Integral Terminal Sliding Mode Control for a Class of Nonaffine Nonlinear Systems with Uncertainty

Abstract

This paper is concerned with an integral terminal sliding mode tracking control for a class of uncertain nonaffine nonlinear systems. Firstly, the nonaffine nonlinear systems is approximated to facilitate the desired control design via a novel dynamic modeling technique. Next, for the unmeasured disturbance of nonlinear systems, integral terminal sliding mode disturbance observer is presented. The developed disturbance observer can guarantee the disturbance approximation error to converge to zero in the finite time. Subsequently, based on approximated nonlinear model and the designed disturbance observer, the integral terminal sliding mode tracking control is presented for nonaffine nonlinear systems with uncertainty. Different from traditional terminal sliding-mode control, this paper accomplishes finite convergence time for nonaffine nonlinear systems and avoids the singular problem in the controller design. Furthermore, the control system is forced to start on the terminal sliding hyperplane, so that the reaching time of the sliding modes is eliminated. Finally, two numerical simulation results are given to illustrate the effectiveness of the proposed method.

1. Introduction

In recent years, there has been significant progress in the area of designing controllers for nonlinear systems [1–10]. Most of the controllers developed in this context have the common assumption that the system to be controlled is affine; that is, the plant is linear according to the input variables, and the nonlinearities are linearly parameterized by unknown parameters. However, developing a systematic synthesis method for nonaffine nonlinear systems is still a challenging problem for now.

Sliding-mode control (SMC) is a well-known efficient control scheme which has been widely applied for both linear and nonlinear systems [8, 10–16]. This control is also considered as an effective approach for control of the systems with uncertainties. However, the main disadvantage of SMC scheme is that the system states cannot reach the equilibrium point in finite time. So a new control scheme called terminal sliding-mode control (TSMC) is proposed to overcome this drawback utilizing nonlinear sliding surface. Nonlinear switching hyperplanes in TSMC can improve the transient performance substantially. Besides, compared with the conventional SMC with linear sliding manifold, TSMC offers some superior properties such as faster, finite time convergence and higher control precision. However, there exists an intrinsic singular problem in TSMC due to using fractional power functions as the sliding hyperplane [17–23]. Thus, overcoming the singular problem becomes a considerable topic for TSMC. Unfortunately, most of nonsingular TSMC methods [22, 23] are available only for affine nonlinear systems, especially robotic manipulators received the most attention. Nevertheless, these nonsingular TSMC schemes lack strict theoretical analysis and are only suitable for affine nonlinear systems. It is a worthwhile note that only a few researches discuss the control of nonaffine nonlinear systems even if allowing singularity in the TSMC. Therefore, nonsingular TSMC of nonaffine nonlinear systems needs to be investigated by a new technique.

The uncertainty is inherent in practical systems. Designing controller capability of handing uncertainty is of practical interest and is academically challenging. Neural networks (NNs) have been proposed recently as an adaptive controller for nonlinear systems. By the use of their universal approximation capability, the adaptive controller based on neural networks can be designed without significant prior knowledge of the system dynamics [24–28]. Although it is widely accepted that the parameterized neural network is capable of approximating linear or nonlinear mapping by adequately choosing network structures and training methods, a challenging problem for designers is to select an appropriate structure for balancing the number of rules and the approximation accuracy. If the network size is chosen too small, it is impossible to assure the approximation to converge to an acceptable level due to the limited nodes. On the other hand, if overdetermined nodes are given, the computational burden is huge and the waste of computation resource implies its impracticality for real-world applications.

Based on the above works, this paper is to develop an integral terminal sliding mode control approach for a class of nonaffine nonlinear systems with uncertainty, parameters perturbation, and external disturbances. The organization of this paper is as follows. Following the introduction, the problem formulation is described briefly, some assumptions which will play a basic role in our analysis are introduced in Section 2. To facilitate the desired control design, a novel dynamic modeling technique has been proposed for the nonaffine nonlinear systems in Section 3. The integral terminal sliding mode disturbance observer is presented in Section 4. Section 5 proposes integral terminal sliding mode control based on disturbance observer, and then integral terminal sliding mode control is designed for uncertain nonlinear systems with control singularity. Simulation studies are shown in Section 6 to demonstrate the effectiveness of our proposed approaches. Finally, conclusions are drawn in Section 7.

2. Problem Formulation

Consider a class of uncertain nonaffine nonlinear systems that can be expressed in the following form: 𝑥̇(𝑡)=𝑓₁(𝑥(𝑡),𝑢(𝑡))+Δ𝑓₁(𝑥(𝑡)(𝑡(𝑡),𝑦(𝑡)(𝑡),,𝑢))+𝑑=𝑥 (1) where𝑥(𝑡)=[𝑥₁(𝑡),…,𝑥_𝑛(𝑡)]∈ℝ^𝑛is the state vector of the system in the normal form which is assumed available for measurement,𝑦(𝑡)∈ℝ^𝑛 is the output vector,𝑢(𝑡)∈ℝ^𝑛 is control input vector, and𝑓₁(𝑥(𝑡),𝑢(𝑡))∈ℝ^𝑛 is known smooth vector fields.Δ𝑓₁(𝑥(𝑡),𝑢(𝑡)) is assumed to be continuous of𝑥(𝑡) denoting the the system uncertainty, which contains structural and modeling error.𝑑(𝑡)∈ℝ^𝑛is external disturbance.

