On stabilizing linear systems with input saturation via soft variable structure control laws

A large number of control methods deal with stabilizing single-input linear systems with input saturation. The linear full-state feedback is one of the most used, yielding either nonsaturating or high gain controls, whereby the saturating effects of the latter are reduced by means of anti-windup structures. Among the nonlinear controls, the time optimal control, in this case a bang–bang type control, yields the fastest time response, but its switching surface is generally not characterizable. Related to the speed of the time response is the convergence rate, which can be determined using invariant sets. The most used ones are the ellipsoidal sets, since they can be analyzed using powerful tools such as the Lyapunov equation and designed via convex optimization. For this reason, they are also used for designing soft variable structure controls. The paper presents a nonconservative design method for a stabilizing control of this type employing implicit Lyapunov functions (iLF). A nonsaturating control law is given, including some infinitely densely nested and contractive invariant sets of the equilibrium state. The control law is then optimized by maximizing the iLF-based lower bound of the convergence rate. The maximal convergence control is shown to be of bang–bang type, with a parameter dependent switching scheme. To overcome possible difficulties of a switching controller, a saturating high gain control is also presented.