Control problems for set-valued motions of systems with uncertainty and nonlinearity. (English) Zbl 1451.93210

Filippova, Tatiana F.; Matviychuk, Oxana G.

Tarasyev, Alexander (ed.) et al., Stability, control and differential games. Proceedings of the international conference on stability, control and differential games (SCDG2019), Yekaterinburg, Russia, September 16–20, 2019. Cham: Springer. Lect. Notes Control Inf. Sci. – Proc., 379-389 (2020).

Summary: The control and estimation problems are considered for a nonlinear control system with uncertainty in the initial data and with quadratic nonlinearity in vectors of system velocities. It is assumed also that the values of unknown initial states and admissible controls are constrained by related ellipsoids. Basing on the application of the results and methods of the theory of ellipsoidal calculus applied for estimation of set-valued motions of studied systems, the problems of guaranteed control for the tubes of trajectories of a nonlinear system with uncertainty are investigated. Algorithms for guaranteed moving of the set-valued state of the control system to the smallest neighborhood of a given target set are proposed, the results are illustrated by a model example.
For the entire collection see [Zbl 1444.93003].

MSC:

93C41	Control/observation systems with incomplete information
93C10	Nonlinear systems in control theory
93B03	Attainable sets, reachability

Keywords:

nonlinear control system; uncertainty; set-valued state

References:

