#### Abstract

Chaos control has become an important area of research and consequently many approaches have been proposed to control chaos. This paper proposes a linear regulation method. Different from the existing approaches is that it can provide region of attraction while estimating the bounding behaviour of the norm of the states. The proposed method also possesses design flexibility and can be easily used to cater for special requirement such that control signal should be generated via single input, single state, static feedback and so forth. The applications to the Tigan system, the Genesio chaotic system, the novel chaotic system, and the Lorenz chaotic system justify the above claims.

#### 1. Introduction

Chaotic phenomena have been gradually recognized as one of the inherent properties of nonlinear dynamical systems since the work of Poincaré, Lorenz, Mandelbrot, and so forth. Indeed, it is an astounding fact that chaos theory has been applied in practically all the scientific disciplines. The wide-spread recognition of chaos has henceforth sparked extensive research interests in its control. Chaos control has received much attention since the well-known OGY method [1] and the Pyragas method [2]. Indeed the number of publications devoted to chaos control is huge; for example, around 700 references are compiled by Fradkov to the papers published in peer reviewed journals only in 1997–2000 [3]. As a consequence, chaos control has become an important research domain in the theory of nonlinear dynamical systems (e.g., see monographs [4–8]).

Many techniques have been devised for controlling chaos via, for example, nonlinear and robust control, sliding mode control, adaptive control, partial control, control by weak signals, and finite time control ([9–16] and the references therein). In these approaches, the unstable periodic orbits are determined and a control signal is then generated which will stabilize the chaotic system to an equilibrium, locally or globally. In this paper, yet another method is proposed to stabilize a chaotic system. This approach possesses the following two appealing features:(1)control signal is linear with constant gain matrix. In specific, , where is constant;(2)for a particular design , the bounding behaviour of is explicitly determined.Feature (1) means control through static state feedback; therefore, this can simplify the implementation compared with adaptive approaches; feature (2), however, is even desirable as it implies that the bounding evolution of can be estimated* a priori*. Therefore, it is expected that both the domain of attraction and the rate of convergence of can be tuned explicitly. These results form the contributions of the paper. The roadmap is as follows: Section 2 formulates the problem to be addressed and states the main results; Section 3 provides several numerical examples to validate the results. And finally Section 4 concludes the paper.

#### 2. Linear Regulation of Chaos

Consider a chaotic system . After a possibly coordinate transformation, it is brought into the following representation:where is a constant matrix; is a locally integrable nonlinear function that is not necessarily Lipchitz. Many control design methods can be utilized to regulate the chaotic system. Here one seeks a particularly simple, linear state feedback control law:where is a constant matrix. By assuming that all system states are measurable, the problem is to find a constant gain , such that the chaotic systemtogether with initial condition is stabilized and the bounding behaviour of is determined. Before presenting the main result, the following assumption is made.

*Assumption 1. *For nonlinear function , there exists an integer , such that

This assumption essentially says that is bounded and Lebesgue measurable, henceforth not necessarily Lipschitzian. The main result can now be stated as follows.

Theorem 2 (chaos control by linear regulation with bound estimation). *With the above assumption, the system (1) controlled by the state feedback control is exponentially stable if is designed such that all the eigenvalues of matrix have a strictly negative real part with the initial condition which satisfies**where the constants and are specified by**Furthermore, the state is explicitly bounded by*

*Proof. *See the Appendix.

Several observations follow immediately from the above important result:(1)Firstly, for a particular design , the chaotic system can be exponentially stabilized for any initial condition within .(2)Secondly, a design determines the constants and , and this subsequently determines the region of attraction as well as the bounding properties of in terms of (7). Henceforth, the evolution of and allowable initial conditions can be determined* a priori*.(3)Thirdly, if a large region of attraction and a fast roll-off rate of are desired, one can choose the static gain making the real part of small enough, and this may lead to high gain control, but a “wise” combination of states feedback can provide much improved performance. This will be illustrated in the next section.(4)Fourthly, the choice of static gain can be of importance for implementation; for example, a choice implies static state feedback, while implies single input, single state, static feedback control. This issue will be further discussed in the numerical studies in the next section.(5)Finally, although not shown here, it can indeed be shown that the proposed design possesses robustness to modelling uncertainty. Hence exponential stability of chaotic systems can be guaranteed with bounded disturbances.

