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Robust exponential stabilization for large-scale uncertain impulsive systems with coupling time-delays. (English) Zbl 1154.34041

Robust exponential stability of a large-scale uncertain impulsive systems with coupling time-delays is investigated. The robust and exponential stability criteria are established for the following system \[ \dot{x}_i(t)=A_ix_i(t)+f_i(t,x_i(t))+ \sum\limits_{j=1}^{N}B_{ij}x_j(t-\tau _j(t))+u_{ci}(t), \;t\in (t_k,t_{k+1}], \]
\[ \Delta x_i(t)=(C_{ik}-I)x_i(t)+u_{di}(t), \;t=t_k, \;k\in \mathbb{N}, i=1,2,\ldots N; \] where \(x_i=(x_{i1},x_{i1},\ldots, x_{in})^T \in \mathbb{R}^n\), represents the state vector of the \(i\)–th subsystem; \(\Delta x_i(t_k)=x(t_k^+)-x(t_k)\); \(f_i : \mathbb{R}_+\times \mathbb{R}^n \rightarrow \mathbb{R}^n\) is a smooth nonlinear vector function with \(f_i(t,0)\equiv 0\). By utilizing Lyapunov’s method and Razumikhin technique, the feeback hybrid controllers in terms of linear matrix inequalities are proved under which the robust exponential stability is achieved for a closed-loop large-scale uncertain impulsive systems with coupling time-delays. These criteria can be easily used for the design of a feedback controller. For illustration of this result one example is given. The numerical simulation procedure is coded and executed in the MATLAB environment.

MSC:

34K35 Control problems for functional-differential equations
34K20 Stability theory of functional-differential equations
34K45 Functional-differential equations with impulses
93C23 Control/observation systems governed by functional-differential equations
93D15 Stabilization of systems by feedback

Software:

Matlab
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References:

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