this case, a prediction model (similar to the Kalman filter [RA-2]) is implemented and the parameters are esti- mated by minimizing a criterion which is a function of the predicted output. One of the main purposes of identification is to determine the dynamics of a process so that a proper control law may be designed and implemented to cause the system to perform according to some set of criteria. For example, better knowledge of a production industry plant or an economic system may be obtained for improved control. The identification procedure can also be utilized for a diagnostic examination to analyze the properties of a system, such as the determination of rate coeffi- cients in chemical reactions and reactivity coefficients in nuclear reactors. This goal has practicality in biology, economics, medicine, and many other related fields. Of course, identification of a process may simply be carried out to verify the structure of a theoretical model which was posed. And by continuously monitoring a process, a system identifier can learn parameters which vary slowly through time. Thus, at each instant of time, the system behavior is approximated and an effective controller can be implemented for that instant. A con- troller constructed by way of this type of parameter 14 learning procedure is called a: parameter adaptive con- troller [KU], [SK]. And sometimes when system parameters vary, a reliability index of that system.may change. A Check on the reliability of the system can be maintained by identifying the system parameters. 1.4. Object of the Thesis This first chapter has staged an introduction to two areas of systems theory: singularly perturbed systems and parameter estimation. The rest of this treatise is to serve as a tutorial to the unification of these two studies. Chapter II will establish the algebraic language of this text and the mathematical structure of the sin- gularly perturbed system. The third chapter will concen- trate on the solution to the identification problem at hand, including the mathematical and systems approaches and techniques utilized. Chapter IV will provide tangible reckoning of the newly-constructed procedures through computer-oriented examples. The final chapter will summarize the results of this thesis and provide directional comments for further research in this area. CHAPTER II FORMULATION OF THE PROBLEM In this chapter the mathematics behind the singularly perturbed system will be introduced. The construction of the two-time scale concept will be included along with the algebraic "evolution" of that system. Sections 2.2 and 2.3 will focus on the decoupling; i.e., separation, of that system into fast and slow subsystems. Section 2.4 will deal with properties and theorems for the singularly perturbed system. The discussion will end with the fifth section representing the formal problem statement. 2.1. The Singularly Perturbed System It is the nature of systems engineering to commence a discussion with analytic statements regarding the vari- ables of the problem under consideration. In this case, the statements may consist of vector-form ordinary dif- ferential equations interrelating the variables. The general form of these is 3%: n = 'F1(nlu't) t ”(to) = no (201-1) y F2(n:u:t) (2.1.2) 15 16 where n is an n'-dimensional time-differentiable state vector, y is a q-dimensional output vector, u is an r-dimensional input vector, and t is the scalar-time variable with the initial time instant being to' (Until otherwise stated, let to = 0.) As discussed in the introduction to this thesis, the fundamental concept embedded in singular perturbation theory is that of slow and fast states. During the onset of the process the slow variables remain relatively con- stant compared to the fast variables which die out quickly; i.e., reach their quasi-steady-states. Thus, if there are n slow states x and m fast states 2, the n'-state vector n can be partitioned as n A [’2‘] (2.1.3) with n + m = n'. Rewriting (2.1.1) and (2.1.2) in terms of x and 2 yields 2% x = f(x,z.u.t) . X(0) = x0 (2-1-4) 6 - - ° 1 s d—t z - F3(X.2.u.t) . 2(0) - z (2. . ) Y = F4(X.2.u.t) (2.1.6) where f, F3, and F4 are merely the adjusted functionals of F1 and F2. (Generally, f and F3 are also functions 17 of a parameter c which represents small "parasitic" masses, capacitances, etc. of the system.) Now, by assuming that t is the time frame for the characteristics that are slow, and allowing T to repre- sent the time frame for the fast characteristics, it is reasonable to assume that the ratio of t to T is some small positive number s [KO-l]. That is, if t is in seconds and T is in milliseconds, then s is 0.001. Assum- ing that T = 0 corresponds to the instant t = 0, it is found that t E . (2.1.7) If I were now changed to microseconds, then 6 would decrease in magnitude. And when a is shrunk, the fixed t period will correspond to quite a long I period. So, as e is decreased toward zero, one fixed t period will correspond to an infinitely long T interval. Thus, fol- lowing the example, if 6 were decreased to 10-9, one t period (1 second) would contain one billion T periods (nanoseconds). Feeling familiarized with this two-time scale concept, it seems reasonable that the states x and 2 should inter- act according to t and T, respectively. That is, the states x are %~times slower than 2, and likewise are their respective derivatives. Accordingly, F3 can be 18 rescaled as g = eF3, so that g and f are of the same order of magnitude [KO-1]. Thus, equations (2.1.4) and (2.1.5) become x = f(x,z,u,t) , x(0) = x0 (2.1.8) 2|“ 8 5% z = g(x,z,u,t) , z(O) = z0 . (2.1.9) (Recent results by Chow et a1. [CHO-l] show this state description is utilizable for systems with lightly damped high frequency modes.) Note that as e + 0 here, ad? XS = f(xstzstutt) I XS(O) = X0 (2'1'10) 0 = g(xs,zs,u,t) (2.1.11) where xs(t) and zs(t) are the quasi-steady-states of x(t) and z(t), respectively. Here then, equation (2.1.11) is algebraic and can now be backward substituted into (2.1.10) to yield a new differential equation in x8. It is worth noting that for the long-term studies in the classical quasi-steady-state approach, the deriva- tive of z with respect to t in equation (2.1.5) is set equal to zero which then yields the system of equations (2.1.10) and (2.1.11). Thus, dzs/dt = 0 which requires 2 to be a constant. However, equation (2.1.11) defines 28 as a time-varying quantity. Even though this procedure is justifiable in yielding approximate solutions, it does 19 leave this obvious inconsistency. It is through the introduction of the two-time scale concept discussed above that this inconsistency is circumvented. For in equation (2.1.11) e(dzs/dt)==0 results from letting c + 0 rather than from sz/dt = 0. If the time scale is changed to I using (2.1.7), equations (2.1.8) and (2.1.9) become = e f(x,z,u,tr) (2.1.12) elm N g(x,z,u,cr) . (2.1.13) 310 N Now as e + 0, equation (2.1.12) implies that x remains constant in the fast time period. Therefore, during this initial fast time period the only fast variations are in 2. Accordingly, z = zf + 25 (2.1.14) and thus equation (2.1.13) becomes (with zf = z - 2s and e + 0 and dzs/dt = 0): g% 2f = g(X°,z: + zf(1),u(1),0), Zf(0) = 20 ’ 25(0) (2.1.15) often called.the "boundary layer system." Finally, it is recognized that (2.1.10) and (2.1.11) represent the slow model and (2.1.15) represents the fast model, with 20 (I! x(t) xs(t) (2.1.16) zs(t) + zf(§) = zs(t) + zf(T) . (2.1.17) Ill z(t) (From hereon,‘zf will be expressed in the T-domain, so that the investigation of the system characteristics at t = 2 seconds (say, with e = 0.001) will then involve examination of zf at T = 2000 milliseconds.) These representations (2.1.16) and (2.1.17) are merely the zero-order approximations of the asymptotic expansions in e of the solutions x and z for the system (2.1.8) and (2.1.9) [OM-3], [GA]. The solutions x and z are therein expressed as x(t) = xo(t) + ex1(t) + 62x2(t) + ... + xo(r) + ex1(t) + €2§2(1) + ... (2.1.18) z(t) = zo(t) + ezl(t) + 8222(t) + ... + 50(1) + 221(1) + e222(r) + ... . (2.1.19) Thus, (with io(r) O--see (GA, pg. 29]) "D IID x(t) xs(t) xo(t) (2.1.20) IID z(t) 28(t) + zf(1) é 20(t) + 20(1) . (2.1.21) 21 2.2. Exact Decomposition of the System It is natural to ask at this point if the system (2.1.8), (2.1.9) with (2.1.6) can be decoupled into separate subsystems. In facilitating this task, the linear time-invariant matrix version of this system will be used: -d—x=Ax+Az+Bu x(0)=x° (221) dt 11 12 1 ' ' ° 5 EL 2 = A x + A z + B u z(0) = z0 (2 2 2) dt 21 22 2 ' ° ° y = Clx + C22 + Eu (2.2.3) where A11 is n x n A12 is n x m B1 is n x r A21 is m x n A22 is m x m B2 is m x r C1 is q x n C2 is q x m E is q x r and where the argument t for x, z, u, and y has been suppressed for ease of notation. From hereon, equations (2.2.1)-(2.2.3) will be called system CS for coupled system. Kokotovic et al. [KO-1] provide an iterative scheme to separate the slow and fast subsystems, wherein the newly-determined subsystem matrices are obtained in terms of A11, A12, A21, and A22 without ill-conditioned modal transformations. And an alternative algorithm 22 