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

Authors: Song Xu; Hui Lv; Heng Liu; Aijing Liu
Journal: Complexity
 (2019)

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## 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.


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## Body

## 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: Section2 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

Theα-th fractional-order integral is(1)Dt−αft=1Γα∫0tt−τα−1fτdτ,where Γ⋅ represents Euler’s function.Caputo’sα-th derivative is given as(2)Dtαft=1Γn−α∫0tt−τn−α−1fnτdτ,where n−1≤α<n.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 as(3)Eα,βz=∑k=0∞zkΓαk+β,with α,β>0 and z∈ℂ.
The Laplace transform of (3) can be given as(4)ℒtβ−1Eα,β−atα=sα−βsα+a,where a∈ℝ+ is a constant.Lemma 1 (see [1]).
Letβ∈ℂ, 0<α<2,μ∈ℝ, and(5)πα2<μ<minπ,πα,and then it holds(6)Eα,βz=−∑j=1n1Γβ−αjzj+o1zn+1,z⟶∞,where n∈ℕ and μ≤argz≤π.Lemma 2 (see [1]).
Suppose that0<α<2 and β∈ℝ. For μ>0 and πα/2<μ≤minπ,πα, it holds(7)Eα,βz≤a1+z,with a>0 and μ≤argz≤π and z≥0.Lemma 3 (see [35]).
Letxt be a smooth function. Suppose that xt=0 is an equilibrium of(8)Dtαxt=fx,where fx is a continuous nonlinear function. If(9)g1x≤Vt,xt≤g2x,DtβVt,xt≤−g3x,with 0<β<1 and g1⋅,g2⋅, and g3⋅ being class-k functions, then system (8) is asymptotically stable.Lemma 4 (see [7, 36]).
Suppose thatxt∈ℝn is a smooth function. For 0<α<1, it holds(10)12DtαxTtxt≤xTtDtαxt.Lemma 5 (see [1]).
Letxt∈ℂ10,T with T>0 and 0<α≤1, then it holds(11)Dt−αDtαxt=xt−x0,DtαDt−αxt=xt.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(12)Dtαyt=ft,yt,y0=y0.Based on Lemma 5, (12) can be rewritten as(13)yt=y0+1Γα∫0tt−τα−1fτ,yτdτ.Defineh=T/N,N∈Z,tn=nh,n=0,1,…,N. Thus, (13) is estimated as [1](14)yhtn+1=y0+hαΓα+2ftn+1,yhptn+1+hαΓα+2∑j=0naj,n+1ftj,yhtj,with aj,n+1=nα+1−n−αn+1α for j=0 and aj,n+1=n−j+2α+1+n−jα+1−2n−j+1α+1 for 1≤j≤n, yhptn+1=y0+1/Γα∑j=0nbj,n+1ftj,yhtj, p=α+1, and bj,n+1=hα/αn+1−jα−n−jα.The approximation error can be obtained asmaxytj−yhtj=ohp [1].


