# Fractional-Order Fast Terminal Sliding Mode Control for a Class of Dynamical Systems

Authors: Guoliang Zhao
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

This paper introduces a novel fractional fast terminal sliding mode control strategy for a class of dynamical systems with uncertainty. In this strategy, a fractional-order sliding surface is proposed, the corresponding control law is derived based on Lyapunov stability theory to guarantee the sliding condition, and the finite time stability of the closeloop system is also ensured. Further, to achieve the equivalence between convergence rate and singularity avoidance, a fractional-order nonsingular fast terminal sliding mode controller is studied and the stability is presented. Finally, numerical simulation results are presented to illustrate the effectiveness of the proposed method.


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## Body

## 1. Introduction

In the past thirty years or so, fractional calculus has been found in numerous diverse and widespread fields of science and engineering [1–4] since the first fractional-order (FO) controller was introduced by Oustaloup in 1988 [1]. Thereafter, many different FO control strategies are studied, for example, fractional-order PID controller [5], FO lead-lag compensators [6], and FO optimal controllers [7, 8]. Fractional-order control has made great progress in recent years, many researchers have endeavored to study different nonlinear systems, and several control methods and stabilities are proposed. To name a few, asymptotic stability is extended to Mittag-Leffler stability for nonlinear fractional neutral singular systems, and the newly proposed stability implies asymptotic stability [9, 10]. A sliding mode controller with a new fractional-order integral switching surface for an uncertain chaotic fractional order economic system is developed; the sliding mode motion exists on every point of the switching surface and any state outside the surface is driven to the surface in a finite time [11]. Stabilization and synchronization of a class of fractional-order chaotic systems with a novel fractional-order sliding mode approach are thoroughly studied by Aghababa [12]. For a class of Lipschitz nonlinear fractional-order systems, a nonfragile observer is employed to tackle chaos synchronization problems [13]. For the open loop oscillatory systems, a novel conformal mapping based fractional-order methodology is developed to tune the classical PID controllers. The conventional pole placement tuning via linear quadratic regulator (LQR) method is also extended in some extent [14]. The primary resonance phenomenon of Duffing oscillator with fractional-order derivative is researched by Shen et al. [15].Recently, researchers have been usingH∞ control in fractional-order systems. In particular, Shen and Lama [16] proposed a state feedback H∞ controller for commensurate fractional-order systems with a prescribed H∞ performance. In the work of [17], a positive reduced-order fractional system is constructed, and the associated error system is stable with a prescribed H∞ performance for a stable positive fractional-order system. For the fractional-order systems with time-delay, Wang and Gao designed a fractional-order proportional-derivative (PD) controller with H∞ performance [18]. Most recently, a novel fractional-order sliding mode controller for output tracking of a time-varying reference signal is designed and the proposed method is applied to a fractional-order gyroscope model [19]. Consensus problems of fractional-order systems with nonuniform input and communication delays over directed static networks are studied based on a frequency-domain approach and generalized Nyquist stability criterion; some results are obtained to ensure the consensus of the fractional-order systems [20]. For the consensus problems of delayed fractional-order systems and multiagent systems, see [21, 22] and the references therein. Furthermore, intelligent systems can also be incorporated into the fractional-order controller design. By using the chaotic ant swarm system, fractional-order PID controller with objective function composed of overshoot, steady-state error, raising time, and settling time is studied [23].Terminal sliding-mode control (TSMC) is a well-known efficient control scheme in the field of variable structure control, which has been widely applied to linear and nonlinear systems [24–28]. TSMC uses nonlinear sliding surfaces and the nonlinear switching hyperplanes can improve the transient performance substantially. Besides, compared with the conventional SMC with linear sliding surface, TSMC offers some superior properties such as finite time convergence and higher control precision. Dadras firstly introduces the fractional-order TSMC (FO-TSMC) to the integer-order nonlinear systems. However, there remain some unsolved problems in the paper [29], such as the chattering problem of control effort and the strict proof of convergence time to a given vicinity around the origin on the sliding surface.In this paper, a new fractional-order fast terminal sliding surface is proposed to ensure fast convergence, a reaching law is developed to eliminate the chattering and theoretical proof of the finite time convergence is given. Moreover, a fractional-order nonsingular fast terminal sliding mode controller is proposed to achieve the fast convergence and eliminate chattering; integer-order plants with uncertainties and unmodeled dynamics are simulated to verify the newly proposed controller.The rest of this paper is organized as follows. In Section2, some preliminaries and definitions of fractional-order calculus are introduced. In Section 3, the system model under investigation and the problem formulation are presented. In Section 4, the FO-TSMC is briefly reviewed; a fractional-order fast terminal sliding mode surface design and finite time stability analysis are done in Section 5. In Section 6, a fractional-order nonsingular fast terminal sliding mode surface design and stability analysis are researched. In Section 7, the effectiveness of the proposed control schemes are illustrated by numerical examples and the results are compared with the FO-TSMC. Finally, some concluding remarks are included in Section 8.


