H ∞ control for a hyperchaotic finance system with external disturbance based on the quadratic system theory

This paper considers the control problem for a hyperchaotic system with energy-bounded disturbance under the delayed feedback controller. Using the quadratic system theory, an augmented Lyapunov functional, some integral inequalities and rigorous mathematical derivations, a sufficient condition is first established by using linear matrix inequalities under which the closed-loop system can achieve some desirable performances including the boundedness, the performance and the asymptotic stability. Subsequently, several convex optimization problems are formulated to obtain the optimal performance indices. Finally, numerical simulations are presented to illustrate the effectiveness of the obtained results.


Introduction
The economic/finance systems are the special nonlinear systems (Cesare & Sportelli, 2005;Chen & Chen, 2007;Chen & Ma, 2001a, 2001bSalarieh & Alasty, 2008). It has been recognized that the chaos often occurs in such nonlinear systems (Chen & Chen, 2007;Chen & Ma, 2001a;Salarieh & Alasty, 2008). In fact, the financial crisis is essentially a kind of chaotic behaviour. In the past more than two decades, the economic/finance systems have received wide research attention and the main topics are the dynamic analysis, the feedback control and the synchronization (Chen, 2008;Chen, Liu et al., 2014;Dadras & Momeni, 2010;Jahanshahi et al., 2019;Son & Park, 2011;Tacha et al., 2016;Wang et al., 2012;Xu et al., 2020;Zhao et al., 2011). For example, in Chen (2008), the complex dynamics and the chaos control have been investigated for a finance system under time-delayed feedbacks by numerical simulations, and in Tacha et al. (2016), the problems of dynamic analysis and adaptive control have been addressed for a modified finance system. In Zhao et al. (2011), several control strategies have been employed to consider the synchronization problem for the chaotic finance system. In particular, in our recent work (Xu et al., 2020), the quadratic system theory has been utilized to control the chaos of the finance dynamics under the time-delayed feedback controller.
On the other hand, the finance systems might inevitably be affected by external disturbances stemmed from CONTACT Erfeng Xu yali9699@126.com environmental interference (Zhao & Wang, 2014). If the external disturbance is ignored in designing the controller, the resultant closed-loop system might have poor performance. In Zhao and Wang (2014), the delayed feedback controller has been designed for a chaotic finance system subject to external disturbance such that the closed-loop system is asymptotically stable with a prescribed H ∞ performance level. In Xu et al. (2018), the finite-time H ∞ control problem has been considered for a disturbed chaotic finance system by using the delayed feedback controller. In Harshavarthini et al. (2020), the finite-time resilient fault-tolerant control problem has been studied for a nonlinear finance system. However, it is worth mentioning that most above references are mainly concerned with the finance models composed of three first-order or fractional-order differential equations. In Yu et al. (2012), by adding an additional state to the model in Chen and Ma (2001a) to represent the average profit margin, a more reasonable finance model has been proposed. It has been shown that such a four-dimensional system displays the more complex hyperchaotic behaviour. In the past several years, the synchronization and control problems have also attracted considerable research attention for various hyperchaotic finance systems Hajipour et al., 2018;Vargas et al., 2015;Zheng, 2016). For example, the adaptive algorithm has been proposed in Vargas et al. (2015) to address the synchronization problem for a hyperchaotic system with unknown parameters. In Zheng (2016), the impulsive control scheme has been utilized to study the stabilization and synchronization of an uncertain hyperchaotic finance system. Nevertheless, it should be pointed out that the external disturbances are not sufficiently incorporated in the considered hyperchaotic finance systems. Moreover, it is observed that the time-delay phenomenon has been ignored in controlling the hyperchaotic finance systems.
Motivated by the above discussions, in this paper, we will be concerned with the H ∞ control problem for a hyperchaotic system with energy-bounded disturbance via the delayed feedback controller. Using the quadratic system theory (Amato et al., 2007), the augmented Lyapunov functional and some integral inequalities, a sufficient condition is first proposed in the framework of linear matrix inequalities (LMIs) under which the closedloop dynamics can achieve some desirable performances. Then, several optimization problems are given to handle the different performance requirements. Finally, simulations results are given to illustrate the effectiveness of the obtained results. The main contributions of this work are as follows: (1) the H ∞ control problem is addressed, for the first time, for a hyperchaotic finance system under the delayed feedback controller and an LMI-based sufficient condition is established; (2) the quadratic system theory is utilized to investigate a hyperchaotic finance system based on which the performances of the closed-loop dynamics are specifically characterized.
Notation. "T" denotes the transpose of a matrix. R n is the n-dimensional Euclidean space. The real matrix P > 0 (P ≥ 0) denotes that P is symmetric and positive definite (semi-definite). · is the 2-norm of a vector. λ(·) M is the maximum eigenvalue value of a matrix. I is an identity matrix. The symmetric terms in a symmetric matrix are denoted by * . Matrices are assumed to have compatible dimensions.

