Passive control for a class of uncertain time-delay systems (Q2764371)

The linear uncertain time-delay system is considered NEWLINE\[NEWLINE\begin{aligned} \dot x(t) &= (A_0+\Delta A_0)x(t)+ (A_1+\Delta A_1) x(t- h)+ (B_w+\Delta B_w) w(t),\\ z(t) &= (C_0+\Delta C_0) x(t)+ (C_1+\Delta C_1) x(t- h)+(D_w+\Delta D_w) w(t),\end{aligned}\tag{1}NEWLINE\]NEWLINE where \(x(t)\in \mathbb{R}^n\), \(z(t)\in\mathbb{R}^p\), \(w(t)\in \mathbb{R}^p\) is the exogenous input, NEWLINE\[NEWLINE\left[\begin{matrix} \Delta A_0 &\Delta A_1 &\Delta B_w\\ \Delta C_0 &\Delta C_1 &\Delta D_w\end{matrix}\right]= \left[\begin{matrix} E_1\\ E_2\end{matrix}\right] F(t)[G_1\;G_2\;G_3],\quad F^T(t) F(t)\leq I.\tag{2}NEWLINE\]NEWLINE The Lyapunov-Krasovsky functional NEWLINE\[NEWLINEL(x)= x^T Px+ \int^t_{t- h} x^T(\tau) Qx(\tau) d\tauNEWLINE\]NEWLINE is used in the investigation.NEWLINENEWLINENEWLINETheorem. Suppose that there exist symmetric positive-definite matrices \(P\) and \(Q\) such that the following holds NEWLINE\[NEWLINE\left[\begin{matrix} P\overline A_0+\overline A_0P+ Q & P\overline A_1 &\overline C^T_0- P\overline B_w\\ \overline A^T_1P &-Q &\overline C^T_1\\ \overline C_0-\overline B^T_w P &\overline C_1 &-(\overline D_w+ D^T_w)\end{matrix}\right]< 0.NEWLINE\]NEWLINE Then, the uncertain time-delay system (1) is quadratically stable. NEWLINE\[NEWLINE\begin{aligned} \overline A_0= A_0+\Delta A_0,\;\overline A_1 &= A_1+\Delta A_1,\;\overline B_w= B_w+ \Delta B_w,\;C_0= C_0+\Delta C_0,\\ \overline C_1 &= C_1+\Delta C_1,\quad\overline D_w= D_w+ \Delta D_w.\end{aligned}NEWLINE\]NEWLINE Analogous robust stability conditions are obtained, i.e. conditions of stability on arbitrary perturbations of matrices, that satisfy conditions (2).