# Adaptive Neural Network Variable Structure Control for Liquid-Filled Spacecraft under Unknown Input Saturation

Authors: Hongwei Wang; Shufeng Lu; Xiaojuan Song
Journal: International Journal of Aerospace Engineering
 (2020)

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## Abstract

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.


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## Body

## 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 [1–3]. 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 [11–16]. 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 [17–19].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 [24–26] 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 saturationThe remainder of this paper is organized as follows. In Section2, 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 Figure1, the inertial coordinate frame is indicated by O−XYZ. The body-fixed coordinate frame is indicted by O1−X1Y1Z1; the point O1 represents the center mass of the rigid body of the spacecraft. And the spherical pendulum coordinate frame is indicated by O2−X2Y2Z2. The pivot point (or attachment point) of the spherical pendulum is represented by point O2, which is located on the O1X1 axis of the spacecraft body-fixed coordinate frame. The length of the spherical pendulum is represented by l, point C represents the centers of masses of the system, and point P represents the position of the spherical pendulum.Figure 1
Schematic diagram and dynamic model of liquid-filled spacecraft. (a) Simplified schematic diagram. (b) Equivalent model of liquid sloshing.
(a)(b)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,y,z<<l, such that x≈−l. The location of point P with respect to O2 can be expressed as
(1)r=−l2−y2−z2yzT=−lyzT.The angular momentum of the system about pointC can be written as
(2)H=Jω+μmbrp×ω×rp+ṙp,where μ=mb/mb+mb, mb and mp represent the masses of spacecraft main body and spherical pendulum, respectively.As such, equation (2) can be written as matrix form
(3)H=Jbω+hp,where Jb=J−μmbrp×rp×, hp=μmbrp×ṙ.The law of conservation of angular momentum is applied to equation (3), and the kinetic equations of the system can be written as
(4)Jbω̇+δTη¨=−ω×Jbω+δTη̇−Jbω+dt+u.where
(5)δ=μmb−z0rx−ly−rx−l0,rx represents the distance between point O1 and point O2. η=yzT indicates the sloshing variable of the spherical pendulum.The dynamic equation for the relative motion of the pendulum is derived under the assumption that pointC is stationary in space. Therefore, the dynamic equation of the pendulum about point C can be written as
(6)μmbap=To+Tg,where ap=r¨2=1−μr¨p denotes the absolute acceleration of point P, and r¨p=r¨+ω̇×rp+2ω×ṙ+ω×ω×rp,T0=−∂R/∂ẏ∂R/∂żT with R=c1ẏ2+c2ż2ẏ,ż≠0, in which c1 and c2 represent the viscosity coefficients of liquid fuel. Tg=∂U/∂y∂U/∂zT with U=mpgl2−y2−z2.


### 2.2. The Kinematic Equation

In order to avoid the singularity of parameters, the familiar four unitary quaternionsq0qT∈R4 is employed to describe the attitude of the spacecraft as follows:
(7)q̇0q̇=12STq0,qω=12−qq0I3+q×ω,where ω∈R3 represents the angular velocity of the spacecraft. q×∈R3×3 denotes the skew symmetric matrix of q. The four unitary quaternions are subjecting to the constraint relationship q02+qTq=1.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 ψ is introduced for convenience in equation (6). Finally, it is possible to obtain the mathematical model of the liquid-filled spacecraft as follows:
(8a)Jmbω̇=−ω×Jmbω+δTψ+δTCψ+Kη−CM−1δω+dt+u,(8b)η̇=ψ−M−1δω,(8c)ψ̇=−Cψ+Kη−CM−1δω,where Jmb=Jb−δTM−1δ=J+ΔJ with ΔJ is the uncertain parameter matrix,
(9)M=μmb1001,C=M−12c1002c2,K=mpM−1g/l00g/l.If the termsΔJω̇ and ω×ΔJω are the disturbances for the maneuver control system, equation (8a) can be rewritten as
(10)ω̇=J−1−ω×Jω+δTψ+δTCψ+Kη−CM−1δω+Td+u,where Td=−ΔJω̇−ω×ΔJω+dt is considered as the lumped disturbance.All three components of the control torqueut are constrained by a priori fixed constant given by
(11)u∈Su=uit≤u¯m,i=1,2,3,where u¯m 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 attitudeq and angular velocity ω. The desired attitude and angular velocity are set to qd=000T and ωd=000T. 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 thatqt and ωt will converge to the origin asymptotically under parametric uncertainty, external disturbances, and input saturation


## 2.1. The Dynamic Equation

As shown in Figure1, the inertial coordinate frame is indicated by O−XYZ. The body-fixed coordinate frame is indicted by O1−X1Y1Z1; the point O1 represents the center mass of the rigid body of the spacecraft. And the spherical pendulum coordinate frame is indicated by O2−X2Y2Z2. The pivot point (or attachment point) of the spherical pendulum is represented by point O2, which is located on the O1X1 axis of the spacecraft body-fixed coordinate frame. The length of the spherical pendulum is represented by l, point C represents the centers of masses of the system, and point P represents the position of the spherical pendulum.Figure 1
Schematic diagram and dynamic model of liquid-filled spacecraft. (a) Simplified schematic diagram. (b) Equivalent model of liquid sloshing.
(a)(b)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,y,z<<l, such that x≈−l. The location of point P with respect to O2 can be expressed as
(1)r=−l2−y2−z2yzT=−lyzT.The angular momentum of the system about pointC can be written as
(2)H=Jω+μmbrp×ω×rp+ṙp,where μ=mb/mb+mb, mb and mp represent the masses of spacecraft main body and spherical pendulum, respectively.As such, equation (2) can be written as matrix form
(3)H=Jbω+hp,where Jb=J−μmbrp×rp×, hp=μmbrp×ṙ.The law of conservation of angular momentum is applied to equation (3), and the kinetic equations of the system can be written as
(4)Jbω̇+δTη¨=−ω×Jbω+δTη̇−Jbω+dt+u.where
(5)δ=μmb−z0rx−ly−rx−l0,rx represents the distance between point O1 and point O2. η=yzT indicates the sloshing variable of the spherical pendulum.The dynamic equation for the relative motion of the pendulum is derived under the assumption that pointC is stationary in space. Therefore, the dynamic equation of the pendulum about point C can be written as
(6)μmbap=To+Tg,where ap=r¨2=1−μr¨p denotes the absolute acceleration of point P, and r¨p=r¨+ω̇×rp+2ω×ṙ+ω×ω×rp,T0=−∂R/∂ẏ∂R/∂żT with R=c1ẏ2+c2ż2ẏ,ż≠0, in which c1 and c2 represent the viscosity coefficients of liquid fuel. Tg=∂U/∂y∂U/∂zT with U=mpgl2−y2−z2.


