Elsevier

Automatica

Volume 42, Issue 9, September 2006, Pages 1563-1568
Automatica

Brief paper
The minimal disturbance invariant set: Outer approximations via its partial sums

https://doi.org/10.1016/j.automatica.2006.04.019Get rights and content

Abstract

This paper is concerned with outer approximations of the minimal disturbance invariant set (MDIS) of a discrete-time linear system with an additive set-bounded disturbance. The k-step disturbance reachable sets (Minkowski partial sums) are inner approximations of MDIS that converge to MDIS. Enlarged by a suitable scaling, they can lead to outer approximations of MDIS. Three families of approximations, each based on partial sums, are considered. Theoretical properties of the families are proved and interrelated. Algorithmic questions, including error bounds, are addressed. The results are illustrated by computational data from several examples.

Introduction

We consider sets defined by the Minkowski sum,Fk=BW+ABW++Ak-1BWRn,that play a well-known role in the disturbance response of the systemx(t+1)=Ax(t)+Bw(t),w(t)WRm.Specifically, the following facts are known (Aubin, 1991, Kolmanovsky and Gilbert, 1998). If A is asymptotically stable, W is compact and 0W, then there exists a compact set FRn such that: 0FkF for k{1,2,,}=N+ and FkF as k. Further, F is: the limit set for all solutions of (2), disturbance-invariant (d-invariant) in that it satisfies the inclusion AF+BWF, minimal over the class of all d-invariant sets. For computational reasons, it is assumed hereafter that W is polyhedral. Thus, Fk is polyhedral. Even so, with the exception of a few trivial cases (Kolmanovsky and Gilbert, 1998, Mayne and Schroeder, 1997), F defies exact, concrete characterization. Consequently, efforts have been made to approximate F in various ways (Bertsekas and Rhodes, 1971, Blanchini, 1999, Kolmanovsky and Gilbert, 1998, Rakovic et al., 2005, Schweppe, 1973). Since the Fk are inner approximations of F, it is tempting to enlarge them so they then become relatively tight outer approximations. The simplest way of doing this is to use Fk approximations: sets σFk such that FσFk and σ>1. See for example, Kolmanovsky and Gilbert (1998) where an algorithmic method isproposed for approximately minimizing σ. Such “minimal”Fk approximations do not in general retain the d-invariance property of F. This is unfortunate since d-invariance of sets that approximate F may be useful in a variety of robust control schemes. See, for example: Bemporad, Casavola, and Mosca (1997), Mayne and Schroeder (1997), Gilbert and Kolmanovsky (1999), Mayne, Rawling, Rao, and Scokaert (2000) and others. In view of the considerable prior attention given to d-invariant sets, the recent results of Rakovic et al., 2004, Rakovic et al., 2005 are quite remarkable. A broad family of d-invariant Fk approximations is characterized and algorithmic means are given for achieving a specified accuracy of approximation.

Motivated, in part, by ideas in Rakovic et al. (2005), this paper takes a broader view of outer approximations based on Fk by exploring in comprehensive and integrated ways three families of approximations: Fk approximations, d-invariant Fk approximations, O(σFk) approximations where σ>1 and O(σFk) is the maximal d-invariant subset of σFk. Contributions include: algorithms for optimizing Fk approximations that have advantages over the methods in Kolmanovsky and Gilbert (1998) and Hirata and Ohta (2003); an alternative setting for d-invariant Fk approximations that sharpens the results in Rakovic et al., 2004, Rakovic et al., 2005 and simplifies related proofs; comparisons of the d-invariant Fk approximations and the d-invariant O(σFk) approximations, including summaries of key data on representative example problems.

Contents of the paper are arranged as follows. This section concludes with assumptions, notations and a review of basic results that are used subsequently without further comment. Section 2 characterizes the class of all Fk approximations and presents a new algorithm for minimizing σ when FσFk. Section 3 gives the main algebraic results on d-invariant Fk approximations. Proofs of the theorems are included because they are short and provide insight. Section 4 discusses the alternative d-invariant approximation, O(σFk), and its features. Algorithmic issues, applicable to all three families of approximations, are addressed in Section 5. Section 6 presents several illustrative examples. Concluding remarks appear in Section 7.

