OBSERVER DESIGN FOR IRREGULAR DESCRIPTOR SYSTEMS

Transcription

1 OBSERVER DESIGN FOR IRREGULAR DESCRIPTOR SYSTEMS A Thesis Presented to the Academic Faculty by Mahendra Kumar Gupta In Partial Fulfillment of Requirements for the Degree Doctorate Indian Institute of Technology Patna Copyright 216 by Mahendra Kumar Gupta

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3 OBSERVER DESIGN FOR IRREGULAR DESCRIPTOR SYSTEMS Approved by: Dr. N. K. Tomar Supervisor Department of Mathematics Indian Institute of Technology Patna Dr. S. Bhaumik Co-supervisor Department of Electrical Engineering Indian Institute of Technology Patna Dr. Y. M. Tripathi Chairperson and Member Doctoral Committee Department of Mathematics Indian Institute of Technology Patna Dr. P. K. Srivastava Committee Member Department of Mathematics Indian Institute of Technology Patna Dr. S. K. Parida Committee Member Department of Electrical Engineering Indian Institute of Technology Patna Date Approved:

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5 Acknowledgments I would like to express my deepest gratitude to my supervisor Dr. Nutan Kumar Tomar, Department of Mathematics, IIT Patna for his guidance, motivation, help and continuous encouragement throughout my research work, without which this thesis would not have come into existence. I would like to thank my co-supervisor Dr. Shovan Bhaumik, Department of Electrical Engineering, IIT Patna for his valuable advises throughout my PhD duration. I would like to express my sincere thanks to the members of my Doctoral Committee: Dr. Y. M. Tripathi (Chairperson and Head of the Department), Dr. P. K. Srivastava, Dr. S. K. Parida for their affection and consideration shown in throughout my training. I am highly obliged to my teachers Dr. S. K. Gupta (IIT Roorkee), Dr. R. K. Pandey, Dr. Atul Thakur and other faculty members of Department of Mathematics, IIT Patna for their outstanding teaching. I am thankful to Central Library and other office staff of IIT Patna for their necessary help and cooperation in related matters. I am specially thankful to our collaborator Prof. Mohamed Darouach, CRAN-CNRS UMR 739, Université de Lorraine, France, for his guidance during my research. My heartfelt thanks are to Prof. Abraham Berman, Technion, Israel for his teaching and affection during my one month stay in Israel. My greatest thanks goes to some of my closest friends Suman Kumar, Vikas Kumar Mishra, Farha Sultana, and my hostel room mate Raj Kumar Mistri who continuously cheered me up and supported me whenever I felt disheartened. Mathematical discussions with Vikas and Raj Kumar have benefitted me a lot in my PhD. I am extremely thankful to my friends Bapi, Satyajit, Anshika, Devendra, Shashi Prakash, Nibedita and all research scholars of Department of Mathematics, IIT Patna for making me feel very happy throughout my stay. I am specially thankful to my senior friends Anand, Manoj, Debasis, and Dipendu for their support and best wishes. Finally, I extend the most sincere and special appreciation to my mother Smt. Narangi Devi, father Shri. Suraj Mal Gupta, my elder brother Chandra Prakash Gupta, bhabhi Rashmi Agrawal for their endless love, care, support, understanding, endless patience, and encouragement when it was most required. I can not imagine a life without them who sacrificed their happiness to make my life better and helped me to achieve my goal. I sincerely thank to my loving sisters Vijay and Mamta, their families and other relatives for their best wishes and blessings. I would like to acknowledge the financial support provided by UGC, New Delhi, India through JRF and SRF scheme (Award Letter No. F. 2-1/211(SA-I)) and DST, New Delhi for attending the summer school in Israel. Bihta, Patna (Mahendra Kumar Gupta) v

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7 Certificate Date: 22 April 216 This is to certify that the Thesis entitled OBSERVER DESIGN FOR IRREG- ULAR DESCRIPTOR SYSTEMS submitted by Mahendra Kumar Gupta to Indian Institute of Technology Patna, is a record of bonafide research work under our supervision and we consider it worthy of consideration for the degree of Doctor of Philosophy of the Institute. Dr. N. K. Tomar Supervisor Department of Mathematics Indian Institute of Technology Patna Dr. S. Bhaumik Co-supervisor Department of Electrical Engineering Indian Institute of Technology Patna

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9 Declaration I certify that a. The work contained in this Thesis is original and has been done by myself under the general supervision of my supervisor/s. b. The work has not been submitted to any other Institute for degree or diploma. c. I have followed the Institute norms and guidelines and abide by the regulation as given in the Ethical Code of Conduct of the Institute. d. Whenever I have used materials (data, theoretical analysis, figures and text) from other sources, I have given due credit to them by citing them in the text of the Thesis and giving their details in the reference section. Further, I have taken permission from the copyright owners of the sources, whenever necessary. Mahendra Kumar Gupta

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11 List of Publications Publications in journals 1. M.K. Gupta, N.K. Tomar, S. Bhaumik, Full- and reduced-order observer design for rectangular descriptor systems with unknown inputs, J. Frankl. Inst., Elsevier, 352, , M.K. Gupta, N.K. Tomar, S. Bhaumik, On detectability and observer design for rectangular linear descriptor system, Int. J. Dyn. Control, Springer, 215, DOI 1.17/s x. 3. M.K. Gupta, N.K. Tomar, S. Bhaumik, Observer design for descriptor systems with Lipschitz nonlinearities: an LMI approach, Nonlinear Dyn. Syst. Theory, InforMath Publishing, 14(3), pp , M.K. Gupta, N.K. Tomar, S. Bhaumik, Index one generalized observer design for linear descriptor systems, IEEE Transactions on Systems, Man, and Cybernetics: Systems. (Communicated, revised once) 5. M.K. Gupta, N.K. Tomar, V. K. Mishra, S. Bhaumik, Observer design for descriptor systems with applications to chaos-based secure communication, Asian Journal of Control, Wiley. (Communicated) 6. M.K. Gupta, N.K. Tomar, M. Darouach, Unknown inputs observer design for descriptor systems with monotone nonlinearities, Automatica, Elsevier. (Communicated) 7. On causal detectability for descriptor systems. (In preparation for journal submission) Publications in conference proceedings 8. M.K. Gupta, N.K. Tomar, S. Bhaumik, On observability of irregular descriptor systems, In Int. Conf. on Advances in Control and Optimization of Dynamical systems, IIT Kanpur, India, March 13-15, 214, 3(1), pp , IFAC. Presented by M.K. Gupta. xi

