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

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.

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 using H_∞ 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 Section 2, 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]). The 𝛼th-order Riemann-Liouville fractional derivative of function 𝑓(𝑡) with respect to 𝑡₀ and 𝑡 is given by 𝐷^𝛼𝑓(𝑡)=1Γ()𝑑𝑚−𝛼𝑑𝑡^𝑚𝑚𝑡_𝑡0𝑓(𝜏)(𝑡−𝜏)^{𝛼−𝑚+1}𝑑𝜏, (1) and the Riemann-Liouville definition of the 𝛼th-order fractional integration is given by _𝑡0𝐼^𝛼_𝑡𝑓(𝑡)=1Γ(𝛼)^𝑡_𝑡0𝑓(𝜏)(𝑡−𝜏)^1−𝛼𝑑𝜏, (2) 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 derivative _𝑎𝐷^𝛼_𝑡𝑓(𝑡)(𝑛−1≤𝛼<𝑛) of a function 𝑓(𝑡) is integrable, then _𝑎𝐷_𝑡^−𝛼_𝑎𝐷^𝛽_𝑡𝑓(𝑡)=_𝑎𝐷_𝑡^𝛽−𝛼𝑓(𝑡)−^𝑛_𝑗=1𝑎𝐷_𝑡^𝛽−𝑗𝑓(𝑡)_𝑡=𝑎(𝑡−𝑎)^𝛼−𝑗Γ.1+𝛼−𝑗 (3)

Lemma 3 (see [30]). Fractional derivatives and integrals have the following properties, namely,
Semigroup Property: _𝑎𝐷_𝑡𝑎^−𝛼𝐷_𝑡^−𝛽𝑓(𝑡)=_𝑎𝐷_𝑡^{−𝛼−𝛽}𝑓(𝑡). (4) Composition Rules: 𝑑^𝑛𝑑𝑥^𝑛𝑎𝐷^𝛼_𝑡𝑓(𝑡)=_𝑎𝐷_𝑡^𝑛+𝛼𝑓(𝑡),_𝑎𝐷^𝛼_𝑡𝑑^𝑛𝑑𝑥^𝑛𝑓(𝑡)=_𝑎𝐷_𝑡^𝑛+𝛼𝑓(𝑡)−^𝑛−1_𝑗=0𝑓^(𝑗)(𝑎)(𝑡−𝑎)^{𝑗−𝛼−𝑛}Γ,1+𝑗−𝛼−𝑛 (5)(6) where 𝛼>0,  𝑡≥𝑎,  𝛽>0, and 𝑛−1≤𝛼<𝑛.

Lemma 4 (see [4]). The fractional integration operator _𝑎𝐼^𝛼_𝑡 with 𝛼>0 is bounded, 𝐼^𝛼𝑦≤𝐾𝑦_𝑝,0≤𝐾<∞,1≤𝑝≤∞. (7)

3. Problem Statement and Preliminaries

Consider the following nonlinear systems 𝑥̈=𝐹(𝑥,𝑡)+𝐺(𝑥,𝑡)𝑢(𝑡)+𝑑(𝑥,𝑡), (8) where 𝑥=[𝑥₁,𝑥₂,…,𝑥_𝑛]^𝑇, 𝑢(𝑡) is the control effort, 𝑑(𝑥,𝑡) is the external disturbance, 𝐹(𝑥,𝑡)=𝐹_𝑛(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)=𝐺_𝑛(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 uncertainty 𝐿(𝑥,𝑡) is defined as 𝐿(𝑥,𝑡)=Δ𝐹(𝑥,𝑡)+Δ𝐺(𝑥,𝑡)𝑢(𝑡)+𝑑(𝑥,𝑡), (9) where 𝐿(𝑥,𝑡) satisfies the condition that ‖𝐷^1+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 𝑒(𝑡)=𝑥_𝑑(𝑡)−𝑥(𝑡). (10)

