Controllers and observer synthesis for linear systems with multiple time-varying delays in range

Most of physical systems present time-varying delays in their inner dynamics. This causes instability, oscillation and even poor closed performance. Also, the present disturbance can cause instability. This article is addressing techniques to develop stability criteria for closed-loop and states estimation analysis of multiple time-varying delays systems. By selecting a suitable Lyapunov–Krasovskii functional (LKF), the derivative of double integration terms are upper bounded by using reciprocally convex matrix inequality. The closed-loop stability criteria are derived fulfilling $H_{\infty }$ performance index for multiple time-varying delays systems. Similar technique is also adopted to estimate unmeasured states fulfilling $H_{\infty }$ norm bound. The developed criteria are demonstrated to a numerical example. It is shown that H${}_{\infty }$ memory based controller has better performance on rejecting the introduction disturbance with having lower peak and shallow valley than other techniques.