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External polyhedral estimates of reachable sets of discrete-time systems with integral bounds on additive terms. (English) Zbl 1471.93027

Summary: We deal with the reachability problem for linear and bilinear discrete-time uncertain systems under integral non-quadratic constraints on additive input terms and set-valued constraints on initial states. The bilinearity is caused by an interval type uncertainty in coefficients of the system. Algorithms for constructing external parallelepiped-valued (shorter, polyhedral) estimates of reachable sets are presented. For linear time-invariant systems, two techniques for constructing touching external estimates with constant orientation matrices are described and compared. For time-dependent bilinear systems, parallelepiped-valued estimates are constructed. For bilinear systems with constant coefficients, nonconvex estimates are proposed in the form of unions of parallelepipeds. Evolution of all estimates is determined by systems of recurrence relations.

MSC:

93B03 Attainable sets, reachability
93C55 Discrete-time control/observation systems
93C05 Linear systems in control theory
93C10 Nonlinear systems in control theory
52B12 Special polytopes (linear programming, centrally symmetric, etc.)
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[1] R. Baier; T. Donchev, Discrete approximation of impulsive differential inclusions, Numer. Funct. Anal. Optim., 31, 653-678 (2010) · Zbl 1205.34012 · doi:10.1080/01630563.2010.483878
[2] V. A. Baturin; E. V. Goncharova; F. L. Pereira; J. B. Sousa, Measure-controlled dynamic systems: Polyhedral approximation of their reachable set boundary, Autom. Remote Control, 67, 350-360 (2006) · Zbl 1126.93314 · doi:10.1134/S0005117906030027
[3] F. L. Chernousko; D. Ya. Rokityanskii, Ellipsoidal bounds on reachable sets of dynamical systems with matrices subjected to uncertain perturbations, J. Optim. Theory Appl., 104, 1-19 (2000) · Zbl 0968.93011 · doi:10.1023/A:1004687620019
[4] T. F. Filippova, Estimates of reachable sets of impulsive control problems with special nonlinearity, AIP Conf. Proc., 1773 (2016), 100004.
[5] T. F. Filippova, The HJB approach and state estimation for control systems with uncertainty, IFAC-PapersOnLine, 51 (2018), Issue 13, 7-12.
[6] T. F. Filippova, Differential equations for ellipsoidal estimates of reachable sets for a class of control systems with nonlinearity and uncertainty, IFAC-PapersOnLine, 51 (2018), Issue 32,770-775.
[7] A. Girard, C. Le Guernic and O. Maler, Efficient computation of reachable sets of linear time-invariant systems with inputs, in: Hybrid Systems: Computation and Control, Lecture Notes in Comput. Sci., 3927, Springer, Berlin, 2006,257-271. · Zbl 1178.93024
[8] K. G. Guseinov; O. Ozer; E. Akyar; V. N. Ushakov, The approximation of reachable sets of control systems with integral constraint on controls, NoDEA Nonlinear Differential Equations Appl., 14, 57-73 (2007) · Zbl 1141.93010 · doi:10.1007/s00030-006-4036-6
[9] M. I. Gusev, On convexity of reachable sets of a nonlinear system under integral constraints, IFAC-PapersOnLine51 (2018), Issue 32,207-212.
[10] L. Jaulin, M. Kieffer, O. Didrit and É. Walter, Applied Interval Analysis: With Examples in Parameter and State Estimation, Robust Control and Robotics, Springer-Verlag, London, 2001. · Zbl 1023.65037
[11] E. K. Kostousova, State estimation for dynamic systems via parallelotopes: Optimization and parallel computations, Optimiz. Methods and Software, 9, 269-306 (1998) · Zbl 0919.93017 · doi:10.1080/10556789808805696