based on the modal transformation matrices is presented in [KO-7]. Since these algorithms are cleanly presented and available in those papers, a transformation technique akin to that in [KO-4] will be discussed here. Consider the matrix T: T= (2.2.4) L I2 where L and M are any matrices of the proper sizes, along with I1 and 12, to yield a square matrix T. It is easily verified by checking T-lT = TT-1 = I3 that I1 eM T'1 = (2.2.5) --L Iz-eLM‘ . Assume now that L and M satisfy A - A22L + EL(A11 - A12L) = 0 (2.2.6) 21 A - M(A22 + ELAlz) + €(A11 - AlZL)M = 0 . (2.2.7) 12 (By checking matrix sizes, it must be that L is m x n and M is n X m, I1 is n X n and 12 is m X m, and thus I3 and T are (n + m) X (n + m) matrices.) Introducing the change of variables 23 £1 = X (2.2.8) 52 = z + Lx = z + Lg1 (2.2.9) into system CS yields g% 51 = (All - A12L)g1 + A1252 + Blu (2.2.10) 931 g = (A - A L + eLA - eLA L)g + dt 2 21 22 11 12 1 + (A22 + 51.1112);2 + (32 + eLBl)u (2.2.11) y = (c1 - CZL)£1 + C252 + Eu . (2.2.12) By using (2.2.6) in (2.2.11), the system simplifies to .g% g1 = (All - AlzL)g1 + A12€2 + Blu (2.2.13) 6 ._ e d? 52 — (A22 + eLA12)g2 + (32 + eLBl)u (2.2.14) y = (c1 - CZL)§1 + c252 + Eu . (2.2.15) Another change of variables v2 = 52 (2.2.16) v1 51 - 8M52 = 51 - eMv2 (2.2.17) turns (2.2.13)-(2.2.15) into d _ - HE v1 ' ‘A11 ' A12L’V1 + [€(A11 ' AlzL’M + A12 - M(A22 + eLA12)]v2 + [B1-M(B2-+eLB1)]u (2.2.18) 24 d e a; v2 = (A22 + eLA12)v2 + (B2 + eLB1)u (2.2.19) y = (C1 - CZL)v1 + [e(C1 - CZL)M + C2]v2 + Eu . (2.2.20) By applying (2.2.7) in (2.2.18), the simplification becomes (2.2.21) d e a? v2 (A22 + eLA12)v2 + (B2 + sLB1)u (2.2.22) Y = (C1 - CZL)V1 + [c~:(c1 - chm + C2]v2 + Eu . (2.2.23) What has thus been constructed is a state transformation - q r - c F - 1 v1 51 eME2 x eM(z+Lx) x = = = '1' v2 - £2 . z+Lx . L z . (2.2.24) turning the coupled singularly perturbed system CS into the decoupled slow and fast subsystems (2.2.21)-(2.2.23). Assuming that A is non-singular, the choice of 22 L and M can be made through their asymptotic expansion representation as 25 1 L = A22 A21 + 0(6) (2.2.25) _ -1 M — A12 A22 + 0(8) (2.2.26) which, for small 6, satisfy equations (2.2.6) and (2.2.7). Definition: A matrix P is of order 6, 0(8), if there exists positive constants 6* and c such that the norm IIPII satisfies IIPII §_cc for all ee[0,e*]. If these two expressions for L and M are substituted in (2.2.21)-(2.2.23), the resulting system is -1 - A A - A120(s)]v1 + d _ 6? v1 ' [A11 12 A22 21 -1 + [131 - (A12 A22 + 0(6))B2 - -1 -1 ' EA12 A22(A22 A21 + 0‘5”31 ' -1 - eO(e)(A22 A21 + 0(a))B1]u (2.2.27) 5 il»v = [A + cA-l A A + sO(e)A ]v + dt 2 22 22 21 12 12 2 -1 + [B2 + eA22 A21 B1 + cO(e)Bllu (2.2.28) 1 y = [c1 - c291;2 A21 + 0(6))1v1 + + [2:[C1 - C2(A212A21 + 0(a)) x x (A12 A3; + 0(e))] + C2]v2 + Eu . (2.2.29) 26 Defining A A A - A A'1 A (2 2 30) o 11 12 22 21 ' ° A _ -1 c 3 c - c A"1 A (2 2 32) o 1 2 22 21 ' ' and recalling the fact that A0(e) = 0(6) for any matrix A, then for small 5 system (2.2.27)-(2.2.29) simplifies to §% v1 = [A0 + 0(a)]v1 + [so + 0(a)]u (2.2.33) 8 g% v2 = [A22 + 0(a)]v2 + [32 + 0(6)]u (2.2.34) y = [C0 + 0(8)]V1 + [C2 + 0(5)]v2 + Eu . (2.2.35) This is the c-asymptotic expansion representation of the decomposed system (2.2.21)-(2.2.23). The zero-order approximation of this system is thus d _ '3? v1 — on1 + Bou (2.2-36) 5 IL V = A v + B u (2 2 37) dt 2 22 2 2 ' ° y = Cov1 + sz2 + Eu . (2.2-38) If the expressions for L and M in equations (2.2.25) and (2.2.26) are expanded [KO-4] to 27 _ —1 -2 2 M = A A'1 + e(A A A"2 - 12 22 o 12 22 - A. A"2 A A A'l) + 0(52) (2 2 40) 12 22 21 12 22 ' ° ' then the first-order approximation of the decoupled system becomes: 6 = [A - 6A A"2 A A ]v + 1 o 12 22 21 o 1 + [B - e(A A"2 A B + A A'1 B )]u o 12 22 21 1 12 22 2 (2.2.41) e6 = [A + eA’1 A‘ A ]v + [B -+eA'1 A B ]u 2 22 22 21 12 2 2 22 21 1 (2.2.42) _ -2 y — [cO - ec2 A22 A21 Ao]v1 + -1 . + [c2 + see A12 A22]v2 + Eu . (2.2.43) 2.3. Approximate Decomposition of the System Pausing for a moment, it is interesting to examine what would result by approximately decomposing system CS. This procedure is done by setting 6 = 0 in equa- tion (2.2.2). This yields 222 + Bzu (2.3.1) 0 II A ‘E + A 21 or NI l _-1 — — -A22(A21x + Bzu) (2.3.2) 28 where the bar indicates that e = 0. Also §'= C E + C E’+ EH . (2.3.3) 1 2 Substituting (2.3.2) into system CS leaves the slow subsystem. d _ _ 0 3? x8 - ons + BouS , xS(0) - x (2.3.4) Y5 = Coxs + Eons (2.3.5) where §'= x E} §’= ys, and 3': us are the slow parts I S of the variables x, z, y, and u, respectively, and E 3 E — c A-1 o 2 225 2 . (2.3.6) To derive the fast subsystem, it is assumed that the slow variables are constant during the fast transients, so that dE/dt = 0 and i = constant during that fast period. Subtracting (2.3.1) from (2.2.2) and (2.3.3) from (2.2.3) produces z(JL z - ——’ ) = A (x - E) + A (z - E) + dt dt 21 22 + 32m - '3) (2.3.7) y - y = C1(x - E) + C2(z - E) + E(u - Ii) . (2.3.8) Since x is predominantly slow, x 2'}, and letting 29 zf = z - 2' (2.3.9) uf =u-E=u- (18 (2.3.10) yf =y -§=y 'Ys (2.3.11) then equations (2.3.7) and (2.3.8) become d _ e E? zf(t) - A22 zf(t) + B2 uf(t) , 213(0) = z° - 2(0) (2.3.12) yf(t) = C2 zf(t) + Euf(t) . (2.3.13) Introducing the fast "stretching" time scale T = t/e produces 54T— zfm = A22 zfm + azufm. sz) = z°-E(0) (2.3.14) yf(T) = C22f(T) + Euf(T) . (2.3.15) Notice that (2.3.1) and (2.3.4) represent the linearized time-invariant matrix versions of (2.1.11) and (2.1.10), just as (2.3.14) is to (2.1.15). And notice that (2.2.36)- (2.2.38), the decomposition via asymptotic expansion, is the same system as (2.3.4), (2.3.5), (2.3.12), and (2.3.13) with v1 and v2 being identified as x8 and zf, respectively. Looking back at (2.3.2), it becomes appar- ent that E = 28, and thus 30 l _ _ -1 — .- zs(t) ‘ A22 A21 "s‘t’ A22 B2 u‘s‘t’ 9 2 3 16 — A3 xs(t) + 33 us(t) ( . . ) so that _ _ -1 o _ -1 23(0) ‘ A22 A21 x A22 B2 “s‘o’ _ o where A3 is m X n and B3 is m x r. It is now time to collect together the equations of the system to be examined in the remainder of this thesis. Therefore, equations (2.3.4), (2.3.5), (2.3.14), (2.3.15), and (2.3.16) will comprise that system, hereby dubbed system DS, the decoupled system. 2.4. Properties of the System There is enough foundation at this point to discuss properties of the singularly perturbed system. The system CS has already been shown to possess a two-time scale characteristic. This effect is evident in the eigenvalue structure of that system. Lemma 1: Suppose A22 exists and has all L.H.P. eigen- values, none on the imaginary axis. Then, as e + 0, the first n eigenvalues of system CS tend to the eigenvalues of the reduced system 31 (2.3.4), while the remaining m eigenvalues tend to infinity as the eigenvalues of é-AZZ . Proof: By rewriting equations (2.2.33) and (2.2.34) as fvl- pAo+0(e) 0 . _v1. FBO+0(e) V2 0 %A22+%§0‘€’ V2 %Bz+%0(€’ 3 Av + Bu (2.4.1) it is clear that the eigenvalues of system CS are con- tained in the eigenvalues of this system matrix A--which consist of the eigenvalues of A6 + 0(a) and the eigen- values of é-AZZ + %~0(e) . (Also see [KO-4].) Q.E.D. Thus, system DS consists of two subsystems: the slow subsystem containing n small eigenvalues (in magni- tude) and a fast subsystem with m large eigenvalues. And the smaller 8 is, the greater is the separation of these two groups of eigenvalues. In an asymptotically stable system the fast modes corresponding to the large eigenvalues are important only during a short period (measured in T-units). And after that period those modes become negligible and the behavior of the system can be described merely by its slow modes (using the t-domain). (This is related to the concepts of Dominant Pole 32 Theory [SH] which holds that the system eigenvalues of small magnitude dominate the system behavior.) Neglecting the fast modes (parasitics) is equivalent to assuming that they are infinitely fast; that is, allowing 2 + 0 in system CS. The last paragraph mentioned the concept of stability. Basically, a system is asymptotically stable if when the system is started near an equilibrium point, the state of the system approaches that equilibrium point as t + m, where an equilibrium point is a constant vector solution of the state differential equation. Theorem 2.1: If the real parts of all eigenvalues of A0 and of A22 are negative, then there exists an 5* > 0 such that for all e < 5* the system CS is asymp- totically stable. Proof: Referring to (2.4.1), the system matrix A 0 there becomes, for sufficiently small 6, [ C) 1. ] 0 E A22 and thus, if eigenvalues of A0 and A22 are in the left- half of the complex plane then the system CS is asymp- totically stable. This can be considered also as _ l l _ o(AO + 0(a)) — o(Ao) + 0(a) and 0(8 A22 + 5 0(5)) - '% o(A22) + %20(e), and thus for small positive 6 these spectra become simply o(Ao) and %~o(A22), respectively. (Here o(P) stands for the spectrum of P--the set of all 33 eigenvalues of P.) Hence, if o(Ao) and o(AZZ) are in the left-half plane then the system CS is asymptotically stable. Q.E.D. Thus, stability of system DS implies stability of system CS. Other