12 9. M.K. Gupta, N.K. Tomar, S. Bhaumik, Detectability and observer design for linear descriptor system, In 22nd Mediterranean Conference on Control and Automation, University of Palermo, Palermo, Italy, June 16-19, 214, pp , IEEE. Presented by N.K. Tomar. Other publications (not included in the thesis) 1. M.K. Gupta, N.K. Tomar, Proportional-derivative observer design for rectangular linear descriptor systems (In Hindi), Bharatiya Vaigyanik Evam Audyogik Anusandhan Patrika, CSIR, 23(1), pp , June, M.K. Gupta, N.K. Tomar, S. Bhaumik, PD observer design for linear descriptor systems, In Int. Conf. on Mathematical Sciences, Sathyabama University, Chennai, July 17-19, 214, pp. 4-43, Elsevier. Presented by M.K. Gupta. 12. Sonam Chandra, M.K. Gupta, N.K. Tomar, Synchronization of Rössler chaotic system for secure communication via descriptor observer design approach, In 215 Int. Conf. on Signal Processing, Computing and Control, Solan, India, Sept , 215, pp , IEEE. Presented by M.K. Gupta. xii

13 Contents Acknowledgment List of Publications Contents List of Figures List of Symbols List of Abbreviations Abstract v xi xiii xvii xix xxi xxiii 1 Introduction General introduction Brief on observers for state space systems Basic concepts from matrix theory About linear descriptor systems Solution concepts Solutions in classical sense Solutions in distributional sense Solutions with respect to inputs and outputs Observability and detectability concepts Review of observer design methods for linear descriptor systems Review of observer design methods for semilinear descriptor systems Motivation Objectives of the thesis xiii

14 1.9 Organization of the thesis About simulation Luenberger observers for linear descriptor systems Introduction Detectability results Luenberger observer design: A new approach Full-order observers Pole placement approach Linear matrix inequality (LMI) approach Reduced-order observers Causal detectability result Numerical examples Luenberger observers for linear descriptor systems with unknown inputs Introduction Full-order observer design Reduced-order observer design Numerical examples Generalized observers for linear descriptor systems Introduction Design of generalized observer of index one Observer design approach I Observer design approach II Physical significance of Condition (B) for square systems Proof of (B1) (B) Numerical examples Observer design for semilinear descriptor systems with Lipschitz nonlinearities Introduction Basic assumptions and transformations Observer design Application to chaos-based secure communication xiv

15 6 Observer design for semilinear descriptor systems with monotone nonlinearities Introduction Observer design Observer design with nonlinearities in output equation Particular cases Examples Conclusions and scope for future research 129 Bibliography 133 xv

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17 List of Figures 2.1 (a)-(c) Plot of true and estimated value of states by full-order observer in Example 2.2. (d) Plot of errors in estimation of states by full-order observer in Example (a)-(c) Plot of true and estimated value of states by reduced-order observer in Example 2.2. (d) Plot of errors in estimation of states by reduced-order observer in Example (a)-(d) Plot of true and estimated value of states by full-order observer in Example (a)-(d) Plot of true and estimated value of states by reduced-order observer in Example (a)-(c) Plot of true and estimated value of states by full-order observer in Example 2.4. (d) Plot of errors in estimation of states by full-order observer in Example (a)-(c) Plot of true and estimated value of states by reduced-order observer in Example 2.4. (d) Plot of errors in estimation of states by reduced-order observer in Example Plot of true and estimated values of states by full-order observer in Example Plot of true and estimated values of states by reduced-order observer in Example Plot of true and estimated values of states by full-order observer in Example Plot of true and estimated values of states by reduced-order observer in Example Plot of true and estimated values of states by Approach I in Example Plot of true and estimated values of states by Approach II in Example Lorenz attractor for the system (5.18) for parameters used in Table Time response of the original state x 1 and estimated state ˆx xvii

18 5.3 Time response of the original state x 2 and estimated state ˆx Time response of the original state x 3 and estimated state ˆx Time response of the original transmitted signal s(t) and recovered signal ŝ(t) A nonlinear series LCR circuit (a)-(c) Plot of true and estimated values of states. (d) Estimated errors (Example 6.1) xviii

19 List of Symbols Symbol Description Zero matrix with compatible dimensions I Identity matrix with compatible dimensions I n Identity matrix of order n R Set of real numbers C Set of complex numbers C + {s s C, Re(s) } i.e. closed right half complex plane R n Set of n dimensional real vectors R(λ) Set of polynomials with coefficients in R R m n Set of m n matrices with entries in R R m n (λ) Set of m n matrices with entries in R(λ) A T The transpose of the matrix A A 1 The inverse of the invertible matrix A A + Generalized inverse of matrix A A > (A ) A is positive definite (positive semidefinite) A < (A ) A is negative definite (negative semidefinite) det(a) Determinant of matrix A λ An eigenvalue λ(a) Set of all eigenvalues of matrix A λ(e, A) Set of all finite generalized eigenvalues of matrix pair (E, A) Re(λ) Real part of the complex number λ x Euclidean norm of x and x = x T x Symmetric part block-diag(a 1, A 2,..., A p ) Block diagonal matrix with matrices A 1, A 2,..., A p as diagonal elements xix