## 2. Basic Definition and Preliminaries of Fractional-Order Calculus

Fractional-order integration and differentiation are the generalization of the integer-order ones. Efforts to extend the specific definitions of the traditional integer-order to the more general arbitrary order context led to different definitions for fractional derivatives [4]. Riemann-Liouville definition is adopted in this paper.Definition 1 (see [2]).
Theαth-order Riemann-Liouville fractional derivative of function f(t) with respect to t0 and t is given by
(1)Dαf(t)=1Γ(m-α)dmdtm∫t0t‍f(τ)(t-τ)α-m+1dτ,
and the Riemann-Liouville definition of the αth-order fractional integration is given by
(2)t0Itαf(t)=1Γ(α)∫t0t‍f(τ)(t-τ)1-αdτ,
where m-1≤α<m, m is the first integer larger than α, and Γ is the Gamma function.Lemma 2 (see [2]).
If the Riemann-Liouville fractional derivativeaDtαf(t)(n-1≤α<n) of a function f(t) is integrable, then
(3)aDt-α(Dtβaf(t))=Datβ-αf(t)-∑j=1n‍[aDtβ-jf(t)]t=a(t-a)α-jΓ(1+α-j).Lemma 3 (see [30]).
Fractional derivatives and integrals have the following properties, namely,
Semigroup Property:(4)aDt-αaDt-βf(t)=Dat-α-βf(t).
Composition Rules:
(5)dndxn[Datαf(t)]=Datn+αf(t),(6)aDtα[dndxnf(t)]=Datn+αf(t)-∑j=0n-1‍f(j)(a)(t-a)j-α-nΓ(1+j-α-n),
where α>0,  t≥a,  β>0, and n-1≤α<n.Lemma 4 (see [4]).
The fractional integration operatoraItα with α>0 is bounded,
(7)∥Iαy∥≤K∥y∥p,0≤K<∞,1≤p≤∞.


## 3. Problem Statement and Preliminaries

Consider the following nonlinear systems(8)x¨=F(x,t)+G(x,t)u(t)+d(x,t),
where x=[x1,x2,…,xn]T, u(t) is the control effort, d(x,t) is the external disturbance, F(x,t)=Fn(x,t)+ΔF(x,t),  ΔF(x,t) is an uncertain term representing the un-modeled dynamics or structural variation of the system, G(x,t)=Gn(x,t)+ΔG(x,t), and ΔG(x,t) is the input uncertainty. In our study of this nonlinear system, we will impose one assumption.Assumption.
The lumped uncertaintyL(x,t) is defined as
(9)L(x,t)=ΔF(x,t)+ΔG(x,t)u(t)+d(x,t),
where L(x,t) satisfies the condition that ∥D1+α(L(x,t))∥≤δ and δ is a given positive constant.Remark 5.
It is needed to emphasis that this hypothesis is not restrictive as it can be satisfied by many dynamical systems in the literature [29], such as inverted pendulum, chaotic gyro [31], magnetic bearing [32, 33], and ship roll motion [34]. We will remove this assumption later in Section 6.
The tracking error can be defined as(10)e(t)=xd(t)-x(t).The control objective considered in this paper is to design a robust fractional-order terminal sliding mode controller to stabilize state variables in finite time. In the next section, we will discuss how to design the appropriate switching surface and the fractional-order fast terminal sliding mode controller.


## 4. Review of Fractional-Order Terminal Sliding Mode Control (FO-TSMC)

Theorem 6 (see [29]).
Let the sliding surface be defined as(11)s=Dαe¨+λe(t)p/q,α∈(0,1),
where p and q are positive odd integers. Consider the system (8) in which the lumped uncertainties satisfy the constraint that ||Dα+1(L(x,t))||≤δ. If the terminal sliding mode control law is designed as follows:
(12)u(t)=Gn-1[(λpqe(p/q)-1e˙+ks+Ksgn(s))x¨d(t)-Fn(x,t)+D-1-αmmmml×(λpqe(p/q)-1e˙+ks+Ksgn(s))+δ],
where k and K are positive constants, sgn(s) is the signum function of sliding surface s, and λ is a positive design constant, then the tracking error converges to zero in finite time.The derivative of (11) with respect to time can be represented by
(13)s˙=D1+αe¨+λpqe(t)(p/q)-1e˙.Remark 7.
In this paper, to eliminate the chattering, the control input is modified as(14)u(t)=Gn-1[(λpqe(p/q)-1e˙+ks+Ksigρ(s))x¨d(t)-Fn(x,t)mmmmn+D-1-α(λpqe(p/q)-1e˙+ks+Ksigρ(s))+δ],
where ρ is a positive design parameter, sigρ(s)=|s|ρsgn(s).


