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

This paper introduces a novel fractional fast terminal slidingmode control strategy for a class of dynamical systemswith 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.


Introduction
In the past thirty years or so, fractional calculus has been found in numerous diverse and widespread fields of science and engineering [1][2][3][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 leadlag 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 fractionalorder 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 fractionalorder systems with time-delay, Wang and Gao designed a fractional-order proportional-derivative (PD) controller with H ∞ performance [18].Most recently, a novel fractionalorder 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 fractionalorder 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, fractionalorder 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][25][26][27][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 fractionalorder 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.

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 th-order Riemann-Liouville fractional derivative of function () with respect to  0 and  is given by and the Riemann-Liouville definition of the th-order fractional integration is given by where  − 1 ≤  < ,  is the first integer larger than , and Γ is the Gamma function.
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 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.

Review of Fractional-Order Terminal Sliding Mode Control (FO-TSMC)
Theorem 6 (see [29]).Let the sliding surface be defined as where  and  are positive odd integers.Consider the system (8) in which the lumped uncertainties satisfy the constraint that ‖ +1 ((, ))‖ ≤ .If the terminal sliding mode control law is designed as follows: where  and  are positive constants, sgn() is the signum function of sliding surface , and  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 Remark 7. In this paper, to eliminate the chattering, the control input is modified as where  is a positive design parameter, sig  () = ||  sgn().

Fractional Order Fast Terminal Sliding-Mode Control (FO-FTSMC)
The fractional-order sliding surface can be defined as follows: where and 0 <  2 = / < 1, where  and  are odd numbers.Note that A fast terminal sliding mode type reaching law is defined as where If the terminal mode control law is designed as follows: where 0 <  < 1, the following theorem holds.
Theorem 8. Let the sliding surface be defined by (15), where  and  are positive odd integers.Consider the system (8) in which the lumped uncertainties satisfy the constraint that ‖ +1 ((, ))‖ ≤ .If the terminal sliding mode control law is set as (18), the tracking error converges to zero in finite time.

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  − , which results in the unboundedness of the control input as  approaches to zero.In the controller design, a nonsingular item | ė | 2− 2 is adopted to avoid this problem.To achieve equivalence between convergence rate and singularity avoidance, fractional-order nonsingular fast terminal slidingmode control (FO-NFTSMC) is proposed, and the method is described as where  1 > 0,  2 > 0, 1 <  2 < 2, and  1 >  2 .We choose the terminal sliding mode control law as where and  1 ,  2 are the design parameters.
Proof.Let the Lyapunov function be defined as (19).Taking derivative of (19) with respect to time and using (41) yields where  1 = ) < 0 when  1 and  2 are appropriately chosen.Thus, we can verify that V < 0 as ‖‖ ̸ = 0, ė () ̸ = 0.The finite time reach ability will be shown by simulation.The proof is completed.
Remark 12.The sliding surface proposed by Liang is defined as  = + 1 sig  1 + 2 sig  2 ė = 0 [24]; we replace the error  by   ; thus, (40) is obtained.We can see that the new sliding surface can perform better if appropriate design parameters are chosen.

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.

Magnetic Bearing System.
Consider the uncertain magnetic bearing system [29,33] with the following form: where  is the mass of rotor,  is the friction, and   and   are the position and current stiffness, respectively; (, ) is the external disturbance and   is control current (control input).Moreover, the variable state  denotes the deviation of the nominal air gap which is also referred as the rotor position in the magnetic bearing system.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   = sin().It can be seen that the dynamics of system states are stabilized to a given motion.The corresponding () and control input () 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.
where  1 is the angle (in radians) of the pendulum from the vertical,  is the mass of the cart,  is the mass of the pole,  is the gravitational constant,  is the half length of the pole, and  is the force applied to the cart.The parameters employed in this simulation of inverted pendulum are given as follows:  = 1 kg,  = 0.1 kg,  = 0.5 m, and  = 9.8 m/s 2 .In order to eliminate the chattering, a modified reaching law

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   where , q , q ∈ R 2 are the vectors of joint angular position, velocity, and acceleration, respectively.() is the 2 × 2 symmetric positive-definite inertia matrix, (, q ) is the 2 × 1 vector containing Coriolis and centrifugal forces, () is the 2 × 1 gravitational torque,  is 2 × 1 vector of applied joint torques that are actually the control inputs and   denotes the 2 × 1 bounded input disturbance vector.Moreover, the involved matrices can be expressed as (46) The system parameter values are  1 = 0.In order to compare the performance of the three controllers, the trajectories are listed in Figure 4(a), control efforts  1 ,  2 are shown in Figure 4(b), and the sliding surface  1 ,  2 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    ⋅ ė ( = 1, 2) have the ability to attenuate the chattering whether  ė is large or small.The simulation results show that the newly proposed method can eliminate the chattering by carefully tuning the parameters.

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.

Figure 1 :
Figure 1: (a) The time response of the magnetic bearing system states (  = sin()).(b) The time response of sliding surface and control input for magnetic bearing system states (  = sin()).

Figure 2 :
Figure 2: (a) The time response of the magnetic bearing system states (  = 0).(b) The time response of sliding surface and control input for magnetic bearing system states (  = 0).

Figure 3 :
Figure 3: (a) The time response of the single inverted pendulum states (  = 0).(b) The time response of sliding surface and control input for single inverted pendulum system states (  = 0).

Figure 4 : 2 Figure 5 :
Figure 4: (a) The time response of the two-link manipulator system states.(b) The control input for two-link manipulator system states.