A New Fast Nonsingular Terminal Sliding Mode Control for a Class of Second-Order Uncertain Systems

Abstract

This paper considers the robust and adaptive nonsingular terminal sliding mode (NTSM) control for a class of second-order uncertain systems. First, a new fast NTSM was proposed which had global fast convergence rate in the sliding phase. Then, a new form of robust NTSM controller was designed to handle a wider class of second-order uncertain systems. Moreover, an exponential-decline switching gain was introduced for chattering suppression. After that, a double sliding surfaces control scheme was constructed to combine the NTSM control with the adaptive technique. The benefit is that a strict demonstration can be given for the stagnation problem in the stability analysis of NTSM. Finally, a case study for tracking control of a variable-length pendulum was performed to verify the proposed controllers.

Academic Editor:Asier Ibeas

School of Automation, Beijing Institute of Technology, Zhongguancun Street, Haidian District, Beijing, China

Received 12 July 2016; Revised 30 October 2016; Accepted 15 November 2016

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

It is well known that the sliding mode control (SMC) has invariant property to the matched uncertainties. It has become a popular method to control nonlinear uncertain systems [1, 2]. In the earlier studies of SMC, the research was focused on the linear sliding mode (LSM). In [3, 4], the concept of terminal attractor was proposed and used to design the sliding mode which was known as the terminal sliding mode (TSM). The TSM had advantage of finite time convergence, and it has been widely used in many applications, such as robots, spacecraft, and DC-DC buck converters [5-7].

Compared to the LSM, the TSM has lower convergence speed when the states are far from the origin. Considering this problem, Yu et al. designed a fast terminal sliding mode (FTSM) which had global fast convergence in the sliding phase [8]. The FTSM technique has been developed for controlling nonlinear second-order systems with uncertain terms in [9-11]. Moreover, the concept of terminal attractor was also applied to the reaching law by Yu and Man [12]. As the negative fractional power exists in the TSM control signals, it may result in singularity of the control input. This is hard to accept in real implementations. So, the control input was switched to a general sliding mode control in order to avoid the singularity when the states converged into a small vicinity of origin in [13]. However, this switching method is a suboptimal solution which lost the advantage of TSM. In [14, 15], a new form of TSM known as nonsingular TSM (NTSM) was proposed and applied to robotic manipulators and piezoelectric actuators. It also has a lower convergence speed when the states are far from the origin. Although the NTSM control has solved the singularity problem, it needs an extra demonstration to show that the motion of sliding surface will not stagnate at some nonzero points in the reaching phase. According to the existing literatures, most of the TSM control methods were designed for the second-order systems. In [16-18], the recursive TSM control was developed for higher order nonlinear systems.

For the robust TSM controllers, the boundary information of system uncertainties is usually required to be known in advance. However, it is difficult to obtain in many practical implementation processes. Therefore, the adaptive technique was employed to cooperate with the TSM and FTSM control in [19-21]. Moreover, the adaptive NTSM control method was developed for uncertain systems with input nonlinearity in [22, 23]. Note that a strict proof for the stagnation problem of NTSM can be given in the robust NTSM control, for example, [14, 15]. However, it was difficult to obtain for the case of adaptive NTSM control in [22, 23].

Previous studies have developed the TSM control and thus have great significance. On the other hand, there are still some points valuable for further research. Most of the NTSM controllers were designed for a class of uncertain systems in which the coefficient of control input was a known function (or with an uncertain item). In the case that this coefficient is totally unknown except its boundary, few NTSM controllers were reported. So, the purpose of this paper is to develop the NTSM control for the latter case.

The main contribution of this paper can be summarized as follows.

(1) A fast NTSM (FNTSM) is designed to combine the advantages of LSM and NTSM together. Thus, the convergence rate of the NTSM is enhanced when the initial states are far away from the origin.

(2) A new robust NTSM controller is proposed for the case that the coefficient of control input is unknown. It is also applicable to the case that the coefficient is known. Moreover, an exponential-decline switching gain is designed to attenuate the chattering phenomenon.

(3) An adaptive NTSM control scheme with double sliding surfaces is proposed to give a strict proof for the stagnation problem of NTSM. The system uncertainty is firstly compensated on an integral sliding surface; after that the system trajectory is forced to the designed fast NTSM.

