American Institute of Mathematical Sciences

Advanced
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 (2010), 319.   Google Scholar

[2]

R. Baier and F. Lempio, Computing Aumann's integral,, in Modeling Techniques for Uncertain Systems (Sopron, 18 (1994), 71.   Google Scholar

[3]

N. S. Bakhvalov, N. P. Zhidkov and G. M. Kobel'kov, Numerical Methods,, Nauka, (1987).   Google Scholar

[4]

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

[5]

F. L. Chernousko, State Estimation for Dynamic Systems,, CRC Press, (1994).   Google Scholar

[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 (2013), 47.   Google Scholar

[7]

T. Filippova, Differential equations of ellipsoidal state estimates in nonlinear control problems under uncertainty,, Discrete Contin. Dyn. Syst. 2011, (2011), 410.   Google Scholar

[8]

M. I. Gusev, External estimates of the reachability sets of nonlinear controlled systems,, Avtomat. i Telemekh., 73 (2012), 39.   Google Scholar

[9]

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

[10]

E. K. Kostousova, Polyhedral estimates for attainability sets of linear multistage systems with integral constraints on the control,, Computational Technologies, 8 (2003), 55.   Google Scholar

[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, 9 (2006), 84.   Google Scholar

[12]

E. K. Kostousova, On polyhedral estimates for trajectory tubes of dynamical discrete-time systems with multiplicative uncertainty,, Discrete Contin. Dyn. Syst. 2011, II (2011), 864.   Google Scholar

[13]

E. K. Kostousova, On tight polyhedral estimates for reachable sets of linear differential systems,, AIP Conf. Proc., 1493 (2012), 579.   Google Scholar

[14]

N. N. Krasovskii and A. I. Subbotin, Positional Differential Games,, Nauka, (1974).   Google Scholar

[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., 42 (2010), 5.   Google Scholar

[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, 191 (2000), 69.   Google Scholar

[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 (1990), 788.   Google Scholar

[18]

A. B. Kurzhanski and I. Vályi, Ellipsoidal Calculus for Estimation and Control,, Birkhäuser, (1997).   Google Scholar

[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).   Google Scholar

[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.   Google Scholar

[21]

B. T. Polyak and P. S. Scherbakov, Robust Stability and Control,, Nauka, (2002).   Google Scholar

[22]

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

[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., 33 (1994), 173.   Google Scholar

[24]

V. V. Vasin and I. I. Eremin, Operators and Iterative Processes of Fejér Type. Theory and Applications,, Ross. Akad. Nauk Ural. Otdel., (2005).   Google Scholar

[25]

A. Yu. Vazhentsev, Internal ellipsoidal approximations for problems of the synthesis of a control with bounded coordinates,, Izv. Akad. Nauk Teor. Sist. Upr., (2000), 70.   Google Scholar

[26]

V. M. Veliov, Second order discrete approximations to strongly convex differential inclusions,, Systems Control Lett., 13 (1989), 263.   Google Scholar

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 (2010), 319.   Google Scholar

[2]

R. Baier and F. Lempio, Computing Aumann's integral,, in Modeling Techniques for Uncertain Systems (Sopron, 18 (1994), 71.   Google Scholar

[3]

N. S. Bakhvalov, N. P. Zhidkov and G. M. Kobel'kov, Numerical Methods,, Nauka, (1987).   Google Scholar

[4]

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

[5]

F. L. Chernousko, State Estimation for Dynamic Systems,, CRC Press, (1994).   Google Scholar

[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 (2013), 47.   Google Scholar

[7]

T. Filippova, Differential equations of ellipsoidal state estimates in nonlinear control problems under uncertainty,, Discrete Contin. Dyn. Syst. 2011, (2011), 410.   Google Scholar

[8]

M. I. Gusev, External estimates of the reachability sets of nonlinear controlled systems,, Avtomat. i Telemekh., 73 (2012), 39.   Google Scholar

[9]

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

[10]

E. K. Kostousova, Polyhedral estimates for attainability sets of linear multistage systems with integral constraints on the control,, Computational Technologies, 8 (2003), 55.   Google Scholar

[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, 9 (2006), 84.   Google Scholar

[12]

E. K. Kostousova, On polyhedral estimates for trajectory tubes of dynamical discrete-time systems with multiplicative uncertainty,, Discrete Contin. Dyn. Syst. 2011, II (2011), 864.   Google Scholar

[13]

