H ` Control and Its Applications

Due to its fascinating theoretical as well as its proven practical applicability, the so-calledH -control problem has drawn considerable attention since its explicit formulation by Zames (1981). Although the misnomer H is associated in the literature with a variety of concrete problem formulations, the book under review is con"ned to the disturbance attenuation problem for linear timeinvariant "nite-dimensional systems: For a given system, design a controller that feeds a measured signal back to a control input such that the e!ect of a disturbance entering the system onto a controlled output signal is reduced as far as possible where the quality of the attenuation is measured in terms of the controlled system's energy gain. Despite this innocent formulation, the real attraction of this problem results from numerous other concrete and practically important tasks that can be subsumed to this simple scenario, such as shaping the transfer matrix of a control system through a suitable controller (loop-shaping) or rendering a system not only nominally but even robustly stable against a well-de"ned class of uncertainties (robust stabilization). The theoretical parts of the book under review are entirely devoted to questions of how to compute the optimal attenuation level, of how to design controllers that achieve these levels as closely as desired, and of characterizing when the attenuation level can be rendered arbitrarily small, what amounts to the solvability of the so-called almost disturbance decoupling problem. As the main theme, the author demonstrates in a detailed and complete fashion how to answer these questions both for discreteand continuous-time systems by using as a main tool the so-called special coordinate basis for a control system in which the various controlledand conditioned-invariant subspaces from geometric control theory admit a very simple description. Two realistic design examples serve to demonstrate how the explicit controller design algorithms perform both in simulation and in real-time implementation. After an overview over the relevance of H -control and an exact problem formulation in the time domain in Chapter 1, the author turns to a computation-oriented description of canonical forms for matrices and matrix pairs as well as for system matrices in Chapter 2. The center stage is taken by a re"nement of the so-called special coordinate basis with a careful discussion of how to extract in these speci"c coordinates all relevant algebraic objects that are de"ned in geometric control, such as the "niteand in"nite-zero structure, largest controlled invariant and controllability subspaces as well as their duals, and the relation to the underlying system's invertibility properties. Since all these intimate relations do appear only in a rather scattered fashion in the literature, the authors' merit is to provide a pretty complete picture with detailed arguments of how to prove the various connections what renders the subject accessible for a novice to geometric control. We regret the decision of the author to demonstrate only by means of an example how the actual transformation of a general system has to be performed. Chapter 3 is devoted to the so-called suboptimality tests, veri"able conditions for the existence of controllers that achieve a certain desired level of disturbance attenuation. The author provides a collection of various tests in terms of the solvability of quadratic matrix inequalities and Riccati equations or inequalities literally as they appear in the literature, among them the famous two Riccati equation solution from the breakthrough work (Doyle, Glover, Khargonekar & Francis 1989). Chapter 4 comprises a thorough investigation of the transformation properties of geometric subspaces and Riccati equations if transforming the underlying system from a discrete-time into a continuous-time description. Although it is conceptually a bit unclear why the inherent symmetry is broken and the corresponding Caley transformation and its inverse are treated separately, this chapter provides a very complete and fully proved reference list of relations that are useful for a variety of problems that involve the translation of continuousto discrete-time results. As an impressive demonstration, it is revealed in Chapters 8}10 how these preparations render the proofs of H -results for discrete-time systems almost into a routine exercise. The Chapters 5}7 dealing with continuous-time systems together with the discrete-time counterparts of Chapters 8}10 form the central part of the book in demonstrating the power of the geometric approach in combination with the special coordinate basis to arrive at deep insights into how the achievable attenuation level is in#uenced by the various geometric properties of the underlying system. The main topic in Chapters 5 and 8 is to give an explicit formula for the optimal attenuation level in the