Many future space missions will involve spacecrafts with flexible appendages such as large antennas,sun shields,solar arrays,and solar sails. These complex flexible nonlinear systems (CFNSs) are the systems with both nonlinear and flexible characteristics. Compared with the rigid body spacecrafts without flexible appendages,the design and implementation of attitude and orbit control systems (AOCSs) for the spacecrafts with such structures involve significant difficulties,because of the combination of uncertainty on several vehicle parameters,the frequency and damping of each flexible mode,and the potentially significant coupling between axes^{[1, 2, 3]}. At the early stage,researchers developed the low-dimensional models for the CFNSs by ignoring the impacts of flexible modalities or considering flexible modalities as external disturbances. To improve the performances of the close-loop systems, several nonlinear control techniques based on those models such as compound control^[4],$H_\infty$ state-feedback control^[5], geometric control^[6],etc.,have been proposed for solving the attitude tracking control problem of spacecrafts. Furthermore,a finite-time control theory^[7] has been designed to provide fast convergence. However,in terms of attitude tracking tasks,large angle or fast angular rate is generally considered in the rotation process. Since these results do not consider flexible characteristics of the CFNSs,the vibration in the flexible structures introduced by the orbiting attitude slewing operation may degrade the attitude pointing accuracy. This is the reason why the flexible appendages cannot be ignored anymore,especially when,in some cases,they are also required to rotate^[8].

Currently,the spacecraft dynamic model is described as a relationship between the 6 DOF vector of acceleration at the composite center of the mass. Barring the small angle and angular rate assumptions,the commonly adopted linearized model is not suitable to design attitude control laws. Moveover,according to the modal analysis theory,the order of the flexible appendages is usually enormously high leading to that it is impossible to depict the CFNSs with accurate mathematical models. Generally,the modal truncation method is used to reduce the order. However,to cut off the order means to lose some information of the CFNSs,which leads to some unpredictable challenges. Taking the Hubble telescope launched in 1990 for an example^[9],a PID controller was designed with this method. But in space,frequency sympathetic vibration caused by disturbance occurred,which made the system performance worse.~~Several~~recent~~studies have considered the application of linear robustness analysis techniques to this problem,but significant difficulties were also encountered with promising results reported,such as the extremely complicated problem of computing tight upper and lower bounds on singular value^[10]. One important reason for the difficulties in modeling and control is that the present modeling and control theory is based on accurate dynamic analysis as well as modeling and control are considered separately. As a result,the current modeling method needs to be modified by an engineering modeling method,i.e., a characteristic modeling method^[9]. The theory of characteristic modeling,as a kind of data-based model-free approach for control system design,requires the analysis of the dynamic and control performance of the system instead of an accurate plant dynamic analysis. Based on this methodology,a golden section adaptive control law^[11],which aims at designing an engineering-oriented adaptive controller by using only a few parameters,has been successfully used in various industries.

Sliding mode control (SMC)^[12] has been widely studied and used for many years due to its simplicity and robustness in system variations and disturbances. The idea of the SMC is the peculiar sliding motion introduced by discontinuous control,which enables the system trajectory to reach and stay in a prescribed switching manifold indefinitely. It is also well known that SMC is an efficient method to deal with nonlinear systems with exogenous disturbances and parametric uncertainties^[13]. Since modern industrial control systems are digitally implemented^[14],which means that the industrial SMC would be executed in discrete time, the need for research in DTSMC is evident. However,even though the long history of SMC development and main results have been reported, only a few of them among available results are devoted to discrete-systems^[15]. It should be pointed out that DTSMC cannot be obtained from its continuous counterpart by means of simple equivalence,theoretically. Although the investigation of the discretization behaviors of terminal SMC systems^[16] revealed the underpinning generation mechanism periodic orbits,which will help develop effective discrete-time terminal SMC with less chattering and better robustness,it has not been gotten fully used for actual controlled plants.

In this paper,by introducing a novel sliding mode surface based on the characteristic modeling,a finite-time DTSMC method is proposed for the tracking control problem of the rigid spacecraft with flexible structures. Firstly,according to the characteristic modeling method,the characteristic model of the CFNSs is established and the parameters of the model are obtained with identification algorithm. Secondly,a characteristic model-based DTSMC law is designed and Lypapunov stability law is applied to tackle the problem how states can reach the bounded sliding surface in finite time. Finally,the proposed controller is used for the attitude tracking control problem of the CFNSs.