After combining the uncertainty and disturbance together, the nonlinear system (1) can be rewritten as 𝑥̇(𝑡)=𝑓₁(𝑥(𝑡)(𝑡(𝑥(𝑡)(𝑡)),𝑦(𝑡)(𝑡),,𝑢))+𝐷,𝑢,𝑡=𝑥 (2) where𝐷(𝑥,𝑢,𝑡)=Δ𝑓₁(𝑥(𝑡),𝑢(𝑡))+𝑑(𝑡) is called compound disturbance.

To achieve the proposed control objective, the following assumptions are required.

Assumption 1. There exist known positive constants 𝜖̌ such that for all 𝑡∈𝑅⁺,‖𝐷(𝑥,𝑢,𝑡)‖⩽𝜖̌.

Assumption 2. Consider ‖Δ𝑢‖∈[0,𝛿]and0⩽‖𝜕𝑓/𝜕𝑢‖⩽𝛿̌, where 𝛿and𝛿̌are two finite positive constants.

Remark 3. In many actual process control systems and flight control systems,‖Δ𝑢(𝑡)‖∈[0,𝛿]is a physical restriction of many practical systems because their states and outputs (actuators) cannot change too fast because of system “inertia.” So, Assumption 2 is reasonable.

In this paper, the control objective is to design the disturbance-observer-based integral terminal sliding mode tracking control and make the system output follow a given desired output of the nonlinear system in the presence of uncertainty and external disturbance. For the desired tracking signal 𝑦_𝑑, the proposed integral terminal sliding mode control must ensure that all closed-loop signals are convergent in the finite time.

3. Novel Nonaffine Nonlinear Approximation

The problem of controlling the plants characterized by models that are nonaffine in the control input vector is a difficult one. Especially for the tracking control, the liberalization may result in the design of sufficiently accurate controllers in the case of stabilization around the operating point, in the case of tracking of desired trajectories the problem becomes much more difficult, because the linearized model is time-varying. Hence, there is a clear need for the development of systematic control design techniques for nonlinear models that are nonaffine in𝑢and that are suitable for the case of tracking of desired trajectories.

For the nonaffine nonlinear model (2), the Taylor expansion of the nonlinear function𝑓₁(𝑥(𝑡),𝑢(𝑡))with respect to𝑢(𝑡)around𝑢(𝑡−𝜏)can result in 𝑥̇=𝑓₁(𝑥,𝑢(𝑡−𝜏))+𝑔(𝑥,𝑢(𝑡−𝜏))Δ𝑢+𝑂(⋅)+𝐷, (3) whereΔ𝑢=𝑢−𝑢(𝑡−𝜏), 𝑔(𝑥,𝑢(𝑡−𝜏))=(𝜕𝑓₁(𝑥,𝑢)/𝜕𝑢)∣_{𝑢=𝑢(𝑡−𝜏)}, and the remainder 𝑂(⋅)=[Δ𝑢]^𝑇𝑓_𝑑𝑑Δ𝑢/2 is bounded by ‖𝑂(⋅⩽𝑟)‖^𝑝‖Δ𝑢‖²2, (4) where 𝑓_𝑑𝑑=(𝜕²𝑓₁(𝑥,𝑢)/𝜕²𝑢)∣_𝑢=𝜁  and𝜁 is a point between 𝑢 and 𝑢(𝑡−𝜏). Let0≤‖𝑓_𝑑𝑑‖≤𝑟_𝑝 with𝑟_𝑝 as a finite positive number. The parameter𝜏>0is the updating input. It may be chosen as the sampling-time in a sampled-data control system or as an integer multiple of the sampling-time. A better choice of the parameter𝜏is the sampling because a larger𝜏may lead to an inaccurate approximation when the system function 𝑓₁(𝑥,𝑢) varies quickly.

Equation (3) can be representation as the following form: 𝑥̇=𝑓(𝑥,𝑢(𝑡−𝜏))+𝑔(𝑥,𝑢(𝑡−𝜏))𝑢+𝑂(⋅)+𝐷(𝑥,𝑢,𝑡), (5) where 𝑓(𝑥,𝑢(𝑡−𝜏))=𝑓₁(𝑥,𝑢(𝑡−𝜏))−𝑓_𝑑(𝑥,𝑢(𝑡−𝜏))𝑢(𝑡−𝜏). Convenient for the following statements, 𝑢(𝑡−𝜏) is defined as 𝜐(𝑡); then (3) can be described as follows: 𝑥̇=𝑓(𝑥,𝜐)+𝑔(𝑥,𝜐)𝑢+𝑂(⋅)+𝐷(𝑥,𝑢,𝑡). (6)

Remark 4. In Assumption 2,‖Δ𝑢‖should not be too large in order to limit the approximation error of model (3) for a computed 𝑢(𝑡). Therefore, to approximation accuracy, Assumption 2 must be satisfied. The significance of the Assumption 2 has been explained in Remark 3.