## 5. Fractional Order Fast Terminal Sliding-Mode Control (FO-FTSMC)

The fractional-order sliding surface can be defined as follows:(15)s=Dαe¨+k1siga1(e)+k2siga2(e),
where 0≤α<1, k1=diag(k11,…,k1n)>0, k2=diag(k21,…,k2n)>0, sigai(e)=|e|aisgn(e)(i=1,2),  a1>1, and 0<a2=p/q<1, where p and q are odd numbers. Note that
(16)ddtsigai(e)=ai|e|ai-1e˙=aisigai-1(e)sgn(e)e˙.
A fast terminal sliding mode type reaching law is defined as
(17)s˙=D1+αe¨+(k-1|e|a1-1+k-2|e|a2-1)e˙,
where k-1=k1a1 and k-2=k2a2. If the terminal mode control law is designed as follows:
(18)u=Gn-1[x¨d(t)+D-α-1((k-1|e|a1-1+k-2|e|a2-1)e˙)mmmmn-Fn(x,t)+D-α-1(γ1s+γ2sigρ(s)+δsgn(s))],
where 0<ρ<1, the following theorem holds.Theorem 8.
Let the sliding surface be defined by (15), where p and q are positive odd integers. Consider the system (8) in which the lumped uncertainties satisfy the constraint that ∥Dα+1(L(x,t))∥≤δ. If the terminal sliding mode control law is set as (18), the tracking error converges to zero in finite time.Proof.
Let the Lyapunov function be defined as(19)V=12sTs.
Taking derivative of (19) with respect to time, using (5), (18), and f(t)=D1+α(D-(1+α)f(t)), we have
(20)V˙=sTs˙=sT(D1+αe¨+(k-1|e|a1-1+k-2|e|a2-1)e˙)=sT(D1+α(x¨d(t)-Fn(x,t)-Gn(x,t)u(t)-L(x,t))mmml+(k-1|e|a1-1+k-2|e|a2-1)e˙)=sT(-D1+α(D-α-1((k-1|e|a1-1+k-2|e|a2-1)e˙))mmml-D1+α(D-α-1(γ1s+γ2sigρ(s)+δsgn(s)))mmml-D1+α(L(x,t))+(k-1|e|a1-1+k-2|e|a2-1)e˙)=sT(-γ1s-γ2sigρ(s)-δsgn(s)-D1+α(L(x,t)))=sT(-γ1s-γ2sigρ(s))-δ|s|-sTD1+α(L(x,t))≤sT(-γ1s-γ2sigρ(s))=-γ1∥s∥2-γ2∥s∥ρ+1.
Choosing γ1>0 and γ2>0 appropriately, we can obtain V˙<0 when ∥s∥≠0, so the sliding motion converges to zero. In order to verify that the tracking error occurs in finite time, we can define a stopping time as follows:
(21)τs=inf{t≥tr∣e(t)=0},
where tr is the time for the system states to reach the sliding surface and s(tr)=0. Taking the concept of fractional integral and derivative operators into (15), we have
(22)D-α(Dαe¨(t))=D-α(-k1siga1e-k2siga2e).
By Lemma 2, the following holds:
(23)e¨(t)-[Dtrtα-1e¨(t)]t=tr(t-tr)α-1Γ(α)=-D-α(k1siga1e+k2siga2e).
Since α-1<0, it follows that
(24)[Dtrtα-1e¨(t)]t=tr(t-tr)α-1Γ(α)=0.
Combining (23) and (24), we have
(25)e¨(t)=-D-α(k1siga1e+k2siga2e).
By the Riemann-Liouville definition, Lemma 2, operator trDt-2, and the associativity law, we deduce