## 2.2. The Kinematic Equation

In order to avoid the singularity of parameters, the familiar four unitary quaternionsq0qT∈R4 is employed to describe the attitude of the spacecraft as follows:
(7)q̇0q̇=12STq0,qω=12−qq0I3+q×ω,where ω∈R3 represents the angular velocity of the spacecraft. q×∈R3×3 denotes the skew symmetric matrix of q. The four unitary quaternions are subjecting to the constraint relationship q02+qTq=1.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 ψ is introduced for convenience in equation (6). Finally, it is possible to obtain the mathematical model of the liquid-filled spacecraft as follows:
(8a)Jmbω̇=−ω×Jmbω+δTψ+δTCψ+Kη−CM−1δω+dt+u,(8b)η̇=ψ−M−1δω,(8c)ψ̇=−Cψ+Kη−CM−1δω,where Jmb=Jb−δTM−1δ=J+ΔJ with ΔJ is the uncertain parameter matrix,
(9)M=μmb1001,C=M−12c1002c2,K=mpM−1g/l00g/l.If the termsΔJω̇ and ω×ΔJω are the disturbances for the maneuver control system, equation (8a) can be rewritten as
(10)ω̇=J−1−ω×Jω+δTψ+δTCψ+Kη−CM−1δω+Td+u,where Td=−ΔJω̇−ω×ΔJω+dt is considered as the lumped disturbance.All three components of the control torqueut are constrained by a priori fixed constant given by
(11)u∈Su=uit≤u¯m,i=1,2,3,where u¯m 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 attitudeq and angular velocity ω. The desired attitude and angular velocity are set to qd=000T and ωd=000T. 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 thatqt and ωt 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:(12)s=ω+kq,where k is a positive scalar. By multiplying both sides of equation (12) by J, taking the derivative with respect to time, and using equations (8a), (8b), (8c), and (11), the following is obtained:
(13)Jṡ=−ω×Jω+δTCψ+Kη−CM−1δω+u+12Jkq0I3+q×ω.The form of control law is chosen as(14)u=ueq+uvs,where ueq represents the equivalent control component and the role of ueq is used to guarantee that ṡt=0 for all time.
(15)ueq=ω×Jω−δTCψ+Kη−CM−1δω−12Jkq0I3+q×ω.Note that the control law (equation (14)) also includes the variable structure component; this component is selected to ensure that the sliding surface st=0 as t→∞ can obtained in finite time. Its design is as follows:
(16)V=12sTJs.The first derivative ofV along the trajectory of the system can be obtained
(17)V̇=sTuvs,by choosing(18)uvs=−∑sα,where α=α1α2α3T with αi>0, ∑s=diagsgns1sgns2sgns3. Here, sgn⋅ denotes sign function and is defined as
(19)sgnβ=1,β>0,0,β=0,−1,β<0.Substituting equation (18) into equation (17)
(20)V̇=−sT∑sα=−∑i=13αsi<0.It follows thats∈L∞ and limt→∞s=0.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, uvs=−∑sα becomes u¯vs=−tanhsα and tanhs is expressed as
(21)tanhs=tanhs1tanhs2tanhs3T.Consider the same Lyapunov function (equation (18)), this yields
(22)V̇=sTu¯vs≤−αsTtanhs≤0.By using the fact thatsTtanhs is a positive-definite function in s∈R3, it is shown that limt→∞s=0.To determine the dynamics of spacecraft on the sliding surface, similar methods are adopted as in [11]. Once st=0 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 st=0 to replace ω, namely, using equation (12) to substitute equation (7). This yields
(23)q̇0=−12qT−kq=k21−q02,(24)q̇=12q0I+q×−kq=−12kq0q.The definition oftf is the instant of time when the sliding mode s=0 is reached, and the following result is easily obtained by integrating the two sides of equation (23).
(25)q0t=1−21−q0tfe−kt−tf1+q0tf+1−q0tfe−kt−tf,t≥tf.Solimt→∞q0=1. Since the constraint relation of quaternions q02+qTq=1 and the condition of reaching the sliding surface ω=−kq, it can be concluded that limt→∞q=limt→∞ω=0.Remark 2.
Another problem associated with the preceding control law is caused by the quaternion direction representation. It is well known thatq0=1 and q0=−1 represent the same orientation. Since the above control method guarantees that the attitude of the spacecraft can be stabilized at q0=1, namely, limt→∞q0=1, this shows that even if the initial position of the spacecraft is located very close to q0=−1, the spacecraft will rotate completely to derive point q0=1. 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 s=ω+kqsgnq0. Then, it is shown that
(26)q̇0=12qTkqsgnq0=k21−q02sgnq0.
Equation (26) means that when the initial condition is q0>1, the system reaches limt→0q0=1. Conversely, when the initial condition is q0<1, the system reaches limt→0q0=−1.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 thatTd=0 in this section. First, consider a simple control law can stabilize the control objective ω. Assuming that variables ω, q, η, and ψ are measurements available for feedback, given the Lyapunov function V2=1/2ωTJω, its first derivative with respect to time is obtained as follows:
(27)V̇2=ωTδTKCTηψ−δTCM−1δω+u.The form of the state feedback variable structure control is chosen to be(28)u=−δTKCTηψ−kdtanhω,where kd is a positive constant. Substituting equation (28) into equation (27) yields
