Robust optimal parametric LQ control with a guaranteed cost bound and applications

Abstract

For an optimal parametric linear quadratic (LQ) control problem, a design objective is to determine a controller of constrained structure such that the closedloop system is asymptotically stable and an associated performance measure is optimized. In the presence of system uncertainty, the system via a parametric LQ design is further required to be robust in terms of maintaining the closed-loop stability with a guaranteed cost bound. This problem is referred to as 'robust optimal parametric LQ control with a guaranteed cost bound' and is addressed in this work. A new design method is proposed to find an optimal controller for simultaneously guaranteeing robust stability and performance over a specified range of parameter variations. The results presented generalizesome previous work in this area. A versatile numerical algorithm is also given for computing the robust optimal gains. The usefulness of the design method is demonstrated by numerical examples and a design of the robust control of a VTOL helicopter. I. IntroductionThis work addresses the simultaneous robust stability and performance problem via optimal parametric linear quadratic (LQ) control of linear uncertain systems. The present work is motivated by the guaranteed cost control approach of Chang and Peng ( 1972) and the recent work of Bernstein and Haddad for robust stochastic LQ control problems (Bernstein 1987, Bernstein and Haddad 1988 a, b, and the references therein). Chang and Peng (1972) considered the robust optimal LQ control problems with a full-state feedback and showed that the solutions of a resultant modified Riccati equation are guaranteed to give both robust stability and performance over a specified range of parameter variations. Among various later extensions of this approach, Bernstein and Haddad recently extended this approach to the robust control problems for linear stochastic systems with structural real-valued parameter uncertainty (Bernstein and Haddad 1988 a, b), and for the systems with state-dependent and control-dependent white noise (Bernstein 1987). Instead of using the absolutevalue bound (Chang and Peng 1972), they utilized a quadratic Lyapunov bound suggested by Petersen and Hollot (1986) in conjunction with the guaranteed cost approach to provide a simultaneous robust stability and performance via fixed-order dynamic feedback (Bernstein and Haddad 1988 a), and further considered this problem along with a unified treatment and extension of several quadratic Lyapunov bounds developed previously for feedback control design (Bernstein and Haddad 1988 b). Explicit solutions for the robust control were then derived using the optimal projection control synthesis approach of Hyland and Bernstein (1984).In the present work, we propose to extend Chang and Peng's approach (1972) further into the robust optimal parametric LQ control problems for linear determin-

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