(26)e(t)-[Dtrte(t)]t=tr(t-tr)2-e(tr)=-trDt-2-α(k1siga1e+k2siga2e).
According to Lemma 4, we have
(27)Dt-2-αtr(k1siga1e+k2siga2e)≤K1∥ea1∥+K2∥ea2∥,
where K1=Kk1 and K2=Kk2. It follows that
(28)∥e(t)-[Dtrte(t)]t=tr(t-tr)2∥-∥e(tr)∥≤K1∥ea1∥+K2∥ea2∥.
Noting that e(ts)=0, it yields
(29)∥[Dtrte(t)]t=tr2∥(ts-tr)≤∥e(tr)∥.
We will study (29) in two cases, if ∥e˙(tr)∥≠0, it yields
(30)ts≤2∥e(tr)∥∥e˙(tr)∥+tr.
If ∥e˙(tr)∥=0,  e(tr)=0, we have ts=tr. In the following, consider the case that e(tr)≠0. We know that ∥e˙(tr)∥=0,  ∥e(t)∥≤∥e(tr)∥. For e(t)<0, we have e(t)→0 as t→∞; for e(t)>0, we have e(t)→0 as t→∞. For simplicity, we just consider the case ∥e˙(tr)∥=0,  e(tr)>0, and t≥tr. The proof of another case is similar. By composition rules of Lemma 3 and (15), we have
(31)Dαe¨(t)=D1+αe˙(t)-e˙(tr)(t-tr)-α-nΓ(1-α-n)=D1+αe˙(t)=-k1siga1e-k2siga2e≤-k1siga1e=-k1ea1.
Since ∥e˙(tr)∥=0, it follows that
(32)∑j=12[Dtrtα+1-je˙(t)]t=tr(t-tr)α+1-jΓ(2-j)=0,(33)D-(α+1)(D1+αe˙(t))mm=e˙(t)-∑j=12[Dtrtα+1-je˙(t)]t=tr(t-tr)α+1-jΓ(2-j)=e˙(t).
Based on ∥e˙(tr)∥=0 and e(tr)>0, by Definition 1, we have
(34)D-(α+1)(-k1ea1)mm=-k1Γ(α+1)∫trt‍ea1(τ)(t-τ)αdτ≤-k1Γ(α+1)∫trt‍ea1(τ)(ts-tr)αdτ,
where 0<(t-τ)α≤(ts-tr)α whenever t≥tr. Thus, (33) can be simplified as
(35)e˙(t)≤-k1Γ(α+1)∫trt‍ea1(τ)(ts-tr)αdτ,
integrating (35) with respect to t on interval [tr,ts] and combining with e(ts)=0 and (34), the following inequality holds:
(36)-e(tr)≤-k1Γ(α+1)(ts-tr)-α∫trts‍∫trt‍ea1(τ)dτdt.
For the double integral in (36), we apply the mean value theorem, there exist ξ1∈(tr,ts) and e(ξ1)>0, satisfy
(37)-e(tr)≤-k1ea1(ξ1)(ts-tr)2-α2Γ(α+1).
Finally, we get
(38)ts≤tr+(2Γ(α+1)e(tr)k1·ea1(ξ1))1/(2-α).
From (38) we can see that ts cannot be infinity; actually, the right hand of (38) is finite when tr<t<ts. It can be concluded that the system trajectories attain to reference trajectories in finite time.Remark 9.
The fractional sliding surface (15) is slightly different from the FTSMC [24] which is expressed as
(39)s=e˙+k1siga1(e)+k2siga2(e),
where k1>0,  k2>0,  a1>1, and 0<a2<1. Note that sigaie=|e|aisgn(e)(i=1,2). Letting a2=p/q, the fractional-order fast terminal sliding-mode control (FO-FTSMC) in (15) does not have the singularity problem except e(t)=0.