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 𝑠=𝐷^𝛼𝑒̈+𝜆𝑒(𝑡)^𝑝/𝑞,𝛼∈(0,1), (11) where 𝑝 and 𝑞 are positive odd integers. Consider the system (8) in which the lumped uncertainties satisfy the constraint that ||𝐷^𝛼+1(𝐿(𝑥,𝑡))||≤𝛿. If the terminal sliding mode control law is designed as follows: 𝑢(𝑡)=𝐺_𝑛⁻¹𝑥̈_𝑑(𝑡)−𝐹_𝑛(𝑥,𝑡)+𝐷^−1−𝛼×𝜆𝑝𝑞𝑒^{(𝑝/𝑞)−1}𝑒̇+𝑘𝑠+𝐾sgn(𝑠)+𝛿, (12) 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 𝑠̇=𝐷^1+𝛼̈𝑝𝑒+𝜆𝑞𝑒(𝑡)^{(𝑝/𝑞)−1}𝑒.̇ (13)

Remark 7. In this paper, to eliminate the chattering, the control input is modified as 𝑢(𝑡)=𝐺_𝑛⁻¹𝑥̈_𝑑(𝑡)−𝐹_𝑛(𝑥,𝑡)+𝐷^−1−𝛼𝜆𝑝𝑞𝑒^{(𝑝/𝑞)−1}𝑒̇+𝑘𝑠+𝐾sig^𝜌(𝑠)+𝛿, (14) where 𝜌 is a positive design parameter, sig^𝜌(𝑠)=|𝑠|^𝜌sgn(𝑠).

