American Institute of Mathematical Sciences

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2015, 2015(special): 723-732. doi: 10.3934/proc.2015.0723

On control synthesis for uncertain dynamical discrete-time systems through polyhedral techniques

Elena K. Kostousova 1,

1. 

N.N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16, S.Kovalevskaja street, Ekaterinburg, 620990, Russian Federation

Received  July 2014 Revised  January 2015 Published  November 2015

Problems of feedback terminal target control for linear discrete-time systems without and with uncertainties are considered. We continue the development of methods of control synthesis using polyhedral (parallelotope-valued) solvability tubes. The cases without uncertainties, with additive parallelotope-bounded uncertainties, and also with interval uncertainties in coefficients of the system are considered. Also the same systems under state constraints are considered. Nonlinear recurrent relations are presented for polyhedral solvability tubes for each of the mentioned cases. Two types of control strategies, which can be calculated on the base of the mentioned tubes, are proposed. Controls of the second type can be calculated by explicit formulas. Results of computer simulations are presented.
Keywords: parallelotopes, polyhedral estimates, parallelepipeds, control synthesis, Discrete-time systems, interval analysis., uncertain systems.
Mathematics Subject Classification: Primary: 93C41, 93C55, 93B52; Secondary: 93B40, 52B1.
Citation: Elena K. Kostousova. On control synthesis for uncertain dynamical discrete-time systems through polyhedral techniques. Conference Publications, 2015, 2015 (special) : 723-732. doi: 10.3934/proc.2015.0723
References:
[1]

I. M. Anan'evskii, N. V. Anokhin and A. I. Ovseevich, Synthesis of a bounded control for linear dynamical systems using the general Lyapunov function, Dokl. Akad. Nauk, 434, no. 3 (2010), 319-323 [Russian], Transl. as Dokl. Math., 82 (2010), 831-834.

[2]

R. Baier and F. Lempio, Computing Aumann's integral, in Modeling Techniques for Uncertain Systems (Sopron, 1992), (eds. A. B. Kurzhanski and V. M. Veliov), Progr. Systems Control Theory, vol. 18, Birkhäser, Boston (1994), 71-92.

[3]

N. S. Bakhvalov, N. P. Zhidkov and G. M. Kobel'kov, Numerical Methods, Nauka, Moscow, 1987 [Russian].

[4]

B. R. Barmish and J. Sankaran, The propagation of parametric uncertainty via polytopes, IEEE Trans. Automat. Control., AC-24 (1979), 346-349.

[5]

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

[6]

A. N. Daryin and A. B. Kurzhanski, Parallel algorithm for calculating the invariant sets of high-dimensional linear systems under uncertainty, Zh. Vychisl. Mat. Mat. Fiz., 53, no.1 (2013), 47-57 [Russian], Transl. as Comput. Math. Math. Phys., 53, no.1 (2013), 34-43.

[7]

T. Filippova, Differential equations of ellipsoidal state estimates in nonlinear control problems under uncertainty, Discrete Contin. Dyn. Syst. 2011, Dynamical systems, differential equations and applications, 8th AIMS Conference, Suppl. vol. I (2011), 410-419.

[8]

M. I. Gusev, External estimates of the reachability sets of nonlinear controlled systems, Avtomat. i Telemekh., no. 3 (2012), 39-51 [Russian], Transl. as Autom. Remote Control, 73 (2012), 450-461.

[9]

E. K. Kostousova, Control synthesis via parallelotopes: optimization and parallel computations, Optim. Methods Softw., 14 (2001), 267-310.

[10]

E. K. Kostousova, Polyhedral estimates for attainability sets of linear multistage systems with integral constraints on the control, Computational Technologies, 8 (2003), 55-74 [Russian; also available from: http://www.ict.nsc.ru/jct/search/article?l=eng].

[11]

E. K. Kostousova, On polyhedral estimates in problems of the synthesis of control strategies in linear multistep systems, Algorithms and Software for Parallel Computations, Ross. Akad. Nauk Ural. Otdel., Inst. Mat. Mekh., Ekaterinburg, vol.9, (2006), 84-105 [Russian].

[12]

E. K. Kostousova, On polyhedral estimates for trajectory tubes of dynamical discrete-time systems with multiplicative uncertainty, Discrete Contin. Dyn. Syst. 2011, Dynamical systems, differential equations and applications, 8th AIMS Conference, Suppl. vol. II (2011), 864-873.

[13]

E. K. Kostousova, On tight polyhedral estimates for reachable sets of linear differential systems, AIP Conf. Proc., 1493 (2012), 579-586; doi: http://dx.doi.org/10.1063/1.4765545.

