Complexity

Volume 2019, Article ID 7567695, 13 pages

https://doi.org/10.1155/2019/7567695

## Robust Control of Disturbed Fractional-Order Economical Chaotic Systems with Uncertain Parameters

^{1}School of Economics and Management, Anhui Jianzhu University, Hefei 230009, China^{2}School of Economics and Management, Huainan Normal University, Huainan 232038, China^{3}Department of Applied Mathematics, Huainan Normal University, Huainan 232038, China^{4}School of Science, Guangxi University for Nationalities, Nanning 530006, China^{5}School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China

Correspondence should be addressed to Heng Liu; moc.liamg@221gnehuil

Received 16 May 2019; Revised 22 August 2019; Accepted 5 September 2019; Published 31 October 2019

Guest Editor: Lazaros Moysis

Copyright © 2019 Song Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper focuses on the robust control of fractional-order economical chaotic system (FOECS) with parametric uncertainties and external disturbances. The dynamical behavior of FOECS is studied by numerical simulation, and circuit implementations of FOECS are also given. Based on fractional-order Lyapunov stability theorems, a robust adaptive controller, which can guarantee that all signals remain bounded and the tracking error tends to a small region, is designed. The proposed method can be used to control a large range of fractional-order systems with system uncertainties. Fractional-order adaptation laws are constructed to update the estimation of adaptive parameters. Finally, the robustness and effectiveness of our control method are indicated by simulation results.

#### 1. Introduction

It has been shown that fractional-order nonlinear systems (FONSs) have been investigated by a lot of engineers and physicists because FONSs have wide potential applications in many domains [1–3]. In fact, the fractional calculus brings some advantages in modeling nonlinear systems. The fractional calculus can model real-world models in the whole-time domain, and it has memory. It should be mentioned that the integer-order one does not have these abilities. Thus, the fractional calculus will play a great role in modeling many actual systems, for example, stochastic diffusion, molecular spectroscopy, control theory, viscoelastic dynamics, quantum mechanics, and many research results can be seen in [4–8] and the references therein. On the contrary, it is well known that chaotic system is a supremely intricate nonlinear system that has been widely investigated due to its successful applications in signal processing, combinatorial optimization, secure communication, and many others. Especially, a chaotic system has the property that it is sensitive to the changing of initial conditions and the variations of the system parameters. Consequently, a large number of meritorious results on control and synchronization of fractional-order chaotic dynamics of nonlinear systems have become a hot research topic and a lot of interesting results have been reported, for example, in [9–13].

In the last two decades, the study of economical system has become more and more popular [14–24]. A lot of works have been done to describe properties of economical date and the dynamic behavior of economical systems. Recently, many researchers have made a lot of efforts to investigate main features of economic theory, e.g., overlapping waves of structural changes or commercial demand and irregular and erratic economic fluctuations. In fact, economists usually consider a model that has a simple behavior and composed of only endogenous variables. Thus, they can consider exogenous shock variables based on weather variables, political events, and other human factors. To describe the complicated economical behavior, some mathematical models were also introduced, for example, the van der Pol model [25] and the IS-LM model [26]. Actually, there are many kinds of nonlinear systems that show chaotic behavior [27]. Thus, if economical systems show chaotic phenomenon, it is hard for people to provide feasible economic decision making. That is to say, it is advisable to study and control economical chaotic systems.

The remainder of our work is organized as follows: Section 2 presents the development of research in this field, the existing research gaps, and the contributions of this study. Section 3 gives some preliminaries about the fractional calculus and the description and dynamical behavior of fractional-order economical chaotic systems (FOECSs). Section 4 presents the controller design procedure and the stability analysis. Simulation results are shown in Section 5. Finally, Section 6 gives a brief conclusion of this paper.

