An $L_{\mathrm{\infty }}$ performance control for time-delay systems with time-varying delays: delay-independent approach via ellipsoidal $\mathcal{D}$-invariance

This paper is concerned with a delay-independent output-feedback controller synthesis suppressing the $L_{\mathrm{\infty }}$-gain of linear time-delay systems with time-varying delays. We first proposed a continuous-time version of the existing discrete-time ellipsoidal $\mathcal{D}$-invariant set and established its existence condition in terms of some linear matrix inequalities (LMIs). This existence condition was further extended to characterizing the $L_{\mathrm{\infty }}$-gain of linear time-delay systems with time-varying delays. Because of the delay-independent property of the proposed $\mathcal{D}$-invariant set, the $L_{\mathrm{\infty }}$-gain analysis does not depend on the choice of delays including their magnitudes and time derivatives. Based on this analysis method, we also constructed an output-feedback controller synthesis for ensuring the $L_{\mathrm{\infty }}$-gain of time-delay systems bounded by a performance level $\rho$. In an equivalent fashion to the $L_{\mathrm{\infty }}$-gain analysis method, this controller synthesis is independent of the delays in the sense that the obtained controller coefficients do not depend on the delay characteristics. Finally, numerical results were given to demonstrate the effectiveness and validity of the proposed results.