(29)V̇2=−ωTδTCM−1δω−kdωTtanhω≤−δTCM−1δω2−kdωTtanhω≤0.By using Lyapunov’s second method, it can be concluded that limt→∞ωt=0.It is worth mentioning that the main goal is to stabilizeωt and qt; 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 s=0, it is obtained limt→∞ωt=limt→∞qt=0. 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:
(30)u=−kpq−δTKCTηψ−kdtanhs−kz2q0I3+q×tanhq−k12z,where kp,kz are positive constants; zt∈R3 is filter vector, and it can be calculated by the solution of linear differential equation żt=qt−k12z.Remark 4.
The filter statezt satisfies the stable first-order linear vector differential equation and is forced by attitude vector qt. In terms of frequency response characteristics, this filter dynamics can be considered as low-pass systems for all bounded qt signals. Also, it should be noted that the value of the filter gain k1 plays a significant role in defining the low-pass characteristics of the filter state zt. 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(31)V3=12ωTJω+kpqTq+1−q02+kz∑i=13logcoshqi−k12zi+12γk2=12ωTJω+2kp1−q0+kz∑i=13logcoshqi−k12zi+12γk2,where parameter kt is a time-varying function and an adaptive updating law is designed for kt subsequently.
The first derivative of equation (31) is given by
(32)V̇3=ωTJω̇−2kpq̇0−kzq̇−k12żTtanhq−k12z+1γkk̇=ωT−ω×Jω+δTψ+δTCψ+Kη−CM−1δω−kpq−δTKCTηψ−kdtanhs−kz/2q0I3+q×tanhq−k12z−2kpq̇0+kzq̇−k12żTtanhq−k12z+1γkk̇=−ωTδTCM−1δω−kzk12żTtanhż−kdωTtanhs+1γkk̇.
The updating law fork is designed to satisfy
(33)k̇=−γkdqTtanhs,where γ is a positive constant.
According to equation (32), equation (33) can be rewritten as
(34)V̇3=−ωTδTCM−1δω−kzk12żTtanhż−kdsTtanhs≤0.
The integral on both sides of equation (34) yields supt≥0Vt≤V0. This shows that V3t is a bounded function, such that ω,q,zt,żt is uniformly bounded. Due to the boundedness of all signals in the time derivative of equation (34), it is easy to demonstrate that V¨3t is bounded, and the uniform continuity of V̇3t can be obtained. Applying the Barbalat lemma to obtain V̇t→0 as t→∞which further leads to the conclusion that limt→∞ω=limt→∞s=0, and from this result, it can be concluded that limt→∞kq=0. Next, since the third derivative of zt function for time is uniformly bounded, this proves that z¨tis uniformly bounded. Additionally, since limt→∞żt=0, the following results are obtained:
(35)limt→∞∫0tz¨τdτ=limt→∞żt=0.
Using Barbalat’s lemma, it is shown thatlimt→∞z¨t=0; since z¨t=q̇−k2żt, it follows that limt→∞q̇=0 as t→∞. According to the above obtained conclusion and the boundness of signals on the right side of equation (7), it can verify that q¨t is uniformly bounded; hence, applying the uniform continuity for limt→∞q̇=0, the following results are obtained:
(36)limt→∞∫0tq̇τdτ=limt→∞qt=0,and reusing Barbalat’s lemma, the result limt→∞q=0 is obtained. Because limt→∞ω=0, using equations (8a), (8b), and (8c), one can conclude that ω¨t is bounded. This means that ω̇t is uniformly bounded. By implying Barbalat’s lemma, it is possible to conclude that limt→∞ω̇=0. Further, note that Jω̇→0 as t→∞ such that ηψT→0 as t→∞. Thus, the trajectory of the system converges to
(37)limt→∞t,st,ηψ=0.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 ofk also affects the system response since k is used to achieve the relative weight between ωt and qt. A smaller k will result in a slow response in qt. 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]
(38)v=gu=u+Δu,where v=v1v2v3T represents the control torque to be designed and gu is defined as
(39)gu=umax×tanhuumax.Assuming thatΔu 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 Δu=gu−u 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 fx∈R, it can be approximated as
(40)fx=f^x,θ∗+ε=θ∗Tϕx+ε,φn=exp−x−ci2σi2,i=1,2,⋯,n,where x∈Rn and ε∈R represent the input signals of the neural network and approximation error, respectively.Andϕx=φ1x,φ2x,φ3x,⋯,φlxT∈Rl is a known continuous smooth vector value function; φxi are selected as the commonly used Gaussian function. l is called the number of neural network nodes. θ∗∈Rn is the optimal weight vector, and ci∈Rn and σi>0 are the center vector and radial basis vector of ϕx, respectively. The RBFNN output f^x,θ∗ is the approximation value of fx. Optimal weight vector is defined as
(41)θ∗=θ1∗x,θ2∗x,θ3∗x,⋯,θl∗x∈Rl.In general,θ∗ is the vector selected from θ to minimize fx−f^x,θ. Mathematically, that is
(42)θ∗=argminθ^supx∈Ωfx−f^x,θ,where θ^ is the estimation value of θ∗, Ω∈Rl is compact set.For MIMO systems, the nonlinear functionΔu can be approximated using the RBF network as follows:
(43)Δu=θ∗Tϕx+ε,where x=qTωTT and θ∗∈Rl×3, θ∗=θ1∗Tx,θ2∗Tx,θ3∗TxT, ε=ε1ε2ε3T.Assumption 7.
According to the universal approximation property of RBF network, the maximum mean square value of approximation error vector is bounded and satisfiesε≤ε0, in which ε0 is a small constant.Assumption 8.
The ideal NN optimal weightθ∗ is bounded, such that θ∗≤θm and θm is a positive constant.


