Designing of the robust stabilizing controller for uncertain delay systems -- LMI approach (Q2726034)

The task of robust stabilization of a control system with delay is considered NEWLINE\[NEWLINE\begin{multlined} \dot x(t)= (A_0+\Delta A_0(t)) x(t)+ \sum^N_{i=1} (A_i+\Delta A_i(t)) x(t- dx_i(t))+\\ (B_0+\Delta B_0(t)) u(t)+ \sum^M_{l=1} (B_l+\Delta B_l) u(t- du_l(t)),\end{multlined}NEWLINE\]NEWLINE NEWLINE\[NEWLINEx(t)= 0,\quad t< 0,\quad x(0)= x_0.NEWLINE\]NEWLINE It is supposed that delays are limited and system perturbations have the form NEWLINE\[NEWLINE\Delta A_i= H_iF(t) E_i,\quad\Delta B_l(\cdot)= H_{N+ 1+l} F(t) E_{N+1+l},\quad\|F(t)\|\leq 1.NEWLINE\]NEWLINE The control is searched in the feedback form \(u(t)= Kx(t)\). The Lyapunov-Krasovsky functional NEWLINE\[NEWLINEV(x(t))= x^T Px+{1\over N} \sum^N_{i=1} \int^t_{t- dx_i} x^T(\tau) x(\tau) d\tau+ {1\over M} \sum^M_{l=1} \int^t_{t-du_l} x^T(\tau) K^T Kx(\tau) d\tauNEWLINE\]NEWLINE is used to solve the stabilization task. Sufficient stabilization conditions are obtained in the form of matrix inequalities.