Control of Nonlinear Systems Via State Feedback State-Dependent Riccati Equation Techniques

Nonlinear regulation and nonlinear H-infinity control via state- dependent Riccati equation (SDRE) techniques are considered. Relationships between SDREs and Hamilton-Jacobi/Bellman inequalities/equations are examined, and a necessary condition for existence of solutions involving nonlinear stabilizability is derived. A single additional necessary criterion is given for the SDRE methods to yield the optimal control or guaranteed induced L2 gain properties. Pointwise stabilizability and detectability of factorizations prove necessary and sufficient, respectively, for well-posedness of standard numerical implementations of suboptimal SDRE regulators, but neither proves necessary if analytical solutions are allowed. For scalar analytic systems or those with full rank constant control input matrices, stabilizability and nonsingularity of the state weighting matrix function result in local and global asymptotic stability, respectively, due to equivalence between nonlinear and factored controllability in these cases. A proof of asymptotic stability for sampled data analytic SDRE controllers is also given, but restrictive assumptions make the main utility of these results guidance in choosing appropriate system factorizations. Conditions for exponential stability are also derived. All results are extendable to SDRE nonlinear H-infinity control with additional assumptions. The SDRE theory is illustrated by application to momentum control of a dual-spin satellite and comparison with other current methods.