### 3.2. The FOECS

The FOECS can be described by(15)Dtαζ1t=ζ3t+ζ2t−aζ1t+a1ζ4t,Dtαζ2t=1−bζ2t−ζ12t+a2ζ4t,Dtαζ3t=−ζ1t−cζ3t+a3ζ4t,Dtαζ4t=−ζ1tζ2tζ3t,where 0<α≤1 is the fractional order, ζ1t represents the interest rate of the market, ζ2t is the investment demand, ζ3t corresponds to the price index, ζ4t 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 a1,a2,anda3 are impact parameters. Let ζt=ζ1t,ζ2t,ζ3t,ζ4tT be the state vector.Leta=2.10,b=0.01,c=2.61,a1=8.41,a2=6.40,a3=2.21, and the initial condition be ζ0=1.0,5.0,4.1,3.0T.It has been shown in [37] that, under above parameters, when α>0.88, the system (15) exhibits chaotic behavior. Figure 1 shows the chaotic behavior of system (15) for α=0.90.Figure 1
Chaotic phenomenon of (15) with α=0.92.
(a)
(b)
(c)
(d)Now, let us consider the circuit implementation of the FOECS. Just as the results in [38], the approximation of 1/s0.9 can be given by(16)1s0.9≈2.2675s+1.292s+215.4s+0.01292s+2.154s+359.4.Thus, an unit circuit is designed to implement the function1/s0.9 which is shown in Figure 2. Here, the chain fractance consists of three resistors Ra,Rb,andRd and three capacitors Ca,Cb,andCd. The transfer function FCs of this chain can be given by(17)FCs=C0/Cas+1/RaCa+C0/Cbs+1/RbCb+C0/Cds+1/RdCd.Figure 2
Unit circuit for realizing1/s0.9.LetC0=1μF; from (16) and (17), we can gain Ra=62.84MΩ,Rb=250kΩ,Rd=205kΩ,Ca=1.23μF,Cb=1.835μF,Cd=1.1μF,and FCt≈1/t0.9.To implement the FOECS, an electronic circuit is constructed by using R-C components, analog multipliers, and operational amplifiers. The circuit diagram is depicted in Figure3, whose mathematical equations are given as follows:(18)Dtαζ1t=1R6C1ζ3t+1R1R2C1ζ1tζ2t+1R5R12C1ζ4t−1R3R4C1ζ1t,Dtαζ2t=1−1R7C2ζ2t−1R8C2ζ12t+1R9R13C2ζ4t,Dtαζ3t=−1R10C3ζ1t−1R11R14C3ζ3t+1R15R16C3ζ4t,Dtαζ4t=−1R17C4ζ1tζ2tζ3t,where C1,C2,andC3 are three fractional capacitors, and they are implemented by Ra,Rb,Rd, Ca,Cb,andCd. In order to satisfy the FOECS, we choose the values of resistors and capacitors as R1=R2=1kΩ,R3=R5=4kΩ,R4=120Ω,R6=R8=R10=R17=1MΩ, R7=100MΩ,R16=90Ω,R9=R11=R12=30Ω,R13=R15=5kΩ,R14=12kΩ, and C1=C2=C3=C4=1μF.Figure 3
Simulation circuit of the FOECS.Remark 1.
In the circuit implementation of the FOECS, the order of the system is set asα=0.9. Noting that the system parameters in (15) are very hard to be approximated by the circuit, the parameters in (18) are different with those in (15) (in fact, we used the approximate quantity of the parameters in (15)).Remark 2.
Based on [18], we know that the financial variables of financial systems could have long memory, such as stock market prices, gross domestic product, interest rates, and foreign exchange rates. Compared with the integer-order calculus, the fractional-order calculus introduced into the system (15) can more accurately describe the long-term memory effect in the financial system.


