Controller Design for Polynomial Nonlinear Systems with Affine Uncertain Parameters

By means of polynomial decomposition, a control scheme for polynomial nonlinear systems with affine time-varying uncertain parameters is presented. The idea of polynomial decomposition is to convert the coefficients of polynomial into a matrix with free variables, so that the nonnegativity of polynomials with even orders can be checked by linear matrix inequality (LMI) solvers or bilinear matrix inequality (BMI) solvers. Control synthesis for polynomial nonlinear system is based on Lyapunov stability theorem in this paper. Constructing Lyapunov function and finding feedback controller are automatically finished by computer programming with algorithms given in this paper. For multidimension systems with relatively high-order controller, the controller constructed with full monomial base will be in numerous terms. To overcome this problem, the reduced-form controller with minimum monomial terms is derived by the proposed algorithm. Then a suboptimal control aiming at minimum cost performance with gain constraints is advanced. The control scheme achieves effective performance as illustrated by numerical examples.