CONTROL AND ESTIMATION FOR A CLASS OF IMPULSIVE DYNAMICAL SYSTEMS

Tatiana F. Filippova     (Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., Ekaterinburg, 620990; Ural Federal University, 19 Mira str., Ekaterinburg, 620002, Russian Federation)

Abstract


The nonlinear dynamical control system with uncertainty in initial states and parameters is studied. It is assumed that the dynamic system has a special structure in which the system nonlinearity is due to the presence of quadratic forms in system velocities. The case of combined controls is studied here when both classical measurable control functions and the controls generated by vector measures are allowed.  We present several theoretical schemes and the estimating algorithms allowing to find the upper bounds for reachable sets of the studied control system.   The research develops the techniques of the ellipsoidal calculus and of the theory of evolution equations for set-valued states of dynamical systems having in their description the uncertainty of set-membership kind.  Numerical results of system modeling based on the proposed methods are included.


Keywords


Control systems, Nonlinearity of quadratic type, Uncertainty, Impulse control, Ellipsoidal calculus, Tube of trajectories

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References


Asselborn L., Groß D., Stursberg O. Control of uncertain nonlinear systems using ellipsoidal reachability calculus. IFAC Proc. Volumes, 2013. Vol. 46, no. 23. P. 50–55. DOI: 10.3182/20130904-3-FR-2041.00204

Aubin J.-P., Frankowska H. Set-Valued Analysis. Basel: Birkhäuser, 1990. 461 p.

August E., Lu J., Koeppl H. Trajectory enclosures for nonlinear systems with uncertain initial conditions and parameters. In: Proc. of the 2012 American Control Conf. , June 27–29, 2012, Montréal, Canada. QC. IEEE Computer Soc., 2012. P. 1488–1493.

Blanchini F., Miani S. Set-Theoretic Methods in Control. Ser. Syst. Control: Foundations & Applications. Birkhäuser, Basel, 2015. XV+487 p. DOI: 10.1007/978-0-8176-4606-6

Boscain U., Chambrion T., Sigalotti M. On some open questions in bilinear quantum control. In: Proc. of the European Control Conf. (ECC), July 17–19, 2013, Zurich, Switzerland. IEEE Xplore, 2013. P. 2080–2085. DOI: 10.23919/ECC.2013.6669238

Ceccarelli N., Di Marco M., Garulli A., Giannitrapani A. A set theoretic approach to path planning for mobile robots. In: Proc. 43rd IEEE Conf. on Decision and Control (CDC) Dec. 14–17, 2004, Nassau, Bahamas. IEEE Xplore, 2004. P. 147–152. DOI: 10.1109/CDC.2004.1428621

Chernousko F.L. State Estimation for Dynamic Systems. CRC Press: Boca Raton, 1994. 320 p.

Chernousko F.L., Rokityanskii D.Ya. Ellipsoidal bounds on reachable sets of dynamical systems with matrices subjected to uncertain perturbation. J. Optim. Theory Appl., 2000. Vol. 104, No. 1. P. 1–19. DOI: 10.1023/A:1004687620019

Demyanov V.F., Rubinov A.M. Quasidifferential Calculus. New York: Optimization Software Inc., 1986.

Filippova T.F. Set-valued solutions to impulsive differential inclusions. Math. Comput. Model. Dyn. Syst., 2005. No. 11. P. 149–158.

Filippova T.F. Differential equations of ellipsoidal state estimates in nonlinear control problems under uncertainty. Discrete Contin. Dyn. Syst., Vol. Suppl.-2011. 2011 P. 410–419. DOI: 10.3934/proc.2011.2011.410

Filippova T.F. Approximation techniques in impulsive control problems for the tubes of solutions of uncertain differential systems. Advances in Applied Mathematics and Approximation Theory. Springer Proc. Math. Stat., vol. 41. New York: Springer, 2013. P. 385-396. DOI: 10.1007/978-1-4614-6393-1_25

Filippova T.F. State estimation for a class of nonlinear dynamic systems with uncertainty through dynamic programming technique. In: Proc. of the 6th Int. Conf. PhysCon2013, August 26–29, 2013, San Lois Potosi, Mexico, 2013. P. 1–6.

Filippova T.F. Estimating reachable sets of control systems with uncertainty on initial data and with nonlinearity of a special kind. In: Proc. of the Int. Conf. Stability and Oscillations of Nonlinear Control Systems (Pyatnitskiy’s Conference), June 1–3, 2016, Moscow, Russia. IEEE Xplore, 2016. P. 1–4. DOI: 10.1109/STAB.2016.7541183

Filippova T.F. Ellipsoidal estimates of reachable sets for control systems with nonlinear terms. IFAC-PapersOnLine, 2017. Vol. 50, No. 1. P. 15355–15360. DOI: 10.1016/j.ifacol.2017.08.2460

Filippova T.F. Dynamics and estimates of star-shaped reachable sets of nonlinear control systems. J. Chaotic Modeling and Simulation (CMSIM), 2017. No. 4. P. 469–478.