This paper is organized as follows: Section II describes the 6 DOF spacecraft simulation model,AOCS and characteristic modeling criteria used in this study. The main researches are presented in Section III,in which the characteristic model-based DTSMC is proposed to achieve the highly accurate tracking performance. Simulation results are provided in Section IV. Finally,some conclusions are presented in Section V.

A. Spacecraft Attitude Dynamics and Kinematics

The main body model is obtained thanks to Euler/Netwon equations applied to the rigid body. Considering the worst case,the corresponding uncertainty around their nominal value is 50 %. The dynamic model of the rigid body is

where $\boldsymbol{\omega}_s = \left[\omega_x,\ \omega_y,\ \omega_z \right]^{\rm T} \in R ^{3 \times 1}$ is the angular velocity, $\boldsymbol{I}_s$ $\in$ $R ^{3 \times 3}$ is the rotational inertia of the spacecraft,$\boldsymbol{T}_T \in R ^{3 \times 1}$ is the aggregate external torque,$\boldsymbol{\eta}_{sa} \in R^{m \times 1}$ is the vibration modal coordinates array of the flexible appendage,$\boldsymbol{F}_{sa}$ $\in$ $R^{3 \times m}$ is the coupling coefficient matrix between the flexible appendage and the rigid body,and $(\cdot)^{\times}$ denotes a $3 \times 3$ skew-symmetric matrix,that is

The dynamic model of the flexible appendage cantilevered on the main body is commonly described by the so-called cantilever hybrid model^[17]. This model gives a relationship between the 6 DOF acceleration vector and the 6 DOF forces vector applied by the main body to the appendage:

where $\boldsymbol{\omega}_{sa} \in R^{m \times m}$is the modal frequency matrix of the flexible appendage vibration,and $\xi_{sa}$ is the vibration damping ratio of appendage.

Quaternion is generally used in the onboard attitude presentation, which is defined by $q_0=\cos(\frac{\alpha}{2})$,$\boldsymbol{q}=$ $[\gamma_1 \sin(\frac{\alpha}{2})$, $\gamma_2\sin(\frac{\alpha}{2})$, $\gamma_3\sin(\frac{\alpha}{2})]^{\rm T}$,where $[\gamma_1,\gamma_2,\gamma_3]$ is the principle axis from the current attitude to the reference attitude and $\alpha$ is the principle angle. The kinematics model is established referring to the aforementioned frame and quaternion:

Considering the tilt of the cantilevered appendage,the varying angle $\theta_{sa}$ affects all terms of the spacecraft dynamic model. Due to this problem,the coupling matrix between the flexible appendage and the rigid body is time varying as follows:

B. Characteristic Modeling

Modeling based on plant dynamic characteristics and performance requirements rather than only accurate plant dynamic analysis is the key idea of the characteristic modeling. Unlike other intelligent modeling methods such as the T-S fuzzy modeling^[18],the characteristic modeling compresses all information of the system into several characteristic parameters. A characteristic model has the following features^[19].

1) A plant characteristic model is equivalent to its practical plant in output for the same input,i.e.,the output error can maintain within a permitted range in a dynamic process as well as their outputs are equal in the steady state.

2) The order and form of a characteristic model mainly depends on control performance requirements.

3) Compared with an original dynamic equation,the structure of a characteristic model should be simpler,easier,and more convenient for realization in engineering.

4) Unlike the reduced-order model of a high-order system,a characteristic model compresses all the information of the high-order system into several characteristic parameters,which means no information is lost. Generally,a characteristic model is represented by a slowly time-varying difference equation.

Consider the nonlinear system

where $x$ and $u$ denote the state and input of the system, respectively. Choosing

then (5) can be rewritten as

Assume that the properties of the nonlinear system (7) are as follows^[20]:

a) There is only a single input and a single output.

b) The power of $u(t)$ is 1.

c) If $x_i=0$ and $u_i=0$,we have $f(\cdot)=0$.

d) $f(\cdot)=0$ is continuously differentiable with all variables, and partial derivative values are bounded.

e) $| f ( x(t+\triangle t),u(t+\triangle t) ) | - | f( x(t),u(t) ) | < M \triangle t$,where the constant $M > 0$ and $\triangle t$is sampling time.

f) The states and control value are bounded.