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

To remove the drawbacks discussed in Remark11 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 kt 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:(44a)ut=−kpq−δTKCTη^ψ^−kdtanhs−kz2q0I3+q×tanhq−k12z−θ∧Tϕx.Modified updating law fork is designed to satisfy
(44b)k̇=−γkkqkqTtanhs+∑i=13Lisisgnk+Likqisgnsi.Adaptive updating law for liquid sloshing displacement variable is designed as(44c)η^̇ψ^̇=Aη^ψ^−I−CM−1δω−Γ−1KCδω.Neural network compensator updating law is given by(44d)θ^̇=PϕxωT−δθsθ^,where
(45)A=0I−K−C.Γ∈R4×4 and P∈Rl×l are positive definite symmetric matrices.γ>0 denotes an adaptive attitude gain.Assumption 9.
It is assumed that the positive constantLi strictly dominates the unknown disturbance Td and small constant ε0, namely, Tdi≤dm≤dm+ε0+1/2δθθm2≤Li is satisfied,∀T≥0,i=1,2,3, where δθ is a small positive constant.Theorem 2.
Consider the attitude maneuver control system governed by equations (7) and (10), if control input ut 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 t≥0. In addition, they asymptotically converge to the set defined by limt→∞st,zt,eηeψ,θ~=0.Proof.
Consider the following Lyapunov function candidate:(46)V4=12ωTJω+2kp1−q0+12eηeψTΓeηeψ+kz∑i=13logcoshqi−k12zi+12γk2+12trθ~TP−1θ~,where tr⋅ represents the trace of a matrix ⋅.
The first derivative with respect to time can be obtained by computingV4 along the trajectory of the system
(47)V̇4=ωT−ω×Jω+δTψ+δTKCTηψ−δTCM−1δω+dt−kpq−δTKCTη^ψ^−kdtanhs−kz/2q0I3+q×tanhq−k12z+θ~Tϕx+ε+eηeψTΓAηψ−I−CM−1δω−η^̇ψ^̇+kzq̇−k12żTtanhq−k12z+1γkk̇+trθ~TP−1θ~̇≤−λmδTCM−1δω2+s−kqdm+ε0−Li∑i=13sisgnk+kqisgnsi−kqsTtanhs−kzk12żTtanhż−eηeψTQeηeψ+δθtrθ~Tsθ∗−θ~.
It is worth mentioning that the following inequality holds:(48)trθ~Tsθ∗−θ~≤sθ~θ∗−sθ~2≤12sθ∗2−12sθ~2.
Substituting equation (48) into equation (47), it can obtain
(49)V̇4≤−λmδTCM−1δω2+s−kqdm+ε0−kzk12żTtanhż−kqsTtanhs−eηeψTQeηeψ+12δθθm2s−12δθsθ~2−Li∑i=13si−Li∑i=13kqi≤−λmδTCM−1δω2−kzk12żTtanhż−Li−dm−ε0∑i=13kqit−s1+Li−dm−ε0−12δθθm2∑i=13si−eηeψTQeηeψ−12δθθ~2∑i=13si
Note the fact that(50)ωTTd+ε≤ωdm+ε0≤dm+ε0∑i=13si+kqi,where s1=kptanhs and λm⋅ is the minimum eigenvalues of the matrix ⋅. Γ can be computed as the solution of ΓA+AΓT=−2Q for any fixed Q=QT>0.
From Assumption9, it is easy to show that
(51)V̇4≤0.
It can be seen thatωt, qt, zt, żt, θ~, and eηeψT are all uniformly bounded. It can also be concluded that V¨t is bounded, which means that V̇t is uniform continuity function and limt→∞V̇t=0 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:
(52)limt→∞ktqt=0,zt,st=0,eηeψ,θ~=0..
This completes the proof of Theorem2.Remark 11.
According to the proof of Theorem2, 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 Theorem2 only guarantees that ktqt=0 as t→∞; this will not necessarily ensure that qt will converge to zero. If kt converges to zero faster than qt, then qt will converge to a nonzero constant value. In order to guarantee that qt converge to zero, it is necessary to prevent kt from converging to zero. This can be achieved by using a sufficiently small γ to ensure kt changes slowly such that it does not deviate too much from its initial value. In addition, since kt is used to achieve relative weights between ωt and qt, a smaller kt will result in slow response of qt. Conversely, a higher kt will lead to oscillatory response in system. Subsequent numerical simulation examples show that appropriate initial value k0 and/or small enough parameter γ can be found in most cases. This ensures that the sliding mode s converges to the origin under parametric uncertainty, external disturbances, and input saturation. In other words, initial value k0 and parameter γ 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 q0I3+q×≤1, q≤1, andtanhs−k12z≤1. If the term
(53)δTKCTη^ψ^is not considered in equation (46), note that the control input satisfies the input saturation limitations, namely, ut≤kp+kq+kz/2≤u¯m. This indicates that the RBFNN saturation compensator is used to compensate for saturation uncertainty caused by term
(54)δTKCTη^ψ^.


