This study addresses the problem of attitude maneuver control for a three-axis stabilized liquid-filled spacecraft using an adaptive neural network variable structure control algorithm in the presence of parametric uncertainty, external disturbances, and control input saturation. The liquid fuel is equivalent to a spherical pendulum model, and the coupled dynamic model of liquid-filled spacecraft is derived using the conservation law of angular momentum moment. Then, adaptive variable structure control technique is designed, which contains hyperbolic tangent function that preserves control smoothness at all times. The proposed control algorithm has the properties that state variables converge to the origin asymptotically under parametric uncertainty and external disturbance. Furthermore, the controller derived here is extended by adding a feed-forward saturation compensation scheme to reduce the influence of unknown control input saturation on the system. Also, the saturation compensation scheme is derived by using a radial basis function neural network to approximate the unknown saturation nonlinearity. The associated stability proof of the resulting closed-loop system is presented based on Lyapunov analysis, and asymptotic convergence of the state variables is guaranteed via the Barbalat lemma. Numerical simulations are presented to illustrate the spacecraft performance obtained by using the proposed controllers.

1. Introduction

Modern spacecraft often contain large amounts of liquid fuel to achieve highly accurate pointing, fast slewing, and other fast maneuvers from large initial conditions during orbital transfer, rendezvous, and docking. Understanding the coupling effect between the sloshing dynamics of the fuel and a vehicle’s control systems is one of the major challenges for the aerospace industry. Liquid sloshing is usually considered to be the motion of the liquid surface inside its container. When a spacecraft with partially filled liquid performs attitude or trajectory maneuvers, liquid fuel in the tank inevitably sloshes. Liquid sloshing often has a significant impact on spacecraft attitude as well as can even induce instability and gives a great challenge to the design of controller with high precision [13]. Taking the famous SpaceX Falcon 1 in 2007 as an example, the fuel slosh caused by its second stage spin is out of control. Fuel sloshing can affect the dynamic response, which leads to mass distribution and energy dissipation. In order to reduce these effects and design high-performance controllers for liquid-filled spacecraft, it is necessary to study the coupled model and controller design for liquid-filled spacecraft. Many researchers have studied the effects of liquid sloshing on spacecraft attitude control systems. In the control system, a linear equivalent mechanical model (EMM), such as a pendulum model and a spring-mass model, is often used to replace liquid sloshing [2, 4]. Yue [5] studied the chaotic dynamics of a flexible spacecraft with a liquid-filled tank, where the liquid sloshing inside the tank was represented by a pendulum model [6]. Yue and Zhu [7] employed the equivalent spherical pendulum model to derive a mathematical model of a spacecraft and designed a hybrid control method based on input shaping technology and feedback linearization method to control the implementation of an attitude maneuver. With consideration of parameter uncertainties and unknown external disturbances, Song and Lu [8] used a two-mode spring-mass model to establish a mathematical model for the liquid-filled spacecraft, and a hybrid control method was developed by combining adaptive sliding mode control with the input shaping technology to achieve the attitude maneuver task. Zhang and Wang [9] studied the attitude regulation of a liquid-filled spacecraft under low-frequency sinusoidal disturbances. The dynamic model of the liquid-filled spacecraft was represented by a rigid body with a single spherical pendulum. Kang and Cochran [10] treated the sloshing mass of a spacecraft inside a partially filled liquid tank as a spherical pendulum model and investigated the problem of control stability of a spinning spacecraft. Among these equivalent models, the pendulum model is more suitable and is widely used in the study of launch vehicles. In this paper, the sloshing dynamics are modeled by a simple pendulum.

Variable structure control (VSC) has been extensively investigated in the design of robust controllers with high accuracy for linear and nonlinear systems and has been applied to various engineering fields. This usage is due to the fact that VSC has the advantage of dealing with unknown external disturbances, model uncertainties, and fast dynamic control for dynamic systems. For these advantages, VSC has received growing attention in the field of spacecraft attitude control research [1116]. Boškovic et al. [11] proposed two globally stable VSC algorithms for robust stabilization of rigid spacecraft under control input saturation, parametric uncertainty, and external disturbances. Wallsgrove and Akella [12] developed a globally stabilizing saturated VSC for a rigid spacecraft with unknown bounded disturbances. VSC also provides a systematic approach to the design of on/off control laws for multivariable systems, which is a desirable approach for the thruster-actuator modes commonly used in satellites. However, since variable structure controllers usually use high-frequency switching to drive the trajectories of the system on the sliding surface, this inevitably results in the chattering of the signal in the system in practical applications. Therefore, it is necessary to eliminate this undesirable characteristic by replacing the sign function with a saturation function in the design of such a controller [1719].

