Article information
2017 , Volume 22, ¹ 2, p.19-36
Kostousova E.K.
On feedback target control for uncertain discrete-time systems through polyhedral techniques
Problems of feedback target control for linear and bilinear discrete-time systems under uncertainties and state constraints are considered. We continue the development of methods of control synthesis using polyhedral (parallelotope-valued) solvability tubes. The paper deals with two types of problems, where the controls appear either additively or in the system matrix (i. e., in the coefficients of the system). Both problems are considered for systems with parallelotope-bounded additive uncertainty and with interval uncertainties in the coefficients. Moreover the systems are considered under constraints on the state. The techniques for calculation of the polyhedral solvability tubes by the recurrent relations are presented. Control strategies, which can be constructed on the base of the mentioned polyhedral tubes by explicit formulas, are proposed. Illustrative examples are given.
[full text] Keywords: discrete-time systems, uncertain systems, state constraints, control synthesis, solvability tubes, parallelotopes, parallelepipeds, interval analysis
Author(s): Kostousova Elena Kirillovna Dr. , Senior Scientist Position: Leading research officer Office: N.N. Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences Address: 620990, Russia, Ekaterinburg, 16, S.Kovalevskaya str.
Phone Office: (343) 375-35-01 E-mail: kek@imm.uran.ru SPIN-code: 8563-6036 References: [1] Krasovskii, N.N., Subbotin, A.I. Game-Theoretical Control Problems. New York: Springer-Verlag; 1988: 517. [2] Kurzhanski, A.B., Valyi, I. Ellipsoidal Calculus for Estimation and Control. Boston: Birkhauser; 1997: 321. [3] Kurzhanski, A.B., Varaiya, P. Dynamics and Control of Trajectory Tubes: Theory and Computation. Series “Systems & Control: Foundations & Applications”. Basel: Birkhauser; 2014; (85): 445. [4] Kurzhanski, A.B., Nikonov, O.I. On the problem of synthesizing control strategies. Evolution equations and multivalued integration. Soviet Mathematics Doklady. 1990; 41(2):300–305. [5] Kurzhanski, A.B., Mel’nikov, N.B. On the problem of control synthesis: the Pontryagin alternating integral and the Hamilton–Jacobi equation. Sb. Math. 2000; 191(6):849–881. [6] Aubin, J.-P., Bayen, A.M., Saint-Pierre, P. Viability Theory: New Directions. Berlin: Springer-Verlag; 2011: 803. [7] Cardaliaguet, P., Quiancampoix, M., Saint-Pierre, P. Set-valued numerical analysis for optimal control and differential games. Eds. Bardi, M., Raghavan, T.E.S., Parthasarathy, Ņ. Stochastic and Differential Games: Theory and Numerical Methods. Annals of the International Society of Dynamic Games. Boston: Birkhauser; 1999: (4):177–247. [8] Chernousko, F.L. State estimation for dynamic systems. Boca Raton: CRC Press; 1994: 304. [9] Lotov, A.V. A numerical method for constructing sets of attainability for linear controlled systems with phase constraints. U.S.S.R. Computational Mathematics and Mathematical Physics. 1976; 15(1):63–74. [10] Barmish, B.R., Sankaran, J. The propagation of parametric uncertainty via polytopes. IEEE Transactions on Automatic Control. 1979; AC-24(2):346–349. [11] Kumkov, S.I., Patsko, V.S., Pyatko, S.G., Fedotov, A.A. Construction of the solvability set in a problem of guiding an aircraft under wind disturbance. Proceedings of the Steklov Institute of Mathematics. 2005; ( Suppl 1):S163–S174. [12] Pakhotonsky, V.Yu., Uspensky, A.A., Ushakov, V.N. Construction of stable bridges in differential games with phase restrictions. Journal of Applied Mathematics and Mechanics. 2003; 67(5):681–692. [13] Mitchell, I.M., Bayen, A.M., Tomlin, C.J. A time-dependent Hamilton-Jacobi formulation of reachable sets for continuous dynamic games. IEEE Transactions on Automatic Control. 2005; 50(7):947–957. [14] Botkin, N., Turova, V. Numerical construction of viable sets for autonomous conflict control systems. Mathematics. 2014; 2(2):68–82. Doi:10.3390/math2020068. [15] Chernousko, F.L. Ellipsoidal state estimation for dynamical systems. Nonlinear Analysis. 