×

Geometric control theory and linear switched systems. (English) Zbl 1298.93121

Summary: This paper consider some special topics related to controllability and stabilizability of linear switching systems. While providing a short overview on the most important facts related to the topic it is shown how fundamental role is played by the finite switching property in obtaining the controllability and stabilizability results. This paper tries to present some aspects that remain hidden in the vast amount of previous works but that may contribute to a more deep understanding of the results.

MSC:

93B27 Geometric methods
93B30 System identification
93C05 Linear systems in control theory
93B05 Controllability
93D21 Adaptive or robust stabilization
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Liberzon, D.; Agrachev, A., Lie-algebraic stability criteria for switched systems, SIAM J Control Optim, 40, 1, 253-269 (2001) · Zbl 0995.93064
[2] Ge, S.; Sun, Z.; Lee, T., Reachability and controllability of switched linear systems, (Proceedings of the 2001 American Control Conference. Proceedings of the 2001 American Control Conference, Arlington, VA, USA (2001)), 1898-1903, vol. 3 · Zbl 0689.65018
[3] Xie, G.; Zheng, D.; Wang, L., Controllability of switched linear systems, IEEE Trans Autom Control, 47, 8, 1401-1405 (2002) · Zbl 1364.93075
[4] Cheng, D.; Chen, H. F., Accessibility of switched linear systems, (Proceedings of the 42nd IEEE Conference on Decision and Control. Proceedings of the 42nd IEEE Conference on Decision and Control, Maui, Hawaii, USA (2003)), 5759-5764, vol. 6
[5] Lin, H.; Antsaklis, P. J., Stability and stabilizability of switched linear systems: A short survey of recent results, (Proceedings of the 13th IEEE Mediterranean Conference on Control and Automation. Proceedings of the 13th IEEE Mediterranean Conference on Control and Automation, Limassol, Cyprus (2005)), 24-29
[6] Sun, Z.; Ge, S. S.; Lee, T. H., Controllability and reachability criteria for switched linear systems, Automatica, 38, 5, 775-786 (2003) · Zbl 1031.93041
[7] Liberzon, D., Switching inSystemsandControl (2003), Birkhauser: Birkhauser Boston, MA
[8] Sun, Z.; Ge, S. S., Switched Linear Systems. Control and Design (2005), Springer · Zbl 1075.93001
[9] Altafini, C., The reachable set of a linear endogenous switching system, Syst Control Lett, 47, 4, 343-353 (2002) · Zbl 1106.93304
[10] Xie, G.; Wang, L., Necessary and sufficient conditions for controllability of switched linear systems, (Proceedings of the 2002 American Control Conference. Proceedings of the 2002 American Control Conference, Anchorage, Alaska, USA (2002)), 1897-1902, vol. 3 · Zbl 0531.93011
[11] Yang, Z., An algebraic approach towards the controllability of controlled switching linear hybrid systems, Automatica, 38, 1221-1228 (2002) · Zbl 1031.93042
[12] Sontag, E.; Qiao, Y., Further results on controllability of recurrent neural networks, Syst Control Lett, 36, 121-129 (1999) · Zbl 0914.93011
[13] Liberzon, D.; Hespanha, J. P.; Morse, A. S., Stability of switched systems: a Lie-algebraic condition, Syst Control Lett, 37, 117-122 (1999) · Zbl 0948.93048
[14] Bokor, J.; Szabo, Z.; Szigeti, F., Controllability of LTI switching systems using nonnegative inputs, (Proceedings of the European Control Conference 2007. Proceedings of the European Control Conference 2007, Kos, Greece (2007)), 1948-1953
