- Open Access
Alternate control systems
© Feng et al.; licensee Springer. 2014
- Received: 9 August 2014
- Accepted: 19 November 2014
- Published: 3 December 2014
The exponential stability of a class of nonlinear systems by means of alternate control is studied. An exponential stability criterion is given in terms of a set of linear matrix inequalities. Numerical simulations are presented to verify the correction of the obtained results.
- alternate control system
- exponential stabilization
- Chua’s oscillator
In this paper, we first investigate the stability of the alternate control system, then by using the stability criterion obtained we study the stability of Chua’s oscillator. Also, numerical simulations are illustrated to show the effectiveness of the results.
The rest of the paper is organized as follows. In Section 2, we formulate the problem of alternate control system and introduce some notations and lemmas. We then establish, in Section 3, an exponential stability criterion. In Section 4, we discuss the alternate control of Chua’s oscillator. Lastly, we conclude the paper.
where presents state vector, is a continuous nonlinear function satisfying and there exists a diagonal matrix such that for any , is constant matrix, denotes the external input of system (1).
where are constant matrices, denotes the control period, is a constant.
Our target is to design suitable T, τ, , and such that the system (1) can be stabilized at the origin.
It is obvious that the system (3) is a classical switched system where the switching rule only depends on the time.
Remark 1 When , the alternate control system (3) becomes the classical intermittent control system .
In the sequel, we will use the following two lemmas.
Lemma 1 (Sanchez and Perez )
Lemma 2 (Boyd et al. )
Throughout this paper, we use , , and to denote the transpose, the maximum eigenvalue and the minimum eigenvalue of a square matrix P, respectively. is used to denote the Euclidean norm of the vector x. The matrix norm is also referred to the Euclidean norm. We use (<0, ≤0, ≥0) to denote a symmetrical positive (negative, semi-negative, semi-positive) definite matrix P. is defined by .
where , for any .
It follows from (7) and (8) that:
By induction, we have:
From (10) we know that
where , for any .
So we finish the proof. □
where are two constants.
Thus we can choose .
This paper gives a new model of control system, namely alternate control system. A stability criterion is given in terms of linear matrix inequalities. By the new method, the chaotic Chua circuit is controlled.
Obviously, there is no rest time in an alternate control system. By comparing our model with the traditional intermittent control system, we know that our model is a generalization of intermittent control system. The proposed method can be applied to linear and nonlinear systems.
This paper considers systems without delay. For delayed systems [19–21], we know that the methods used to deal with them are different from ones of the systems without delay. We are ready to focus on this aspect in future papers.
The second author is a Senior Member, IEEE.
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), the Fundamental Research Funds for the Central Universities (grant No. XDJK2015D004) and Key Program of Chongqing Three Gorges University (grant No. 14ZD18).
- Yang T: Impulsive Control Theory. Springer, Berlin; 2001.Google Scholar
- Yang T: Impulsive control. IEEE Trans. Autom. Control 1999, 44(5):1081-1083. 10.1109/9.763234View ArticleGoogle Scholar
- Yang T, Chua LO: Impulsive stabilization for control and synchronization of chaotic systems: theory and application to secure communication. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 1997, 44(10):976-988. 10.1109/81.633887MathSciNetView ArticleGoogle Scholar
- Yang Z, Xu D: Stability analysis and design of impulsive control systems with time delay. IEEE Trans. Autom. Control 2007, 52(8):1448-1454.View ArticleGoogle Scholar
- Allerhand LI, Shaked U: Robust state-dependent switching of linear systems with dwell time. IEEE Trans. Autom. Control 2013, 58(4):994-1001.MathSciNetView ArticleGoogle Scholar
- Geromel JC, Deaecto GS, Daafouz J: Suboptimal switching control consistency analysis for switched linear systems. IEEE Trans. Autom. Control 2013, 58(7):1857-1861.MathSciNetView ArticleGoogle Scholar
- Heertjes MF, Sahin IH, van de Wouw N, Heemels WPMH: Switching control in vibration isolation systems. IEEE Trans. Control Syst. Technol. 2013, 21(3):626-635.View ArticleGoogle Scholar
- Tanwani A, Shim H, Liberzon D: Observability for switched linear systems: characterization and observer design. IEEE Trans. Autom. Control 2013, 58(4):891-904.MathSciNetView ArticleGoogle Scholar
- Li C, Huang T, Chen G: Exponential stability of time-controlled switching systems with time delay. J. Franklin Inst. 2012, 349(1):216-233. 10.1016/j.jfranklin.2011.10.016MathSciNetView ArticleGoogle Scholar
- Li C, Feng G, Liao X: Stabilization of nonlinear systems via periodically intermittent control. IEEE Trans. Circuits Syst. II, Express Briefs 2007, 54(11):1019-1023.View ArticleGoogle Scholar
- Zochowski M: Intermittent dynamical control. Physica D 2000, 145: 181-190. 10.1016/S0167-2789(00)00112-3View ArticleGoogle Scholar
- Li N, Cao J: Periodically intermittent control on robust exponential synchronization for switched interval coupled networks. Neurocomputing 2014, 131: 52-58.View ArticleGoogle Scholar
- Huang J, Li C, He X: Stabilization of a memristor-based chaotic system by intermittent control and fuzzy processing. Int. J. Control. Autom. Syst. 2013, 11(3):643-647. 10.1007/s12555-012-9323-xView ArticleGoogle Scholar
- Huang, J, Li, C, Han, Q: Quasi-synchronization of chaotic neural networks with parameter mismatch by periodically intermittent control. In: Proceeding of CSIE 2009, March 31 - April 2, Los Angeles, California, USA, 7 volumes (2009)Google Scholar
- Huang T, Li C, Liu X: Synchronization of chaotic systems with delay using intermittent linear state feedback. Chaos 2008., 18: Article ID 033122Google Scholar
- Sanchez EN, Perez JP: Input-to-state stability (ISS) analysis for dynamic NN. IEEE Trans. Circuits Syst. I, Regul. Pap. 1999, 46(11):1395-1398. 10.1109/81.802844MathSciNetView ArticleGoogle Scholar
- Boyd S, Ghaoui L, Feron EEI, Balakrishnan V: Linear Matrix Inequalities in System and Control Theory. SIAM, Philadephia; 1994.View ArticleGoogle Scholar
- Shilnikov L: Chau’s circuit: rigorous results and future problems. Int. J. Bifurc. Chaos 1994, 4(3):489-519. 10.1142/S021812749400037XMathSciNetView ArticleGoogle Scholar
- Xia W, Cao J: Pinning synchronization of delayed dynamical networks via periodically intermittent control. Chaos 2009., 19(1): Article ID 013120 10.1063/1.3071933Google Scholar
- Yang X, Cao J: Stochastic synchronization of coupled neural networks with intermittent control. Phys. Lett. A 2009, 373(36):3259-3272. 10.1016/j.physleta.2009.07.013MathSciNetView ArticleGoogle Scholar
- Zheng G, Cao J: Robust synchronization of coupled neural networks with mixed delays and uncertain parameters by intermittent pinning control. Neurocomputing 2014, 141: 153-159.View ArticleGoogle Scholar
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