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Matrix diagonal stability in systems and computation
Automatica 38 (2002) 371–378
www.elsevier.com/locate/automatica
Book reviews Matrix diagonal stability in systems and computation Eugenius Kaszkurewicz and Amit Bhaya (Eds.); Birkh%auser, Boston, 2000, ISBN 3-7643-4088-6 This book is a research monograph aimed at graduate students and researchers in the 0elds of control, dynamical systems and the convergence of algorithms. The monograph is centered around the theme of “diagonal-type” Lyapunov functions applied to the problem of stability analysis for nonlinear dynamical systems. Such Lyapunov functions are Lyapunov functions de0ned by diagonal positive de0nite matrices. Although the class of diagonal-type Lyapunov functions is quite restrictive, a major contention of the book is that this class of Lyapunov functions is an extremely important class arising naturally in such diverse problems as mechanical systems and biological systems. An important feature of diagonal-type Lyapunov functions and the corresponding notion of diagonal stability is that they lead to many interesting mathematical problems in the areas of linear algebra, control systems and nonlinear systems. This book collects many of the results arising in connection to diagonal-type Lyapunov functions including a large number of results which have been developed by the authors. The monograph also considers the application of these results to problems of convergence of numerical algorithms, stability of neural networks and circuits, and large scale systems. Chapter 1 of the book introduces some of the main definitions of the book and motivates them with examples from mechanical systems, biological systems, convergence analysis, neural networks and non-smooth variable structure systems. In particular, a linear system de0ned in terms of a system matrix A is de0ned to be diagonally stable if it is stable with a diagonal quadratic Lyapunov function. This notion can be applied to both continuous time linear systems and discrete time linear systems leading to the corresponding notions of Hurwitz diagonal stability and Schur diagonal stability. A feature of these de0nitions is that they are not coordinate-free but depend on the particular state space realization chosen for the linear system under consideration. Chapter 2 of the book introduces some matrix theory concepts and results relating to the notions of diagonal stability introduced in Chapter 1. In particular, this chapter introduces the notion of D-stability for a system matrix A. Indeed a matrix A is said to be Hurwitz D-stable if the matrix AD is Hurwitz stable for any
positive diagonal matrix D. Also a number of related stability notions are de0ned and considered in this chapter. The notion of D-stability is related to issues of robustness. Also, it is shown that diagonal stability implies D-stability but the reverse implication need not hold. The results presented in this chapter involve the various connections between the di;erent notions of diagonal stability and D-stability considered over various classes of matrices. Some important classes of matrices being considered are non-negative matrices in which all of the elements of the matrix are required to be non-negative and M-matrices which are a special class of stable matrices with non-negative o;-diagonal elements. The matrix theory results of this chapter are also applied to control theory questions such as the question of simultaneous stability for a polytope of matrices. This chapter also addresses the numerical problem of 0nding diagonal quadratic Lyapunov functions via the use of convex linear matrix inequality (LMI) optimization problems. Chapter 3 considers various classes of nonlinear dynamical systems and uncertain dynamical systems for which diagonal-type Lyapunov functions can be applied to establish stability. In particular, this chapter concentrates on nonlinear systems with sector bounded nonlinearities. An important result presented in this chapter is Persidskii’s Theorem which applies to a class of nonlinear systems of the form x˙ = Af(x), where the components of f(x) are independent sector bounded nonlinearities. Such nonlinearities are said to have a diagonal structure. Persidskii’s Theorem gives a suAcient condition for the global asymptotic stability of such a system using a diagonal-type Lyapunov function. Various extensions of Persidskii’s Theorem are presented involving di;erent forms of the nonlinear dynamical system and di;erent classes of Lyapunov functions. Also, corresponding discrete-time results are considered including discrete-time systems with delays. This chapter also considers the question of quadratic stability for interval uncertain systems in which the set of possible system matrices for a linear time invariant system is de0ned by an interval matrix and a common quadratic Lyapunov function is sought to establish robust stability. Chapter 4 considers the problem of convergence analysis for various iterative methods for the solution of linear and nonlinear equations. In particular, diagonal-type Lyapunov methods developed for the stability analysis of nonlinear systems are applied to these convergence