Based on (7),the following lemma will be used in the derivation of the main results.

Lemma 1^[20]. For any nonlinear system that can be described as (7),if assumptions a)-d) are satisfied and sampling time $\triangle t$ satisfies certain conditions,the desired signal $r$ and its derivative value are bounded,the error characteristic model of the system can be expressed with the following 2-order difference equation:

where $g_0(k)=-g_1(k)+ {O} (\triangle t)$,$ {O} (\triangle t)$ represents the high-order infinitesimal term of the sampling time,$e(k)=x(k)-r(k)$,and $u(k)$ is the bounded sampling control. If the system is stable and assumptions e)-f) are satisfied,then

1) $f_1(k)$,$f_2(k)$,$g_0(k)$,and $g_1(k)$ are slowly time varying.

2) The ranges of these coefficients can be determined beforehand.

3) In dynamic process,under the same input,selecting suitable sampling period $\triangle t$ can make sure that the output error between the characteristic model and the controlled plant keeps within a permitted limit.

4) In steady state,both outputs are equal.

For the minimum-phase system,in general the error characteristic model is chosen as follows:

A. The Characteristic Model of the Spacecraft

Due to the attitude kinematics and dynamics of (1),(2),and (3), defining $\boldsymbol{x}=\left[q_0,\boldsymbol{q}^{\rm T},\boldsymbol{\eta}_{sa}^{\rm T}, \dot{\boldsymbol{\eta}}_{sa}^{\rm T},\boldsymbol{\omega}^{\rm T}_s\right]^{\rm T}$ and $\boldsymbol{u}=\boldsymbol{T}_T$,the attitude kinematics and dynamics of the CFNSs can be rewritten as:

where

It is clear that (10) can be rewritten as the form of (7) and the assumptions a)-f) are satisfied,so characteristic model of the angular velocity can be described in the form of (8) after it is discretized. In order to verify the possible intervals of the characteristic parameters of the CFNSs,simulation is done with several kinds of control signals. Suppose that the control input $u$ has the following four types:

1) Step signal: $u(k)=10$.

2) 0.2 Hz sinusoid signal: $ u(k)=10\sin(0.4k \pi \triangle t)$.

3) 0.2 Hz square wave signal:$ u(k)=10 \textrm{sgn} \left[\sin (0.4k \pi \triangle t)\right].$

4) White noise.

A SSOS,which has one degree freedom single solar panel driven by a constant velocity to point to the sun,is applied to the simulation in open loop. The main parameters of the flexible satellite are presented in Table Ⅰ.

Table Ⅰ

Table Ⅰ DEFINITION OF THE SSOS Main body and flexible appendage data $\boldsymbol{I}_s$ $\left({\rm kg}\cdot{\rm m}^2\right)$ $\left[\begin{array}{ccc} 6 393.31 & 26.95 & -21.09 \\ 26.95 & 4 737.30 & 1 868.48\\ -21.09 & 1 868.48 & 8 361.13 \end{array}\right]$ $\boldsymbol{\omega}_{sa}$ $({\rm Hz})$ ${\rm diag}\{1.02,1.24,1.92,2.86,3.88,...\}$ $\xi_{sa}$ $0.005$ $\boldsymbol{F}_{s0}^{\rm T}$ $\left[\begin{array}{ccc} 0.33591 & 18.3213 & -29.8864 \\ -0.010055 & -20.8362 & -26.3452 \\ -29.8864 & 0.075222 & 0.5616 \\ 20.0637 & -0.36422 & -0.7904 \\ 0.024013 & 10.2219 & 27.968 \\ \vdots & \vdots & \vdots \end{array}\right]$

Table Ⅰ DEFINITION OF THE SSOS

Here we use the recursive least-square method to estimate parameters. Set the forgetting factor $\lambda_s=0.97$. The recursive least-square method is as follows:

where $\hat{\boldsymbol{\theta}}(k-1) = [\hat{f}_1(k),\ \hat{f}_2(k),\ \hat{g}_0(k),\ \hat{g}_1(k)]^{\rm T}$ and $\boldsymbol{\phi}(k-1) = [y(k-1),\ y(k-2),\ u(k-1),\ u(k-2)]^{\rm T}$.

