# Pseudo-State Sliding Mode Control of Fractional SISO Nonlinear Systems

Authors: Bao Shi; Jian Yuan; Chao Dong
Journal: Advances in Mathematical Physics
 (2013)

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## Abstract

This paper deals with the problem of pseudo-state sliding mode
control of fractional SISO nonlinear systems with model inaccuracies. Firstly,
a stable fractional sliding mode surface is constructed based on the Routh-Hurwitz conditions for fractional differential equations. Secondly, a sliding
mode control law is designed using the theory of Mittag-Leffler stability. Further,
we utilize the control methodology to synchronize two fractional chaotic
systems, which serves as an example of verifying the viability and effectiveness
of the proposed technique.


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## Body

## 1. Introduction

Fractional calculus has a long history of three hundred years, over which a firm theoretical foundation has been established. In the past few decades, with deep understanding of the power of fractional calculus and rapid development of computer technology, an enormous number of interesting and novel applications have emerged in physics, chemistry, engineering, finance, and other sciences [1–3]. In particular, engineers and scientists from various fields have developed plenty of fractional dynamical systems, that is, systems which are better characterized by noninteger order mathematics models. As has been stated in [3], a suitable way to more efficient control of fractional dynamical systems is to design fractional controllers.Several pioneering attempts to develop fractional control methodologies have been made, such as TID controller [4], CRONE controller [5], fractional PID controller [6], and fractional lead-lag compensator [7]. Basic ideas and technical formulations of the above four fractional control schemes with comparative comments have been presented in [8].Very recently, by applying fractional calculus to advanced nonlinear control theory, several fractional nonlinear control schemes have been proposed, such as fractional sliding mode control, fractional adaptive control, and fractional optimal control. Exactly, to design fractional sliding mode controls, various fractional sliding surfaces have been constructed in [9–21]. In particular, adaptive sliding mode controls have been proposed in [12, 19–21], fractional terminal sliding mode controls in [15, 16], and sliding mode controls for linear fractional systems with input and state delays in [17]. In [22], the authors have presented two ideas to extend the conventional model reference adaptive control (MRAC) by using fractional parameter adjustment rule and fractional reference model. In [23], a fractional model reference adaptive control algorithm for SISO plants has been proposed, which can guarantee the stability and ability to reject disturbances. In [24–26], adaptive controllers have been designed to control and synchronize fractional chaotic systems. In [27], Agrawal has proposed a general formulation for a class of fractional optimal control problems, which is specialized for a system with quadratic performance index subject to a fractional system dynamic constraint. In [28], the authors have generalized the optimality conditions of [27]. Besides the above fractional control methodologies, fractional optimal synergetic control has been presented in [29] and active disturbance rejection control in [30].Motivated by the above contributions, this paper proposes a sliding mode control design for fractional SISO nonlinear systems in the presence of model inaccuracies. By constructing a stable fractional sliding mode surface on the basis of Routh-Hurwitz conditions, a sliding mode control law is designed. Further, stability analysis is performed using Mittag-Leffler stability theory. Comparing this with methods in the previous papers, we utilize the fractional derivative of the sliding mode surface instead of first-order derivative, to obtain the equivalent control law. Moreover, to carry out the stability analysis of the closed-loop fractional nonlinear system, we use the fractional derivative of the Lyapunov function candidate in terms of Theorem 2 in [18].The rest of the paper is organized as follows. Section2 reviews some basic definitions for fractional calculus. Section 3 proposes the sliding control design for fractional SISO nonlinear systems. Numerical simulations of synchronization of the fractional Genesio-Tesi system and the fractional Arneodo system are presented in Section 4. Finally, Section 5 concludes this paper with some remarks on future study.


