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Switching between stabilizing controllers. (English) Zbl 1011.93533
Summary: This paper deals with the problem of switching between several linear time-invariant (LTI) controllers–all of them capable of stabilizing a specific LTI process–in such a way that the stability of the closed-loop system is guaranteed for any switching sequence. We show that it is possible to find realizations for any given family of controller transfer matrices so that the closed-loop system remains stable, no matter how we switch among the controller. The motivation for this problem is the control of complex systems where conflicting requirements make a single LTI controller unsuitable.

##### MSC:
 93D15 Stabilization of systems by feedback 93B12 Variable structure systems
QDES
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##### References:
 [1] Bainov, D.D.; Simeonov, P.S., Systems with impulse effects: stability, theory and applications, (1989), Academic Press New York · Zbl 0676.34035 [2] Boyd, S.P.; Barratt, C.H., Linear controller design: limits of performance, (1991), Prentice-Hall New Jersey · Zbl 0748.93003 [3] Boyd, S.; Ghaoui, L.E.; Feron, E.; Balakrishnan, V., Linear matrix inequalities in system and control theory, SIAM studies in applied mathematics, Vol. 15, (1994), SIAM Philadelphia, PA · Zbl 0816.93004 [4] Branicky, M.S., Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE transactions on automatic control, 43, 475-482, (1998) · Zbl 0904.93036 [5] Clegg, J.C., A nonlinear integrator for servo-mechanisms, AIEE transactions, part II, applications and industry, 77, 41-42, (1958) [6] Dayawansa, W.P.; Martin, C.F., A converse Lyapunov theorem for a class of dynamical systems which undergo switching, IEEE transactions on automatic control, 44, 751-760, (1999) · Zbl 0960.93046 [7] Eker, J.; Malmborg, J., Design and implementation of a hybrid control strategy, IEEE control and systems magazine, 19, 12-21, (1999) [8] Francis, B.A., A course in $$H∞$$ control theory, Lecture notes in control and information sciences, Vol. 88, (1985), Springer New York [9] Gurvits, L. (1996). Stability of linear inclusions—Part 2. Technical Report, NECI. [10] Hespanha, J. P. (1998). Logic-based switching algorithms in control. Ph.D. thesis, Yale University, New Haven, CT. [11] Hespanha, J. P., Liberzon, D., Morse, A. S., Anderson, B. D. O., Brinsmead, T. S., & de Bruyne, F. (2001). Multiple model adaptive control, part 2: Switching, International Journal of Robust and Nonlinear Control (Special Issue on Hybrid Systems in Control), 11, 479-496. · Zbl 0994.93025 [12] Hespanha, J. P., & Morse, A. S. (1996). Towards the high performance control of uncertain processes via supervision. In Proceedings of the 30th annual conference on information sciences and systems 1, 405-410. [13] Hollot, C. V., Beker, O., & Chait, Y. (2001). Plant with integrator: An example of reset control overcoming limitations of linear feedback. In Proceedings of the 2001 American control conference, Vol. 2 (pp. 1519-1520). · Zbl 1006.93030 [14] Horowitz, I.; Rosenbaum, P., Non-linear design for cost of feedback reduction in systems with large parameter uncertainty, International journal of control, 24, 6, 977-1001, (1975) · Zbl 0312.93019 [15] Khalil, H.K., Nonlinear systems, (1992), Macmillan New York · Zbl 0626.34052 [16] Liberzon, D.; Hespanha, J.P.; Morse, A.S., Stability of switched linear systemsa Lie-algebraic condition, Systems and control letters, 37, 117-122, (1999) · Zbl 0948.93048 [17] Liberzon, D.; Morse, A.S., Basic problems in stability and design of switched systems, IEEE control systems and magazine, 19, 59-70, (1999) · Zbl 1384.93064 [18] Megretski, A.; Rantzer, A., System analysis via integral quadratic constraints, IEEE transactions on automatic control, 42, 819-830, (1997) · Zbl 0881.93062 [19] Molchanov, A.P.; Pyatnitskiy, Y.S., Criteria of asymptotic stability of differential and difference inclusions encountered in control theory, Systems and control letters, 13, 59-64, (1989) · Zbl 0684.93065 [20] Morse, A. S. (1995). Control using logic-based switching. In A. Isidori (Ed.), Trends in control: An European perspective (pp. 69-113). London: Springer. [21] Morse, A.S., Supervisory control of families of linear set-point controllers—part 1exact matching, IEEE transactions on automatic control, 41, 1413-1431, (1996) · Zbl 0872.93009 [22] Morse, A.S., Supervisory control of families of linear set-point controllers—part 2robustness, IEEE transactions on automatic control, 2, 1500-1515, (1997) · Zbl 0926.93010 [23] Narendra, K.S.; Balakrishnan, J., A common Lyapunov function for stable LTI systems with commuting A-matrices, IEEE transactions on automatic control, 39, 2469-2471, (1994) · Zbl 0825.93668 [24] Packard, A. (1995). Personal communication. [25] Shorten, R. N., & Narendra, K. S. (1997). A sufficient condition for the existence of a common Lyapunov function for two second order linear systems. In Proceedings of the 36th conference on decision and control, 4, 3521-3522. [26] Tay, T.T.; Moore, J.B.; Horowitz, R., Indirect adaptive techniques for fixed controller performance enhancement, International journal of control, 50, 1941-1959, (1989) · Zbl 0691.93030 [27] Tsitsiklis, J.N.; Blondel, V.D., The Lyapunov exponent and joint spectral radius of pairs of matrices are hard—when not impossible—to compute and to approximate, Mathematics of control, signals and systems, 10, 1, 31-40, (1997) · Zbl 0888.65044 [28] Vegte, J.V., Feedback control systems, (1994), Prentice-Hall New Jersey [29] Ye, H.; Michel, A.N.; Hou, L., Stability theory for hybrid dynamical systems, IEEE transactions on automatic control, 43, 461-474, (1998) · Zbl 0905.93024 [30] Yoshihiro Mori, T. M., & Kuroe, Y. (1997). A solution to the common Lyapunov function problem for continuous-time systems. In Proceedings of the 36th conference on decision and control, Vol. 3 (pp. 3530-3531). [31] Youla, D.C.; Jabr, H.A.; Bongiorno, J.J., Modern Wiener-Hopf design of optimal controllers—part II. the multivariable case, IEEE transactions on automatic & control, 21, 319-338, (1976) · Zbl 0339.93035
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