## 6. Fractional-Order Nonsingular Fast Terminal Sliding-Mode Control

During the control design of TSMs, there often exists singularity problem. The reason is that the designed controllers often have nonlinear terms, such asx-α, which results in the unboundedness of the control input as x approaches to zero. In the controller design, a nonsingular item |e˙|2-a2 is adopted to avoid this problem. To achieve equivalence between convergence rate and singularity avoidance, fractional-order nonsingular fast terminal sliding-mode control (FO-NFTSMC) is proposed, and the method is described as
(40)s=Dαe+k1siga1e+k2siga2e˙=0,
where k1>0,  k2>0,  1<a2<2, and a1>a2. We choose the terminal sliding mode control law as
(41)u=Gn-1[(D1+αe+k-1|e|a1-1·e˙)x¨d-Fn(x,t)+k-2-1|e˙|2-a2mmmml·(γ1s+γ2sigρs)+δsgn(s)+k-2-1|e˙|2-a2mmmml×(D1+αe+k-1|e|a1-1·e˙)x¨d-Fn(x,t)+k-2-1|e˙|2-a2],
where k-1=k1a1,  k-2=k2a2, k-2-1=1/k-2, and γ1,γ2 are the design parameters.Theorem 10.
Let the sliding surface be defined by (40), where k1>0,  k2>0, 1<a2<2, and a1>a2. Consider the system (8) in which the lumped uncertainties satisfy the constraint that ∥L(x,t)∥≤δ. If the terminal sliding mode control law is set as (41), the tracking error converges to zero in finite time.Proof.
Let the Lyapunov function be defined as (19). Taking derivative of (19) with respect to time and using (41) yields
(42)V˙=sTs˙=sT(D1+αe+k-1|e|a1-1·e˙+k-2|e˙|a2-1·e¨)=sT(-Gn(x,t)u(t)-L(x,t))k-2|e˙|a2-1·(x¨d(t)-Fn(x,t)mmmmmmmlm-Gn(x,t)u(t)-L(x,t))k-2|e˙|a2-1)lll+sT(D1+αe+k-1|e|a1-1·e˙)=sTk-2|e˙|a2-1·(-δsgn(s)-L(x,t))lll+sT|e˙|(-γ1s-γ2sigρ(s))-sT(|e˙|-1)lll×(D1+αe+k-1|e|a1-1·e˙)≤-sT|e˙|(γ12s+γ22sigρ(s))=-|e˙|(γ12∥s∥2+γ22∥s∥ρ+1),
where γ1=γ11+γ12,  γ2=γ21+γ22, γ11>0, γ12>0, γ21>0,  γ22>0, and sT|e˙|(-γ11s-γ21sigρ(s))-sT(|e˙|-1)(D1+αe+k-1|e|a1-1·e˙)<0 when γ1 and γ2 are appropriately chosen. Thus, we can verify that V˙<0 as ∥s∥≠0,  e˙(t)≠0. The finite time reach ability will be shown by simulation. The proof is completed.Remark 11.
Equation (40) is equivalent to Yang's NFTSM [24] when α=0, Dαe=e. For 1<a2<2 and a1>a2, we can verify that no singularity problem occurred in (41). Furthermore, ∥Dα+1(L(x,t))∥≤δ is replaced by ∥L(x,t)∥≤δ which is easy to be satisfied in practice.Remark 12.
The sliding surface proposed by Liang is defined ass=e+k1siga1e+k2siga2e˙=0 [24]; we replace the error e by Dαe; thus, (40) is obtained. We can see that the new sliding surface can perform better if appropriate design parameters are chosen.


## 7. Simulation Results

In this section, we will give three illustrative examples to show the applicability and efficiency of the proposed controller. The first example is a magnetic bearing system, the second one is a single inverted pendulum, and the last one is a two-link manipulator.


### 7.1. Magnetic Bearing System

Consider the uncertain magnetic bearing system [29, 33] with the following form:
(43)mx¨(t)=-cx˙+kxx+kiix+d(x,t),
where m is the mass of rotor, c is the friction, and kx and ki are the position and current stiffness, respectively; d(x,t) is the external disturbance and ix is control current (control input). Moreover, the variable state x denotes the deviation of the nominal air gap which is also referred as the rotor position in the magnetic bearing system.The following parameters are employed in the numerical simulations:m=2.565kg,  c=0.001,  kx=25.2,  ki=40,  α=0.9, and x0=[2,-5]T. The parameters for FO-TSMC are defined as p=23,  q=25,  k=1,  K=5,  λ=1, and ρ=1. The parameters for FO-FTSMC are defined as k1=10,  k2=10,  a1=2,  a2=3/5,  γ1=10,  γ2=10, and ρ=2/5. The parameters for FO-NFTSMC are defined as k1=15,  k2=6,  a1=3,  a2=5/3,  γ1=15,  γ2=6, and ρ=2/5.State responses of the closed loop system are shown in Figure1. Figure 1(a) represents the state time response for tracking a desired trajectory xd=sin(t). It can be seen that the dynamics of system states are stabilized to a given motion. The corresponding s(t) and control input u(t) are illustrated in Figure 1(b). We can infer that, among the three fractional-order controllers, FO-FTSMC can track the reference signal in the shortest time, while the control input is slightly larger than FO-NFTSMC and FO-TSMC. The time taken by FO-NFTSMC is shorter than FO-TSMC, while the control input is slightly smaller than FO-NFTSMC and FO-TSMC.(a) The time response of the magnetic bearing system states (xd=sin(t)). (b) The time response of sliding surface and control input for magnetic bearing system states (xd=sin(t)).
(a)
(b)Figure2(a) shows the state responses for regulating the states to xd=0. Time response of sliding surface and the control effort of the three controllers are presented in Figure 2(b). Results show that the trajectories of FO-FTSMC and FO-NFTSMC converge faster than that of FO-TSMC. The boundary layer for FO-TSMC is ∥s∥≤2×10-2, while FO-FTSMC and FO-NFTSMC keep the boundary layer with 2×10-3 and 2×10-2.(a) The time response of the magnetic bearing system states (xd=0). (b) The time response of sliding surface and control input for magnetic bearing system states (xd=0).
(a)
(b)