Problem formulation
In Chen andMa (2001a, 2001b), a chaotic finance system is proposed. Such a finance model contains four sub-blocks (i.e. production, money, stock and labour force) and is formulated by the following three first-order differential equations: where the states x 1 (t), x 2 (t) and x 3 (t) are, respectively, the interest rate, the investment demand and the price index; a > 0, b > 0 and c > 0 are, respectively, the saving amount, the cost per investment and the demand elasticity of commercial markets.
By adding an additional state in the model (1), a more reasonable finance model is proposed in Yu et al. (2012), which is described as follows: where the state x 4 (t) denotes the average profit margin, and d, e are positive scalars.
In Yu et al. (2012), it has been identified that the model (2) displays sophisticated hyperchaotic behaviour when the system parameters are selected as a = 0.9, b = 0.2, c = 1.5, d = 0.2 and e = 0.17. Moreover, it has been verified that, under the assumption (abce + be + cd − ce)/(cd − ce) > 0, the model (2) has three equilibrium points (3) Moreover, it has been recognized that the finance systems are unavoidable influenced by external disturbances (Jahanshahi et al., 2019;Zhao & Wang, 2014). Adding the disturbance ω(t) ∈ R l and the control input u(t) ∈ R m to (2) yields thaṫ where B and D are matrices, and In this paper, the external disturbance ω(t) is assumed to be energy-bounded and satisfies the condition Remark 2.1: The finance systems are inevitably disturbed by external environments, such as the plagues and the wars. For example, due to the impact of the 2019 novel coronavirus (2019-nCoV), the market confidence will be reduced and correspondingly, the lower investment demand and the lower interest rate will occur. In this case, the impact of 2019-nCoV can be seen as the external disturbance and should be added to the finance systems to reflect the real finance dynamics. In addition, it is worth mentioning that the external disturbances might disappear within the finite time. Therefore, it is reasonable to suppose that the external disturbance is energybounded.
As in Zhao and Wang (2014), this paper adopts the delayed feedback controller where K 1 , K 2 are the controller gains, x * is an unstable equilibrium point, and τ > 0 is the time delay. For the given equilibrium point Denoting that r(t) x(t) − x * , and using (4)-(6), one has the closed-loop systeṁ Note that the nonlinearityf (r) can be written as follows: where G 1 = diag{G 1 , 0, 0}, G 2 = diag{−1, 0, 0, 0}, G 3 = 0 4×4 and G 4 = −dG 1 withG 1 = 0 1/2 1/2 0 . Using (8), the closed-loop dynamics (7) can be further written aṡ The initial condition associated with (9) is denoted by The main purpose of our paper is to design the delayed feedback controller (5) such that the closed-loop dynamics (9) has the following properties: (1) all state trajectories are bounded for all admissible initial conditions and external disturbances; (2) the H ∞ performance requirement where γ > 0 is a prespecified scalar and V(t) is an Lyapunov functional; 3) when ω(t) = 0, the asymptotic stability is guaranteed for all admissible initial conditions.
For purpose of the subsequent local analysis, we introduce the following box: wherer j > 0 (j = 1, 2, 3, 4) are scalars. The above box can be represented as where "Co" denote the convex hull and
Remark 3.1: Recently, the H ∞ control problem has been addressed in Xu et al. (2018) for a chaotic finance system with external disturbance in the framework of finite time. However, it is noted that the results in Xu et al. (2018) are based on the linearized model and one cannot perform the accurate analysis and design. Very recently, the quadratic system theory has been adopted in Xu et al. (2020) to stabilize the finance system (1). However, the disturbance is ignored in Xu et al. (2020). In fact, when the external disturbance is considered, one has to first determine the admissible initial conditions and disturbances to ensure the boundedness of the state trajectories and then discuss the corresponding H ∞ performance. Therefore, the proposed Theorem 3.1 in this paper is not the simple extension of the result in Xu et al. (2020).