discussions on stability and stabili- zation can be found in [KO-2], [KO-4], [PO-l], [WI], and [GRU]. Controllability of the system CS can now be estab- lished, too. Definition: A pair of matrices (A,B) is a controllable pair (and thus the system K = Ax + BU is controllable) if rank [B,AB,A2B,...,An-1B]==n, where A is n X n and B is n> 0 such that for all e < 8* the system CS is controllable. Proof: From (2.4.1) it follows that for 8 small the controllability of the reduced and boundary layer systems, that is of the pairs (AO,B°) and (A22,B2), implies the controllability of the original system CS. (That is, the subsystem (2.2.33) is a regular perturbation of the reduced system (2.2.36) and the subsystems (2.2.33) 34 and (2.2.34) are connected through u, but have different eigenvalues.) (See also [KO-2], [KO-4].) Q.E.D. Thus, controllability of system DS implies controllability of system CS. It should be noted here that a matrix K exists such that A22 + 82K is non-singular. And the controllability of the system CS is not influenced by u = Kz + w. Thus, even if A.1 doesn't exist, Theorem 2~25till hOldS) but 22 with the matrix A22 + BZK replacing A22 in the definition of A0 and B0 in equations (2.2.30) and (2.2.31). The last concept to be discussed in observability. Definition: A pair of matrices (A,C) is an observable pair (and thus the system K # Ax + BU, Y = CX + EU is observable) if the rank 2 n-llT [C,CA,CA ... CA = n, where A is n X n and C is q X n and rank A = n. An analogous argument leads to the proof of the last important theorem: Theorem 2.3: If the pairs (AO,CO) and (A22,C2) are observable, then there exists an 8* > 0 such that for all e < 5* the system CS is observable. 35 Thus, observability of system DS implies observability of system CS. (For additional reading on observability see Javid [JA].) In summary, the formalized system DS is compiled here: Slow Decoupled Subsystem (SDSS): git xs(t) = ons(t) + Bough). xsw) = xo 6 Rn y8(t) = Coxs(t) + Eou8(t) zs(t) = A3xs(t) + B3us(t) Fast Decoupled Subsystem (FDSS): 39T- zf('r) = A22 zfm + Bzufh), zf(0) = zo-zs(0) eRm yf(r) C22f(r) + Euf(T) . 2.5. The Identification Problem It is now of interest to examine these two decoupled subsystems with respect to parameter estimation concepts. The problem under investigation can now be stated simply as: Given 5 priori knowledge that a system exhibits the behaviors characteristic of slow and fast phenomena, determine the "inner workings" of that system from the input and output data records available. 36 What do the words "inner workings" refer to? In the present case of this problem, it is initially assumed that the process under scrutiny is of the bi-structural form of system DS. The "inner workings" of that system are then the internal mechanisms as defined by the system matrices and the time-scale parameter c. What they describe are the functionings of the decomposed slow and fast subsystems. So once the matrices A0, 30' Co' E0, A3, 33' A22, B2, C2, E, and the initial values x°,zo and the parameter c are known, then the system DS is totally describable, and is then ready for further explorations such as in the area of optimal control. Therefore, for the remainder of this thesis, the goal will be the determination of these parametric quantities. In order to discover these quantities, it will be necessary to choose the proper experimental design [GOO], [IS]. This includes the selection of the input signals [LE], for care need be taken to use inputs which will act to "excite" all the fast and/or slow states so that accurate determination of the parameters will be made. Also an appropriate identification scheme must be used which: (a) has good discriminating ability in order to identify the faster components over the slower ones; (b) is "good" in the sense that it yields a model 37 consistent with the data; and (c) yields estimates which converge, in some statistical sense, even in the presence of noise. Finally, since the system DS is operating in two time scales, it is important to consider relevant sampling time(s) on the process to be identified. It will be the intent of the next chapter to utilize identification theory to solve the problem of estimating the parameters of the decoupled singularly perturbed systems SDSS and FDSS from a process operating in coupled form (CS) . CHAPTER III PROBLEM SOLUTION The scope of this chapter will be to provide the theoretical solution to the problem of identifying the decoupled singularly perturbed subsystems. The first section will take into account the salient characteristics of the singularly perturbed structure and discuss what type of an identification method might be used to exploit these features. Also discussed in that section is model representation. This is expanded in Section 3.2 where the models for the identification-solution method are dealt with. The algorithms involved in the identifica- tion procedure are unveiled in the next section, and the last section presents the application of the identifi- cation method to the specific problem of a singularly perturbed system. 3.1. Considerations for a Solution to the Problem At this point, let us examine system DS again: snss: 33E xs(t) ons(t) +Bous(t); xs(0) =x(0) (3.1.1) ys(t) Coxs(t) + Eous(t) zs(t) A3xs(t) + B3us(t) 38 39 a _ . FDSS. 8 3E zf(t) — A222f(t) + Bzufuz). zf(0) 2(0) - 23(0) yf(t) = szf(t) + Euf (3.1.2) The block diagram of the mechanics of the singularly perturbed process may be seen in Figure 3.1. In this SDSS FDSS . “f Yf Figure 3.1. Singularly Perturbed Process figure the coupled system CS is visualized as two decom- posed subsystems operating in parallel. What is desired is some method and set of procedures for determining the dynamics of each subsystem from the input-output data information that is given. Before addressing the issue of the method of iden- tification, it is relevant to note something about the model to be identified. Up to this point, the systems 40 of equations under consideration have been represented in state space equation forms. Since, in the framework of identification, models of the fast and slow subsystems are to be found, it is totally reasonable to try to deter- mine any structural format as long as it is equivalent in an input-output sense. The following are two different representations of the same observable system. The first is the observable input-output canonical form [GU], [BE] for a multi-input, multi-output (MIMO) system: §(0)y(t) = 6(0)u(t) . (3.1.3) P(D) is a square non-singular (q X q) polynomial matrix in the differential operator D and 0(5) isha (q X r) polynomial matrix in D, and u(t), y(t) are (r X 1) input and (q X 1) output vector functions, respectively. For the case of a single-input, single-output (8180) system, this can be expressed in a scalar linear time-invariant differential equation: (n) (n-l) _ y (t) + an“1 y (t) + ... aoy(t) - = b u(n) (t) ‘1' bn_ n u‘n‘1’(t) + ... + bou(t) . 1 (3.1.4) The second representation is the observable companion form in state space form: 41 3% x(t) Ax(t) + Eu(t) y(t) Cx(t) + Eu(t) (3.1.5) The transformation from (3.1.5) to (3.1.3) can be divided into two steps [BE]. In the first step, by elimi- nating state vector x(t) from equation (3.1.5), an equiva- lent representation of the form (3.1.3) is obtained. During the second step, a unimodular matrix is formed, with which the representation obtained in the first step can be transformed to the desired input-output canonical form satisfying certain requirements on the degrees of the element polynomials in P(D) and 6(D). The transforma- tion from (3.1.3) to (3.1.5) can be obtained either through the Structure Theorem [WOL] or by the algorithm developed by Guidorzi [GU]. Other transformation procedures between the two representations can be found in Ogata [0G], and algorithms on the relationship of the initial conditions between the two forms are dealt with by Heinen [HE]. With these equivalency considerations, it is therefore legitimate to choose to find either the differential equations (3.1.3) describing the system behavior or the state space description (3.1.5) of that behavior. As to what kind of method would be suited to the problem at hand, a major consideration is the fact that the two subsystems operate with different time constants. 42 This difference manifests itself through the effect of the fast subsystem dynamics. Since the fast dynamics dissipate rapidly (on order % times faster than the slow dynamics), a key concern should be a procedure which can identify a continuous-time system in a limited time frame. Of the multitude of schemes in the literature, several methods show promise of accomplishing identifica- tion in a finite time period. Obviously, though, any identification attempted--in particular, adaptive control procedures--are performed in a finite time interval, even if they are only theoretically valid on the infinite time frame. There exists a well-defined procedure of identification which has been proved to be valid and successful on a finite time interval [PEA-1,2L which will be discussed at length later. In most identification methods, the initial condi- tions of the system have to be determined along with the system parameters, even though it is the set of system parameters that is of primaryinterest. With the problem at hand--of identifying the fast and slow subsystems-- finding the initial conditions for each separate subsystem adds to the complexities of the problem. The basic issue is to determine the system dynamics of FDSS and SDSS. The same afore—mentioned finite-time procedure (called 43 H-identification), due to Pearson, uses a noncausal filter which eliminates the initial conditions during the iden- tification process so that the system parameters can be identified alone. 