Robust delay-derivative-dependent state-feedback control for a class of continuous-time system with time-varying delays
Introduction
Nowadays, there always exists time delay in many practical systems, such as networked controlled systems and communication networks. It has been proved that in dynamic systems, the existence of time delay is frequently the source of instability and poor performance. Over the past few decades, many important results on the Lyapunov stability and stabilization have been given for dynamic systems with time delays [1], [2], [3], [4], [5], [6], [7], [8], [9] and other real application with time delays [34], [35], [36], [37]. It has been noted that due to the vast applications of dynamic systems with time delays, delay-dependent strategy and delay-independent approach under time-varying delays, uncertainties and disturbances are employed to the stability analysis [10], [11], [12], [13], [14], [15], [16], [17], [18], [19]. The delay-dependent strategies generally received much more attentions for its less conservatism than delay-independent ones, especially for the dynamic systems with a small delay range.
It is well known that the state-feedback control design for continuous-time dynamic systems with time-varying delays has been a difficult issue of control theory, it is to design a controller such that the state of the dynamic system stays a prescribed reference signal, which has been paid much attentions for its extensively applications in many areas such as engineer and finance. Nowadays, some important results of this issue have been made in recent years [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31]. It is worth noting that the systems in the above literature, there still exists some conservatism for continuous-time dynamic systems with state-feedback control and time-varying delays waiting for the further improvements. In order to obtain less conservative stability conditions, augment Lyapunov–Krasovskii functionals with the triple integral terms were employed [22], [23], [24], [25], [26]. It should be noted that, the term always bounded as , for example, the result in [23], [24] is conservatism with time delay τ is approximated with . Accordingly, with convex polyhedron method [17], this approximation was tackled as and . Despite these efforts, some conservatism still exist with middle stage of the derivation. To the best of authors׳ knowledge, few integral inequalities are directly available on lower bound of Lyapunov–Krasovskii functional. Therefore, aforementioned works still leave plenty of room for improvement.
Motivated by the above discussions, the issue of delay-derivative-dependent state-feedback control is designed for a class of continuous-time system with time-varying delays. New criteria with reduced conservatism are presented by employing an improved reciprocally convex combination method and an augment Lyapunov–Krasovskii functional. The obtained results are expressed in terms of linear matrix inequalities (LMIs), which can be easily tested by recently developed algorithms solving LMIs. Finally, two numerical examples are also presented including liquid monopropellant rocket motor system to illustrate the effectiveness and the advantage of obtained results.
Section snippets
Preliminaries
Consider a class of continuous-time system with time-varying delay as follows:where is the state vector, is the control input, is the disturbance input. is the signal to be estimated; is the initial condition and continuously differentiable. Ai, Bi, Ci and Di are known real constant matrices with appropriate dimensions. is the time-varying function
Main results
Theorem 3.1 Given the constant scalars , , the continuous-time system (1) is asymptotically stable if there exist mode-dependent symmetric matrices , , , i=1,2, , and any appropriately dimensioned matrices such that the following matrix inequalities hold: where
Illustrative example
Example 1 Consider the system 1 with the following parameter:and . For given the time-varying delay , it yields that , by using and solving LMIs in Theorem 3.2, one has the control feedback controller and maximum admissible upper bound τM as shown in Tables 1 and 2. It can be seen from Table 1, Table 2 that, a smaller values of γ is obtained. Therefore, a better disturbance
Conclusions
The problem of delay-derivative-dependent state-feedback control for a class of continuous-time system with time-varying delays is concerned in this paper. An augment Lyapunov–Krasovskii functional and an improved reciprocally convex combination used to expand the region of solve solution, and derived an improved sufficient condition. At last, two examples including liquid monopropellant rocket motor system have been given to demonstrate the effectiveness and the less conservatism of the
Acknowledgement
This work was supported by the National Basic Research Program of China (No. 61473061,11461082), the Natural Science Foundation of Hubei Province(No.2015CFC880) and the Open Project of the Key Laboratory of Biological Resources Protection and Utilization of Hubei Province(PKLHB1331).
Jun Cheng received B.S. degree from Hubei University for Nationalities, and Ph.D. Degree from University of Electronic Science and Technology of China. He is currently a staff with Hubei University for Nationalities. His research interests include Markovian jump systems, switched systems, neural networks and time-delay systems. He is a very active reviewer for many international journals.
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Jun Cheng received B.S. degree from Hubei University for Nationalities, and Ph.D. Degree from University of Electronic Science and Technology of China. He is currently a staff with Hubei University for Nationalities. His research interests include Markovian jump systems, switched systems, neural networks and time-delay systems. He is a very active reviewer for many international journals.
Hailing Wang, was born on November 26, 1980. She received her Ph.D. from City University of Hong Kong in 2012. Her research interests include dynamical systems, nonlinear dynamics, limit cycles, bifurcation and chaos. She is an associate professor of School of Science, Hubei University for Nationalities, on December 2013 to present. She has reviewed for many Journals, such as Applied Mathematical Modelling, Applied Mathematics and Computation, Ecological Modelling, Physics Letters A, International Journal of Bifurcation and Chaos, Journal of Statistical Mechanics: Theory and Experiment, Journal of Biological Systems, Discrete Dynamics in Nature and Society, Chaos, Solitons & Fractals, Abstract and Applied Analysis.
Shiqiang Chen is a professor in School of Science, Hubei University for Nationalities. He received the M.S. degree in Computer Software and Theory from Institute of Computer Software, Guizhou University, China, in 2005. His research interests mainly focus on network measurement and monitoring, network communication, internet of things and network security technology.
Zhijun Liu was born on October 8, 1974. He received his Ph.D. degree in computational mathematics, Dalian University for Technology in January 2007. He is a professor of School of Science, Hubei University for Nationalities, on December 2011 to present. His main research interests include the theory and its application of nonlinear dynamical systems. He is an Associate Editor for Journal of Applied Mathematics, The Scientific World Journal, and a reviewer of American Mathematical Society׳s Mathematics Reviews.
Jun Yang received the B.S. degree from Leshan Normal University, Leshan, China, in 2004 and the Ph.D. degree from University of Electronic Science and Technology of China, Chengdu, China, in 2009, all in Applied Mathematics. He is currently a Associate Professor with Civil Aviation Flight University of China, Guanghan, China. His current research interests include system and control theory, sampled-data control, fuzzy control systems and functional differential equations.