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Nonlinear Controller Design Via Approximate Solution of Hamilton-Jacobi Equations

Costas Kravaris, Dimitra Mousavere, and Antonis Pappas. Department of Chemical Engineering, University of Patras, 1 Karatheodori St., Patras, 26500, Greece

This work develops a numerical algorithm for the calculation of optimal nonlinear state feedback for nonlinear systems. The optimality criterion consists of quadratic error terms and quadratic input penalty terms, integrated over an infinite time horizon. The optimization problem is approached via Hamilton-Jacobi equations, whose solution determines the optimal nonlinear state feedback law.

In order to reduce computational effort and complexity in the numerical calculations, the proposed algorithm involves the application of the Newton-Kantorovich iteration to the pertinent nonlinear equations. At each step of the iteration, a Zubov partial differential equation is approximately solved via power series. At step N of the iteration, the method generates the (N+1)-th order truncation of the Taylor series expansion of the optimal state feedback function.

A specific application of the method is in the control of nonlinear chemical processes that exhibit unstable inverse dynamics (non-minimum-phase behaviour). The proposed formulation is not restricted to minimum-phase systems and therefore is directly applicable to nonlinear processes with inverse response characteristics. As a comparison, a synthetic output formulation is also considered, with ISE-optimal choice of synthetic output. Again, the Hamilton-Jacobi equation is needed for the calculation of the optimal synthetic output.

The theoretical results are applied to the control of a nonisothermal CSTR where a series-parallel Van de Vusse reaction takes place. The two formulations Hamilton-Jacobi with input penalty versus ISE-optimal synthetic output are compared via simulation studies. Performance-wise, the results are comparable, but the optimal synthetic output formulation involves much heavier computational effort.