Remark 5. The traditional model simplification method does not global. It can be seen that (6) is a time-varying simplified model. The method which is proposed in this subsection can achieve the global approximation for nonlinear systems (2). So the proposed simplified model method can effectively solve the tracking control problem using affine nonlinear control strategy, such as sliding mode control, output-feedback control, and backstepping control.

Remark 6. By (3) and Assumption 2, it can be seen that 𝑢(𝑡−𝜏) should be around the input 𝑢. If the time-delay 𝜏 is selected too large, the precision of approximation of simplified model will be reduced. So the selection of𝜏often requires experience. Theoretically, the smaller the 𝜏 the better precision of global approximation, the best precision of global approximation if 𝜏=0. But𝑢is control law to be solved, so it is unable to be realized. In order to obtain exact time-varying trim point, here, further improvement of above proposed method is given as follows. Considering lag property of the filtering as 𝜐̇=−𝑘_𝜁𝜐+𝑘_𝜁𝑢, (7) then lim_𝜆→∞𝜐=𝑢. This is a very good solution to the problem that𝑢(𝑡−𝜏)may not be around𝑢. Here,𝑘_𝜁→∞is only a rigorous expression for mathematics meanings, in general, 𝑘_𝜁∈[5,50]. The filter (7) is not unique. The filtering 𝜐 can be completely replaced by other filtering equation, such as higher-order differentiator [30] and integral filter [31].

4. Sign Integral Terminal Sliding Mode Disturbance Observer

In this section, the design process of the sign integral terminal sliding mode disturbance observer will be given. Firstly, the following auxiliary sign integral terminal sliding mode vector 𝑠_𝑧 is introduced: 𝑠_𝑧=𝑒_𝑧+𝜗𝑒_𝑧𝑖, (8) where 𝜗>0 is a design parameter, 𝑒_𝑧=𝑥−𝑧=[𝑒_𝑧1,…,𝑒_𝑧𝑛]^𝑇 is auxiliary error, 𝑧 and 𝑒_𝑧𝑖 are given by 𝑧̇=𝑓₁()+𝑒̇𝑥,𝑢𝐷,_𝑧𝑖=sign𝑒_𝑧, (9)(10) where𝑒_𝑧𝑖is the integration ofsign(𝑒_𝑧)and has the initial value−𝑒_𝑧(0)/𝜗; andsign(𝑒_𝑧𝑖)=[sign(𝑒_𝑧1),…,sign(𝑒_𝑧𝑛)]^𝑇for𝑒_𝑧𝑖(𝑖=1,…,𝑛) being the𝑖th element of the auxiliary error vector𝑒_𝑧.

If𝑠_𝑧(𝑡)can keep at zero, such that𝑒_𝑧(𝑡)=−𝜗𝑒_𝑧𝑖(𝑡), then dynamics (10) will be 𝑒̇_𝑧𝑖(𝑡)=sign𝑒_𝑧(𝑡). (11) Therefore,𝑒_𝑧𝑖(𝑡)will converge to zero in the finite time𝑇_𝑓: 𝑇_𝑓=𝑒_𝑧(0)𝜗. (12) Note that when the auxiliary sign integral terminal sliding vector𝑠_𝑧satisfies𝑠_𝑧(𝑡)=0, the convergence of the 𝑒_𝑧(𝑡) is accomplished in the same time finite time (12) due to the fact that 𝑒_𝑧(𝑡)=−𝜗𝑒_𝑧𝑖(𝑡).

Next, to keep the system on the sign integral terminal sliding surface 𝑠_𝑧(𝑡)=0, the sliding mode disturbance estimate 𝐷 will be set to 𝐷=𝑘_𝑧𝑠_𝑧+𝜅sign𝑠_𝑧+𝜗sign𝑒_𝑧, (13) where 𝑘_𝑧>0 and 𝜅>𝜖̌>0 are design parameters.

Theorem 7. Considering the uncertain nonaffine nonlinear system (1) and supposing that Assumption 1 is available, the sign integral terminal sliding mode observer is designed according to (8)–(13). Then, auxiliary errors 𝑒_𝑧(𝑡) and 𝑒_𝑧𝑖(𝑡) are guaranteed with finite-time convergence stability.