Notations and definitions are standard: |I| is the cardinality of IN+ and int(X) is the interior of XRn, X is a C set (Blanchini, 1999) if it is compact, convex, 0int(X) and int(X), αX={αx:xX} for αR, QX={Qx:xX} for QRm×n, X+Y={x+y:xX,yY} the Minkowski sum of X and Y. Polyhedral C sets are assumed to have representations of the form X={x:eXix1iIX}, where eXiR1×n and eXi0 for all iIXN+ and |IX| is minimal so that |IX| measures the complexity of X. If α0 and QRn×n is non-singular, the operations αX, QX and X+Y preserve polyhedral and C set properties of their operands. The p-norm unit ball is Bp={x:xp1}. Forp=1 and , BpRn is a polyhedral C set of the form Bp={x:epix1iIp}, IpN+.

In this paragraph, X,YRn are assumed to be compact. The support function of X,hX(η)maxxXηx, is defined for all ηR1×n; hαX(η)=hX(αη),hQX(η)=hX(ηQ),hX+Y(η)=hX(η)+hY(η); if XγB2,hX(η)γη2; if 0int(X),hX(η)>0 for all η0; YX if and only if hY(eXi)1 for all iIX; if |IX| is minimal, hX(eXi)=1 for all iIX. Suppose YX. Then the Hausdorff distance (Schneider, 1993) between X and Y becomes d(X,Y)=min{α:XY+αBp}. Thus, Y+d(X,Y)Bp is the smallest closed p-neighborhood of Y that contains X. It is easy to confirm that ZY implies d(X,Y)d(X,Z).

Assume hereafter: A is asymptotically stable (spectral radius ρ(A)<1); A,B is controllable; W is a C set. This implies F is a C set and Fk is a C set if and only if kkmin{:rank[BABA-1B]=n}. Unless otherwise noted, W={x:eWix1iIW}. Then Fk is a polyhedral C set for all kkmin.

Section snippets

The class of Fk approximations

The class of all Fk approximations is given by {σFk:σσ̲k} whereσ̲kmin{σ:FσFk},kkmin.It is easy to show σ̲k exists and σ̲k1 as k. Thus, σ̲kFk, the tightest Fk approximation of F, is defined for all kkmin and σ of σFk can be made arbitrarily close to 1 by the choice of k. While σ̲kFk is not generally d-invariant, it is useful in determining the size of disturbance induced errors in (2). For example, suppose e(t)=Cex(t) denotes a system error in (2); then with x(0)Fk,e(t)σ̲kCeFk for

The class of d-invariant Fk approximations

Assume in this section that B is the identity matrix and WRn is a polyhedral C set. The definitions of d-invariance and Fk approximations provide a necessary and sufficient condition for a d-invariant Fk approximation:σAFk+WσFk,σ>1.By substituting (1) into (9), it is seen that condition (9) is equivalent toAkWσ-1(σ-1)W,σ>1.This leads to the proof of the following theorem. LetμkmaxiIWhW(eWiAk),σk(1-μk)-1.

Theorem 1

The following results hold for allkN+. (i) μk>0andμj0, asj. (ii) There exists a

The d-invariant approximation O(σFk)

While O(σFk), the maximal d-invariant subset of σFk, has been used in Kolmanovsky and Gilbert (1998), Hirata and Ohta (2003) as a test for the validity of FσFk, its potential as a d-invariant approximation of Fk has not been fully explored. This is unfortunate because O(σFk) approximations have important advantages: they apply when σkFk does not exist (rankB<n or rankB=n and μk1); when σkFk does exist, they are often better approximations in the sense that O(σFk)σFk with σ<σk; unlike σkFk

Approximation errors and their algorithmic implications

The error in Fk approximations of F is best measured by d(σFk,F), the Hausdorff distance between σFk and F. Since FσFk, d(σFk,F)=min{α:σFkF+αBp}. Because concrete characterizations of F are rare, there is no general way of evaluating d(σFk,F). Nevertheless, a computationally feasible, fairly tight, upper bound for d(σFk,F) is available. Since FkF,d(σFk,F)d(σFk,Fk).Further, d(σFk,Fk)=min{α:σFkFk+αBp}=min{α:(σ-1)FkαBp}, a characterization that, for p=1 and p=, allows the

Numerical examples

The numerical results presented in this section involve four examples that are representative of many more that have been studied. Details are given in Table 1. In all examples, except for Ic, B is the identity matrix. Results of the experiments are shown in Table 2, Table 3. Each table shows σk,σ̲k,dk and d̲k for values of k selected from a full range of k values. The entry is used for σk when d-invariant Fk approximations do not exist. Other quantities are also included: |Ik|, the number of

Conclusions

This paper addresses the outer approximations of F that are based on its partial sums Fk. Three classes of polyhedral approximations are investigated: Fk-approximations (FσFk, σσ̲k); d-invariant Fk-approximations (FσFk, σσk when μk<1); O(σFk)d-invariant approximations (FO(σFk)σFk, σ>σ̲k). Approximations within each class are algorithmically implementable and bounds on the resulting approximation error are available. While any level of approximation accuracy can in principle be

Chong-Jin Ong received his B.Eng (Hons) and M.Eng degrees in mechanical engineering from the National University of Singapore in 1986 and 1988, respectively, and the M.S.E. and Ph.D. degrees in mechanical and applied mechanics from University of Michigan, Ann Arbor, in 1992 and 1993, respectively.

