Fuzzy Disturbance Observer-Based Sliding Mode Control for Liquid-Filled Spacecraft With Flexible Structure Under Control Saturation

This paper investigates a fuzzy disturbance observer (FDO)-based terminal sliding mode control (TSMC) strategy for the liquid-filled spacecraft with flexible structure(LFS-FS) under control saturation. Firstly, a novel FDO is designed to estimate the lumped uncertainty, including the inertia uncertainty, external disturbance, the coupling of liquid slosh and flexible structure(LF), as well as the parts that exceed control saturation. The merits of the FDO lie in that estimation error can be arbitrarily small by adjusting the designed parameters and the prior information is not required. Then, based on the estimation of FDO, a finite-time TSMC is designed, which has more satisfactory control performance, such as chattering reduction and fast convergence speed. The stability of the closed-loop system is proved strictly by Lyapunov theory. Finally, numerical simulations are presented to demonstrate the effectiveness of the proposed method.


I. INTRODUCTION
Over the past few decades, the space technology development is recognized as an important part for national security, earth observation, planetary exploration and so on. Since it has been rapidly developed for the control strategy of rigid spacecraft, many methods have been introduced to enhance the stability of the control effect, such as sliding mode control(SMC) [1]- [4], adaptive control [5], [6], robust control [7]- [9]. However, The missions of the spacecraft are becoming increasingly complex, flexible appendages [10] and liquid storage chamber [11] are widely adopted on the spacecraft, which not only change the control mode but also make the control more difficult. For example, there are some features of highly flexibility and low damping for flexible appendages such as solar panels and long antennas [12]. Hence, the control performance and stability of LFS-FS would be deteriorated due to the coupling between the rigid body and the flexible appendages [13]- [15]. In addition, because the space task becomes more complex, longer The associate editor coordinating the review of this manuscript and approving it for publication was Jianyong Yao . residence time and much more liquid fuel are needed for the spacecraft [16], [17]. However, due to the sloshing of the liquid fuel, the maneuverability of spacecraft is inevitably influenced, which causes the bad control performance and even the failure task [18], [19]. Therefore, the attitude stability control for LFS-FS with model uncertainties and external disturbance have been one of the important research topics.
What's more, as is known to all, the control torque produced by the actuator is usually not infinite in practical applications, which will no doubt cause the control saturation problems and lead substantial degradation to the system [20]- [24]. In [25], American researcher RD Robinett discussed a generalized feedback control law design method under control saturation constraints as early as 1997. Then in [26], [27], the Boskovic in Yale taked the control saturation problem into account in the design of the control law for spacecraft and designed a control algorithm based on variable structure. In [28], Hu et al. investigated a finite-time controller under input saturation based on a second-order disturbance observer and the adaptive control.
Although Boskovicv et al. [26], [27], Hu et al. [21], Xiao et al. [22], Hu et al. [28], Xiao et al. [29], Hu et al. [30], Zou et al. [31], Ruiter [32], and Zhu et al. [33], Xia et al. [34], Lu and Xia [35] have developed some controllers to effectively deal with the constrained actuator output, hardly any study discussed the control algorithm for LFS-FS under control saturation constrain and thus cannot meet the demands of aerospace task for the future. For the above reasons, this paper will propose a control method for the stability control problem of LFS-FS with actuator saturation.
The disturbance observer, is widely used for nonlinear control problems with external disturbances and uncertainties. Reference [36], [37] proposed extended state observerbased adaptive and robust control methods, which estimated the unmeasurable system states and the additive disturbances effectively. A novel adaptive sliding mode disturbance observer is proposed in [38], which could achieve the precise and fast trajectory tracking control for space manipulator after capturing an uncertain space target. And the FDO has universal approximation capability for the system with unknown disturbances and uncertainties [39]. Therefore, it is widely employed to counteract the system disturbances and improve controller robustness. A novel control method for the flight simulator was proposed in [40], in which a FDO was designed to compensate the external disturbances. The unknown uncertainty and disturbance of nonlinear system were estimated by the FDO adopted in [41], [42], and the simulations are carried out to verify the effectiveness and the applicability of the FDO for the nonlinear control system with uncertainty.
Furthermore, in nonlinear control designs, SMC is a wellknown and powerful control scheme that has been widely used. In [43]- [45] the SMC was adopted to design an effective and stable controller for nonlinear systems. The hierarchical SMC was employed to design the controller for the space vehicle in [46]. In [6], a second-order SMC method for spacecraft was presented to guarantee control error can converge to zero in the finite time. SMC has some advantages such that it is not sensitive to parameters change. And, the TSMC adds terminal items based on SMC, which makes the control system have faster convergence speed and ensures that the system converges in a finite time.
In this work, the integration of FDO and TSMC is proposed to solve the attitude stability problem for the LFS-FS under control saturation, uncertainties and disturbances, which can ensure the control system of the LFS-FS is stable. The main contributions are summarized as follows: 1) From the theoretic aspect, an FDO-based TSMC strategy is proposed in this paper. The FDO, designed by the fuzzy logic system (FLS), can approximate the uncertainties and various types of disturbances, which provides freedom from derivative of disturbance bound assumption. What's more, the FDO-based TSMC strategy is proposed to guarantee the asymptotically stability, and reject the lumped uncertainty with lower chattering and higher accuracy, which has been rigorously proved by the Lyapunov theory. 2) From the engineering application aspect, the integrated design of FDO and TSMC can provide the fast and high precision attitude stabilization control of the complex spacecraft, which is coupled with liquid slosh and flexible vibration, even in the presence of external disturbances and control saturation. What's more, the simulations and comparison analysis are presented to verify the effectiveness and better performance of the proposed method, which can reduce the chattering and accelerate the convergence speed effectively.
The outline of this paper is as follows. At first, the mathematical model of LFS-FS under control input constraint is introduced in Section II. Then, FLS is introduced and the designed FDO is adopted to estimate the lumped uncertainty in Section III. In Section IV, the TSMC based on the estimations of FDO is proposed and the Lyapunov function is adopted to prove that the controller system is stable. Simulation results and analysis are shown in Section V. The paper ends with the conclusions in Section VI.

