Dissipative control for linear time-delay systems (Q2783668)

Consider the linear time-delay system described by the equations NEWLINE\[NEWLINE\begin{aligned} \dot x(t) &= Ax(t)+ A_dx(t- d)+ B_1w(t),\\ z(t) &= C_1x(t)+ C_{1d}x(t- d)+ D_{11}w(t).\end{aligned}\tag{1}NEWLINE\]NEWLINE The quadratic energy supply function \(E\) associated with system (1) is defined by NEWLINE\[NEWLINEE(w,z,T)=\langle z,Qz\rangle_T+ 2\langle z,Sw\rangle_T+ \langle w,Rw\rangle_T,NEWLINE\]NEWLINE where \(Q\), \(S\) and \(R\) are real matrices of appropriate dimensions with \(Q\) and \(R\) symmetric. The system (1) is said to be strictly \((Q,S,R)\)-dissipative, if for any \(T\geq 0\) and some scalar \(\alpha> 0\), under zero initial state, the following condition is satisfied NEWLINE\[NEWLINEE(w,z,T)\geq \alpha\langle w,w\rangle_T.NEWLINE\]NEWLINE Theorem 1. Given matrices \(Q\), \(S\) and \(R\) with \(Q_-=-Q\geq 0\). If there exist matrices \(P>0\) and \(V>0\) such that the following holds NEWLINE\[NEWLINEJ= \left[\begin{matrix} PA+ A^T P+ V & PA_d & PB_1- C^T_1 S & C^T_1Q^{{1\over 2}}_-\\ A^T_d P & -V & -C^T_{d1}S & C^T_{d1}Q^{{1\over 2}}_-\\ B^T_1 P-S^T C_1 & -S^T C_{1d} & -(R+- D^T_{11}S+ S^T D_{11}) & D^T_{11} Q^{{1\over 2}}_-\\ Q^{{1\over 2}}_- C_1 & Q^{{1\over 2}}_- C_{1d} & Q^{{1\over 2}}_- D_{11} & -I\end{matrix}\right]< 0,NEWLINE\]NEWLINE then the time-delay system (1) is quadratically stable and strictly \((Q,S,R)\)-dissipative.NEWLINENEWLINENEWLINEThe proof is developed using the Lyapunov-Krasovsky functional NEWLINE\[NEWLINEL(x,t)= x^T Px+ \int^t_{t-a} x^T(\tau) Vx(\tau) d\tau.NEWLINE\]