Modern Control System Theory and Design, 2nd Edition

8.12.  FOUNDATIONS OF H^∞ CONTROL THEORY

Based on the preceding results, the concepts of modern H^∞ control theory will be presented. The very basic problem that H^∞ as presented by Zames in Reference 16 focuses on is sensitivity reduction of feedback control systems as an optimization problem, and it is separated from the problem of stabilization. The technique is concerned with the effects of feedback on uncertainty, where the uncertainty may be in the form of an additive disturbance U(s) as illustrated in Figure 8.29. H^∞ control theory approaches the problem from the point of view of classical sensitivity theory, which has been presented in Chapter 5 and subsections 8.11A–8.11D, with the difference that feedback will not only reduce but also optimize sensitivity in an appropriate sense.

H_∞ control theory is a complex subject. The purpose of presenting it in this book, which is designed for the undergraduate student and the practicing engineer, is to introduce it and motivate the reader with an interest in this field to review some of the recent research papers which have been written on this subject [18–20]. In its basic form, H^∞ control theory attempts to minimize the supremum function over the entire frequency range

where S(jω) is the sensitivity function and W(jω) is a weighting function. We can view the magnitude of the product S(jω)W(jω) as the magnitude of the weighted sensitivity. ...