3.2. Model of a System to be Identified Since H-identification will be used in this disser- tation, it is now time to examine more closely this pro- cedure and how it relates to the singularly perturbed system identification problem. This procedure is a least squares equation error parameter identification technique (see Figure 3.2), but differs from other known applica- tions of least squares in a number of ways. The first of these is, as mentioned above, that only input-output n u { Process + ~43 Figure 3.2. Least-Squares Equation Error Model 1P 44 data is presumed to be given over a fixed finite time interval with no attempt to estimate unknown initial conditions. The second characteristic of H-identification is sufficiently general to include a variety of nonlinear, time varying, differential delay, possible unstable, multivariable system models. The next is that the formu- lation leads to an explicitly defined function of the parameters which simplifies the computations significantly. Also, this approach is a "one shot" identification scheme, as Opposed to other methods which are iterative in time. The last, and most germane feature, is the way in which the unknown disturbances are modeled on the finite obser— vation time interval. While Maximum Likelihood and other statistical methods of identification represent the dis- turbances by stochastic processes with underlying Markov process representations, the model for unknown disturbance signals in this approach is the deterministic homogeneous differential operator equation: 0. . T(D,6)d(t) = 2 Gina-1 d(t) = 0 , i=0 "D 6 o 1,0_<_t_<_t1<°° (3.2.1) (with the order being a s [0,0 max preselected) a 1' max where the 51's and the initial conditions are completely arbitrary. That is, the disturbances can be approximated 45 by the arbitrary solution of a homogeneous ordinary dif- ferential equation on a specified finite time interval. Actually, this model can be regarded as generating a stochastic process if the 61's and the initial conditions, d(i)(0) (i = 1,...,a), comprise 2a independent random variables with essentially infinite variances. The above model is actually quite suitable, since the data set is presumed to consist of input-output data observed on a finite observation time interval. Thus, the shorter the time interval, the more reasonable is the above dis- turbance model for a modest value of a. With respect to the finite time-interval length, it has been verified by simulation studies [PEA-2,3] that this time interval can be surprisingly short in many cases; i.e., on the order of the dominant system time constant, or less. At this juncture, it is appropriate to introduce the model formulation for the identification procedure. To refresh the memory, equation (3.1.3) is rewritten as: P(D)y(t) + 5(0)u(t) = o , 0 5 t 5 t (3.2.2) 1 0 Since the system parameters are contained within P(D) and 5(D), it would be best to express them in terms of the parameters in question as: P(D,w) + 5(n.w)u(t) = 0 , 0 5 t 5 t1 (3.2.3) 46 where 5 n ~ n-' P(D,(1)) = £0 pimp 1 (3.2.4) 1: " n ~ n-i Q(D,w) = '20 01(0)): (3.2.5) 1: and D 2 g% with w = (m1,...,w8) being the vector of system parameters. By defining a vector valued function f(w) with components fi(w) selected to reflect the ways in which the parameters enter into P and 6, it is then easy to define v(t) and V(t) (depending on the data pair [u(t),y(t)]) and operators P and Q such that equation (3.2.3) becomes P(D)v(t) + Q(D)V(t)f(w) = 0 , 0 :_t i t (3.2.6) 1 O This is the case for systems which are separable in the parameters--as are all linear systems-~wherein the generic decomposition of equation (3.2.3): P(D)v(t) + Q(D)g(t,w) = 0 , 0 5 t‘: t1 (3.2.7) admits to equation (3.2.6) via g(t,w) E V(t)f(w) . (3.2.8) Here V(t) is a matrix valued function of the data and f(-) is a continuously differentiable vector-valued func- tion of m with the single valued property 47 f(w) = f(w*) if and only if m = A* (3.2.9) for all m and w*. Now, if the actual input u(t) and output y(t) are corrupted by additive disturbances d1(t) and d2(t), respectively, so that u(t) and y(t) are observed accord- ing to: y(t) = §(t) + d1(t) , 0 5_t : t1 (3.2.10) u(t) = 6(t) + d2(t) , 0 5 t 5 t1 (3.2.11) then the model (3.2.3) becomes fi(0,w)[y(t) - d1(t)1 + 0(D.w)(u(t) - 42m] = 0 . 0 5 t 5,t (3.2.12) 1 0 And if d1(t), d2(t) are assumed to be solutions of the differential equation (3.2.1) on [0,t1], then operating on both sides of equation (3.2.12) with T(D,6) yields: T(0.6)§(D.w)y(t) + T(D.6)0(D.w)u(t) 0 ,05t5t 1. (3.2.13) This is analogous to T(D,6)P(D)v(t) + T(D,6)Q(D)V(t)f(w) = 0 , Oitfitl (3.2.14) 48 by following the same decomposition scheme that trans- formed equation (3.2.3) into equation (3.2.6). By expanding out T(D,6), equation (3.2.14) takes the form: up o-l D (D)v(t) + 61D P(D)v(t) +...+ 60P(D)v(t) + + DaQ(D)V(t)f(w) + a Da-lQ(D)V(t)f(w) +...+ l + daQ(D)V(t)f(w) O . (3.2.15) Writing this in vector form yields: DaP(D)v(t) + [Du-1P(D)v(t),...,P(D)v(t),DaQ(D)V(t), 5 El Da'10(D)V(t).....0(0)V(t)] Ga = o . f(w) 61f(w) (3.2.16) 6af(w) This can be simplified into the following model form: P(D)v(t) + 6(0)V(t)f(e) = o , 0 :.t 5 t (3.2.17) 1 where 6 = (61,...,6a,w1,...,m8) . (3.2.18) The vector function f(e) satisfies the same single-valued property (3.2.9) if the original function f(w) in (3.2.6) does. This should be the case with a model that has been properly parametrized. 49 To summarize, the basic model including disturbances is represented in equation (3.2.17). This is, of course, valid only for models which are separable in the param- eters. Otherwise, it would take the generic form: P(D)v(t) + 6(0)‘g’(t,e) = 0 , o 5 t 5 t1 (3.2.19) This generic model can be viewed in Figure 3.3. 1d(t) u(t) System y(t) S(w) P(D)v(t) + 6(D)§(t,e) = o 6 = (61(1)) Figure 3.3. Basic Model 3.3. The Algorithms of H-identification Having established the basic model to be identified, it is time to deal with the algebraic mechanics behind H-identification. Definition: The basic model (3.2.19) is H-identifiable if and only if the parameter vector 6 can be identified via the input u(t) and the output y(t) on a finite time interval [0,t1] 50 without estimating (implicitly or explicitly) the initial condition of the model. Now, let a square non-singular polynomial matrix F(D) be chosen such that F‘lw) [15(0), 6(0)] (3.3.1) is a proper transfer function matrix. The form of F(D) is m . F(D) = 2 Fibm‘l , m > n . (3.3.2) i=0 Then an auxiliary error function is implicitly designated via F(D)z(t,e) = P(D)v(t) + 6(D)§(t,e) . (3.3.3) To get a better handle on the nature of z(t,6), it is wise to first examine the homogeneous solution to equa- tion (3.3.3): F(D)z(t) = o . (3.3.4) The solution to this can be expressed as: Ax(t) , x(0) = xo 6 Rn 341-: x(t) z(t) Cx(t) (3.3.5) 51 where (A,C) is an appropriate observable pair with minimal dimension state space 3} Then z(t) takes the form: z(t) = CeAtxo . (3.3.6) Therefore, the particular solution to equation (3.3.3) will take the form: t z(t,6) = CeA x + v(t) + u(t,6) , 05 t_<_ t1 . 0 (3.3.7) By operating on both sides of this by F(D), it is seen that the particular solutions v(t) and u(t,9) are the zero state solutions to P(D)v(t) = P(D)v(t) (3.3.8) F(D)u(t.e) = 6(D)§(t.e) . (3.3.9) respectively. In the case of separability :h1 the parameters, the particular solution z(t,6) would be: At z(t,6) = Ce x0 + v(t) + M(t)p(6) (3.3.10) whereupon M(t) would then be found (through the degeneracy of equation (3.3.9» as the zero state solution to: F(D)M(t) = 6(D)V(t) . (3.3.11) However, for either case, the particular solution z(t,6) contains unknown parameters 6 and x0. Since it is desired 52 to identify 0 without actually estimating x0, the term CeAtxo need be eliminated. This can be accomplished via an annihilation filter H [PEA-1,2]. Laying some groundwork first, let T denote the Hilbert space of all vector valued square integrable functions on [0,t1]. And let v(t) and §(t,e) range over the space of piecewise continuous functions on [0,t1]. Also, let To denote the subspace of T containing solutions to equation (3.3.6). That is, At n Ce x0, xo 6 R , 0 §|t : t To = {x(t)IX(t) (3.3.12) Definition: The filter H is a linear Operator with domain T and range (T - T0) with the property: t 1 J{ [H(t,T)]W(T)dT o T CeAtW-leA TCT]w(I)dT t 1 T u(t) - CeAtW—ljf eA Tchp(-r)dr ° (3.3.13) IID (at) 9- H(x)(t)) 1 [I6(t-T) IID 0 where (A,C) is the observable pair for the system in equation (3.3.5) and t 1 T w 9 J{ eA tCTCeAtdt (3.3.14) 0 53 is the observability Gramian for that same system and—-only here--6 is the Dirac function. Now, H is a self-adjoint projection operator, as can readily be seen from H[H((p(t))] = H(w(t)) and adj [H(w(t))] = H(w(t)) (see Appendix A). But the sig- nificance to the filter H is that its null space is To; that is, H Atx0) — V x e R , 0 i t 5 t1 . (3.3.15) H(Ce o l I o s Thus, Operating on the solution z(t,6) ixiequation,03.3.7) yields: ~ A z(t,6) H(Z(t)9)) = ”(V(tH + H(u(t,9)) = A ~ ~ = v(t) + u(t,6) , 0 i t §_t1 . (3.3.16) Therefore, since the initial condition response is arbi- trary and has nO physical significance, the solution z(t,6) is projected down into the subspace (T - To), via H, thus annihilating the initial condition response on [0,t1]. In so doing, v(t) and u(t,6) are also pro- jected down into that same subspace yielding: t 1 T v(t) = v(t) - ceAtw‘1 jf eA Tchde (3.3.17) 0 t1 T u(t,6) = u(t,6) — CeAtW-l eA Tch(r,e)dr . 