Suboptimal adaptive control of dynamic systems with state constraints based on Barrier Lyapunov functions

Abstract

This study designed a suboptimal output control strategy to characterise an attractive and invariant set for the state trajectories of perturbed linear systems with noisy measurements and state constraints. The state constraints were defined by a given polytope formed of by n-dimensional vectors. An adaptive linear controller enforced the existence of an attractive and invariant set (centred at the origin) for the trajectories of the perturbed system. A barrier Lyapunov function (BLF) and the attractive ellipsoid method (AEM) derived the adjustment law of the adaptive gain. The controller design used the linear matrix inequality technique to solve two optimisation problems. The first solution provided a maximal set where the initial conditions must belong without violating the state constraints. The second optimisation solution characterised the invariant minimal attractive set for the system trajectories. An academic example verified how the proposed adaptive control generated the system trajectories that converged to the minimal attractive ellipsoid while keeping them inside the polytope defining the state constraints. The simulation script showed the advantages of the adaptive BLF controller (ABLC) against classical AEM controller. A second numerical example considered a direct current motor showing the advantages of the ABLC against the sliding mode technique.