## 3.1. Design of Sliding Mode Control for Spacecraft

The sliding surface in vector form is defined as follows:(12)s=ω+kq,where k is a positive scalar. By multiplying both sides of equation (12) by J, taking the derivative with respect to time, and using equations (8a), (8b), (8c), and (11), the following is obtained:
(13)Jṡ=−ω×Jω+δTCψ+Kη−CM−1δω+u+12Jkq0I3+q×ω.The form of control law is chosen as(14)u=ueq+uvs,where ueq represents the equivalent control component and the role of ueq is used to guarantee that ṡt=0 for all time.
(15)ueq=ω×Jω−δTCψ+Kη−CM−1δω−12Jkq0I3+q×ω.Note that the control law (equation (14)) also includes the variable structure component; this component is selected to ensure that the sliding surface st=0 as t→∞ can obtained in finite time. Its design is as follows:
(16)V=12sTJs.The first derivative ofV along the trajectory of the system can be obtained
(17)V̇=sTuvs,by choosing(18)uvs=−∑sα,where α=α1α2α3T with αi>0, ∑s=diagsgns1sgns2sgns3. Here, sgn⋅ denotes sign function and is defined as
(19)sgnβ=1,β>0,0,β=0,−1,β<0.Substituting equation (18) into equation (17)
(20)V̇=−sT∑sα=−∑i=13αsi<0.It follows thats∈L∞ and limt→∞s=0.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, uvs=−∑sα becomes u¯vs=−tanhsα and tanhs is expressed as
(21)tanhs=tanhs1tanhs2tanhs3T.Consider the same Lyapunov function (equation (18)), this yields
(22)V̇=sTu¯vs≤−αsTtanhs≤0.By using the fact thatsTtanhs is a positive-definite function in s∈R3, it is shown that limt→∞s=0.To determine the dynamics of spacecraft on the sliding surface, similar methods are adopted as in [11]. Once st=0 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 st=0 to replace ω, namely, using equation (12) to substitute equation (7). This yields
(23)q̇0=−12qT−kq=k21−q02,(24)q̇=12q0I+q×−kq=−12kq0q.The definition oftf is the instant of time when the sliding mode s=0 is reached, and the following result is easily obtained by integrating the two sides of equation (23).
(25)q0t=1−21−q0tfe−kt−tf1+q0tf+1−q0tfe−kt−tf,t≥tf.Solimt→∞q0=1. Since the constraint relation of quaternions q02+qTq=1 and the condition of reaching the sliding surface ω=−kq, it can be concluded that limt→∞q=limt→∞ω=0.Remark 2.
Another problem associated with the preceding control law is caused by the quaternion direction representation. It is well known thatq0=1 and q0=−1 represent the same orientation. Since the above control method guarantees that the attitude of the spacecraft can be stabilized at q0=1, namely, limt→∞q0=1, this shows that even if the initial position of the spacecraft is located very close to q0=−1, the spacecraft will rotate completely to derive point q0=1. 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 s=ω+kqsgnq0. Then, it is shown that
(26)q̇0=12qTkqsgnq0=k21−q02sgnq0.
Equation (26) means that when the initial condition is q0>1, the system reaches limt→0q0=1. Conversely, when the initial condition is q0<1, the system reaches limt→0q0=−1.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 thatTd=0 in this section. First, consider a simple control law can stabilize the control objective ω. Assuming that variables ω, q, η, and ψ are measurements available for feedback, given the Lyapunov function V2=1/2ωTJω, its first derivative with respect to time is obtained as follows:
(27)V̇2=ωTδTKCTηψ−δTCM−1δω+u.The form of the state feedback variable structure control is chosen to be(28)u=−δTKCTηψ−kdtanhω,where kd is a positive constant. Substituting equation (28) into equation (27) yields
(29)V̇2=−ωTδTCM−1δω−kdωTtanhω≤−δTCM−1δω2−kdωTtanhω≤0.By using Lyapunov’s second method, it can be concluded that limt→∞ωt=0.It is worth mentioning that the main goal is to stabilizeωt and qt; 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 s=0, it is obtained limt→∞ωt=limt→∞qt=0. 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:
(30)u=−kpq−δTKCTηψ−kdtanhs−kz2q0I3+q×tanhq−k12z,where kp,kz are positive constants; zt∈R3 is filter vector, and it can be calculated by the solution of linear differential equation żt=qt−k12z.Remark 4.
The filter statezt satisfies the stable first-order linear vector differential equation and is forced by attitude vector qt. In terms of frequency response characteristics, this filter dynamics can be considered as low-pass systems for all bounded qt signals. Also, it should be noted that the value of the filter gain k1 plays a significant role in defining the low-pass characteristics of the filter state zt. 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(31)V3=12ωTJω+kpqTq+1−q02+kz∑i=13logcoshqi−k12zi+12γk2=12ωTJω+2kp1−q0+kz∑i=13logcoshqi−k12zi+12γk2,where parameter kt is a time-varying function and an adaptive updating law is designed for kt subsequently.
The first derivative of equation (31) is given by
(32)V̇3=ωTJω̇−2kpq̇0−kzq̇−k12żTtanhq−k12z+1γkk̇=ωT−ω×Jω+δTψ+δTCψ+Kη−CM−1δω−kpq−δTKCTηψ−kdtanhs−kz/2q0I3+q×tanhq−k12z−2kpq̇0+kzq̇−k12żTtanhq−k12z+1γkk̇=−ωTδTCM−1δω−kzk12żTtanhż−kdωTtanhs+1γkk̇.
The updating law fork is designed to satisfy
(33)k̇=−γkdqTtanhs,where γ is a positive constant.
According to equation (32), equation (33) can be rewritten as
(34)V̇3=−ωTδTCM−1δω−kzk12żTtanhż−kdsTtanhs≤0.
The integral on both sides of equation (34) yields supt≥0Vt≤V0. This shows that V3t is a bounded function, such that ω,q,zt,żt is uniformly bounded. Due to the boundedness of all signals in the time derivative of equation (34), it is easy to demonstrate that V¨3t is bounded, and the uniform continuity of V̇3t can be obtained. Applying the Barbalat lemma to obtain V̇t→0 as t→∞which further leads to the conclusion that limt→∞ω=limt→∞s=0, and from this result, it can be concluded that limt→∞kq=0. Next, since the third derivative of zt function for time is uniformly bounded, this proves that z¨tis uniformly bounded. Additionally, since limt→∞żt=0, the following results are obtained:
(35)limt→∞∫0tz¨τdτ=limt→∞żt=0.
Using Barbalat’s lemma, it is shown thatlimt→∞z¨t=0; since z¨t=q̇−k2żt, it follows that limt→∞q̇=0 as t→∞. According to the above obtained conclusion and the boundness of signals on the right side of equation (7), it can verify that q¨t is uniformly bounded; hence, applying the uniform continuity for limt→∞q̇=0, the following results are obtained:
(36)limt→∞∫0tq̇τdτ=limt→∞qt=0,and reusing Barbalat’s lemma, the result limt→∞q=0 is obtained. Because limt→∞ω=0, using equations (8a), (8b), and (8c), one can conclude that ω¨t is bounded. This means that ω̇t is uniformly bounded. By implying Barbalat’s lemma, it is possible to conclude that limt→∞ω̇=0. Further, note that Jω̇→0 as t→∞ such that ηψT→0 as t→∞. Thus, the trajectory of the system converges to
(37)limt→∞t,st,ηψ=0.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 ofk also affects the system response since k is used to achieve the relative weight between ωt and qt. A smaller k will result in a slow response in qt. 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]
(38)v=gu=u+Δu,where v=v1v2v3T represents the control torque to be designed and gu is defined as
(39)gu=umax×tanhuumax.Assuming thatΔu 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 Δu=gu−u 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 fx∈R, it can be approximated as
(40)fx=f^x,θ∗+ε=θ∗Tϕx+ε,φn=exp−x−ci2σi2,i=1,2,⋯,n,where x∈Rn and ε∈R represent the input signals of the neural network and approximation error, respectively.Andϕx=φ1x,φ2x,φ3x,⋯,φlxT∈Rl is a known continuous smooth vector value function; φxi are selected as the commonly used Gaussian function. l is called the number of neural network nodes. θ∗∈Rn is the optimal weight vector, and ci∈Rn and σi>0 are the center vector and radial basis vector of ϕx, respectively. The RBFNN output f^x,θ∗ is the approximation value of fx. Optimal weight vector is defined as
(41)θ∗=θ1∗x,θ2∗x,θ3∗x,⋯,θl∗x∈Rl.In general,θ∗ is the vector selected from θ to minimize fx−f^x,θ. Mathematically, that is
(42)θ∗=argminθ^supx∈Ωfx−f^x,θ,where θ^ is the estimation value of θ∗, Ω∈Rl is compact set.For MIMO systems, the nonlinear functionΔu can be approximated using the RBF network as follows:
(43)Δu=θ∗Tϕx+ε,where x=qTωTT and θ∗∈Rl×3, θ∗=θ1∗Tx,θ2∗Tx,θ3∗TxT, ε=ε1ε2ε3T.Assumption 7.
According to the universal approximation property of RBF network, the maximum mean square value of approximation error vector is bounded and satisfiesε≤ε0, in which ε0 is a small constant.Assumption 8.
The ideal NN optimal weightθ∗ is bounded, such that θ∗≤θm and θm is a positive constant.


