Nonlinear control analysis and synthesis using sum-of-squares programming.

摘要

This thesis considers Lyapunov based control analysis and synthesis methods for continuous time nonlinear systems with polynomial vector fields. We take an optimization approach of finding polynomial Lyapunov functions through the use of SOS programming and the application of the Positivstellensatz theorem.; There are three main areas considered in this thesis: local stability analysis, local performance analysis, and global and local controller and observer synthesis.; For local stability analysis, we present SOS programs that enlarge a provable region of attraction for polynomial systems. We propose using pointwise maximum and minimum of polynomials to reduce the number of decision variables and to obtain larger inner bounds on the region of attraction. This idea is illustrated most notably with a Van der Pol equations example. We also extend this region of attraction inner bound enlargement problem to polynomial systems with uncertain dynamics by considering both parameter-dependent and independent Lyapunov functions. Besides using the pointwise maximum of such functions, we also propose gridding the uncertain parameter space to further reduce the size of the SOS program. The significance of the gridding method is made apparent with two examples. A related stability region analysis problem of finding a tight outer bound for attractive invariant sets is also studied. We also present some computation statistics on a region of attraction benchmark example with arbitrary data and increasing problem size.; We study two local performance analysis problems for polynomial systems. The first is on finding outer bounds for the reachable set due to disturbances with L2 and Linfinity bounds. A SOS based refinement of the outer bound is proposed and illustrated with a previously studied example. The second problem is on finding an upper bound for the L2 → L2 gain and its refinement. Interesting results are obtained when this method is applied to an adaptive control example.; For controller synthesis, we present SOS programs for finding global and local Control Lyapunov Functions. For observer synthesis, we formulate SOS programs that search for polynomial observers using Lyapunov based methods. Examples are provided to demonstrate these synthesis methods.; It is hoped that the optimization based methods in this thesis will complement existing nonlinear analysis and design methods.