372
Book reviews / Automatica 38 (2002) 371–378
analysis problems. When the e;ects of 0nite arithmetic and variable delays are considered, the models arising in the convergence analysis problem become nonlinear and time-varying. The chapter presents suAcient conditions for convergence of a special class of iterative algorithms referred to as synchronous block-iterative methods. It also presents convergence results for more general iterative algorithms referred to as asynchronous block iterative methods. Some results are also presented for the special case in which the equations being solved have a special structure referred to as “almost linear”. Finally, this chapter considers the convergence of iterative algorithms referred to as team algorithms which are hybrids of a number of di;erent standard algorithms for the solution of nonlinear equations. Chapter 5 applies diagonal Lyapunov methods to a number of speci0c applications. The 0rst application which is considered is Hop0eld–Tank neural networks. These are nonlinear dynamical systems with a diagonal structure. The main results given for such networks are suAcient conditions for global asymptotic stability of a unique equilibrium point. Such neural networks are considered in both continuous time and discrete time. This chapter also considers a connection between passive RLC circuits and diagonally stable matrices. Other applications considered in this chapter are digital 0lters subject to quantization errors and biological systems referred to as Trophic Chains. Chapter 6 considers various interconnected systems and large scale systems in which diagonal Lyapunov functions can be applied. The 0rst problem considered is that of diagonal stability for a large scale system which consists of the interconnection of a number of subsystems. For this class of problems, a vector Lyapunov approach is considered and a comparison principle is used to establish a suAcient condition for global asymptotic stability using a diagonal-type Lyapunov function. Another application considered in this chapter involves the
problem of absolute stability for an interconnected system of the Lur’e type. The results obtained on this problem are then extended to a collection of linearly interconnected Lur’e systems. Also considered is a problem of state feedback stabilization for a linear interval system in which the coeAcients of the system matrices lie within given intervals. Finally, the theory is applied to a problem of decentralized power-frequency control in power systems. Overall, I found the monograph to be a useful collection of results on the somewhat specialized area of diagonal Lyapunov functions and their applications to nonlinear and uncertain dynamical systems. The book also contains related linear algebra results. Because of the level of specialization, the book would be most useful to researchers and postgraduate students working on research problems closely related to those considered in the book. However, the book would also provide a useful reference to any researcher working in the areas of control, dynamical systems or linear algebra. Ian R. Petersen School of Electrical Engineering, University of New South Wales at the Australian Defence Force Academy, Canberra, ACT 2600, Australia E-mail address: [email protected] About the reviewer Ian R. Petersen was born in Victoria, Australia in 1956. He received a Ph.D in Electrical Engineering in 1984 from the University of Rochester. From 1983 to 1985 he was a Postdoctoral Fellow at the Australian National University. In 1985 he joined the School of Electrical Engineering, University of New South Wales at the Australian Defence Force Academy and he is currently a Full Professor. He served as an Associated Editor for the IEEE Transactions on Automatic Control and Systems and Control Letters. Currently he is an Associate Editor for Automatica and SIAM Journal on Control and Optimization. He was made a Fellow of the IEEE in 1999. His main research interests are in robust control theory and its applications.