2. Robust NTSM Controller Design

Consider a second-order nonlinear system as [figure omitted; refer to PDF] where [straight phi](x) and γ(x) denote bounded uncertain functions, γ(x)≠0.

Assumption 1.

The uncertain functions [straight phi](x) and γ(x) satisfy [figure omitted; refer to PDF] where _{[straight phi]0} , _{[straight phi]1} , _{[straight phi]2} , _Km , and _KM are positive constants. _Km and _KM are assumed to be known. Robust and adaptive controllers are designed for the two cases that [straight phi]- is known or not, respectively. In the rest of the paper, [straight phi](x) and γ(x) are simply denoted as [straight phi] and γ.

2.1. FNTSM Design

In the study of [14], a kind of NTSM was designed as [figure omitted; refer to PDF]

It is equivalent to the following form which can be converted to a conventional TSM: [figure omitted; refer to PDF]

As illustrated in [8], the TSM has slower convergence rate than the LSM when the initial states are far away from the origin. It can be explained by its eigenvalue. For TSM _{x 1} +^{β-1/χx11/χ} =0, it is easy to obtain the eigenvalue as [figure omitted; refer to PDF]

As 1/χ-1<0, the eigenvalue tends to be negative infinity at the origin. This fact implies that the convergence rate is infinitely large. This is the major merit of TSM. However, when the initial state is far away from the origin, the convergence rate is smaller than the case χ=1, that is, a LSM.

In order to enhance the convergence rate of NTSM (3), a new FNTSM is designed as [figure omitted; refer to PDF]

The FNTSM (6) is a switching sliding surface which transfers the system dynamic from a LSM (q=1) to a NTSM (1<q<2) (Figure 1; e.g., [straight epsilon]=1). Thus, it has global fast convergence rate and reserves the advantage of NTSM.

Figure 1: The proposed sliding surface on the phase plane.

[figure omitted; refer to PDF]

Here, a simple example is given to compare the NTSM (3) and the FNTSM (6), [figure omitted; refer to PDF]

Considering system (7), the control input was taken as the NTSM controller form of [14] [figure omitted; refer to PDF]

With the same reaching law, the parameters for the sliding surfaces were chosen as [figure omitted; refer to PDF]

Note that the FNTSM became equivalent to the NTSM within the region (_x1 )<=1 because of _qF =11/7 inside this region. As shown in Figure 2, the convergence rate of the proposed FNTSM controller was faster than the NTSM controller's, while the control input of FNTSM controller was much smaller. In particular, for the FNTSM (6), the switching of sliding surface is not continuous. When the state _x1 converged into the prescribed region, the sliding surface and control signal had slight discontinuous changes. As a result, the control signal switched to a NTSM control from a LSM control. As shown in Figure 2(b), there was a small saltation of the control signal (from 0.967 to 0.582) at t=2.86 s because of the switching of sliding surface.

Figure 2: Comparison between the proposed FNTSM and the NTSM.

(a) System response

[figure omitted; refer to PDF]

(b) Control input

[figure omitted; refer to PDF]

2.2. Controller Design

Most of the NTSM controllers were designed for system (1) with a known γ (or with an uncertain item). However, these controllers are not theoretically applicable to totally unknown γ. In this subsection, a new form of NTSM controller is designed for system (1) in the case that [straight phi]- is known. A proof is given to show that the stagnation problem of sliding surface will not occur. Moreover, the new NTSM controller is also applicable to the case that γ is known.

Generally, the NTSM control design contains the sliding surface and the reaching law. A discontinuous reaching law with switching function is widely adopted to drive the system trajectory onto the sliding surface, which leads to the chattering. In the following theorem, an exponential-decline switching gain (EDSG) is designed to attenuate the chattering phenomenon.

The EDSG is formulated as [figure omitted; refer to PDF] where _α1 >[straight phi]-, a>0, _b1 ≥1, and _b2 >0. The EDSG accelerates the convergence rate of sliding surface at the beginning of reaching phase and attenuates the amplitude of chattering without deteriorating the system robustness.

Theorem 2.

For system (1), the following controller forces the system dynamic to the sliding surface (6) in finite time: [figure omitted; refer to PDF] where _α1 =[straight phi]-+η, _α2 ≥0, and η is a small positive constant.

Proof.