[1]	Agarwal, R.P., Hedia, B., Beddani, M.: Structure of solution sets for impulsive fractional differential equations. J. Fract. Calc. Appl. 9(1), 15-34 (2018)
[2]	Alamo, T., Bravo, J.M., Camacho, E.F.: Guaranteed state estimation by zonotopes. Automatica 41(6), 1035-1043 (2005) · Zbl 1091.93038
[3]	Apreutesei, N.C.: An optimal control problem for a prey-predator system with a general functional response. Appl. Math. Lett. 22(7), 1062-1065 (2009). https://doi.org/10.1016/j.aml.2009.01.016 · Zbl 1179.49023
[4]	August, E., Lu, J., Koeppl, H.: Trajectory enclosures for nonlinear systems with uncertain initial conditions and parameters. In: Proceedings of the 2012 American Control Conference, pp. 1488-1493, Fairmont Queen Elizabeth, Montréal, Canada, June 2012
[5]	Boscain, U., Chambrion, T., Sigalotti, M.: On some open questions in bilinear quantum control. In: European Control Conference (ECC), pp. 2080-2085, Zurich, Switzerland, July 2013
[6]	Ceccarelli, N., Di Marco, M., Garulli, A., Giannitrapani, A.: A set theoretic approach to path planning for mobile robots. In: Proceedings of the 43rd IEEE Conference on Decision and Control, pp. 147-152, Atlantis, Bahamas, December 2004
[7]	Cichoń, M., Satco, B.R.: Measure differential inclusions—between continuous and discrete. Adv. Differ. Equ. (1), 56, 1-18 (2014) · Zbl 1350.49014
[8]	Chernousko, F.L.: State Estimation for Dynamic Systems. CRC Press, Boca Raton (1994)
[9]	Filippova, T.F.: Differential equations of ellipsoidal state estimates in nonlinear control problems under uncertainty. Discret. Contin. Dyn. Syst. Supplement 2011, Dynamical Systems, Differential Equations and Applications 1, 410-419, Springfield, American Institute of Mathematical Sciences (2011) · Zbl 1306.93011
[10]	Filippova, T.F.: Estimates of reachable sets of impulsive control problems with special nonlinearity. In: Proceedings of the AIP Conference, vol. 1773, pp. 100004, 1-8 (2016). https://doi.org/10.1063/1.4964998
[11]	Filippova, T.F.: Differential equations of ellipsoidal state estimates for bilinear-quadratic control systems under uncertainty. J. Chaotic Model. Simul. 1, 85-93 (2017)
[12]	Filippova, T.F.: Ellipsoidal estimates of reachable sets for control systems with nonlinear terms. IFAC-PapersOnLine: 20th IFAC World Congress, Toulouse, France, July 9-14, 2017 50(1), 15355-15360 (2017)
[13]	Filippova, T.F.: Differential equations for ellipsoidal estimates of reachable sets for a class of control systems with nonlinearity and uncertainty. IFAC-PapersOnLine 51(32), 770-775 (2018)
[14]	Filippova, T.F.: Estimation of star-shaped reachable sets of nonlinear control systems. In: Lirkov, I., Margenov, S., Wasniewski, J. (eds.) LSSC 2017, vol. 10665, pp. 210-218. LNCS. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-73441-5_22 · Zbl 1447.93015
[15]	Filippova, T.F.: Description of dynamics of ellipsoidal estimates of reachable sets of nonlinear control systems with bilinear uncertainty. In: Nikolov, G., Kolkovska, N., Georgiev, K. (eds.) Numerical Methods and Applications, NMA 2018. Lecture Notes in Computer Science, vol. 11189, pp. 97-105. Springer, Cham (2019) https://doi.org/10.1007/978-3-030-10692-8_11 · Zbl 1420.93007
[16]	Filippova, T.F., Berezina, E.V.: On state estimation approaches for uncertain dynamical systems with quadratic nonlinearity: theory and computer simulations. In: Lirkov, I., Margenov, S., Wasniewski, J. (eds.) Large-Scale Scientific Computing, LSSC 2007. Lecture Notes in Computer Science, vol. 4818, pp. 326-333. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78827-0_36 · Zbl 1229.93016
[17]	Filippova, T.F., Matviychuk, O.G.: Reachable sets of impulsive control system with cone constraint on the control and their estimates. In: Lirkov, I., Margenov, S., Wasniewski, J. (eds.) Large-Scale Scientific Computing, LSSC 2011. Lecture Notes in Computer Science, vol. 7116, pp. 123-130. Springer, Berlin, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29843-1_13 · Zbl 1354.93022
[18]	Filippova, T.F., Matviychuk, O.G.: Estimates of reachable sets of control systems with bilinear-quadratic nonlinearities. Ural. Math. J. 1(1), 45-54 (2015). https://doi.org/10.15826/umj.2015.1.004 · Zbl 1396.93017
[19]	Gusev, M.I.: Internal approximations of reachable sets of control systems with state constraints. Proc. Steklov Inst. Math. 287(suppl. 1), 77-92 (2014). https://doi.org/10.1134/S0081543814090089 · Zbl 1305.93033
[20]	Koller, T., Berkenkamp, F., Turchetta, M., Krause, A.: Learning-based model predictive control for safe exploration. In: Proceedings of the IEEE Conference on Decision and Control CDC2018, 8619572, pp. 6059-6066 (2018)
[21]	Kostousova, E.K., Kurzhanski, A.B.: Guaranteed estimates of accuracy of computations in problems of control and estimation. Vychisl. Tekhnologii 2(1), 19-27 (1997)
[22]	Krasovskii, N.N.: Theory of Motion Control: Linear Systems. Nauka, Moscow (1968) · Zbl 0172.12702
[23]	Krasovskii, N.N.: On control under incomplete information. J. Appl. Math. Mech. 40(2), 179-187 (1976) · Zbl 0355.90080
[24]	Krasovskii, N.N., Subbotin, A.I.: Game-Theoretical Control Problems. Springer Series in Soviet Mathematics, Springer, New York (1988)
[25]	Kurzhanski, A.B.: Control and Observation under Uncertainty. Nauka, Moscow (1977)
[26]	Kurzhanski, A.B., Filippova, T.F.: On the theory of trajectory tubes a mathematical formalism for uncertain dynamics, viability and control. In: Kurzhanski, A.B. (ed.) Advances in Nonlinear Dynamics and Control: A Report from Russia. Progress in Systems and Control Theory, vol. 17, pp. 122-188. Birkhäuser, Boston (1993) · Zbl 0912.93040
[27]	Kurzhanski, A.B., Valyi, I.: Ellipsoidal Calculus for Estimation and Control. Birkhäuser, Boston (1997) · Zbl 0865.93001
[28]	Kurzhanski, A.B., Varaiya, P.: Dynamics and control of trajectory tubes: theory and computation. Syst. Control. Found. Appl. 85. Basel, Birkhäuser (2014) · Zbl 1336.93004
[29]	Malyshev, V.V., Tychinskii, Yu.D.: Construction of attainability sets and optimization of maneuvers of an artificial Earth satellite with thrusters in a strong gravitational field. Proc. RAS Theory Control. Syst. 4, 124-132 (2005)
[30]	Matviychuk, O.G.: Estimates of the reachable set of nonlinear control system under uncertainty. In: Proceedings of the AIP Conference, vol. 2025, p. 100005, 1-9 (2018). https://doi.org/10.1063/1.5064934
[31]	Mazurenko, S.S.: A differential equation for the gauge function of the star-shaped attainability set of a differential inclusion. Doklady Math. 86(1), 476-479 (2012) · Zbl 1357.93008
[32]	Schweppe, F.C.: Uncertain Dynamical Systems. Prentice-Hall, Englewood Cliffs, NJ (1973)
[33]	Scott, J.K., Raimondo, D.M., Marseglia, G.R., Braatz, R.D.: Constrained zonotopes: a new tool for set-based estimation and fault detection. Automatica 69, 126-136 (2016) · Zbl 1338.93119