×

Robust control for unstructured perturbations. An introduction. (English) Zbl 0941.93504

Lecture Notes in Control and Information Sciences. 168. Berlin: Springer. vi, 118 p. (1992).
This short book tries to expose the reader to some of the newest trends in the so-called \(H\)-infinity approach to control theory. This theory is now becoming part of the larger area called robust control. Robust control assures acceptable performance (including stability) independent of parameter variations in the system under control. A typical assumption is that the parameters either vary slowly in time or change abruptly but on long intervals remain almost constant. A proper and simplifying mathematical assumption in robust control literature is that parameters are constant but unknown. Even with this simplifying assumption the mathematical problems arising in robust control are very difficult and only very special cases have been fully clarified. There are basically two approaches in robust control: one assumes perturbations in the frequency response which leads to Hardy spaces of bounded analytic functions (\(H\)-infinity) as the parameter spaces, while the second assumes perturbations of physical parameters which leads to differential or difference equations with perturbed coefficients. Perhaps the most popular result of the analysis type in the second approach is Kharitonov’s, which reduces the stability of a linear differential equation with independently perturbed coefficients to stability of only four extreme cases. In the first approach, the most significant initial results in the 1980s were the establishment of the existence of solutions to the basic problem of minimizing the \(H\)-infinity norm of the closed-loop system transfer function, the demonstration of strong relations with Nevanlinna-Pick interpolation theory, and the equivalent description of the basic problem in the state space and the time domain.
Based on these basic \(H\)-infinity results, the authors give an introduction to recent developments in the first (\(H\)-infinity) approach.
The introduction is short and contains some historical background, engineering motivations, and math preliminaries.
The (essentially) first chapter, Chapter 2, “Robust stabilization”, is fundamental since robust stability is the primary requirement in modern robust control theory. In addition to \(H\)-infinity perturbations of the plant model, the famous simultaneous stabilization problem of finitely many plants is also discussed.
Chapter 3, “Nevanlinna-Pick interpolation theory”, introduces this very useful theory and shows how to use it effectively for computing the solution to the robust stabilization problem for SISO systems.
The same Nevanlinna-Pick interpolation theory is used in Chapter 4 to solve the problem of minimizing the \(H\)-infinity norm of the closed-loop system transfer function.
Chapters 5, 6, and 7 are on multivariable (MIMO) systems.
In Chapter 5, the \(H\)-infinity sensitivity minimization is reduced to the matrix Nehari problem with the help of tools like \(Q\)-parameterization of stabilizing controllers and inner-outer factorization of rational and analytic matrices.
In Chapter 6, the solution to the problem of Chapter 5 is given using two approaches: the Hankel norm (in the frequency domain) and the reduction to two Riccati equations (in the time domain). A matlab software is also presented as a computational aid. The final Chapter 7 is on multiobjective design where, for instance, the \(H\)-infinity norm minimization may be combined with the robust stabilization problem. The so-called \(U\)-parameterization is proposed as being more convenient than the well-known \(Q\)-parameterization.

MSC:

93-02 Research exposition (monographs, survey articles) pertaining to systems and control theory
93B36 \(H^\infty\)-control
93D21 Adaptive or robust stabilization

Software:

Matlab
PDFBibTeX XMLCite
Full Text: DOI