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

The fractional-order sliding surface can be defined as follows: 𝑠=𝐷^𝛼𝑒̈+𝑘₁sig^𝑎1(𝑒)+𝑘₂sig^𝑎2(𝑒), (15) where 0≤𝛼<1, 𝑘₁=diag(𝑘₁₁,…,𝑘_1𝑛)>0, 𝑘₂=diag(𝑘₂₁,…,𝑘_2𝑛)>0, sig^𝑎𝑖(𝑒)=|𝑒|^𝑎𝑖sgn(𝑒)(𝑖=1,2),  𝑎₁>1, and 0<𝑎₂=𝑝/𝑞<1, where 𝑝 and 𝑞 are odd numbers. Note that 𝑑𝑑𝑡sig^𝑎𝑖(𝑒)=𝑎_𝑖|𝑒|^𝑎𝑖−1𝑒̇=𝑎_𝑖sig^𝑎𝑖−1(𝑒)sgn(𝑒)𝑒.̇ (16) A fast terminal sliding mode type reaching law is defined as 𝑠̇=𝐷^1+𝛼𝑒̈+𝑘₁|𝑒|^𝑎1−1+𝑘₂|𝑒|^𝑎2−1𝑒,̇ (17) where 𝑘₁=𝑘₁𝑎₁ and 𝑘₂=𝑘₂𝑎₂. If the terminal mode control law is designed as follows: 𝑢=𝐺_𝑛⁻¹𝑥̈_𝑑(𝑡)+𝐷^−𝛼−1𝑘₁|𝑒|^𝑎1−1+𝑘₂|𝑒|^𝑎2−1𝑒̇−𝐹_𝑛(𝑥,𝑡)+𝐷^−𝛼−1𝛾₁𝑠+𝛾₂sig^𝜌(𝑠)+𝛿sgn(𝑠), (18) 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 ‖𝐷^𝛼+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 1𝑉=2𝑠^𝑇𝑠. (19) Taking derivative of (19) with respect to time, using (5), (18), and 𝑓(𝑡)=𝐷^1+𝛼(𝐷^−(1+𝛼)𝑓(𝑡)), we have 𝑉̇=𝑠^𝑇𝑠̇=𝑠^𝑇𝐷^1+𝛼𝑒̈+𝑘₁|𝑒|^𝑎1−1+𝑘₂|𝑒|^𝑎2−1𝑒̇=𝑠^𝑇𝐷^1+𝛼𝑥̈_𝑑(𝑡)−𝐹_𝑛(𝑥,𝑡)−𝐺_𝑛()𝑢(𝑡)()+𝑥,𝑡−𝐿𝑥,𝑡𝑘₁|𝑒|^𝑎1−1+𝑘₂|𝑒|^𝑎2−1𝑒̇=𝑠^𝑇−𝐷^1+𝛼𝐷^−𝛼−1𝑘₁|𝑒|^𝑎1−1+𝑘₂|𝑒|^𝑎2−1𝑒̇−𝐷^1+𝛼𝐷^−𝛼−1𝛾₁𝑠+𝛾₂sig^𝜌(𝑠)+𝛿sgn(𝑠)−𝐷^1+𝛼(𝐿(𝑥,𝑡))+𝑘₁|𝑒|^𝑎1−1+𝑘₂|𝑒|^𝑎2−1𝑒̇=𝑠^𝑇−𝛾₁𝑠−𝛾₂sig^𝜌(𝑠)−𝛿sgn(𝑠)−𝐷^1+𝛼(𝐿(𝑥,𝑡))=𝑠^𝑇−𝛾₁𝑠−𝛾₂sig^𝜌(𝑠)−𝛿|𝑠|−𝑠^𝑇𝐷^1+𝛼(𝐿(𝑥,𝑡))≤𝑠^𝑇−𝛾₁𝑠−𝛾₂sig^𝜌(𝑠)=−𝛾₁‖𝑠‖²−𝛾₂‖𝑠‖^𝜌+1. (20) Choosing 𝛾₁>0 and 𝛾₂>0 appropriately, we can obtain 𝑉̇<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: 𝜏_𝑠=inf𝑡≥𝑡_𝑟∣𝑒(𝑡)=0, (21) where 𝑡_𝑟 is the time for the system states to reach the sliding surface and 𝑠(𝑡_𝑟)=0. Taking the concept of fractional integral and derivative operators into (15), we have 𝐷^−𝛼𝐷^𝛼𝑒(̈𝑡)=𝐷^−𝛼−𝑘₁sig^𝑎1𝑒−𝑘₂sig^𝑎2𝑒. (22) By Lemma 2, the following holds: 𝑒(̈𝑡)−_𝑡𝑟𝐷_𝑡^𝛼−1𝑒(̈𝑡)_{𝑡=𝑡𝑟}𝑡−𝑡_𝑟^𝛼−1Γ(𝛼)=−𝐷^−𝛼𝑘₁sig^𝑎1𝑒+𝑘₂sig^𝑎2𝑒. (23) Since 𝛼−1<0, it follows that _𝑡𝑟𝐷_𝑡^𝛼−1𝑒(̈𝑡)_{𝑡=𝑡𝑟}𝑡−𝑡_𝑟^𝛼−1Γ(𝛼)=0. (24) Combining (23) and (24), we have 𝑒(̈𝑡)=−𝐷^−𝛼𝑘₁sig^𝑎1𝑒+𝑘₂sig^𝑎2𝑒. (25) By the Riemann-Liouville