A. MATHEMATICAL SYSTEM MODEL
The diagram of LFS-FS is shown in Figure 1. The dynamic equations can be described by [16], [47]  where the unit quaternion q = [q 0 q T v ] T = [q 0 q 1 q 2 q 3 ] T , q 0 is the scalar part of q and q v is the vector part, and satisfies q T v q v + q 2 0 = 1; ω ∈ R 3 is the angular velocity of the body fixed frame with respect to the inertia reference frame, χ and η are the modal coordinate vectors of flexible structure and liquid slosh respectively; u, d represent the control torque and external disturbance, respectively; The inertia matrix J is described as J = J 0 + J , where the nominal part and the uncertain part are represented by J 0 and J ; K i , C i (i = 1, 2) denote the stiffness and damping matrices, respectively; M , N ∈ R 4×3 are the coupling matrixs of the liquid slosh and flexible structure between rigid dynamics, respectively. And the expressions of M and the quality matrix M 0 of liquid slosh are given as follows.
where m i , b i (i = 1, 2) represent the quality of the sloshing liquid mass and the distance between each sloshing mass and the center mass of the total spacecraft, respectively. The q × v ∈ R 3×3 is a cross product matrix,which can be defined as follows.
Consider the situation of actuator input saturation, the dynamic model (3) is expressed as where sat(·) represents the nonlinear saturation characteristic of the actuators, defined as where u max and u min are the upper bound and lower bound of actuator torque respectively.
Design a virtual variable u = ( u 1 , u 2 , u 3 ) ∈ R 3 , in which u i , i = 1, 2, 3 is given as Thus (6) is rewritten as Define C = ω × J 0 ,D = I 3 and the lumped uncertainty In this work, our goal is to achieve the attitude stabilization in the presence of inertia uncertainty, external disturbances and LF with bounded control input. The main objective is to design a FDO-based TSMC scheme to guarantee the attitude stabilization of complex spacecraft system (1) (2) and (9), which means the q and ω will converge to an arbitrary small set containing the origin.

