An understanding of invariant set theory is essential in the design of controllers for constrained systems, since state and control constraints can be . . , x p } . The notation {xk E R I P is used to inQcate that each element of the sequence {xk)o P is an element of a positively invariant set for the closed-loop system. The paper briefly reviews some concepts in invariant set theory and shows that the various sets can be computed using a single recursive algorithm. The ideas presented in the first part of the paper are applied to the fundamental design goal of guaranteeing feasibility in predictive control. New necessary and sufficient conditions based on the control horizon, prediction horizon and terminal constraint set are given in order to guarantee that the predictive control problem will be feasible for all time, given any feasible initial state. Keywords: invariant sets, positively invariant sets, control invariant sets, stabilisable sets, controllable sets, predictive control, feasibility. Most systems are subject to state and control constraints and the design of controllers for such systems is a very active area of research. In particular, invariant set theory [3] has been shown to be crucial in understanding the behaviour of constrained systems, since constraints can be satisfied if and only if the initial state is contained in a set which is positively invariant for the closed-loop system. Section 2 provides a unified framework for capturing the most important ideas in invariant set theory. Invariant set theory has been very successful in providing sufficient nominal and robust feasibility and stability conditions in Model Predictive Control (MPC) [ 113. One of the primary reasons for MPC's success is the ease with which constraints on both the control inputs and states can be incorporated in the controller synthesis. However, there still seems to be a few "missing links" in the understanding of the effect of the various design parameters on the feasible domain of an MPC scheme. The main aim of Section 3 is to apply the invariant set framework of Section 2 to MPC, generalise some existing results and present some new necessary and sufficient conditions for the analysis and synthesis of MPC controllers with guaranteed nominal feasibility. It is hoped that the formal framework presented in this paper will facilitate a better understanding of the most basic problem in the design of MPC controllers.  Consider the following discrete-time, nonlinear, timeinvariant (NLTI) dynamic system: where k E Z, Xk is the system state and uk is the control input. It is assumed that f ( q , u k ) is uniquely defined over X x U with f(0,O) = 0. Full state measurement and no disturbances or model uncertainty is assumed. The system is subject to pointwise-in-time constraints on the control inputs and/or the states: The set U is assumed to be compact and simply connected, while X is assumed to be closed and simply connected. It is assumed that ( 0 , O ) E W" x U". An admissible control input, sequence or law is one that satisfies the input constraints. From this point on, it is understood that the control law and states are subject to the constraints in (2). 2.1 Invariant and Positively Invariant Sets Definition 2.1 (Positively invariant set [3] ). The nonempty set st c Rn is positively invariant for the autonomous system xk+l = f ( x k ) if and only if Vxo E R, the system evolution satisfies Xk E R,Vk E W+. The set R is invariant if and only if xo E R implies that Xk E R,Vk E Z. Cl is used to denote any arbitrary subset of R". Given Cl, R" is contained in A, i.e. A\B 5 {x : x E A,x B } = A n BC.
In REF a nonlinear model predictive control framework is presented that guarantees nominal feasibility using invariant sets by providing necessary and sufficient conditions on the control horizon, the prediction horizon and the constraint set.
Invariant sets for constrained nonlinear discrete-time systems with application to feasibility in model predictive control
Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187)
Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187)
Computer Science