Construction of optimal Lyapunov function for systems with structured uncertainties

The robust stability problem for nominally linear system with nonlinear, time-varying structured perturbations p_j, j=1,...,q, is considered. The system is of the form x˙=A_Nx+^qΣ_j=1p_jA _jx. When Lyapunov direct method is utilized to solve the problem, usually some quadratic form is chosen as a Lyapunov function. The note presents a procedure of selection of optimal Lyapunov function from the class of quadratic forms. Under some weak conditions the procedure is effective in solving the robust stability problem. The procedure is simple, requires only such numerical routines as inverting positive definite symmetric matrices and determining the eigenvalues and eigenvectors of symmetric matrices. It is expected that when optimal Lyapunov function is used for a robust linear feedback controller design, the resulting controller will have the smallest in some sense gains. The examples demonstrate the effectiveness of the method. The method is effective for large scale systems, it worked well for 24 dimensional system