0 (3.3.18) 54 Now, H—identification minimizes the inner product norm of equation (3.3.16) yielding the functional J1(8) for the least squares minimization problem: t1 J1(8) = <2(t,6),2(t,6)> = [ ;T(t,6)2(t,6)dt . ° (3.3.19) Thus, any value of 6 which satisfies the basic model (3.2.19) is also a solution to J1(6) = 0 . (3.3.20) Conversely, any value of 6 which satisfies equation (3.3.20) is a candidate for a value Of 6 satisfying the basic model (3.2.19). By substituting equations (3.3.16)-(3.3.18) into equation (3.3.19), J1(9) unfolds as: t1 t1 J1(0) = f vT(t)v(t)dt+2 f vT(t)u(t)9)dt+ O 0 t1 + J{ uT(t)8)v(t)dt - nTW-ln - O - 2nTw'1y(e) - yT(e)w‘1y(e) (3.3.21) where t1 T n = J{ eA tCTv(t)dt (3.3.22) 0 t1 T y(0) = J{ eA tcTu(t,e)dt . (3.3.23) 55 In the case of separable-in-the—parameters models, u(t,6) = M(t)p(6) (3.3.24) so that y(6) = Np(e) (3.3.25) where t1 T N =f eA tCTM(t)dt (3.3.26) 0 The functional J1(6) can then reduce to an explicit func- tion Of 9: 32(0) = a + 2pr(0) + pT(e)4p(e) (3.3.27) where t1 a = [ VT(t)\)(t)dt - nT '10 (3.3.28) 0 t1 b = J{ M¢(t)v(t)dt - NTw’ln (3.3.29) 0 t1 0 = f MT(t)M(t)dt - NT 'lN . (3.3.30) 0 Thus, once the data is collected and (a,b,¢) are found, no further integrations are needed involving the data over [0,t1]. It is left to just minimize J1(6) or J2(6) with respect to 8. 56 Theorem 3.1: A minimizing value 9* for the positive definite J2(e) is a least squares estimate Of e which is unique if (as a sufficient condition) the data makes 0 positive definite, which occurs if the columns of6(D)V(t) are linearly independent functions on [0,t1]. Proof: By letting A = 9(6), due to its single-valued property, and setting J2(8) as: — T T J2(A) = a + 2b 1 + A 41 , (3.3.31) then the minimization becomes equivalent to v3é(1) = b + 01 = 0 (3.3.32) NIH which is the normal equation for ham=§w)+mO1,o5t5H. use» A unique solution to this normal equation (3.3.32) is found if and only if the columns Of H(t) are linearly independent on [0,t1]. For then H(t) has full rank, and thus 4 is positive definite and therefore non-singular, allowing a unique solution to the normal equation [SEB]. The subspace (T - To) contains the columns Of H(t), which can be represented as the projection of the function F-1(D)5(D)V(t)1 in that subspace. Since To is the null space for F(D), it follows that linear dependence, or independence, of the columns of M(°) cannot be altered 57 by Operating on that projection of F-1(D)6(D)V(t)1 with F(D). Q.E.D. Thus, 4 is non-singular if the columns of 6(D)V(t) are linearly independent functions. However, this is mainly of theoretical interest since it is not assumed that the data is differentiable. A final point concerning H-identification is that the theory is still valid for any initial time to (0 i-to < t1), whereby any reference to t = 0 in the algorithm is replaced by t = to. 3.4. Identifying Decoupled Subsystems via H-identification The aim of this section is to describe how H-identi- fication is used to identify the decoupled subsystems FDSS and SDSS. Re-examination of Figures 3.1 and 3.3 will help to facilitate this. Their combination is demon- strated in Figure 3.4. 0531 (t) Ext) + %;Y(t) DSSZ u(t) _ d(t) Figure 3.4. Basic Identification Model for Singularly Perturbed Systems. 58 In Figure 3.4, one of the decoupled subsystems (D881) is considered as the main system containing the parameter vector w to be identified, whereas the second decoupled subsystem (D882) is considered as the distur- bance model (containing parameter vector 6) with output d(t). These two mechanisms are acting in parallel, and the Observed output y(t) is as in equation (3.2.10): y(t) = §(t) + d(t) . (3.4.1) This matches equation (2.3.11) in that y(t) = ys(t) + yf(t) ; (3.4.2) that is, there is equivalency between the two sets Of signals via: {§(t). d(t)} = {ys(t). yf(t)} (3.4.3) provided there is no other outside noise disturbances acting on the system CS or its component subsystems. That H-identification should apply well here depends On several factors. First, and most significant, is that the output of the disturbance subsystem, d(t), is in fact a solution of a homogeneous differential equation. Without loss of generality, suppose that FDSS is the subsystem DSSZ considered as the disturbance mechanism. Examining the differential equations for FDSS as seen in 59 equation (3.1.2) shows that its output yf satisfies equa- tion (3.2.1) as d(t) where the order a of T(D,6) is at least as large as the sum Of the order of zf and uf. Therefore, d(t) g yf(t) will have some T(D,6) in existence to annihilate it. Secondly, since the goal is to identify --not one--but two (sub)systems, the need to determine the initial conditions of each is eliminated via the filter H, thus alleviating such difficulties. Thirdly, of course, is that H-identification has been proved to be quite successful [PEA-1,2,4] over very small time intervals [0,t1]. This is of great significance to the problem Of identifying the fast subsystem since the effects Of FDSS die out quite rapidly, necessitating the need for such a "fast" algorithm. At this point it might seem that H-identification, certainly used in the vein of points one and two just above, is application to any system that admits to a decoupling into subsystems. This might be the case; however, the application in this thesis of H-identifica- tion to the particular problem of the decoupled singularly perturbed system is special in the following way: the majority Of simulation studies carried out by Pearson et a1. indicate satisfactory performance of H-identifica- tion when the modes of the disturbance model D852 and the modes Of the system.model DSSl are located some 60 distance mo = %f , or greater, from each other in the complex plane. Since the nature Of the singularly per- turbed system is that it has subsets of modes located apart from each other in the complex plane, this is in line with the assumptions Of successful H-identification. As far as the issue of exogeneous noise, n(t), acting on the system CS, it seems reasonable that this will only affect identification if the modes Of n(t) are close to the modes of the subsystem to be identified. This concern will be addressed within Chapter V of this thesis. As a final point, once the parameters Of the main subsystem DSSl are found, that knowledge can then be incorporated to facilitate the identification of the remaining system DSSZ. Details of this will be discussed in the next chapter. In the next chapter, the computational considerations concerning the implementation of the algorithms will be dealt with, along with some examples and results to verify the direct numerical applications Of H-identification to singularly perturbed systems. CHAPTER IV COMPUTATIONAL CONSIDERATIONS AND RESULTS This chapter will deal with the implementation Of the H-identification procedure for singularly perturbed systems. Section 4.1 will modify the algorithms of H—identification which simplifies the procedure. The second section will discuss specific computational aspects involved with implementing the algorithm and running it on the computer. The third section will focus on the direct application Of the implemented algorithm for the singularly perturbed system, and provide specific examples and their associated results. The last section will examine these results and discuss their significance. 