×

Linear quadratic optimal sampled data control of linear systems with unknown switched modes and stochastic disturbances. (English) Zbl 1348.93186

Summary: It is nontrivial to control a dynamic system that is switched consistently with a completely unknown switched modes. This problem is further complicated if the system is subject to stochastic disturbance. This paper studies the linear quadratic optimal control problem of linear continuous systems with stochastic disturbance and unknown switched process. By integrating one-step adaptive estimator with optimal control theory, a linear quadratic optimal stabilization controller based on sampled feedback is developed for systems that are continuous in nature yet switched consistently with unknown modes. It is shown that with the proposed control scheme, both parameter estimate error and system stabilization error are ensured to be bounded, and the existence of the upper bound is explicitly confirmed. The results compliment and extend the existing works on digital feedback control of switched linear continuous systems with unknown switched processes and stochastic disturbances.

MSC:

93C57 Sampled-data control/observation systems
49N10 Linear-quadratic optimal control problems
93E20 Optimal stochastic control
93C73 Perturbations in control/observation systems
93E12 Identification in stochastic control theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Branicky, Multiple Lyapunov functions and other analysis tools for switched and hybrid system, IEEE Transactions on Automatic Control 43 (4) pp 475– (1998) · Zbl 0904.93036 · doi:10.1109/9.664150
[2] Caines, On the adaptive control of jump parameter systems via nonlinear filtering, SIAM Journal Control and Optimization 33 (6) pp 1758– (1995) · Zbl 0843.93076 · doi:10.1137/S0363012992238679
[3] Lin, Stability and stabilizability of switched linear systems: a survey of recent results, IEEE Transactions on Automatic Control 54 (2) pp 7308– (2009) · Zbl 1367.93440 · doi:10.1109/TAC.2008.2012009
[4] Liberzon, Switching in Systems and Control (2003) · doi:10.1007/978-1-4612-0017-8
[5] Liberzon, Stability of switched linear systems: a Lie-algebraic condition, System & Control Letters 37 (3) pp 117– (1999) · Zbl 0948.93048 · doi:10.1016/S0167-6911(99)00012-2
[6] Agrachev, Lie-algebraic stability criteria for switched systems, SIAM Journal on Control and Optimization 40 pp 253– (2001) · Zbl 0995.93064 · doi:10.1137/S0363012999365704
[7] Margaliot, Lie-algebraic stability conditions for nonlinear switched systems and differential inclusins, System & Control Letters 55 (1) pp 8– (2006) · Zbl 1129.93521 · doi:10.1016/j.sysconle.2005.04.011
[8] Cai, Smooth Lyapunov functions for hybrid systems part i: existence is equivalent to robustness, IEEE Transactions on Automatic Control 52 (7) pp 1264– (2007) · Zbl 1366.93430 · doi:10.1109/TAC.2007.900829
[9] Sun, Controllability and and reachability criteria for switched linear systems, Automatica 38 (5) pp 775– (2002) · Zbl 1031.93041 · doi:10.1016/S0005-1098(01)00267-9
[10] Ji, Design of switching sequences for controllability realization of switching linear systems, Automatica 43 (4) pp 662– (2007) · Zbl 1114.93019 · doi:10.1016/j.automatica.2006.10.010
[11] Ji, A constructive approach to reachability realization of discrete-time switched linear systems, System & Control Letters 56 (11) pp 669– (2007) · Zbl 1120.93005 · doi:10.1016/j.sysconle.2007.06.010
[12] Bemporad, Observability and controllability of piecewise affine and hybrid systems, IEEE Transactions on Automatic Control 45 (10) pp 1864– (2000) · Zbl 0990.93010 · doi:10.1109/TAC.2000.880987
[13] Hespanha, Nonlinear norm-observability notions and stability of switched systems, IEEE Transactions on Automatic Control 52 (2) pp 154– (2005) · Zbl 1365.93349 · doi:10.1109/TAC.2004.841937