2005; 63(5–7):872–879. [16] Filippova, T.F. Differential equations for ellipsoidal estimates for reachable sets of a nonlinear dynamical control system. Proceedings of the Steklov Institute of Mathematics. 2010; (Suppl 3):S75–S84. [17] Gusev, M.I. External estimates of the reachability sets of nonlinear controlled systems. Automation and Remote Control. 2012; 73(3):450–461. [18] Daryin, A.N., Kurzhanski, A.B. Parallel algorithm for calculating the invariant sets of high-dimensional linear systems under uncertainty. Computational Mathematics and Mathematical Physics. 2013; 53(1):34–43. [19] Vazhentsev, A.Yu. On internal ellipsoidal approximations for problems of control synthesis with bounded coordinates. Journal of Computer and Systems Sciences International. 2000; 39(3):399–406. [20] Kostousova, E.K. Control synthesis via parallelotopes: optimization and parallel computations. Optimization methods & software. 2001; 14(4):267–310. [21] Kostousova, E.K. Polyhedral estimates for attainability sets of linear multistage systems with integral constraints on the control. Computational Technologies. 2003; 8(4):55–74. (In Russ.) [22] Kostousova, E.K. On polyhedral estimates in problems for the synthesis of control strategies in linear multistep systems. Algoritmy i programmnye sredstva parallelnykh vychisleniy. Ekaterinburg: Rossiiskaya Akademiya Nauk, Ural’skoe Otdelenie, Institut Matematiki i Mekhaniki. 2006; (9):84–105. (In Russ.) [23] Kostousova, E.K. On boundedness and unboundedness of polyhedral estimates for reachable sets of linear differential systems. Reliable Computing. 2013; (19):26– 44. Available at: http://interval.louisiana.edu/reliable-computing-journal/volume-19/reliablecomputing-19-pp-026-044.pdf (accessed 20.02.2017). [24] Kostousova, E.K. On control synthesis for uncertain differential systems using a polyhedral technique. Lecture Notes in Computer Science. 2014; (8353):98–106. [25] Kostousova, E.K. On target control synthesis under set-membership uncertainties using polyhedral techniques. IFIP Advances in Information and Communication Technology (IFIP AICT). 2014; (443):170–180. [26] Kostousova, E.K. On the polyhedral method of solving problems of control strategy synthesis. Proceedings of the Steklov Institute of Mathematics. 2016; 292(Suppl 1):S140–S155. [27] Moore, R.E. Methods and applications of interval analysis. Philadelphia: SIAM;1979: 190. [28] Shary, S.P. Konechnomernyy interval'nyy analiz [Finite-dimensional interval analysis]. Novosibirsk: XYZ; 2016: 617. Available at: http://www.nsc.ru/interval/Library/InteBooks/SharyBook.pdf (accessed 20.02.2017). (In Russ.) [29] Kornoushenko, E.K. Interval coordinatewise estimates for the set of accessible states of a linear stationary system. I. Automation and Remote Control. 1980; (41):598–606. [30] Kornoushenko, E.K. Interval coordinatewise estimates for the set of accessible states of a linear stationary system. II. Automation and Remote Control. 1981; (41):1633–1639. [31] Ivlev, R.S., Sokolova, S.P. Constructing vector control for an /intervally/ specified multivariable object. Computational Technologies. 1999; 4(4):3–13. (In Russ.) [32] Jaulin, L., Kieffer, M., Didrit, O., Walter, E. Applied Interval Analysis: With Examples in Parameter and State Estimation, Robust Control and Robotics. London: Springer-Verlag; 2001: 379. [33] Ramdani, N., Nedialkov, N.S. Computing reachable sets for uncertain nonlinear hybrid systems using interval constraint propagation techniques. Nonlinear Analysis: Hybrid Systems. 2011; 5(2):149–162. [34] Lhommeau, M., Jaulin, L., Hardouin, L. Capture basin approximation using interval analysis. International Journal Adaptive Control and Adaptive Signal Processing. 2011. Vol. 25, No. 3. P. 264–272. [35] Filippova, T.F., Lisin, D.V. On the estimation of trajectory tubes of differential inclusions. Proceedings of the Steklov Institute of Mathematics. 2000; (Suppl 2):S28–S37. [36] Bakhvalov, N.S., Zhidkov, N.P., Kobel’kov, G.M. Chislennye metody [Numerical methods]. Moscow: Nauka; 1987: 600. (In Russ.)
Bibliography link: Kostousova E.K. On feedback target control for uncertain discrete-time systems through polyhedral techniques // Computational technologies. 2017. V. 22. ¹ 2. P. 19-36
|