4.1. Modification Of H-identification Recently, a formulation of H-identification was described [PEA-5] which reveals the underlying least squares functional to be minimized over the system param- eters unrestrained by the parameters characterizing the disturbance modes. At the same time, this formulation accrues significant benefits in streamlining and simplify- ing the computations needed to Obtain the least squares functional from the Observed input-output data. 61 62 Taking the system description (3.2.3) without disturbances: §(D.w)y(t) + 6(0.w)u(t) = o and reshaping this yields: R(D)k(t.m) = 0 ) 0 g't : t1 . (4.1.1) (4.1.2) By including disturbances d(t) acting upon the system, equation (4.1.2) admits to a general form as: R(D)k(t,(0) = S(D,(0)d(t) , 0 5 t 5 t 1 0 Application of T(D,6) on both sides yields: T(D,0)R(D)k(t,w) = 0 which in vector form presents itself as 0 = [0“R(0), Da-1R(D),...,R(D)] ’ k(t,w) ' 61k(t,w) _dak(t,m)d (4.1.3) (4.1.4) (4.1.5) Then the equation error function z(t,6) is the solution tO F(D) z(t, e) = [0°‘R(0),0°"1R(0), . . . ,R(D)] ' k(t,w) ' 61k(t,w) _fiak (ttw). (4.1.6) 63 A with 6 = (6,w) = (61,...,6a,w1,...,w8). Here, F(D) is chosen so that F‘1(s)R(s)s“ (4.1.7) is a proper transfer function matrix. As in Chapter III, the filter H is applied to z(t,6) and the functional J1(6) becomes: t 1 J1(6) = = f ET(t,e)E(t,e)dt . ° (4.1.8) Now, if c(t,6) satisfies equation (4.1.6) with zero initial conditions, then J1(6) becomes (see Appendix B): t1 J1(e) = [ cT(t.e)c(t.e)dt — ¢T(e)w'1¢(e) ° (4.1.9) where t 1 T 0(0) J{ eA tCTC(t,6)dt . (4.1.10) 0 In the case of separable-in-the-parameters models, k(t,w) becomes: k(t,w) 5 U(t)h(w) (4.1.11) where U(t) is a matrix valued function of the Observed input-output data on [0,t1] and h(w) is a continuously 64 differentiable vector valued function of the system param- eters. Then, J1(9) will become an explicitly defined function of 0: Q Q h(w) 32(9) = [hT(w) ,(STHT((0)] oo od “do “ad H‘w’d h(u)) = [hT(w),6THT(w)]Q (4.1.12) H(w)6 where Fh(w) H(w) = ' 0 (4.1.13) 0 ', h(w) - 0 columns . and the Gramian is t1 , t1 (2 = f 1? (t)§(t)dt = [ YT(t)Y(t)dt - NT '11: ° ° (4.1.14) with Y(t) as the zero-state solution to F(D)Y(t) = [DaR(D),Da-1R(D),...,R(D)]U(t) (4.1.15) and 1 T N = Jf eA tCTY(t)dt . (4.1.16) 65 The matrix 0 is effectively a time correlation matrix with some bias removal terms which arise from the appli- cation of the annihilating filter H. Also, 9 is symmetric and non-negative definite, so that J2(6) satisfies the positive definite property: J2(6) 3 0 . (4.1.17) Thus, once 0 is computed, any hill-climbing technique can be used on J2(8) without further integrations of the data on [0,t1]. Examining J2(6), it is seen that the disturbance parameters, 6, enter quadratically in J2(6). Thus, a necessary condition for a minimal value Of J2(6) would be the vanishing of the gradients: %% = 0 and 39 = 0 . (4.1.18) Thus, solving the first of these yields: A 5 = - [HT(m)oddH(w)]'1HT (w)fldoh(w) (4.1.19) assuming the needed inverse exists functionally in w. Substituting 6 into J2(6) yields an explicit function Of the system parameters: J3(w)=hT(w)[ooo-oodn(w)(HT(w)0ddn(w)1‘1HT(w)0do]h(m) . (4.1.20) 66 In general, J3(w), though positive definite, is nonlinear, nonquadratic, and not necessarily convex in w. Although not explicitly present, the effect Of the disturbance parameters is manifest in the inverse of HT(w)QddH(w). At this point, the computations needed for 0 are undertaken. First, the solution Y(t) to equation (4.1.15) can be partitioned as: A ("1' A Y(t) = [Yo(t),Y1(t),...,Ya(t)], o _ t1. (4.1.21) A P. A (Thus, DYi+1(t) = Yi(t), o 5_t 5 t1, 0 _ a - 1.) Then N can be partitioned as: t N - [N N N 1 - 1 ATtCTlY (t) y (t)]dt - 0' 11°00! (I "’ e O ,..., a . ° (4.1.22) Then 0 is partitioned into (a + 1)2 blocks as t 1 _ T _ T -1 . . aij — J{ Yi(t)Yj(t)dt NiW Nj , 0 §_1,3 : a . ° (4.1.23) Whence 00d, ado, and add for equation (4.1.12) are defined as: QOd = [QOl"'°'Qoo] _ T Qdo -_Qod _ Q11... filo add = ; -. 1 . (4.1.24) 0&1... Qua 67 Let a matrix function Z(t1) be defined as: T -A t1 Z(t1) = e N (4.1.25) with a similar partitioning: Z(t1) = [Zo(t1)l--°tza(t1)] = e l [NO'°°°'NC1] (As occurred before, Dzi+1(t) = Zi(t), 0 5_t : t1, 0 5.i 5.0 - 1.) Then the bias removal term becomes: T At A t ~_ NTW-lN = ZT(t )e 1w'le 12(t ) = ZT(t )w 12(t ) 1 1 1 1 (4.1.27) where ... A -ATt1 -At1 W = e We . (4.1.28) is the observability Gramian for the pair (-A,C). Through the partitioning of Z and N, each 21 satisfies t -ATtl 1 ATT T Zi(t1) = e e C Yi(r)dr , i = 0,1,...,a ° (4.1.29) which is the solution (for t = t1) to the differential equation - _ _ T T _ Zi(t) - A zi(t) + c Yi(t) , zi(0) — o . (4.1.30) 68 Let (A,B,C,E) be a minimal realization [GOO] for the transfer function F-1(s)R(s)sa. Then using equation (4.1.15), Yo(t) is the solution to Xo(t) = Axo(t) + BU(t) Xo(0) = 0 Yo(t) = Cxo(t) + BU(t) 0 5 t 5 t1 (4.1.31) and thus _ -1 Yi+l(t) D Yi(t) 0 3 t 5 t1 . (4.1.32) This leads to the following theorem [PEA-5]: Theorem 4.1: Let (A,B,C,E) be a minimal realization for F-1(s)R(s)sa, with det A # 0. Then a least squares estimate Of w, in the separable case, is Obtained by minimizing J3(w). The matrices Yi(t), Zi(t) (0 3 i :ja) comprising 0 are efficiently determined from the zero state solution to: (i = 0) xo(t) = Axon.) + BU(t) Yo(t) = CXo(t) + BU(t) 2 (t) = -A¢z (t) + cTy (t) (4.1.33) 0 O O (1 = l,...,a) Xi(t) = A-1[Xi_1(t) - BD-iU(t)] -1 Yi(t) = CXi(t) + ED U(t) (-AT)‘1[ T Zi(t) zi_1(t) - c Yi(t)] (4.1.34) 69 where D-1 denotes the i-fold pure integration Operator with zero initial conditions. Proof: Since J3(w) has already been derived together with equations (4.1.33), it remains to establish equa- tions (4.1.34). Since Xi(t) can be defined iteratively _ -1 from Xi+1(t) - D Xi(t) (i = O,...,a-1), the first and second relations in equations (4.1.34) are therefore immediately seen to be valid from equations (4.1.33). . _ -1 And since zi+1(t) - D Zi(t), the third relation in equa- tions (4.1.34) is also immediately valid from equations (4.1.33). Q.E.D. Equations (4.1.34) represent a significant saving in computation not only because the datamatrix is gen- erally sparse, but also because the number Of distinct time functions in U(t) is less than the number Of non- zero entries. Furthermore, all that is needed Of the 2 function is Z(t1). And aside from the pure integra- tions D-iU(t), only one other set of integrations (equa- tions (4.1.33)) is needed. It is the application of Theorem 4.1 that will be used as the specific algorithm for H-identification. 4.2. Computational Aspects Before the actual implementation Of the algorithm can be undertaken, a choice for F(D) must be made. Apart 70 from det F(D) # 0, the selection of F(D) is quite unre- stricted and the modes of F(D) can, in theory, be selected as either stable or unstable since all computations are confined to the finite interval [0,t1]. However, strongly unstable modes in F(D) are undesirable since the control Of the integration errors will be more difficult. Now, if n is the order Of the system to be identified (DSSl as in Figure 3.4) and a is the order of the disturbance process (DSSZ), then the order of F(D) must be c with c Z'n + a such that F-1(s)R(s)sa is prOper. Pole-zero cancellation is permitted in F-1(s)R(s)sa, but any such cancelled modes must be included in the W matrix because such modes, although not controllable, are observable and must be included in the annihilating filter. The choice of F(D) with c assumed even is c/2 F(D) = (T (D2 + k2w§)I (4.2.1) k=1 with A 2n mo _ t_1 , (4.2.2) This selection Of F(D) simplifies the computations sig- nificantly. The fundamental solution to the homogeneous equation F(D)z(t) = 0 (that is, the modes for CeAt) involves the functions {sin(kwot), cos(kwot)}, k = 1,2,...,%, which are orthogonal over the observation 71 time interval [0,t1]. Hence, the Gramian matrix W is diagonal, as is W, such that: t1 W=W=7I. (4.2.3) (This same frequency mo was mentioned near the end of Chapter III as the minimum resolving distance between the two sets of modes of D551 and D582.) Notice that the resonance frequencies of the filter F-1(D) coincide with the null frequencies Of the filter H so that the composite filter HF-1(D)R(D)Da tends to preserve the useful information in the data at all frequencies. In the case of linear, time-invariant SISO systems, n(n) = (0“,Dn'1,....1] (4.2.4) yielding a matrix transfer function (as 1 X (n + 1)). +.. c/2 1 [sn+a’sn a 1,...,sa], 2 2 2) (s + k mo k=1 (4.2.5) F-1(s)R(s)sa = In order for (A,B,C,E) to be a minimal realization for F-1(s)R(s)sa, the following four conditions are needed [GOO]: (1) C(sI - A)’1B + E = F-1(s)R(s)sa (4.2.6a) (11) rank [CT,(CA)T,...,(CAn-1)T]==rank A (4.2.6b) 72 1 (iii) rank [B,AB,...,An- B] = rank A (4.2.6C) (iv) rank CB °° - - CAn-IB ' ‘ ' = rank A . (4.2.6d) CAD—1B -- Ice By taking A (as c X c) in the canonical form A = F0 ; ' : ; I 0 1 _-ac -a1_ (4.2.7) where c/2 sc + alsc-1 + ... + a = (s2 + kzwz) (4.2.8) c k= 0 and C (as 1 X c) in the form C = [l,0,...,0] , (4.2.9) the matrices B (as c X (n + 1)) and E (as l X (n + 1)) can easily be found satisfying conditions (4.2.6). For example, with n = 2 and a = 1, then c = 4 so that: A = F 0 1 0 0" B = u 1 0 0 ' 0 0 1 0 0 1 0 0 0 o 1 -Sw§ 0 1 L-4w: 0 —562 0‘ L 0 -503 0 . c = [1,0,0,01 E = [0,0,0] (4.2.10a) 73 F-1(s)R(s)sa = C(sI - A)‘18 + E = _ 1 3 2 1 ‘ C/2 2 2 2 (5 Is Is ] (s + k wo) (4.2.10b) k=1 As to the choice in t1, some fraction of the longest expected system time constant is suggested [PEA-2,3]. All numerical examples were run on a Prime 750 Com- puter using Fortran IV language (see Appendix C). The IMSL Library [IMSL] was utilized for integration (DGEAR), interpolation (ICSEVU, ICSCCU), and minimization (ZSRCH, ZXMIN) routines. The IMSL library contains a comprehen- sive range of high quality validated algorithms. The library is internally self-consistent and well-documented for the user. The effectiveness of the IMSL library is discussed in Jacobs [JAC]. The routine DGEAR was signif- icant for the problem of integrating singularly perturbed systems of equations since it is a backward differentia- tion formula based on Gear's stiff methods [GE], [SHA]. The routine ZSRCH systematically searches a spatial region for good starting points to serve as initial guesses to ZXMIN. This is necessary since J3(m) is not necessarily convex. Hence more than one initial guess may be neces- sary before the absolute minimum of J3(w) is achieved. 