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21 List of Abbreviations Abbreviation C-observable DAE I-observable KCF LCR LHS LMI LTI PD PID R-observable RHS RSE S-observable SVD WCF Full form Completely Observable Differential Algebraic Equations Impulse Observable Kronecker Canonical Form Inductor Capacitor Resister Left Hand Side Linear Matrix Inequality Linear Time Invariant Proportional Derivative Proportional Integral Derivative Reachable Observable Right Hand Side Restricted System Equivalent Strongly Observable Singular Value Decomposition Weierstrass Canonical Form xxi

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23 Abstract Control systems have greatly influenced the development of human civilization and opened a new horizon of enormous possibilities in various fields of application. Almost all technological applications require a solid understanding of the physical and dynamical features of the concerned control systems. Mathematical models are the key to understand all such features of control systems. Physical systems of interest in control theory are sometimes modeled by ordinary differential equations along with some algebraic constraints. Such systems are often called descriptor systems. This thesis presents the results from researching observer design for descriptor systems using algebraic approach. More evidently, our approach is based on the restricted system equivalent theory under certain structural assumptions on system operators. The thesis is focused on three topics as elucidated in the following. The first topic involves the consideration of linear time-invariant descriptor systems in irregular form which may be under-determined or over-determined. We design Luenberger observers for such systems with or without considering the unknown inputs in dynamic as well as in output equations. Under certain rank conditions on the system operators, we propose a unified theory for designing full- and reduced-order Luenberger observers. Our approach reduces the effort significantly in comparison with the use of other approaches which require the same assumptions on the system operators. Furthermore, necessary and sufficient condition for the existence of Luenberger observers is derived directly in terms of the system coefficient matrices. The second topic is the construction of generalized observers for a general class of linear time-invariant descriptor systems. It is observed that some physical systems do not satisfy the necessary condition for the existence of Luenberger observers. In such cases, the role of generalized observers is important. The thesis presents a new sufficient condition for the existence of generalized observers of index one. In case of square systems, the proposed condition corresponds to the existence of an output feedback such that the closed loop system becomes regular and of index at most two. Most of the systems that arise in practice are nonlinear to some extent, at least over portions of their operational range. The first step in dealing with a nonlinear system is usually, if possible, to linearize it around some nominal operating point. A better approximation to nonlinear system is the semilinear systems, i.e. a system consists of a linear part as well as a nonlinear part, and can be derived from a general nonlinear system by making a local approximation about some nominal trajectory. The construction of observers for semilinear rectangular descriptor systems is the third topic of this thesis. In literature (mostly) all the observer design methods for semilinear descriptor systems have xxiii

24 been restricted to the case when the nonlinear part of the system satisfies the Lipschitz property. This thesis considers observer design for semilinear descriptor systems with the monotone nonlinearities. The approach is based on the theory of linear matrix inequalities. Apart from these three main theoretical aspects, applications of the work are described in constrained mechanical systems, electrical circuits, and synchronization of chaotic systems. Various other numerical examples are considered to illustrate the proposed theory. xxiv

25 Chapter 1 Introduction 1.1 General introduction A continuous-time LTI descriptor system, in general, has the form Eẋ(t) = Ax(t) + Bu(t), (1.1a) y(t) = Cx(t), (1.1b) where x(t) R n, u(t) R k, y(t) R p are the semistate vector, the input vector, and the output vector, respectively. E, A R m n, B R m k, and C R p n are known constant matrices. System (1.1) is called regular if m = n and there exists λ C such that the matrix pencil (λe A) is invertible, where C denotes the set of complex numbers. If a descriptor system is not regular, then it is called an irregular descriptor system. Obviously, rectangular systems of the form (1.1) fall into the category of irregular descriptor systems. System (1.1) is called normal if m = n and the matrix E is invertible. Therefore, normal systems are particular type of regular systems. A square system with E I n is an important special case and we write it separately as ẋ(t) = Ax(t) + Bu(t), (1.2a) y(t) = Cx(t), (1.2b) where A R n n and B R n k. A system of the form (1.2) is called state space system and the vector x(t) R n is called state vector. Note that the conversion of a normal system into a state space system, using the inverse of matrix E, depends on the condition number of the matrix E. Hence, normal systems are always recommended to be considered as regular descriptor systems. In the literature, other terms synonymous with descriptor are singular, 1 generalized state 1

