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

Feedback controllers for port-Hamiltonian systems reveal an intrinsic inverse optimality property since each passivating state feedback controller is optimal with respect to some specific performance index. Due to the nonlinear port-Hamiltonian system structure, however, explicit (forward) methods for optimal control of port-Hamiltonian systems require the generally intractable analytical solution of the Hamilton-Jacobi-Bellman equation. Adaptive dynamic programming methods provide a means to circumvent this issue. However, the few existing approaches for port-Hamiltonian systems hinge on very specific sub-classes of either performance indices or system dynamics or require the intransparent guessing of stabilizing initial weights. In this paper, we contribute towards closing this largely unexplored research area by proposing a time-continuous adaptive feedback controller for the optimal control of general time-continuous input-state-output port-Hamiltonian systems with respect to general Lagrangian performance indices. Its control law implements an online learning procedure which uses the Hamiltonian of the system as an initial value function candidate. The time-continuous learning of the value function is achieved by means of a certain Lagrange multiplier that allows to evaluate the optimality of the current solution. In particular, constructive conditions for stabilizing initial weights are stated and asymptotic stability of the closed-loop equilibrium is proven. Our work is concluded by simulations for exemplary linear and nonlinear optimization problems which demonstrate asymptotic convergence of the controllers resulting from the proposed online adaptation procedure.