On existence of stabilizing switching laws within a class of unstable linear systems.

*(English)*Zbl 1421.93108Summary: The equivalence of two conditions, condition (3) and condition (4) stated in problem statement section, regarding the existence of stabilizing switching laws between two unstable linear systems first appeared in [E. Feron, Quadratic stabilization of switched system via state and output feedback. Tech. Rep., CICS-F-468, MIT, Cambridge, MA, USA (1996)].
Although Feron never published this result, it has been referenced in almost every survey on switched systems; see, for example, [D. Liberzon and A. S. Morse, IEEE Control Sys. 19, No. 5, 59–70 (1999; Zbl 1384.93064)].
This paper proposes another way to prove the equivalence of two conditions regarding the existence of stabilizing switching laws between two unstable linear systems. One is effective for theoretical derivation, while the other is implementable, and a class of stabilizing switching laws have been explicitly constructed by M. A. Wicks, P. Peleties and R. A. DeCarlo [“Construction of piecewise Lyapunov functions for stabilizing switched systems”, in: Proceedings of the 33rd IEEE conference on decision and control. Los Alamitos, CA: IEEE Computer Society. 3492–3497 (1994; doi:10.1109/CDC.1994.411687)]. With the help of the equivalent relation, a condition for the existence of controllers and stabilizing switching laws between two unstabilizable linear control systems is then proposed. Then, the study is further extended to the issue concerning the construction of quadratically stabilizing switching laws among \(N\) unstable linear systems and \(N\) unstabilizable linear control systems. The obtained results are employed to study the existence of control laws and quadratically stabilizing switching laws within a class of unstabilizable linear control systems. The numerical examples are illustrated and simulated to show the feasibility and effectiveness of the proposed methods.

##### MSC:

93D05 | Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory |

93C30 | Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems) |

93C05 | Linear systems in control theory |

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\textit{S. S. D. Xu} and \textit{C.-C. Chen}, Abstr. Appl. Anal. 2013, Article ID 681523, 11 p. (2013; Zbl 1421.93108)

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##### References:

[1] | Wicks, M. A.; Peleties, P.; DeCarlo, R. A., Construction of piecewise Lyapunov functions for stabilizing switched systems, Proceedings of the 33rd IEEE Conference on Decision and Control |

[2] | Narendra, K. S.; Balakrishnan, J., A common Lyapunov function for stable LTI systems with commuting A-matrices, IEEE Transactions on Automatic Control, 39, 12, 2469-2471, (1994) · Zbl 0825.93668 |

[3] | Feron, E., Quadratic stabilization of switched system via state and output feedback, CICS-F-468, (1996), Cambridge, Mass, USA: MIT, Cambridge, Mass, USA |

[4] | Morse, A. S., Control Using Logic-Based Switching, (1997), London, UK: Springer, London, UK |

[5] | Narendra, K. S.; Balakrishnan, J., Adaptive control using multiple models, IEEE Transactions on Automatic Control, 42, 2, 171-187, (1997) · Zbl 0869.93025 |

[6] | Ooba, T.; Funahashi, Y., Two conditions concerning common quadratic Lyapunov functions for linear systems, IEEE Transactions on Automatic Control, 42, 5, 719-721, (1997) · Zbl 0892.93056 |

[7] | Branicky, M. S., Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Transactions on Automatic Control, 43, 4, 475-482, (1998) · Zbl 0904.93036 |

[8] | Hespanha, J. P.; Morse, A. S., Stabilization of nonholonomic integrators via logic-based switching, Automatica, 35, 3, 385-393, (1999) · Zbl 0931.93055 |

[9] | Liberzon, D.; Morse, A. S., Basic problems in stability and design of switched systems, IEEE Control Systems Magazine, 19, 5, 59-70, (1999) · Zbl 1384.93064 |

[10] | Skafidas, E.; Evans, R. J.; Savkin, A. V.; Petersen, I. R., Stability results for switched controller systems, Automatica, 35, 4, 553-564, (1999) · Zbl 0949.93014 |

[11] | Decarlo, R. A.; Branicky, M. S.; Pettersson, S.; Lennartson, B., Perspectives and results on the stability and stabilizability of hybrid systems, Proceedings of the IEEE, 88, 7, 1069-1082, (2000) |

