Optimization of quadratic performance indexes for nonlinear control systems
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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 (stability, fast and smooth set-point tracking, disturbance rejection, etc.), optimality is also requested with respect to a physically meaningful quadratic performance index. In particular, the value of the performance index can be calculated exactly by solving Zubov's partial differential equation (PDE). It can be shown that Zubov's 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's PDE enables the development of a series solution method that can be easily implemented with the aid of a symbolic software package such as MAPLE. 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 should be pointed out, that for the optimally calculated controller parameter values, an explicit estimate of the size of the closed-loop stability region can be provided by using results from Zubov's stability theory.