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

To remove the drawbacks discussed in Remark11 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 kt 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:(44a)ut=−kpq−δTKCTη^ψ^−kdtanhs−kz2q0I3+q×tanhq−k12z−θ∧Tϕx.Modified updating law fork is designed to satisfy
(44b)k̇=−γkkqkqTtanhs+∑i=13Lisisgnk+Likqisgnsi.Adaptive updating law for liquid sloshing displacement variable is designed as(44c)η^̇ψ^̇=Aη^ψ^−I−CM−1δω−Γ−1KCδω.Neural network compensator updating law is given by(44d)θ^̇=PϕxωT−δθsθ^,where
(45)A=0I−K−C.Γ∈R4×4 and P∈Rl×l are positive definite symmetric matrices.γ>0 denotes an adaptive attitude gain.Assumption 9.
It is assumed that the positive constantLi strictly dominates the unknown disturbance Td and small constant ε0, namely, Tdi≤dm≤dm+ε0+1/2δθθm2≤Li is satisfied,∀T≥0,i=1,2,3, where δθ is a small positive constant.Theorem 2.
Consider the attitude maneuver control system governed by equations (7) and (10), if control input ut 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 t≥0. In addition, they asymptotically converge to the set defined by limt→∞st,zt,eηeψ,θ~=0.Proof.
Consider the following Lyapunov function candidate:(46)V4=12ωTJω+2kp1−q0+12eηeψTΓeηeψ+kz∑i=13logcoshqi−k12zi+12γk2+12trθ~TP−1θ~,where tr⋅ represents the trace of a matrix ⋅.
The first derivative with respect to time can be obtained by computingV4 along the trajectory of the system
(47)V̇4=ωT−ω×Jω+δTψ+δTKCTηψ−δTCM−1δω+dt−kpq−δTKCTη^ψ^−kdtanhs−kz/2q0I3+q×tanhq−k12z+θ~Tϕx+ε+eηeψTΓAηψ−I−CM−1δω−η^̇ψ^̇+kzq̇−k12żTtanhq−k12z+1γkk̇+trθ~TP−1θ~̇≤−λmδTCM−1δω2+s−kqdm+ε0−Li∑i=13sisgnk+kqisgnsi−kqsTtanhs−kzk12żTtanhż−eηeψTQeηeψ+δθtrθ~Tsθ∗−θ~.
It is worth mentioning that the following inequality holds:(48)trθ~Tsθ∗−θ~≤sθ~θ∗−sθ~2≤12sθ∗2−12sθ~2.
Substituting equation (48) into equation (47), it can obtain
(49)V̇4≤−λmδTCM−1δω2+s−kqdm+ε0−kzk12żTtanhż−kqsTtanhs−eηeψTQeηeψ+12δθθm2s−12δθsθ~2−Li∑i=13si−Li∑i=13kqi≤−λmδTCM−1δω2−kzk12żTtanhż−Li−dm−ε0∑i=13kqit−s1+Li−dm−ε0−12δθθm2∑i=13si−eηeψTQeηeψ−12δθθ~2∑i=13si
Note the fact that(50)ωTTd+ε≤ωdm+ε0≤dm+ε0∑i=13si+kqi,where s1=kptanhs and λm⋅ is the minimum eigenvalues of the matrix ⋅. Γ can be computed as the solution of ΓA+AΓT=−2Q for any fixed Q=QT>0.
From Assumption9, it is easy to show that
(51)V̇4≤0.
It can be seen thatωt, qt, zt, żt, θ~, and eηeψT are all uniformly bounded. It can also be concluded that V¨t is bounded, which means that V̇t is uniform continuity function and limt→∞V̇t=0 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:
(52)limt→∞ktqt=0,zt,st=0,eηeψ,θ~=0..
This completes the proof of Theorem2.Remark 11.
According to the proof of Theorem2, 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 Theorem2 only guarantees that ktqt=0 as t→∞; this will not necessarily ensure that qt will converge to zero. If kt converges to zero faster than qt, then qt will converge to a nonzero constant value. In order to guarantee that qt converge to zero, it is necessary to prevent kt from converging to zero. This can be achieved by using a sufficiently small γ to ensure kt changes slowly such that it does not deviate too much from its initial value. In addition, since kt is used to achieve relative weights between ωt and qt, a smaller kt will result in slow response of qt. Conversely, a higher kt will lead to oscillatory response in system. Subsequent numerical simulation examples show that appropriate initial value k0 and/or small enough parameter γ can be found in most cases. This ensures that the sliding mode s converges to the origin under parametric uncertainty, external disturbances, and input saturation. In other words, initial value k0 and parameter γ 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 q0I3+q×≤1, q≤1, andtanhs−k12z≤1. If the term
(53)δTKCTη^ψ^is not considered in equation (46), note that the control input satisfies the input saturation limitations, namely, ut≤kp+kq+kz/2≤u¯m. This indicates that the RBFNN saturation compensator is used to compensate for saturation uncertainty caused by term
(54)δTKCTη^ψ^.