[12] | Shorten, R. N.; Narendra, K. S., On common quadratic Lyapunov functions for pairs of stable LTI systems whose system matrices are in companion form, IEEE Transactions on Automatic Control, 48, 4, 618-621, (2003) · Zbl 1364.93579 |

[13] | Liberzon, D., Switching in Systems and Control, xiv+233, (2003), Boston, Mass, USA: Birkhäuser, Boston, Mass, USA |

[14] | Zhao, J.; Dimirovski, G. M., Quadratic stability of a class of switched nonlinear systems, IEEE Transactions on Automatic Control, 49, 4, 574-578, (2004) · Zbl 1365.93382 |

[15] | Sun, Z.; Ge, S. S., Analysis and synthesis of switched linear control systems, Automatica, 41, 2, 181-195, (2005) · Zbl 1074.93025 |

[16] | Ji, Z.; Wang, L.; Xie, G., Quadratic stabilization of switched systems, International Journal of Systems Science, 36, 7, 395-404, (2005) · Zbl 1121.93063 |

[17] | Margaliot, M.; Hespanha, J. P., Root-mean-square gains of switched linear systems: a variational approach, Automatica, 44, 9, 2398-2402, (2008) · Zbl 1153.93339 |

[18] | Wu, L.; Zheng, W. X., Weighted \(H^\infty\) model reduction for linear switched systems with time-varying delay, Automatica, 45, 1, 186-193, (2009) · Zbl 1154.93326 |

[19] | Ji, Z.; Lin, H.; Lee, T. H., A new perspective on criteria and algorithms for reachability of discrete-time switched linear systems, Automatica, 45, 6, 1584-1587, (2009) · Zbl 1166.93308 |

[20] | Huang, Z. H.; Xiang, C.; Lin, H.; Lee, T. H., Necessary and sufficient conditions for regional stabilisability of generic switched linear systems with a pair of planar subsystems, International Journal of Control, 83, 4, 694-715, (2010) · Zbl 1209.93133 |

[21] | Zhang, L.; Gao, H., Asynchronously switched control of switched linear systems with average dwell time, Automatica, 46, 5, 953-958, (2010) · Zbl 1191.93068 |

[22] | Lien, C.-H.; Yu, K.-W.; Chung, Y.-J.; Chang, H.-C.; Chung, L.-Y.; Chen, J.-D., Exponential stability and robust \(H_\infty\) control for uncertain discrete switched systems with interval time-varying delay, IMA Journal of Mathematical Control and Information, 28, 1, 121-141, (2011) · Zbl 1216.93041 |

[23] | Karimi, H. R., Robust delay-dependent \(H_\infty\) control of uncertain time-delay systems with mixed neutral, discrete, and distributed time-delays and Markovian switching parameters, IEEE Transactions on Circuits and Systems I, 58, 8, 1910-1923, (2011) |

[24] | Lien, C.-H.; Chen, J.-D.; Yu, K.-W.; Chung, L.-Y., Robust delay-dependent H control for uncertain switched time-delay systems via sampled-data state feedback input, Computers and Mathematics with Applications, 64, 5, (2012) · Zbl 1356.93027 |

[25] | Zhao, X.; Zhang, L.; Shi, P.; Liu, M., Stability and stabilization of switched linear systems with mode-dependent average dwell time, IEEE Transactions on Automatic Control, 57, 7, 1809-1815, (2012) · Zbl 1369.93290 |

[26] | Zhao, X.; Zhang, L.; Shi, P.; Liu, M., Stability of switched positive linear systems with average dwell time switching, Automatica, 48, 6, 1132-1137, (2012) · Zbl 1244.93129 |

[27] | Sun, Z., Robust switching of discrete-time switched linear systems, Automatica, 48, 1, 239-242, (2012) · Zbl 1244.93075 |

[28] | Müller, M. A.; Liberzon, D., Input/output-to-state stability and state-norm estimators for switched nonlinear systems, Automatica, 48, 9, 2029-2039, (2012) · Zbl 1257.93088 |

[29] | Zhang, G.; Han, C.; Guan, Y.; Wu, L., Exponential stability analysis and stabilization of discrete-time nonlinear switched systems with time delays, International Journal of Innovative Computing, Information and Control, 8, 3, 1973-1986, (2012) |