Remark 3.2:
In Chen et al. (2013);de Souza and Coutinho (2014), the local stabilization/control problem has been studied for nonlinear quadratic time-delay systems. However, the systems addressed in Chen et al. (2013); de Souza and Coutinho (2014) contain the state delay but not the input delay. Therefore, the results proposed in Chen et al. (2013); de Souza and Coutinho (2014) cannot applicable for the hyperchaotic finance system subject to the delayed feedback controller. Moreover, it should be pointed out the external disturbance is not considered in de Souza and Coutinho (2014) and the boundedness of system trajectories is not discussed.
For the case of non-delayed feedback, the controller can be denoted as Correspondingly, the closed-loop system can be written asṙ Using the Lyapunov functionV(t) = x T (t)X −1 x(t), where X > 0, the following result can be readily established.

Remark 3.3:
The main results of the paper are based on the LMIs. Due to the use of the quadratic system theory, more LMIs are introduced in our obtained results, which will lead to longer computation time in solving optimization problems. The LMIs and decision variables in above optimization problems can be readily calculated. For example, 35 LMIs and 106 + 8 m scalar variables are involved in solving Prob.3.
Next, we will consider the H ∞ control problem under the delayed controller u(t) = K 2 (x(t − τ ) − x * ). To this end, we have to estimate the largest disturbance tolerance level β M . By solving Problem 1 withr 1 = 0.9, r 2 = 2.1,r 3 = 3,r 4 = 5, = 4.3 and Y 1 = 0, we have β M = 2.0336 * 10 3 . Letting β = 1.8 * 10 3 < β M and solving Problem 2 with the same choosing of the scalarsr 1 ,r 2 , r 3 ,r 4 and as above, one obtains the minimum H ∞ performance level γ m = 0.1467 and the following controller gain: Using the above controller gain, the state responses and the truncated H ∞ performance level γ t of the error   In the simulation, the external disturbance is selected as ω(t) = 13.4 * e −0.05t to ensure that ∞ 0 ω T (t)ω(t)dt ≤ β = 1.8 * 10 3 . From Figure 2, it is seen that the stability of the error dynamics (7) can be guaranteed when the disturbance ω(t) disappears under the proposed control scheme. Moreover, it is clear from Figure 3 that the truncated H ∞ performance level γ t is less than γ m = 0.1467.

Conclusions
Based on the quadratic system theory, an augmented Lyapunov functional and some integral inequalities, an LMI-based sufficient condition has been obtained in this paper for a hyperchaotic system with energy-bounded disturbance under the delayed feedback controller, which can guarantee that the closed-loop dynamics has some desirable performances including the boundedness, the H ∞ performance and the asymptotic stability. Then, several convex optimization problems have been given to handle different system performance requirements. Finally, numerical simulations have been presented to demonstrate the effectiveness of our proposed results.
The existence of the chaotic behaviour in finance systems will result in inherent indefinitenes of the macroeconomic operation. Therefore, it is imperative to propose some effective control schemes to stabilize the chaotic finance dynamics. This paper has attempted to control a hyperchaotic finance system with external disturbance in a more accurate local framework. Our proposed control scheme can be seen as an alternative for the governments in formulating measures to revive the economy.