## 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]
(38)v=gu=u+Δu,where v=v1v2v3T represents the control torque to be designed and gu is defined as
(39)gu=umax×tanhuumax.Assuming thatΔu 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 Δu=gu−u 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 fx∈R, it can be approximated as
(40)fx=f^x,θ∗+ε=θ∗Tϕx+ε,φn=exp−x−ci2σi2,i=1,2,⋯,n,where x∈Rn and ε∈R represent the input signals of the neural network and approximation error, respectively.Andϕx=φ1x,φ2x,φ3x,⋯,φlxT∈Rl is a known continuous smooth vector value function; φxi are selected as the commonly used Gaussian function. l is called the number of neural network nodes. θ∗∈Rn is the optimal weight vector, and ci∈Rn and σi>0 are the center vector and radial basis vector of ϕx, respectively. The RBFNN output f^x,θ∗ is the approximation value of fx. Optimal weight vector is defined as
(41)θ∗=θ1∗x,θ2∗x,θ3∗x,⋯,θl∗x∈Rl.In general,θ∗ is the vector selected from θ to minimize fx−f^x,θ. Mathematically, that is
(42)θ∗=argminθ^supx∈Ωfx−f^x,θ,where θ^ is the estimation value of θ∗, Ω∈Rl is compact set.For MIMO systems, the nonlinear functionΔu can be approximated using the RBF network as follows:
(43)Δu=θ∗Tϕx+ε,where x=qTωTT and θ∗∈Rl×3, θ∗=θ1∗Tx,θ2∗Tx,θ3∗TxT, ε=ε1ε2ε3T.Assumption 7.
According to the universal approximation property of RBF network, the maximum mean square value of approximation error vector is bounded and satisfiesε≤ε0, in which ε0 is a small constant.Assumption 8.
The ideal NN optimal weightθ∗ is bounded, such that θ∗≤θm and θm is a positive constant.


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

To remove the drawbacks discussed in Remark11 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 kt 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:(44a)ut=−kpq−δTKCTη^ψ^−kdtanhs−kz2q0I3+q×tanhq−k12z−θ∧Tϕx.Modified updating law fork is designed to satisfy
(44b)k̇=−γkkqkqTtanhs+∑i=13Lisisgnk+Likqisgnsi.Adaptive updating law for liquid sloshing displacement variable is designed as(44c)η^̇ψ^̇=Aη^ψ^−I−CM−1δω−Γ−1KCδω.Neural network compensator updating law is given by(44d)θ^̇=PϕxωT−δθsθ^,where
(45)A=0I−K−C.Γ∈R4×4 and P∈Rl×l are positive definite symmetric matrices.γ>0 denotes an adaptive attitude gain.Assumption 9.
It is assumed that the positive constantLi strictly dominates the unknown disturbance Td and small constant ε0, namely, Tdi≤dm≤dm+ε0+1/2δθθm2≤Li is satisfied,∀T≥0,i=1,2,3, where δθ is a small positive constant.Theorem 2.
Consider the attitude maneuver control system governed by equations (7) and (10), if control input ut 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 t≥0. In addition, they asymptotically converge to the set defined by limt→∞st,zt,eηeψ,θ~=0.Proof.
Consider the following Lyapunov function candidate:(46)V4=12ωTJω+2kp1−q0+12eηeψTΓeηeψ+kz∑i=13logcoshqi−k12zi+12γk2+12trθ~TP−1θ~,where tr⋅ represents the trace of a matrix ⋅.
The first derivative with respect to time can be obtained by computingV4 along the trajectory of the system
(47)V̇4=ωT−ω×Jω+δTψ+δTKCTηψ−δTCM−1δω+dt−kpq−δTKCTη^ψ^−kdtanhs−kz/2q0I3+q×tanhq−k12z+θ~Tϕx+ε+eηeψTΓAηψ−I−CM−1δω−η^̇ψ^̇+kzq̇−k12żTtanhq−k12z+1γkk̇+trθ~TP−1θ~̇≤−λmδTCM−1δω2+s−kqdm+ε0−Li∑i=13sisgnk+kqisgnsi−kqsTtanhs−kzk12żTtanhż−eηeψTQeηeψ+δθtrθ~Tsθ∗−θ~.
It is worth mentioning that the following inequality holds:(48)trθ~Tsθ∗−θ~≤sθ~θ∗−sθ~2≤12sθ∗2−12sθ~2.
Substituting equation (48) into equation (47), it can obtain
(49)V̇4≤−λmδTCM−1δω2+s−kqdm+ε0−kzk12żTtanhż−kqsTtanhs−eηeψTQeηeψ+12δθθm2s−12δθsθ~2−Li∑i=13si−Li∑i=13kqi≤−λmδTCM−1δω2−kzk12żTtanhż−Li−dm−ε0∑i=13kqit−s1+Li−dm−ε0−12δθθm2∑i=13si−eηeψTQeηeψ−12δθθ~2∑i=13si
Note the fact that(50)ωTTd+ε≤ωdm+ε0≤dm+ε0∑i=13si+kqi,where s1=kptanhs and λm⋅ is the minimum eigenvalues of the matrix ⋅. Γ can be computed as the solution of ΓA+AΓT=−2Q for any fixed Q=QT>0.
From Assumption9, it is easy to show that
(51)V̇4≤0.
It can be seen thatωt, qt, zt, żt, θ~, and eηeψT are all uniformly bounded. It can also be concluded that V¨t is bounded, which means that V̇t is uniform continuity function and limt→∞V̇t=0 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:
(52)limt→∞ktqt=0,zt,st=0,eηeψ,θ~=0..
This completes the proof of Theorem2.Remark 11.
According to the proof of Theorem2, 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 Theorem2 only guarantees that ktqt=0 as t→∞; this will not necessarily ensure that qt will converge to zero. If kt converges to zero faster than qt, then qt will converge to a nonzero constant value. In order to guarantee that qt converge to zero, it is necessary to prevent kt from converging to zero. This can be achieved by using a sufficiently small γ to ensure kt changes slowly such that it does not deviate too much from its initial value. In addition, since kt is used to achieve relative weights between ωt and qt, a smaller kt will result in slow response of qt. Conversely, a higher kt will lead to oscillatory response in system. Subsequent numerical simulation examples show that appropriate initial value k0 and/or small enough parameter γ can be found in most cases. This ensures that the sliding mode s converges to the origin under parametric uncertainty, external disturbances, and input saturation. In other words, initial value k0 and parameter γ 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 q0I3+q×≤1, q≤1, andtanhs−k12z≤1. If the term
(53)δTKCTη^ψ^is not considered in equation (46), note that the control input satisfies the input saturation limitations, namely, ut≤kp+kq+kz/2≤u¯m. This indicates that the RBFNN saturation compensator is used to compensate for saturation uncertainty caused by term
(54)δTKCTη^ψ^.