26 space, 2 semistate, 3 implicit, 4 and DAEs 5 7 (Differential Algebraic Equations). These different terms exist because of separate investigations into these systems by different disciplines. For example, since the matrix E is generally singular, mathematicians have long been calling these systems as singular systems. But, after the seminal work of D. G. Luenberger, 8,9 the name descriptor systems is most frequently used in control systems and engineering because (1.1) provides a natural description of the underlying physical problem. In general (1.1a) is a linearized version of a set of nonlinear implicit differential equations F (ẋ, x, u) = with singular F ẋ, and in the simplest case (1.1a) is just a combination of differential equations along with algebraic constraints. Thus, numerical analysts prefer the term DAEs or implicit systems for (1.1). The origin of DAEs theory may be traced back to the works of K. Weierstrass 1 and L. Kronecker 11 on matrix pencils. The root of the terms generalized state space and semistate lies in electrical engineering. Indeed, when dynamical reactive elements and static resistive devices come together in a circuit, they yield models in the form (1.1). In state space model (1.2), the state vector x(t) can be evaluated with respect to any initial vector x(), i.e. the vector x(t) in any free state space model ((1.2) with u ) satisfies the semigroup property. Because of the fact that all initial conditions are not valid for (1.1), the vector x(t) in (1.1) is no longer a state vector. In particular, the name semistate came into the existence by the work of R. W. Newcomb. 12 In this thesis, we will use the term descriptor systems for (1.1) and the vector x(t) will be called semistate vector. Although state space models are very significant from theoretical and numerical point of view, yet these models have certain drawbacks from modeling perspectives. In modeling of any real world problem, the employment of various physical laws naturally yields DAEs (descriptor form), while a reduction to ODEs (state space form) is difficult. On the other hand, in the process of transforming a descriptor form into a state space form, the system may lose some intrinsic properties. For example, impulses which are important in circuit theory can not be treated properly by using state space models. Further, the requirement of state space form restricts the use of automatic modeling tools, such as Modelica, Spice, Simulink, Dymola, etc. Usually, these tools yield systems in descriptor form. Moreover, the study of over- and under-determined systems of the form (1.1) enables us to model complex dynamical systems with algebraic constraints. There is a wide class of problems which are most appropriately modeled in the form of rectangular descriptor systems. Some particular examples are electrical circuits, chemical processes, biological systems, 19 constrained mechanical systems, 2 multi-body dynamics, 21,22 and large-scale systems where various sub-models are linked together. 23,24 The above discussion reveals the need to understand the basic concepts for general descriptor systems of the form (1.1). The major contribution of the thesis is the development of sufficient (and sometimes necessary also) conditions for the existence of observers 2

27 for rectangular descriptor systems. For given (1.1), a dynamical system whose inputs are u(t) and y(t) and whose output ˆx(t) is an approximation to the semistate vector x(t) is called an observer for (1.1). In many applications, for example feedback controller design, knowledge of all the semistates is required. But, measurement of all semistates may be stalled by cost or sensor inaccessibility. Moreover, it is not possible to solve the system because, in practical situations, initial conditions of the system are not known in advance. In such cases, the vector x(t) can be estimated from the output of an observer. The rest of the chapter is organized as follows: Section 1.2 presents a brief review on observer design problems for state space linear systems. Section 1.3 provides basic concepts from matrix theory that will be applied throughout the thesis in the development of our results. Sections 1.4 presents some important characteristics of LTI descriptor systems. Sections 1.5 and 1.6 are devoted to literature review on observer design for linear and semilinear descriptor systems, respectively. Sections 1.7 and 1.8 describe the motivation and contributions of the thesis, respectively. A detailed overview of the thesis is presented in Section 1.9. Some comments on the simulation part of the thesis are given in Section Brief on observers for state space systems In this section, we consider state space systems of the form (1.2). The method to design an observer for state space systems was first considered by Luenberger Later observer design problems for such systems have received a great attention in the literature In general, the Luenberger observer design problem for the system (1.2) is to construct matrices F, S, L, M, N, and H of compatible dimensions such that the following state space system becomes an observer for (1.2), i.e. ˆx x as t holds for arbitrary initial conditions z(), x(), and arbitrarily given input u(t). ż(t) = F z(t) + Su(t) + Ly(t), (1.3a) ˆx(t) = Mz(t) + Ny(t) + Hu(t). (1.3b) Each state space system can not be observed. To design an observer of the form (1.3), observability of the given system (1.2) is a sufficient condition. The problem of observability is related with the computation of initial condition x() R n by observing the output y over an interval of time. More precisely, system (1.2) is observable if for any t and any initial state x(t ) = x there exists a finite time t 1 > t such that knowledge of u(t) and y(t) for t t t 1 suffices to determine x. There is in fact no loss in assuming that u(t) is identically zero throughout the interval. Mathematically, the observability of system (1.2) can be characterized by any of the following equivalent conditions. 3

28 λi n A (O1) rank = n λ C; C λi n A (O2) rank = n λ λ(a); C (O3) rank of the observability matrix C T (CA) T... (CA n 1 ) T T = n; (O4) for any polynomial p(λ) = λ n + a 1 λ n a n 1 λ + a n, a i R, i = 1, 2,..., n, there exists a constant matrix K R n p such that det(λi A + KC) = p(λ); (O5) K R n p such that eigenvalues of matrix (A KC) can be arbitrarily replaced. Every state space system does not satisfy the observability condition. But, observers can also be designed under the detectability condition. The detectability condition on (1.2) is the necessary and sufficient for designing its observer of the form (1.3). The detectability property of (1.2) can be verified by any of the following equivalent characterizations. λi A (D1) rank = n λ C + ; C (D2) K R n p such that (A KC) is a stable matrix, i.e. eigenvalues of matrix (A KC) are in the open left half complex plane. 1.3 Basic concepts from matrix theory In this section, we recall some basic concepts from matrix theory. Such concepts will be used throughout the thesis. A detailed discussion on these concepts can be found in any standard textbook on the matrix theory. We particularly refer to the textbooks by F. R. Gantmacher; 46,47 G. H. Golub and C. F. Van Loan; 48 R. Piziak and P. L. Odell. 49 Definition 1.1. The least non-negative integer ν such that rank(e ν ) = rank(e ν+1 ) is called index for a matrix E. Definition 1.2. The non-negative integer ν such that N ν = and N ν 1 is called nilpotency index for a nilpotent matrix N. 4