Proof. Based on (8)–(13), the sign integral terminal sliding mode dynamic equation (8) also can be expressed by 𝑠_𝑧̇=𝑥̇−𝑧̇+𝜗𝑒̇_𝑧𝑖=𝑓₁(𝑥,𝑢)+𝐷−𝑓₁(𝑥,𝑢)−𝐷−𝜗sign𝑒_𝑧+𝜗𝑒̇_𝑧𝑖=𝐷−𝑘_𝑧𝑠_𝑧−𝜅sign𝑠_𝑧. (14) Choose the Lyapunov function candidate: 𝑉_𝑠=12𝑠^𝑇_𝑧𝑠_𝑧. (15) The time derivative of 𝑉_𝑠 along the trajectories of the equation in (14) is 𝑉̇_𝑠=𝑠^𝑇_𝑧𝑠_𝑧̇=𝑠^𝑇_𝑧𝐷−𝑘_𝑧𝑠_𝑧−𝜅sign𝑠_𝑧⩽−𝑘_𝑧𝑠^𝑇_𝑧𝑠_𝑧+𝑠_𝑧𝜖̌−𝜅𝑠^𝑇_𝑧sign𝑠_𝑧. (16) Under Assumption 1, considering design parameter 𝜅>𝜖̌, that is, 𝜅>‖𝐷‖, (16) can be modified as 𝑉̇_𝑠⩽−𝑘_𝑧𝑠^𝑇_𝑧𝑠_𝑧=−2𝑘_𝑧𝑉_𝑠. (17) From (17), we can get the conclusion that if 𝑠_𝑧≠0, then 𝑉̇_𝑠<0 is true. Thus, the auxiliary sliding vector 𝑠_𝑧 of the sign integral terminal sliding mode disturbance observer is always kept on the surface 𝑠_𝑧(𝑡)=0. At the result, the auxiliary errors 𝑒_𝑧(𝑡) and 𝑒_𝑧𝑖(𝑡) are guaranteed with finite-time convergence stability. This ends the proof.

Remark 8. Comparing with the existing results [14], the proposed sign integral terminal sliding mode disturbance observer can guarantee the disturbance estimate error to converge to zero in the finite time. In addition, the advantages of proposed sign integral terminal sliding mode will be explained in Remarks 11 and 12.

Remark 9. It is worth noting that the known upper boundary of the dynamic error is required in the design of disturbance observer. However, upper boundary 𝜖̌ is difficult to be obtained in practice. So, the adaptive gain𝜅in (13) is considered. There are many research results on adaptive gain [16]. For simple convenience, the detail is omitted.

5. Fractional Integral Terminal Sliding Mode Control

In this section, we develop the tracking control scheme for the case where all states are available using fractional integral terminal sliding mode control approach. Before the discussion, the tracking error is defined as 𝑒(𝑡)=𝑦(𝑡)−𝑦_𝑑(𝑡). Instead of using a linear sliding function, we introduce fractional integral terminal sliding mode below. First, the fractional integral terminal sliding mode function is defined as follows: 𝑠(𝑡)=𝑒(𝑡)+𝛼̆𝑒_𝐼(𝑡),𝑒_𝐼̇(𝑡)=𝑒^𝑞/𝑝(𝑡), (18) where 𝛼̆>0; 𝑞and 𝑝 are positive odd integers with 𝑝>𝑞; 𝑒_𝐼=∫^𝑡₀𝑒^𝑞/𝑝(𝜏)𝑑𝜏 is the integration of the 𝑞/𝑝-fractional power of the tracking error 𝑒 with the initial value −𝑒(0)/𝛼̆; the nonlinear 𝑒^𝑞/𝑝 is obtained by the operation 𝑒₀^𝑞/𝑝=[𝑒₁^𝑞/𝑝sign(𝑒₁),…,𝑒_𝑛^𝑞/𝑝sign(𝑒_𝑛)]^𝑇.

If the surface 𝑠(𝑡)=0, based on definition (18), the fractional integral terminal sliding mode function can be expressed in the form 𝑠(𝑡)=𝑒(𝑡)+𝛼̆^𝑡₀𝑒^𝑞/𝑝(𝜏)𝑑𝜏. (19) At the same time, the integrator 𝑒_𝐼(𝑡) can be modified as 𝑒_𝐼̇=−𝛼̆^𝑞/𝑝𝑒_𝐼^𝑞/𝑝(𝑡). (20)

From solving the error dynamic equation (20), the convergence time of𝑒_𝐼is obtained as follows: 𝑇_𝑓=𝑒_𝐼^{1−𝑞/𝑝}𝛼̆^𝑞/𝑝=‖𝑒(01−𝑞/𝑝)‖^{1−𝑞/𝑝}𝛼̆.1−𝑞/𝑝 (21) Meanwhile, the time spent for the convergence of the tracking error 𝑒(𝑡) is also 𝑇_𝑓.

Next, to keep the system on the integral terminal sliding surface 𝑠(𝑡)=0, we need to design control input vector𝑢for the system (2). Considering the time-varying simplified model (6), we modify the time derivative of𝑠(𝑡)along the dynamics (6) as 𝑠(̇𝑡)=𝑒̇+𝛼̆𝑒_𝐼=𝑓+𝑔𝑢+𝑂(⋅)+𝐷−𝑦̇_𝑑+𝛼̆𝑒_𝐼̇, (22) where 𝑓, 𝑔, and 𝑂(⋅) are defined in Section 3.

According to Assumption 2, we will first consider the case when the control gain is nonsingular; thatis, |𝑔|≠0. Following that, our focus will be on the control design in the case when the control gain is singular; thatis, |𝑔|=0.

