MPC of constrained discrete-time linear periodic systems — A framework for asynchronous control: Strong feasibility, stability and optimality via periodic invariance

Abstract

a b s t r a c tState-feedback model predictive control (MPC) of discrete-time linear periodic systems with timedependent state and input dimensions is considered. The states and inputs are subject to periodically time-dependent, hard, convex, polyhedral constraints. First, periodic controlled and positively invariant sets are characterized, and a method to determine the maximum periodic controlled and positively invariant sets is derived. The proposed periodic controlled invariant sets are then employed in the design of least-restrictive strongly feasible reference-tracking MPC problems. The proposed periodic positively invariant sets are employed in combination with well-known results on optimal unconstrained periodic linear-quadratic regulation (LQR) to yield constrained periodic LQR control laws that are stabilizing and optimal. One motivation for systems with time-dependent dimensions is efficient control law synthesis for discrete-time systems with asynchronous inputs, for which a novel modeling framework resulting in low dimensional models is proposed. The presented methods are applied to a multirate nano-positioning system.

Least-restrictive robust MPC of periodic affine systems with application to building climate control

“…The determination of maximum robust periodic controlled invariant sets for a very similar problem setting was considered in [5]. The determination of maximum periodic controlled invariant sets for general linear periodic systems was considered in [14,15]. The maximum robust periodic controlled invariant sets of this paper can be computed with straightforward modifications of the methods presented in [5,14,15].…”

“…Periodic systems are useful extensions of time-invariant systems, and provide a natural framework for modeling various real-world phenomena and control problems [3,25,26,29]. The presented methods are an extension of theory developed for periodic linear systems in [14,15]. Periodic affine systems are an important generalization of periodic linear systems, but many tools for rigorously dealing with periodic affine systems can be derived from the tools for periodic linear systems.…”

“…Thus constraints can be rigorously enforced, indefinitely, for any {oldewurtel,jones}@control.ee.ethz.ch prediction horizon length, even one shorter than one period length. Closely related maximum periodic controlled invariant sets for periodic linear systems were considered in [5,14,15]. MPC was not considered in [5].…”

“…MPC was not considered in [5]. Unlike [14,15], in this paper robustness is considered. Robustness in MPC is usually enforced either by considering min-max openloop predictions [2] or by performing closed-loop predictions [1,2,11,16,24].…”

On stability and stabilization of periodic discrete-time systems with an application to satellite attitude control

“…and Zulli (1995), model predictive control Böhm (2011), Gondhalekar and Jones (2011), and control with saturation Zhou et al (2011). In the monograph (Bittanti and Colaneri, 2009, Chapter 13), a thorough exposition of existing results on stabilization techniques, including also frequency domain considerations or lifting techniques, is presented.…”