Structure-preserving local optimal control of mechanical systems

While system dynamics are usually derived in continuous time, respective model-based optimal control problems can only be solved numerically, ie, as discrete-time approximations. Thus, the performance of control methods depends on the choice of numerical integration scheme. In this paper, we present a first-order discretization of linear quadratic optimal control problems for mechanical systems that is structure preserving and hence preferable to standard methods. Our approach is based on symplectic integration schemes and thereby inherits structure from the original continuous-time problem. Starting from a symplectic discretization of the system dynamics, modified discrete-time Riccati equations are derived, which preserve the Hamiltonian structure of optimal control problems in addition to the mechanical structure of the control system. The method is extended to optimal tracking problems for nonlinear mechanical systems and evaluated in several numerical examples. Compared to standard discretization, it improves the approximation quality by orders of magnitude. This enables low-bandwidth control and sensing in real-time autonomous control applications.