An important problem encountered in practice is the limitations of control inputs because of the physical constraints of onboard actuators. Input magnitude saturation is one of the major control issues in spacecraft attitude control systems, and the occurrence of input saturation may lead to unacceptable performance degradation or even instability. The design of controllers with input saturation has both practical interest and theoretical significance. This is a challenging problem since the input magnitude is upper-bounded by a priori fixed constant. Some researchers have solved the attitude control problem with actuator saturation. Zou and Kumar [20] proposed a robust adaptive fuzzy controller for spacecraft in the presence of unknown mass moment of inertia matrix, external disturbances, actuator failure, and control input constraints. Zou and Kumar [21] considered the problem of the control input saturation and proposed a distributed attitude coordination control scheme for spacecraft formation flying based on a neural network. Zhu et al. [22] presented an adaptive sliding mode controller for the attitude stabilization of spacecraft under control constraints. Ma et al. [23] proposed a fault-tolerant attitude tracking control scheme for a flexible spacecraft under actuator failure, model uncertainty, external disturbances, and actuator saturation. In addition, the authors of [2426] proposed a novel saturation compensation scheme for nonlinear systems based on a radial basis function (RBF) neural network. This method uses a smooth function to approximate the saturation function, and an RBF neural network is employed to approximate unknown control input saturation. Based on the RBF neural network, a feed-forward saturation compensator is developed to reduce the influence of unknown input saturation for spacecraft in [27, 28].

Inspired by the aforementioned works, in this work, the problem of the attitude maneuver of a liquid-filled spacecraft is investigated under input saturation, parametric uncertainty, and external disturbances. Variable structure control (VSC) and adaptive control approaches are applied to design the control algorithms. The main elements of this paper are stated as follows: (1)The dynamic equation of a three-axis stabilized liquid-filled spacecraft is established by using the law of conservation of momentum, and the liquid fuel inside the partially filled fuel tank is modeled as a spherical pendulum(2)A continuous hyperbolic tangent function is incorporated into the design of the controller, rather than the typical switching function of variable structure control, to ensure control signal smoothness at all times. Based on the RBF neural network, an actuator saturation compensation scheme is presented to compensate for the effects of input saturation on the spacecraft attitude control system(3)State variables of the system converge to the origin asymptotically by the proposed controller under parametric uncertainty and external disturbances. Also, the corresponding control torque converge to the origin subject to input saturation

The remainder of this paper is organized as follows. In Section 2, the mathematical model of the liquid-filled spacecraft is established, and the attitude control problem is formulated. Subsequently, the adaptive output feedback VSC is proposed, and a rigorous stability analysis of the resulting closed-loop system is presented in Section 3. Simulation results are then presented and analysed in Section 4 followed by conclusions in Section 5.

2. Mathematical Modelling

The mathematical model of rigid-liquid coupling spacecraft can be described by the following dynamic and kinematic equations.

2.1. The Dynamic Equation

As shown in Figure 1, the inertial coordinate frame is indicated by . The body-fixed coordinate frame is indicted by ; the point represents the center mass of the rigid body of the spacecraft. And the spherical pendulum coordinate frame is indicated by . The pivot point (or attachment point) of the spherical pendulum is represented by point , which is located on the axis of the spacecraft body-fixed coordinate frame. The length of the spherical pendulum is represented by , point represents the centers of masses of the system, and point represents the position of the spherical pendulum.

In view of the limitation of the length of the pendulum and the size of the tank, the sloshing of the liquid fuel is assumed to a small amplitude, namely, , such that . The location of point with respect to can be expressed as

The angular momentum of the system about point can be written as where , and represent the masses of spacecraft main body and spherical pendulum, respectively.

As such, equation (2) can be written as matrix form where , .

The law of conservation of angular momentum is applied to equation (3), and the kinetic equations of the system can be written as where represents the distance between point and point . indicates the sloshing variable of the spherical pendulum.

The dynamic equation for the relative motion of the pendulum is derived under the assumption that point is stationary in space. Therefore, the dynamic equation of the pendulum about point can be written as where denotes the absolute acceleration of point , and , with , in which and represent the viscosity coefficients of liquid fuel. with .