5.1. Case of Nonsingular Control Gain

In this subsection, we assume that |𝑔|≠0 for the nonaffine nonlinear systems (2) with simplified model (6). Then, we consider the following the control vector 𝑢 as 𝑢=𝑔⁻¹−𝑘𝑠−𝑓−𝐷−𝛼̆𝑒_𝐼̇+𝑦̇_𝑑−𝑟_𝑠, (23) where 𝑘>0 is a design parameter;𝐷is defined in (13); to restrain the dynamic error 𝑂(⋅) from (6), the robust term 𝑟_𝑠 is designed as 𝑟_𝑠=𝜍𝑠‖𝑠‖,‖𝑠‖≠0,0,𝑠=0, (24) where 𝜍>‖𝑂(⋅)‖ is a design constant.

The above design procedure of the terminal sliding mode control can be summarized in the following theorem, which contains the results for disturbance-observer-based terminal sliding mode tracking control of uncertain nonaffine systems with external disturbance.

Theorem 10. Considering the uncertain nonaffine system (1) with the external disturbance and assuming that Assumptions 1 and 2 are available, nonaffine nonlinear approximation is given as (6) and the terminal sliding mode disturbance observer is designed as (8)–(13). If the proposed terminal sliding mode tracking control and the robust term are chosen as (23) and (24), then all signals of the closed-loop system are convergent in the finite time (21).

Proof. Choose the Lyapunov function candidate: 1𝑉=2𝑠^𝑇𝑠. (25) Under Assumption 2, substituting (23) and (24) into (22), the time derivative of𝑉along the trajectories of (22) is 𝑉̇=𝑠^𝑇𝑠̇=𝑠^𝑇𝑓+𝑔𝑢+𝐷+𝑂(⋅)−𝑦̇_𝑑+𝛼̆𝑒_𝐼̇=𝑠^𝑇𝑓−𝑘𝑠−𝑓−𝐷−𝛼̆𝑒_𝐼̇+𝑦̇_𝑑−𝑟_𝑠+𝐷+𝑂(⋅)−𝑦̇_𝑑+𝛼̆𝑒_𝐼̇⩽−𝑘𝑠^𝑇s+‖𝑠‖‖𝑂(⋅)‖−𝜍‖𝑠‖⩽−𝑘𝑠^𝑇𝑠=−2𝑘𝑉. (26) Since 𝑠(0)=0 and (26), the system is always kept on the fractional integral terminal sliding surface 𝑠(0)=0. As a result, the tracking error 𝑒_𝐼(𝑡) and error 𝑒(𝑡) converge to zero in finite time (21). This concludes the proof.

Remark 11. The characteristics of the proposed fractional integral terminal sliding mode control including (1) the finite convergence time can be easily adjusted according to (21); (2) the singular problem does not occur on the control law in contrast to traditional TSMC; (3) the system starts on the sliding mode surface 𝑠=0; that is, fast response is obtained.

Remark 12. Aside from the characteristics in Remark 11, the convergence time of the fractional integral terminal sliding mode control is calculable and analyzable in contrast to the high-order SMC. In comparison, the dynamic SMC only assures asymptotic stability.

Remark 13. In order to reduce chattering which is caused by discontinuous sign function, 𝜍𝑠/‖𝑠‖ in robust term (24) can be replaced by the continuous function 𝑟_𝑠 defined by 𝑟_𝑠=𝜍𝑠‖𝑠‖+𝜖 (27) with 𝜖=𝜖₀+𝜖₁‖𝑒‖, where 𝜖₀ and 𝜖₁ are two positive constants.

5.2. Case of Singular Control Gain

In Section 5.1, we assume that |𝑔(𝑥)|≠0 for the simplified model (6). However, there exists the feasibility of |𝑔(𝑥)|=0 at a moment in the practical system which leads to the control singularity. Thus, we propose the fractional integral terminal sliding mode control for the simplified model (6) with control singularity case in this subsection. Considering the control singularity, the control input vector 𝑢 is given by 𝑢=𝑔(𝑥,𝜐)𝑔^𝑇(𝑥,𝜐)𝑔(𝑥,𝜐)+𝜆⁻¹𝑢̌, (28) where 𝜆>0 is a design constant and 𝑢̌ will be given later.

