Stabilization of Linear Systems by Pulsewidth Modulation of Switching Actuators

The overall dynamics of linear systems under pulsewidth modulated actuators is nonlinear, an inevitable fact that must be considered in designing feedback control for these systems. For small sampling periods, however, this nonlinearity is not severe, so that controller design can rely on the same techniques conventionally adopted for the linear systems. When a linear controller designed by these techniques is applied to the originally nonlinear system, it may not perform adequately at higher sampling periods. In particular, it may fail to stabilize the closed-loop system as predicted for a linear approximation of the original system. This paper introduces a class of nonlinear feedback laws outperforming their existing linear counterparts in terms of stability and transient response. The control laws in this class are nonlinearly modified forms of the linear quadratic (LQ) regulators and yield a closed-loop behavior similar to that of a linear system under a LQ regulator. As a result, the closed-loop dynamics is locally stable in essence, and will be globally stable under additional assumptions. As a case study, an inverted pendulum on a cart is stabilized using the proposed nonlinear control law. The superior performance of this control law over its linear counterparts is shown by numerical simulations.