29 Definition 1.3. Let A R m n. Then B R n m is a left inverse of A if and only if BA = I n. Similarly, C R n m is a right inverse for A if and only if AC = I m. Lemma 1.1. Let A R m n. Then (a) A has a left inverse if and only if A has full column rank n. (b) A has a right inverse if and only if A has full row rank m. It is well known that the rank of a matrix is unaltered if it is pre- and/or post- multiplied by an invertible matrix. The next lemma provides a generalization of this fact. Lemma 1.2. The rank of a given matrix A R m n is unchanged if it is pre- (post-) multiplied by a full column (row) rank matrix. The following result is related to the rank of a block matrix. Lemma 1.3. Let A, B, and C be any matrices of compatible dimensions. Then the following inequality holds A B rank rank(a) + rank(c). C Moreover, the equality holds if A is full row rank and/or C is full column rank. The following discussion is concerned with the singular value decomposition (SVD) of any matrix A R m n. In matrix theory, the SVD is one of the most numerically reliable decompositions because it uses the orthogonal matrices. Singular Value Decomposition (SVD). Let A R m n with rank A = α. Then there exist orthogonal matrices M R m m and N R n n such that Σ A A = M N T, where Σ A R α α is a diagonal matrix with positive decreasing elements. If A is complex, then M and N are unitary instead of orthogonal. In linear control systems theory, the SVD is an important factorization of a rectangular real or complex matrix with many applications in signal s and/or system s approximation. Mathematically, the SVD is very useful in determining rank, range, null space, row compression, column compression, etc. of a matrix. Row and Column Compression Form. Let A R m n with rank A = α. Then there exist orthogonal matrices P 1 R m m and Q 1 R n n such that P 1 A = Σ α and AQ 1 = ˆΣα 5

30 where Σ α R α n is a full row rank matrix and ˆΣ α R m α is a full column rank matrix. In view of the SVD, matrices P 1 and Q 1 may be obtained as P 1 = M T and Q 1 = N. Matrices P 1 and Q 1 are called row compression and column compression of the matrix A, respectively. If A is full column rank, then there exists a nonsingular matrix P 2 I n such that P 2 A =. This matrix P 2 may be obtained via the SVD of matrix A as NΣ 1 A P 2 = M T. Similarity, if A is full row rank then there exists a nonsingular I matrix Q 2 such that AQ 2 = I n. Next, we recall the following definitions and theorems regarding polynomial matrices. Definition 1.4. Let P (λ) be a polynomial matrix. Then the normal rank of P (λ) is defined as normal-rank P (λ) = max{rank P (λ)} λ C Definition 1.5. A matrix pencil λe A is said to be column (row) unimodular if λe A has full column (row) rank for all finite λ C. Definition 1.6. A matrix pencil λe A is said to be column (row) regular if it has full column (row) normal-rank. Definition 1.7. The matrix pair (E, A), or the descriptor system (1.1), is said to be regular if m = n and normal-rank(λe A) = n. Definition 1.8. Let E, A R m n. Then a fixed finite λ 1 C is called a finite generalized eigenvalue of the matrix pair (E, A), if rank(λ 1 E A) < normal-rank(λe A). Obviously, if matrix pair (E, A) is regular then λ 1 C is a finite generalized eigenvalue of (E, A) if and only if it is a root of the polynomial det(λe A). Throughout the thesis, the set of all finite generalized eigenvalues for matrix pair (E, A) will be denoted by λ(e, A). Moreover, λ(a) denotes the set of all eigenvalues of a square matrix A. We end this section by recalling the following result for regular descriptor systems. Theorem 1.1. The following statements are equivalent for square descriptor systems. 1. The matrix pair (E, A) is regular. 2. det(λe A) for all except a finite number of λ C. 3. det(λe A) is not identically zero as a function of λ. 4. The descriptor system (1.1a) is uniquely solvable for a consistent initial condition x() and sufficiently smooth u(t). 6

31 I n1 5. nonsingular matrices P and Q such that QEP =, QAP = N where the matrix N is a nilpotent matrix and n 1 + n 2 = n. A 1 I n2, 1.4 About linear descriptor systems This section is devoted to various basic concepts of linear descriptor systems. concepts will be used in the subsequent chapters. These Solution concepts For a continuous control function u, state space system (1.2a) has a continuously differentiable unique solution with respect to any given initial condition x() R n. Even the non-smoothness of u does not prevent the system (1.2a) of having a solution. But, for descriptor systems the solvability problem is somewhat more rigorous. The first discussion on solutions of DAEs of the form (1.1a) (by considering Bu as arbitrary non-homogeneity f) seems to be one by F. R. Gantmacher. 47 To understand classical solutions of the system (1.1a), we first recall the theory of Gantmacher which requires two notions, namely the equivalence of systems and the Kronecker canonical form (KCF) of the matrix pencil (λe A) R m n (λ). Definition 1.9. Let Q R m m, P R n n be nonsingular matrices and QEP = Ẽ, QAP = Ã, QB = B, CP = C and x = P x, then the following system Ẽ x(t) = Ã x(t) + Bu(t), (1.4a) y(t) = C x(t), (1.4b) is called a restricted system equivalent (RSE) to the system (1.1). A matrix pencil (λe A) can be written in its simplest form called the KCF by means of the above equivalence. In the following, we explain the existence and representation of the KCF. Kronecker Canonical Form (KCF). Let a matrix pencil (λe A) R m n (λ) be given. Then there exist invertible matrices U R m m and V R n n such that for some nonnegative integers δ, δ and multi-indices η, ρ, σ, ɛ U(λE A)V = block-diag( δ δ, λe η A η, λi ρ J ρ, λn σ I σ, λe ɛ A ɛ ). (1.5) 7

32 If we take multi-indices: l(η) η = (η 1, η 2,..., η l(η) ) N l(η), η = η i ; i=1 l(ρ) ρ = (ρ 1, ρ 2,..., ρ l(ρ) ) N l(ρ), ρ = ρ i ; i=1 l(ɛ) ɛ = (ɛ 1, ɛ 2,..., ɛ l(ɛ) ) N l(ɛ), ɛ = ɛ i ; i=1 l(σ) σ = (σ 1, σ 2,..., σ l(σ) ) N l(σ), σ = σ i. Then the structure of block matrices in (1.5) is as follows: i=1 1. δ δ is a zero block matrix of order δ δ ; 2. λe η A η is a block diagonal matrix and each block takes the form I λe ηi A ηi = λ where is a column vector of appropriate size; T T R η i+1 η i (λ), I 3. λi ρ J ρ is a block diagonal matrix and each block takes the form λ λi ρi J ρi = λi ρi. R ρ i ρ i (λ), λ C,.. 1 λ i.e. each J ρi is in the Jordan form; 4. λn σ I σ is a block diagonal matrix and each block takes the form λn σi I σi = λ. I σi R σ i σ i (λ),.. 1 i.e. each N σi is a nilpotent matrix of nilpotency index σ i ; 8