It is clear that 𝑔^𝑇(𝑥,𝜐)𝑔(𝑥,𝜐)𝑔^𝑇(𝑥,𝜐)𝑔(𝑥,𝜐)+𝜆⁻¹=1−𝜆𝑔^𝑇(𝑥,𝜐)𝑔(𝑥,𝜐)+𝜆⁻¹. (29) Substituting (28) and (29) into (6), we obtain 𝑥̇=𝑓(𝑥,𝜐)+𝑢̌+𝐷−𝑔^𝑇(𝑥,𝜐)𝑔(𝑥,𝜐)+𝜆⁻¹𝑢̌+𝑂(⋅). (30) According to (30), the compound disturbance can be modified as 𝐷=𝐷−𝑔^𝑇(𝑥,𝜐)𝑔(𝑥,𝜐)+𝜆⁻¹𝑢̌+𝑂(⋅). (31) Due to the unknown compound disturbance 𝐷, the sign integral terminal sliding mode disturbance observer needs to be developed to estimate it. Thus, the similar auxiliary sliding mode is expressed: 𝑠_𝑧=𝑒_𝑧+𝑎₀𝑒_𝑧𝑖,𝑧̇=𝑓+𝑢̌+𝑒̇𝐷,_𝑧𝑖=sign𝑒_𝑧𝑖, (32)(33)(34) where 𝑠_𝑧,𝑒_𝑧,𝑒_𝑧𝑖, and𝑧are defined in Section 4. Based on (32)–(34), the sign integral terminal sliding mode disturbance estimate 𝐷 is given by 𝐷=𝑘_𝑧𝑠_𝑧+𝜅sign𝑠_𝑧+𝜗sign𝑒_𝑧, (35) where 𝐷 is the estimate of compound disturbance 𝐷.

Based on the sign integral terminal sliding mode disturbance observer, the fractional integral terminal sliding mode tracking control is designed as 𝑢̌=−𝑘𝑠−𝑓−𝐷−𝛼̆𝑒^𝑞/𝑝+𝑦̇_𝑑, (36) where the parameters 𝑘,𝛼̆,𝑞, and𝑝are defined in (23) and (18), respectively.

The above design procedure and analysis can be summarized in the following theorem, which contains the results for the simplified model (6) with the control singularity case.

Theorem 14. Considering the uncertain nonaffine system (1) with the external disturbance and supposing that Assumptions 1 and 2 are available, nonaffine nonlinear approximation is given as (6) and the sign integral terminal sliding mode disturbance observer is designed as (32)–(35). If the proposed fractional integral terminal sliding mode control law is chosen as (36), then sliding mode surface will always keep at 𝑠(𝑡)=0.

Proof. Considering the time-varying simplified model (6) with singular control gain, we modify the time derivative of 𝑠(𝑡)along the dynamics (30) as 𝑠(̇𝑡)=𝑒̇+𝛼̆𝑒_𝐼=𝑓+𝑔𝑢+𝑂(⋅)+𝐷−𝑦̇_𝑑+𝛼̆𝑒_𝐼̇. (37) Choose the Lyapunov function candidate: 1𝑉=2𝑠^𝑇𝑠. (38) Substituting (36) into (37), the time derivative of 𝑉 along the trajectories of the equation in (37) is 𝑉̇=𝑠^𝑇𝑠̇=𝑠^𝑇𝑓+𝑢̆+𝐷−𝑦̇_𝑑+𝛼̆𝑒_𝐼̇=𝑠^𝑇𝑓−𝑘𝑠−𝑓−𝑘_𝑧𝑠_𝑧−𝐷−𝛼̆𝑒^𝑞/𝑝+𝑦̇_𝑑+𝐷−𝑦̇_𝑑+𝛼̆𝑒_𝐼̇⩽−𝑘𝑠^𝑇𝑠=−2𝑘𝑉. (39) According to 𝑠(0)=0 and (39), we can know that the system is always kept on the fractional integral terminal sliding surface 𝑠(0)=0. So, the tracking error 𝑒_𝐼(𝑡) and error 𝑒(𝑡) converge to the equilibrium point in the finite time (21). This concludes the proof.

Remark 15. From (31), the integrated effect of control singularity is treated as a part of the external disturbance which is approximated using the sign integral terminal sliding mode disturbance observer (33)–(35). Although the uncertain nonlinear system (1) has the feasibility of control singularity, Lyapunov analysis shows that the system is asymptotically convergent in the finite time under the proposed disturbance-observer-based fractional integral terminal sliding mode control. In general, the design parameter𝑘_𝑧should be chosen as a large positive constant to guarantee the design requirement of the proposed sliding mode disturbance observer.

6. Simulation Results

6.1. Simulation Example for Duffing-Holmes System

To verify the validity of the proposed nonaffine nonlinear approximation in Section 3, consider Duffing-Holmes system 𝑥̇₁=𝑥₂,𝑥̇₂=−𝑝₁𝑥₁−𝑝₂𝑥₂−𝑥³₁+ℎ𝑥₂,𝑢, (40) where 𝑝₁=0.2sin(10𝑡); 𝑝₂=0.2(1+cos(5𝑡)); ℎ(𝑥₂,𝑢)=𝑢³+(2+cos(𝑥₂))𝑢 is a nonaffine control term.

The initial conditions are chosen as [𝑥₁(0),𝑥₂(0)]^𝑇=0, and the simulation time is chosen as 𝑡=30s. we define control input value 𝑢 as 𝑢(𝑡)=()()0.1sin2𝑡,𝑡⩽7,sin2𝑡,7<𝑡⩽30. (41) As compared with the existing approximate method, we adopt the method of [29] at the local working point 𝑢=0 for nonaffine nonlinear systems (2). Then, we get the affine nonlinear approximation as follows: 𝑥̇₁=𝑥₂,𝑥̇₂=−𝑝₁𝑥₁−𝑝₂𝑥₂−𝑥³₁+2+cos𝑥₂𝑢. (42) By using the proposed approximation method in Section 3, the affine nonlinear approximation is 𝑥̇₁=𝑥₂,𝑥̇₂=−𝑝₁𝑥₁−𝑝₂𝑥₂−𝑥³₁−2𝜁³+3𝜁²𝑢+2+cos𝑥₂̇𝑢,𝜁=−𝑘_𝜁𝜁+𝑘_𝜁𝑢,𝜁(0)=0. (43) The designed constant in (43) is chosen as 𝑘_𝜁=100. Simulation results are shown in Figures 1 and 2, respectively.