definition, Lemma 2, operator _𝑡𝑟𝐷_𝑡⁻², and the associativity law, we deduce 𝑒(𝑡)−_𝑡𝑟𝐷_𝑡𝑒(𝑡)_{𝑡=𝑡𝑟}𝑡−𝑡_𝑟2−𝑒𝑡_𝑟=−_𝑡𝑟𝐷_𝑡^−2−𝛼𝑘₁sig^𝑎1𝑒+𝑘₂sig^𝑎2𝑒. (26) According to Lemma 4, we have _𝑡𝑟𝐷_𝑡^−2−𝛼𝑘₁sig^𝑎1𝑒+𝑘₂sig^𝑎2𝑒≤𝐾₁𝑒^𝑎1+𝐾₂𝑒^𝑎2, (27) where 𝐾₁=𝐾𝑘₁ and 𝐾₂=𝐾𝑘₂. It follows that 𝑒(𝑡)−_𝑡𝑟𝐷_𝑡𝑒(𝑡)_{𝑡=𝑡𝑟}𝑡−𝑡_𝑟2−𝑒𝑡_𝑟≤𝐾₁𝑒^𝑎1+𝐾₂𝑒^𝑎2. (28) Noting that 𝑒(𝑡_𝑠)=0, it yields _𝑡𝑟𝐷_𝑡𝑒(𝑡)_{𝑡=𝑡𝑟}2𝑡_𝑠−𝑡_𝑟≤𝑒𝑡_𝑟. (29) We will study (29) in two cases, if ‖𝑒(̇𝑡_𝑟)‖≠0, it yields 𝑡_𝑠≤2𝑒𝑡_𝑟𝑒̇𝑡_𝑟+𝑡_𝑟. (30) If ‖𝑒(̇𝑡_𝑟)‖=0,  𝑒(𝑡_𝑟)=0, we have 𝑡_𝑠=𝑡_𝑟. In the following, consider the case that 𝑒(𝑡_𝑟)≠0. We know that ‖𝑒(̇𝑡_𝑟)‖=0,  ‖𝑒(𝑡)‖≤‖𝑒(𝑡_𝑟)‖. 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(̇𝑡_𝑟)‖=0,  𝑒(𝑡_𝑟)>0, and 𝑡≥𝑡_𝑟. The proof of another case is similar. By composition rules of Lemma 3 and (15), we have 𝐷^𝛼𝑒(̈𝑡)=𝐷^1+𝛼𝑒(̇𝑡)−𝑒̇𝑡_𝑟𝑡−𝑡_𝑟^{−𝛼−𝑛}Γ(1−𝛼−𝑛)=𝐷^1+𝛼𝑒(̇𝑡)=−𝑘₁sig^𝑎1𝑒−𝑘₂sig^𝑎2𝑒≤−𝑘₁sig^𝑎1𝑒=−𝑘₁𝑒^𝑎1. (31) Since ‖𝑒(̇𝑡_𝑟)‖=0, it follows that ²_{𝑗=1𝑡𝑟}𝐷_𝑡^{𝛼+1−𝑗}𝑒(̇𝑡)_{𝑡=𝑡𝑟}𝑡−𝑡_𝑟^{𝛼+1−𝑗}Γ𝐷2−𝑗=0,_−(𝛼+1)𝐷^1+𝛼𝑒(̇𝑡)=𝑒(̇𝑡)−²_{𝑗=1𝑡𝑟}𝐷_𝑡^{𝛼+1−𝑗}𝑒(̇𝑡)_{𝑡=𝑡𝑟}𝑡−𝑡_𝑟^{𝛼+1−𝑗}Γ=𝑒(̇𝑡).2−𝑗 (32)(33) Based on ‖𝑒(̇𝑡_𝑟)‖=0 and 𝑒(𝑡_𝑟)>0, by Definition 1, we have 𝐷^−(𝛼+1)−𝑘₁𝑒^𝑎1=−𝑘₁Γ(𝛼+1)^𝑡_𝑡𝑟𝑒^𝑎1(𝜏)(𝑡−𝜏)^𝛼𝑑𝜏≤−𝑘₁Γ(𝛼+1)^𝑡_𝑡𝑟𝑒^𝑎1(𝜏)𝑡_𝑠−𝑡_𝑟^𝛼𝑑𝜏, (34) where 0<(𝑡−𝜏)^𝛼≤(𝑡_𝑠−𝑡_𝑟)^𝛼 whenever 𝑡≥𝑡_𝑟. Thus, (33) can be simplified as 𝑒(̇𝑡)≤−𝑘₁Γ(𝛼+1)^𝑡_𝑡𝑟𝑒^𝑎1(𝜏)𝑡_𝑠−𝑡_𝑟^𝛼𝑑𝜏, (35) integrating (35) with respect to 𝑡 on interval [𝑡_𝑟,𝑡_𝑠] and combining with 𝑒(𝑡_𝑠)=0 and (34), the following inequality holds: −𝑒𝑡_𝑟≤−𝑘₁Γ()𝑡𝛼+1^𝑠−𝑡_𝑟^{−𝛼𝑡𝑠}_𝑡𝑟^𝑡_𝑡𝑟𝑒^𝑎1(𝜏)𝑑𝜏𝑑𝑡. (36) For the double integral in (36), we apply the mean value theorem, there exist 𝜉₁∈(𝑡_𝑟,𝑡_𝑠) and 𝑒(𝜉₁)>0, satisfy −𝑒𝑡_𝑟≤−𝑘₁𝑒^𝑎1𝜉₁𝑡_𝑠−𝑡_𝑟^2−𝛼().2Γ𝛼+1 (37) Finally, we get 𝑡_𝑠≤𝑡_𝑟+2Γ(𝛼+1)𝑒(𝑡_𝑟)𝑘₁⋅𝑒^𝑎1(𝜉₁)^1/(2−𝛼). (38) From (38) we can see that 𝑡_𝑠 cannot be infinity; actually, the right hand of (38) is finite when 𝑡_𝑟<𝑡<𝑡_𝑠. 