Stability of Phase-Based Gain Modulation with Designer-Chosen Switch Functions

Phase-Based Gain-Modulation (PBGM) control is realized by modulating controller gains in response to the phase of the system state or tracking error. PBGM controllers have been applied to robotic hands, parallel manipulators and flexible mechanisms to give increased damping, reduced tracking error and friction compensation. A novel method is presented to establish Lyapunov stability for PBGM control. Prior PBGM stability results incorporated a constraint which limited the range of provably stable systems. The present result removes this constraint, establishing Lyapunov stability for a substantially broader class of systems. Additionally, the new approach decouples the selection of the Lyapunov function from the controller design, permitting the controls designer to independently specify a switch function which determines the application of gain modulation. The present results are applied to analyze PBGM control of the Sarcos dextrous manipulator, illuminating the stability properties of control experiments previously reported in the literature. Numerical methods for design calculations are also presented.