(a)

(b)

Figure 1 

State response by the proposed approximation.

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(b)

Figure 2 

State response using the approach developed in [29].

From Figures 1 and 2, it is observed that two methods have the same approximation during working point 𝑢=0. On the other hand, when the working point stay away from 𝑢=0, changes. Therefore, from the global approximation, the proposed method is better than the method of [29].

6.2. Simulation Example for 6DOF Near Space Vehicle Dynamics with Coordinated Turn

To verify the validity of the fractional integral terminal sliding mode control, the differential equations governing the near space vehicle (NSV) dynamics with coordinated turn are given by ̇̇̇̇𝑥=𝑉cos𝛾cos𝜒,𝑦=𝑉cos𝛾sin𝜒,𝑧=𝑉sin𝛾,𝑉=𝑓_𝑣=1𝑀,̇𝑇cos𝛼−𝐷−𝑀𝑔sin𝛾𝜒=𝑓_𝜒=1,̇𝑀𝑉cos𝛾𝐿sin𝜇+𝑌cos𝜇+𝑇sin𝜇sin𝛼𝛾=𝑓_𝛾=1,𝑀𝑉𝐿cos𝜇−𝑌sin𝜇−𝑀𝑔cos𝛾+𝑇cos𝜇sin𝛼 (44)(45)(46)(47)(48)(49) where the three position variables (𝑥,𝑦,𝑧) in the inertial frame, airspeed (𝑉), fight path angle (𝛾), and flight path heading (𝜒) are the six state variables; thrust (𝑇), attack angle (𝛼), and roll angle (𝜇) are the control variables; the drag force (𝐷), lift force (𝐿), and lateral force (𝑌) are expressed as follows: 𝐷=𝑞𝑆𝐶_𝐷,𝐿=𝑞𝑆𝐶_𝐿,𝑌=𝑞𝑆𝐶_𝑌, (50) where 𝑞=0.5𝜌𝑉² is dynamic pressure, 𝑆 is reference area, 𝐶_𝐷=𝐶_𝐷𝛼+𝐶_𝐷𝛿𝑒𝛿_𝑒+𝐶_𝐷𝛿𝑎𝛿_𝑎+𝐶_𝐷𝛿𝑟𝛿_𝑟+𝐶_𝐷𝛿𝑐𝛿_𝑐,𝐶_𝐿=𝐶_𝐿𝛼+𝐶_𝐿𝛿𝑒𝛿_𝑒+𝐶_𝐿𝛿𝑎𝛿_𝑎+𝐶_𝐿𝛿𝑐𝛿_𝑐, and 𝐶_𝑌=𝐶_𝑌𝛽𝛽+𝐶_𝑌𝛿𝑒𝛿_𝑒+𝐶_𝑌𝛿𝛼𝛿_𝛼+𝐶_𝑌𝛿𝑟𝛿_𝑟. In this paper, we will assume that the parameters 𝐶_𝐷𝛼, 𝐶_𝐷𝛿𝑒,𝐶_𝐷𝛿𝑎,𝐶_𝐷𝛿𝑟,𝐶_𝐷𝛿𝑐𝛿_𝑐, 𝐶_𝐿𝛼, 𝐶_𝐿𝛿𝑒, 𝐶_𝐿𝛿𝑎, 𝐶_𝐿𝛿𝑐, 𝐶_𝑌𝛽,𝐶_𝑌𝛿𝑒, 𝐶_𝑌𝛿𝛼, and 𝐶_𝑌𝛿𝑟 are uncertain, while the description of the variables are shown in [32].