#### 3. Linear Regulation of Chaos: Robustness Issue

In the above analysis, linear regulation of chaotic systems is investigated. To have a complete discussion on a design methodology, robustness issue must be considered. In this section, additive uncertainty associated with both state matrix and nonlinear function is analyzed.

To proceed, consider the following uncertain system:where and are the additive uncertainty associated with state matrix and nonlinear function, respectively. To look, in detail, at the effect of uncertainty on design, suppose the following condition holds:As is uncertainty associated with , hence it should have . Considering and the assumption (ii) , it is reasonable to assume , while . Therefore, inequality (9) can now be written asIt is seen immediately from Theorem 2 that, for a stabilizing design , the feasible initial condition now becomesAnd the state is now bounded byA consideration of gives the following result.

Theorem 3 (robust regulation of chaos). *For a stabilizing design , the existence of uncertainty reduces the radius of feasible initial conditions and causes an even conservative estimation of the bounding property for ; in specific, the initial conditions now satisfy and the state is bounded by*

*Proof. *A collection of results leading to (8)–(12) gives the desired solution.

*Remark 4. *The existence of uncertainty leads to conservative estimation and increases of uncertainty bound lead to decreased stability margin. This result is thus compatible with the small gain theorem.

#### 4. Linear Regulation of Chaos: Numerical Study

##### 4.1. Tigan System

First consider the newly discovered three-dimensional chaotic attractor, the Tigan system [17], given bywith , , and . Write (14) intowhere and . From , one can choose and . Then notice that has an unstable eigenvalue at , and we will now design a stabilizer . To make have eigenvalues with strictly negative real parts is easy, but for implementation, one should choose to be simple; for example, , that is, , a single input, single state, static feedback control, which is the simplest of all possible control actions. Now the eigenvalues of the controlled state equation are prescribed at , , , and then from one can designate and . Finally the range of the initial conditions can be determined from inequality (5), ; that is, . With initial condition (thus ), the transient behaviour of is shown in Figure 1, where it is seen clearly that all states are exponentially stabilized with the simple linear feedback control. The corresponding evolution of is shown in Figure 2, and also the theoretical bound computed from (7) is shown, cross validating the theoretical results presented in the main theorem.

Now two very important observations follow.

*(1) Observation I: Enlargement of Domain of Attraction*. From Figure 2, it is seen that the estimation of the bound is very conservative, and it is wondered whether the region of attraction is too restricted. This is indeed the case! Take, for example, (thus ), for the same controller, and the transient behaviour of is shown in Figure 3. It is seen clearly that even though the initial condition is far outside of the allowable initial conditions, the same controller still exponentially stabilizes the chaotic system.

*(2) Observation II: Design Freedom for States Regulation and Detuning*. It is also seen that the above control , although it is the simplest of all possible control actions, results in restricted domain of attraction for initial conditions and “sluggish” regulation of states. Although it has been shown that domain of attraction for initial conditions can be extended far outside, the transient response of should still be improved. To resolve the problem, it is expected that one should choose even aggressive control action, for example, high gain control. However, as remarked in the last section, a wise combination of states feedback can result in improved performance; for example, choose ; the chaotic system is exponentially stabilized within 0.4 s, compared with 5 s using ; see Figure 4.

##### 4.2. Genesio Chaotic System

Now consider the Genesio chaotic system [18] represented bywith , , . That is, and .