Consider the following Lyapunov function for the sliding surface: [figure omitted; refer to PDF]

Differentiating (12) with respect to time yields [figure omitted; refer to PDF]

Substituting the controller (11) into (13) yields [figure omitted; refer to PDF]

With γ≥_Km and ^{(_x2 )q-1} >0, (14) turns to [figure omitted; refer to PDF]

As A(t)_α1 ≥[straight phi]-+η, it is obtained as [figure omitted; refer to PDF]

Considering the last expression of (16), if the motion of sliding surface does not stagnate at the points (_x1 ≠0, _x2 =0) in the reaching phase, it would converge to zero in finite time according to the extended Lyapunov description of finite time stability in Remark 2 of [14].

Here, a demonstration is given to show that the motion of sliding surface does not stagnate at the points (_x1 ≠0, _x2 =0).

For system (1) with the controller (11), taking _x2 =0, we obtain [figure omitted; refer to PDF] where s=_x1 .

In the case of _x1 >0, the following inequality is satisfied for _{x 2} : [figure omitted; refer to PDF]

In the other case of _x1 <0, it follows that [figure omitted; refer to PDF]

It is concluded that the points (_x1 ≠0, _x2 =0) are not attractors. Thus, the motion of sliding surface will not stagnate at these points, and it converges to zero in finite time.

This completes the proof.

Remark 3.

Note that the term ^{(_x2 )2-q} sign(s) of controller (11) is constructed for the stability analysis. It is usually taken as ^{(_x2 )2-q} sign(_x2 ) in the existing NTSM controllers, for example, the controller (8). However, it is not applicable to the stability analysis of the discussed system (1).

Remark 4.

In the case that γ is known, the proposed controller (11) is also applicable by taking _Km =γ.

3. Adaptive NTSM Control Scheme Design

In practical situations, it is usually difficult to obtain the prior knowledge of [straight phi]-. Therefore, an adaptive controller is designed for system (1) in this section. In the studies of adaptive NTSM control [22, 23], the switching gain was directly adjusted by the adaptive technique. However, it was difficult to prove that the points (_x1 ≠0, _x2 =0) were not attractors as in Theorem 2. So, the adaptive switching gain was modified by the constant control gain in the extreme case that the stagnation problem may happen.

In this section, a double sliding surfaces control scheme is designed to combine the NTSM control with the adaptive technique. The first layer is a common integral sliding mode (ISM) in which the impact of uncertain function [straight phi] is compensated. The second layer is a NTSM in which the system trajectory converges to the origin. The purpose of using double sliding surfaces is to provide a strict proof for the stagnation problem of NTSM.

In this section, the control law is designed as [figure omitted; refer to PDF] where _ua denotes an adaptive controller which forces the system motion to the first sliding surface. After that, a NTSM controller _un drives the system motion to the second sliding surface.

The first sliding surface ISM is designed as [figure omitted; refer to PDF] where θ>0 and (_x2 (0),z(0)) are the initial values of (_x2 ,z). The exponential-decline term is introduced to eliminate the reaching phase [24]. Then the system dynamic is on the first sliding surface at the initial time.

Differentiating (21) with respect to time yields [figure omitted; refer to PDF] where Λ=θ^e-θt (_x2 (0)+z(0)). Once the first sliding surface is established, the original system (1) will approach to the system motion of an integral chain system as _{x 2} =-_un .

Here, the NTSM controller _un for system _{x 2} =-_un is designed as [figure omitted; refer to PDF] where _α3 is a positive constant, _p1 is a constant satisfying 0<_p1 <1, and _s2 denotes the FNTSM (6).

The continuous controller (23) had been verified in [14], and it also had been proved that the points (_x1 ≠0, _x2 =0) were not attractors. For an integral chain system without uncertainty (e.g., _{x 2} =-_un ), the discontinuous control is not necessary. Although the proposed NTSM controller (11) in Theorem 2 can be used to design _un , it is not employed as it is discontinuous.

With the following assumption, an adaptive controller is designed in Theorem 6 for system (22) by combining the TSM reaching law with the familiar first-order SMC adaptive control (e.g., [25-27]).

Assumption 5.

Assume [straight phi]+(γ-1)_un is unknown and satisfies the following inequality: [figure omitted; refer to PDF] where _c0 , _c1 , and _c2 are unknown positive constants.

Theorem 6.

For system (22), the sliding surface _s1 converges to zero asymptotically with the following adaptive controller: [figure omitted; refer to PDF] where (_k0 ~_k4 )≥0, 1≥_p2 >0, and _c^i (0)>0 (i=0,1,2).