## 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 c evenly spaced in −π/6,π/6×−π/6,π/6. Nominal inertia matrix of the spacecraft is given by
(55)J=503000385−50−5420kg/m2.The external disturbances are assumed to be(56)dt=0.02sin0.1t+20.03cos0.4t+30.03sin0.3t+1,and an uncertainty ΔJ on the inertia matrix J is given by ΔJ=0.5J; sensor noise is set as 0.5rand3,1, where 0.5rand3,1 represents the random Gaussian white noise vector. The controller gains are listed as kp=20, kq=5, kz=10, k1=2, P=30I15, L1=L2=L3=10, σ=2, γ=0.01, Q=I4, and δθ=0.05.The relevant parameters of liquid fuel sloshing are selected asmb=100kg, mp=0.1mb, l=0.228m, rx=1.2m, and c1=c2=0.05. It is assumed that the spacecraft is at the earth’s orbit altitude of 800 km and the local gravitational acceleration is selected as g=7.689m/s2.Initial angular velocity isω0=000T, initial displacement of the pendulum is η^0=00T, and initial quaternions value is
(57)q0qT=0.1736−0.52640.26230.7876T.Initial estimated neural network optimal weight is given byθ^0=0,0,0T, and initial attitude gain is given by k0=2.5. Bound of control torque is um=30Nm.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 Figures2–8.Case 2.
Attitude maneuver control using variable structure controller without considering RBFNN, the simulation results are shown in Figures9–14.Figure 2
Time histories of angular velocities in Case1.Figure 3
Time histories of attitude quaternions in Case1.Figure 4
Time histories of liquid sloshing displacement in Case1.Figure 5
Time histories of attitude gaink in Case 1.Figure 6
Time histories of filter vector in Case1.Figure 7
Time histories of estimated NN optimal weight in Case1.Figure 8
Time histories of control input in Case1.Figure 9
Time histories of angular velocities in Case2.Figure 10
Time histories of attitude quaternions in Case2.Figure 11
Time histories of liquid sloshing displacement in Case2.Figure 12
Time histories of attitude gainkt in Case 2.Figure 13
Time histories of filter vector in Case2.Figure 14
Time histories of control input in Case2.The simulation results of Case1 are shown in Figures 2–8. The simulation results of Case 2 are shown in Figures 9–14. 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 k. It can be reported from Figure 5 that the control objective is achieved and kt converges to approximately 0.5 in about 60 seconds.Figures6 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 En=∫0tutdt is introduced here and is calculated over the total simulation time of t=200s. It is obvious that the robust saturated control strategy completes the attitude maneuver successfully with less control energy.Case 3.
The roles of auxiliary functionk0 on the system response are demonstrated by choosing three different initial k0 values. The simulation results are shown in Figures 15–18 by choosing k0=0.5. The simulation results are shown in Figures 19–22 by choosing k0=5. The simulation results are shown in Figures 23–26 by choosing k0=10.Figure 15
Time histories of angular velocities in Case3 when k0=0.5.Figure 16
Time histories of attitude quaternions in Case3 when k0=0.5.Figure 17
Time histories of filter vector in Case3 when k0=0.5.Figure 18
Time histories of attitude gaink in Case 3 when k0=0.5.Figure 19
Time histories of angular velocities in Case3 when k0=5.Figure 20
Time histories of attitude quaternions in Case3 when k0=5.Figure 21
Time histories of filter vector in Case3 when k0=5.Figure 22
Time histories of attitude gaink in Case 3 when k0=5.Figure 23
Time histories of angular velocities in Case3 when k0=10.Figure 24
Time histories of attitude quaternions in Case3 when k0=10.Figure 25
Time histories of filter vector in Case3 when k0=10.Figure 26
Time histories of attitude gaink in Case 3 when k0=10.The simulation results are shown in Figures15–18 by choosing k0=0.5, the simulation results are shown in Figures 19–22 by choosing k0=5, and simulation results are shown in Figures 23–26 by choosing k0=10. Here, the role of attitude gain k0 is explained in the language of classical second-order linear system. A smaller k0 value is conceptually equivalent to a higher damping ratio. When k0 value decreases, the emphasis of control will deviate from the q term and move towards the ω term, which will lead to overdamped sluggish behavior in Figures 15–18. The higher k0 value corresponds conceptually to the lower damping ratio. When k increases, the focus of control is shifted away from ω term and moves towards q term, which results in the underdamped oscillation behavior shown in Figures 19–26. Therefore, k0 acts as desired weighting to achieve the acceptable response between angular velocity ωt and quaternions qt, and these analyses are consistent with the explanation in Remark 12. Designers can appropriately choose k0 value and parameter γ 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 parameterk1 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.



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