## 3.1. Fractional Calculus

Theα-th fractional-order integral is(1)Dt−αft=1Γα∫0tt−τα−1fτdτ,where Γ⋅ represents Euler’s function.Caputo’sα-th derivative is given as(2)Dtαft=1Γn−α∫0tt−τn−α−1fnτdτ,where n−1≤α<n.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 as(3)Eα,βz=∑k=0∞zkΓαk+β,with α,β>0 and z∈ℂ.
The Laplace transform of (3) can be given as(4)ℒtβ−1Eα,β−atα=sα−βsα+a,where a∈ℝ+ is a constant.Lemma 1 (see [1]).
Letβ∈ℂ, 0<α<2,μ∈ℝ, and(5)πα2<μ<minπ,πα,and then it holds(6)Eα,βz=−∑j=1n1Γβ−αjzj+o1zn+1,z⟶∞,where n∈ℕ and μ≤argz≤π.Lemma 2 (see [1]).
Suppose that0<α<2 and β∈ℝ. For μ>0 and πα/2<μ≤minπ,πα, it holds(7)Eα,βz≤a1+z,with a>0 and μ≤argz≤π and z≥0.Lemma 3 (see [35]).
Letxt be a smooth function. Suppose that xt=0 is an equilibrium of(8)Dtαxt=fx,where fx is a continuous nonlinear function. If(9)g1x≤Vt,xt≤g2x,DtβVt,xt≤−g3x,with 0<β<1 and g1⋅,g2⋅, and g3⋅ being class-k functions, then system (8) is asymptotically stable.Lemma 4 (see [7, 36]).
Suppose thatxt∈ℝn is a smooth function. For 0<α<1, it holds(10)12DtαxTtxt≤xTtDtαxt.Lemma 5 (see [1]).
Letxt∈ℂ10,T with T>0 and 0<α≤1, then it holds(11)Dt−αDtαxt=xt−x0,DtαDt−αxt=xt.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(12)Dtαyt=ft,yt,y0=y0.Based on Lemma 5, (12) can be rewritten as(13)yt=y0+1Γα∫0tt−τα−1fτ,yτdτ.Defineh=T/N,N∈Z,tn=nh,n=0,1,…,N. Thus, (13) is estimated as [1](14)yhtn+1=y0+hαΓα+2ftn+1,yhptn+1+hαΓα+2∑j=0naj,n+1ftj,yhtj,with aj,n+1=nα+1−n−αn+1α for j=0 and aj,n+1=n−j+2α+1+n−jα+1−2n−j+1α+1 for 1≤j≤n, yhptn+1=y0+1/Γα∑j=0nbj,n+1ftj,yhtj, p=α+1, and bj,n+1=hα/αn+1−jα−n−jα.The approximation error can be obtained asmaxytj−yhtj=ohp [1].


## 3.2. The FOECS

The FOECS can be described by(15)Dtαζ1t=ζ3t+ζ2t−aζ1t+a1ζ4t,Dtαζ2t=1−bζ2t−ζ12t+a2ζ4t,Dtαζ3t=−ζ1t−cζ3t+a3ζ4t,Dtαζ4t=−ζ1tζ2tζ3t,where 0<α≤1 is the fractional order, ζ1t represents the interest rate of the market, ζ2t is the investment demand, ζ3t corresponds to the price index, ζ4t 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 a1,a2,anda3 are impact parameters. Let ζt=ζ1t,ζ2t,ζ3t,ζ4tT be the state vector.Leta=2.10,b=0.01,c=2.61,a1=8.41,a2=6.40,a3=2.21, and the initial condition be ζ0=1.0,5.0,4.1,3.0T.It has been shown in [37] that, under above parameters, when α>0.88, the system (15) exhibits chaotic behavior. Figure 1 shows the chaotic behavior of system (15) for α=0.90.Figure 1
Chaotic phenomenon of (15) with α=0.92.
(a)
(b)
(c)
(d)Now, let us consider the circuit implementation of the FOECS. Just as the results in [38], the approximation of 1/s0.9 can be given by(16)1s0.9≈2.2675s+1.292s+215.4s+0.01292s+2.154s+359.4.Thus, an unit circuit is designed to implement the function1/s0.9 which is shown in Figure 2. Here, the chain fractance consists of three resistors Ra,Rb,andRd and three capacitors Ca,Cb,andCd. The transfer function FCs of this chain can be given by(17)FCs=C0/Cas+1/RaCa+C0/Cbs+1/RbCb+C0/Cds+1/RdCd.Figure 2
Unit circuit for realizing1/s0.9.LetC0=1μF; from (16) and (17), we can gain Ra=62.84MΩ,Rb=250kΩ,Rd=205kΩ,Ca=1.23μF,Cb=1.835μF,Cd=1.1μF,and FCt≈1/t0.9.To implement the FOECS, an electronic circuit is constructed by using R-C components, analog multipliers, and operational amplifiers. The circuit diagram is depicted in Figure3, whose mathematical equations are given as follows:(18)Dtαζ1t=1R6C1ζ3t+1R1R2C1ζ1tζ2t+1R5R12C1ζ4t−1R3R4C1ζ1t,Dtαζ2t=1−1R7C2ζ2t−1R8C2ζ12t+1R9R13C2ζ4t,Dtαζ3t=−1R10C3ζ1t−1R11R14C3ζ3t+1R15R16C3ζ4t,Dtαζ4t=−1R17C4ζ1tζ2tζ3t,where C1,C2,andC3 are three fractional capacitors, and they are implemented by Ra,Rb,Rd, Ca,Cb,andCd. In order to satisfy the FOECS, we choose the values of resistors and capacitors as R1=R2=1kΩ,R3=R5=4kΩ,R4=120Ω,R6=R8=R10=R17=1MΩ, R7=100MΩ,R16=90Ω,R9=R11=R12=30Ω,R13=R15=5kΩ,R14=12kΩ, and C1=C2=C3=C4=1μF.Figure 3
Simulation circuit of the FOECS.Remark 1.
In the circuit implementation of the FOECS, the order of the system is set asα=0.9. Noting that the system parameters in (15) are very hard to be approximated by the circuit, the parameters in (18) are different with those in (15) (in fact, we used the approximate quantity of the parameters in (15)).Remark 2.
Based on [18], we know that the financial variables of financial systems could have long memory, such as stock market prices, gross domestic product, interest rates, and foreign exchange rates. Compared with the integer-order calculus, the fractional-order calculus introduced into the system (15) can more accurately describe the long-term memory effect in the financial system.