2.2. The Kinematic Equation

In order to avoid the singularity of parameters, the familiar four unitary quaternions is employed to describe the attitude of the spacecraft as follows: where represents the angular velocity of the spacecraft. denotes the skew symmetric matrix of . The four unitary quaternions are subjecting to the constraint relationship .

Here, the sloshing of the liquid is assumed to be negligibly small. Hence, the second-order small quantities in the dynamic system coupling equations (equations (4) and (6)) can be neglected for linearization. As a result, the new vector matrix is introduced for convenience in equation (6). Finally, it is possible to obtain the mathematical model of the liquid-filled spacecraft as follows: where with is the uncertain parameter matrix,

If the terms and are the disturbances for the maneuver control system, equation (8a) can be rewritten as where is considered as the lumped disturbance.

All three components of the control torque are constrained by a priori fixed constant given by where is the largest amplitude of torque provided by the actuator.

2.3. Control Problem

In this work, the rest-to-rest attitude maneuver control problem is considered. The control aim is to achieve desired rotations for attitude and angular velocity . The desired attitude and angular velocity are set to and . The subsequent work is to design a robust saturated controller for maneuver control system (equation (10)) such that for any initial attitude quaternions and angular velocity (1)all state variables in resulting closed-loop systems are uniformly bounded(2)the controlled spacecraft achieves the objective that and will converge to the origin asymptotically under parametric uncertainty, external disturbances, and input saturation

3. Robust Adaptive Variable Structure Control Design

The primary objective of the VSC is to design a suitable algorithm for changing the structure of the control system during its operation [11, 14, 29]. Based on the existing results, an adaptive output feedback VSC is presented to achieve closed-loop stability with lumped disturbances and unknown input saturation.

3.1. Design of Sliding Mode Control for Spacecraft

The sliding surface in vector form is defined as follows: where is a positive scalar. By multiplying both sides of equation (12) by , taking the derivative with respect to time, and using equations (8a), (8b), (8c), and (11), the following is obtained:

The form of control law is chosen as where represents the equivalent control component and the role of is used to guarantee that for all time.

Note that the control law (equation (14)) also includes the variable structure component; this component is selected to ensure that the sliding surface as can obtained in finite time. Its design is as follows:

The first derivative of along the trajectory of the system can be obtained

by choosing where with , . Here, denotes sign function and is defined as

Substituting equation (18) into equation (17)

It follows that and .

Remark 1. One of the main problems associated with the above-mentioned control law (equation (18)) is input chattering. This is induced by the imperfection of existing switching devices, namely, the sign function. Therefore, sign functions are usually replaced by saturation functions or approximate sign functions. It is important to note that the formulation in this paper uses the continuous hyperbolic tangent function instead of the typical switching function of variable structure control. As such, becomes and is expressed as

Consider the same Lyapunov function (equation (18)), this yields

By using the fact that is a positive-definite function in , it is shown that .

To determine the dynamics of spacecraft on the sliding surface, similar methods are adopted as in [11]. Once is reached, the motion system will remain on the sliding surface. Therefore, the dynamics of spacecraft on the sliding surface can be obtained by using to replace , namely, using equation (12) to substitute equation (7). This yields

The definition of is the instant of time when the sliding mode is reached, and the following result is easily obtained by integrating the two sides of equation (23).

So . Since the constraint relation of quaternions and the condition of reaching the sliding surface , it can be concluded that .

Remark 2. Another problem associated with the preceding control law is caused by the quaternion direction representation. It is well known that and represent the same orientation. Since the above control method guarantees that the attitude of the spacecraft can be stabilized at , namely, , this shows that even if the initial position of the spacecraft is located very close to , the spacecraft will rotate completely to derive point . It should be mentioned that the aforementioned case is not an energy-efficient control scheme. To avoid this “unwinding phenomenon” described in [30], the sliding surface is modified to . Then, it is shown that Equation (26) means that when the initial condition is , the system reaches . Conversely, when the initial condition is , the system reaches .

Remark 3. Another important problem related to the preceding control law is that it does not explicitly consider control input saturation. In addition, it is worth noting that disturbances always exist in the maneuver control system; however, the above control laws also do not take into account the rejection of disturbances.

An adaptive robust saturated control algorithm is proposed here to solve the above problem for liquid-filled spacecraft attitude maneuver. The proposed control scheme maintains the favorable characteristic of sliding mode control and uses the available control authority to achieve control objectives. This will be discussed in the following sections.

