Symplectic Integration Schemes for Systems With Nonlinear Dissipation

In previous work a variational integration scheme for mechanical systems containing linear dissipation was developed (Limebeer et al., 2020). The key idea is to use a transmission line as the solution to a power-balancing control problem—in the physics literature controllers of this type are known as “heat baths.” The power-balanced closed-loop system is thereby made amenable to analysis with Hamilton's principle. As a consequence, variational principles can be used to find symplectic integration schemes for dissipative systems. The outcome of this work is a coherent physical explanation as to why variational/symplectic integration algorithms for the open dissipative system can perform as well as their conservative counterparts. An important application of these integration algorithms is the solution of long-time-horizon optimal control problems. In this article, these ideas are extended to systems with smooth memoryless nonlinear dissipation. This extension employs a nonlinear lossless transmission line, which is again the solution to a power-balancing control problem. As in the case of linear dissipation, the resulting variational integration schemes are demonstrated to have excellent numerical properties, which are significantly less demanding computationally than their implicit nonvariational counterparts.