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Finite-time stability of linear time-varying systems with jumps. (English) Zbl 1162.93375
Summary: This paper deals with the finite-time stability problem for continuous-time linear time-varying systems with finite jumps. This class of systems arises in many practical applications and includes, as particular cases, impulsive systems and sampled-data control systems. The paper provides a necessary and sufficient condition for finite-time stability, requiring a test on the state transition matrix of the system under consideration, and a sufficient condition involving two coupled differential-difference linear matrix inequalities. The sufficient condition turns out to be more efficient from the computational point of view. Some examples illustrate the effectiveness of the proposed approach.

##### MSC:
 93C57 Sampled-data control/observation systems 93D09 Robust stability 93C05 Linear systems in control theory
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##### References:
 [1] Amato, F., Ariola, M., Carbone, M., & Cosentino, C. (2006). Finite-time output feedback control of linear systems via differential linear matrix conditions. In Proc. conf. on decision and control (pp. 5371-5375) [2] Amato, F., Ariola, M., & Cosentino, C. (2005). Finite-time control of linear time-varying systems via output feedback. In Proc. American Control Conference (pp. 4722-4726) [3] Amato, F.; Ariola, M.; Cosentino, C., Finite-time stabilization via dynamic output feedback, Automatica, 42, 337-342, (2006) · Zbl 1099.93042 [4] Amato, F.; Ariola, M.; Dorato, P., Finite time control of linear systems subject to parametric uncertainties and disturbances, Automatica, 37, 1459-1463, (2001) · Zbl 0983.93060 [5] Antsaklis, P.J., Hybrid systems II, (1995), Springer [6] Boel, R.; Stremersch, G., Discrete event systems: analysis and control, (2000), Springer · Zbl 0996.00058 [7] Boyd, S.; El Ghaoui, L.; Feron, E.; Balakrishnan, V., Linear matrix inequalities in system and control theory, (1994), SIAM Press · Zbl 0816.93004 [8] Callier, F.M.; Desoer, C.A., Linear system theory, (1991), Springer Verlag · Zbl 0508.93011 [9] Dorato, P. (1961). Short time stability in linear time-varying systems, In Proc. IRE International convention record part 4 (pp. 83-87) [10] Gahinet, P.; Nemirovski, A.; Laub, A.J.; Chilali, M., LMI control toolbox, (1995), The Mathworks Inc [11] Kamenkov, G., On stability of motion over a finite interval of time, Journal of applied mathematics and mechanics, 17, 529-540, (1953), (in Russian) [12] Lebedev, A., On stability of motion during a given interval of time, Journal of applied mathematics and mechanics, 18, 139-148, (1954), (In Russian) [13] Lebedev, A., The problem of stability in a finite interval of time, Journal of applied mathematics and mechanics, 18, 75-94, (1954), (in Russian) · Zbl 0055.32102 [14] Sun, W.; Nagpal, K.; Khargonekar, P.P., $$\mathcal{H}_\infty$$ control and filtering for sampled-data systems, Institute of electrical and electronic engineering transactions on automatic control, 38, 1162-1175, (1993) · Zbl 0784.93027 [15] Weiss, L.; Infante, E.F., Finite time stability under perturbing forces and on product spaces, Institute of electrical and electronic engineering transactions on automatic control, 12, 54-59, (1967) · Zbl 0168.33903 [16] Yang, T., Impulsive control theory, (2001), Springer
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