[15] Korobov, V. I., A geometric criterion of local controllability of dynamical systems in the presence of constraints on the control, Diff Equ, 15, 1136-1142 (1980) · Zbl 0449.93007
[16] Frankowska, H.; Olech, C.; Aubin, J. P., Controllability of convex processes, SIAM J Control Optim, 24, 6, 1192-1211 (1986) · Zbl 0607.49025
[17] Jurdjevic, V., Geometric Control Theory (1997), Cambridge University Press · Zbl 0940.93005
[18] Grasse, K. A.; Sussmann, H. J., Nonlinear Controllability and Optimal Control chapter: Global controllability by nice controls (1990), Dekker: Dekker New York, 33-79
[19] Agrachev, A. A.; Sachkov, Y. L., Control Theory from the Geometric Viewpoint, Encyclopaedia of Mathematical Sciences, 87 (2004), Springer · Zbl 1062.93001
[20] Aubin, J.; Cellina, A., Differential Inclusions (1984), Springer- Verlag: Springer- Verlag Berlin
[21] Wolenski, P. R., The exponential formula for the reachable set of a Lipschitz differential inclusion, SIAM J Control Optim, 28, 5, 1148-1161 (1990) · Zbl 0736.34015
[22] Dontchev, A. L.; Lempio, F., Difference methods for differential inclusions: a survey, SIAM Rev, 34, 2, 263-294 (1992) · Zbl 0757.34018
[23] Smirnov, G. V., Introduction to the Theory of Differential Inclusions, Graduate Studies in Mathematics, 41 (2002), American Mathematical Society Providence, RI · Zbl 0992.34001
[24] Wonham, W. M., Linear Multivariable Control: A Geometric Approach (1985), Springer-Verlag: Springer-Verlag NewYork · Zbl 0609.93001
[25] Sussmann, H., Some properties of vector field systems that are not altered by small perturbations, J Diff Equ, 20, 292-315 (1976) · Zbl 0346.49036
[26] Sontag, E., Remarks on the preservation of various controllability properties under sampling, (Mathematical Tools and Models for Control, Systems Analysis and Signal Processing Travaux Rech. Coop. Programme 567. Mathematical Tools and Models for Control, Systems Analysis and Signal Processing Travaux Rech. Coop. Programme 567, Paris: CNRS (1983)), 623-637, Vol. 3 (Toulouse/Paris 1981/1982)
[27] Sontag, E.; Sussmann, H., Accessibility under sampling., (Proceedings of the 21st IEEE Conference on Decision Control. Proceedings of the 21st IEEE Conference on Decision Control, Orlando, Florida (1982))
[28] Sontag, E., Orbit theorems and sampling, (Algebraic and Geometric Methods in Nonlinear Control Theory. Algebraic and Geometric Methods in Nonlinear Control Theory, Math. Appl. Dordrecht: Reidel (1986)), 441-483, vol. 29
[29] Sun, Z.; Ge, S., Sampling and control of switched linear systems, (Proceedings of the 41st IEEE Conference on Decision and Control. Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas, Nevada, USA (2002)), 4413-4418, vol. 4 · Zbl 1017.83016
[30] Sontag, E., An approximation theorem in nonlinear sampling, (Mathematical Theory of Networks and Systems. Lecture Notes in Control and Information Science, 58 (1984), Springer: Springer London), 806-812
[31] Petreczky, M., Reachability of linear switched systems: differential geometric approach, Syst Control Lett, 565, 112-118 (2006) · Zbl 1129.93446
[32] Stikkel, G.; Bokor, J.; Szabó, Z., Necessary and sufficient condition for the controllability of switching linear hybrid systems, Automatica, 40, 6, 1093-1098 (2004) · Zbl 1109.93011
[33] Balas, G.; Bokor, J.; Szabo, Z., Invariant subspaces for LPV systems and their applications, IEEE Trans Autom Control, 48, 11, 2065-2068 (2003) · Zbl 1364.93105
[34] Xie, G.; Zheng, D.; Wang, L., Controllability of switched linear systems, IEEE Trans Autom Control, 47, 8, 1401-1405 (2002) · Zbl 1364.93075