[14] Bemporad, Control of systems integrating logic, dynamics, and constrains, Automatica 35 (3) pp 407– (1999) · Zbl 1049.93514 · doi:10.1016/S0005-1098(98)00178-2
[15] Cassandras, Optimal control of a class of hybrid systems, IEEE Transactions on Automatic Control 46 (3) pp 398– (2001) · Zbl 0992.93052 · doi:10.1109/9.911417
[16] Xu, Optimal control of switched systems based on parameterization of the switching instants 49 (1) pp 2– (2004) · Zbl 1365.93308
[17] Polyakov, Optimal stochastic SD control with preview, IEEE Transactions on Automatic Control 52 (8) pp 1532– (2007) · Zbl 1366.93705 · doi:10.1109/TAC.2007.900845
[18] Liu, Stability condition for sampled data based control of linear continious switched systems, System & Control Letters 60 (10) pp 794– (2011)
[19] Xiang, Stability analysis and control synthesis of switched impulsive systems, International Journal of Robust and Nonlinear Control 22 (13) pp 1440– (2012) · Zbl 1287.93079 · doi:10.1002/rnc.1757
[20] Xiang, H control synthesis of switched discrete-time fuzzy systems via hybrid approach, Optimal Control Applications and Methods 34 (6) pp 635– (2013) · Zbl 1282.93118 · doi:10.1002/oca.2043
[21] Xiang, Asymptotic stability, 2 gain, boundness analysis, and control synthesis for switched systems: a switching frequency approach, International Journal of Adaptive Control and Signal Processing 26 (4) pp 350– (2012) · Zbl 1274.93235 · doi:10.1002/acs.1287
[22] Xiang, H finite-time control for switched nonlinear discrete-time systems with norm-bounded disturbance, Journal of the Franklin Institute 348 (2) pp 331– (2011) · Zbl 1214.93043 · doi:10.1016/j.jfranklin.2010.12.001
[23] Ji, Controllability, stabilizability and continuous-time Markovian jump linear quadratic control, IEEE Transactions on Automatic Control 35 (7) pp 777– (1990) · Zbl 0714.93060 · doi:10.1109/9.57016
[24] Chen, Identification and Stochastic Adaptive Control (1991) · doi:10.1007/978-1-4612-0429-9
[25] Xiang, Robust observer design for nonlinear uncertain switched systems under asynchronous switching, Nonlinear Analysis: Hybrid Systems 6 (1) pp 754– (2012)
[26] Xiang, H-infinity filtering for switched nonlinear systems under asynchronous switching, International Journal of System Science 42 (5) pp 751– (2011) · Zbl 1233.93094 · doi:10.1080/00207721.2010.488763
[27] Xiang, Fault detection for switched nonlinear systems under asynchronous switching, International Journal of Control 84 (8) pp 1362– (2011) · Zbl 1235.93245 · doi:10.1080/00207179.2011.598191
[28] Yao, Quadratic index control of stochastic linear continuous-time system based on sampled-data, Science in China: Series E 33 (4) pp 326– (2003)
[29] Zhang, Linear quadratic optimal sampled-data control of linear continuous switched systems with stochastic disturbance pp 2209– (2012)
[30] Vu L Morgansen K Stability of feedback switched systems with state and switching delays Proceedings of the American Control Conference St. Louis, MO 2009 1754 1759
[31] Sun, Stability and L2-gain analysis for switched delay systems: a delay-dependent method, Automatica 42 pp 1769– (2006) · Zbl 1114.93086 · doi:10.1016/j.automatica.2006.05.007
[32] Xie, Stabilization of switched linear systems with time-delay in detection of switching signal, Journal of Mathematical Analysis and Applications 305 pp 277– (2005) · Zbl 1140.93463 · doi:10.1016/j.jmaa.2004.11.043
[33] Duncan, Adaptive continous-time linear quadratic Gaussian control, IEEE Transactions on Automatic Control 44 (9) pp 1653– (1999) · Zbl 0963.93080 · doi:10.1109/9.788532
[34] Liu F Song YD Ma XP Sampled-data adaptive control of linear continuous switched systems with stochastic disturbance 31st Chinese Control Conference (CCC) Hefei, China 2012 1479 1483
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.