## 4. Controller Design and Stability Analysis

Taking uncertain parameters and disturbances into consideration, the controlled FOECS can be described as(19)Dtαζ1t=ζ3t+ζ2t−aζ1t+a1ζ4t+u1t+d1t+φ1Tζtθ1,Dtαζ2t=1−bζ2t−ζ12t+a2ζ4t+u2t+d2t+φ2Tζtθ2,Dtαζ3t=−ζ1t−cζ3t+a3ζ4t+u3t+d3t+φ3Tζtθ3Dtαζ4t=−ζ1tζ2tζ3t+u4t+d4t+φ4Tζtθ4,where uit∈ℝ represents the controller, dit∈ℝ denotes the external disturbance, φiζt:ℝ4↦ℝmi is a known bounded nonlinear function, and θi∈ℝmi is an unknown constant vector with i=1,2,3,4 and mi∈ℕ.Remark 3.
In fact, the economical systems usually suffer from system uncertainties and unknown disturbances, for example, unknown price and cost fluctuation, market’s human intervention, and system model uncertainties. Note that we assume thatφ1⋅,φ2⋅,φ3⋅, and φ4⋅ can have different dimensions, and it is easy to know that the uncertain terms considered in system (19) stand for a large range of system uncertainties. Thus, the economical model considered in some recent literature, for example, [10, 16, 18, 37–39], is a special case of our model (19).
Our objective here is to design a proper controlleruit such that the state vector ζt could track the desired signal ζdt=ζd1t,ζd2t,ζd3t,ζd4tT∈ℝ4. The tracking error is defined as et=e1t,e2t,e3t,e4tT=ζt−ζdt. To proceed, we need the following assumption.Assumption 1.
The nonlinear functiondit is bounded, i.e., there exists a constant di∗>0 such that(20)dit≤di∗.Assumption 2.
The referenced signalζdt is a known continuous bounded function and has bounded known continuous first-order derivative.Lemma 6 (see [40]).
If a is a positive constant, it holdsx−xtanhx/a≤0.2785a=a′.According to the definition ofet, it follows from (15) that(21)Dtαe1t=ζ3t+ζ2t−aζ1t+a1ζ4t+u1t+φ1Tζtθ1−Dtαζd1t+d1t,Dtαe2t=1−bζ2t−ζ12t+a2ζ4t+u2t+d2t−Dtαζd2t+φ2Tζtθ2,Dtαe3t=−ζ1t−cζ3t+a3ζ4t+u3t+d3t+φ3Tζtθ3−Dtαζd3t,Dtαe4t=−ζ1tζ2tζ3t+u4t+d4t+φ4Tζtθ4−Dtαζd4t.Thus, the controller can be designed as(22)u1t=−k1e1t−ζ3t−ζ2t−aζ1t−a1ζ4t−φ1Tζtθ^1t+Dtαζd1t−tanhe1ta1,u2t=−k2e2t+ζ12t−a2ζ4t−tanhe2ta2−φ2Tζtθ^2t+Dtαζd2t−1+bζ2t,u3t=−k3e3t+ζ1t−tanhe3ta3−φ3Tζtθ^3t+Dtαζd3t+cζ3t−a3ζ4t,u4t=−k4e4t+ζ1tζ2tζ3t−tanhe4ta4−φ4Tζtθ^4t+Dtαζd4t,where ki>0 is a design parameter, ai>0 is a constant, and d^i∗t and θ^it are the estimation of d∗i and θi, respectively.Remark 4.
