Multiplier IQCs for uncertain time delays (Q5933495)

A repeated time-delay uncertainty with upper bound \(\Delta\triangleq \delta(s) I_q\) is considered, where \(\delta(s)= e^{-s\tau}\) and \(0\leq\tau\leq\overline\tau\). A stability criterion for the time-delay system is obtained. For given \(\overline\tau> 0\) suppose that the time-delay uncertainty is expressed as \(\Delta(j\omega)\triangleq \delta(j\omega)I_q\), where \(\delta(j\omega)= e^{-j\omega\tau}\) and \(0\leq \tau\leq \overline\tau\). Let \(\dot w\in L^q_{2l}[0, \infty)\). Assume that:NEWLINENEWLINENEWLINE(i) the feedback system is finite-gain stable when \(\tau> 0\),NEWLINENEWLINENEWLINE(ii) \(G(s)\) is strictly proper,NEWLINENEWLINENEWLINE(iii) there exists an \(M(j\omega)\) with \(\text{herm}(M(j\omega))\geq 0\) for \(|\omega|< 2{\pi\over\overline\tau}\) such that for some \(\varepsilon> 0\), NEWLINE\[NEWLINE\Biggl[\begin{matrix} G(j\omega)\\ I_q\end{matrix}\Biggr]^*\prod_M(j\omega) \Biggl[\begin{matrix} G(j\omega)\\ I_q\end{matrix}\Biggr]\leq \varepsilon I,\quad\forall \omega\in \mathbb{R}.NEWLINE\]NEWLINE Then the feedback system is robust against the time delay \(0\leq \tau\leq\overline\tau\) in the sense that the mapping \((v,w,\dot w)\to (u,y)\) is finite-gain stable.