He joined the National University of Singapore in 1993 and is now an Associate Professor with the Department of Mechanical Engineering. His research interests are in model predictive control, robust

References (14)

There are more references available in the full text version of this article.

Cited by (60)

  • Computation of invariant sets via immersion for discrete-time nonlinear systems

    2023, Automatica
    Citation Excerpt :

    The early literature has been devoted to tractable fixed-point algorithms for linear systems with polyhedral constraints, see, e.g., Blanchini (1999) and Gilbert and Tan (1991) and the references therein. In the presence of bounded disturbances in linear systems, robust invariant sets are defined and corresponding fixed-point algorithms have been developed, see, e.g., Kolmanovsky and Gilbert (1998), Ong and Gilbert (2006), Rakovic, Kerrigan, Kouramas, and Mayne (2005) and Trodden (2016). Recently, the authors in Wang, Jungers, and Ong (2021) have proposed a fixed-point algorithm for linear systems subject to a class of non-convex constraints.

  • Computation of the maximal invariant set of discrete-time linear systems subject to a class of non-convex constraints

    2021, Automatica
    Citation Excerpt :

    In Dorea and Hennet (1999), Gilbert and Tan (1991) and Pluymers et al. (2005), recursive algorithms have been proposed to compute polyhedral invariant sets of linear systems. For linear systems with bounded disturbances, robust invariant sets can be computed using different algorithms (Artstein & Raković, 2008; Kolmanovsky & Gilbert, 1998; Ong & Gilbert, 2006; Rakovic et al., 2005; Raković et al., 2007; Trodden, 2016). For linear systems with control, the computation of (control) invariant sets is more complicated and a few algorithms have been proposed to compute inner or outer approximations (Darup & Cannon, 2017; Gutman & Cwikel, 1987; Rungger & Tabuada, 2017).

  • Tube-based robust output feedback model predictive control for autonomous rendezvous and docking with a tumbling target

    2020, Advances in Space Research
    Citation Excerpt :

    However, determining the mRPI is difficult except that the closed-loop system dynamics are nilpotent. Many scholars (Kolmanovsky and Gilbert, 1998; Rakovic et al., 2005; Ong and Gilbert, 2006; Martinez, 2015; Trodden, 2016) have proposed several methods to approximate the mRPI set, with the most concise and practical method for linear discrete uncertain systems having been contributed by Rakovic et al. (2005). However, this method can only be applied to two- or three-dimensional cases, because the minkowski sum is impossible to evaluate in higher-dimensional system.

  • Invariant convex approximations of the minimal robust invariant set for linear difference inclusions

    2018, Nonlinear Analysis: Hybrid Systems
    Citation Excerpt :

    It is well known that the state of a stable linear system with additive disturbances converges to an invariant set which is known as the minimal robust positively invariant (mRPI) set, see [1,2].

View all citing articles on Scopus

Chong-Jin Ong received his B.Eng (Hons) and M.Eng degrees in mechanical engineering from the National University of Singapore in 1986 and 1988, respectively, and the M.S.E. and Ph.D. degrees in mechanical and applied mechanics from University of Michigan, Ann Arbor, in 1992 and 1993, respectively.

He joined the National University of Singapore in 1993 and is now an Associate Professor with the Department of Mechanical Engineering. His research interests are in model predictive control, robust control and machine learning.

Elmer Gilbert received the degree of Ph.D. from the University of Michigan in 1957. Since then, he has been in the Department of Aerospace Engineering at the University of Michigan and is now Professor Emeritus. His current interests are in optimal control, nonlinear systems, robotics and machine learning. He has published numerous papers and holds eight patents. He received IEEE Control Systems Field Award in 1994 and the Bellman Control Heritage Award of the American Automatic Control Council in 1996. He is a member of the Johns Hopkins Society of Scholars, a Fellow of the American Association for the Advancement of Science, a Fellow of the Institute of Electrical and Electronics Engineers and a member of the National Academy of Engineering (USA).

This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Tor Arne Johansen under the direction of Editor Frank Allgöwer.

View full text