4 Roughly speaking, the value J3(&) 5,10- is sufficient [PEA—2] to be assured that ||8 - (0*II2 is also small and 74 that convergence has occurred. However, this threshold might depend on the [0,t1] interval. And the routine ZXMIN uses a quasi-Newton method to find the unconstrained minimum of J3(w). The only restrictions to H-identification comes from under-ordering the disturbance model, and from dis- turbance models with too large a value of a, for then the formulation may not be suitable because a long time interval will be required when there are a large number of unknown parameters due to many frequency components in the disturbance. 4.3. Direct Application and Results The most basic illustration of the use of H-identifi- cation as applied to singularly perturbed systems is the minimum-dimensioned linear, time-invariant example: dt x(t) = A11x(t) + A122(t) + Blu(t) 6: Bit- z(t) = A21x(t) + A222(t) + B2u(t) y(t) = C1x(t) + C22(t) (4.3.1) where all variables (matrices) are scalars (l X 1). (Note that the effect of the input, u(t), is through the state equations directly. This is a standard formulation with u(t).) 75 Thus, this SISO system represents the process Operat- ing in coupled form CS. Known design inputs u(t) will be given along with the Observed output y(t) on [0,t1]. What is desired are the representations of the two sub- systems (approximate): snss: g% xs(t) = ons(t) + Bous(t) ys(t) = Coxs(t) + Dous(t) (4.3.2) FDSS: e g% zf(t) = A22zf(t) + B2uf(t) C22f(t) + Duf(t) . (4.3.3) yf(t) Each of these subsystems is input-output equivalent to: §(t) + a1§(t) = bou(t) + b1u(t) . (4.3.4) Therefore, it remains to determine, for each subsystem, the scalar quantities a1, b0, b1. According to the formulations in Section 4.1, it is found that: o = R(D)k(t,w) = R(D)U(t)h(w) = = [0,1] §(t) 0 -fi(t) 0 1 0 §(t) o -fi(t) a1 b0 b1 o 5 t 5t1 , (4.3.5) 76 This last equation is the model of the subsystem DSSl to be identified, with the other subsystem output considered as disturbance, d(t), such that: A _ ~ y(t) = yf(t) + ys(t) = y(t) + d(t) . (4.3.6) With the a priori knowledge of the dimensions n and a of D881 and D882, respectively--along with the Observed input and output--U(t), h(w), and H(w) can be formed. Also, O can be chosen to make F-1(s)R(s)sa proper, whereby (A,B,C,E) can be selected. Then the necessary integra- tions (4.1.33) and (4.1.34) can be executed, and 0 formed. At this point, all integrations of the data are complete, and minimization of J3(w) is all that remains. Once successful minimization is attained, the model D881 is known, and this information can then be used to find the parameters of the other subsystem DSSZ. Then, if so desired, the newly learned models can be transformed into state—space configuration (as discussed in Sec- tion 3.1). Tests for further determining the multiplier c in the state-space form of FDSS are discussed in Mendel [ME]. For each example attempted, some a priori knowledge on the separation Of modes between the fast and slow subsystems was assumed. This information was used to help in the design of the input signals. The input 77 signals were Of step, ramp, parabolic, and sinusoidal types. Various combinations of inputs and final times, t1, were applied until consistency (to three of four significant places) in one or more parameters appeared. These newly found parameters were then fixed as known constants, making the parameter vector, h(w), smaller. This iterative procedure was continued until all param- eters within h(m) were learned. At this point, the estimated model for DSSl was attained. TO learn the parameters of DSSZ, the newly found model 0851 was simulated, and its output effect, y1(t), was subtracted from the output, y(t), Of the original coupled system CS, yielding y2(t) (plus some small disturbance effect due to the unknown initial condi- tions on DSSl). Then H-identification was performed anew to learn the estimated parameters of DSSZ. As far as which subsystem to identify first and over what interval(s) Of time to do this identification, a heuristic study pointed directly to a unique procedure: Procedure 4.1: (i) Observe the coupled system output, y(t), over some time interval [to,t1], where to > 0 is some time instant after the fast response has effectively died out. 78 (ii) Use H-identification to estimate the param- eters of the slow subsystem. Then the esti- mated output §s(t), for a given input, can be determined--up to the initial condition response--over any time interval desired. (iii) Observe the coupled system output, y(t), over some interval [0,ta], where 0 < ta i-to‘ (iv) Form the output measurement y(t) - §s(t) over [0,ta]. (V) With this formed output, use H-identification to estimate the parameters of the fast subsystem. Actually, the Observations Of y(t) over [0,t1] can be made all at once, with the record of y(t) over [0,ta] being stored for the later computations in step (iv). For most of the examples considered (shown in the tables to follow) , to _<_ 1 (second) worked well as a time instant after which the fast response had effectively dissipated. And the instant ta was generally taken as to. It was hoped that FDSS could be successfully esti- mated over [0,ta] first, and then SDSS estimated over ltdtl]. For then this would point toward an adaptive control procedure for singularly perturbed systems. 79 A very interesting discovery was made while running the computer analysis. It was found that the subsystems being identified were not the zeroth-order approximation (equations (2.2.36)-(2.2.38)) nor even the first-order approximation (equations (2.2.41)-(2.2.43)). What was being identified was the gxggg decoupled subsystems (equations (2.2.21)-(2.2.23)): SDSS: v1 = (All - AlzL)v1 + [B1 - M(B2 + eLB1)]u (4.3.7) FDSS: 51°72 = (A22 + eLA12)v2 + (132 + eLBl)u (4.3.9) y2 = [6(C1 — CzL’M + Czlvz (4.3.10) where A y(t) = y1(t) + y2(t) = ys(t) + yf(t) (4.3.11) and L and M satisfy equations (2.2.6) and (2.2.7). The specific examples tested are listed in Table 4.1. A range of values for e was considered, from 8 =‘% to e = 5%5 . For each example in Table 4.1, the model Of CS was (one fast state and one slow state): 1': x . = A + Blu] e z z x [y] = [3,2] (4.3.12) 80 . mODHM> 05H“ x. com (I) 89H) H- $6.82 $5.: 8TH- _HH a lloHHom a H H o 2: II 8.7 H.. «$6.8: immmai 87.7 EH fl b..wa G H H 6 cm . I S H- H.. $6.03 immai SLH- T; a 6 mm m H H 8 I. 3H- H) .mSSC AWLQHC ONLH) TH— _H H me H. . H H s OH .I H.H.. H) $66: $6.: SLH) H pr m H . H o m I «H. H.. image .mai ALH- H m N H H O “NH. m.H) H.. N 36.2 36.: TH. m H H H o ummu.~h.0n.amv soawHHn.On.Hm. m .H l O .I i. i .i *OHQEMXM .< .< m d w AH u E .H u av monmem .H.v OHQMB 81 _m.H.OH mummm.m Hoo.o mooo.o AHHO.H.HO0.0.000.H. n Hm.~.oH mummm.m voo.o moo.o ANNO.H.voo.o.Hoo.H. O _m.m.oH Glamm.m mooo.o mooo.o Ao¢O.H.ooo.o.ooo.Hv m _m.m.oH mlmoo.~ Hoo.o hooo.o .moa.a.ooo.o.aoo.a. e Hm.h.HH mummo.h moo.o Hoo.o AmN~.H.Hoo.o:.Hoo.H. m HOH.HH vlmmm.m mao.o hoo.o Amam.a.aoo.o.emm.ov m _ma.m_ mumwv.m mmH.o mmo.o .mwm.~.m~o.o.~mm.ov H _Hu.ou. _AGCmu_ __«3ue__ __H%4m_w__ .Hm.om.Hm. * mHmsmxm Eoumhmbom SOHmIImuasmom .N.w manna 82 _H.o.o_ mlmom.m Hom.m mmo.o .HNO.¢.¢HH.OI.¢.vaV b Hm.o.oa «(woa.m Hom.v oeo.o Hamm.m.vmo.o.m.¢oav w Hm.o.oH m)fimm.m N¢0.0 mooo.o .owm.m.mao.ot.¢o.omv m H0.0.0H mlflm¢.m moo.o Hooo.o Ammm.m.ooo.o.oo.o~. v HH.oH elmfio4N omo.o Noo.o Amhh.m.ooo.o.mo.oav m _H.OH wimh~.a moa.o hHo.o Amom.m.ooo.o.oao.mv N Hm.oH m)fl~m.¢ mv¢.o mma.o Aaaw.m.mmo.o.mmm.nv H M s m 3 l__¥3__ H ~O ~H m n H OH _AGV b_ __«B I <__ __¥3 I 3__ A Q m my * OH wam Emumhmnsm ummmllmuasmom .m.v OHQMB 83 Table 4.4. Example and Result (n = 1, m = 2) . . , . g% x -1 0 11 x 1 P11 1 1 _ _ _ ;L_ . '26'6? z1 - o 1.05 1 21 + 20 [u]. 1 d 1 '5-0' a-E 22 -O.9 0 ‘1 22 '2—0' L g b . L. - L d ”x - [Y] = [2,311] 21 1511u (11.12.13) = (-2,—19,-21) SDSS*: 98 + 2ys = 06 + 289u “ ° " 7113 ' 542493 *. = — — FDSS ' Yf + 4°Yf + 399Y£ one + 5491 “f ' 5491 “f (a1,bo,b1)slow = (2.003,-0.001,5.234) (a1,a2 ,bo ,61,b2)fast = (40.09,399.8,0.002,1.293,98.06) ||8 - w*|| slow A 4 = 0.001 m - w = 0.006 IIw-Icllslow Ii IISlOW IIG - 5*II fast A * = 0.002 m - w = 0.885 |J3(w)|slow = 4.46E-5 IJ3(w)Ifast = 3.08E-4 (ta'to'tl) = (0.8,0.8,8.0) 84 and the model of each subsystem was: y + aly = boa + blu . (4.3.13) The results for these examples can be found in Tables 4.2 and 4.3. It has already been reported [PEA-2,3] that the algorithm does not work well for systems with too many modes--particularly high frequency modes. Therefore, the size of each example was kept to a minimum to test the actual effectiveness of the algorithm and procedure. As also reported in these same articles, there is essentially no effect by over-ordering the disturbance model. In many examples this was done.. Table 4.4 indi- cates the results of an example with one slow state and two fast states. 