Control and Synchronization of Fractional-Order Financial System Based on Linear Control

Control and synchronization of the financial systems with fractional-order are discussed in this paper. Based on the stability theory of fractional-order differential equations, Routh-Hurwitz stability condition, and by using linear control, simpler controllers are designed to achieve control and synchronization of the fractional-order financial systems. The proposed controllers are linear and easy to implement, which have improved the existing results. Theoretical analysis and numerical simulations are shown to demonstrate the validity and feasibility of the proposed method.


Introduction
Chaos, as a very interesting nonlinear phenomenon, has been intensively investigated in many fields over the last four decades.Since the pioneering work 1 that chaotic dynamics could exist in an economical model, research on the dynamical behavior of economical model has become one of the most interesting and important topics which have received increasing attention.Many continuous chaotic models have been proposed to study complex economic dynamics, such as the forced Vander-Pol model 2 , the IS-ML model 3 , Behrens-Feichtinger model 4 , and Cournot-Puu model 5 .Just as all the other chaotic systems in engineering, financial chaotic system has complex dynamical behaviors and possess some special features, such as excessive sensitivity to initial conditions, the complex patterns of phase portraits, positive Lyapunov exponents, and bounded and fractal properties of the motion in the phase space.These features are inherent properties of the system itself, rather than caused by external disturbances, which denote some economic behaviors in the fields of finance, stocks, and social economics.In addition, as a nonlinear system, there exist many attractors with control used for discussing fractional-order financial systems in 22, 23 , the linear control is economic and easy to implement, through which control and synchronization of fractionalorder financial systems will be obtained only by choosing suitable feedback gains.The main job of this paper lies in two aspects.One is to remove chaotic phenomenon from fractional financial system by controlling, which makes prediction impossible in the financial world.The other is to realize harmonious and sustainable development between drive financial systems and response ones by investigating synchronization.The obtained results have a certain value to the theoretical guidance and application.
The remainder of this paper is organized as follows.In Section 2, preliminary results are presented and fractional-order financial system is described.In Section 3, some sufficient criteria for control of the fractional-order financial system are given.In Section 4, we discussed synchronization of the fractional-order financial system via linear error feedback.In Section 5, numerical simulations are given to illustrate the effectiveness of the main results.Finally, conclusions are drawn in Section 6.

Preliminaries and System Description
Fractional calculus is a generalization of integration and differentiation to a noninteger-order integrodifferential operator D α t defined by

2.1
Fractional calculus is being used in various fields gradually, such as biophysics, nonlinear dynamics, informatics, and control engineering 29 .There are some definitions for fractional derivatives.The commonly used definitions are Grunwald-Letnikov GL , Riemann-Liouville RL , and Caputo C definitions.
The Grunwald-Letnikov GL derivative with fractional-order α is given by where • means the integer part.The Riemann-Liouvill RL fractional derivatives are defined by where Γ • is the gamma function, Γ τ ∞ 0 t τ−1 e −t dt.The Caputo C fractional derivative is defined as follows: It should be noted that the advantage of Caputo approach is that the initial conditions for fractional differential equations with Caputo derivatives take on the same form as those for integer-order differential, which have well-understood physical meanings.Therefore, in the rest of this paper, the notation D α * is chosen as the Caputo fractional derivative operator The fractional-order chaotic financial system can be described by Consider the following fractional-order system: 2.7

3.1
It is easy to obtain that system 2.5 has three equilibrium points if

3.2
The Jacobian matrix of system 2.5 at equilibrium point x * , y * , z * is The characteristic equation of the Jacobian matrix J is given by
According to the above analysis, when the maximum fractional-order α among α 1 , α 2 , and α 3 is less than 0.8436, there exist two stable equilibrium points; when α 1 α 2 α 3 0.8436, system 2.5 will admit a limit cycle; when fractional-order α 1 , α 2 , and α 3 are all greater than 0.8436, there are no stable equilibria, with all the equilibrium points being unstable, which implies that there may exist chaos for system 2.5 .By calculating the values of Lyapunov exponents of system 2.5 , it could be found that system 2.5 exhibits chaotic behaviors if maximum fractional-order α among α 1 , α 2 , and α 3 is greater than or equal to 0.86 23 .