PII: S 0 0 0 5 - 1 0 9 8 ( 0 1 ) 0 0 2 1 2 - 6
Power system dynamics and stability Jan Machowski, Janusz W. Bialek and James R. Bumby; Wiley, New York, 1997, ISBN 0-471-95643-0 Electrical power systems belong to the largest dynamic systems installed by men. Their local extension in Europe reaches from Portugal to Poland. This whole area is synchronous connected. The dynamic behaviour lies in the time range from microseconds to hours which is described in Chapter 1. The high frequency phenomena are travelling waves caused by lighting or switching actions. This so-called transients are not described in the book as well as the
electromagnetic phenomena caused by energy transfers between the inductances and capacitors of the network. All this proceeds with damped oscillations of frequencies over 50 Hz or 60 Hz. The electromechanical phenomenon is an energy transfer between the rotating masses of the generators via the network which is normally called network dynamic or power system dynamic. The described phenomena are normally power swings which are stimulated by small or large disturbances and damped by system immanent damping e;ects. In special network structures like extended systems and by regulators, negative damping may cause instability, as well as serious faults like short
www.elsevier.com/locate/automatica
Book reviews Matrix diagonal stability in systems and computation Eugenius Kaszkurewicz and Amit Bhaya (Eds.); Birkh%auser, Boston, 2000, ISBN 3-7643-4088-6 This book is a research monograph aimed at graduate students and researchers in the 0elds of control, dynamical systems and the convergence of algorithms. The monograph is centered around the theme of “diagonal-type” Lyapunov functions applied to the problem of stability analysis for nonlinear dynamical systems. Such Lyapunov functions are Lyapunov functions de0ned by diagonal positive de0nite matrices. Although the class of diagonal-type Lyapunov functions is quite restrictive, a major contention of the book is that this class of Lyapunov functions is an extremely important class arising naturally in such diverse problems as mechanical systems and biological systems. An important feature of diagonal-type Lyapunov functions and the corresponding notion of diagonal stability is that they lead to many interesting mathematical problems in the areas of linear algebra, control systems and nonlinear systems. This book collects many of the results arising in connection to diagonal-type Lyapunov functions including a large number of results which have been developed by the authors. The monograph also considers the application of these results to problems of convergence of numerical algorithms, stability of neural networks and circuits, and large scale systems. Chapter 1 of the book introduces some of the main definitions of the book and motivates them with examples from mechanical systems, biological systems, convergence analysis, neural networks and non-smooth variable structure systems. In particular, a linear system de0ned in terms of a system matrix A is de0ned to be diagonally stable if it is stable with a diagonal quadratic Lyapunov function. This notion can be applied to both continuous time linear systems and discrete time linear systems leading to the corresponding notions of Hurwitz diagonal stability and Schur diagonal stability. A feature of these de0nitions is that they are not coordinate-free but depend on the particular state space realization chosen for the linear system under consideration. Chapter 2 of the book introduces some matrix theory concepts and results relating to the notions of diagonal stability introduced in Chapter 1. In particular, this chapter introduces the notion of D-stability for a system matrix A. Indeed a matrix A is said to be Hurwitz D-stable if the matrix AD is Hurwitz stable for any
positive diagonal matrix D. Also a number of related stability notions are de0ned and considered in this chapter. The notion of D-stability is related to issues of robustness. Also, it is shown that diagonal stability implies D-stability but the reverse implication need not hold. The results presented in this chapter involve the various connections between the di;erent notions of diagonal stability and D-stability considered over various classes of matrices. Some important classes of matrices being considered are non-negative matrices in which all of the elements of the matrix are required to be non-negative and M-matrices which are a special class of stable matrices with non-negative o;-diagonal elements. The matrix theory results of this chapter are also applied to control theory questions such as the question of simultaneous stability for a polytope of matrices. This chapter also addresses the numerical problem of 0nding diagonal quadratic Lyapunov functions via the use of convex linear matrix inequality (LMI) optimization problems. Chapter 3 considers various classes of nonlinear dynamical systems and uncertain dynamical systems for which diagonal-type Lyapunov functions can be applied to establish stability. In particular, this chapter concentrates on nonlinear systems with sector bounded nonlinearities. An important result presented in this chapter is Persidskii’s Theorem which applies to a class of nonlinear systems of the form x˙ = Af(x), where the components of f(x) are independent sector bounded nonlinearities. Such nonlinearities are said to have a diagonal structure. Persidskii’s Theorem gives a suAcient condition for the global asymptotic stability of such a system using a diagonal-type Lyapunov function. Various extensions of Persidskii’s Theorem are presented involving di;erent forms of the nonlinear dynamical system and di;erent classes of Lyapunov functions. Also, corresponding discrete-time results are considered including discrete-time systems with delays. This chapter also considers the question of quadratic stability for interval uncertain systems in which the set of possible system matrices for a linear time invariant system is de0ned by an interval matrix and a common quadratic Lyapunov function is sought to establish robust stability. Chapter 4 considers the problem of convergence analysis for various iterative methods for the solution of linear and nonlinear equations. In particular, diagonal-type Lyapunov methods developed for the stability analysis of nonlinear systems are applied to these convergence
