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Delay-dependent robust stabilization of uncertain systems with multiple state delays. (English) Zbl 0973.93043

Consider uncertain time-delay systems described by the following state equations \[ \dot x(t)= [A_0+\Delta A_0(x, t)] x(t)+ [B+\Delta B(x,t)] u(t)+ \sum^l_{i=1} [A_i+\Delta A_i(x, t)] x(t- d_i(t)),\tag{1} \]
\[ 0< d_i(t)\leq \tau_i,\quad i= 1,\dots, l, \]
\[ \Delta A_i(x, t)= D_iF_i(x, t) E_i,\quad \Delta B(x, t)= D_0 F_0(x, t) E_0, \] where \(F_j(x,t)\) are unknown real time-varying matrices with Lebesgue measurable elements bounded by \[ F^T_j(x, t) F_j(x, t)\leq I\quad\text{for all }t. \] Theorem: Consider the uncertain delay system (1) with \(u(t)\equiv 0\). Given scalars \(\tau_i\) and \(\tau\) satisfying \(0< \tau_i\leq \tau\), then for any time delays \(0< d_i(t)\leq \tau_i\), this system is robustly stable if there exists matrices \(X>0\), \(P_{ij}> 0\), and scalars \(\varepsilon_i> 0\), \(d_j> 0\), \(\rho_{ij}> 0\) satisfying the following \[ S_1(\tau_i):= \left[\begin{matrix} S_{10} & H_1 & H_2\\ H^T_1 &-J_1 & 0\\ H^T_2 & 0 & -J_2\end{matrix}\right]< 0,\quad \left[\begin{matrix} X & XA^T_j & XE^T_j\\ A_jX & P_{ij}- \rho_{ij}D_i D^T_j & 0\\ E_j X & 0 & \rho_{ij} I\end{matrix}\right]> 0, \] where \[ S_{10} = \sum^l_{i=0} A_iX+ X \sum^l_{i=0} A^T_i+ \sum^l_{i=0} \alpha_i D_i D^T_i+ \sum^l_{i=1} \tau_i\varepsilon_i D_i D^T_i+ \sum^l_{i=1} \tau_i A_i W_i A^T_i+ \sum^l_{i=1} (l+ 1) \tau_i X, \]
\[ W_i = \sum^l_{j=0} P_{ij},\quad H_1= [XE^T_0, XE^T_1,\dots, XE^T_l],\quad J_1= \text{diag}(\alpha_0 I,\alpha_1I,\dots, \alpha_lI), \]
\[ H_2 = [\tau_1 A_1 W_1 E^T_1, \tau_2 A_2 W_2 E^T_2,\dots, \tau_l A_l W_l E^T_l], \]
\[ J_2 = \text{diag}(\tau_1(\varepsilon_1 I- E_1 W_1 E^T_1),\;\tau_2(\varepsilon_2 I- E_2 W_2 E^T_2),\dots, \tau_l(\varepsilon_l I- E_l W_l E^T_l)). \] Moreover, a stabilizing control law has the form \(u(t)= YX^{-1} x(t)\) for a suitable \(Y\).

MSC:

93D21 Adaptive or robust stabilization
93C23 Control/observation systems governed by functional-differential equations
93D09 Robust stability
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