Chaos Control
In this subsection, linear state feedback controller is designed to control fractional-order chaotic financial system to its equilibrium.
The controlled fractional-order chaotic financial system is given by where k 1 , k 2 , and k 3 denote feedback gains and x * , y * , z * is the desired equilibrium point.
Obviously, system 3.7 has one equilibrium point x * , y * , z * .The Jacobian matrix of system 3.7 at equilibrium point x * , y * , z * is The characteristic equation of the Jacobian matrix 3.8 is 3.9 Our goal is to find suitable feedback gains such that all the state trajectories of system 3.7 are controlled to its equilibrium point, that is to say, roots of 3.9 should satisfy the conditions in the lemma.Theorem 3.1.When a 3, b 0.1, c 1, P 1 0, 10, 0 ; system 3.7 stabilizes to equilibrium point P 1 , if state feedback gains k 1 , k 2 , and k 3 satisfy the following conditions: Proof.Substituting the parameters a, b, and c into 3.9 , one obtains It is very easy to obtain the roots of 3.11 :

3.12
Note that λ 1 is a negative real number, λ 2,3 are a pair of conjugate imaginary roots, and the real parts of imaginary root are negative, that is, arg λ 1 π, arg λ 2,3 > π/2.Therefore, the trajectory of the controlled fractional-order system 3.7 is asymptotically stable at equilibrium point P 1 .
In Theorem 3.1, we designed three simple linear feedback controllers to ensure controlled system 3.7 stabilized to P 1 .In practice, two linear feedback controllers or single linear feedback controller in controlled system 3.7 will do the same thing.Then we have the following corollaries.Corollary 3.2.Controlled system 3.7 will approach asymptotically to P 1 with one of the following conditions about feedback gains:

3.13
Corollary 3.3.Controlled system 3.7 will approach asymptotically to P 1 if feedback gains satisfy and k 3 satisfy one of the following conditions, system 3.7 will approach and stabilizes to equilibrium point P 2 asymptotically: , k 2 0, k 3 0.

3.16
According to the Routh-Hurwitz criterion, real parts of these eigenvalues λ 1,2,3 of 3.16 are all negative if that is, k 2 < −0.9.That implies that the trajectory of the controlled fractional-order system 3.7 is asymptotically stable at equilibrium point P 2 .
2 Substituting the parameters a 3, b 0.1, c 1, and k 2 k 3 0 into 3.9 , one obtains By applying the Routh-Hurwitz criterion, if the following conditions for feedback gains are met, then real parts of these eigenvalues λ 1,2,3 are all negative.It follows that k 1 < 1/220 121 − √ 62161 .Thus, the trajectory of the controlled fractional-order system 3.7 is asymptotically stable at equilibrium point P 2 .
Remark 3.5.Actually, we adopt single linear feedback controller to stabilize P 2 of controlled system in Theorem 3.4, namely, stabilizing P 2 by adding single linear feedback controller on the first state or the second state, but we cannot do it via adding single linear feedback controller on the third state; the reasons are described as follows.
Substituting the parameters a 3, b 0.1, c 1, and k 1 k 2 0 into 3.9 , one obtains , and k 3 satisfy one of the following conditions, system 3.7 will approach and stabilize to equilibrium point P 2 asymptotically: , k 2 0, k 3 0.

3.22
Proof.The proof is the same as that of Theorem 3.4 and so we omit it here.

Chaos Synchronization
In this section, we will investigate synchronization of fractional-order financial system 2.5 .Three and two simple linear feedback controllers are designed to achieve synchronization, which simplify the existing synchronization schemes and reduce the synchronization cost.
Drive system and response system are described as follows, respectively: where u 1 , u 2 , and u 3 denote the external control inputs, to be designed later.It follows from systems 4.1 and 4.2 that the following error dynamical system is where e 1 x m − x s , e 2 y m − y s , and e 3 z m − z s .Our aim is to find suitable control laws u i i 1, 2, 3 for stabilizing the error dynamics system 4.3 .To this end, the following theorem is proposed.Theorem 4.1.For any given initial conditions, synchronization between systems 4.1 and 4.2 will occur if control schemes are defined as follows:

4.4
where k 1 , k 2 , and k 3 are feedback gains and satisfy the following conditions:

4.6
Error dynamical system 4.6 can be rewritten as the following matrix form: where Suppose that λ is one of the eigenvalues of matrix A and the corresponding nonei- Taking conjugate transposal on both sides of 4.9 , one obtains Aε H λε H . 4.10 Equation 4.9 multiplied left by 1/2ε H plus 4.10 multiplied right by 1/2ε, we have Because a chaotic system has bounded trajectories, there exists a positive constant L, such that |x| < L, |y| < L. Thus,

4.12
From 4.11 , we have

4.13
where It is obvious that real parts of all eigenvalues λ are negative and matrix P should be negative definite, namely, the following inequalities hold:

4.15
Simplifing the above inequalities, one has

4.16
Therefore, based on stability theorem of fractional-order systems, error system 4.6 is asymptotically stable at the origin, which implies that synchronization between systems 4.1 and 4.2 will be achieved.
Based on the above analysis, it is easy to obtain that two linear feedback controllers could also achieve synchronization between systems 4.1 and 4.2 .Then, we have the following corollary.Corollary 4.2.For any given initial condition, if control schemes are described as u 1 k 1 e 1 , u 3 k 3 e 3 and feedback gains satisfy .17 then the response system 4.2 can synchronize the drive system 4.1 .
Proof.The proof is similar to that of Theorem 4.1.After some computations, we have

4.19
Matrix P must be negative definite; if the following inequalities hold: Therefore, real parts of all eigenvalues λ are negative; according to the stability theorem of fractional-order systems, error system 4.6 is asymptotically stable.This means that the slave system 4.2 can asymptotically synchronize the master system 4.1 .Remark 4.3.References 22, 23 presented control of the model by using sliding mode control and nonlinear control, respectively, but with no consideration of synchronization.In contrast, control cost of linear control is low.
Remark 4.4.Response time of control and synchronization could be adjusted with suitable state feedback gains and error feedback gains in allowed limits.

Numerical Simulations
In this section, to verify theoretical results obtained in the previous section, the corresponding numerical simulations will be performed and an improved predictor-corrector algorithm is applied see the appendix .In all simulations, fractional-order α 1 , α 2 , and α 3 is chosen as 0.86, 0.92, and 0.95 to ensure the existence of chaos in system 2.5 .

Chaos Synchronization
When the parameters of system 2.5 are chosen as a 3, b 0.1, and c 1, select the initial values of the drive and the response systems as x m 0 , y m 0 , z m 0 4, −3, 2 and x s 0 , y s 0 , z s 0 −3, 2, 4 , respectively.For error feedback gain k 1 , k 2 , k 3 12, 9, −2 , simulation result of the synchronization between systems 4.1 and 4.2 is shown in Figure 7.The synchronization error states between systems 4.1 and 4.2 and displayed in Figure 8.When error feedback gains k 1 8, k 2 0, and k 3 −2, it can be seen that the derive system 4.1 and the response system 4.2 achieve the synchronization in Figure 9. Figure 10 displays the error state time response between systems 4.1 and 4.2 .

Conclusion
In this paper, based on the stability theory of fractional-order systems and Routh-Hurwitz stability condition, some sufficient conditions for control and synchronization of the fractional-order chaotic financial system by linear feedback control have been derived.Finally, numerical simulations are provided to verify the effectiveness of the results obtained.The results   Set h T/N, t n nh n 0, 1, 2, . . ., N .Then the above equation can be discretized as follows: h α Γ α 2 n j 0 a j,n 1 f t j , x h t j , A.3 where The error estimate e is Max |x t j −x h t j | O h ρ j 0, 1, . . ., N , where ρ Min 2, 1 α .CDJXS11172237 , the Specialized Research Fund for the Doctoral Program of Higher Education of China no.20093401120001; no.102063720090013 , the Natural Science Foundation of Anhui Province no.11040606M12 , and the Natural Science Foundation of Anhui Education Bureau no.KJ2010A035 .

4 . 5 Proof.
Combining 4.3 with 4.4 , the error system 4.3 is given by D q 1 * e 1 y m − a − k 1 e 1 x s e 2 e 3 , D q 2 * e 2 x m x s e 1 − b k 2 e 2 , D q 3 * e 3 −e 1 − c k 3 e 3 .