State-dependent Riccati equation solution of the toy nonlinear optimal control problem

The SORE approach to optimal control has been shown in previous studies to be a systematic design technique for a broad class of nonlinear regulator problems. In this paper we examine the "toy non-linear" optimal control problem posed as a counter-example to the SORE method. and presented bq Doyle and Packard in the nonlinear control workshop at the 1997 American Control Conference [1]. We first review the SORE design technique, and the conditions required to assure local asymptotic stability of the method. We show that applying SDRE to the toy nonlinear problem using a single parameterization may fail to produce a stable solution. Examining the problem in detail, we determine these characteristics which pose difficulty for the SORE technique, and propose solutions for each. We then demonstrate that by choosing an appropriate parameterization for this problem, and by adding a stabilizing; term to the system dynamics in the controller design equation, the SDRE method generates a well behaved solution of this problem.