### 7.2. Single Inverted Pendulum

The problem to be considered is a pole-balancing of an inverted pendulum. The system is represented by(44)x˙1=x2,x˙2=mlx22sinx1cosx1-(M+m)gsinx1mlcos2(x1)-4/3l(M+m)nn-cosx1mlcos2(x1)-4/3l(M+m)u,
where x1 is the angle (in radians) of the pendulum from the vertical, M is the mass of the cart, m is the mass of the pole, g is the gravitational constant, l is the half length of the pole, and u is the force applied to the cart. The parameters employed in this simulation of inverted pendulum are given as follows: M=1kg,  m=0.1kg,  l=0.5m, and g=9.8m/s2. In order to eliminate the chattering, a modified reaching law k1s+K2sigρ(s) is used for FO-TSMC. The parameters for FO-TSMC are defined as p=3,  q=5,  K1=15,  K2=3,  λ=10,  α=0.9, and ρ=4/5. The parameters for FO-FTSMC are defined as k1=10,  k2=10,  a1=2,  a2=3/5,  α=0.9,  γ1=20,  γ2=20, and ρ=4/5. The parameters for FO-NFTSMC are defined as k1=3,  k2=3,  a1=2,  a2=5/3,  α=0.9,  γ1=3,  γ2=3, and ρ=3/5. Figures 3(a) and 3(b) show that the controller can track the trajectories well and the chattering of the FO-TSMC is eliminated.(a) The time response of the single inverted pendulum states (xd=0). (b) The time response of sliding surface and control input for single inverted pendulum system states (xd=0).
(a)
(b)


### 7.3. Two-Link Manipulator System

A two-link rigid robotic manipulator is used to evaluate the performance of the proposed adaptive FO-FTSMC and FO-NFTSMC. The dynamic equation of the robotic manipulator is given by Feng et al. [28] as
(45)M(q)q¨+C(q,q˙)+G(q)=τ(t)+τd,
where q,q˙,q¨∈ℝ2 are the vectors of joint angular position, velocity, and acceleration, respectively. M(q) is the 2×2 symmetric positive-definite inertia matrix, C(q,q˙) is the 2×1 vector containing Coriolis and centrifugal forces, G(q) is the 2×1 gravitational torque, τ is 2×1 vector of applied joint torques that are actually the control inputs and τd denotes the 2×1 bounded input disturbance vector. Moreover, the involved matrices can be expressed as
(46)M(q)=[a11(q)a12(q)a21(q)a22(q)],C(q,q˙)=[-β12(q2)q˙12-2β12(q2)q˙1q˙2β12(q2)q˙22],τd=[τd1τd2],G(q)=[(m1+m2)gr1cos(q2)+m2gr2cos(q1+q2)m2gr2cos(q1+q2)],τ=[τ1τ2],a11(q)=(m1+m2)r12+m2r22+2m2r1r2cos(q2)+J1,a12(q)=a21(q)=m2r22+m2r1r2cos(q2),a22(q)=m2r22+J2.
The system parameter values are m1=0.5kg,  m2=1.5kg, J1=5kgm,  J2=5kgm, r1=1m, and r2=0.8m. The parameters for FO-TSMC are defined as p=3,  q=5,  K1=[5,10]T,  K2=[5,10]T,  λ=[10,30]T, and α=0.9. The parameters for FO-FTSMC are defined as k1=15,  k2=6,  a1=2,  a2=3/5, γ1=[5,10]T, and γ2=[5,12]T. The parameters for FO-NFTSMC are defined as k1=[5,10]T,  k2=[10,30]T,  a1=[2,2]T,  a2=[5/3,5/3]T,  γ1=[5,10]T,  γ2=[10,35]T,  ρ=3/5, τd1=2sin(t)+0.5sin(200πt), and τd2=cos(2t)+0.5sin(200πt). The reference signal is set as xd=[1.25-7e-t/5+7e-4t/20,1.25+e-t-e-4t/4]T.In order to compare the performance of the three controllers, the trajectories are listed in Figure4(a), control efforts τ1,τ2 are shown in Figure 4(b), and the sliding surface s1,s2 are depicted in Figure 5. FO-NFTSMC and FO-FTSMC can keep the boundary layer within 0.1 while FO-TSMC is relatively large; the main possible cause of this phenomenon is that the sliding surface of FO-TSMC will need a larger control effort to guarantee the reaching law to track the desired trajectory. For the FO-NFTSMC and FO-FTSMC, the coupled term sigaie·e˙(i=1,2) have the ability to attenuate the chattering whether ee˙ is large or small. The simulation results show that the newly proposed method can eliminate the chattering by carefully tuning the parameters.(a) The time response of the two-link manipulator system states. (b) The control input for two-link manipulator system states.
(a)
(b)Figure 5
The time response of sliding surface for two-link manipulator system.