372
Book reviews / Automatica 38 (2002) 371–378
analysis problems. When the e;ects of 0nite arithmetic and variable delays are considered, the models arising in the convergence analysis problem become nonlinear and time-varying. The chapter presents suAcient conditions for convergence of a special class of iterative algorithms referred to as synchronous block-iterative methods. It also presents convergence results for more general iterative algorithms referred to as asynchronous block iterative methods. Some results are also presented for the special case in which the equations being solved have a special structure referred to as “almost linear”. Finally, this chapter considers the convergence of iterative algorithms referred to as team algorithms which are hybrids of a number of di;erent standard algorithms for the solution of nonlinear equations. Chapter 5 applies diagonal Lyapunov methods to a number of speci0c applications. The 0rst application which is considered is Hop0eld–Tank neural networks. These are nonlinear dynamical systems with a diagonal structure. The main results given for such networks are suAcient conditions for global asymptotic stability of a unique equilibrium point. Such neural networks are considered in both continuous time and discrete time. This chapter also considers a connection between passive RLC circuits and diagonally stable matrices. Other applications considered in this chapter are digital 0lters subject to quantization errors and biological systems referred to as Trophic Chains. Chapter 6 considers various interconnected systems and large scale systems in which diagonal Lyapunov functions can be applied. The 0rst problem considered is that of diagonal stability for a large scale system which consists of the interconnection of a number of subsystems. For this class of problems, a vector Lyapunov approach is considered and a comparison principle is used to establish a suAcient condition for global asymptotic stability using a diagonal-type Lyapunov function. Another application considered in this chapter involves the
problem of absolute stability for an interconnected system of the Lur’e type. The results obtained on this problem are then extended to a collection of linearly interconnected Lur’e systems. Also considered is a problem of state feedback stabilization for a linear interval system in which the coeAcients of the system matrices lie within given intervals. Finally, the theory is applied to a problem of decentralized power-frequency control in power systems. Overall, I found the monograph to be a useful collection of results on the somewhat specialized area of diagonal Lyapunov functions and their applications to nonlinear and uncertain dynamical systems. The book also contains related linear algebra results. Because of the level of specialization, the book would be most useful to researchers and postgraduate students working on research problems closely related to those considered in the book. However, the book would also provide a useful reference to any researcher working in the areas of control, dynamical systems or linear algebra. Ian R. Petersen School of Electrical Engineering, University of New South Wales at the Australian Defence Force Academy, Canberra, ACT 2600, Australia E-mail address: [email protected] About the reviewer Ian R. Petersen was born in Victoria, Australia in 1956. He received a Ph.D in Electrical Engineering in 1984 from the University of Rochester. From 1983 to 1985 he was a Postdoctoral Fellow at the Australian National University. In 1985 he joined the School of Electrical Engineering, University of New South Wales at the Australian Defence Force Academy and he is currently a Full Professor. He served as an Associated Editor for the IEEE Transactions on Automatic Control and Systems and Control Letters. Currently he is an Associate Editor for Automatica and SIAM Journal on Control and Optimization. He was made a Fellow of the IEEE in 1999. His main research interests are in robust control theory and its applications.
PII: S 0 0 0 5 - 1 0 9 8 ( 0 1 ) 0 0 2 1 2 - 6
Power system dynamics and stability Jan Machowski, Janusz W. Bialek and James R. Bumby; Wiley, New York, 1997, ISBN 0-471-95643-0 Electrical power systems belong to the largest dynamic systems installed by men. Their local extension in Europe reaches from Portugal to Poland. This whole area is synchronous connected. The dynamic behaviour lies in the time range from microseconds to hours which is described in Chapter 1. The high frequency phenomena are travelling waves caused by lighting or switching actions. This so-called transients are not described in the book as well as the
electromagnetic phenomena caused by energy transfers between the inductances and capacitors of the network. All this proceeds with damped oscillations of frequencies over 50 Hz or 60 Hz. The electromechanical phenomenon is an energy transfer between the rotating masses of the generators via the network which is normally called network dynamic or power system dynamic. The described phenomena are normally power swings which are stimulated by small or large disturbances and damped by system immanent damping e;ects. In special network structures like extended systems and by regulators, negative damping may cause instability, as well as serious faults like short