Local stabilization of uncertain time-delay systems with saturating actuators (Q2769557)

The system of a delay differential equation is considered NEWLINE\[NEWLINE\dot x(t)= [A+\Delta A] x(t)+ [A_d+\Delta A_d] x(t- \tau)+ [B+\Delta B] f(u(t)),\;t\geq t_0\geq 0,\tag{1}NEWLINE\]NEWLINE with a feedback control NEWLINE\[NEWLINEu(t)= Kx(t)\in \mathbb{R}^m.\tag{2}NEWLINE\]NEWLINE Here \(\Delta A\), \(\Delta A_d\), \(\Delta B\) are the uncertainties matrices and have the following structure NEWLINE\[NEWLINE\Delta A= D_a F_a(t) E_a,\quad\Delta A_d= D_d F_d(t) E_d,\quad\Delta B= D_b F_b(t) E_b,NEWLINE\]NEWLINE where \(F_a(t)\), \(F_d(t)\), \(F_b(t)\) are unknown matrices satisfying NEWLINE\[NEWLINEF^T_i(t) F_i(t)\leq I,\qquad i= a,d,b;NEWLINE\]NEWLINE \(f(u)= [\sigma_1(u_1),\dots, \sigma_m(u_m)]^T\), \(\sigma_i(u_i)= \text{sign }u_i\), \(\min\{|u_i|,\Delta_i\}\), \(\Delta_i\geq 0\). Sufficient conditions for asymptotic stability of the zero state are obtained for the closed-loop system (1), (2).NEWLINENEWLINENEWLINETwo methods are given to design a feedback controller. Numerical examples are considered.