E. K. Kostousova, On tight polyhedral estimates for reachable sets of linear differential systems,, AIP Conf. Proc., 1493 (2012), 579.   Google Scholar

[14]

N. N. Krasovskii and A. I. Subbotin, Positional Differential Games,, Nauka, (1974).   Google Scholar

[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., 42 (2010), 5.   Google Scholar

[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, 191 (2000), 69.   Google Scholar

[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 (1990), 788.   Google Scholar

[18]

A. B. Kurzhanski and I. Vályi, Ellipsoidal Calculus for Estimation and Control,, Birkhäuser, (1997).   Google Scholar

[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).   Google Scholar

[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.   Google Scholar

[21]

B. T. Polyak and P. S. Scherbakov, Robust Stability and Control,, Nauka, (2002).   Google Scholar

[22]

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

[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., 33 (1994), 173.   Google Scholar

[24]

V. V. Vasin and I. I. Eremin, Operators and Iterative Processes of Fejér Type. Theory and Applications,, Ross. Akad. Nauk Ural. Otdel., (2005).   Google Scholar

[25]

A. Yu. Vazhentsev, Internal ellipsoidal approximations for problems of the synthesis of a control with bounded coordinates,, Izv. Akad. Nauk Teor. Sist. Upr., (2000), 70.   Google Scholar

[26]

V. M. Veliov, Second order discrete approximations to strongly convex differential inclusions,, Systems Control Lett., 13 (1989), 263.   Google Scholar

[1]

Peter Giesl, Zachary Langhorne, Carlos Argáez, Sigurdur Hafstein. Computing complete Lyapunov functions for discrete-time dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 299-336. doi: 10.3934/dcdsb.2020331
[2]

Hui Lv, Xing'an Wang. Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 127-142. doi: 10.3934/naco.2020020
[3]

Xuefeng Zhang, Yingbo Zhang. Fault-tolerant control against actuator failures for uncertain singular fractional order systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 1-12. doi: 10.3934/naco.2020011
[4]

Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 755-774. doi: 10.3934/dcdsb.2020133
[5]

Angelica Pachon, Federico Polito, Costantino Ricciuti. On discrete-time semi-Markov processes. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1499-1529. doi: 10.3934/dcdsb.2020170
[6]

Stefan Siegmund, Petr Stehlík. Time scale-induced asynchronous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1011-1029. doi: 10.3934/dcdsb.2020151
[7]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444
[8]

Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339
[9]

Veena Goswami, Gopinath Panda. Optimal customer behavior in observable and unobservable discrete-time queues. Journal of Industrial & Management Optimization, 2021, 17 (1) : 299-316. doi: 10.3934/jimo.2019112
[10]

Ming Chen, Hao Wang. Dynamics of a discrete-time stoichiometric optimal foraging model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 107-120. doi: 10.3934/dcdsb.2020264
[11]

Xu Zhang, Chuang Zheng, Enrique Zuazua. Time discrete wave equations: Boundary observability and control. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 571-604. doi: 10.3934/dcds.2009.23.571
[12]

Haixiang Yao, Ping Chen, Miao Zhang, Xun Li. Dynamic discrete-time portfolio selection for defined contribution pension funds with inflation risk. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020166
[13]

Javier Fernández, Cora Tori, Marcela Zuccalli. Lagrangian reduction of nonholonomic discrete mechanical systems by stages. Journal of Geometric Mechanics, 2020, 12 (4) : 607-639. doi: 10.3934/jgm.2020029
[14]

Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046
[15]

Gabrielle Nornberg, Delia Schiera, Boyan Sirakov. A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3857-3881. doi: 10.3934/dcds.2020128
[16]

Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020375
[17]

Adrian Viorel, Cristian D. Alecsa, Titus O. Pinţa. Asymptotic analysis of a structure-preserving integrator for damped Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020407
[18]

Zsolt Saffer, Miklós Telek, Gábor Horváth. Analysis of Markov-modulated fluid polling systems with gated discipline. Journal of Industrial & Management Optimization, 2021, 17 (2) : 575-599. doi: 10.3934/jimo.2019124
[19]

Vandana Sharma. Global existence and uniform estimates of solutions to reaction diffusion systems with mass transport type boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021001
[20]

Duy Phan, Lassi Paunonen. Finite-dimensional controllers for robust regulation of boundary control systems. Mathematical Control & Related Fields, 2021, 11 (1) : 95-117. doi: 10.3934/mcrf.2020029

 Impact Factor: 

Tools

Metrics

  • PDF downloads (55)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences