Design of a Non-Singular Adaptive Integral-Type Finite Time Tracking Control for Nonlinear Systems With External Disturbances

This paper proposes an adaptive non-singular fast terminal sliding mode control (FTSMC) with integral surface for the finite time tracking control of nonlinear systems with external disturbances. An appropriate parameter-tuning adaptation law is derived to tackle the disturbances. A new fast terminal sliding scheme with self-tuning algorithm is proposed to synthesize the adaptive non-singular fast integral terminal sliding approach. The proposed approach has the following features: 1) It does not require the derivative of the fractional power terms with respect to time, thereby eschewing the singularity problem typically associated with TSMC; 2) It guarantees the existence of the switching phase under exogenous disturbances with unknown bounds; 3) Because of the integral terms in the sliding surface, the power functions are hidden behind the integrator; 4) It ensures chattering-free dynamics. The effectiveness of the proposed approach is assessed using both a simulation and an experimental study. The obtained results showed that the FTSM control technique guarantees that when the switching surface is reached, tracking errors converge to zero at a fast convergence rate. Additionally, the integral term offers one extra degree-of-freedom and since the time-derivative of fractional power terms is not needed in the controller, the proposed switching surface provides a comprehensive framework for singularity avoidance.


I. INTRODUCTION A. BACKGROUND AND MOTIVATION
Sliding Mode Control (SMC) is a powerful tool for solving the robust stability and tracking problem of nonlinear dynamical systems operating under various kinds of uncertainties and disturbances [1]- [7]. The key advantages of SMC are robustness to parametric uncertainties, low sensitivity to external disturbances, order reduction, fast convergence, and ease of implementation [8]- [11]. Due to these advantages, SMC has been extensively used in applications including robotics, chaotic systems, wind The associate editor coordinating the review of this manuscript and approving it for publication was Shihong Ding . power systems, etc. [12]- [16]. Although SMC guarantees robustness and performance, it suffers from the chattering phenomenon mainly caused by the high frequency switching of the SMC exciting the system's unmolded dynamics [17], [18]. A number of approaches such as the boundary-layer approach via sigmoid and saturation functions [19], disturbance-estimation and observer-based techniques [20], high-order SMC scheme [21], [22], and artificial intelligence (AI) strategies [23] have been proposed in the literature to either reduce or eliminate the chattering phenomena.
An adaptive nonlinear SMC technique with fast chatteringfree and non-overshooting responses is developed in [24] for nonlinear multi-input multi-output (MIMO) systems. In [25], VOLUME 9, 2021 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ an adaptive chattering-free SMC approach based on the proportional-integral switching manifold is proposed to stabilize MIMO systems with matched and mismatched uncertainties. In [26], under the assumption that the bound of parameter uncertainties and its first-derivative are unknown, a chattering-free SMC for a perturbed chaotic system is suggested. In [27], a chattering-free adaptive robust SMC is proposed for the synchronization of two chaotic systems with disturbances and unknown uncertainties. In [28], a dynamic output-feedback SMC strategy which eschews the chattering phenomenon and the high-gain control problem is developed for the stabilization of linear MIMO systems with uncertainties. An adaptive fuzzy chattering-free SMC is proposed in [29] for nonlinear single-input single-output (SISO) systems. In [30], a chattering-free robust control scheme based on the adaptive second-order SMC and resonant control laws is suggested for LCL shunt active power filters. In [31], an augmented chattering-free proportional-integral switching surface-based SMC design is proposed of a coalmine power grid with chaotic structure. In [32], an adaptive chattering-free SMC technique is proposed for the precision motion of a piezoelectric nano-positioning system subject to perturbations.
In conventional SMC approaches, the employed switching manifold is often linear and guarantees the asymptotic stabilization of the controlled system and the convergence of the state trajectories to zero in infinite interval. Terminal Sliding Mode (TSM) Control, on the other hand, enables finite time convergence and a smaller steady-state tracking error [33], [34]. However, the singularity problem of TSM leads to unbounded control inputs. Additionally, when the states/tracking errors are far away from zero, TSM shows a slow convergence rate compared to conventional SMC [35], [36]. Therefore, the singularity and chattering problems of TSM method must be properly addressed. The TSM control scheme is actually very sensitive around the equilibrium point and due to the fractional power terms and their negative fractional power derivatives, this method can yield unexpectedly large values leading to the singularity problem.

B. LITERATURE REVIEW
Various non-singular TSM approaches have been investigated in recent years to mitigate singularity problem. A non-singular TSM control method was proposed in [37] for rigid manipulators; however, the adaptation law was not designed in [37]. An adaptive Fast Terminal Sliding Mode (FTSM) method which removes the singular problems of the original TSM is proposed in [38] for an electromechanical actuator system. Parametric uncertainties are approximated in that method [38] using the integration of the filtered states. In [39] and [40], using the non-singular TSM concept, a finite time attitude tracker is designed to drive the angular velocity and attitude tracking errors of a spacecraft to the origin in finite time. However, those techniques considered in [39] and [40] cannot adaptively estimate the bounds of the perturbations. In [41], a modified time-varying non-singular TSM approach is presented for rigid manipulators with perturbations where the system's performance is enhanced by adding a time-varying gain in the sliding manifold. In [42], a passive finite time fault-tolerant controller according to the robust non-singular FTSM is planned for robotic manipulators with actuator faults and parametric uncertainties. An online time-delay estimator-based fault estimation algorithm is presented in [42] to approximate the actuator faults. In [43], a non-singular FTSM is combined with an adaptation technique for the stabilization of an aircraft with varying gravity center. However, the designed method [43] is only valid for a specific nonlinear model. In [44], a non-singular FTSM control method based on the tracker differentiator and extended state observer is proposed for uncertain permanent magnet synchronous motor systems. The approach [44], however, exhibits small chattering dynamics. Xin et al. [45] studied the adaptive robust non-singular FTSM control for second-order uncertain systems; however, the designed method of [45] is not presented for high-order dynamic systems. In [46], a robust adaptive gain non-singular TSM approach is provided for a tracker design of formation flying of spacecraft in a framework based on leader-follower approach, but some considerable chattering can be observed in the results of [46]. The theory of [47] studies non-singular adaptive TSM control technique for an attitude tracker design of spacecraft with faults on actuators; but again, the chattering phenomenon is obvious in the presented outcomes. In [48], an adaptive non-singular second-order FTSM controller is planned for n-link robotic manipulators; however, some high-frequency oscillations are experienced in the controller inputs. Nevertheless, to the best of our knowledge, the non-singular switching manifolds had been designed via a power function which is a ratio of odd positive integers. By employing an integral term in the proposed sliding surface, there is no restriction on the exponents and the power functions are 'hidden' behind the integrator in the sliding surface.

C. CONTRIBUTIONS
This paper proposes an adaptive non-singular fast terminal sliding mode control (FTSMC) with integral surface for the finite time tracking control of nonlinear systems with external disturbances. Its main contributions are as follows: • A non-singular FTSM strategy and proportional-integral switching surface that guarantees the finite time convergence of the switching surfaces to zero with fast convergence rate.
• A design that does not require the time-derivative of the fractional power terms in the controller, thereby avoiding the singularity problem.
• By using a bipolar sigmoid function with tunable gains instead of a signum function, an appropriate adaptation law is derived to tackle the external disturbances without any knowledge about the bounds of the perturbations.
• A new adaptation law to mitigate the chattering problem.

D. PAPER ORGANIZATION
The remainder of the paper is organized as follows. Section 2 formulates the control problem and describes the dynamics of the disturbed nonlinear system under consideration. The design process for the adaptive non-singular fast integral terminal sliding controller technique is provided in Section 3. Simulation studies and experimental results are presented in Section 4 to validate the efficiency of proposed approach. Lastly, some concluding remarks are presented in Section 5.

