Eric C. Kerrigan

Invariant approximations of the minimal robust positively Invariant set

This note provides results on approximating the minimal robust positively invariant (mRPI) set (also known as the 0-reachable set) of an asymptotically stable discrete-time linear time-invariant system. It is assumed that the disturbance is bounded, persistent and acts additively on the state and that the constraints on the disturbance are polyhedral. Results are given that allow for the computation of a robust positively invariant, outer approximation of the mRPI set. Conditions are also given that allow one to a priori specify the accuracy of this approximation.

Optimal control of constrained, piecewise affine systems with bounded disturbances

The solution to the problem of optimal control of piecewise affine systems with a bounded disturbance is characterised. Results that allow one to compute the value function, its domain (robustly controllable set) and the optimal control law are presented. The tools that are employed include dynamic programming, polytopic set algebra and parametric programming. When the cost is time (robust time-optimal control problem) or the stage cost is piecewise affine (robust optimal and robust receding horizon control problems), the value function and the optimal control law are both piecewise affine and each robustly controllable set is the union of a finite set of polytopes. Conditions on the cost and constraints are also proposed in order to ensure that the optimal control laws are robustly stabilising.

Invariant sets for constrained nonlinear discrete-time systems with application to feasibility in model predictive control

An understanding of invariant set theory is essential in the design of controllers for constrained systems, since state and control constraints can be satisfied if and only if the initial state belongs to 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.

Designing model predictive controllers with prioritised constraints and objectives

This paper shows how a class of objective functions can be incorporated into a prioritised, multiobjective optimisation problem, for which a solution can be obtained by solving a sequence of single-objective, constrained, convex programming problems. The objective functions considered in this paper typically arise in model predictive control (MPC) of constrained, linear systems. The framework presented in this paper can be used to design a flexible, multiobjective MPC controller that takes priorities into account during the online computation of the control input.

On the Minimal Robust Positively Invariant Set for Linear Difference Inclusions

This paper provides a new and efficient method for the computation of an arbitrarily close outer robust positively invariant (RPI) approximation to the minimal robust positively invariant (mRPI) set for linear difference inclusions. It is assumed that the linear difference inclusion is absolutely asymptotically stable (AAS) in the absence of an additive state disturbance, which is the case for parametrically uncertain or switching linear discrete-time systems controlled by a stabilizing linear state feedback controller.

Robust feasibility in model predictive control: necessary and sufficient conditions

A number of results are derived for analysing the robust feasibility of a given model predictive control (MPC) scheme which ignores model mismatch and/or disturbances during control input computation. The main contribution of this paper is the development of computationally tractable tests for determining the robust feasibility of an MPC controller for linear or piecewise affine systems, where the constraints are given by the union of convex polyhedra and the disturbance acts additively on the state. Practical tests are also presented which allow one to give robust feasibility guarantees for all optimal and suboptimal MPC control actions.

Robustly stable feedback min-max model predictive control

This paper is concerned with the practical real-time implementability of robustly stable model predictive control (MPC) when constraints are present on the inputs and the states. We assume that the plant model is known, is discrete-time and linear time-invariant, is subject to unknown but bounded state disturbances and that the states of the system are measured. In this paper we introduce a new stage cost and show that the use of this cost allows one to formulate a robustly stable MPC problem that can be solved using a single linear program, which implies that the receding horizon control (RHC) law is piecewise affine, and can be explicitly pre-computed, so that the linear program does not have to be solved on-line.

A GEOMETRIC APPROACH TO REACHABILITY COMPUTATIONS FOR CONSTRAINED DISCRETE-TIME SYSTEMS

Abstract The problem of reachability computations for systems with simultaneous constraints on the states, control inputs and disturbances is treated. A problem formulation is provided for the general case and, for the special case where the disturbance constraints are independent of the states and inputs, it is discussed how the solution can be approached using standard geometric concepts. It is also discussed how the procedure can be implemented for a class of piecewise affine systems with polyhedral constraints using computational geometry software.

Optimized robust control invariant sets for constrained linear discrete-time systems

Abstract In this paper we introduce the concept of optimized robust control invariance for a discrete-time, linear, time-invariant system subject to additive state disturbances. A novel characterization of a family of the robust control invariant sets is given. The existence of a constraint admissible member of this family can be checked by solving a single linear programming problem. The solution of the same linear programming problem yields the corresponding feedback controller.

RELATIONSHIPS BETWEEN AFFINE FEEDBACK POLICIES FOR ROBUST CONTROL WITH CONSTRAINTS

Abstract This paper is concerned with the analysis of a recently-proposed robust control policy for linear discrete-time systems subject to bounded state disturbances with mixed constraints on the states and inputs, which parameterizes the input as an affine function of the past disturbance sequence. The paper shows that this disturbance feedback policy is equivalent to the class of affine state feedback policies with memory of prior states, and thus subsumes the wellknown classes of open-loop and pre-stabilising control policies. Furthermore, the parameterization transforms the non-convex problem of finding an admissible state feedback policy to an equivalent and tractable convex problem.