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 𝑠=𝑒̇+𝑘₁sig^𝑎1(𝑒)+𝑘₂sig^𝑎2(𝑒), (39) where 𝑘₁>0,  𝑘₂>0,  𝑎₁>1, and 0<𝑎₂<1. Note that sig^𝑎𝑖𝑒=|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 as 𝑥^−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 𝑠=𝐷^𝛼𝑒+𝑘₁sig^𝑎1𝑒+𝑘₂sig^𝑎2𝑒̇=0, (40) where 𝑘₁>0,  𝑘₂>0,  1<𝑎₂<2, and 𝑎₁>𝑎₂. We choose the terminal sliding mode control law as 𝑢=𝐺_𝑛⁻¹𝑥̈_𝑑−𝐹_𝑛(𝑥,𝑡)+𝑘₂⁻¹|𝑒|̇^2−𝑎2⋅𝛾₁𝑠+𝛾₂sig^𝜌𝑠+𝛿sgn(𝑠)+𝑘₂⁻¹|𝑒|̇^2−𝑎2×𝐷^1+𝛼𝑒+𝑘₁|𝑒|^𝑎1−1⋅𝑒̇, (41) where 𝑘₁=𝑘₁𝑎₁,  𝑘₂=𝑘₂𝑎₂, 𝑘₂⁻¹=1/𝑘₂, 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 𝑉̇=𝑠^𝑇𝑠̇=𝑠^𝑇𝐷^1+𝛼𝑒+𝑘₁|𝑒|^𝑎1−1⋅𝑒̇+𝑘₂|𝑒|̇^𝑎2−1⋅𝑒̈=𝑠^𝑇𝑘₂|𝑒|̇^𝑎2−1⋅𝑥̈_𝑑(𝑡)−𝐹_𝑛(𝑥,𝑡)−𝐺_𝑛(𝑥,𝑡)𝑢(𝑡)−𝐿(𝑥,𝑡)+𝑠^𝑇𝐷^1+𝛼𝑒+𝑘₁|𝑒|^𝑎1−1⋅𝑒̇=𝑠^𝑇𝑘₂|𝑒|̇^𝑎2−1⋅−𝛿sgn(𝑠)−𝐿(𝑥,𝑡)+𝑠^𝑇|𝑒|̇−𝛾₁𝑠−𝛾₂sig^𝜌(𝑠)−𝑠^𝑇𝑒|̇)×𝐷(|−1_1+𝛼𝑒+𝑘₁|𝑒|^𝑎1−1⋅𝑒̇≤−𝑠^𝑇|𝑒|̇𝛾₁₂𝑠+𝛾₂₂sig^𝜌(𝑠)=−|𝑒|̇𝛾₁₂‖𝑠‖²+𝛾₂₂‖𝑠‖^𝜌+1, (42) where 𝛾₁=𝛾₁₁+𝛾₁₂,  𝛾₂=𝛾₂₁+𝛾₂₂, 𝛾₁₁>0, 𝛾₁₂>0, 𝛾₂₁>0,  𝛾₂₂>0, and 𝑠^𝑇|𝑒|̇(−𝛾₁₁𝑠−𝛾₂₁sig^𝜌(𝑠))−𝑠^𝑇(|𝑒|̇−1)(𝐷^1+𝛼𝑒+𝑘₁|𝑒|^𝑎1−1⋅𝑒)̇<0 when 𝛾₁ and 𝛾₂ are appropriately chosen. Thus, we can verify that 𝑉̇<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, 𝐷^𝛼𝑒=𝑒. For 1<𝑎₂<2 and 𝑎₁>𝑎₂, we can verify that no singularity problem occurred in (41). Furthermore, ‖𝐷^𝛼+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 as 𝑠=𝑒+𝑘₁sig^𝑎1𝑒+𝑘₂sig^𝑎2𝑒̇=0 [24]; we replace the error 𝑒 by 𝐷^𝛼𝑒; 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: 𝑚𝑥̈(𝑡)=−𝑐𝑥̇+𝑘_𝑥𝑥+𝑘_𝑖𝑖_𝑥+𝑑(𝑥,𝑡), (43) where 𝑚 is the mass of rotor, 𝑐 is the friction, and 𝑘_𝑥 and 𝑘_𝑖 are the position and current stiffness, respectively; 𝑑(𝑥,𝑡) is the external disturbance and 𝑖_𝑥 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,  𝑘_𝑥=25.2,  𝑘_𝑖=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 Figure 1. Figure 1(a) represents the state time response for tracking a desired trajectory 𝑥_𝑑=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)