In this work, an adaptive controller (22) which seems complicated is designed. In this controller, all inputs, i.e., uit,i=1,2,3,4, have the same structure. Take u1t as an example; it contains four parts, i.e., a nonlinear function −ζ3t−ζ2t−aζ1t−a1ζ4t consisting of system variables, a feedback term of the tracking error −k1e1t−tanhe1t/a1, an adaptive term −φ1Tζtθ^1t, and a smooth function of the referenced signal Dtαζd1t. It can be observed that all inputs are easy to be implemented.
Substituting (22) into (21) gives(23)Dtαeit=−kieit+dit−tanheitai−ϕiTζtθ˜it,with i=1,2,3,4, and(24)θ˜it=θ^it−θi,is the estimation error of the unknown constant vector θi.
Multiplyingeit to both sides of (23), and according to Assumption 1 and Lemma 6, we have(25)eitDtαeit=−kiei2t+eitdit−eittanheitai−eitφiTζtθ˜it≤−kiei2t+eitdi∗−eittanheitai−eitφiTζtθ˜it≤−kiei2t+eit−eittanheitai−eitφiTζtθ˜it=−kiei2t−ai′−eitφiTζtθ˜it,where ai′>0.
Let the fractional-order adaptation laws be(26)Dtαθ^it=σi1eitφiTζt−σi1σi2θ^i,(27)Dtαd^i∗t=λi1d^i∗tai′−λi1λi2d^i∗t,with i=1,2,3,4,λi1,λi2,σi1,σi2>0, and d˜i∗t=d^i∗t−di∗ is the estimation error of the unknown constant di∗. The structure of the proposed method is shown in Figure 4.
Based on above discussion, we can give the following theorem.Figure 4
The structure of the proposed method.Theorem 1.
Based on Assumptions 1 and 2, the controller (22) of the system (19) together with the fractional-order adaptation laws (26) and (27) can guarantee that all variables remain bounded and the tracking error et converges to a small neighborhood of the origin.Proof.
Let the Lyapunov function be(28)Vt=12∑i=14ei2t+12∑i=141σi1θ˜iTtθ˜it+12∑i=141λi1d˜i∗2t.
From (25)–(27) and Lemma 4, one has(29)DtαVt=12∑i=14Dtαei2t+12∑i=141σi1Dtαθ˜iTtθ˜it+12∑i=141λi1Dtαd˜i∗2t≤∑i=14eitDtαeit+∑i=141σi1θ˜iTtDtαθ˜it+∑i=141λi1d˜i∗tDtαd˜i∗t=−∑i=14kiei2t−∑i=14eitφiTζtθ˜it+∑i=141σi1θ˜iTtDtαθ˜it+∑i=141λi1d˜i∗tDtαd˜i∗t−∑i=14ai′.
It follows from (29) that(30)DtαVt≤−∑i=14ai′−∑i=14eitϕiTζtθ˜it+∑i=14θ˜iTteitϕiTζt−σi2θ^it−∑i=14kiei2t+∑i=14d˜i∗tai′d˜i∗t−λi2d^i∗t=−∑i=14kiei2t−∑i=14λi2d˜i∗td^i∗t−∑i=14σi2θ˜iTtθ^it=−∑i=14kiei2t−∑i=14λi2d˜i∗td˜i∗t+di∗−∑i=14σi2θ˜iTtθ˜it+θi=−∑i=14kiei2t−∑i=14λi2d˜i∗2t−∑i=14λi2d˜i∗tdi∗−∑i=14σi2θ˜iTtθi−∑i=14σi2θ˜iTtθ˜it≤−∑i=14kiei2t−∑i=14λi22d˜i∗2t+∑i=14λi22di∗2+∑i=14σi22θiTθi−∑i=14σi22θ˜iTtθ˜it≤−k¯∑i=14ei2t−λ¯22i=∑14d˜i∗2t−σ¯22∑i=14θ˜iTtθ˜it+A≤−BVt+A,where k¯=min1≤i≤4ki, σ¯2=min1≤i≤4λi2, σ¯2=min1≤i≤4σi2, B=min2k¯,λ¯2λ¯1,σ¯2σ¯1, λ¯1=max1≤i≤4λi1, σ¯1=max1≤i≤4σi1, and A=∑i=14λi2/2di∗2+∑i=14σi2/2θiTθi.