Proof.

Define a new Lyapunov function for _s1 as [figure omitted; refer to PDF] where _c~i =_ci -_c^i . Its derivative is [figure omitted; refer to PDF]

Substituting controller (25) into (27) yields [figure omitted; refer to PDF]

With the adaptive laws in (25) and _c^i (0)>0, it ensures that _c^i >0. Since γ≥_Km , it follows that [figure omitted; refer to PDF]

Applying the Barbalat lemma [28], the sliding surface _s1 converges to zero asymptotically.

This completes the proof.

The boundary layer approach is a common method for chattering reduction. It is well known that the chattering reduction is achieved with a small cost of control precision. More specifically, the sliding surface is driven to a vicinity of zero. In [29], a concept of real sliding surface was introduced which was similar to the boundary layer. It also means that the sliding surface is forced to a small region rather than zero. Based on this concept, a modified adaptive law was proposed to avoid the overestimation of adaptive gain in [29]. The adaptive gain slowly decreased to the necessary value once the real sliding surface was established.

In the case that the boundary layer approach is adopted for the proposed adaptive controller (25), the chattering amplitude can be further attenuated if the adaptive gains (_c^0 ,_c^1 ,_c^2 ) decrease to a low level. Thus, the above two methods are combined to design a modified adaptive controller as [figure omitted; refer to PDF] where the sat(·) function and the adaptive laws are taken as [figure omitted; refer to PDF] where a positive constant O denotes the boundary lay. (_k-0 ~_k-2 ) and (_αc0 ~_αc2 ) are positive constants.

According to Theorem 6, the sliding surface _s1 converges to zero asymptotically. Thus, it converges into (_s1 )<=O in finite time. Then, the adaptive gains begin to decrease, and _{V 2} becomes sign indefinite. As soon as the sliding surface escapes from the region (_s1 )<=O, the adaptive gains increase again so that the sliding surface is driven back into the region (_s1 )<=O. Finally, the sliding surface _s1 remains in a larger region than (_s1 )<=O which is the so-called real sliding surface. The second expressions of the modified adaptive laws ensure the positive values of the adaptive gains, and they are valid in a short time.

Remark 7.

For the tracking control issues, it is necessary to modify the adaptive laws of _c^1 and _c^2 to reduce the chattering amplitude. However, it is not necessary when the states (_x1 ,_x2 ) converge to zero.

4. Simulation

In this section, the designed robust and adaptive NTSM controllers were tested for the tracking control of a variable-length pendulum [25] (Figure 3) formulated as [figure omitted; refer to PDF]

Figure 3: Variable-length pendulum model.

[figure omitted; refer to PDF]

The angular coordinate x is impelled to track a reference signal _xc by the torque u. A known mass m moves alone the rod without friction, and the distance R(t) between the center o and the mass m is unmeasured. It is assumed that the states ^{(xx )T} can be measured for the controller design. The parameters of (32) and the reference signal are taken as [figure omitted; refer to PDF]

Defining the tracking error as _e1 =x-_xc , the tracking control is converted to the stabilization of the following error system: [figure omitted; refer to PDF]

4.1. Robust Control

In this subsection, three robust controllers (A, B, and C) were evaluated. According to Theorem 2, controllers A and B (without EDSG) were designed using the FNTSM of [11] and the proposed FNTSM (6), respectively. The controller C was the proposed FNTSM-EDSG controller in Theorem 2.

Controller A (based on FNTSM of [11]): [figure omitted; refer to PDF]

Controller B (based on FNTSM (6)): [figure omitted; refer to PDF]

Controller C (based on FNTSM (6) + EDSG): [figure omitted; refer to PDF]

The simulation results are given in Figure 4. As shown in Figures 4(a) and 4(b), the angular coordinate converged to the reference signal within t=4 s. The sliding surfaces converged to zero within t=1.5 s (Figure 4(c)). It is clear that controller B cost less convergence time than the controller A, while the control inputs of these controllers had slight difference (Figures 4(b) and 4(d)). This fact implies that the controller designed by the proposed FNTSM was more efficient. As expected, the chattering amplitude of controller C (±5) was half of controller B (±10) (Figure 4(d)). However, the performance of controller C had small difference with that of controller B. The result confirms that the EDSG is effective for chattering suppression without deteriorating the system robustness.

Figure 4: Simulation result of the robust control.