[35] Xie, G.; Wang, L., Equivalence of some controllability notions for linear switched systems and their geometric criteria, (Proceedings of the 2003 American Control Conference. Proceedings of the 2003 American Control Conference, Denver, Colorado, USA (2003)), 5158-5190
[36] Szigeti, F., A differential algebraic condition for controllability and observability of time varying linear systems, (Proceedings of the 31st IEEE Conference on Decision and Control. Proceedings of the 31st IEEE Conference on Decision and Control, Tucson, USA (1992)), 3088-3090
[37] Cheng, D., Controllability of switched bilinear systems, IEEE Transactions on Automatic Control, 50, 4, 511-515 (2005) · Zbl 1365.93039
[38] Veliov, V.; Krastanov, M., Controllability of piecewise linear systems, Syst Control Lett, 7, 335-341 (1986) · Zbl 0609.93006
[39] Krastanov, M.; Veliov, V., On the controllability of switching linear systems, Automatica, 41, 4, 663-668 (2005) · Zbl 1175.93033
[40] Xie, G.; Wang, L., Reachability realization and stabilizability of switched linear discrete-time systems, J Math Anal Appl, 280, 2, 209-220 (2003) · Zbl 1080.93015
[41] Jia, Z.; Fengb, G.; Guoc, X., A constructive approach to reachability realization of discrete-time switched linear systems, Syst Control Lett, 56, 11-12, 669-677 (2007)
[42] Ji, Z.; Wang, L.; Guo, X., Design of switching sequences for controllability realization of switched linear systems, Automatica, 43, 4, 662-668 (2007) · Zbl 1114.93019
[43] Aubin, J. P.; Cellina, A., Differential Inclusions (1984), Springer- Verlag: Springer- Verlag Berlin
[44] Kalman, R. E., Contributions to the theory of optimal control, Boletin de la Sociedad Matematica Mexicana, 5, 102-119 (1960)
[45] Brammer, R. F., Controllability in linear autonomous systems with positive controllers, SIAM J Control Optim, 10, 2, 329-353 (1972) · Zbl 0242.93007
[46] Cabot, A.; Seeger, A., Multivalued exponentiation analysis. part I: Maclaurin exponentials, Set-Valued Anal, 14, 4, 347-379 (2006) · Zbl 1113.26026
[47] Molchanov, A. P.; Pyatnitskiy, Y. S., Criteria of asymptotic stability of differential and difference inclusions encountered in control theory, Syst Control Lett, 13, 1, 59-64 (1989) · Zbl 0684.93065
[48] Liu, D.; Molchanov, A., Criteria for robust absolute stability of time-varying nonlinear continuous-time systems, Automatica, 38, 11, 627-637 (2002) · Zbl 1013.93044
[49] Liberzon, D.; Agrachev, A., Lie-algebraic stability criteria for switched systems, SIAM J Control Optim, 40, 1, 253-269 (2001) · Zbl 0995.93064
[50] Mancilla-Aguilar, J.; Garcìa, R.; Sontag, E.; Wang, Y., Representation of switched systems by perturbed control systems, (Proceedings of the 43rd IEEE Conference on Decision and Control. Proceedings of the 43rd IEEE Conference on Decision and Control, Paradise Island, Bahamas (2004)), 3259-3264 · Zbl 1288.94034
[51] Geromel, J.; Colaneri, P., Stabilization of continuoustime switched systems, (Proceedings of the 16th IFAC World Congress. Proceedings of the 16th IFAC World Congress, Prague, Czech Republic (2005)) · Zbl 1330.93190
[52] Wirth, F., A converse Lyapunov theorem for linear parameter-varying and linear switching systems, SIAM J Control Optim, 44, 1, 210-239 (2005) · Zbl 1098.34040
[53] Wirth, F., A converse Lyapunov theorem for switched linear systems with dwell times, (Proceedings of the 44th IEEE Conference on Decision and Control and European Control Conference. Proceedings of the 44th IEEE Conference on Decision and Control and European Control Conference, Seville, Spain (2005)), 4572-4577
