- 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.
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.
2 Problem formulation and preliminaries
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 .
3 Main results
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. □
4 Numerical example
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).
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