#### 2. Literature Review

The chaotic dynamics in economical systems were first founded in 1985, and after that, many control and synchronization methods for economical chaotic systems have been reported [16, 28–30]. In [16], a robust adaptive controller was given to control chaos in FOECSs, where the matched system uncertainties were considered, whereas in [16], the sign(·) function was used in the controller design which will lead to chattering phenomenon. In order to control a representative chaotic fractional finance system, an adaptive fuzzy control approach was given in [31] where fuzzy systems were used to approximate nonlinear functions. A new aspect of robust synchronization of a FOECS has been addressed in [32]. The fixed points and chaotic and periodic motions are given in [33], and dynamical behavior of a FOECS with time delay was studied in [34]. Dadras and Momeni [28] provided an adaptive control method to study the synchronization problem of FOECSs based on a sliding surface. The system studied in [28] was known, but it is impossible to accurately model an actual system in real life. And in the controller design, in order to make the sliding mode exist at every point of the sliding mode surface, the control law was constructed by using the sign function so that the chattering was unavoidable. In the final stability analysis, the integer-order Lyapunov stability theory was applied. Therefore, compared with [16], Dadras and Momeni [28] did not completely study the fractional-order economic chaotic system with the fractional-order stability theory. It should be mentioned that in the above literature considering the control or synchronization of FOECSs, the system model should be known in advance. However, it is well known that most systems suffer from system uncertainties and disturbances in nature. In actual life, we know that economical systems may suffer from weather changes, the limited size of transport, political influence, monetary policy, and many other human factors. Consequently, we should take system uncertainties and external disturbances into consideration when we investigate the control of FOECSs.

Due to limitations of available theoretical tools for analyzing the stability of nonlinear fractional systems, the number of research studies in this field is still low in comparison to that of integer-order systems. Based on the above discussion, in this paper, we investigate the control of FOECSs with unmatched system uncertainties and external disturbances. The fractional Lyapunov stability method is utilized to design the robust controller and analyze the system’s stability. Compared with some related works, the main contribution of our work can be concluded as follows. (1) A robust adaptive controller is designed for FOECSs with unmatched system uncertainties. The system uncertainties model we considered is representative, and many models used in the literature, for example, in [16], can be seen as a special case of our model. (2) The stability analysis is proven strictly. The stability analysis method we used is very similar to that of the integer-order systems. It should be pointed out that our main result (see, Theorem 1) of our method provides a framework which can be easily referenced to analyze stability of fractional-order systems.

#### 3. Preliminaries

##### 3.1. Fractional Calculus

The *α*-th fractional-order integral iswhere represents Euler’s function.

Caputo’s *α*-th derivative is given aswhere .

For fractional calculus, we give the following results to facilitate the controller design as well as the stability analysis.

*Definition 1 (see [1]). *The Mittag-Leffler function is given aswith and .

The Laplace transform of (3) can be given aswhere is a constant.

Lemma 1 (see [1]). *Let , , andand then it holdswhere and .*

Lemma 2 (see [1]). *Suppose that and . For and , it holdswith and and .*

Lemma 3 (see [35]). *Let be a smooth function. Suppose that is an equilibrium ofwhere is a continuous nonlinear function. Ifwith and , and being class- k functions, then system (8) is asymptotically stable.*

Lemma 4 (see [7, 36]). *Suppose that is a smooth function. For , it holds*

Lemma 5 (see [1]). *Let with and , then it holds*

In the following parts, we will use an algorithm to solve fractional-order differential equations. A brief explanation of this algorithm is given below.

Consider

Based on Lemma 5, (12) can be rewritten as

Define Thus, (13) is estimated as [1]with for and for , , , and .

The approximation error can be obtained as [1].

##### 3.2. The FOECS

The FOECS can be described bywhere is the fractional order, represents the interest rate of the market, is the investment demand, corresponds to the price index, is the confidence of the market, *a* stands for the saving amount, *b* is the cost in each investment, *c* represents the demand elasticity, and are impact parameters. Let be the state vector.

Let , and the initial condition be .

It has been shown in [37] that, under above parameters, when , the system (15) exhibits chaotic behavior. Figure 1 shows the chaotic behavior of system (15) for .