## 7.1. Magnetic Bearing System

Consider the uncertain magnetic bearing system [29, 33] with the following form:
(43)mx¨(t)=-cx˙+kxx+kiix+d(x,t),
where m is the mass of rotor, c is the friction, and kx and ki are the position and current stiffness, respectively; d(x,t) is the external disturbance and ix is control current (control input). Moreover, the variable state x denotes the deviation of the nominal air gap which is also referred as the rotor position in the magnetic bearing system.The following parameters are employed in the numerical simulations:m=2.565kg,  c=0.001,  kx=25.2,  ki=40,  α=0.9, and x0=[2,-5]T. The parameters for FO-TSMC are defined as p=23,  q=25,  k=1,  K=5,  λ=1, and ρ=1. The parameters for FO-FTSMC are defined as k1=10,  k2=10,  a1=2,  a2=3/5,  γ1=10,  γ2=10, and ρ=2/5. The parameters for FO-NFTSMC are defined as k1=15,  k2=6,  a1=3,  a2=5/3,  γ1=15,  γ2=6, and ρ=2/5.State responses of the closed loop system are shown in Figure1. Figure 1(a) represents the state time response for tracking a desired trajectory xd=sin(t). It can be seen that the dynamics of system states are stabilized to a given motion. The corresponding s(t) and control input u(t) are illustrated in Figure 1(b). We can infer that, among the three fractional-order controllers, FO-FTSMC can track the reference signal in the shortest time, while the control input is slightly larger than FO-NFTSMC and FO-TSMC. The time taken by FO-NFTSMC is shorter than FO-TSMC, while the control input is slightly smaller than FO-NFTSMC and FO-TSMC.(a) The time response of the magnetic bearing system states (xd=sin(t)). (b) The time response of sliding surface and control input for magnetic bearing system states (xd=sin(t)).
(a)
(b)Figure2(a) shows the state responses for regulating the states to xd=0. Time response of sliding surface and the control effort of the three controllers are presented in Figure 2(b). Results show that the trajectories of FO-FTSMC and FO-NFTSMC converge faster than that of FO-TSMC. The boundary layer for FO-TSMC is ∥s∥≤2×10-2, while FO-FTSMC and FO-NFTSMC keep the boundary layer with 2×10-3 and 2×10-2.(a) The time response of the magnetic bearing system states (xd=0). (b) The time response of sliding surface and control input for magnetic bearing system states (xd=0).
(a)
(b)


## 7.2. Single Inverted Pendulum

The problem to be considered is a pole-balancing of an inverted pendulum. The system is represented by(44)x˙1=x2,x˙2=mlx22sinx1cosx1-(M+m)gsinx1mlcos2(x1)-4/3l(M+m)nn-cosx1mlcos2(x1)-4/3l(M+m)u,
where x1 is the angle (in radians) of the pendulum from the vertical, M is the mass of the cart, m is the mass of the pole, g is the gravitational constant, l is the half length of the pole, and u is the force applied to the cart. The parameters employed in this simulation of inverted pendulum are given as follows: M=1kg,  m=0.1kg,  l=0.5m, and g=9.8m/s2. In order to eliminate the chattering, a modified reaching law k1s+K2sigρ(s) is used for FO-TSMC. The parameters for FO-TSMC are defined as p=3,  q=5,  K1=15,  K2=3,  λ=10,  α=0.9, and ρ=4/5. The parameters for FO-FTSMC are defined as k1=10,  k2=10,  a1=2,  a2=3/5,  α=0.9,  γ1=20,  γ2=20, and ρ=4/5. The parameters for FO-NFTSMC are defined as k1=3,  k2=3,  a1=2,  a2=5/3,  α=0.9,  γ1=3,  γ2=3, and ρ=3/5. Figures 3(a) and 3(b) show that the controller can track the trajectories well and the chattering of the FO-TSMC is eliminated.(a) The time response of the single inverted pendulum states (xd=0). (b) The time response of sliding surface and control input for single inverted pendulum system states (xd=0).
(a)
(b)


