Exponential Stability for a Class of Uncertain Linear Systems with a Single Time-Delay (or Multiple Time-Delays)  

作　　者：Weiqiu Pan Tianzeng Li Weiqiu Pan;Tianzeng Li(School of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong, China)

机构地区：[1]School of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong, China

出　　处：《Journal of Applied Mathematics and Physics》2021年第3期413-426,共14页应用数学与应用物理（英文）

摘　　要：This paper investigates the issue of exponential stability for a class of uncertain linear systems with a single time-delay (or multiple time-delays). We consider that the uncertainties are the parameter disturbance and the external disturbance, both of which are stochastic. The external disturbances involve not only the current state <em>x</em>(<em>t</em>) but also the delayed state <em>x</em>(<em>t</em> - <span style="white-space:nowrap;"><em>&#964;</em></span>). By means of the Lyapunov-Krasovskii functional, the sufficient conditions on exponential stability for the uncertain linear systems with a single time-delay (or multiple time-delays) are performed in the form of the linear matrix inequality (LMI). Selecting the suitable matrices <em>P</em> (or <img src="Edit_b2ad88c4-c55b-4ba5-9f0c-a3cd77848964.bmp" alt="" /> ) and <em>Q</em> (or <img src="Edit_fc454108-e7f0-490d-96ec-bc6d29b72e71.bmp" alt="" /> ) and parameter <span style="white-space:nowrap;"><em>&#946;</em></span> (or <img src="Edit_ec124eb5-7b2c-4ade-809e-53c7cc39e9a0.bmp" alt="" /> ), we can also get the bounds of the state variables for the single time-delay (or multiple time-delays) systems. In order to stabilize the solution of the single time-delay (or multiple time-delays) systems at the equilibrium point, we designed the state feedback control. Thus, the corresponding stabilization criteria are given. Finally, Numerical simulations show that a small disturbance can make a great change to the state variables of the systems. When the feedback gain control is added, the state variables of the systems can quickly stabilize at the equilibrium point. This also shows the effectiveness of the proposed method.This paper investigates the issue of exponential stability for a class of uncertain linear systems with a single time-delay (or multiple time-delays). We consider that the uncertainties are the parameter disturbance and the external disturbance, both of which are stochastic. The external disturbances involve not only the current state <em>x</em>(<em>t</em>) but also the delayed state <em>x</em>(<em>t</em> - <span style="white-space:nowrap;"><em>&#964;</em></span>). By means of the Lyapunov-Krasovskii functional, the sufficient conditions on exponential stability for the uncertain linear systems with a single time-delay (or multiple time-delays) are performed in the form of the linear matrix inequality (LMI). Selecting the suitable matrices <em>P</em> (or <img src="Edit_b2ad88c4-c55b-4ba5-9f0c-a3cd77848964.bmp" alt="" /> ) and <em>Q</em> (or <img src="Edit_fc454108-e7f0-490d-96ec-bc6d29b72e71.bmp" alt="" /> ) and parameter <span style="white-space:nowrap;"><em>&#946;</em></span> (or <img src="Edit_ec124eb5-7b2c-4ade-809e-53c7cc39e9a0.bmp" alt="" /> ), we can also get the bounds of the state variables for the single time-delay (or multiple time-delays) systems. In order to stabilize the solution of the single time-delay (or multiple time-delays) systems at the equilibrium point, we designed the state feedback control. Thus, the corresponding stabilization criteria are given. Finally, Numerical simulations show that a small disturbance can make a great change to the state variables of the systems. When the feedback gain control is added, the state variables of the systems can quickly stabilize at the equilibrium point. This also shows the effectiveness of the proposed method.

关 键 词：Time-Delay System UNCERTAINTIES Lyapunov-Krasovskii Functional The Linear Matrix Inequality (LMI) 

分 类 号：O17[理学—数学]