OPTIMAL FUZZY CONTROL In control design, it is often of interest to synthesize a controller to satisfy, in an optimal fashion, certain performance criteria and constraints in addition to stability. The subject of optimal control addresses this aspect of control system design. | Fuzzy Control Systems Design and Analysis A Linear Matrix Inequality Approach Kazuo Tanaka Hua O. Wang Copyright 2001 John Wiley Sons Inc. CHAPTERS ISBNs 0-471-32324-1 Hardback 0-471-22459-6 Electronic OPTIMAL FUZZY CONTROL In control design it is often of interest to synthesize a controller to satisfy in an optimal fashion certain performance criteria and constraints in addition to stability. The subject of optimal control addresses this aspect of control system design. For linear systems the problem of designing optimal controllers reduces to solving algebraic Riccati equations AREs which are usually easy to solve and detailed discussion of their solutions can be found in many textbooks 1 . However for a general nonlinear system the optimization problem reduces to the so-called Hamilton-Jacobi HJ equations which are nonlinear partial differential equations PDEs 2 . Different from their counterparts for linear systems HJ equations are usually hard to solve both numerically and analytically. Results have been given on the relationship between solution of the HJ equation and the invariant manifold for the Hamiltonian vector field. Progress has also been made on the numerical computation of the approximated solution of HJ equations 3 . But few results so far can provide an effective way of designing optimal controllers for general nonlinear systems. In this chapter we propose an alternative approach to nonlinear optimal control based on fuzzy logic. The optimal fuzzy control methodology presented in this chapter is based on a quadratic performance function 4-7 utilizing the relaxed stability conditions. The optimal fuzzy controller is designed by solving a minimization problem that minimizes the upper bound of a given quadratic performance function. In a strict sense this approach is a suboptimal design. One of the advantages of this methodology is that the design conditions are represented in terms of LMIs. Refer to 8 for a more thorough treatment of optimal fuzzy .