3.2. State Feedback Variable Structure Control Design with Time-Varying Sliding Surface

In order to give a clear design process of the control scheme, it is assumed that in this section. First, consider a simple control law can stabilize the control objective . Assuming that variables , , , and are measurements available for feedback, given the Lyapunov function , its first derivative with respect to time is obtained as follows:

The form of the state feedback variable structure control is chosen to be where is a positive constant. Substituting equation (28) into equation (27) yields By using Lyapunov’s second method, it can be concluded that .

It is worth mentioning that the main goal is to stabilize and ; the question that arises in this case is whether a control law similar to equation (28) can be found to achieve this objective. In the analysis of the previous section, the equivalent control based on sliding control is designed. When the sliding surface , it is obtained . Therefore, a control law is proposed, which can be composed of proportional attitude feedback term and augmented variable structure component. The state feedback variable structure control law can be obtained in the following form: where are positive constants; is filter vector, and it can be calculated by the solution of linear differential equation .

Remark 4. The filter state satisfies the stable first-order linear vector differential equation and is forced by attitude vector . In terms of frequency response characteristics, this filter dynamics can be considered as low-pass systems for all bounded signals. Also, it should be noted that the value of the filter gain plays a significant role in defining the low-pass characteristics of the filter state . The choice of this design parameter depends on the level of signal noise encountered in the practical application [31].

Theorem 1. For spacecraft attitude maneuver control systems governed by equations (7) and (10), the designed control law given in equation (30) can guarantee the global asymptotic stability of the closed-loop system.

Proof. Consider the following Lyapunov function conditions where parameter is a time-varying function and an adaptive updating law is designed for subsequently.
The first derivative of equation (31) is given by The updating law for is designed to satisfy where is a positive constant.
According to equation (32), equation (33) can be rewritten as The integral on both sides of equation (34) yields . This shows that is a bounded function, such that is uniformly bounded. Due to the boundedness of all signals in the time derivative of equation (34), it is easy to demonstrate that is bounded, and the uniform continuity of can be obtained. Applying the Barbalat lemma to obtain as which further leads to the conclusion that , and from this result, it can be concluded that . Next, since the third derivative of function for time is uniformly bounded, this proves that is uniformly bounded. Additionally, since , the following results are obtained: Using Barbalat’s lemma, it is shown that ; since , it follows that as . According to the above obtained conclusion and the boundness of signals on the right side of equation (7), it can verify that is uniformly bounded; hence, applying the uniform continuity for , the following results are obtained: and reusing Barbalat’s lemma, the result is obtained. Because , using equations (8a), (8b), and (8c), one can conclude that is bounded. This means that is uniformly bounded. By implying Barbalat’s lemma, it is possible to conclude that . Further, note that as such that as . Thus, the trajectory of the system converges to

Remark 6. The state feedback control law requires all states to be measurable. This is a very difficult condition to achieve in practical application. In addition, the value of also affects the system response since is used to achieve the relative weight between and . A smaller will result in a slow response in . Also, it is worth mentioning that the control law (equation (30)) does not consider the rejection of disturbance and the control input saturation. In this case, adaptive output feedback control strategy is used to solve this kind of problem.

3.3. Adaptive Output Feedback Variable Structure Control under Actuator Saturation
3.3.1. Radial Basis Function Neural Network (RBFNN)

Due to the universal approximation characteristics and learning ability of neural networks (NN), NN have proven to be a powerful tool for controlling complex dynamic nonlinear systems with parametric uncertainties. Due to its simple structure and good approximation ability, RBFNN is employed to approximate the continuous function in the present work. First, the input of the saturated nonlinear control is approximated by a smooth function defined as [27] where represents the control torque to be designed and is defined as

Assuming that is unknown and from the perspective of compensation, the saturated component can be considered as a system disturbance and compensated by feed-forward scheme. Since is a continuous function that can be approximated by a radial basis function neural network (RBFNN), it is well known that as long as the NN is large enough, the neural network can approximate any continuous function with arbitrary precision. For a continuous nonlinear function , it can be approximated as where and represent the input signals of the neural network and approximation error, respectively.

And is a known continuous smooth vector value function; are selected as the commonly used Gaussian function. is called the number of neural network nodes. is the optimal weight vector, and and are the center vector and radial basis vector of , respectively. The RBFNN output is the approximation value of . Optimal weight vector is defined as

In general, is the vector selected from to minimize . Mathematically, that is where is the estimation value of , is compact set.

For MIMO systems, the nonlinear function can be approximated using the RBF network as follows: where and , , .

Assumption 7. According to the universal approximation property of RBF network, the maximum mean square value of approximation error vector is bounded and satisfies , in which is a small constant.