## 2. Basic Definitions for Fractional Calculus

Fractional calculus is a generalization of integration and differentiation to noninteger order fundamental operatoraDtα, where a and t are the bounds of the operation and a∈R. The continuous integrodifferential operator is defined as [1]
(1)aDtα={dαdtαα>01α=0∫at(dτ)αα<0.The three most frequently used definitions for fractional calculus are the Grünwald-Letnikov definition, the Riemann-Liouville definition, and the Caputo definition [1–3].Definition 1.
The Grünwald-Letnikov derivative definition of orderα is described as
(2)aDtαf(t)=limh→01hα∑j=0∞‍(-1)j(αj)f(t-jh).Definition 2.
The Riemann-Liouville derivative definition of orderα is described as
(3)aDtαf(t)=1Γ(n-α)dndtn∫atf(τ)dτ(t-τ)α-n+1,n-1<α<n.However, applied problems require definitions of fractional derivatives allowing the utilization of physically interpretable initial conditions, which containsf(a), f′(a), and so forth. Unfortunately, the Riemann-Liouville approach fails to meet this practical need. It is M.  Caputo who solved this conflict.Definition 3.
The Caputo definition of fractional derivative can be written as(4)aDtαf(t)=1Γ(n-α)∫atf(n)(τ)dτ(t-τ)α-n+1,n-1<α<n.In the rest of the paper, we use the Caputo approach to describe the fractional systems and the Grünwald-Letnikov approach to propose numerical simulations. To simplify the notation, we denote the fractional-order derivative of orderα as Dα instead of 0Dtα in this paper.


