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

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

Model predictive control of linear periodic systems - a unified framework including control of multirate and multiplexed systems

State-feedback model predictive control (MPC) of discrete-time linear periodic systems with possibly time-dependent state and control input dimension is considered. States and inputs are subject to hard, mixed, polytopic constraints. It is described how discrete-time linear systems, both time-invariant and periodic, with multirate or multiplexed control inputs can be modeled as such periodic systems. This makes linear periodic systems with possibly time-dependent dimensions a unified, coherent and succinct state-space modeling framework for a large variety of control problem for linear plants, periodic or non. In this paper it is shown how important theoretical results for state-feedback MPC of constrained linear time-invariant (LTI) systems are conceptually equivalent to what is required for linear periodic systems. Specifically the determination of (maximum) periodic controlled and positively invariant sets and the solution of reverse periodic discrete-time algebraic Riccati equations are considered indispensable. A general definition, and a method for the determination, of maximum periodic controlled and positively invariant sets are proposed here. Thus least-restrictive, strongly feasible MPC problems resulting in infinite-horizon optimal state-feedback control laws are designed. The proposed methods are applied to a multirate twin-actuator nano-positioning system