According to (30), we have(31)DtαVt+yt=−BVt+A,where yt>0.
The Laplace transform of (31) is given as(32)Vs=sα−1sα+BV0+Assα+B−Yssα+B=sα−1sα+BV0+sα−1+αAsα+B−Yssα+B.
According to (4), we can solve (32) as(33)Vt=V0Eα,1−Btα+AtαEα,1+α−Btα−yt∗t−1Eα,0−Btα,where ∗ represents the convolution operator. Noting t−1Eα,0−Btα≥0 and yt≥0, we have yt∗t−1Eα,0−Btα≥0.
Then, we obtain(34)Vt≤V0Eα,1−Btα+AtαEα,1+α−Btα.
Becausearg−Btα=−π, −Btα≥0, and 0<α≤1, based on Lemma 2, it holds(35)Eα,1−Btα≤C1+Btα,where C>0. Then, we have(36)limt⟶∞V0Eα,1−Btα=0.That is to say, with respect to any μ>0, one can find t1>0 so that(37)V0Eα,1−Btα<μ.
It follows from Lemma 1 that(38)tαEα,α+1−Btα≤AB+μ.
Noting that we can find proper design parameters such thatA/B≤μ, according to (34), (37), and (38), we know(39)Vt<3μ.
Based on the definition ofVt, it is easy to know that the boundedness of all signals can be guaranteed. On the contrary, the tracking error can be arbitrarily small.Remark 5.
In this paper, we design adaptation laws (26) and (27) to update θ^it and d^i∗t, respectively. These laws update the parameters by using fractional differential equations. In fact, this kind of laws can also be seen in some literature, e.g., in [41–44]. However, in the abovementioned literature, the fractional adaptation laws only have one term, i.e., the positive term. Noting that the fractional derivatives of the updated parameters are greater than zero, and thus, according to the properties of the fractional derivative, the updated parameters are monotonically increasing. That is to say, the boundedness of the updated parameters cannot be guaranteed. However, in this work, by introducing the second term −σi1σi2θ^i in (26), we can guarantee the boundedness of the updated parameter.Remark 7.
In the proof of Theorem 1, we can see thatet≤6μ. To drive the tracking error et as small as possible, we can set A/B as small as possible. To obtain this objective, according to the form of A and B, we can choose large ki, λi1, and σi1 and small σi2 and λi2.Remark 8.
To analyze the fractional-order system’s stability, a commonly used Lyapunov function isVt=eTtet. It should be emphasized that its fractional derivative is [45](40)DtαVt=∑i=1∞Γ1+αΓ1+iΓ1−i+αDtietDtα−iet+2eTtDtαet.
In fact, this complicated form is very hard to be used. However, in this paper, by using our method, it is not necessary to use (40) in the stability analysis.