33 5. λe ɛ A ɛ is a block diagonal matrix and each block takes the form λe ɛi A ɛi = λ I I R ɛ i ɛ i +1 (λ). The block diagonal form (1.5) is unique up to the orders of the blocks and is called the KCF of the matrix pencil (λe A). The KCF was derived by L. Kronecker in But, it is still an active area of research, mainly because of numerical issues to obtain it efficiently. Recently, Wong sequences have been employed to find numerically reliable real valued transformation matrices U and V to find the KCF of any matrix pencil corresponding to real matrix pair (E A). 5 In the KCF representation (1.5), the numbers ρ i s are called the degrees of finite elementary divisors of the matrix pencil (λe A). Moreover, the σ i s are the degrees of infinite elementary divisors, the η i s are the row minimal indices, and the ɛ i s are the column minimal indices. All the elementary divisors and minimal indices are unique for a matrix pencil. It is notable that each zero column in the KCF is actually one 1 ɛ- block and each zero row is one 1 η-block. Thus in some of the research articles, authors do not write δ δ KCF block separately in the Note that if any square system is regular, then δ = δ = l(η) = l(ɛ) = in KCF (1.5). Thus, in case of the regular descriptor system, KCF reduces to the following famous Weierstrass canonical form (WCF) 1 I n1 UEV =, UAV = N A 1 I n2, (1.6) where A 1 R n 1 n 1, n 1 + n 2 = n, and the matrix N R n 2 n 2 is nilpotent. Furthermore, the index of any regular descriptor system is given by the index of nilpotency of the matrix N. If the matrix N does not appear (void) in WCF (1.6), then the index of the system is assumed to be zero. Index of a square system is zero means that the system is normal Solutions in classical sense In this subsection, classical solutions of system (1.1a) are considered. Definition 1.1. For a given input function u(t), a classical solution of system (1.1a) is a continuously differentiable function x that satisfies (1.1a) for all t. 9

34 In view of the KCF, any system of the form (1.1a) is RSE to the following system δ δ x 1 = B 1 u, (1.7a) E ηi N σi E ɛi x 2 = A ηi x 2 + B 2 u, (1.7b) x 3 = J ρi x 3 + B 3 u, (1.7c) x 4 = x 4 + B 4 u, (1.7d) x 5 = A ɛi x 5 + B 5 u, (1.7e) T where B1 T B2 T B3 T B4 T B5 T = UB. Now, solutions of (1.1a) can be characterized easily by solutions of (1.7). Solution of system (1.7a). The system (1.7a) is solvable if and only if the input part B 1 u. If input satisfies B 1 u =, then arbitrary continuously differentiable function x 1 solves (1.7a). Hence (1.7a) either does not have a solution or has an infinite number of solutions. If δ = (that is the case when δ number of 1 ɛ-blocks exist), then the corresponding δ number of variables does not appear and therefore can be chosen freely. Likewise, if δ = (that is the case when δ number of 1 η-blocks exist), then the equation B 1 u = has to be satisfied for the existence of solutions. Solution of system (1.7b). From the structure of E ηi and A ηi, the system (1.7b) can be written as where e =... 1 ẋ 2 (t) = Kx 2 (t) + B 21 u(t), (1.8a) = ex 2 (t) + B 22 u(t), (1.8b) is the last row of I ηi. The matrix K R η i η i is a submatrix of A ηi without the last row, and B 2i (i = 1, 2) are appropriate parts of the matrix B 2. Clearly, (1.8a) is in state space form and has a unique solution x 2 (t) = e Kt x 2 + for a given initial condition x 2 () = x 2. constraint given by (1.8b). t e K(t s) B 21 u(s)ds, (1.9) But the solution (1.9) has to satisfy the Solution of system (1.7c) The system (1.7c) is again a state space system that has unique solution x 3 (t) = e Jρ i t x 3 + for a given initial condition x 3 () = x 3. t e Jρ i (t s) B 3 u(s)ds, (1.1) 1

35 Solution of system (1.7d) Since N σi is nilpotent with nilpotency index σ i, we have σ i 1 x 4 (t) = Nσ k i B 4 u (k) (t). (1.11) Hence, for the existence of solution, initial conditions must satisfy k= σ i 1 x 4 () = Nσ k i B 4 u (k) (). (1.12) Solution of system (1.7e) The system (1.7e) can be written as k= ẋ 51 =. + e T x 52 + B 5 u, (1.13).. 1 where x 52 is the last component of the variable vector x 5. It can be seen that for any x 52 and any input B 5 u there exists a solution x 51 that can be uniquely determined by the initial values x 51 (). Hence, for all initial values and all controls there exist solutions which are of course not unique as x 52 can arbitrarily be chosen. Thus, the following result can be stated from the above analysis. 52 Theorem 1.2. The descriptor system (1.1a) has a classical solution for all sufficiently smooth Bu(t) if and only if KCF (1.5) does not contain the η-blocks (including zero rows). Any solution of the system (1.1a) with fixed input u(t) is uniquely determined by the initial value x() if and only if KCF (1.5) does not contain the ɛ-blocks (including zero columns). Note that solutions for the system (1.1a) may exist in the presence of η-blocks for a given sufficiently smooth u(t) that satisfies the constraints appeared due to (1.7a), (1.7b), and (1.7d). Moreover, the following two statements can be seen easily. (i) η-blocks (including zero rows) do not appear in the KCF if and only if rank (λe A) = m. (ii) ɛ-blocks (including zero columns) do not appear in the KCF if and only if rank (λe A) = n. Next, we formally define consistent initial vectors for the system (1.1a). Definition A vector x R n is said to be consistent initial vector for system (1.1a) if there exists u such that (1.1a) possesses at least one classical solution. 11

