A linear matrix inequality approach to design state feedback control for non-polynomial nonlinear systems

Abstract

This thesis aims to design a state feedback controller for non-polynomial systems with bounded control inputs. The problem formulation begins by transforming the non-polynomial systems into polynomial systems. This can be done by defining non-polynomial terms as new state variables with algebraic constraints satisfying the non-polynomial properties. This method avoids the approximation of the recast polynomial systems. Then we design the state feedback control based on the extended Lyapunov stability theorem and the quadratic performance criterion. Two state feedback control laws are proposed: Theorem 1 for static Lyapunov matrix variables and Theorem 2 for polynomial ones. For Theorem 1, the design conditions are derived in terms of linear matrix inequality constraints. An upper bound on the optimal quadratic cost function can be readily obtained using available LMI solvers. Similarly, the design condition in Theorem 2 is derived as linear matrix inequality constraints. However, with a prior fixed degree of Lyapunov matrix variable, these constraints become state-dependent linear inequality matrices, which can be solved by using the sum of squares technique. Numerical examples are provided to demonstrate the effectiveness of the proposed control design.