Iterative feedback tuning for Hamiltonian systems based on variational symmetry

This paper proposes a novel iterative feedback tuning (IFT) for Hamiltonian systems, which can describe a practically important class of nonlinear systems. Hamiltonian systems have a special property called variational symmetry, and it can be used to estimate the input-output mapping of the variational adjoint for certain input-output mappings of the systems. First, we derive a modified variational symmetry to adapt to the gradient estimation of an optimal control-type cost function with respect to adjustable parameters of a controller. Second, we provide an IFT algorithm based on the property, which generates the optimal parameters minimizing the cost function by iteration of experiments. The proposed algorithm requires less number of experiments to estimate the gradient than the conventional IFT methods for nonlinear systems. We also provide a method to optimize the elements of the dissipation matrix, which does not directly appear in the Hamiltonian function, by equipping a dynamic feedback of the generalized coordinate. Moreover, we provide an IFT algorithm considering parameter constraints so that the parameters can be optimized within a prescribed search range. Finally, a numerical simulation of a two-link robot manipulator including a comparison with the conventional IFT methods demonstrates the effectiveness of the proposed method.