Disturbance Attenuation

Abstract

This chapter may be viewed as the heart of this book, for it joins together almost all the results exposed in the previous chapters, with a special accent on the Popov-Yakubovich theory developed in Chapter 3. In fact, what will follow is, in a way, the time-variant (discrete) version of the H∞ theory whose natural framework is the time-invariant case. As is well known, the H∞ theory has been deeply investigated in the last decade and many mathematical tools proved efficiency in solving different aspects of the cited theory. Our option here concerns the game-theoretic situation directly derived, as a particular case, from the Popov-Yakubovich-like result presented in Theorem 3.2.2. Such a result invokes an operator based approach which, in our opinion, provides better understanding of the structural aspects of the solution to the so-called disturbance attenuation problem, as well as easier ways for deriving the formulae. In fact our motivation in developing the subsequent theory was the following. Given a (generalized) time-variant discrete system, assume that a stabilizing controller exists such that the resultant closed-loop input-output operator has its norm bounded by a prescribed positive number γ, that is, such a controller provides γ-disturbance attenuation. Starting with this general hypothesis and taking into consideration a minimal set of initial assumptions made on the given system, our major objective consists of deriving “as much as possible” necessary conditions expressed in a very suitable form, i.e. by means of the Kalman-Szegö-Popov-Yakubovich systems. Such expression of the necessary conditions, which turn out to be also sufficient, points out a striking fact: the existence of a stabilizing controller that simultaneously provides γ-disturbance attenuation has remarkable implications concerning the existence of solutions to some nonlinear system, in fact as Kalman-Szegö-Popov-Yakubovich system. Finally, it is worthwhile emphasizing that our approach can be viewed also as a Popov-Yakubovich version for solving (indirectly) a general Nehari problem.