II. PROBLEM FORMULATION
Consider the nonlinear system with external disturbances defined by: = 0) and f (x, t) ∈ R are two bounded known smooth and nonlinear functions, and d(x, t) ∈ R is a nonlinear function representing the uncertainties and disturbances, and is assumed to fulfill |d(x, t)| ≤ D, where D denotes a known constant scalar. The nonlinear dynamic system (1) is assumed to track the references x 1d (t) ∈ R and x 2d (t) ∈ R, where x 2d (t) =ẋ 1d (t), and x 2d (t) denotes a time-differentiable function. One can consider the tracking errors as: withė 1 (t) = e 2 (t). Considering (1) and (2), one obtains: Our main objectives is to design a robust SMC-based tracking control approach that: 1) guarantees the convergence of the states of (1) to the desired trajectory; 2) ensures that when the switching surface is reached, the tracking errors converge to zero at a fast convergence rate; 3) warrants the existence of the switching phase in the presence of exogenous disturbances d(x, t) with unknown bounds; 4) properly mitigates the singularity and chattering problems.
Remark 1: The condition (5) has a structure like a boundary layer, which is an extension of the sliding function described in [49].
the absolute value of the term S i (t) is bigger than the fractional power term of |e i (t)| ≤ ε i . Then, a fast convergence rate in both cases is achieved by increasing the magnitude of the parameters ε i and ρ i .
Theorem 1: Consider the second-order nonlinear dynamic system (1) and the switching surface (4). Assume that the equivalent controller is proposed as: Then, the state trajectories of (1) converge to the desired trajectory along the sliding mode s(t) = 0.
Remark 2: It should be stated that the design of equivalent controller (6) is inspired by the work of [51]. However, that paper considered a stabilization control problem whereas here we consider a tracking control procedure and the termẋ 2d (t) is considered in the equivalent control law (6).
Theorem 2: Consider the disturbed nonlinear second-order dynamics (1) and the switching surface (4). Assume that the controller is described as where κ and γ are two arbitrary positive scalars and χ is a constant which fulfils χ ≥ D. Then, the tracking purpose of a reference signal x d (t) is satisfied and the switching surface s(t) converges to the origin within a finite time.
Proof: Construct the Lyapunov function as: Differentiating the Lyapunov function along the trajectory of (3) and (4), it follows: −ẋ 2d (t) + c 1 S 1 (e 1 (t)) + c 2 S 2 (e 2 (t))]. (15) Substituting (13) into (15), one findṡ Thus, in relation to the finite time stability theory, the manifold s(t) converges to zero and the tracking errors approach to the origin in a finite time.
In practice, the upper bound of d(x, t) is unknown and the consequent determination of χ is hard. In the next theorem, an adaptation method is planned to approximate the disturbances unknown bounds.
Note that the discontinuous sign function proposed in controller (17) causes the chattering phenomenon and results in unwanted responses in the controlled dynamics. Hence, the function sgn in (17) is replaced by a continuous hyperbolic tangent function tanh with adaptive law to enable the modification of the steepness and amplitude of the control law. As it is displayed in FIGURE 1, the steepness of the hyperbolic tangent function governs how it approximates the sign function. That is, if the steepness constant increases, the obtained function will approach the sign function thereby giving rise to the chattering problem. However, if the steepness and the gain of tanh is decreased, then we have chattering-free dynamics. Therefore, the adaptive chattering-free continuous controller is defined by: whereφ(t) is the adaptive steepness coefficient of tanh which is considered as
Theorem 4: Consider the dynamics (1) and the switching manifold (4). If the controller signal is chosen as (25) and the adaptation rules are designed aṡ
Remark 3: In order to completely eliminate the chattering problem resulting from the discontinuous sign function (sgn(s(t))), the control signal (25) and the adaptation tuning equations (27)-(28) are adapted via a hyperbolic tangent. Hence, the controller input and adaptation tuning equations are obtained as: + c 2 S 2 (e 2 (t)) + γ s(t) + κtanh(Is(t)) |s(t)| η +χ(t)tanh(φ(t)s(t))}, where I is the steepness constant of the hyperbolic tangent. Note selecting a I to be very small can lead to steady-state errors, whereas assigning too large of a value for I can give rise to the chattering problem. Thus, the value of I is required to be chosen suitably based on each specific system. Considering the tracking control problem, the performance indices are presented as the objective functions which will be used in the next section: (    (III) Integral of Square Value of control signal (ISV):

IV. SIMULATION AND EXPERIMENTAL RESULTS
To assess the performance of the proposed approach, we carry out a simulation and experimental study. Additionally, for comparison purposes, we consider the approach proposed in [14].
The time histories of the system state variables x 1 and x 2 are displayed in FIGURE 2 and FIGURE 3. It can be seen from these two figures that the suggested approach offers more accurate and much fast transient response than the approach in [14]. The comparison of the control signals is exhibited in FIGURE 4 which indicates that the planned controller yields superior vibration control. The time histories of the switching surfaces are shown in FIGURE 5. Evidently, the quick convergence of the proposed switching curve to zero in comparison with the switching surface of [14] can be observed. FIGURE 6 shows the phase plots of the states and sliding surfaces, which demonstrates that the proposed sliding surface and the system states have faster transient response compared to the method of [14]. Table 1 presents the IAE, ITAE and ISV numerical results. Using the proposed control technique, the performance indices of tracking errors and control signals are smaller compared to results of the technique of [14]. This result demonstrates the enhanced tracking performance of the proposed technique over the other existing ones. In summary, the simulation results on the VDP circuit validate the efficiency and usefulness of the newly introduced control scheme.
Two other simulations are conducted to examine the robustness of the suggested control method to different   external disturbances. To that end, the external disturbances are set to triangle and square waveforms with respectively. The simulations are rerun without retuning the controller gains. In both cases, the simulation results of the tracking performance and sliding surfaces are nearly identical to the ones for the sinusoidal disturbance d(t) = 0.3 sin(0.2π (0.1πt), and thus are not reproduced here. Time responses of the control inputs are given in FIGURE 7, which shows some chattering issues in the new conditions.

