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On control synthesis for uncertain dynamical discretetime systems through polyhedral techniques
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 
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 highdimensional 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 discretetime 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 HamiltonJacobi 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 setvalued 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 NelderMead 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 BrunnMinkowski Theory,, Cambridge Univ. Press, (1993). Google Scholar 
[23] 
A. M. Taras'yev, A. A. Uspenskiy and V. N. Ushakov, Approximation schemas and finitedifference operators for constructing generalized solutions of HamiltonJacobi 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 highdimensional 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 discretetime 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 HamiltonJacobi 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 setvalued 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 NelderMead 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 BrunnMinkowski Theory,, Cambridge Univ. Press, (1993). Google Scholar 
[23] 
A. M. Taras'yev, A. A. Uspenskiy and V. N. Ushakov, Approximation schemas and finitedifference operators for constructing generalized solutions of HamiltonJacobi 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 
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