[54] Clarke, F. H.; Ledyaev, Y.; Sontag, E.; Subbotin, A., Asymptotic controllability implies feedback stabilization, IEEE Trans Autom Control, 42, 10, 1394-1407 (1997) · Zbl 0892.93053
[55] Cheng, D.; Lin, Y.; Wang, Y., Accessibility of switched linear systems, IEEE Trans on Autom Control, 51, 9, 1486-1491 (2006) · Zbl 1366.93048
[56] Ancona, F.; Bressan, A., Patchy vector fields and asymptotic stabilization, ESAIM-Control, Optimization and Calculus of Variations, 4, 445-471 (1999) · Zbl 0924.34058
[57] Rifford, L., Semiconcave control-Lyapunov functions and stabilizing feedbacks, SIAM J Control Optim, 41, 3, 659-681 (2002) · Zbl 1034.93053
[58] Szabo, Z.; Bokor, J.; Balas, G., Generalized piecewise linear feedback stabilizability of controlled linear switched systems, (Proceedings of the 47th IEEE Conference on Decision and Control. Proceedings of the 47th IEEE Conference on Decision and Control, Cancun, Mexico (2008)), 3410-3414
[59] Xie, G.; Wang, L., Controllability implies stabilizability for discrete-time switched linear systems, (Hybrid Systems: Computation and Control. Book Series Lecture Notes in Computer Science, 3414 (2005), Springer: Springer Berlin/Heidelberg), 667-682 · Zbl 1078.93037
[60] Xie, G.; Wang, L., Periodical stabilization of switched linear systems, J Comput Appl Math, 181, 1, 176-187 (2005) · Zbl 1107.93032
[61] Kellett, C.; Teel, A., Weak converse Lyapunov theorems and control-Lyapunov functions, SIAM J Control Optim, 42, 6, 1934-1959 (2004) · Zbl 1151.34307
[62] Lin, H.; Antsaklis, P., A necessary and sufficient condition for robust asymptotic stabilizability of continuous-time uncertain switched linear systems, (Proceedings of the 43rd IEEE Conference on Decision and Control. Proceedings of the 43rd IEEE Conference on Decision and Control, Paradise Island, Bahamas (2004)), 3690-3695, vol. 4
[63] Lin, H.; Antsaklis, P., A converse Lyapunov theorem for uncertain switched linear systems, (Proceedings of the 44th IEEE Conference on Decision and Control and European Control Conference. Proceedings of the 44th IEEE Conference on Decision and Control and European Control Conference, Seville, Spain (2005)), 3291-3296
[64] Blanchini, F.; Savorgnan, C., Stabilizability of switched linear systems does not imply the existence of convex Lyapunov functions, (Proceedings of the 45th IEEE Conference on Decision and Control. Proceedings of the 45th IEEE Conference on Decision and Control, San Diego, California, USA (2006)), 119-124
[65] Branicky, M., Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Trans Autom Control, 43, 4, 475-482 (1998) · Zbl 0904.93036
[66] Johansson, M.; Rantzer, A., Computation of piecewise quadratic Lyapunov functions for hybrid systems, IEEE Trans Autom Control, 43, 4, 555-559 (1998) · Zbl 0905.93039
[67] Hu, T.; Ma, L.; Lin, Z., Stabilization of switched systems via composite quadratic functions, IEEE Trans Autom Control, 53, 11, 2571-2585 (2008) · Zbl 1367.93509
[68] Corona, D.; Giua, A.; Seatzu, C., Stabilization of switched systems via optimal control, (Proceedings of the 16th IFAC World Congress. Proceedings of the 16th IFAC World Congress, Prague, Czech Republic (2005)) · Zbl 1291.93278
[69] Seatzu, C.; Corona, D.; Giua, A.; Bemporad, A., Optimal control of continuous-time switched affine systems, IEEE Trans Autom Control, 51, 5, 726-741 (2006) · Zbl 1366.49038
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.