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

We examine the “toy nonlinear” optimal control problem posed as a counter-example to the state dependent Riccati equation (SDRE) method, and presented by Doyle et al. (1997). We first review the SDRE 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 parametrization may fail to produce a stable solution. Examining the problem in detail, we determine those characteristics which pose difficulty for the SDRE technique, and propose solutions for each. We then demonstrate that by choosing an appropriate parametrization 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