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Optimal Control of Port-Hamiltonian Systems A Time-Continuous Learning Approach

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Document pages: 17 pages

Abstract: Feedback controllers for port-Hamiltonian systems reveal an intrinsic inverseoptimality property since each passivating state feedback controller is optimalwith respect to some specific performance index. Due to the nonlinearport-Hamiltonian system structure, however, explicit (forward) methods foroptimal control of port-Hamiltonian systems require the generally intractableanalytical solution of the Hamilton-Jacobi-Bellman equation. Adaptive dynamicprogramming methods provide a means to circumvent this issue. However, the fewexisting approaches for port-Hamiltonian systems hinge on very specificsub-classes of either performance indices or system dynamics or require theintransparent guessing of stabilizing initial weights. In this paper, wecontribute towards closing this largely unexplored research area by proposing atime-continuous adaptive feedback controller for the optimal control of generaltime-continuous input-state-output port-Hamiltonian systems with respect togeneral Lagrangian performance indices. Its control law implements an onlinelearning procedure which uses the Hamiltonian of the system as an initial valuefunction candidate. The time-continuous learning of the value function isachieved by means of a certain Lagrange multiplier that allows to evaluate theoptimality of the current solution. In particular, constructive conditions forstabilizing initial weights are stated and asymptotic stability of theclosed-loop equilibrium is proven. Our work is concluded by simulations forexemplary linear and nonlinear optimization problems which demonstrateasymptotic convergence of the controllers resulting from the proposed onlineadaptation procedure.

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