Constrained control of discrete-time linear periodic system

Abstract

The aim of this paper is twofold. In the first part, we provide a method for constructing invariant sets for discrete-time linear periodic systems with state and input constraints. The main advantage of the method is that it generates invariant sets at any step of the underlying set iteration. In the second part a novel interpolating controller between a local unconstrained optimal control law and a global maximum state contractive controller is proposed. At each time instant, two linear programming problems are solved on-line. Proofs of recursive feasibility and asymptotic stability are given.

Reference governors and maximal output admissible sets for linear periodic systems

“…As a result, periodic systems possess N invariant sets that are cyclically related, as discussed below. In [21], [22], these N sets have been characterized for asymptotically stable periodic systems subject to state constraints. However, the case of Lyapunov stable systems, which is central to reference governors, and systems with output constraints were not investigated.…”

“…To investigate, we begin with an example scenario: assume the current timeslot is τ = N − 1. Using Corollary 1 and the structure of A k in (20), we can express H τ x , H τ v , h τ in (22) in terms of H 0x , H 0v , h 0 as:…”

Fast model predictive control for linear periodic systems with state and control constraints

“…Definition The largest periodic invariant set is generally called the maximal admissible set (MAS) , . It is well known that if the system is asymptotically stable, then the MAS can be computed in polyhedral form using linear programming . In the rest of the paper, the MAS for the system with constraints will be denoted as {Ω 0 ,Ω 1 ,…,Ω p −1 } see Figure .…”

“…Definition For the system and the constraints , define as the set of all states, that can be steered to the set {Ω 0 ,Ω 1 ,…,Ω p −1 } in no more than N steps along admissible trajectories, that is, the states that satisfy . is called an N ‐step controlled set and can be constructed by using procedures in . Because {Ω 0 ,Ω 1 ,…,Ω p −1 } is a periodic invariant set, it follows that is a periodic controlled invariant set.…”

“…Among them, one controller is chosen for performance while the rest are used to extend the domain of attraction. Satisfaction of input and state constraints is guaranteed using polyhedral periodic invariant sets . Even though interpolating control has been shown to be an effective solution for the constrained LTI control problem to the best of the authors' knowledge, there is until now no interpolating scheme based on quadratic programming (QP) for periodic systems.…”