36 In view of the WCF, a regular descriptor system (1.1a) is RSE to the following system x 1 = A 1 x 1 + B 1 u (slow subsystem) (1.14a) Nx 2 = x 2 + B 2 u (fast subsystem) (1.14b) where x 1 R n 1, x 2 R n 2, n 1 + n 2 = n, the matrix N R n 2 n 2 is nilpotent with nilpotency index (say) ν. Define C ν as the set of ν-times continuously differentiable functions. The solution of the system (1.14) for a given x 1 () = x 1 and u C ν at time t > is denoted by where x 1 (t) x(t, u, x 1 ) =, (1.15) x 2 (t) x 1 (t) = e A 1t x 1 () + t ν 1 x 2 (t) = N i B 2 u (i) (t). i= e A 1(t τ) B 1 u(τ)dτ, Clearly, in view of the RSE the solution of given regular system (1.1a) is V x 1 x 2 (1.16a) (1.16b). Thus, regularity is equivalent to the existence and uniqueness of solution to the system with respect to a consistent initial condition and sufficiently smooth input vector u. remarkable that the solution x 1 (t) represents the cumulative effect of u(τ), τ t, while on the contrary, x 2 (t) responds so rapidly that it insistently reflects the properties of u(t) at time t. That is why, we call systems (1.14a) and (1.14b) the slow and the fast subsystems, respectively. The consistent initial conditions for fast subsystem is to be determined by the solution itself and the control function u(t) has to be ν-times continuously differentiable for the unique classical solution. It is Solutions in distributional sense We start our discussion with the following example of a free system (i.e. u ) that illustrates the existence of distributional solutions in DAEs. Example ẋ = x. (1.17) 1 12

37 Equivalently, ẋ 2 = x 1, (1.18) = x 2. (1.19) If the past history of the system is like x 1 ( ) = and x 2 ( ), then the solution of (1.17) for t : x 1 (t) = x 2 ( )δ(t) (1.2) x 2 (t) =, (1.21) has impulsive nature at t =. Here δ(t) is the Dirac delta distribution. In free systems, impulsive solutions arise only due to inconsistent past history; and the situation of inconsistent past history may arise naturally in physical systems. For example, (1.17) is a mathematical model of a simple electrical network consisting of a capacitor only; x 2 denoting the potential and x 1 the current. If x 2 ( ) and the switch is closed at t =, then for t, x 2 = but x 1 (t) = x 2 ( )δ(t), where δ(t) is the Dirac delta distribution representing the impulsive nature of the solution. The concept of initial value in the physical sense can be understood only when the past history of the system has been included in our consideration. This viewpoint has been well addressed by G. Doetsch 53 in the context of ODEs, and well reviewed in the context of DAEs in a recent review article by S. Trenn. 52 So, precise mathematical description of initial condition for (1.1a) requires x( ) instead of exact value of x() and (1.1a) holds for the interval, ). There are different theories to understand the impulsive response of DAEs for inconsistent initial conditions x( ) All these theories conclude that Dirac delta distributions occur in solutions of the system if inconsistent initial conditions are taken into account. Such solutions of (1.1a) are called impulsive solutions. Thus, impulsive solutions can be understood as solutions with distributions (generalized functions). Finally, we conclude the discussion by pointing out the following observations from the works of Hou and Müller; 51,59 Ishihara and Terra. 6 ˆ The initial condition x( ) is admissible to (1.1a) if there exists a u such that the system has a solution x. Here, u and x both are understood in the sense of distributions. It is clear from the KCF of (1.1a) that for free systems (i.e. u in (1.1a)), we can always find admissible initial condition x( ). 13

38 ˆ Every x( ) is admissible for free system (1.1a) if and only if rank λe A = rank E A. ˆ Every x( ) is admissible to (1.1a) if and only if rank λe A B = rank E A B. ˆ Since Ex appears in solutions of (1.1a), the concept of admissibility of Ex( ) is introduced in place of the admissibility of x( ). 61 Initial condition Ex( ) is admissible to (1.1a) if and only if rank λe A B = rank λe A B Ex. ˆ The pair (Ex( ), u) is admissible to (1.1a), i.e. Ex( ) and u are such that there exists at least one trajectory x satisfying (1.1a), if and only if rank λe A = rank λe A Ex Bu. Now, let the system (1.1a) be regular. Then using the regularity and the WCF, the system (1.1a) can be transformed by Laplace transformations into the following form sx 1 = A 1 x 1 + B 1 u + x 1 (1.22a) snx 2 = x 2 + B 2 u + Nx 2 (1.22b) The solution of (1.22) can easily be found as (for details, see the book by G. R. Duan 62 ) x 1 (t) = e A 1t x 1 + t Moreover, the solution of system (1.22b) is ν 1 x 2 (t) = N i B 2 (u (i) (t) + i= i 1 k= e A 1(t τ) B 1 u(τ)dτ. (1.23) ) δ (k) u (i 1 k) () ν 1 δ (i 1) N i x 2. (1.24) From the expression (1.24), it is clear that the presence of non-smooth inputs can also lead to distributional solutions. i=1 14

