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## Über dieses Buch

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®

## Inhaltsverzeichnis

### Chapter 1. Practical Application of Descriptor Systems

Abstract
With the development of computer technology, the theory of descriptor systems began to play an important role in the theory of automatic control. In this chapter, we consider some practical applications connected with the development of mathematical models of processes and control plants in descriptor form.
Alexey A. Belov, Olga G. Andrianova, Alexander P. Kurdyukov

### Chapter 2. Basics of Discrete-Time Descriptor Systems Theory

Abstract
The mathematical theory of discrete-time descriptor systems is different from the theory of normal ones. In spite of some similarities, discrete-time descriptor systems provide radically different behavior such as noncausality. This chapter deals with some important basic aspects of linear discrete-time descriptor systems.
Alexey A. Belov, Olga G. Andrianova, Alexander P. Kurdyukov

### Chapter 3. Anisotropy-Based Analysis of LDTI Descriptor Systems

Abstract
In this chapter, we provide some background material on an anisotropy-based approach to the analysis of linear discrete-time systems (LDTI). Concepts of mean anisotropy of Gaussian random sequences and anisotropic norms for linear systems are introduced in [1, 2, 2] and briefly described below. This chapter deals with generalization of anisotropy-based analysis of the class of descriptor systems using generalized algebraic Riccati equations and convex optimization techniques.
Alexey A. Belov, Olga G. Andrianova, Alexander P. Kurdyukov

### Chapter 4. Optimal Control

Abstract
In this chapter, we solve an optimal anisotropy-based control problem for descriptor systems. Consider state feedback and output feedback control problems. The solution of the first is based on generalized discrete-time algebraic Riccati equations (GDARE). The solution of the second is based on a two-stage control procedure that consists of causalization and stabilization.
Alexey A. Belov, Olga G. Andrianova, Alexander P. Kurdyukov

### Chapter 5. Suboptimal Control

Abstract
In this chapter, a problem of suboptimal anisotropy-based state feedback control design for linear discrete-time (LDTI) descriptor systems is solved. The problem may be formulated as follows: to find such control law that the closed-loop system is admissible, and its anisotropic norm is bounded by a given positive value. A similar problem for normal discrete-time systems is solved in [1]. The conditions of anisotropic norm boundedness for descriptor systems, connected with the existence of the solution of GDARE (or LMI) and the special type inequality are given in Sect. . But application of these results to design problems is not a trivial task. There exist a number of papers where suboptimal $$\mathscr {H}_\infty \,$$-based state feedback control is designed. For example, in [1] necessary and sufficient conditions of such control law existence are formulated in terms of generalized discrete-time algebraic Riccati inequalities (GDARI) with one matrix variable; in [1, 1] conditions are given in terms of strict LMIs. Using the methods of control law design, proposed in the mentioned papers, we solve the suboptimal anisotropy-based control problem for descriptor systems.
Alexey A. Belov, Olga G. Andrianova, Alexander P. Kurdyukov

### Chapter 6. Anisotropy-Based Analysis for LDTI Descriptor Systems with Nonzero-Mean Input Signals

Abstract
In classical anisotropy-based theory, the input signal is supposed to be a signal with zero mean and certain “spectral color.” But in real technical systems, an input signal can be a stochastic signal with nonzero mean. That is why the extension of anisotropy-based theory on the class of signals with nonzero mean has a practical interest. In this chapter, we consider the concept of anisotropy of the random vector with nonzero mean and give a definition of mean anisotropy of the Gaussian nonzero-mean sequence, generated by the shaping filter in descriptor form. We also develop an anisotropic norm computation procedure in the frequency domain when the input signal has nonzero mean.
Alexey A. Belov, Olga G. Andrianova, Alexander P. Kurdyukov

### Chapter 7. Robust Anisotropy-Based Control

Abstract
Interest in stability analysis and control for descriptor systems with parametric uncertainties has grown recently due to their frequent presence in dynamical systems. Uncertainties in such systems are often causes of instability and bad performance. It is known that control of uncertain descriptor systems is much more complicated than that of the normal ones. The aim of this chapter is to provide conditions of anisotropic norm boundedness for descriptor systems with norm-bounded parametric uncertainties and to develop methods of control design that make the closed-loop uncertain system admissible with prescribed anisotropy-based performance.
Alexey A. Belov, Olga G. Andrianova, Alexander P. Kurdyukov

### Backmatter

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