(b)

Figure 1 

(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)).

Figure 2(a) shows the state responses for regulating the states to 𝑥_𝑑=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)

(b)

Figure 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).

7.2. Single Inverted Pendulum

The problem to be considered is a pole-balancing of an inverted pendulum. The system is represented by 𝑥̇₁=𝑥₂,𝑥̇₂=𝑚𝑙𝑥²₂sin𝑥₁cos𝑥₁−(𝑀+𝑚)𝑔sin𝑥₁𝑚𝑙cos²𝑥₁()−−4/3𝑙𝑀+𝑚cos𝑥₁𝑚𝑙cos²𝑥₁−4/3𝑙(𝑀+𝑚)𝑢, (44) 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+𝐾₂sig^𝜌(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)

(b)

Figure 3 

(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).

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 𝑀𝑞𝑞̈+𝐶𝑞,𝑞̇+𝐺𝑞=𝜏(𝑡)+𝜏_𝑑, (45) where 𝑞,𝑞,̇𝑞̈∈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 𝜏_𝑑 denotes the 2×1 bounded input disturbance vector. Moreover, the involved matrices can be expressed as 𝑀𝑞=𝑎₁₁𝑞𝑎₁₂𝑞𝑎₂₁𝑞𝑎₂₂𝑞,𝐶𝑞,𝑞̇=−𝛽₁₂𝑞₂𝑞̇²₁−2𝛽₁₂𝑞₂𝑞̇₁𝑞̇₂𝛽₁₂𝑞₂𝑞̇²₂,𝜏_𝑑=𝜏_𝑑1𝜏_𝑑2,𝐺𝑞=𝑚₁+𝑚₂𝑔𝑟₁cos𝑞₂+𝑚₂𝑔𝑟₂cos𝑞₁+𝑞₂𝑚₂𝑔𝑟₂cos𝑞₁+𝑞₂,𝜏𝜏=¹𝜏₂,𝑎₁₁𝑞=𝑚₁+𝑚₂𝑟²₁+𝑚₂𝑟²₂+2𝑚₂𝑟₁𝑟₂cos𝑞₂+𝐽₁,𝑎₁₂𝑞=𝑎₂₁𝑞=𝑚₂𝑟²₂+𝑚₂𝑟₁𝑟₂cos𝑞₂,𝑎₂₂𝑞=𝑚₂𝑟²₂+𝐽₂. (46) 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, 𝜏_𝑑1=2sin(𝑡)+0.5sin(200𝜋𝑡), and 𝜏_𝑑2=cos(2𝑡)+0.5sin(200𝜋𝑡). The reference signal is set as 𝑥_𝑑=[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 Figure 4(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 sig^𝑎𝑖𝑒⋅𝑒̇(𝑖=1,2) have the ability to attenuate the chattering whether 𝑒𝑒̇ is large or small. The simulation results show that the newly proposed method can eliminate the chattering by carefully tuning the parameters.

(a)

(b)

Figure 4 

(a) The time response of the two-link manipulator system states. (b) The control input for two-link manipulator system states.

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.

Acknowledgment

This work was supported by the Education Department of Heilongjiang Province Science and Technology Research Projects.