III. FUZZY DISTURBANCE OBSERVER
To give a better understanding of FDO, the FLS is briefly introduced firstly in this section. After that, we designed a new fuzzy disturbance observer to estimate the whole uncertainties , which could realize an arbitrarily small estimation error by adjusting the designed parameters.

A. DESCRIPTION OF THE FUZZY LOGIC SYSTEM
The FLS contains five parts: fuzzifier, rule bases, inference engine, type-reducer, and defuzzifier. The fuzzy inference engine uses IF-THEN rules to map from an input compact set The fuzzy rules can be described as . . , A i n are fuzzy variables and Y is a fuzzy singleton.
Using center-average defuzzifier and singleton fuzzifier, the output of FLS is represented as where p represents the number of fuzzy logic rules, θ T = (h 1 , h 2 , · · ·, h p ) T is the vector of adjustable parameters, denotes the vector of fuzzy basis function, and the µ A j i (x i ) denotes the membership function. Assumption 1 [48], [49]: Let f (x) be a continuous function defined on a compact set M x . Then, for any constant ε > 0, there exists a fuzzy logic system (12) such that sup

B. FDO DESIGN FOR SPACECRAFT ATTITUDE CONTROL SYSTEM
In this work, we design the fuzzy disturbance observer for spacecraft attitude control system. Based on the (10) and (12), the FDO is constructed aṡ The estimation of the lumped uncertainty is given aŝ thus the FDO can be written as: where z denotes the observer state, the designed constant σ > 0,ˆ is the estimate value of the lumped uncertainty , θ is achieved by an adaptation law which is expressed aṡ where the constants κ 0 > 0,γ 0 > 0 are the designed values. Define the estimated error e = J 0 ω − z, the dynamics of the observation estimation errors is described aṡ According to the Assumption 1, satisfies the following form where θ * is the optimal parameter vector. The ε denotes approximation error of FLS, which is bounded byε. Substituting the (14) and (18) into (17),ė can be rewritten asė whereθ =θ − θ * denotes the parameter estimation error. The e is exponentially convergent, whenθ approaches 0.
Proof: The following Lyapunov function candidate is chosen: Taking the time derivative of (21), and combining (19) and (16),V F satisfies: It is noted that the following inequalities satisfy Then,V F in (22) can be transformed into following forṁ Integrating both sides of (24) from 0 to T yields which is equivalent to (26) The robust performance of (26) can be explained by Barbalat's lemma [50]. If ε ∈ L 2 , i.e. ∞ 0 ε 2 dt < ∞, then e ∈ L 2 and m ∈ L 2 . This means lim t→∞ e = 0 and lim t→∞ m = 0. Even though ε / ∈ L 2 , e 2 is bounded by ε 2 . Therefore, the observation error can be reduced to any small value with adjusting the σ and γ 0 . Thus,ˆ can estimate with arbitrarily small error.