B. EXAMPLE 2: INVERTED PENDULUM SYSTEM
The inverted pendulum is a famous test system for assessing control strategies [55], [56]. This system is widely used for educational goals, and belongs to the under-actuated mechanical systems. It also has specific various real-life usages such as robotics, position control, aerospace vehicles control, etc. [55]. The main control objective is to balance or stabilize the pendulum in the inverted position. The schematic representation of the inverted pendulum system is shown   in FIGURE 8. In that figure, l p , m c , m p , θ, x and u represent respectively the pendulum length, cart mass, pendulum mass, pendulum angular position, cart position and horizontal driving force. The nonlinear motion equations are found by means of Lagrangian equations [57], [58], from which the ''cyclic'' coordinate x can simply be removed to be left with the second-order equation as [57], [59], and: or equivalently: Defining x 1 = θ and x 2 =θ, yields the following state-space presentation [60]:   It can be inferred from these figures that the position and velocity states suitably track the reference signals using the proposed method. The control input of the system is displayed in FIGURE 11, which shows smooth and fchattering-free dynamics. From FIGURE 12, it can be observed that the planned sliding surface is smooth and approaches to zero quickly. FIGURE 13 demonstrates the phase plots of the states and sliding surfaces. It can be obviously seen that the proposed sliding surface and system states converge to the equilibrium faster than those of the method of [14]. All these figures verify that the offered control technique has much better robust performance compared to the method of [14]. The IAE, ITAE and ISV comparative results are given in Table 2. It can be observed from Table 2 that the performance indices values are much less for the proposed control technique in comparison with the other method. Then, by comparing the simulation results, one can conclude that the tracking performance of the suggested control method is superior to that of the method of [14].
Two other simulations are executed to study the robustness of the proposed method to various external disturbances. For this purpose, the exterior disturbances are set to triangle and square waveforms with Control signals in the presence of (a) triangle wave disturbance, (b) square wave disturbance.    toolboxes. We execute experiments on the practical cartinverted pendulum system depicted in FIGURE 15. The angular position of the pendulum and the linear position of the cart are measured using two E40S encoders by Autonics Company. The employed card in this practical system is the PCI-1751, which is connected to the computer via D/A and A/D converters. The obtained pendulum's angular position and cart's linear position are shown in FIGURE 16 and  FIGURE 17, respectively. The position of the inverted pendulum is changes from π to 0, and stabilizes around the equilibrium. These experimental results prove the performance of the proposed approach. The control signal is illustrated in FIGURE 18.
The above practical results confirm the good tracking performance and effectiveness of the proposed control scheme.

V. CONCLUSION
This paper proposed an adaptive non-singular fast terminal sliding mode control (FTSMC) with integral surface for the finite time tracking control of nonlinear systems with external disturbances. It main objective is to establish strong robust performance, fast finite time convergence and chattering-free dynamics. The proposed approach substitutes the signum function with a bipolar function with tunable coefficients and derives an appropriate adaptive parameter-tuning law to tackle the unknown bounded disturbances and alleviate the undesired chattering problem. It also does not require the time-derivative of the fractional power terms in the controller, thereby eschewing the singularity problem. Implementation of the proposed approach to a VDP circuit and an inverted pendulum confirmed its good tracking performance. Additionally, a comparison study to the approach proposed in [14] highlighted its superior performance and dynamic response. Our future work will focus on augmenting the proposed approach with continuous finite time convergence differentiators and implementing the proposed design for chaos suppression, chaotic synchronization and filter design.
AFEF FEKIH (Senior Member, IEEE) received the B.Sc., M.Sc. and Ph.D. degrees in electrical engineering from the National Engineering School of Tunis, Tunisia, in 1995, 1998, and 2002, respectively. She is currently a Full Professor with the Department of Electrical and Computer Engineering and the Chevron/BORSF Professor in engineering with the University of Louisiana at Lafayette, Lafayette, USA. She has authored or coauthored more than 200 publications in international journals, chapters, and conference proceedings. Her research interests include control theory and applications, including nonlinear and robust control, optimal control, fault tolerant control with applications to power systems, wind turbines, and unmanned vehicles and automotive engines. She is a member of the His research interests include nonlinear control, underwater vehicles and robotics, multi-body dynamic modeling, obstacle avoidance, and intelligent navigation. VOLUME 9, 2021