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# Alternate control systems

- Yuming Feng
^{1, 2}, - Chuandong Li
^{1}Email author, - Tingwen Huang
^{3}and - Weijing Zhao
^{4}

**2014**:305

https://doi.org/10.1186/1687-1847-2014-305

© Feng et al.; licensee Springer. 2014

**Received:**9 August 2014**Accepted:**19 November 2014**Published:**3 December 2014

## Abstract

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.

## Keywords

- alternate control system
- exponential stabilization
- Chua’s oscillator

## 1 Introduction

*etc.*Intermittent control methods are special cases of switching control methods and have been studied by many researchers,

*e.g.*, [10–15]. Within the intermittent control, one adds a continuous control during the first part of the period while in the other part of the period there is no control. This method is available for some cases, but it costs time. For other cases in which the time is very important, this method is not of use. So we advise to add two different controls alternately to the system. We call this system alternate control system. Figure 1 and Figure 2 show the working principles of intermittent control system and alternate control system, from which we conclude that alternate control system is a generalization of intermittent one.

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 $x\in {R}^{n}$ presents state vector, $f:{R}^{n}\to {R}^{n}$ is a continuous nonlinear function satisfying $f(0)=0$ and there exists a diagonal matrix $L=diag({l}_{1},{l}_{2},\dots ,{l}_{n})\ge 0$ such that ${\parallel f(x)\parallel}^{2}\le {x}^{T}Lx$ for any $x\in {R}^{n}$, $A\in {R}^{n\times n}$ is constant matrix, $u(t)$ denotes the external input of system (1).

where ${K}_{1},{K}_{2}\in {R}^{n\times n}$ are constant matrices, $T>0$ denotes the control period, $\tau \in (0,T)$ is a constant.

Our target is to design suitable *T*, *τ*, ${K}_{1}$, and ${K}_{2}$ 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 ${K}_{2}(t)=0$, the alternate control system (3) becomes the classical intermittent control system [10].

In the sequel, we will use the following two lemmas.

**Lemma 1** (Sanchez and Perez [16])

*Given any real matrices*${\mathrm{\Sigma}}_{1}$, ${\mathrm{\Sigma}}_{2}$, ${\mathrm{\Sigma}}_{3}$

*of appropriate dimensions and a scalar*$\u03f5\ge 0$

*such that*$0<{\mathrm{\Sigma}}_{3}={\mathrm{\Sigma}}_{3}^{T}$,

*the following inequality holds*:

**Lemma 2** (Boyd *et al.* [17])

*The LMI*

*where*$Q(x)={Q}^{T}(x)$, $R(x)={R}^{T}(x)$,

*and*$S(x)$

*depend affinely on*

*x*,

*is equivalent to*

Throughout this paper, we use ${P}^{T}$, ${\lambda}_{M}(P)$, and ${\lambda}_{m}(P)$ to denote the transpose, the maximum eigenvalue and the minimum eigenvalue of a square matrix *P*, respectively. $\parallel x\parallel $ is used to denote the Euclidean norm of the vector *x*. The matrix norm $\parallel \cdot \parallel $ is also referred to the Euclidean norm. We use $P>0$ (<0, ≤0, ≥0) to denote a symmetrical positive (negative, semi-negative, semi-positive) definite matrix *P*. $f(x({t}_{1}^{-}))$ is defined by $f(x({t}_{1}^{-}))={lim}_{t\to {t}_{1}^{-}}f(x(t))$.

## 3 Main results

**Theorem 1**

*If there exist a symmetric and positive definite matrix*$P\in {R}^{n\times n}$,

*positive scalar constants*${g}_{1}>0$, ${\u03f5}_{1}>0$, ${\u03f5}_{2}>0$,

*and scalar constant*${g}_{2}\in R$

*such that the following hold*:

- (1)
$PA+{A}^{T}P+P{K}_{1}+{K}_{1}^{T}P+{\u03f5}_{1}{P}^{2}+{\u03f5}_{1}^{-1}L+{g}_{1}P\le 0$,

- (2)
$PA+{A}^{T}P+P{K}_{2}+{K}_{2}^{T}P+{\u03f5}_{2}{P}^{2}+{\u03f5}_{2}^{-1}L-{g}_{2}P\le 0$,

- (3)
${g}_{1}\tau -{g}_{2}(T-\tau )>0$,

*then the origin of the system*(3)

*is exponentially stable*,

*and*

*where* $\gamma =\frac{{g}_{1}\tau -{g}_{2}(T-\tau )}{2T}$, *for any* $t>0$.

*Proof*Let us construct the following Lyapunov function:

It follows from (7) and (8) that:

By induction, we have:

*i.e.*, $\frac{t-\tau}{T}<m\le \frac{t}{T}$, then we have

*i.e.*, $\frac{t}{T}<m+1\le \frac{t+T-\tau}{T}$, then we have that

where $mT\le t<mT+\tau $.

From (10) we know that

where $mT+\tau \le t<(m+1)T$.

where $\gamma =\frac{{g}_{1}\tau -{g}_{2}(T-\tau )}{2T}$, for any $t>0$.

So we finish the proof. □

## 4 Numerical example

*α*and

*β*are parameters and $g(x)$ is the piecewise linear characteristics of Chua’s diode, which is defined by

where $a<b<0$ are two constants.

Thus we can choose $L=diag({\alpha}^{2}{(a-b)}^{2},0,0)$.

## 5 Conclusions

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.

## Authors’ information

The second author is a Senior Member, IEEE.

## Declarations

### 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), the Fundamental Research Funds for the Central Universities (grant No. XDJK2015D004) and Key Program of Chongqing Three Gorges University (grant No. 14ZD18).

## Authors’ Affiliations

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This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.