Robust control of linear systems subject to uncertain time-varying parameters.
This book provides a systematic treatment of the theory of robust control of dynamical systems. The robust stability problem for linear systems subject to time-varying memoryless uncertainties is thoroughly investigated, using an approach based on the Lyapunov state-space stability theory; the relations with input-output operator theory methods are also investigated. The author has chosen to consider only the problems for which a definitive solution has been found, having in mind researchers or engineers working in industries who want to apply these methodologies to practical control problems. The presentation is self contained, all proofs are given as well as real world examples. At the end of each chapter a short summary, a brief history of the development of the research concerning such topics and further references for alternative approaches other than the ones considered are given. The topic is so ``hot'' that new results have appeared after the book's publication, but up to that time the bibliography is quite complete. One could only suggest to add a few, classical results on slowly varying systems. The self contained presentation makes it suitable to be used for educational post-graduate purposes. The structure of the book is the following. Chapter 1 contains an introduction, in Chapter 2 the qualitative behaviour of solutions of an autonomous system of linear differential equations with time dependent coefficient matrix is discussed: necessary and sufficient conditions for existence and uniqueness of solutions, conditions guaranteeing the stability of the equilibrium, the role of the state transition matrix. Several notions of stability are stated, in particular uniform asymptotic stability and exponential stability. Chapter 3 is devoted to the Lyapunov stability analysis of an autonomous linear system whose coefficient matrix depends on a vector \(p\) of uncertain parameters whose time behaviour is unknown in a given hyper-box. The problem of quadratic \(D\)-stability, quadratic \(L_2\) performance and quadratic stability with the satisfaction of a linear quadratic (LQ) performance criterion (guaranteed cost) are considered. The case of norm bounded uncertainties are considered and connections existing between quadratic stability and \(H\) control theory are also investigated. Some examples of realistic applications are given. Chapter 4 deals with the case when some (or all) of the parameters are constant or slowly time-varying. Less conservative conditions for the stability of this kind of systems are provided using techniques based on the use of parameter dependent quadratic Lyapunov functions. Illustrative examples are provided In Chapter 5 the quadratic stabilization problem via state and/or output feedback for linear systems depending on parametric uncertainties under various assumptions on the parameter time behaviour is discussed. The case of quadratic stabilization with performances, such as \(D\)-stability, \(L_2\) performance bound and linear quadratic (LQ) performance criterion are also considered. Chapter 6 presents the recent results on the robust stability problem for discrete time linear systems subject to time-varying memoryless uncertainties. A non-trivial real life application is given: the problem of controlling a plasma wind tunnel, to simulate the re-entry conditions of space vehicles by reproducing desired trajectories in pressure and temperature on a test model. The fundamental mathematical definitions and results used in the book are given in the Appendix.
scientific article; zbMATH DE number 5015855