Invariance of symmetric convex sets for discrete-time saturated systems

Abstract

The characterization of invariance and contractiveness for discrete-time saturated linear systems is considered. The geometrical approach used to analyze the problem leads to conditions valid for generic symmetric convex sets. The application of the results to the ellipsoidal case generalizes known results and leads to computational improvements.

Necessary and sufficient conditions for invariance of convex sets for discrete-time saturated systems

“…Such conditions allow also to implicitly characterize local setinduced Lyapunov functions. The paper extends the results in [5], that deals only with symmetric saturations and sets and with sufficient results. The necessary and sufficient condition holds also for asymmetric saturations and it can be extended to systems with affine nonlinearity at the input.…”

New condition for invariance of ellipsoidal sets for discrete-time saturated systems

“…which is equivalent to (11). Thus, the condition (11) and (14) are equivalent, for each J ∈ I and E ∈ K. Therefore, the condition (11) and (8) are also equivalent, for each J ∈ I and E ∈ K.…”

“…In the last decade, the stability of linear systems subject to actuator saturation attracted a big interest. There are many researchers working on this topic currently, see [1], [4], [8], [12], [13], [16], [19]. The maximal output admissible sets have important applications in the analysis and design of closed-loop systems with state and control constraints [10].…”

“…In this paper, we consider the problem of characterizing and computing contractive sets for discrete-time saturated systems. This sets provide estimations of the domain of attraction and determine set-induced Lyapunov function, see [1], [2], [8]. As a popular candidate invariant set, the ellipsoids have been widely used as an estimation of the domain of attraction for such systems, see [1], [8].…”

“…Sufficient conditions are given for an ellipsoid to be invariant or contractive. They allow to characterize the level set of a local quadratic Lyapunov function, see [1], [8], [15]. The work in this paper is the extension of the results in [9], specifying the convex set as an ellipsoid.…”