# American Institute of Mathematical Sciences

2009, 2009(Special): 466-475. doi: 10.3934/proc.2009.2009.466

## State estimation for linear impulsive differential systems through polyhedral techniques

 1 Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16, S.Kovalevskaja Street, Ekaterinburg GSP-384, 620219, Russian Federation

Received  July 2008 Revised  April 2009 Published  September 2009

The paper is devoted to the state estimation problem in control theory under uncertainty. The approach for estimating the reachable sets of the linear impulsive differential systems is presented. The reachable sets are approximated by the ones for special discrete time systems. The degree of convergence is established. The families of external and internal polyhedral (parallelepiped-valued and parallelotope-valued) estimates of the reachable sets of the auxiliary systems are introduced. Evolution of estimates is determined by systems of recurrence relations. The families which ensure the exact representations of the reachable sets of the auxiliary systems as well as the families of the touching and tight estimates are found. This technique gives the possibility to construct the guaranteed estimates (including $\epsilon$-touching and $\epsilon$-tight ones) for the reachable sets of the primary systems. The results of numerical simulations are presented.
Citation: Elena K. Kostousova. State estimation for linear impulsive differential systems through polyhedral techniques. Conference Publications, 2009, 2009 (Special) : 466-475. doi: 10.3934/proc.2009.2009.466
 [1] Elena K. Kostousova. External polyhedral estimates of reachable sets of discrete-time systems with integral bounds on additive terms. Mathematical Control & Related Fields, 2021, 11 (3) : 625-641. doi: 10.3934/mcrf.2021015 [2] Robert Baier, Thuy T. T. Le. Construction of the minimum time function for linear systems via higher-order set-valued methods. Mathematical Control & Related Fields, 2019, 9 (2) : 223-255. doi: 10.3934/mcrf.2019012 [3] Roberta Fabbri, Sylvia Novo, Carmen Núñez, Rafael Obaya. Null controllable sets and reachable sets for nonautonomous linear control systems. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1069-1094. doi: 10.3934/dcdss.2016042 [4] Michele Campiti. Korovkin-type approximation of set-valued and vector-valued functions. Mathematical Foundations of Computing, 2021  doi: 10.3934/mfc.2021032 [5] Elena K. Kostousova. On polyhedral estimates for trajectory tubes of dynamical discrete-time systems with multiplicative uncertainty. Conference Publications, 2011, 2011 (Special) : 864-873. doi: 10.3934/proc.2011.2011.864 [6] Elena K. Kostousova. On polyhedral control synthesis for dynamical discrete-time systems under uncertainties and state constraints. Discrete & Continuous Dynamical Systems, 2018, 38 (12) : 6149-6162. doi: 10.3934/dcds.2018153 [7] Mohamed Assellaou, Olivier Bokanowski, Hasnaa Zidani. Error estimates for second order Hamilton-Jacobi-Bellman equations. Approximation of probabilistic reachable sets. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 3933-3964. doi: 10.3934/dcds.2015.35.3933 [8] Zhiang Zhou, Xinmin Yang, Kequan Zhao. $E$-super efficiency of set-valued optimization problems involving improvement sets. Journal of Industrial & Management Optimization, 2016, 12 (3) : 1031-1039. doi: 10.3934/jimo.2016.12.1031 [9] Shay Kels, Nira Dyn. Bernstein-type approximation of set-valued functions in the symmetric difference metric. Discrete & Continuous Dynamical Systems, 2014, 34 (3) : 1041-1060. doi: 10.3934/dcds.2014.34.1041 [10] Robert Baier, Matthias Gerdts, Ilaria Xausa. Approximation of reachable sets using optimal control algorithms. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 519-548. doi: 10.3934/naco.2013.3.519 [11] Roger Metzger, Carlos Arnoldo Morales Rojas, Phillipe Thieullen. Topological stability in set-valued dynamics. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1965-1975. doi: 10.3934/dcdsb.2017115 [12] Geng-Hua Li, Sheng-Jie Li. Unified optimality conditions for set-valued optimizations. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1101-1116. doi: 10.3934/jimo.2018087 [13] Dante Carrasco-Olivera, Roger Metzger Alvan, Carlos Arnoldo Morales Rojas. Topological entropy for set-valued maps. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3461-3474. doi: 10.3934/dcdsb.2015.20.3461 [14] Dietmar Szolnoki. Set oriented methods for computing reachable sets and control sets. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 361-382. doi: 10.3934/dcdsb.2003.3.361 [15] Baskar Sundaravadivoo. Controllability analysis of nonlinear fractional order differential systems with state delay and non-instantaneous impulsive effects. Discrete & Continuous Dynamical Systems - S, 2020, 13 (9) : 2561-2573. doi: 10.3934/dcdss.2020138 [16] Yu Zhang, Tao Chen. Minimax problems for set-valued mappings with set optimization. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 327-340. doi: 10.3934/naco.2014.4.327 [17] Bernd Aulbach, Martin Rasmussen, Stefan Siegmund. Approximation of attractors of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 215-238. doi: 10.3934/dcdsb.2005.5.215 [18] Qingbang Zhang, Caozong Cheng, Xuanxuan Li. Generalized minimax theorems for two set-valued mappings. Journal of Industrial & Management Optimization, 2013, 9 (1) : 1-12. doi: 10.3934/jimo.2013.9.1 [19] Sina Greenwood, Rolf Suabedissen. 2-manifolds and inverse limits of set-valued functions on intervals. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5693-5706. doi: 10.3934/dcds.2017246 [20] Zhenhua Peng, Zhongping Wan, Weizhi Xiong. Sensitivity analysis in set-valued optimization under strictly minimal efficiency. Evolution Equations & Control Theory, 2017, 6 (3) : 427-436. doi: 10.3934/eect.2017022

Impact Factor: