Synthesis of Robust Controllers

Abstract

In the last chapter, we introduced the mathematical framework for the description of model uncertainty. As we have seen, the basic idea is to separate the uncertainty Δ from the plant such that the uncertain plant can be interpreted as a feedback system consisting of P and Δ. Herein, Pdenotes the nominal plant, which results from the original physical plant G by adding weights for performance and uncertainty. The basic block diagram which visualizes this situation is Fig. 12.9. The nominal feedback system is denoted by M and consists of P and the controller K. The uncertain feedback system is obtained if the input W_p is connected by feedback with the output z _P using the uncertainty Δ (cf. Fig 12.10). As we have seen, concerning internal stability it suffices to use the block M ₁₁ so that Fig. 12.11 becomes the fundamental block diagram with respect to robust stability. The question is now how large in the sense of the H _∞ norm the uncertainty may be so that the feedback system is stable. This question can completely be answered and is essentially the small gain theorem (cf. Fig. 12.11, where the uncertainty Δ can be formally viewed as a controller for the “plant” M ₁₁). This result together with some applications is the content of Sect. 13.1 and Sect. 13.2. A sufficient condition for robust performance for such a type of uncertainty is also given in Sect. 13.2.