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In the first part of this appendix, we study the coupled generalized algebraic Riccati equation [ABSTRACT FROM AUTHOR]

In this monograph, on the one hand, we have studied the $\mathcal{H}•\infty$ control problems for dynamical systems governed by a set of differential-algebraic equations, in an inclusive manner. It has been shown that most of the existing results for conventional state-variable systems can be extended, under appropriate assumptions, to descriptor systems. On the other hand, we have examined certain properties of the DAEs, which include the solvability, stability, controllability, observability, and dissipativeness. Some useful preliminary results have been first developed in Chapter 2. We have derived the Lyapunov stability theorem and LaSalle’s Invariant Principle for stability test of descriptor systems. Furthermore, the dissipation property has been investigated in the descriptor systems context. It has been shown that the dissipation property can be characterized by the dissipation inequality, which turn out a Hamilton– Jacobi inequality for the nonlinear case or a generalized algebraic Riccati inequality for the linear case. [ABSTRACT FROM AUTHOR]

As we have shown in Chapter 4, the full-order $\mathcal{H}•\infty$ controllers can be constructed from two GAREs for linear descriptor systems or two Hamilton–Jacobi inequalities for nonlinear descriptor systems. The controllers thus obtained have a state dimension not less than that of the generalized plant. [ABSTRACT FROM AUTHOR]

In this chapter, we introduce balancing for stable nonlinear descriptor systems. The approach used here is an extension of balancing for conventional systems. The classical ideas is extended naturally to the descriptor systems. That is, we will consider the past input and output energy of the descriptor systems. More precisely, the controllability function is defined as the past input energy and the observability function as the future output energy. Under certain sufficient conditions, these functions may be transformed into a form that gives us a measure of importance of a state component. A nonlinear descriptor system with its controllability and observability functions in this form is called a balanced representation. [ABSTRACT FROM AUTHOR]

Consider the standard feedback configuration shown in Fig. 4.1. Let $\mathcal{G}•P$ be a nonlinear system described by the following DAE [ABSTRACT FROM AUTHOR]

For control engineers, the stability of the given control systems has always been the main concern. We often encounter controller synthesis problems of the following form: Given a plant $\vec{G}$, design a controller $\vec{K}$ such that the closed-loop system is stable and satisfies certain given performance criteria. It is thus convenient to have a parameterization of the class of controllers which stabilize the plant and to optimize the performance criteria within this class. This approach has been quite successful in the modern control systems theory, the result being the Youla-Kucera parameterization. The objective of this chapter is to extend the results which gave the Youla parameterization for nonlinear systems, as obtained via a state-space approach, of [46] and [76] to the nonlinear descriptor systems case. By providing such a parameterization for descriptor systems, it is hoped that controller synthesis problems for descriptor systems may also become tractable. In particular, one would like to tackle the $\mathcal{H}•\infty$ control problems in this way. [ABSTRACT FROM AUTHOR]

In this chapter we summarize some basic definitions and preliminary results that will be used throughout this book. Most of the treatments are purely algebraic and the definitions are fairly standard. In particular, we give a generalized version of the LaSalle’s Invariance Principle, which is useful in the proof of internal stability in the subsequent chapters. Furthermore, Bounded Real Lemma, which is a basic tool for solving the $\mathcal{H}•\infty$ control problems, is extended to the descriptor systems in a natural way. [ABSTRACT FROM AUTHOR]

In system theory, a dynamical system is often considered as a set of ordinary differential or difference equations (ODE); these equations describe the relations between the system variables. [ABSTRACT FROM AUTHOR]

The purpose of the second part of the appendix is to review some elements of center manifold theory and extend certain existing results for conventional systems to descriptor systems. The proposed development is rather straightforward but is useful in this monograph. Consider a nonlinear descriptor system described by the following set of DAEs. [ABSTRACT FROM AUTHOR]