11. Advanced Gain-Scheduling Techniques for Uncertain Systems
Pierre Apkarian, Richard J. Adams · Society for Industrial and Applied Mathematics eBooks · 1999
11.1 Introduction The gain-scheduling problem has been the subject of a great deal of research over recent years, from both theoretical and practical viewpoints. This renewed interest probably stems from the development of new techniques and software that allow for a more rigorous and systematic treatment of the gain-scheduling problem. The classical approach to this problem essentially consists of repeated design syntheses associated with some scheduling strategy connecting locally designed controllers. Such schemes, however, lack supporting theories that guarantee the behavior of the scheduled controller. A significant contribution toward the elimination of such weaknesses is the formulation of the gain-scheduling problem in the context of convex semidefinite programming [315], an elegant and solidly based branch of optimization theory [296, 292, 403]. Expressed in terms of linear matrix inequalities (LMIs), the gain-scheduling problem is readily and globally solved using currently available efficient optimization software [153]. LMI techniques now appear as very natural mechanisms for the formulation of gain-scheduling problems as well as for a vast array of other problems in the control field. Reference [64] gives an overview of the scope of application of such techniques. As emphasized in control theory, a key stage in the characterization of gain-scheduled controllers is the search for adequate Lyapunov functions that establish stability and a performance bound for the closed-loop system. The linear-fractional transformation (LFT) gain-scheduling techniques in [312, 19, 252, 253] or the so-called quadratic gain-scheduled techniques in [36, 21] make use of a fixed Lyapunov function, as opposed to one that depends on the scheduled variables, to characterize stability and performance. According to [424], such approaches are potentially very conservative because they allow for arbitrary rates of variation in the scheduled variables. More dramatically, it has been shown in [424] that some systems are not even quadratically stabilizable, that is, are not stabilizable on the basis of a single Lyapunov function.