This book is an overview of the \(H^ \infty\)-control problem as seen from the state-space perspective. The author argues strongly and persuasively that this approach has much to commend it when compared with the polynomial or spectral factorization methods. Indeed, the results are proved relatively painlessly without the need for sophisticated results based on Nehari’s theorem. One can get a long way with simple norm inequalities and Riccati equations. On the whole, the book gives a good introduction to this subject and should be read by anyone wishing to start research in this area.
The specific chapters of this book are as follows. Chapter 1 is an introduction and motivation for the problem, explaining the nature of robustness analysis, uncertain systems and the mixed-sensitivity problem. A summary of the book is also given. In Chapter 2, the basic results of linear systems theory needed for \(H^ \infty\) control are given together with a discussion of \(H^ p\) and \(L_ p\) spaces. \(H^ \infty\) control is then related to the (almost) disturbance decouping problem. The full- information \(H^ \infty\) control problem is considered in Chapters 3 and 4 in the regular and singular cases, respectively. Proofs of results are based on norm inequalities and the Riccati equation. The problem of invariant zeros on the imaginary axis is also discussed. By using Pontryagin’s maximum principle, Chapter 5 extends the previous results to the case of measurement feedback. In Chapter 6 an interesting connection between the \(H^ \infty\) control problem and differential games is given and in Chapter 7 the singular minimum entropy problem is solved apart from when there are invariant zeros on the imaginary axis. The \(H^ \infty\) control problem in finite time is covered in Chapter 8 and discrete time versions of the preceding results are described in Chapters 9 and 10. Applications of the theory are discussed in Chapter 11 with a case study based on the usual ‘benchmark’ inverted pendulum. Chapter 12 concludes the book and gives the author’s perspective on the whole area of \(H^ \infty\) control. Perhaps this approach will hold out the best hope for generalizations to nonlinear systems?