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

• Published in: Automatica (Oxford) Vol. 47; no. 2; pp. 326 - 333
• Main Authors: Gondhalekar, Ravi, Jones, Colin N.
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Ltd 01.02.2011; Elsevier
• Subjects: Applied sciences; Asynchronous control; Automation; Computer science; control theory; systems; Constrained control; Constraints; Control system analysis; Control system synthesis; Control systems; Control theory. Systems; Exact sciences and technology; Invariants; Laws; Linear periodic systems; Mathematical models; Model predictive control; Nanostructure; Optimal control; Optimization; Set invariance; Model predictive control; Asynchronous control; Constrained control; Set invariance; Linear periodic systems; State feedback; Asynchronous; Invariant subspace; Control program; Feedback regulation; Control synthesis; Invariant set; Dynamical system; Strong solution; Modeling; Periodic control; Control constraint; Time dependence; Nanometer scale; LQ control; Quadratic control; Constrained optimization; Closed feedback; Polyhedron; Input to state stability; Discrete time; Feasibility; State dependence
• ISSN: 0005-1098; 1873-2836
• DOI: 10.1016/j.automatica.2010.10.021

State-feedback model predictive control (MPC) of discrete-time linear periodic systems with time-dependent 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.