IV. COTROLLER DESIGN FOR SPACECRAFT SYSTEM A. FDO-BASED TSMC DESIGN
In the subsection, a FDO-based terminal sliding mode controller is designed for the LFS-FS with inertia uncertainty and external disturbance under actuator input saturation, which can guarantee the stabilization of the spacecraft system with high speed and precision.
The following sliding mode surface is designed: where s = [ s 1 s 2 s 3 ] T ∈ R 3 , k > 0 is user-designed constant, and β(q v ) = [ β(q 1 ) β(q 2 ) β(q 3 ) ] T satisfies the following form: wheres i = ω i + k · sig r (q i ), sig r (q i ) = |q i | r sign(q i ), i = 1, 2, 3, 0 < r < 1, ν > 0 is a small constant, and a 11 and a 22 satisfy a 11 = (2 − r)ν r−1 , a 22 = (r − 1)ν r−2 . Then, base on the (1) and (3),the time derivative of (27) is obtained as where denotes the lumped uncertainty of the system, g(t) denotes the normal part of the system, which satisfies the following form Thus, the u is designed as: where δ 1 , δ 2 are positive designed constants. Lemma 1 [51]: Consider nonlinear systemẋ = f (x, u), where x is the state vector and u is the control vector. Assuming that there are continuous differentiable positive definite functions V (x), scalar λ > 0, 0 < a < 1, 0 < ϑ < ∞, which makes inequalityV (x) ≤ −λV α (x) + ϑ valid, then the systemẋ = f (x, u) is actually stable in finite time, and the convergence time is where V (x 0 ) is the initial value of V (x).

B. STABILITY ANALYSIS
Theorem 2:Consider the nonlinear sapcecraft system (1)- (2) with control saturation (6). If designing the terminal sliding mode controller (32) augmented by the fuzzy disturbance observer (13) and adaptive law (18) under κ 0 , γ 0 > 0, 0 < r < 1, and δ 1 , δ 2 > 0, then the attitude quaternion q v and the attitude angular veloticy ω will converge to small neighborhoods of origion by selecting suitable controller parameters. Proof: Define Lyapunov function candidate as where (21) which verifies the stability of the FDO. The mathematical proof has been shown in section III-B.
Taking the time derivative of (35), it follows thaṫ By applying the following inequalities (38) Thus, the derivativeV 1 can be derived aṡ When the parameters are chosen as δ 1 > σ > 0, the inequality (39) becomeṡ According to the Lemma1, it means that the system is actually stable in finite time and the system states will reach the sliding surface s = 0 (27) in finite time, which implies that thus we can draw the conclusion thatV 2 = q vqv < 0, the q v is exponentially stable. This means that the system states will reach to the desired equilibrium point asymptotically under the designed control law. Therefore, the proof is completed.

V. SIMULATION RESULTS AND ANALYSIS
In this section, the effectiveness and performance of the FDO (31) and TSMC in (32) for LFS-FS (1)(2)(6) will be verified by simulation.

A. PARAMETER SETTING
It is assumed that the inertia matrix and nominal inertia matrix for the coupled spacecraft are selected as The physical parameters are chosen as [47], [48], [52]:

B. SIMULATION ANALYSIS
To further illustrate the effectiveness of the proposed method, simulation of two cases are made for the comparison. In the first case, the robustness of the proposed control scheme is verified by applying it to the conditions with input saturation and without input saturation respectively. In the second case, to prove the necessity of studying LFS-FS under control saturation, the adaptive variable structure control presented in [21] is applied to the control input saturation problem considered above, the simulation results show the advantages of the controller in this paper.
Case 1: In this case, we verify that the proposed algorithm can effectively overcome the drawback caused by the control saturation. Firstly, the angular velocity and attitude quaternion trajectories are shown in (a) of Figure 2 and Figure 3. As we can see, the coupled spacecraft system achieves stabilization smoothly in less than 20 s with a high accuracy of 10 −3 in static error. The trajectories of the control input is shown in Figure 4 (a), which demonstrates the upper bounder of required torque is aboud 10Nm. Besides, Figure 5 (a) shows the effectiveness of the FDO. The reconstructed lumped uncertainty is caught almost immediately.
Then, to testify the robustness of the proposed controller when faced with input constraint conditions. The restrictions on the actuator output torque are set as u max = 2Nm and u min = −2Nm, which are much smaller than the required ones. Comparing the (a) and (b) of Figure 2 and Figure 3, it can be seen that the state trajectory and the convergence time for and are substantially unchanged, and the static error and the vibration are kept within a small scale, indicating that the controller is robust to the input limitation and can effectively overcome the lag of control effect caused by it. Time responses of the demand control torques are depicted in Figure 4 (b). Note that the control torques are strictly limited to [−2, 2]Nm. Figure 5 shows the result of FDO which reflects that with the consideration of the constrained part of the control input, the reconstructed lumped uncertainties became larger in the initial stage, but when the system is gradually stable under the control, the observer's value would be restored to a small one.