[12] E. K. Kostousova, Outer polyhedral estimates for attainability sets of systems with bilinear uncertainty, J. Appl. Math. Mech., 66 (2002), 547-558. Erratum in: ibid., 66 (2002), 857. · Zbl 1054.93008
[13] E. K. Kostousova, Polyhedral estimates for attainability sets of linear multistage systems with integral constraints on the control (in Russian), Vychisl. Tekhnol., 8 (2003), no. 4, 55-74. Also available from: http://www.ict.nsc.ru/jct/content/t8n4/Kostousova.pdf. · Zbl 1075.93502
[14] E. K. Kostousova, On the boundedness of outer polyhedral estimates for reachable sets of linear systems, Comput. Math. Math. Phys., 48 (2008), 918-932. Erratum in: ibid., 48 (2008), 1915-1916. · Zbl 1164.93309
[15] E. K. Kostousova, State estimation for linear impulsive differential systems through polyhedral techniques, Discrete Contin. Dyn. Syst. (2009), Issue Suppl., 466-475. · Zbl 1184.93011
[16] E. K. Kostousova, On polyhedral estimates for trajectory tubes of dynamical discrete-time systems with multiplicative uncertainty, Discrete Contin. Dyn. Syst. (2011), Issue Suppl., 864-873. · Zbl 1306.93012
[17] E. K. Kostousova, On boundedness and unboundedness of polyhedral estimates for reachable sets of linear differential systems, Reliable Computing, 19, 26-44 (2013)
[18] E. K. Kostousova, External polyhedral estimates of reachable sets of linear and bilinear discrete-time systems with integral bounds on additive terms, in Proceedings of 2018 14th International Conference Stability and Oscillations of Nonlinear Control Systems (Pyatnitskiys Conference), STAB, IEEE Xplore Digital Library, (2018), 1-4.
[19] V. M. Kuntsevich; A. B. Kurzhanski, Attainability domains for linear and some classes of nonlinear discrete systems and their control, J. Automation and Inform. Sci., 42, 1-18 (2010) · doi:10.1615/JAutomatInfScien.v42.i1.10
[20] A. B. Kurzhanski; A. N. Daryin, Dynamic programming for impulse controls, Annual Reviews in Control, 32, 213-227 (2008) · doi:10.1016/j.arcontrol.2008.08.001
[21] A. B. Kurzhanski and I. Vályi, Ellipsoidal Calculus for Estimation and Control, Birkhäuser, Boston, 1997. · Zbl 0865.93001
[22] A. B. Kurzhanski and P. Varaiya, Dynamics and Control of Trajectory Tubes, Theory and Computation, (Systems & Control: Foundations & Applications, Book 85), Birkhäuser/Springer, Cham, 2014. · Zbl 1336.93004
[23] C. Le Guernic, Calcul Efficace de l’Ensemble Atteignable des Systèmes Linéaires avec Incertitudes, Master’s thesis, Université Paris VII, 2005.
[24] A. V. Lotov, Method for constructing an external polyhedral estimate of the trajectory tube for a nonlinear dynamic system, Doklady Mathematics, 95, 95-98 (2017) · Zbl 1366.65067 · doi:10.1134/S1064562417010045
[25] O. G. Matviychuk, Estimation techniques for bilinear control systems, IFAC-PapersOnLine, 51 (2018), Issue 32,877-882.
[26] S. Mazurenko, Partial differential equation for evolution of star-shaped reachability domains of differential inclusions, Set-Valued Var. Anal., 24, 333-354 (2016) · Zbl 1338.93065 · doi:10.1007/s11228-015-0345-4
[27] B. T. Polyak; S. A. Nazin; C. Durieu; E. Walter, Ellipsoidal parameter or state estimation under model uncertainty, Automatica J. IFAC, 40, 1171-1179 (2004) · Zbl 1056.93063 · doi:10.1016/j.automatica.2004.02.014
[28] V. V. Sinyakov, Method for computing exterior and interior approximations to the reachability sets of bilinear differential systems, Differ. Equ., 51, 1097-1111 (2015) · Zbl 1326.93012 · doi:10.1134/S0012266115080145
[29] V. M. Veliov, On the relationship between continuous- and discrete-time control systems, CEJOR Cent. Eur. J. Oper. Res., 18, 511-523 (2010) · Zbl 1206.93043 · doi:10.1007/s10100-010-0167-2
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