Brief paperThe minimal disturbance invariant set: Outer approximations via its partial sums☆
Introduction
We consider sets defined by the Minkowski sum,that play a well-known role in the disturbance response of the systemSpecifically, the following facts are known (Aubin, 1991, Kolmanovsky and Gilbert, 1998). If A is asymptotically stable, is compact and , then there exists a compact set such that: for and as . Further, is: the limit set for all solutions of (2), disturbance-invariant (d-invariant) in that it satisfies the inclusion , minimal over the class of all d-invariant sets. For computational reasons, it is assumed hereafter that is polyhedral. Thus, is polyhedral. Even so, with the exception of a few trivial cases (Kolmanovsky and Gilbert, 1998, Mayne and Schroeder, 1997), defies exact, concrete characterization. Consequently, efforts have been made to approximate in various ways (Bertsekas and Rhodes, 1971, Blanchini, 1999, Kolmanovsky and Gilbert, 1998, Rakovic et al., 2005, Schweppe, 1973). Since the are inner approximations of , it is tempting to enlarge them so they then become relatively tight outer approximations. The simplest way of doing this is to use approximations: sets such that and . See for example, Kolmanovsky and Gilbert (1998) where an algorithmic method isproposed for approximately minimizing . Such “minimal” approximations do not in general retain the d-invariance property of . This is unfortunate since d-invariance of sets that approximate 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 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 by exploring in comprehensive and integrated ways three families of approximations: approximations, d-invariant approximations, approximations where and is the maximal d-invariant subset of . Contributions include: algorithms for optimizing approximations that have advantages over the methods in Kolmanovsky and Gilbert (1998) and Hirata and Ohta (2003); an alternative setting for d-invariant approximations that sharpens the results in Rakovic et al., 2004, Rakovic et al., 2005 and simplifies related proofs; comparisons of the d-invariant approximations and the d-invariant 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 approximations and presents a new algorithm for minimizing when . Section 3 gives the main algebraic results on d-invariant approximations. Proofs of the theorems are included because they are short and provide insight. Section 4 discusses the alternative d-invariant approximation, , 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: is the cardinality of and is the interior of , X is a C set (Blanchini, 1999) if it is compact, convex, and , for , for , the Minkowski sum of X and Y. Polyhedral C sets are assumed to have representations of the form , where and for all and is minimal so that measures the complexity of X. If and is non-singular, the operations , QX and preserve polyhedral and C set properties of their operands. The p-norm unit ball is . For and , is a polyhedral C set of the form , .
In this paragraph, are assumed to be compact. The support function of , is defined for all ; ; if ; if for all ; if and only if for all ; if is minimal, for all . Suppose . Then the Hausdorff distance (Schneider, 1993) between X and Y becomes . Thus, is the smallest closed p-neighborhood of Y that contains X. It is easy to confirm that implies .
Assume hereafter: A is asymptotically stable (spectral radius ); is controllable; is a C set. This implies is a C set and is a C set if and only if . Unless otherwise noted, . Then is a polyhedral C set for all .
Section snippets
The class of approximations
The class of all approximations is given by whereIt is easy to show exists and as . Thus, , the tightest approximation of , is defined for all and of can be made arbitrarily close to 1 by the choice of k. While is not generally d-invariant, it is useful in determining the size of disturbance induced errors in (2). For example, suppose denotes a system error in (2); then with for
The class of d-invariant approximations
Assume in this section that B is the identity matrix and is a polyhedral C set. The definitions of d-invariance and approximations provide a necessary and sufficient condition for a d-invariant approximation:By substituting (1) into (9), it is seen that condition (9) is equivalent toThis leads to the proof of the following theorem. Let Theorem 1 The following results hold for all. (i) and, as. (ii) There exists a
The d-invariant approximation
While , the maximal d-invariant subset of , has been used in Kolmanovsky and Gilbert (1998), Hirata and Ohta (2003) as a test for the validity of , its potential as a d-invariant approximation of has not been fully explored. This is unfortunate because approximations have important advantages: they apply when does not exist ( or and ); when does exist, they are often better approximations in the sense that with ; unlike
Approximation errors and their algorithmic implications
The error in approximations of is best measured by , the Hausdorff distance between and . Since , . Because concrete characterizations of are rare, there is no general way of evaluating . Nevertheless, a computationally feasible, fairly tight, upper bound for is available. Since ,Further, , a characterization that, for and , 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 and for values of k selected from a full range of k values. The entry is used for when d-invariant approximations do not exist. Other quantities are also included: , the number of
Conclusions
This paper addresses the outer approximations of that are based on its partial sums . Three classes of polyhedral approximations are investigated: -approximations (, ); d-invariant -approximations (, when ); d-invariant approximations (, ). 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
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2023, AutomaticaCitation 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, AutomaticaCitation 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 ResearchCitation 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.
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2018, Nonlinear Analysis: Hybrid SystemsCitation 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].
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).
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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.