The authors present a study of the H-infinity control problem and related topics for descriptor systems, described by a set of nonlinear differential-algebraic equations. They derive necessary and sufficient conditions for the existence of a controller solving the standard nonlinear H-infinity control problem considering both state and output feedback. One such condition for the output feedback control problem to be solvable is obtained in terms of Hamilton–Jacobi inequalities and a weak coupling condition; a parameterization of output feedback controllers solving the problem is also provided. All of these results are then specialized to the linear case. The derivation of state-space formulae for all controllers solving the standard H-infinity control problem for descriptor systems is proposed. Among other important topics covered are balanced realization, reduced-order controller design and mixed H2/H-infinity control. "H-infinity Control for Nonlinear Descriptor Systems" provides a comprehensive introduction and easy access to advanced topics.

Control of Discrete-Time Descriptor Systems takes an anisotropy-based approach to the explanation of random input disturbance with an information-theoretic representation. It describes the random input signal more precisely, and the anisotropic norm minimization included in the book enables readers to tune their controllers better through the mathematical methods provided. The book contains numerous examples of practical applications of descriptor systems in various fields, from robotics to economics, and presents an information-theoretic approach to the mathematical description of coloured noise. Anisotropy-based analysis and design for descriptor systems is supplied along with proofs of basic statements, which help readers to understand the algorithms proposed, and to undertake their own numerical simulations. This book serves as a source of ideas for academic researchers and postgraduate students working in the control of discrete-time systems. The control design procedures outlined are numerically effective and easily implementable in MATLAB®

This thesis considers the linear-quadratic optimal control problem for differential-algebraic systems. In this first part, a complete theoretical analysis of this problem is presented. The basis is a new differential-algebraic version of the Kalman-Yakubovich-Popov (KYP) lemma. One focus is the analysis of the solution structure of the associated descriptor KYP inequality. In particular, rank-minimizing, stabilizing, and extremal solutions are characterized which gives a deep insight into the structure of the problem. Further contributions include new relations of the descriptor KYP inequality to structured matrix pencils, conditions for the existence of nonpositive solutions, and the application of the new theory to the characterization of dissipative systems and the factorization of rational matrix-valued functions. The second part of this thesis focuses on robustness questions, i.e., the influence of perturbations on system properties like dissipativity and stability is discussed. Characterizations for the distance of a dissipative systems to the set of non-dissipative systems are given which lead to a numerical method for computing this distance. Furthermore, the problem of computing the H-infinity-norm of a large-scale differential-algebraic system is considered. Two approaches for this computation are introduced and compared to each other.

One of the major concentrated activities of the past decade in control theory has been the development of the so-called "HOO-optimal control theory," which addresses the issue of worst-case controller design for linear plants subject to unknown additive disturbances, including problems of disturbance attenuation, model matching, and tracking. The mathematical OO symbol "H " stands for the Hardy space of all complex-valued functions of a complex variable, which are analytic and bounded in the open right half complex plane. For a linear (continuous-time, time-invariant) plant, oo the H norm of the transfer matrix is the maximum of its largest singular value over all frequencies. OO Controller design problems where the H norm plays an important role were initially formulated by George Zames in the early 1980's, in the context of sensitivity reduction in linear plants, with the design problem posed as a mathematical optimization problem using an (HOO) operator norm. Thus formulated originally in the frequency domain, the main tools used during the early phases of research on this class of problems have been operator and approximation theory, spectral factorization, and (Youla) parametrization, leading initially to rather complicated (high-dimensional) OO optimal or near-optimal (under the H norm) controllers.

This book covers some selected problems of the descriptor integer and fractional order positive continuous-time and discrete-time systems. The book consists of 3 chapters, 4 appendices and the list of references. Chapter 1 is devoted to descriptor integer order continuous-time and discrete-time linear systems. In Chapter 2, descriptor fractional order continuous-time and discrete-time linear systems are considered. Chapter 3 is devoted to the stability of descriptor continuous-time and discrete-time systems of integer and fractional orders. In Appendix A, extensions of the Cayley–Hamilton theorem for descriptor linear systems are given. Some methods for computation of the Drazin inverse are presented in Appendix B. In Appendix C, some basic definitions and theorems on Laplace transforms and Z-transforms are given. Some properties of the nilpotent matrices are given in Appendix D.

This book is devoted to one of the fastest developing fields in modern control theory - the so-called H-infinity optimal control theory. The book can be used for a second or third year graduate level course in the subject, and researchers working in the area will find the book useful as a standard reference. Based mostly on recent work of the authors, the book is written on a good mathematical level. Many results in it are original, interesting, and inspirational. The topic is central to modern control and hence this definitive book is highly recommended to anyone who wishes to catch up with important theoretical developments in applied mathematics and control.