[14]

N. N. Krasovskii and A. I. Subbotin, Positional Differential Games, Nauka, Moscow, 1974 [Russian].

[15]

V. M. Kuntsevich and A. B. Kurzhanski, Attainability domains for linear and some classes of nonlinear discrete systems and their control, Problemy Upravlen. Inform., no.1 (2010), 5-21 [Russian], Transl. as J. Automation and Inform. Sci., 42 (2010), 1-18.

[16]

A. B. Kurzhanskii and N. B. Mel'nikov, On the problem of the synthesis of controls: the Pontryagin alternative integral and the Hamilton-Jacobi equation, Mat. Sb. 191, no. 6 (2000), 69-100 [Russian], Transl. as Sb. Math., 191 (2000), 849-881.

[17]

A. B. Kurzhanski and O. I. Nikonov, On the problem of synthesizing control strategies. Evolution equations and set-valued integration, Dokl. Akad. Nauk SSSR, 311, no. 4 (1990), 788-793 [Russian], Transl. as Soviet Math. Doklady, 41 (1990), 300-305.

[18]

A. B. Kurzhanski and I. Vályi, Ellipsoidal Calculus for Estimation and Control, Birkhäuser, Boston, 1997.

[19]

A. B. Kurzhanski and P. Varaiya, Dynamics and Control of Trajectory Tubes. Theory and Computation (Systems & Control: Foundations & Applications, Book 85), Birkhäuser Basel, 2014.

[20]

J. C. Lagarias, J. A. Reeds, M. H. Wright and P. E. Wright, Convergence properties of the Nelder-Mead simplex method in low dimensions, SIAM Journal of Optimization, 9 (1998), 112-147.

[21]

B. T. Polyak and P. S. Scherbakov, Robust Stability and Control, Nauka, Moscow, 2002 [Russian].

[22]

R. G. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge Univ. Press, Cambridge, 1993.

[23]

A. M. Taras'yev, A. A. Uspenskiy and V. N. Ushakov, Approximation schemas and finite-difference operators for constructing generalized solutions of Hamilton-Jacobi equations, Izv. Ross. Akad. Nauk Tekhn. Kibernet., no. 3 (1994) 173-185 [Russian], Transl. as J. Comput. Systems Sci. Internat., 33, no.6 (1995), 127-139.

[24]

V. V. Vasin and I. I. Eremin, Operators and Iterative Processes of Fejér Type. Theory and Applications, Ross. Akad. Nauk Ural. Otdel., Inst. Mat. Mekh., Ekaterinburg, 2005 [Russian].

[25]

A. Yu. Vazhentsev, Internal ellipsoidal approximations for problems of the synthesis of a control with bounded coordinates, Izv. Akad. Nauk Teor. Sist. Upr., no.3 (2000), 70-77 [Russian].

[26]

V. M. Veliov, Second order discrete approximations to strongly convex differential inclusions, Systems Control Lett., 13, no.3 (1989), 263-269.

show all references

References:
[1]

I. M. Anan'evskii, N. V. Anokhin and A. I. Ovseevich, Synthesis of a bounded control for linear dynamical systems using the general Lyapunov function, Dokl. Akad. Nauk, 434, no. 3 (2010), 319-323 [Russian], Transl. as Dokl. Math., 82 (2010), 831-834.

[2]

R. Baier and F. Lempio, Computing Aumann's integral, in Modeling Techniques for Uncertain Systems (Sopron, 1992), (eds. A. B. Kurzhanski and V. M. Veliov), Progr. Systems Control Theory, vol. 18, Birkhäser, Boston (1994), 71-92.

[3]

N. S. Bakhvalov, N. P. Zhidkov and G. M. Kobel'kov, Numerical Methods, Nauka, Moscow, 1987 [Russian].

[4]

B. R. Barmish and J. Sankaran, The propagation of parametric uncertainty via polytopes, IEEE Trans. Automat. Control., AC-24 (1979), 346-349.

[5]

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

[6]

A. N. Daryin and A. B. Kurzhanski, Parallel algorithm for calculating the invariant sets of high-dimensional linear systems under uncertainty, Zh. Vychisl. Mat. Mat. Fiz., 53, no.1 (2013), 47-57 [Russian], Transl. as Comput. Math. Math. Phys., 53, no.1 (2013), 34-43.

[7]

T. Filippova, Differential equations of ellipsoidal state estimates in nonlinear control problems under uncertainty, Discrete Contin. Dyn. Syst. 2011, Dynamical systems, differential equations and applications, 8th AIMS Conference, Suppl. vol. I (2011), 410-419.