Therefore, from , we can choose and . It is also seen that there are unstable eigenvalues at , and we will now design a stabilizer . Here it is remarkable to note that a very simple negative feedback will exponentially stabilize the chaotic system. Again, although the range of initial conditions ( and ) appears to be restricted, it can be significantly extended. For example, a simulation with initial condition and control action shows exponential stability of the chaotic system; see Figure 5(a). From Figure 5(a), the only concern is the transient performance, and as remarked in the above subsection, a good choice of states feedback can significantly improve the regulation performance. Take, for example, ; the situation is shown in Figure 5(b). It is seen that the chaotic system is regulated within 0.4 s, comparing with 20 s using .

**(a)**

**(b)**

##### 4.3. Novel Chaotic System

Now consider the novel chaotic system [19]:with , , , , and . That is, and . The system has five equilibrium points that are all unstable saddle focus nodes. But still, it is found that, again, a single input, single state, static feedback can achieve exponential stability of the chaotic system, as illustrated in Figure 6(a). The transient performance can be further improved by using two control inputs , , and ; that is, . See Figure 6(b).

**(a)**

**(b)**

##### 4.4. Lorentz Chaotic System

Let us take yet another example, the unified Lorentz chaotic system [20]:where . The system reduces to the general Lorenz system for ; it is the general Lü system for and the general Chen system for . In this study, we take and it is the Lorentz chaotic system with and . As again, a single input, single state, static feedback can exponentially stabilize the Lorenz chaotic system, as illustrated in Figure 7(a). The transient performance can be further improved by using two control inputs, for example, , , and . But here, to demonstrate the flexibility of the proposed design approach, it is insisted that single input should be used; for example, while and . Figure 7(b) illustrates the situation and it is seen that although the transient performance is not improved, exponential stability is achieved through only single input.

**(a)**

**(b)**

#### 5. Conclusion

Linear regulation of chaos has been introduced in this paper. It has been shown that the proposed design method can provide region of attraction while simultaneously obtaining the bounding behaviour of the norm of the states. The proposed method has also demonstrated its flexibility in design freedom through its application to four types of chaotic systems, namely, the Tigan system, the Genesio chaotic system, the novel chaotic system, and the Lorenz chaotic system. The design flexibility has been clearly revealed by the following facts: (1) all the chaotic systems are exponentially stabilized through single input, single state, static feedback control; (2) transient performance can be improved through appropriate combination of states feedback; and (3) single input, static feedback control is still achievable for exponential stability.

#### Appendix

#### Proof of the Main Theorem

The proof of the result utilizes the following lemmas.

Lemma A.1 (Gronwall-Bellman lemma [21–23]). *Let the following conditions hold:*(i)*, , and () are locally integrable, where , and ;*(ii)* is locally integrable on .**If satisfies**then*

Lemma A.2 (generalized Gronwall-Bellman lemma [24, 25]). *Let*(i)* with and ; further define an integer ;*(ii)*: an integrable function with , , ;*(iii)*an essentially bounded function such that**Then if the following inequality holds:**then one has*

Lemma A.1 is the standard Gronwall-Bellman lemma and it has long been used for the stabilization of nonlinear systems and nonlinear observer design (see [21–23, 26], e.g.). However, for the purpose of the current paper, the original Gronwall-Bellman lemma must be further generalized. One of the most important generalizations to the discussion of this paper, Lemma A.2, is due to Pachpatte [27, 28], El Alami [24], and, more recently, N’Doye et al. [25]. This result will be used in the following for the proof of the main result below.

Now consider the dynamical system (1); with condition (i) in Lemma A.1 the differential equation can be integrated to beHenceorHere it is time to invoke Lemma A.2, where it can be easily checked that the conditions for the lemma to hold are fulfilled. Hence, provided thatthe behaviour of is bounded byInequality (A.9) can be further reduced towhile (A.10) leads toA consideration of the inequality gives the desired results.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work is supported by the Natural Science Foundation of Jiangsu Province (no. BK20140829) and Jiangsu Postdoctoral Science Foundation (no. 1401017B).