H_infinity Control of Nonlinear Systems: A Convex Characterization

Wei-Min Lu and John C. Doyle, EE, Caltech

The so-called nonlinear H_infinity control problem in state space is considered with an emphasis on developing machinery with promising computational properties. Both state feedback and output feedback H_infinity control problems for a class of nonlinear systems are characterized in terms of continuous positive definite solutions of algebraic nonlinear matrix inequalities which are convex feasibility problems. The issue of existence of solutions to these nonlinear matrix inequalities (NLMIs) is justified. The other features of the suggested approach are that the nonlinear system considered has few structural constraints; the system coefficient functions are just required to be continuous, no other smoothness is needed; our attention is not just paid to local solutions, the system is considered to evolve in some prescribed open convex set; in this framework, it is proved by using set-valued map machinery that if a concerned NLMI has solutions, then it has a continuous solution which is required for the H_infty control problem to be solvable; the algebraic NLMIs are in fact the state-dependent LMIs, therefore, the existing convex optimization methods for solving LMIs can be used in the practical computation for solving NLMIs. Unfortunately, unlike the linear case, the solution of the NLMIs by themselves are not sufficient to guarantee the existence of the required controller, and the computational implications of the required additional constraints on the NLMI solutions are not clear. This issue is discussed more in the talk.