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**Positive 1D and 2D systems.**
*(English)*
Zbl 1005.68175

Communications and Control Engineering Series. London: Springer. xiii, 431 p. (2002).

The study of positive systems has developped in the last decade due to its important applications in various domains as engineering, economics, social sciences, medicine, biology and ecology etc. At the same time a quite new branch of control systems, that of 2D linear systems has experienced a powerful advancement by providing useful models for digital image processing, seismology, gravity and magnetic field mapping etc. This book unifies the two trends in system theory and it provides the first monograph which is devoted to multivariable (MIMO) 1D and 2D positive linear systems. It represents an extended and modified English language version of a previous Polish language book which was published in 2000. The book is based on the author’s lectures for PhD students delivered at Warsaw University of Technology.

The book contains a quite complete theory of positive systems, including fundamental concepts as reachability, controllability, observability, asymptotic and BIBO stability as well as the realization problem of positive systems. As the title indicates, the book is structured in two parts. First various topics are examined in the context of “classical” 1D systems. The study is realized by drawing a parallel between continuous-time and discrete-time systems or standard and singular ones. In the second part this study is extended to 2D linear systems by emphasizing various models of such systems. Many results have been obtained by the author in the long series of papers indicated in References. Some other results belong to authors such as P.R. Roesser, E. Fornasini, G. Marchesini, M.E. Valcher, L. Farina, S. Rinaldi, J. Klamka and others. Finally the text includes a great number of topics. Each paragraph contains examples that illustrate the theory and the presented algorithms and each chapter ends with a list of significant problems. The differences between the 1D and 2D approaches are emphasized and some open problems are stated.

The text is structured in seven chapters and four appendices. In Chapter 1 basic definitions and properties of positive matrices are introduced as well as Metzler and M-matrices and graphs of different kinds of systems. Chapter 2 provides characterizations of externally and internally positive continuous-time and discrete-time linear systems. For internally positive systems the concepts of asymptotic stability and BIBO stability are examined in detail. Descriptor type linear systems are studied and the notions of weakly positive, externally and internally positive systems are introduced for these systems. The positivity of composite linear systems is analysed as well as the eigenvalue assignment problem for SISO positive systems. Chapter 3 is devoted to the study of the fundamental concepts of reachability, controllability and observability in the framework of positive systems. Necessary and sufficient conditions for these properties are derived and the minimum energy control problem is solved in the case of positive systems. Reachability and controllability of weakly positive systems with state feedback are also studied. Chapter 4 deals with the important realization problem of positive 1D systems. Sufficient conditions for the existence of positive realizations are given as well as some procedures for computation of such realizations in both SISO and MIMO cases.

The aim of the second part of the book represented by Chapters 5, 6 and 7 is the extension of this study to the 2D framework. Chapter 5 includes the presentation of externally and internally positive Roesser, Fornasini-Marchesini and general 2D models of discrete, continuous and continuous-discrete types. In Chapter 6 this study is extended to reachability, controllability and observability of these models and to the solution of minimum energy control problem. Chapter 7 solves the realization problem for the positive 2D systems. Eising’s two level realization of a 2D transfer matrix is used to obtain some algorithms for determining positive realizations. Necessary and sufficient conditions for the existence of positive realizations in Roesser canonical form are obtained and some procedures are derived for the standard and the singular cases.

Appendix A contains some basic results such as Sylvester formula, Weierstrass decomposition of a regular pencil or the presentation of Drazin inverse. In Appendix B four methods for the computation of fundamental matrices for regular pencils are given. Appendix C provides the state formulas for Roesser, Fornasini-Marchesini and general 2D models. Transformations of matrices to their canonical forms for singular 1D and 2D systems form the topic of Appendix D.

The book contains a quite complete theory of positive systems, including fundamental concepts as reachability, controllability, observability, asymptotic and BIBO stability as well as the realization problem of positive systems. As the title indicates, the book is structured in two parts. First various topics are examined in the context of “classical” 1D systems. The study is realized by drawing a parallel between continuous-time and discrete-time systems or standard and singular ones. In the second part this study is extended to 2D linear systems by emphasizing various models of such systems. Many results have been obtained by the author in the long series of papers indicated in References. Some other results belong to authors such as P.R. Roesser, E. Fornasini, G. Marchesini, M.E. Valcher, L. Farina, S. Rinaldi, J. Klamka and others. Finally the text includes a great number of topics. Each paragraph contains examples that illustrate the theory and the presented algorithms and each chapter ends with a list of significant problems. The differences between the 1D and 2D approaches are emphasized and some open problems are stated.

The text is structured in seven chapters and four appendices. In Chapter 1 basic definitions and properties of positive matrices are introduced as well as Metzler and M-matrices and graphs of different kinds of systems. Chapter 2 provides characterizations of externally and internally positive continuous-time and discrete-time linear systems. For internally positive systems the concepts of asymptotic stability and BIBO stability are examined in detail. Descriptor type linear systems are studied and the notions of weakly positive, externally and internally positive systems are introduced for these systems. The positivity of composite linear systems is analysed as well as the eigenvalue assignment problem for SISO positive systems. Chapter 3 is devoted to the study of the fundamental concepts of reachability, controllability and observability in the framework of positive systems. Necessary and sufficient conditions for these properties are derived and the minimum energy control problem is solved in the case of positive systems. Reachability and controllability of weakly positive systems with state feedback are also studied. Chapter 4 deals with the important realization problem of positive 1D systems. Sufficient conditions for the existence of positive realizations are given as well as some procedures for computation of such realizations in both SISO and MIMO cases.

The aim of the second part of the book represented by Chapters 5, 6 and 7 is the extension of this study to the 2D framework. Chapter 5 includes the presentation of externally and internally positive Roesser, Fornasini-Marchesini and general 2D models of discrete, continuous and continuous-discrete types. In Chapter 6 this study is extended to reachability, controllability and observability of these models and to the solution of minimum energy control problem. Chapter 7 solves the realization problem for the positive 2D systems. Eising’s two level realization of a 2D transfer matrix is used to obtain some algorithms for determining positive realizations. Necessary and sufficient conditions for the existence of positive realizations in Roesser canonical form are obtained and some procedures are derived for the standard and the singular cases.

Appendix A contains some basic results such as Sylvester formula, Weierstrass decomposition of a regular pencil or the presentation of Drazin inverse. In Appendix B four methods for the computation of fundamental matrices for regular pencils are given. Appendix C provides the state formulas for Roesser, Fornasini-Marchesini and general 2D models. Transformations of matrices to their canonical forms for singular 1D and 2D systems form the topic of Appendix D.

Reviewer: Valeriu Prepeliţă (Bucureşti)

### MSC:

68U35 | Computing methodologies for information systems (hypertext navigation, interfaces, decision support, etc.) |