In this paper, we will focus on the model of rate dynamics, that is, (47)–(49). So, to put the above equations in the form of (1), we define 𝑥=[𝑥₁,𝑥₂,𝑥₃]^𝑇=[𝑉,𝜒,𝛾]^𝑇, 𝑢=[𝑢₁,𝑢₂,𝑢₃]^𝑇=[𝑇,𝛼,𝜇]^𝑇. Hence the rate equations become 𝑥̇=𝑓(𝑥,𝑢), (51) where 𝑓=[𝑓_𝑣,𝑓_𝜒,𝑓_𝛾]^𝑇 is defined in (47)–(49). By using the proposed approximated model in Section 3, we have 𝑥̇=𝑓+𝑔𝑢_𝜁(⋅),𝑢+𝑂_𝜁=−𝑘_𝜁𝑢_𝜁+𝑘_𝜁𝑢,𝑢_𝜁(0)=0, (52)(53) where 𝑓=𝑓(̆𝑥,𝑢_𝜁)−𝑔_𝑝𝑢_𝜁, 𝑢_𝜁=[𝑢_𝜁1,𝑢_𝜁2,𝑢_𝜁3]^𝑇, 𝑘_𝜁 is 𝑠 design parameter, 𝑓(̆𝑥,𝑢_𝜁)=[𝑓̆_𝑣,𝑓̆_𝜒,𝑓̆_𝛾], 𝑓̆_𝑣,𝑓̆_𝜒,𝑓̆_𝛾, and 𝑔_𝑝 are expressed as𝑓̆_𝑣=1𝑀𝑢_𝜁1cos𝑥₂−𝐷−𝑀𝑔sin𝑥₃,𝑓̆_𝜒=1𝑀𝑥₁cos𝑥₃𝐿sin𝑢_𝜁3+𝑌cos𝑢_𝜁3+𝑇sin𝑢_𝜁3sin𝑢_𝜁2,𝑓̆_𝛾=1𝑀𝑥₁𝐿cos𝑢_𝜁3−𝑌sin𝑢_𝜁3−𝑀𝑔cos𝑥₃+𝑢_𝜁1cos𝑢_𝜁3sin𝑢_𝜁2,𝑔_𝑝=cos𝑢_𝜁2𝑀−𝑢_𝜁1sin𝑢_𝜁2𝑀01𝑀𝑢_𝜁1cos𝑥₃sin𝑢_𝜁2sin𝑢_𝜁31𝑀𝑢_𝜁1cos𝑥₃𝑢_𝜁1cos𝑢_𝜁2sin𝑢_𝜁31𝑀𝑢_𝜁1cos𝑥₃sin𝑢_𝜁2cos𝑢_𝜁31𝑀𝑢_𝜁1sin𝑢_𝜁2cos𝑢_𝜁31𝑀𝑢_𝜁1𝑢_𝜁1cos𝑢_𝜁2cos𝑢_𝜁3−1𝑀𝑢_𝜁1𝑢_𝜁1sin𝑢_𝜁2sin𝑢_𝜁3. (54)

The initial state conditions are arbitrarily chosen as 𝑀=136080Kg, 𝑉(0)=3000m/s, 𝜒(0)=𝛾(0)=0deg,𝑇(0)=240KN,𝛼(0)=1deg, and 𝜇(0)=0deg.

The desired command is considered as 𝑉_𝑑=3000m/s, 𝜒_𝑑=8deg, and 𝛾_𝑑=0deg. In order to ensure the smoothness of airspeed change, we choose the filter as (7), in which the parameter is chosen as 𝑘_𝜁=0.03.

To estimate the uncertainty, we apply integral terminal sliding mode disturbance observer in Section 4 and set the parameters as 𝑘_𝑧=5, 𝜗=2, and 𝜅=0.5. Furthermore, to demonstrate the effectiveness of the proposed integral terminal sliding mode control, the design parameters are chosen as 𝛼̆=1.5, 𝑝=5, 𝑞=3, 𝑘=2, and 𝜍=0.3. The tracking results are shown in Figures 3 and 4. Although, there exists uncertain in the system, the tracking performance is still satisfactory and tracking error converges to zero quickly.

(a)

(b)

(c)

Figure 3 

States𝑉,𝜒, and𝛾follow desired command𝑉𝑑,𝜒𝑑,and𝛾𝑑for near space vehicle system with coordinated turn.

(a)

(b)

(c)

Figure 4 

Control input for near space vehicle system with coordinated turn.

From these simulation results of two cases, we can obtain that the proposed method is valid. And the developed sign integral terminal sliding mode disturbance observer can modify the control performance of the fractional integral terminal sliding mode control.

7. Conclusions

In this paper, the disturbance-observer-based terminal sliding mode tracking control has been proposed for a class of uncertain nonaffine nonlinear systems. To design tracking controller, an on-line approximation has been proposed for a class of nonaffine nonlinear systems. To improve the ability of the disturbance attenuation and system performance robustness, the sign integral terminal sliding mode disturbance observer has been developed to approximate the system disturbance in the finite time. Based on the output of the disturbance observer, the disturbance-observer-based fractional integral terminal sliding mode tracking control has been presented for the uncertain nonlinear system with the time-varying external disturbance. By innovating the fractional error integration, finite-time convergence of tracking errors and integral errors is achieved without singular problem. Furthermore, the finite convergence time is easily calculated in contrast to the traditional high-order sliding mode control. The stability of the closed-loop system has been proved using rigorous Lyapunov analysis. Finally, simulation results have been used to illustrate the effectiveness of the proposed robust terminal sliding mode tracking control scheme. In addition, based on the proposed approach, how to relax Assumption 1 is our future works. At the same time, fault-tolerant control for a class of nonaffine nonlinear systems is also our future works.

Acknowledgments

This work was supported by Natural Science Foundation of Shandong Province under Grant ZR2012F Q030. The authors wish to thank the reviewers for their constructive comments and suggestions which have helped to improve the presentation of the paper.