## 3. Sliding Control of Fractional SISO Nonlinear Systems

Consider fractional SISO nonlinear systems(5)Dαxi=xi+1,i=1,2,…,n-1,Dαxn=f(x)+g(x)u,y=x1,
where α∈(0,1] is the order of the dynamic system, x=(x1,x2…,xn)T∈ℝn is denoted as state vector, and the scalar u,y∈R are system’s input and output, respectively. The dynamic f(x) (possibly nonlinear or time varying) is not exactly known, but estimated as f^(x). The control gain g(x) (possibly time varying or state dependent) is an unknown function.Assumption 4.
The estimation error onf(x) is assumed to be bounded by some known function F(x):
(6)|f(x)-f^(x)|≤F(x).Assumption 5.
g
(
x
) is assumed to be bounded by
(7)0<gmin(x)≤g(x)≤gmax(x),
where both gmin(x) and gmax(x) are known functions.Assumption 6.
The desired tracking signal is denoted asyd(t), and it is assumed that yd(t)∈Cn[0,∞) and yd(t),  Dαyd(t),…,Dnαyd(t) are all bounded.In this paper, our goal is to design a suitable fractional sliding mode controller to make the outputy of the system (5) track the desired signal yd; that is,
(8)limt→∞(y-yd)=0.Now we are ready to give the design steps.Firstly, the tracking error is defined as(9)x~=y-yd.Then, the tracking error vector is defined as(10)x~=x-(yd,Dαyd,…,D(n-1)αyd)T=(x~,Dαx~,…,D(n-1)αx~)T.A fractional sliding surface is proposed as(11)s(t)=c1x~+c2Dαx~+⋯+cn-1D(n-2)αx~+D(n-1)αx~,
where the polynomial P(p)=c1+c2p+⋯+cn-1pn-2+pn-1 satisfies Routh-Hurwitz conditions for fractional differential equations [31].The fractional sliding surface (11) can be rewritten as
(12)s(t)=D(n-1)αx~+∑i=1n-1‍ciD(i-1)αx~=D(n-1)αy-D(n-1)αyd+∑i=1n-1‍ciD(i-1)αx~=D(n-1)αx1-D(n-1)αyd+∑i=1n-1‍ciD(i-1)αx~=xn-D(n-1)αyd+∑i=1n-1‍ciD(i-1)αx~.Taking itsαth order derivative with respect to time yields
(13)Dαs(t)=Dαxn-Dnαyd+∑i=1n-1‍ciDiαx~.Substituting the second equation of system  (5) into  (13), one has
(14)Dαs(t)=f(x)+g(x)u-Dnαyd+∑i=1n-1‍ciDiαx~.In terms of Assumption5, one of the estimated values of g(x) can be chosen as
(15)g^(x)=gmin(x)gmax(x).Bounds ofg(x) are written as
(16)β-1≤g^(x)g(x)≤β,
where β=gmax/gmin.Design the control law as(17)u(t)=1g^(x)[-f^(x)+Dnαyd-∑i=1n-1‍ciDiαx~hhhhhhh-k(x)·sign(s)-σs-f^(x)+Dnαyd-∑i=1n-1‍ciDiαx~]=u^eq(t)+ur(t),
where u^eq(t)=(1/g^(x))[-f^(x)+Dnαyd-∑i=1n-1ciDiαx~], ur(t)=(1/g^(x))[-k(x)·sign(s)-σs], k(x)≥β(F(x)+η)+β(β-1)|u^eq|, and η is a positive constant.To ensure the stability of the fractional system (5), we have the following theorem.Theorem 7.
Under Assumptions4–6, the fractional SISO nonlinear system (5) can be controlled using the sliding mode control law (17) with the fractional sliding surface   (11).Proof.
Consider the following candidate Lyapunov function:(18)V(t)=[s(t)]2,
where s(t) is the fractional sliding surface (11) constructed previously.
Taking itsαth order derivative with respect to time along with the fractional sliding surface (11), one has
(19)DαV=s·Dαs+∑j=1∞‍Γ(1+q)Γ(1+j)Γ(1-j+q)s(j)Dq-js≤s·Dαs+ρ,
where ρ is assumed to be an arbitrarily large positive constant which is a bound on the series of (19).
Substituting (14) into (19) yields
(20)DαV≤s[f(x)+g(x)u-Dnαyd+∑i=1n-1‍ciDiαx~]+ρ.
Substituting the control law (17) into (20) gives
(21)DαV≤s[f-gg^-1f^+(1-gg^-1)(-Dnαyd+∑i=1n-1‍ciDiαx~)hhhhh-gg^-1ksign(s)∑i=1n-1‍]-σs2+ρ=s[∑i=1n-1‍(f-f^)+f^-gg^-1f^+(1-gg^-1)hhhhhh×(-Dnαyd+∑i=1n-1‍ciDiαx~)-gg^-1ksign(s)∑i=1n-1‍]-σs2+ρ=s[(f-f^)+(gg^-1-1)u^eq-gg^-1ksign(s)]-σs2+ρ.
Substituting the estimation error (6), the equivalent control law u^eq, and control gain k of (17) into (21), one derives
(22)DαV≤-η|s|-σs2+ρ<-σV+ρ,
which implies that the sliding mode dynamic is globally stable and the tracking error vector (10) converges to zero according to Theorem 2 in [18]. This proves that the fractional SISO nonlinear system (5) can be controlled using the sliding mode control law (17) with the fractional sliding surface (11).