Assumption 8. The ideal NN optimal weight is bounded, such that and is a positive constant.

3.3.2. Adaptive Output Feedback Variable Structure Control with RBFNN-Based Saturation Scheme

To remove the drawbacks discussed in Remark 11 for the control law (equation (30)), the derived controller in the previous section is extended to the output feedback case to estimate attitude gain function and the liquid sloshing displacement variable by selecting the appropriate adaptive control law. Simultaneously, compensation scheme-based neural network is implemented as feed-forward control method to compensate nonlinear saturation effects.

The adaptive output feedback VSC is proposed as follows:

Modified updating law for is designed to satisfy

Adaptive updating law for liquid sloshing displacement variable is designed as

Neural network compensator updating law is given by where and are positive definite symmetric matrices. denotes an adaptive attitude gain.

Assumption 9. It is assumed that the positive constant strictly dominates the unknown disturbance and small constant , namely, is satisfied,, where is a small positive constant.

Theorem 2. Consider the attitude maneuver control system governed by equations (7) and (10), if control input is given by equation (44a) with the adaptive control law given by equations (44b) and (44c) and NN compensator updating law given by equation (44d). All trajectories of the closed-loop system obtained are uniformly bounded for all . In addition, they asymptotically converge to the set defined by .

Proof. Consider the following Lyapunov function candidate: where represents the trace of a matrix .
The first derivative with respect to time can be obtained by computing along the trajectory of the system It is worth mentioning that the following inequality holds: Substituting equation (48) into equation (47), it can obtain Note the fact that where and is the minimum eigenvalues of the matrix . can be computed as the solution of for any fixed .
From Assumption 9, it is easy to show that It can be seen that , , , , , and are all uniformly bounded. It can also be concluded that is bounded, which means that is uniform continuity function and is obtained using Barbalat’s lemma. Here, the analysis process is similar to the previous section by using Barbalat’s lemma; thus, the following results are revealed: This completes the proof of Theorem 2.

Remark 11. According to the proof of Theorem 2, the adaptive output feedback control scheme developed in equation (46) can achieve the attitude stabilization maneuver of the liquid-filled spacecraft with lumped disturbances and input saturation.

Remark 12. It should be noted that Theorem 2 only guarantees that as ; this will not necessarily ensure that will converge to zero. If converges to zero faster than , then will converge to a nonzero constant value. In order to guarantee that converge to zero, it is necessary to prevent from converging to zero. This can be achieved by using a sufficiently small to ensure changes slowly such that it does not deviate too much from its initial value. In addition, since is used to achieve relative weights between and , a smaller will result in slow response of . Conversely, a higher will lead to oscillatory response in system. Subsequent numerical simulation examples show that appropriate initial value and/or small enough parameter can be found in most cases. This ensures that the sliding mode converges to the origin under parametric uncertainty, external disturbances, and input saturation. In other words, initial value and parameter need to be correctly selected to tune the controller within acceptable transient response and performance.

Remark 13. According to the definition of attitude quaternions in equations (8a), (8b), and (8c) and the property of hyperbolic tangent function, it can be easily obtained that , , and. If the term is not considered in equation (46), note that the control input satisfies the input saturation limitations, namely, . This indicates that the RBFNN saturation compensator is used to compensate for saturation uncertainty caused by term

4. Simulation Results

To demonstrate the effectiveness and performance of the proposed attitude control schemes combined with RBFNN, a liquid-filled spacecraft model governed by equations (8a), (8b), (8c), and (11) is used to conduct numerical simulations. Neural network contains 15 nodes with center vector evenly spaced in . Nominal inertia matrix of the spacecraft is given by

The external disturbances are assumed to be and an uncertainty on the inertia matrix is given by ; sensor noise is set as , where represents the random Gaussian white noise vector. The controller gains are listed as , , , , , , , , , and .

The relevant parameters of liquid fuel sloshing are selected as , , , , and . It is assumed that the spacecraft is at the earth’s orbit altitude of 800 km and the local gravitational acceleration is selected as .

Initial angular velocity is , initial displacement of the pendulum is , and initial quaternions value is

Initial estimated neural network optimal weight is given by , and initial attitude gain is given by . Bound of control torque is .

To demonstrate the characteristics of the proposed control, three simulation cases are given below.

Case 1. Attitude maneuver control using variable structure output feedback control and RBFNN technique, the simulation results are shown in Figures 28.