## 7.3. Two-Link Manipulator System

A two-link rigid robotic manipulator is used to evaluate the performance of the proposed adaptive FO-FTSMC and FO-NFTSMC. The dynamic equation of the robotic manipulator is given by Feng et al. [28] as
(45)M(q)q¨+C(q,q˙)+G(q)=τ(t)+τd,
where q,q˙,q¨∈ℝ2 are the vectors of joint angular position, velocity, and acceleration, respectively. M(q) is the 2×2 symmetric positive-definite inertia matrix, C(q,q˙) is the 2×1 vector containing Coriolis and centrifugal forces, G(q) is the 2×1 gravitational torque, τ is 2×1 vector of applied joint torques that are actually the control inputs and τd denotes the 2×1 bounded input disturbance vector. Moreover, the involved matrices can be expressed as
(46)M(q)=[a11(q)a12(q)a21(q)a22(q)],C(q,q˙)=[-β12(q2)q˙12-2β12(q2)q˙1q˙2β12(q2)q˙22],τd=[τd1τd2],G(q)=[(m1+m2)gr1cos(q2)+m2gr2cos(q1+q2)m2gr2cos(q1+q2)],τ=[τ1τ2],a11(q)=(m1+m2)r12+m2r22+2m2r1r2cos(q2)+J1,a12(q)=a21(q)=m2r22+m2r1r2cos(q2),a22(q)=m2r22+J2.
The system parameter values are m1=0.5kg,  m2=1.5kg, J1=5kgm,  J2=5kgm, r1=1m, and r2=0.8m. The parameters for FO-TSMC are defined as p=3,  q=5,  K1=[5,10]T,  K2=[5,10]T,  λ=[10,30]T, and α=0.9. The parameters for FO-FTSMC are defined as k1=15,  k2=6,  a1=2,  a2=3/5, γ1=[5,10]T, and γ2=[5,12]T. The parameters for FO-NFTSMC are defined as k1=[5,10]T,  k2=[10,30]T,  a1=[2,2]T,  a2=[5/3,5/3]T,  γ1=[5,10]T,  γ2=[10,35]T,  ρ=3/5, τd1=2sin(t)+0.5sin(200πt), and τd2=cos(2t)+0.5sin(200πt). The reference signal is set as xd=[1.25-7e-t/5+7e-4t/20,1.25+e-t-e-4t/4]T.In order to compare the performance of the three controllers, the trajectories are listed in Figure4(a), control efforts τ1,τ2 are shown in Figure 4(b), and the sliding surface s1,s2 are depicted in Figure 5. FO-NFTSMC and FO-FTSMC can keep the boundary layer within 0.1 while FO-TSMC is relatively large; the main possible cause of this phenomenon is that the sliding surface of FO-TSMC will need a larger control effort to guarantee the reaching law to track the desired trajectory. For the FO-NFTSMC and FO-FTSMC, the coupled term sigaie·e˙(i=1,2) have the ability to attenuate the chattering whether ee˙ is large or small. The simulation results show that the newly proposed method can eliminate the chattering by carefully tuning the parameters.(a) The time response of the two-link manipulator system states. (b) The control input for two-link manipulator system states.
(a)
(b)Figure 5
The time response of sliding surface for two-link manipulator system.


## 8. Conclusion

In this paper, the problem of designing FO-FTSMC and FO-NFTSMC for a class of uncertain dynamical systems is investigated. The uncertainties are considered in the system state matrices. Based on the Lyapunov stability criteria, the fractional-order terminal sliding mode control laws are designed and the finite time stability of the closed-loop systems is guaranteed under the proposed controller. By carefully chosing the tuning parameters, the simulation results have demonstrated the effectiveness and robustness of the proposed method. Besides, the two proposed controllers are compared with the FO-TSMC; results show that FO-FTSMC and FO-NFTSMC are superior to FO-TSMC.



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*Source: 384921-2013-12-25.xml*