Abstract
The infinite time reachable set of a strictly stable linear control system is the Hausdorff limit of the finite time reachable set of the origin as time tends to infinity. By definition, it encodes useful information on the long-term behavior of the control system. Its characterization as a limit set gives rise to numerical methods for its computation that are based on forward iteration of approximate finite time reachable sets. These methods tend to be computationally expensive, because they essentially perform a Minkowski sum in every single forward step. We develop a new approach to computing the infinite time reachable set that is based on the invariance properties of the control system and the desired set. These allow us to characterize a polyhedral outer approximation as the unique solution to a linear program with constraints that incorporate the system dynamics. In particular, this approach does not rely on forward iteration of finite time reachable sets.
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References
Baier, R., Büskens, C., Chahma, I.A., Gerdts, M.: Approximation of reachable sets by direct solution methods for optimal control problems. Optim Methods Softw. 22(3), 433–452 (2007)
Graettinger, T.J., Krogh, B.H.: Hyperplane method for reachable state estimation for linear time-invariant systems. J. Optim. Theory Appl. 69(3), 555–588 (1991)
Shao, L., Zhao, F., Cong, Y.: Approximation of convex bodies by multiple objective optimization and an application in reachable sets. Optimization 67(6), 783–796 (2018)
Girard, A., Le Guernic, C., Maler, O. Efficient computation of reachable sets of linear time-invariant systems with inputs. In: Hybrid systems: computation and control. vol. 3927 of Lecture Notes in Comput. Sci. Springer, Berlin; (2006), pp. 257–271
Raković, S.V., Fiacchini, M.: Approximate reachability analysis for linear discrete time systems using homothety and invariance. IFAC Proc. Vol. 41(2), 15327–15332 (2008)
Ben Sassi, M.A., Testylier, R., Dang, T., Girard, A.: Reachability analysis of polynomial systems using linear programming relaxations. In: Chakraborty, S., Mukund, M. (eds.) Automated Technology for Verification and Analysis, pp. 137–151. Berlin, Springer (2012)
Kolmanovsky, I., Gilbert, E.G.: Theory and computation of disturbance invariant sets for discrete-time linear systems. Mat.h Prob. Eng. 4(4), 317–367 (1998)
Artstein, Z., Raković, S.V.: Feedback and invariance under uncertainty via set-iterates. Automatica 44(2), 520–525 (2008)
Rakovic, S.V., Kerrigan, E.C., Kouramas, K.I., Mayne, D.Q.: Invariant approximations of the minimal robust positively invariant set. Optimization 50(3), 406–410 (2005)
Dórea, C.E.T., Hennet, J.C.: \((A, B)\)-invariant polyhedral sets of linear discrete-time systems. J. Optim. Theory Appl. 103(3), 521–542 (1999)
Gaitsgory, V., Quincampoix, M.: Linear programming approach to deterministic infinite horizon optimal control problems with discounting. SIAM J. Control. Optim. 48(4), 2480–2512 (2009)
Korda, M., Henrion, D., Jones, C.N.: Convex computation of the maximum controlled invariant set for polynomial control systems. SIAM J. Control. Optim. 52(5), 2944–2969 (2014)
Rieger, J.: A Galerkin approach to optimization in the space of convex and compact subsets of \({\mathbb{R} }^d\). J. Glob. Optim. 79, 593–615 (2021)
Robinson, S.M.: Bounds for error in the solution set of a perturbed linear program. Linear Algebra Appl. 6, 69–81 (1973)
Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge (2013)
Hu, S., Papageorgiou NS. Handbook of Multivalued Analysis. Vol. I. vol. 419 of Mathematics and its Applications. Kluwer Academic Publishers, Dordrecht; Theory (1997)
Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory. Encyclopedia of Mathematics and its Applications, vol. 151. Cambridge University Press, Cambridge (2014)
Zeidler, E.: Nonlinear Functional Analysis and Its Applications, vol. I. Springer, New York (1986)
Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer, New York (1995)
Joswig, M., Theobald, T.: Polyhedral and Algebraic Methods in Computational Geometry. Universitext, Springer, London (2013)
Balas, E.: Disjunctive Programming. Springer, Cham (2018)
Acknowledgements
We thank Andrew Eberhard for pointing us towards the term disjunctive programming, which we would otherwise never have found in the maze of mathematical literature. We would also like to thank the anonymous reviewers for their valuable feedback and suggestions that have helped to improve this paper.
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Ernst, A., Grüne, L. & Rieger, J. A linear programming approach to approximating the infinite time reachable set of strictly stable linear control systems. J Glob Optim 86, 521–543 (2023). https://doi.org/10.1007/s10898-022-01261-w
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DOI: https://doi.org/10.1007/s10898-022-01261-w
Keywords
- Reachable set
- Discrete-time linear systems
- Numerical approximation
- Polytopes
- Linear optimization
- Disjunctive program