Case 2:
In this case, the fuzzy disturbance observer based terminal sliding mode controller (FDOTSC) proposed in (32) is compared to the adaptive variable structure controller(AVSC) [21]. The red solid line and blue dash line in Figure 6- Figure 10 denote the control performance by (32) and AVSC in [21] respectively. For the AVSC in [21], the control parameters are chosen as δ = 0.02, r = 0.05. The parameters in (32) are consistent with case 1. Figure 6 and Figure 7 show time responses of LFS-FS attitude quaternion and the angular velocity. As we can see, with the controller in (32), the system achieves stabilization smoothly with the time less than 25 s and with a high accuracy of 10 −3 in static error. Although with the AVSC more time is required to achieve the attitude stabilization and the static error exists even after 100 s. Accordingly, the case of FDOTSC in (32) achieves the better performance than the AVSC in [21], which further illustrates the advantages and feasibility of the FDOTSC method.
Time responses of control torques is pictured in Figure 8. The demand control is strictly limited in [-2Nm,2Nm] for both of the controller. Furthermore, note that when the spacecraft system stabilize, the control torque in FDOTC is equal to the lumped uncertainties observed by the FDO, which compensates for the system and makes the performance better.  Because the first two orders of vibration and sloshing modes have the greatest impact on system stability, we just list the time responses of the sloshing liquid η 1 , η 2 and flexible structure χ 1 ,χ 2 , which is shown in Figure 9 and Figure 10. Although, sloshing liquid and flexible structure can cause tracking error and instability during spacecraft maneuvering  process, the FDOTSC compared to AVSC in [21] can effectively track and observe the effects of flexible vibration and liquid sloshing, and achieve fast and stable attitude tracking. From the Figure 9 and Figure 10, as we can see, compared with [21], the FDOTSC proposed by this paper brings less liquid sloshing and flexible vibration. One reason is because the FDO with universal approximation can observe the overall disturbance including the inertia uncertainty, external disturbance and the coupling of LF, another reason is that the FDO-based TSMC controller designed in this work improves system convergence speed and control performance. And the figures also verifies the effectiveness of the proposed method in this work for the problem of liquid slosh and flexible vibration.  Summarizing all the cases above, the controller (FDOTSC) proposed is robust to the control saturation, besides, the flexibility in the selection of control parameters can be used to achieve better performance while satisfying the constraints of control torque and uncertainty. In addition,the control method AVSC designed for spacecraft with small rigid body cannot control the spacecraft with as well. Thus, the studying of control method for FLS under control saturation is of strong utility value. This control method provides a theoretical basis for the practical application of advanced control theory in the design of spacecraft control system.

VI. CONCLUSION
In this paper, the integration of TSMC and FDO for the LFS-FS under control saturation, uncertainties and disturbances is proposed. The mathematical expressions for the LFS-FS under control saturation constrain is firstly introduced and the FDO is designed to estimate the lumped uncertainties caused not only by the external disturbance but also by the fuel slash, flexible structure, inertia uncertainties and control saturation. With the estimated value by the FDO, the TSMC is presented, which makes up for the shortcomings of traditional sliding mode control and ensures that the system converges in a finite time. The stability of the closed-loop system is proved by Lyapunov theory. Simulations of the two cases are carried out to testify the performance of the proposed control scheme. However, the directly inhibition of VOLUME 7, 2019 the flexible vibration and the fluid shaking is not considered in this paper, which is one of subjects for the improvement of control methods for LFS-FS.