## 5. Simulation Studies

In the simulation, the parameters area=2.10,b=0.01,c=2.61,a1=8.41,a2=6.40,anda3=2.21 and the disturbances are chosen as d1t=cosζ1t,d2t=cosζ2t,d3t=cosζ3t,andd4t=cosζ4t which are activated when t=5. Let the initial condition be ζ0=1.5,−2,3,−3T. The fractional order is α=0.95.The parametric uncertain vectors are chosen asθ1t=1.1,−0.5,2,1.5T, θ3t=2,1,−0.5T, and θ2t≡θ4t≡0, and the basis functions are chosen as φ1ζt=ζ1t,ζ2t,ζ3t,ζ4tT and φ3ζt=ζ1tsinζ2t,cosζ3t,sinζ4tT.The controller design parameters are given aski=0.2,λi1=σi1=5,andλi2=σi2=0.005 with i=1,2,3,4.First, let the desired signal beζdt≡0,0,0,0T. The simulation results are given in Figures 5–9. Figure 5 shows the time response of the state variables ζ1t,ζ2t,ζ3t,andζ4t. We can see these variables tend to zero in about 2 s and stay in a very small neighborhood of zero thereafter. The control inputs are depicted in Figure 6. The estimations for θ1t as well θ3t are given in Figures 8 and 9, respectively. According to Figures 7–9, we can see that the boundedness of the updated parameters can be guaranteed, just like the results in Theorem 1 and the statements in Remark 5. These simulation results indicate that our control method has satisfactory control performance and good robustness.Figure 5
Time response of state variables.Figure 6
Time response of control inputs.Figure 7
The estimations ofd1∗,d2∗,d3∗, and d4∗.Figure 8
The estimation ofθ1t.Figure 9
The estimation ofθ3t.Let the desired signal beζdt=cost,sint,cost,sintT. The other parameters and values are chosen as the same as above. The simulation results can be seen in Figures 10–12.Figure 10
Tracking performance of state variables.
(a)
(b)
(c)
(d)Figure 11
Tracking errors.Figure 12
Control inputs and the updated parameters.
(a)
(b)
(c)
(d)Remark 9.
The simulation results of [16] are that the designed controllers were activated at t = 30 s, whereas in our paper, the proposed controllers are activated at about t = 2 s. Compared with the simulation results in [16], it is not difficult to see that the controller we designed does not use the symbolic function cited in [16]. Therefore, there is no chattering phenomenon in our simulation results. Although arctan (10·) was used to approximate sign(·) in the simulation of [16], the controller designed by us can work more effectively in terms of comparing the convergence rate of simulation results. Since we directly use the proposed controller to obtain simulation results, our simulation results can more effectively prove the theory of this paper.


## 6. Conclusions

In this paper, an adaptive robust control method of FOECSs subject to unmatched system uncertainties as well as external disturbances is investigated. It has been shown by our work that (1) the quadratic Lyapunov function can be used in the stability analysis of FOECSs based on the fractional stability criterion and (2) the parameters can be updated by fractional-order differential equations. The proposed method gives an easy way to analyze the stability of fractional-order systems. Simulation results have verified the feasibility of our control method. On the contrary, it is assumed that the unknown parameter is a constant vector, i.e., when the uncertain term is time-varying, the method presented in this paper is powerless. In addition, four control inputs are used, which reduce the real-world applications of our approach. How to solve aforementioned problems is one of our future research directions.



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