## 4. Numerical Simulations

In this section, we apply the fractional sliding mode control method proposed in Section3 to deal with the problem of synchronization between the fractional Arneodo system and the fractional Genesio-Tesi system. To carry out numerical simulations, we utilize the algorithm for numerical calculation of fractional derivatives introduced in [1]. This method is derived from the Grnwald-Letnikov Definition 1 based on the fact that the three Definitions 1, 2, and 3 are equivalent for a wide class of functions.The fractional Arneodo system is represented as(23)Dαy1=y2,Dαy2=y3,Dαy3=5.5y1-3.5y2-y3-y13
with the desired tracking signal yd=y1.The fractional Genesio-Tesi system is described as(24)Dαx1=x2,Dαx2=x3,Dαx3=a(t)x1+b(t)x2+c(t)x3+d(t)x12+u(t),y=x1,
where a(t), b(t), c(t), and d(t) are assumed to be unknown parameters and satisfy |a(t)+6|≤0.1,  |b(t)+2.92|≤0.1, |c(t)+1.2|≤0.1, and |d(t)-1|≤0.1, from which one has
(25)f(x)=a(t)x1+b(t)x2+c(t)x3+d(t)x12,f^(x)=-6x1-2.92x2-1.2x3+x12,F(x)=0.1(|x1|+|x2|+|x3|+x12).Both of the fractional Arneodo system (23) and the fractional Genesio-Tesi system   (24) exhibit chaos with certain values of fractional order α. In the sequel we investigate synchronization between the two fractional chaotic systems using the technique proposed previously. The former system is taken as drive system, while the latter one is considered as response system.In terms of (11), the sliding surface is constructed as
(26)s=c1(x1-y1)+c2(x2-y2)+(x3-y3),
where c1=25 and c2=10 satisfy the Routh-Hurwitz conditions [31].The control law is designed as(27)u=6x1+2.92x2+1.2x3-x12+5.5y1-3.5y2-y3-y13-25(x2-y2)-10(x3-y3)-σs-σs-k·sign(s).Initial conditions for the above two systems are, respectively, chosen asx0=(-0.1,0.5,0.2) and y0=(-0.2,0.5,0.2), while parameters in the control law (27) are selected as η=0.1 and σ=0.5. Numerical simulations of synchronization between the fractional chaotic systems (23) and (24) are presented in Figures 1, 2, 3, and 4, with the simulation time Tsim=10 and time step h=0.0005. For interpretations of the references to color in these figure legends, the reader is referred to the online version of this paper.Synchronization of the fractional Arneodo system and the fractional Genesio-Tesi system with the control input (27) (red line represents the trajectories of the drive system, while blue line represents the trajectories of the response system).
(a)
(b)
(c)Figure 2
Synchronization errors with the control input (27) (red line represents the first error state e1, blue line represents the second error state e2, and black line represents the third error state e3).Figure 3
Time history of fractional sliding mode with the control input (27).Figure 4
Time history of the control input (27).From the simulation results, we see that synchronization performance is excellent but is obtained at the price of high control chattering. It can be eliminated by replacing the discontinuous switching control lawsign(·) by the smooth function tanh(·) in (27); that is,
(28)u=6x1+2.92x2+1.2x3-x12+5.5y1-3.5y2-y3-y13-25(x2-y2)-10(x3-y3)-σs-k·tanh(s).Numerical simulations with the modified control law (28) are presented in Figures 5, 6, 7, and 8. For interpretations of the references to color in these figure legends, the reader is referred to the online version of this paper.Synchronization of the fractional Arneodo system and the fractional Genesio-Tesi system with the control input (28) (red line represents the trajectories of the drive system, while blue line represents the trajectories of the response system).
(a)
(b)
(c)Figure 6
Synchronization errors with the control input (28) (red line represents the first error state e1, blue line represents the second error state e2, and black line represents the third error state e3).Figure 7
Time history of fractional sliding mode with the control input (28).Figure 8
Time history of the control input (28).From the above simulation results, one can easily see that the fractional Arneodo system and the fractional Genesio-Tesi system can be effectively synchronized via the proposed sliding mode control technique. Furthermore, the control chattering caused by the discontinuous control law (17) is successfully eliminated by the modification of (28).


## 5. Concluding Remarks

In this paper, we have investigated the pseudo-state sliding control design for fractional SISO nonlinear systems with model inaccuracies. A stable fractional sliding mode surface has been constructed based on the Routh-Hurwitz conditions for fractional differential equations. Then, a sliding mode control law is designed using the Mittag-Leffler stability theorem. Finally, numerical simulations of synchronization of the fractional Arneodo system and the fractional Genesio-Tesi system have been performed to demonstrate the effectiveness of the proposed control technique.As for the future perspectives, our research activities will be on,(i)
designing   adaptive sliding control to deal with parametric uncertainties inf(·),(ii)
generalizing  the method to fractional MIMO nonlinear systems,(iii)
generalizing  the method to incommensurate nonlinear systems.



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