[8]

M. I. Gusev, External estimates of the reachability sets of nonlinear controlled systems, Avtomat. i Telemekh., no. 3 (2012), 39-51 [Russian], Transl. as Autom. Remote Control, 73 (2012), 450-461.

[9]

E. K. Kostousova, Control synthesis via parallelotopes: optimization and parallel computations, Optim. Methods Softw., 14 (2001), 267-310.

[10]

E. K. Kostousova, Polyhedral estimates for attainability sets of linear multistage systems with integral constraints on the control, Computational Technologies, 8 (2003), 55-74 [Russian; also available from: http://www.ict.nsc.ru/jct/search/article?l=eng].

[11]

E. K. Kostousova, On polyhedral estimates in problems of the synthesis of control strategies in linear multistep systems, Algorithms and Software for Parallel Computations, Ross. Akad. Nauk Ural. Otdel., Inst. Mat. Mekh., Ekaterinburg, vol.9, (2006), 84-105 [Russian].

[12]

E. K. Kostousova, On polyhedral estimates for trajectory tubes of dynamical discrete-time systems with multiplicative uncertainty, Discrete Contin. Dyn. Syst. 2011, Dynamical systems, differential equations and applications, 8th AIMS Conference, Suppl. vol. II (2011), 864-873.

[13]

E. K. Kostousova, On tight polyhedral estimates for reachable sets of linear differential systems, AIP Conf. Proc., 1493 (2012), 579-586; doi: http://dx.doi.org/10.1063/1.4765545.

[14]

N. N. Krasovskii and A. I. Subbotin, Positional Differential Games, Nauka, Moscow, 1974 [Russian].

[15]

V. M. Kuntsevich and A. B. Kurzhanski, Attainability domains for linear and some classes of nonlinear discrete systems and their control, Problemy Upravlen. Inform., no.1 (2010), 5-21 [Russian], Transl. as J. Automation and Inform. Sci., 42 (2010), 1-18.

[16]

A. B. Kurzhanskii and N. B. Mel'nikov, On the problem of the synthesis of controls: the Pontryagin alternative integral and the Hamilton-Jacobi equation, Mat. Sb. 191, no. 6 (2000), 69-100 [Russian], Transl. as Sb. Math., 191 (2000), 849-881.

[17]

A. B. Kurzhanski and O. I. Nikonov, On the problem of synthesizing control strategies. Evolution equations and set-valued integration, Dokl. Akad. Nauk SSSR, 311, no. 4 (1990), 788-793 [Russian], Transl. as Soviet Math. Doklady, 41 (1990), 300-305.

[18]

A. B. Kurzhanski and I. Vályi, Ellipsoidal Calculus for Estimation and Control, Birkhäuser, Boston, 1997.

[19]

A. B. Kurzhanski and P. Varaiya, Dynamics and Control of Trajectory Tubes. Theory and Computation (Systems & Control: Foundations & Applications, Book 85), Birkhäuser Basel, 2014.

[20]

J. C. Lagarias, J. A. Reeds, M. H. Wright and P. E. Wright, Convergence properties of the Nelder-Mead simplex method in low dimensions, SIAM Journal of Optimization, 9 (1998), 112-147.

[21]

B. T. Polyak and P. S. Scherbakov, Robust Stability and Control, Nauka, Moscow, 2002 [Russian].

[22]

R. G. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge Univ. Press, Cambridge, 1993.

[23]

A. M. Taras'yev, A. A. Uspenskiy and V. N. Ushakov, Approximation schemas and finite-difference operators for constructing generalized solutions of Hamilton-Jacobi equations, Izv. Ross. Akad. Nauk Tekhn. Kibernet., no. 3 (1994) 173-185 [Russian], Transl. as J. Comput. Systems Sci. Internat., 33, no.6 (1995), 127-139.

[24]

V. V. Vasin and I. I. Eremin, Operators and Iterative Processes of Fejér Type. Theory and Applications, Ross. Akad. Nauk Ural. Otdel., Inst. Mat. Mekh., Ekaterinburg, 2005 [Russian].

[25]

A. Yu. Vazhentsev, Internal ellipsoidal approximations for problems of the synthesis of a control with bounded coordinates, Izv. Akad. Nauk Teor. Sist. Upr., no.3 (2000), 70-77 [Russian].

[26]

V. M. Veliov, Second order discrete approximations to strongly convex differential inclusions, Systems Control Lett., 13, no.3 (1989), 263-269.

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