[30] | Wu, L.; Zheng, W.; Gao, H., Dissipativity-based sliding mode control of switched stochastic systems, IEEE Transactions on Automatic Control, 58, 3, 785-791, (2012) · Zbl 1369.93585 |

[31] | Attia, S. B.; Salhi, S.; Ksouri, M., Static switched output feedback stabilization for linear discrete-time switched systems, International Journal of Innovative Computing, Information and Control, 8, 5, 3203-3213, (2012) |

[32] | Lu, Q.; Zhang, L.; Shi, P.; Karimi, H. R., Control design for a hypersonic aircraft using a switched linear parameter varying system approach, Journal of Systems and Control Engineering, 227, 1, 85-95, (2012) |

[33] | Rajchakit, M.; Rajchakit, G., Mean square exponential stability of stochastic switched system with interval time-varying delays, Abstract and Applied Analysis, 2012, (2012) · Zbl 1246.93081 |

[34] | Sun, Y., Delay-independent stability of switched linear systems with unbounded time-varying delays, Abstract and Applied Analysis, 2012, (2012) · Zbl 1242.93106 |

[35] | Huang, S.; Xiang, Z.; Karimi, H. R., Stabilization and controller design of 2D discrete switched systems with state delays under asynchronous switching, Abstract and Applied Analysis, 2013, (2013) · Zbl 1271.93131 |

[36] | Ibanez, C. A.; Suarez-Castanon, M. S.; Gutierrez-Frias, O. O., A Switching controller for the stabilization of the damping inverted pendulum cart system, International Journal of Innovative Computing, Information, and Control, 9, 9, 3585-3597, (2013) |

[37] | Lian, J.; Shi, P.; Feng, Z., Passivity and passification for a class of uncertain switched stochastic time-delay systems, IEEE Transactions on Cybernetics, 43, 1, 3-13, (2013) |

[38] | Du, D.; Jiang, B.; Shi, P.; Karimi, H. R., Fault detection for continuous-time switched systems under asynchronous switching, International Journal of Robust and Nonlinear Control, (2013) |

[39] | Wu, Z.; Cui, M.; Shi, P.; Karimi, H. R., Stability of stochastic nonlinear systems with state-dependent switching, IEEE Transactions on Automativc Control, 58, 8, 1904-1918, (2013) · Zbl 1369.93706 |

[40] | Li, X.; Xiang, Z.; Karimi, H. R., Asynchronously switched control of discrete impulsive switched systems with time delays, Information Sciences, 249, 132-142, (2013) · Zbl 1337.93079 |

[41] | Brockett, R. W., Asymptotic stability and feedback stabilization, Differential Geometric Control Theory, 27, 181-191, (1983), Boston, Mass, USA: Birkhäuser, Boston, Mass, USA · Zbl 0528.93051 |

[42] | Yin, S.; Ding, S. X.; Haghani, A.; Hao, H.; Zhang, P., A comparison study of basic data-driven fault diagnosis and process monitoring methods on the benchmark Tennessee Eastman process, Journal of Process Control, 22, 9, 1567-1581, (2012) |

[43] | Yin, S.; Yang, X.; Karimi, H. R., Data-driven adaptive observer for fault diagnosis, Mathematical Problems in Engineering, 2012, (2012) · Zbl 1264.93115 |

[44] | Yin, S.; Luo, H.; Ding, S. X., Real-time implementation of fault-tolerant control systems with performance optimization, IEEE Transactions on Industrial Electronics, (2013) |

[45] | Yin, S.; Ding, S. X.; Sari, A. H. A.; Hao, H., Data-driven monitoring for stochastic systems and its application on batch process, International Journal of Systems Science, 44, 7, 1366-1376, (2013) · Zbl 1278.93259 |

[46] | You, J.; Yin, S.; Karimi, H. R., Robust estimation for discrete Markov system with time-varying delay and missing measurements · Zbl 1296.93196 |

[47] | Leon, S. J., Linear Algebra with Applications, (2002), Upper Saddle River, NJ, USA: Prentice Hall, Upper Saddle River, NJ, USA |

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