Case 2. Attitude maneuver control using variable structure controller without considering RBFNN, the simulation results are shown in Figures 914.

The simulation results of Case 1 are shown in Figures 28. The simulation results of Case 2 are shown in Figures 914. Figures 2 and 9 show the time histories of angular velocities. Figures 3 and 10 show the time histories of the quaternions. Compared with Figures 9 and 10, it can be observed from Figures 2 and 3 that the angular velocities and quaternions converge to the origin in around 20 seconds with good performance and smooth transient response. This demonstrates that the proposed robust control law incorporated with RBFNN can accomplish the desired attitude maneuver under the influence of external disturbances, parametric uncertainty, and sensor noise. Figure 4 shows the time histories of the liquid sloshing displacement. It can be seen that the liquid sloshing displacement is low amplitude from Figure 4. Specifically, the sloshing amplitude is less than 0.1 m. This indicates that the proposed controller can successfully account for the effect of liquid sloshing on the attitude maneuver ability of spacecraft. Figure 5 presents the time histories of attitude gain . It can be reported from Figure 5 that the control objective is achieved and converges to approximately 0.5 in about 60 seconds.

Figures 6 and 12 show the time histories of the filter state. In contrast to Figure 12, the state of the filter also converges to the origin with good transient and steady-state response performance from Figure 6. This demonstrates that the proposed filter is stable and can effectively reject the influence of the sensor noise to ensure the stability of the overall system. Figure 7 shows the time history of the estimated RBFNN optimal weight. It can be seen from Figure 7 that the estimated optimal weight state also converges to zero with good transient and smooth response performance. This shows that the RBFNN saturation compensator is stable and can effectively compensate for saturation nonlinearity. Figures 8 and 14 report the control torques with the RBFNN saturation compensator and without the RBFNN saturation compensator, respectively. It can be seen that all three components of the control signals with the RBFNN saturation compensator exceed the control signals without RBFNN saturation compensator. The energy consumption indicator is introduced here and is calculated over the total simulation time of . It is obvious that the robust saturated control strategy completes the attitude maneuver successfully with less control energy.

Case 3. The roles of auxiliary function on the system response are demonstrated by choosing three different initial values. The simulation results are shown in Figures 1518 by choosing . The simulation results are shown in Figures 1922 by choosing . The simulation results are shown in Figures 2326 by choosing .

The simulation results are shown in Figures 1518 by choosing , the simulation results are shown in Figures 1922 by choosing , and simulation results are shown in Figures 2326 by choosing . Here, the role of attitude gain is explained in the language of classical second-order linear system. A smaller value is conceptually equivalent to a higher damping ratio. When value decreases, the emphasis of control will deviate from the term and move towards the term, which will lead to overdamped sluggish behavior in Figures 1518. The higher value corresponds conceptually to the lower damping ratio. When increases, the focus of control is shifted away from term and moves towards term, which results in the underdamped oscillation behavior shown in Figures 1926. Therefore, acts as desired weighting to achieve the acceptable response between angular velocity and quaternions , and these analyses are consistent with the explanation in Remark 12. Designers can appropriately choose value and parameter to obtain acceptable transient and steady-state response performance and avoid slow or wild oscillation behavior.

5. Conclusions

In this paper, adaptive output feedback VSC is proposed for liquid- filled spacecraft attitude stabilization maneuver under control input saturation, parametric uncertainty, and bounded external disturbances. This control algorithm is based on the variable structure control and adaptive control technique. Additionally, this control algorithm uses continuous hyperbolic tangent functions to guarantee smoothness of the control signals and improve the transient response in the presence of parametric uncertainty and bounded external disturbances. Moreover, the problem of control input saturation is compensated based on RBFNN approaches, where an intelligent saturation compensator is inserted into a feed-forward loop. Moreover, this control scheme uses a low-pass filter variable synthesized through a first-order stable differential equation, which is tuned by the filter parameter and driven by the sliding mode vector, and the influences of noisy measurements on the system response can be reduced by using this filter. Lyapunov analysis is employed to ensure global stability of the resulting closed-loop system. The asymptotic convergence of the system state variables is guaranteed by using Barbalat’s lemma. Numerical examples are presented to illustrate the effectiveness and robustness of the proposed control method.

Data Availability

All the data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there is no conflict of interests regarding the publication of this paper.


The authors sincerely acknowledge the financial support of the National Natural Science Foundation of China (Grant Nos. 11962020, 11862020, and 11502122) and the Inner Mongolia Natural Science Foundation (Grant No. 2019MS05065).