Abstract
This paper deals with the problems of finite-time \(L_2\)–\(L_\infty \) control for stochastic switched systems under asynchronous switching with time-varying delay and nonlinearity. Because of the presence of delay in the switching signal of controllers, the switching of the controllers is asynchronous with the switching of the subsystems. Firstly, based on average dwell time approach, merging switching signal technique and multiple Lyapunov function method, state feedback controllers are designed to guarantee the finite-time boundedness of stochastic switched time-delay systems under asynchronous switching by linearization techniques. Then the finite-time \(L_2\)–\(L_\infty \) performance is analyzed. Finally, a numerical example is given to illustrate the effectiveness of the proposed method.
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Acknowledgements
This work is supported by the National Natural Science Foundation of China under Grants (61573088), (61573087) and (61433004).
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Appendix
Appendix
Because of the switching delay, after the jth subsystem has been switched to the ith subsystem, the controller \(K_j\) is still active. Thus, we rewrite the closed-loop system as
where
For system (47), the piecewise Lyapunov–Krasovskii function candidate is chosen as follows:
where \({P}_{\sigma '(t)}, {Q}_{\sigma '(t)}, {Z}_{\sigma '(t)}\) are positive definite matrices to be determined and
Introducing the free-weighting matrix \(T_{ii}\) and \(T_{ij}\) yields
By It\(\hat{o}\)’s formula [23], we have
where
Combining Assumption 4 and (15), there is \(\varepsilon _{ii}>0\) for
By Lemma 2, we can get
Construct function \(\varGamma _{ii}\), we have
where \(\eta ^T(t)=\left[ \begin{matrix} x^T(t)&x^T(t-\mathrm{d}(t))&\nu ^T(t)&y^T(t) \end{matrix} \right] \) and
Multiplying both sides of \(\varPhi _{ii}\) by \(\varDelta =\mathrm{diag}\{\bar{P}_{ii},\bar{P}_{ii},\bar{P}_{ii},\bar{P}_{ii}\}\) and letting \(M_{ii}= K_{ii}\bar{P}_{ii},T_{ii}={\alpha }P_{ii}\), we derive
where
By using the Schur complement Lemma, letting \(\bar{Q}_{ii}= \bar{P}_{ii}Q_{ii}\bar{P}_{ii},\bar{Z}_{ii}= \bar{P}_{ii}Z_{ii}\bar{P}_{ii},\bar{\varepsilon }_{ii}=\varepsilon ^{-1}_{ii}\), (56) is equivalent to
where
From (51), (57) and Theorem 1, we can obtain
Similarly, we can get
Considering formula (23) and formula (32), we can get
Taking expectations, we can get
Namely
Similarly, we can get
Considering (48) with Theorem 1, we have
Combining (62), (63), (64) and Assumption 3, by some mathematical manipulation, we can get that when \(t \in [t_k+\tau _s, t_{k+1})\),
Similarly, we can get that when \(t \in [t_k, t_k+\tau _s)\),
Considering (48), we can get
Combining (66), (67) and (68), we can get
Simplifying formula (69), we get \(\tau _a>\tau ^*_a\) and \(\tau ^*_a\) is the numerical solution of the following equation.
where
Therefore, if the ADT satisfies (70), by Definition 1, system (11) is finite-time bounded with respect to \([c_1,c_2,T,d,R,\sigma ]\).
The proof is completed. \(\square \)
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He, H., Gao, X. & Qi, W. Finite-Time \(L_2\)–\(L_\infty \) Control for Stochastic Asynchronously Switched Systems with Time-Varying Delay and Nonlinearity. Circuits Syst Signal Process 37, 112–134 (2018). https://doi.org/10.1007/s00034-017-0549-y
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DOI: https://doi.org/10.1007/s00034-017-0549-y