Filippova T.F. Estimation of star-shaped reachable sets of nonlinear control system. In: Lecture Notes in Comput. Sci., vol. 10665: Proc. Large-Scale Scientific Computing. LSSC 2017. Cham: Springer, 2018. P. 210–218. DOI: 10.1007/978-3-319-73441-5_22

Filippova T.F. Differential equations for ellipsoidal estimates of reachable sets for a class of control systems with nonlinearity and uncertainty. IFAC-PapersOnLine, 2018. Vol. 51, No. 32. P. 770–775. DOI: 10.1016/j.ifacol.2018.11.452

Filippova T.F. Description of dynamics of ellipsoidal estimates of reachable sets of nonlinear control systems with bilinear uncertainty. In: Lecture Notes in Comput. Sci., vol. 11189: Numerical Methods and Applications. NMA 2018. Cham: Springer, 2019. P. 97–105. DOI: 10.1007/978-3-030-10692-8_11

Filippova T.F., Lisin D.V. On the estimation of trajectory tubes of differential inclusions. Proc. Steklov Inst. Math., 2000. Vol. Suppl. 2. P. S28–S37.

Filippova T.F., Matviychuk O.G. Algorithms to estimate the reachability sets of the pulse controlled systems with ellipsoidal phase constraints. Autom. Remote Control, 2011. Vol. 72, No. 9. P. 1911–1924. DOI: 10.1134/S000511791109013X

Filippova T.F., Matviychuk O.G. Algorithms of estimating reachable sets of nonlinear control systems with uncertainty. J. Chaotic Modeling and Simulation, 2015. No. 3. P. 205–214.

Kishida M., Braatz R.D. Ellipsoidal bounds on state trajectories for discrete-time systems with linear fractional uncertainties. Optim. Eng., 2015. Vol. 16. P. 695–711. DOI: 10.1007/s11081-014-9255-9

Kostousova E.K., Kurzhanski A.B. Theoretical framework and approximation techniques for parallel computation in set-membership state estimation. In: Proc. of the Symposium on Modelling Analysis and Simulation, July 9–12, 1996, Lille, France, 1996. No. 2. P. 849–854.

Kuntsevich V.M., Volosov V.V. Ellipsoidal and interval estimation of state vectors for families of linear and nonlinear discrete-time dynamic systems. Cybernet. Systems Anal., 2015. Vol. 51. P. 64–73. DOI: 10.1007/s10559-015-9698-9

Kurzhanski A.B. Upravlenie i nablyudenie v usloviyah neopredelennosti [Control and Observation under Conditions of Uncertainty]. Moscow: Nauka, 1977. 392 p. (in Russian)

Kurzhanski A.B., Filippova T.F. On the theory of trajectory tubes — a mathematical formalism for uncertain dynamics, viability and control. In: Advances in Nonlinear Dynamics and Control: a Report from Russia, ed. A. B. Kurzhanski. Progress in Systems and Control Theory, vol. 17. Boston: Birkhäuser, 1993. P. 122–188. DOI: 10.1007/978-1-4612-0349-0_4

Kurzhanski A.B., Valyi I. Ellipsoidal Calculus for Estimation and Control. Systems Control Found. Appl. Basel: Birkhäuser, 1997. 321 p.

Kurzhanski A.B., Varaiya P. Dynamics and Control of Trajectory Tubes: Theory and Computation. Systems Control Found. Appl., vol. 85. Basel: Birkhäuser, 2014. 445 p. DOI: 10.1007/978-3-319-10277-1

Malyshev V.V., Tychinskii Yu.D. Construction of sets of attainability and maneuver optimization for low-thrust artificial satellites of the earth in a strong gravitational field. J. Comput. Syst. Sci. Int., 2005. Vol. 44, No. 4. P. 622–630.

Miller B.M. Method of discontinuous time change in problems of control for impulse and discrete-continuous systems. Autom. Remote Control, 1993. Vol. 54, No. 12. P. 1727–1750.

Panasyuk A.I. Equations of attainable set dynamics. Part 1: Integral funnel equations. J. Optimiz. Theory Appl., 1990. Vol. 64. P. 349–366. DOI: 10.1007/BF00939453

Pereira F.L., Filippova T.F. On a solution concept to impulsive differential systems. In: Proc. of 4th Int. Conf. Tools for Mathematical Modelling (MathTools’03), June 23–28, 2003, St.Petersburg, Russia. 2003. P. 350–355.

Rishel R. An extended Pontryagin principle for control system whose control laws contain measures. SIAM J. Control, 1965. Vol. 3. P. 191–205.

Schweppe F.C. Uncertain Dynamic Systems. New Jersey: Prentice-Hall, Englewood Cliffs, 1973. 563 p.

Veliov V.M. Second order discrete approximations to strongly convex differential inclusions. Systems Control Lett., 1989. Vol. 13. P. 263–269.

Veliov V. M. Second-order discrete approximation to linear differential inclusions. SIAM J. Numer. Anal., 1992. Vol. 29, no. 2. P. 439–451.




DOI: http://dx.doi.org/10.15826/umj.2019.2.003

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