Periodically multiple state-jumps impulsive control systems with impulse time windows
Introduction
Impulsive control is a control paradigm based on impulsive differential equations. An introduction of impulsive control theory can be found in [1], [2]. In an impulsive control system, the plant should have at least one “impulsively” changeable state variable. An impulsive differential equation can be given as where is the state variable, y is the output, is the impulsive control law. This kind of impulsive differential equations can describe systems where the impulse occur at the fixed time.
Impulsive control theory has wild applications, e.g., it can be used in HIV prevention model [3], [4], in pest control model, in nanoelectronics [1], etc. It also plays a very important rule in stabilizing chaotic systems [5], [6], [7], [8].
But in the previous literature of impulsive control (e.g. [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]). The assumption of the occurrence of impulses is fixed or the occurrence can be calculated is popular. We know that any machine/computer cannot put impulses without any error, so the expected times and the actual ones cannot always be the same. For example, we plan to add an input of impulse at time t, our machine/computer may add the impulse in a short time window , where α is a small positive number. This time error is called impulsive time windows and wildly exists in our society. So it is very meaning to study impulsive control systems with impulse time windows. Some related works can be found in [26], [27], [28], [29], [30], [31], [32], [33], [34], [35]. In [32] the authors have proposed impulsive controller with average impulsive interval. Compared with results of [32], our “impulse time windows” removes some restrictions, so our model is a more general one. Fig. 1 shows the distribution diagram of the occurrence of impulses in an impulsive system with impulse time windows. An example of impulse time windows is that a doctor tells an patient who needs injections: “Everyday at 5 to 6 o׳clock in the afternoon come here to take an injection.” This “5 to 6 o׳clock” is an impulse time windows.
In this paper, we first choose a period T, then in the T we input one or two or three impulsive time windows, within which impulses occur, but the exact occurrence time is unknown. With the time moves on, T moves with the time, thus we realize the periodical control of the system.
The rest of the paper is organized as follows. In Section 2, we formulate the problem and introduce some notations and lemmas. We then establish, in Section 3, the main results of the paper. In Section 4, we give a numerical example. Lastly, we conclude the paper.
Section snippets
Problem formulation and preliminaries
Consider a class of nonlinear systems described bywhere presents state vector, is a continuous nonlinear function satisfying and there exists a diagonal matrix such that for any is constant matrix, u(t) denotes the external input of system (1).
In the sequel, we will use the following two lemmas. Lemma 1 Given any real matrices of appropriate dimensions and a scalar such that , theSanchez and Perez [23]
Periodically single state-jumps impulsive control systems with impulse time windows
For stabilizing the origin of the system (1) by means of periodically single state-jumps impulsive control systems with impulse time windows, we mean that in each period we impose an impulse J in the time of , where is unknown and within impulse time window . Note that . Fig. 2 shows the distribution diagram of pulses׳ occurrences.
This method is also called single impulse control with impulse time windows.
So system (1) can be rewritten as follows:
Numerical example
The original and dimensionless form of Chua׳s oscillator [25] is given bywhere α and β are parameters and g(x) is the piecewise linear characteristics of Chua׳s diode, which is defined bywhere are two constants.
In this section, we set the system parameters as and , which make Chua׳s circuit (19) chaotic [25]. Fig. 6 shows the chaotic phenomenon of Chua׳s oscillator with the
Conclusions
This paper studies periodically single (double, triple, multiple, respectively) state-jumps impulsive control systems with impulse time windows. The new model of such control systems is set up. Stability criteria are given in terms of LMIs. By the results presented in this paper, the chaotic Chua׳s circuit is controlled. The proposed method can be applied to linear and nonlinear systems.
Competing interests
The authors declare that they have no competing interests.
Authors׳ contributions
C. Li has proposed the main ideal of paper. Y. Feng has proved the main theory and prepared the paper with latex. T. Huang has provided all the figures of the paper. All authors read and approved the final manuscript.
Acknowledgements
This research is supported by the Natural Science Foundation of China (Grant no: 61374078), NPRP Grant # NPRP 4-1162-1-181 from the Qatar National Research Fund (a member of Qatar Foundation), Scientific & Technological Research Foundation of Chongqing Municipal Education Commission (Grant nos. KJ1401006, KJ1401019) and the Fundamental Research Funds for the Central Universities (Grant no. XDJK2015D004).
Yuming Feng received the B.S. degree and M.S. degree in mathematics from Yunnan University, Kunming, China, in 2003 and in 2006, respectively. From January 2012 to October 2012, he has been serving as a Research Scholar in Udine University, Udine, Italy. From January 2014 to April 2014, he has been serving as a Research Scholar in Texas A&M University at Qatar, Doha, Qatar. Since November 2013, he has been an Associated Professor with Chongqing Three Gorges University. Now he is a Ph.D. student
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Yuming Feng received the B.S. degree and M.S. degree in mathematics from Yunnan University, Kunming, China, in 2003 and in 2006, respectively. From January 2012 to October 2012, he has been serving as a Research Scholar in Udine University, Udine, Italy. From January 2014 to April 2014, he has been serving as a Research Scholar in Texas A&M University at Qatar, Doha, Qatar. Since November 2013, he has been an Associated Professor with Chongqing Three Gorges University. Now he is a Ph.D. student in Southwest University. His current research interest covers impulsive control theory, neural networks, chaos control and synchronization, etc.
Chuandong Li received the B.S. degree in applied mathematics from Sichuan University, Chengdu, China, in 1992 and the M.S. degree in operational research and control theory and the Ph.D. degree in computer software and theory from Chongqing University, Chongqing, China, in 2001 and 2005, respectively. Since 2007, he has been a Professor with Chongqing University. Since November 2006, he has been serving as a Research Fellow with the Department of Manufacturing Engineering and Engineering Management, City University of Hong Kong, Kowloon, Hong Kong, where he will be until November 2008. His current research interest covers iterative learning control of time-delay systems, neural networks, chaos control and synchronization, and impulsive dynamical systems.
Tingwen Huang received the B.S. degree in mathematics from Southwest Normal University, Chongqing, China, in 1990, the M.S. degree in applied mathematics from Sichuan University, Chengdu, China, in 1993, and Ph.D. degree in mathematics from Texas A&M University, College Station, in 2002. From 1994 to 1998, he was a Lecturer with Jiangsu University, Zhenjiang, China. From January to July 2003, he was a Visiting Assistant Professor with Texas A&M University. He is currently with Texas A&M University at Qatar, Doha, Qatar, where he was an Assistant Professor from August 2003 to June 2009 and has been an Associate Professor since July 2009. His research areas include neural networks, complex networks, chaos and dynamics of systems, and operator semigroups and their applications.