Optimized robust control invariance for linear discrete-time systems: Theoretical foundations

Abstract

This paper introduces the concept of optimized robust control invariance for discrete-time linear time-invariant systems subject to additive and bounded state disturbances. A novel characterization of two families of robust control invariant sets is given. The existence of a constraint admissible member of these families can be checked by solving a single and tractable convex programming problem in the generic linear-convex case and a standard linear/quadratic program when the constraints are polyhedral or polytopic. The solution of the same optimization problem yields the corresponding feedback control law that is, in general, set-valued. A procedure for selection of a point-valued, nonlinear control law is provided.

Introduction

The theory of set invariance plays a fundamental role in control of constrained dynamical systems; see for instance the monograph (Aubin, 1991) and the survey paper (Blanchini, 1999). An important role for set invariance is evident in stability theory (La Salle, 1976). Set invariance, inter alia, provides useful tools for the synthesis of: (i) reference governors (Gilbert & Kolmanovsky, 1999), (ii) predictive controllers (Bemporad and Morari, 1999, Findeisen et al., 2003, Mayne, 2001), (iii) robust time-optimal controllers (Bertsekas and Rhodes, 1971, Mayne and Schroeder, 1997) and (iv) robust, tube based, model predictive controllers (Mayne et al., 2005, Raković, 2005). An application of set invariance in non-cooperative dynamic games is reported in Caravani and De Santis, 2000, Caravani and De Santis, 2002 and Raković, De Santis, and Caravani (2005).

Given the importance of set invariance in control theory, the subject has been a topical research area over the last 40 years. A non-exhaustive list of the relevant references includes Blanchini (1994), Blanchini, Mesquine, and Miani (1995), Kolmanovsky and Gilbert (1998), Dórea and Hennet (1999), da Silva Jr. and Tarbouriech (1999), De Santis, Di Benedetto, and Berardi (2004), Raković, Kerrigan, Kouramas, and Mayne (2005) and Raković (2005). Most of these texts address computational issues and algorithmic procedures for the calculation of robust control and positively invariant sets as well as the application of these sets to robust control for constrained systems. One of the prime questions considered in the existing literature is the computation of the maximal robust control invariant (RCI) set (Aubin, 1991, Bertsekas, 1972, Blanchini, 1999, Dórea and Hennet, 1999). An algorithmically efficient technique is based on the computation of $\lambda$ contractive sets (Blanchini, 1994). The theory and computation of minimal and maximal robust positively invariant (RPI) sets for the case of autonomous linear systems are examined in the important paper (Kolmanovsky & Gilbert, 1998). A method for finite time computation of an arbitrarily close outer RPI approximation of the minimal RPI set is discussed in Raković, Kerrigan, et al. (2005). Further contributions include the computation of maximal control and RCI sets for linear discrete-time systems (Dórea & Hennet, 1999) and stabilization of linear discrete-time systems subject to control constraints and an assigned initial condition set (Blanchini et al., 1995).

In this paper we provide a novel characterization of RCI sets. To this end, we introduce two families of RCI sets. The families are parameterized in a way that permits selection of RCI sets via tractable convex optimization problem in the generic linear-convex case. The techniques presented in this paper may be used to obtain improved RCI sets with respect to RPI sets that approximate the minimal RPI set (Raković, Kerrigan, et al., 2005) and, in principle, to obtain a RCI set that approximates the maximal robust control invariant set (see Remark 1, Remark 3 in Section 3).

This paper is organized as follows. Section 2 is concerned with preliminaries. Section 3 provides a novel characterization of two families of RCI sets. Section 4 discusses corresponding computational issues. Sections 5 presents some interesting numerical examples. Section 6 indicates potential extensions and provides concluding remarks.

Nomenclature and basic definitions: Let $\mathbb{N}\triangleq \{0,1,2,\dots \}$, ${\mathbb{N}}_{+}\triangleq \{1,2,\dots \}$, ${\mathbb{N}}_{[q_1,q_2]}\triangleq \{q_1,q_1+1,\dots ,q_2-1,q_2\}$ for a given $q_1\in \mathbb{N}$ and $q_2\in \mathbb{N}$ such that $q_1<q_2$ and ${\mathbb{N}}_q$ denotes ${\mathbb{N}}_{[0,q]}$ for $q\in \mathbb{N}$. Let ${\mathbb{B}}_p^q(r)\triangleq \{x\in {\mathbb{R}}^q\mid |x{|}_p⩽r\}$ be a closed p-norm ball in ${\mathbb{R}}^q$, where $r⩾0$ and $|·{|}_p$ denotes the vector p-norm. Given two sets $\mathcal{A}$ and $\mathcal{B}$, such that $\mathcal{A}\subset {\mathbb{R}}^n$ and $\mathcal{B}\subset {\mathbb{R}}^n$, the Minkowski set addition is defined by $\mathcal{A}\oplus \mathcal{B}\triangleq \{a+b\mid a\in \mathcal{A},b\in \mathcal{B}\}$. Given a vector $v\in {\mathbb{R}}^n$ (which could be a sum of q vectors) and a set $\mathcal{S}\subset {\mathbb{R}}^n$, we write $v\oplus \mathcal{S}$ to denote $\{v\}\oplus \mathcal{S}$. Given the sequence of sets $\{{\mathcal{S}}_i\subset {\mathbb{R}}^n\},i\in {\mathbb{N}}_{[a,b]}$ with $(a,b)\in \mathbb{N}\times \mathbb{N},b>a$, we define ${\oplus }_{i=a}^b{\mathcal{S}}_i\triangleq {\mathcal{S}}_a\oplus \cdots \oplus {\mathcal{S}}_b$. Given a set $\mathcal{S}\subset {\mathbb{R}}^n$ and a real matrix M of compatible dimensions (which could be just a scalar) we define $M\mathcal{S}\triangleq \{\mathit{Ms}\mid s\in \mathcal{S}\}$. The controllability index of a given matrix pair $(A,B)\in {\mathbb{R}}^{n\times n}\times {\mathbb{R}}^{n\times m}$ is denoted by $\mathcal{I}(A,B)$. A polyhedron is the (convex) intersection of a finite number of open and/or closed half-spaces. A polytope is a closed and bounded polyhedron. A set $\mathcal{S}\subset {\mathbb{R}}^n$ is a C set if it is compact, convex and contains the origin. A C set $\mathcal{S}$ is proper if it has a non-empty interior.