Optimization of quadratic performance indexes for nonlinear control systems

The proposed approach aims at the development of a systematic method to optimally choose the controller tunable parameters in a nonlinear control system, where in addition to the traditional set of closed-loop performance specifications, optimality is also requested with respect to the physically meaningful quadratic performance index. In particular, the value of the performance index can be calculated exactly by solving the Zubov partial differential equation (PDE). It can be shown that the Zubov PDE admits a unique and locally analytic solution that is endowed with the properties of a Lyapunov function for the closed-loop system. Moreover, the analyticity property of the solution of Zubov PDE enables the development of a series solution method that can be easily implemented with the aid of a symbolic software package. It can be shown that the evaluation of the above Lyapunov function at the initial conditions leads to a direct calculation of the value of the performance index which now explicitly depends on the controller parameters. Therefore, the employment of static optimization techniques can provide the optimal values of the finite-set of controller parameters. Finally, it shown that an explicit estimate of the size of the closed-loop stability region can be provided by using results from the Zubov stability theory.