Robust adaptive control for a class of uncertain time-delay systems (Q2745598)

The linear system with delay NEWLINE\[NEWLINE\dot x(t)= A_1x(t)+ M_1\delta_1(x(t))+ A_2x(t- \tau(t))+ M_2\delta_2(x(t- \tau(t)))+ Bu\tag{1}NEWLINE\]NEWLINE is considered.NEWLINENEWLINENEWLINEIt is supposed that there exist constant matrices \(N_1\), \(N_2\) satisfying NEWLINE\[NEWLINE\|\delta_1(x(t))\|\leq \|N_1x(t)\|,\quad\|\delta_2(x(t- \tau(t)))\|\leq \|N_2x(t- \tau(t))\|.\tag{2}NEWLINE\]NEWLINE An investigation is carried out with the aid of the Lyapunov-Krasovsky functional NEWLINE\[NEWLINEV= x^T(t) Px(t)+ \int^t_{t- \tau(t)} x^T(p) Rx(p) dp.NEWLINE\]NEWLINE The control is found in the form \(u= Kx(t)\).NEWLINENEWLINENEWLINETheorem. Given positive-definite matrices \(P= P^T>0\), \(Q= Q^T> 0\), \(R= R^T> 0\), constants \(\varepsilon_1> 0\), \(\varepsilon_2> 0\), \(\varepsilon_3> 0\) and the matrix \(K\), under which NEWLINE\[NEWLINE\begin{multlined} P(A_1+ BK)+ (A_1+ BK)^T P+ {1\over \varepsilon_1} PM_1 M^T_1 P+ \varepsilon_1 N^T_1 N_1+ {1\over \varepsilon_2} PM_2 M^T_2P+\\ {1\over \varepsilon_3} PA_2 A^T_2 P+ Q+ R< 0,\quad \varepsilon_2 N^T_2 N_2+ \varepsilon_3 I_n- \sigma^2 R< 0,\end{multlined}NEWLINE\]NEWLINE the authors prove quadratic stability under another existence condition and for \(1-\tau(t)\geq \sigma^2\).NEWLINENEWLINENEWLINEMore complex systems are considered, too.