Observer-based robust control for uncertain systems with time-varying delay (Q2745706)

The following uncertain delay system is considered NEWLINE\[NEWLINE\begin{aligned} & \dot x= Ax(t)+ [A_d+ \Delta A_d(t)]x(t- d(t))+ Bu(t)+ H_1\xi(t),\\ & y(t)= cx(t)+ [C_d+ \Delta C_d(t)] x(t- d(t))+ H_2\xi(t),\\ & x(t)= \Phi(t),\quad t\in [-d, 0],\end{aligned}\tag{1}NEWLINE\]NEWLINE where \(d(t)\in [0, d]\) and \(\xi(t)\) is the uncertain variable satisfying the condition NEWLINE\[NEWLINE\|\xi(t)\|^2\leq \|E_1 x(t)+ E_2 u(t)\|^2.NEWLINE\]NEWLINE Definition. The system (1) is said to be robustly stabilizable based on function observer if there exist a function observer \(\dot z= Fz(t)+ Gy(t)+ Hu(t)\) and a control \(u(t)= Mz(t)+ Ny(t)\) such that \(u(t)- Kx(t)\to 0\) with \(t\to \infty\) and the closed-loop system is robustly stable.NEWLINENEWLINENEWLINESufficient conditions for robust stability of the zero state are obtained in the form of matrix inequalities. A controller design algorithm is provided, and an example is given.