Linear Control of Fractional-Order Financial Chaotic Systems with Input Saturation

In this paper, control of fractional-order financial chaotic systems with saturated control input is investigated by means of statefeedback control method. The saturation problem is tackled by using Gronwall-Bellman lemma and a memoryless nonlinearity function. Based onGronwall inequality andLaplace transform technique, two sufficient conditions are achieved for the asymptotical stability of the fractional-order financial chaotic systems with fractional orders 0 < α ≤ 1 and 1 < α < 2, respectively. Finally, simulation studies are carried out to show the effectiveness of the proposed linear control method.


Introduction
In the past two decades, studies of chaotic systems have received more and more attention in various fields of natural sciences.This is because chaotic systems are rich in dynamics and possess great sensitivity to initial conditions.Up to now, econophysics has been raised to an alternative scientific methodology to comprehend the highly complex dynamics in economic and financial systems.Many economists are working hard to explain the central features of economic data, including erratic macroeconomic fluctuations (business cycles), irregular microeconomic fluctuations, irregular growth, structural changes, and overlapping waves of economic development [1,2].Representative effects, that is, treated as random shocks, are political events, weather variables, and other human factors [3][4][5][6][7].Compared with the opinion discussed above, chaos supports an endogenous explanation of the complexity appeared in economic series.
Since chaos in financial systems was firstly studied in 1985, great impact has been put on the prominent economics recently, because the occurrence of the chaotic phenomenon in the economic system indicates that the macroeconomic operation has in itself the inherent indefiniteness.Studies on the complicated financial systems by using nonlinear method are fruitful [2,8,9].Controlling chaos in fractional-order financial systems is also studied in recent years [10][11][12][13][14][15][16][17][18].In [15], an active sliding mode controller is constructed to synchronize fractional-order financial chaotic systems in masterslave structure.In [16], a necessary condition is introduced to confirm the existence of 1-scroll, 2-scroll, or multiscroll chaotic attractors in a fractional-order financial system and a sliding mode controller is proposed.Active control method is also used in [17], and the variable-order fractional derivative is defined in Caputo type.Wang et al. investigate impulsive synchronization and adaptive-impulsive synchronization of a novel financial hyperchaotic system [18].In above literatures, the stability analysis is carried out based on fractional-order linear system stability theorem and only the situation where fractional order 0 <  ≤ 1 is concerned.
Most of real world technical systems are subjected to input constraints, especially in financial systems.In financial systems, input saturation does exist due to a limited size of weather variables, political events, and other human factors.The existence of input saturation may decrease the control performance or cause oscillations and even lead to instability of the system [19][20][21].It is advisable for us to consider the control of financial systems with input saturation.For the integer-order linear and nonlinear systems, input saturation has received much attention from researchers in the past decade.The sector bounded condition associated with input nonlinearities is useful for analysis and synthesis of control systems subject to input saturation.Then the stability of the system can be formulated using Lyapunov stability theory and invariant theory.
Though many research efforts have been put to the fractional-order financial chaotic systems, the financial systems with saturated control input have rarely been investigated in literatures.Here, with the help of Laplace transform, Mittag-Leffler function, and Gronwall inequality, a linear controller will be derived for fractional-order financial chaotic systems in this paper.There are some main contributions that are worth to be emphasized as follows.
(1) Two sufficient conditions are derived for the asymptotical stability of fractional-order financial chaotic systems with fractional orders 0 <  ≤ 1 and 1 <  ≤ 2, respectively.
(2) A linear controller is given to control the fractionalorder financial chaotic system.(3) A memoryless nonlinearity function is employed to handle the input saturation problem in fractionalorder chaotic systems.

Preliminaries and System Description
where  represents the fractional order and the Euler function The Laplace transform of Caputo fractional derivative can be given as The following definition and lemmas will be used.
Based on the Definition 4, the following lemma holds.
Lemma 6.The autonomous dynamic system is asymptotically stable if the following condition holds: The stability region for 0 <  < 1 is depicted in Figure 1.

Description of Fractional-Order Financial Chaotic Systems.
The fractional-order financial chaotic systems are proposed by [1].The mathematical model describes a fractionalorder financial system including three nonlinear differential equations.The states,  1 (),  2 (), and  3 (), represent the interest rate, the investment demand, and the price index, respectively.The fractional-order model of the system can be described as where  denotes the saving amount,  is the cost per investment, and  is the elasticity of demand of commercial market.0 <  < 2 is the fractional-order derivative.

State-Feedback Controller Design and Stability Analysis
3.1.Fractional Order : 0 <  ≤ 1.Let us rewrite the controlled system (13) as the following compact form: where (), () ∈  3 , represent the state variables and the control input, respectively.Consider that is the vector-valued saturation function with where  0 represents the symmetric saturation level of the th control input.
Noting that in chaotic systems the states are bounded, the nonlinear function () satisfies where Let us define the state-feedback control input as where  ∈  3×3 is the control gain matrix.Then we have where  =  + .
Proof.Taking Laplace transform on (19), we can obtain where () represents the Laplace transform of ().Let  denote the 3 × 3 identity matrix; we have By taking Laplace inverse transform on (21), we get the solution of system ( 14  According to Lemma 2, we know there exist some constants   > 0,  = 1, 2, such that Then ( 23) can be rewritten as From Definition 4 and Lemma Then we have Since ( − 1)‖‖ > , from (33) we know that lim and this ends the proof.
Let  Let  01 =  02 =  03 = 2.The simulation results can be seen in Figures 3 and 4. From the results, we can see that the states variables converge rapidly.The involved system is asymptotic stable.Figure 4 shows the boundedness and smoothness of the saturated control inputs.It can be concluded that good control performance has been achieved.
Case 2 (1 <  < 2).Let the fractional order be  = 1.04.The chaotic behavior is depicted in Figure 5.In the simulation, the control gain matrices are chosen as  = diag[0.9,0.9, 0.9], Let  01 =  02 =  03 = 2.The simulation results can be seen in Figures 6 and 7. From the simulation results we can conclude that good control performance has been achieved.

Conclusions
We investigate the control problem for fractional-order financial chaotic systems subject to input saturation by means of linear control.Two sufficient conditions are given for the stabilization of such systems with fractional orders 0 <  ≤ 1 and 1 <  < 2, respectively.A state-feedback controller is designed and the asymptotical stability of the involved system is guaranteed.It is shown that state-feedback controller can be designed to control the fractional-order financial chaotic systems.Simulation studies confirm the results of this paper.

Figure 3 :Figure 4 :
Figure 3: Time response of the state variables with fractional order  = 0.90.

Figure 6 :Figure 7 :
Figure 6: Time response of the state variables with fractional order  = 1.04.