(a) Angular coordinate

[figure omitted; refer to PDF]

(b) Tracking error

[figure omitted; refer to PDF]

(c) Sliding surfaces

[figure omitted; refer to PDF]

(d) Control torque

[figure omitted; refer to PDF]

4.2. Adaptive Control

In this subsection, the proposed adaptive NTSM control scheme ((23) and (25)) was verified for the tracking control. As a comparison, the adaptive method in [25] was adopted to design the adaptive controller _ua . The controllers were designed as follows.

Controller D (Theorem 6): [figure omitted; refer to PDF]

Controller E (method in [25]): [figure omitted; refer to PDF]

Controllers D and E shared the same NTSM controller _un , and the sliding surface _s2 was taken as _sF2 of controller B.

The simulation results are given in Figures 5 and 6. It is clear that both controllers D and E achieved fast tracking of the reference signal. Controller D had shorter convergence time and better transient performance than controller E, even though the maximum control torque and the chattering amplitude of controller D were smaller. Meanwhile, the sliding surfaces in Figure 6 converged to zero within t=3 s. In Figure 6(a), the motions of ISM started from zero, and there was a short time that ISM did not remain zero due to the affection of uncertainty. It is observed that the FNTSM converged to zero after the ISM, and the stagnation of FNTSM never happened. It verifies that the designed adaptive NTSM control scheme is effective. Since the two controllers shared the same NTSM controller _un , the simulation results confirm that the proposed adaptive method had better transient performance and was more efficient than the method in [25].

Figure 5: Simulation result of the adaptive control.

(a) Angular coordinate

[figure omitted; refer to PDF]

(b) Tracking error

[figure omitted; refer to PDF]

(c) Adaptive gains

[figure omitted; refer to PDF]

(d) Control torque

[figure omitted; refer to PDF]

Figure 6: The response of sliding surfaces.

(a) ISM

[figure omitted; refer to PDF]

(b) FNTSM

[figure omitted; refer to PDF]

Here, the modified adaptive controller ((30) and (31)) was checked for chattering reduction. In the following controller F, the boundary layer approach and the modified adaptive laws were both employed. As a comparison, controller G only employed the boundary layer approach. In order to avoid the unbounded growing of adaptive gains inside the boundary layer, the switching adaptive laws were adopted for controller G. In addition, controllers F and G used the same boundary layer and NTSM controller _un of controller D. Since it is assumed that the states can be measured for the controller design, the measurement noise may deteriorate the performance of controller in real implementations. To illustrate the robustness of the controller, random noise was taken for the states ^{(xx )T} (mean value 0 and standard deviation 0.01).

Controller F: [figure omitted; refer to PDF]

Controller G: [figure omitted; refer to PDF]

The simulation result of controller F is given in Figure 7. It is shown that the angular coordinate tracked the reference signal with a high precision (Figure 7(a)). Due to the adoption of the boundary layer and the modified adaptive laws, the sliding surface _s1 converged to a small domain as (_s1 )<=0.1 (Figure 7(b)). Moreover, the adaptive gains decreased to a low level with approximate reduction ratios as 51.08%, 84.79%, and 16.15%, respectively (Figure 7(c)). As a result, there was a marked reduction in the chattering of controller F compared to that of controller G in Figure 7(d). Thus, the effectiveness of the modified adaptive controller is verified.

Figure 7: Simulation result of the modified adaptive control.

(a) Angular coordinate

[figure omitted; refer to PDF]

(b) Sliding surfaces

[figure omitted; refer to PDF]

(c) Adaptive gains

[figure omitted; refer to PDF]

(d) Control torques

[figure omitted; refer to PDF]

5. Conclusions

This paper studies the NTSM control for a class of second-order systems in which the coefficient of control input is unknown. In this paper, a new FNTSM was designed to integrate the advantages of LSM and NTSM. Based on this FNTSM, new forms of robust controller and adaptive controller were proposed for the discussed uncertain system. The simulation verified the effectiveness of the proposed controllers. The proposed FNTSM shows faster convergence rate than the NTSM, and chattering reduction was achieved by the EDSG. The adaptive controller presented better transient performance and more efficiency than the adaptive method in [25]. Moreover, the modified adaptive controller achieved effective chattering reduction.

Acknowledgments

This research is supported by National Natural Science Foundation of China (no. 61375100, no. 61472037, and no. 61433003).

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