39 Solutions with respect to inputs and outputs This subsection addresses the advantage of output equation (1.1b) to determine solutions of descriptor system (1.1). Before going into the details, we provide the following theorems and definitions. The proofs of the following theorems are simple consequences of the KCF. 51,59 Theorem 1.3. Solution of descriptor system (1.1a) is unique if it exists, if and only if the matrix pencil λe A is column regular. Definition The system (1.1a), or simply the matrix pair (E, A), is said to have no impulsive modes if every admissible initial condition Ex( ) is consistent. Theorem 1.4. The system (1.1a), or simply the matrix pair (E, A), has no impulsive modes if and only if E A rank = normal-rank(λe A) + rank(e). (1.25) E Definition The system (1.1a), or simply the pair (E, A), is said to be impulse free, if it has no impulsive modes and solution is unique for every admissible initial condition Ex( ). Theorem 1.5. The system (1.1a), or simply the matrix pair (E, A), is impulse free if and only if E A rank = n + rank(e). (1.26) E Clearly, for regular descriptor systems both of the concepts having no impulsive modes and impulse free are exactly the same. Theorem 1.6. If system (1.1) is regular, then the following statements are equivalent. 1. The system is impulse free. 2. The system has no impulsive modes. 3. The nilpotent matrix N in Theorem (1.1) is a zero matrix. 4. deg det(se A) = rank E. 5. The system has rank E finite generalized eigenvalues in number. Now, we start our main discussion of this subsection. In the field of observer design for descriptor systems, treating inputs and outputs uniformly brings advantages, the 15

40 output y(t) is always considered as input for observer system. Here, we focus on the determination of the uniqueness and impulse free nature of solutions of (1.1) with the help of output equation. For the existence of solutions, we assume that the initial condition Ex( ) and input u(t) are admissible, i.e. there exists at least one trajectory (may be impulsive) satisfying (1.1a). Since the output y(t) is obtained from the measurement of the underlying physical system, the solution of (1.1a) should not contradict the output equation (1.1b). Combining the output equation (1.1b) with the dynamical part (1.1a), the descriptor system (1.1) can be written in the form of the following system. Ēẋ(t) = Āx(t) + Bū(t), (1.27) where, E A B u(t) Ē =, Ā =, B =, and ū(t) =. C I p y(t) In the lines of Theorem 1.3, we now write the following theorem for the system (1.27). Theorem 1.7. Solution of descriptor system (1.27) is unique if and only if the matrix pencil λē Ā is column regular, i.e. the following condition holds. λe A normal-rank = n (full column rank). (1.28) C The following theorem guarantees the existence of at least one smooth solution of (1.27). Theorem 1.8. The system (1.27), or simply the matrix pair (Ē, Ā), has no impulsive modes if and only if Ē Ā rank = normal-rank(λē Ā) + rank(ē). (1.29) Ē It follows immediately that the system (1.27) has no impulsive modes if and only if E A λe A rank E = normal-rank + rank E. (1.3) C C Combining Theorem 1.7 and Theorem 1.8, we obtain the following theorem that can be understood as an extension of Theorem 1.5 to extended system (1.27). 16

41 Theorem 1.9. The descriptor system (1.27) is impulse free if and only if rank Ē Ā Ē = n + rank Ē, (1.31) or equivalently, E A rank E = n + rank E. (1.32) C Theorem 1.9 ensures the existence of impulse free unique solution of system (1.1) using the output equation. The beauty of Theorem 1.9 lies in the fact that we do not require regularity of the system for the existence and uniqueness of the solution of descriptor system (1.1) provided we have an admissible pair (Ex( ), u(t)) Observability and detectability concepts Contrary to state space systems there are various concepts of observability for descriptor systems. These concepts for regular descriptor systems are well studied and can be found in any textbook on the subject. 62,63 We start this section by recalling some definitions and results related to impulses and impulse observability of general (rectangular) descriptor system (1.1). Definition Let u(t) be identically zero. Then system (1.1), or simply the triple (E, A, C), is said to be impulse observable (I-observable) if y(t) is impulse free for t, only if x(t) is impulse free for t. The following theorem provides test conditions on matrix triple (E, A, C) for I-observability. Theorem 1.1. Let u(t) be identically zero. Then the following statements are equivalent: 1. The system (1.1) is I-observable. 2. If m n, a matrix K of compatible dimensions such that the closed loop matrix pair (E, A + KC) is impulse free. E A 3. rank E = n + rank(e). C From Theorem 1.9 and Theorem 1.1, it is clear that the I-observability condition is equivalent to the existence of an impulse free unique solution of the extended system (1.27) for admissible initial conditions and inputs. 17

42 We now define the detectability concept for system (1.1). This concept is very important for designing observers for (1.1). Definition Let u(t) be identically zero. Then system (1.1), or simply the triple (E, A, C), is said to be detectable if y(t) =, t > implies that x(t) as t. The following theorems provide detectability test criteria for a matrix triple (E, A, C). Theorem Let u(t) be identically zero. Then system (1.1) is detectable if and only if se A rank = n s C +. (1.33) C Theorem Let u(t) be identically zero. The following statements are equivalent for regular descriptor system. 1. The system is detectable. 2. a matrix K of compatible dimensions such that the closed loop matrix pair (E, A + KC) is stable. se A 3. rank = n s C +, s λ(e, A). C From Theorem 1.7 and Theorem 1.12, it is evident that the detectability of system (1.1) ensures the uniqueness of solution for the extended system (1.27). Now, we define other concepts for descriptor systems that are used at various places in the thesis. Definition Let u(t) be identically zero. Then system (1.1), or simply the triple (E, A, C), is said to be reachable observable (R-observable), if y(t) =, t > implies that x(t) = for t >. R-observability property can be checked using the following theorem. Theorem Let u(t) be identically zero. Then system (1.1) is R-observable if and only if se A rank = n s C. (1.34) C Theorem Let u(t) be identically zero. Then the following statements are equivalent for regular descriptor system. 1. The system is R-observable. 18