Robust stabilization of a class of polytopic linear time-varying continuous systems under point delays and saturating controls

Abstract

This paper investigates the stabilization dependent on the delays of, in general, time-varying linear systems with multiple constant point time-delays under saturating state-feedback controls. The matrices describing the state-space dynamics and control belong to polytopes. Also, the controller gain matrix is characterized as belonging to another polytope whose vertices are computed from the knowledge of a closed bounded ball containing a set of values of the state norm and also the component-wise saturating gains and saturation parameterizations. This knowledge defines the polytope vertices through scaling diagonal matrices being associated with the various operation modes in the linear and saturated zones of each input component. In these conditions, the resulting closed-loop system is of polytopic nature whose whole number of vertices is (at most) equal to the product of both numbers of vertices of the above two polytopes characterizing the plant parameterization and the saturation. The closed-loop sufficiency-type stability conditions are obtained from Lyapunov’s stability theory by constructing candidates for each vertex of the polytopic closed-loop system each satisfying, in the most general case, a Riccati matrix differential inequality. Some conditions guaranteeing the stability conditions are obtained from a general Kalman–Yakubovitch–Popov (KYP) Lemma and some weaker stability conditions are also obtained for the time-invariant case from a set of linear matrix inequalities associated with the set of vertices.