Strong robust \(H_\infty\)-control for linear systems with time-delay and uncertainties: LMI approach (Q2705616)

The present paper is devoted to the problem of strong robust \(H_\infty\)-control design for the following autonomous linear delay differential system NEWLINE\[NEWLINE\begin{multlined} \dot x(t)= A_0x(t)+ A_{c1}(t- \tau)+ Dw(t),\;z(t)= C_cx(t)+ C_{c1}x(t- \tau)+\\ Fw(t),\;x(t)= \phi(t)\quad\text{for }t\in [-\tau,0],\end{multlined}\tag{1}NEWLINE\]NEWLINE where \(x\in \mathbb{R}^n\), \(w\in \mathbb{R}^p\), \(z\in\mathbb{R}^q\), \(\tau> 0\) is a constant, \(\phi\in C([-\tau, 0], \mathbb{R})\), and \(A_c\), \(A_{c1}\), \(D\), \(C_c\), \(C_{c1}\), \(F\) are constant matrices of appropriate dimensions. Using the linear matrix inequalities method given by \textit{T. Iwasaki} and \textit{R. E. Skelton} [Automatica 30, No. 8, 1307-1317 (1994; Zbl 0806.93017)], the authors prove that a sufficient condition for system (1) to have the \(H_\infty\)-\(\gamma\)-\(SP\) property is: There exists some positive definite constant matrices \(P,Q\in \mathbb{R}^{n\times n}\) such that the linear matrix inequality NEWLINE\[NEWLINE\begin{pmatrix} PA_c+ A_c' P+ C_c' C_c+ Q & PA_{c1}+ C_c' C_{c1} & PD+ C_c' F\\ A_{c1}' P+ C_{c1}' C_c & -Q+ C_{c1}' C_{c1} & C_{c1}' F\\ D'P+ F'C_c & F'C_{c1} & -\gamma^2I+ F'F\end{pmatrix}< 0NEWLINE\]NEWLINE holds, where \(C'\) denotes the transposed matrix of \(C\), \(\gamma> 0\). The consequence of the last result derived for the particular case when \(C_{c1}= 0\), \(F= 0\) slightly improved the main result of \textit{J. H. Lee}, \textit{S. W. Kim} and \textit{W. H. Kwon} [IEEE Trans. Autom. Control 39, No. 1, 159-162 (1994; Zbl 0796.93026)].NEWLINENEWLINENEWLINEApplying the above result, a necessary and sufficient condition for the existence of state-feedback robust \(H_\infty\)-controllers, which ensure that the closed-loop systems have strong \(H_\infty\)-performance, is also obtained for some linear delay differential systems with uncertainties.