Perspectives and results on the stability and stabilizability of hybrid systems
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...Another remarkable fact is that one may carefully switch between unstable subsystems to make the switched system exponentially stable [22], [48]....
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...The authors hope that the current paper provides useful additional results and represents a meaningful complementary resource to previous survey papers [22], [34], [49], [61]....
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...For example, even when all the subsystems are exponentially stable, the switched systems may have divergent trajectories for certain switching signals [22], [48]....
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...Since there exist excellent reviews on the stability under restricted switching (like dwell time and average dwell time [34], [48]), multiple Lyapunov functions [22], [61] and piecewise quadratic Lyapunov functions [22], we will review these topics very briefly in Section III....
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...There have been several excellent survey papers on the stability of switched systems; see for example the survey papers [22], [34], [49], [61], the recent books [43], [47] and the references cited therein....
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...Piecewise quadratic Lyapunov stability is becoming a standard in the stability analysis of hybrid systems [155, 96, 102, 233, 234]....
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...3 Indeed, sliding motions may be incorporated by defining each such motion and its associated equivalent dynamics [ 94 ] as an additional system to which we can switch....
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...This, however, is not required. See, e.g., [ 94 ] and [95]....
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...Theorem 3.3 [49, Theorem 4.1]: If for each , , a vector field can be selected such that , and any resulting sliding motion dynamics are given by Filippov’s convex combination definition [ 94 ], then the origin 0 of the closed-loop system is stable in the sense of Lyapunov....
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