4.4. Discussion of Results_ A grasp on the effectiveness of H-identification of a singularly perturbed system can now be made through the examination of the information displayed in Tables 4.2 and 4.3. It is seen that the algorithm had much diffi- culty in identifying the separate subsystems when the time-scale factor 8 was large (Example #1: e = %). In this example the relative errors in the slow and fast parameters amounted to 5.3% and 15.8%, respectively. This apparent failure of the algorithm--for this example 85 --stems, most definitely, from the fact that the eigen- values for the fast and slow subsystems are too close to each other, even with the small resolving distance .21 o 10 ‘ The effectiveness of the algorithm to differentiate of w between the subsystems improves as 6 decreases in size. This was anticipated from the beginning of this thesis. As a decreased, the relative error in 6 remained less than 0.1%. (That these errors are considerably small was unexpected; however, not totally surprising. The preliminary investigations of Pearson et al. showed very accurate results in their simulation studies.) The rela- tive errors in the slow subsystem parameter estimates improved as a decreased, due to the fact that the "dis- turbance" system (FDSS) died out rapidly. This parallels the fundamental construct in singular perturbation theory that as e + 0, the coupled system CS basically reduces to the lower-order slow subsystem SDSS. A major reckoning for e 5-%, though, is that almost all parameters 01 (i = 1,2,3) for both subsystems are accurate to two or three significant digits. As to the effect of a decreasing time-scale e on the parameter estimates for the fast subsystem, a slight increase in the relative error is noticed, albeit the absolute error is poor. This could be due, in part, to 86 the extreme speed with which the exponential factors decay, thereby decreasing the richness of information in the output signal yf(t), and/or due to integration errors. As to the example in Table 4.4, wherein the fast- state dimension is increased to two, the parameter esti- mates 8 for both the fast and slow subsystems are accurate to three significant digits, and the relative errors are less than 0.5%. All these results tend to indicate that the two subsystems of a coupled singularly perturbed process can be identified provided the time-scale, e, is much less than unity. In the next chapter of this text, a review of the essentials covered in this thesis will be made, along with relevant conclusions. The chapter will close with some insights on directions for further research in this problem area. CHAPTER V SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS The purpose of this dissertation was to examine the problem of identifying the system parameters of the fast and slow subsystems of a singularly perturbed system. This singularly perturbed system was operating in coupled form with its input and output used to estimate the charac- teristics of the decoupled subsystems. In the beginning of this document, an introduction to the nature of singularly perturbed systems, along with the nature of the identification problem, was pre- sented. Following that, some algebra and theory of sin- gularly perturbed systems was revealed, along with the mechanics of its decemposition into slow and fast subsystems. Next, the solution of the problem at hand was under- taken using a deterministic, least squares, equation error, finite time—interval, identification method. This method utilized a filter to annihilate the initial condi- tion response, and assumed the disturbances to be solu- tions to a homogeneous differential equation. The adapta- tion of this method was then applied to an example set 87 88 of deterministic, linear, time invariant, single-input single-output, stable, observable, controllable, singu- larly perturbed systems. The results of this analysis revealed success in identifying the parameters of the separate subsystems via a unique procedure determined through a heuristic study. The success of the procedure was based, in part, on the time-scale parameter s, for if e was too large the parameter estimates were not significantly close to the true parameter values. It is, therefore, possible to determine reasonable estimates for the parameters of each decoupled subsystem from the input and output (observed over a finite time Iinterval) of a system operating in coupled singularly perturbed form. There are several recommendations for further research in this area. The first of these is to explore the iden- tification problem for singularly perturbed systems that are nonlinear in form. Since the theory of H-identification is valid for nonlinear systems, it might prove applicable to this. Another area of interest would be the problem of identifying the parameters for a singularly perturbed system with more than one time scale. An iterative approach, similar to Procedure 4.1, might solve this problem. Thirdly, the success of identifying the decoupled 89 fast and slow subsystems could be further developed to the broader problem of parameter estimation for any system admitting to a decoupling into subsystems. And finally, and most significantly, a study could be undertaken on the success of parameter estimation for the singularly perturbed system when corrupted by noise. The effects of different noise (stochastic, white, etc.) upon the actual input and output variables might significantly change the effectiveness of the identifica- tion procedure, particularly if the noise contains modes in common with the modes of any subsystem. These directions are significant and warrant further investigations and developments. APPENDIX A APPENDIX A Verification of H(H) = H: t 1 T H[H(z(t))] = H[z(t) - CeAt'W-1 f eA TCTz('r)d'r] °t 1 T H(z(t)) - Hi:CeAt-’Wm1 f eA TCTz('t)d'r] °t 1 H(z(t)) - [{CeAtW-1 f eATTCTz(T)dT} - o 1 T t1 T CeAtW-1f eA TCT{CeATW-1[ eA OCTZ(O)dO}dT] o 0 t1 T . H(z(t)) - CeAt'W-l f eA TCT2(T)dT + 0 t1 1 T -1 ATO T CeAtW-l f eA TCTCeATd‘r W e C z(o)do O t 1 T H(z(t)) - CeAtW-lf eA TCTz(T)d'r + h o t 1 CeAtW-l mm"1 f eATOCTz (a) do 0 t1 T H(z(t)) - CeAtW-l f eA TCTz(I)dI + t1 To . CeAtW-l f eA OCTz(o)do = H(z(t)) . Q.E.D. o 90 91 Verification of adj(H) = H (i.e. H* = H): = <£,y> = (xT(t) _ t1 T (Ce A'"t‘W 1f eA T CT,Tx(T)d1:) O t l = f [meym - 0 t1 T (Ce Atw" '1 of eA T CTx(I)dT) Tdty(t)] 1-’l[ xT(t)y(t) - t1 T 0] xT ('r)CeA d1 (W 1)TeA 1:CTy(t)]dt 0 t1 =f x T(t)y(t)dt - O T .. ftl ftl xT('r)CeATdT(W-1)TeA tCTy(t)dt . T -1 But w = w => w = (w' ) . And .switching t H 1 in second term: 1‘1 =f xT(t)y(t)dt - O t t 1 1 '1' At -1 ATTC T - x (t)Ce dtW e CY(T)dth O O 92 1‘1 T .l. [x (t)y(t)dt - 0 t1 T xT(t)CeA’tW-1 .l. eA TCTy(T)d'r]dt O t 1 T O = , Thus, = . But by definition, = . Thus, = =¢ H* = H. Therefore, H is self-adjoint. Q.E.D. See also HaLmos [HA], wherein it is shown that every pro- jection operator is a self-adjoint operator. APPENDIX B APPENDIX B Derivation of equation (4.1.9): Using operator notation, J1(6) in equation (4.1.8) is given by J1(6) = since 5 = H(c). Since H is self-adjoint and a projection, J1(e) = = t1 T = <;,; - eeAtw'l f eA TCTC(T:9)dT> 0 t1 T = <;,;> - <«';,Cep‘tW-'1 jr eA TCTC(T:9)dT> O = - t1 = <§,2;> - f l;(t,6)CeAtdtW-1¢(9) O = <;,;> - ¢T(e)w'1¢(6) . 93 APPENDIX C 94 Na0x0AN«0x\m(mdm\ 202200 m(&3\002<2\ 202200 unOX0A~HOX\H3Hx\ 202200 0a0x0AMHOX\<>430\ 202200 ¢~Ox0A¢«OX\>mmmIQ\ 202200 ¢0w2chmom2&0\(Z(Z0Aomm(hZA>A000(h\m4mm(\ 202200 (mekuANnuAnnlAO«lADmAmkAhuAOmAnmA¢mAmmANuAau JQZMNFXN (0w2m0A03uANAnmvm4mDAD cmum00. 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I. . . * xnr m” Ao¢.49mkmm3 “4.9m meuH .mH.o.¢.x3.n.m.»omuumw>zmz 4040 Am.m~.n.oaaz.c.awrgzz 4.40 .mfi.m4.n.ono.honmz‘mh00300 0h30£00 o U U ********************************* #0781... ************.¥*.¥ ********.¥*******.¥* 8.....8. 0 123 020 203h00 A+0002u0002 A0.0.h4.>vum AA.AVOOZ40fl> 4+0002n0002 A0.0.h4.h00va AA.0.OO£40*AN.AVOOZ4OIAN.0000240*AA«A000240nh00 0+00Au00A AA.00.0AO0A 0+000Zu0002 A .O.h4.h00qu A00.003.h03.003.013.0.00.h00vh002 4440 7 032Ah200 O 032Ah200 D A00004.A0004000240nA 0 m A 00004 &A 00 0 A A0004 ON 00 0A+00Aum0m AA.00.0AO0A 0A+0002u0002 A0.0.h4.h0000A A00.403.flx3.003.Ax3‘¢«A0.h00wk00£ 4440 032Ah200 0A 032Ah200 A00004.A0004000240nA00004.A00040A0 4.Afl00004 0 00 4.ANA0004 0A 00 .0 A... « s 004~A0004Vm0 " .ZDAP4UAOA00> 1030 00 0000030 024 00020FHZA000 0>APA000 mhA 00 00P4UA02A 0209h00 024 00 4xz40 00 0002A: 0200404 440~OZA00 444 00 PZ4ZH£00P00 01k mmhfimzoo 02Ah300030 *********#*******##****#*********#***$**#*****###u**###fl***##$####$***##$$ 00240.00240.00240.00240\IU400\ ZOitOU ANA«NAVO0Z40.A¢.NA.O0Z40.ANA.4000240.A4.4000240 4400 Amvmxz.Amvhx3.A00003.Amvnx3.A40403.A40003.A¢0013 4400 >.h00.A¢vA03.Am~0000~A4.QVAO 4400 0002.00A.0A.000044A0004 000052A AOH.00020000000 02Ah300030 «um QCKMMDO LIST OF REFERENCES [AH] [AS-1] [AS-2] [BE] [BL] [CHE] [CHO-l] [CHO-Z] [CL] LIST OF REFERENCES Ahlfors, L.V., Complex Analysis, 2nd Ed., McGraw— Hill Book Co., Inc., New York, 1966. 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