The simulation results in the steady state are listed in Table Ⅱ, where errors are the difference between the output of the characteristic model (8) and the dynamic model (10).

Table Ⅱ

Table Ⅱ SIMULATION RESULTS WITH THE INPUTS Row Pitch Yaw Error $2.47 \times 10^{-10}$ $2.01 \times 10^{-8}$ $-7.39 \times 10^{-12}$ $f_1(\infty)$ $1.9875$ $1.9565$ $1.9606$ Input(1) $f_2(\infty)$ $-0.9876$ $-0.9562$ $-0.9605$ $g_0(\infty)$ $-1.01 \times 10^{-7}$ $-4.67 \times 10^{-7}$ $-5.39 \times 10^{-8}$ $g_1(\infty)$ $1.10 \times 10^{-7}$ $5.01 \times 10^{-7}$ $5.85 \times 10^{-8}$ Error $2.49 \times 10^{-9}$ $-2.60 \times 10^{-9}$ $6.26 \times 10^{-12}$ $f_1(\infty)$ $1.9961$ $1.9896$ $1.9629$ Input(2) $f_2(\infty)$ $-0.9961$ $-0.9896$ $-0.9629$ $g_0(\infty)$ $-8.00 \times 10^{-6}$ $-1.10 \times 10^{-5}$ $-4.70 \times 10^{-6}$ $g_1(\infty)$ $8.00 \times 10^{-6}$ $1.10\times 10^{-5}$ $4.55 \times 10^{-6}$ Error $8.00 \times 10^{-7}$ $4.03 \times 10^{-6}$ $1.55 \times 10^{-6}$ $f_1(\infty)$ $1.9174$ $2.0376$ $1.9248$ Input(3) $f_2(\infty)$ $-0.9587$ $-1.0397$ $-0.9267$ $g_0(\infty)$ $-8.70 \times 10^{-6}$ $1.51 \times 10^{-6}$ $-1.09 \times 10^{-7}$ $g_1(\infty)$ $8.70 \times 10^{-6}$ $-1.50\times 10^{-6}$ $8.60 \times 10^{-8}$ Error $\pm5.17 \times 10^{-5}$ $\pm6.76 \times 10^{-5}$ $\pm2.80 \times 10^{-5}$ $f_1(\infty)$ $1.9583$ $1.9874$ $2.0012$ Input(4) $f_2(\infty)$ $-0.9595$ $-0.9877$ $-1.0028$ $g_0(\infty)$ $7.85 \times 10^{-6}$ $-1.18 \times 10^{-7}$ $4.83 \times 10^{-6}$ $g_1(\infty)$ $-7.11 \times 10^{-6}$ $1.16\times 10^{-7}$ $-4.26 \times 10^{-6}$

Table Ⅱ SIMULATION RESULTS WITH THE INPUTS

The simulation results show it is effective to replace the spacecraft model with the corresponding characteristic model. Moreover,to provide the simplicity of design,the characteristic model is chosen as follows:

where $i=1,2,3$ represents the roll,pitch,and yaw axis, respectively,$r_i(k)$ is the desired output,and $e_i(k)=y_i(k)-r_i(k)$.

Furthermore,with the introduction of new state variables $\boldsymbol{x}_i (k)$ $=$ $[e_i(k-1),e_i(k)]^{\rm T}$,the overall system of the spacecraft can be rewritten as

B. Sliding Mode Control for the Spacecraft

In this paper,a novel sliding mode surface is designed as follows:

where $L_1=0.382$ and $L_2=0.618$ are golden section coefficients, $\tau>1$ is the adjustable parameter,and $\hat{f}_{i1}(k)$ and $\hat{f}_{i2}(k)$ are the estimated values of the corresponding coefficients in (12). The coefficients are estimated by the gradient projection algorithm as follows^[20]:

where $\hat{\boldsymbol{\theta}}_{i}(k) = [\hat{f}_{i1}(k), \hat{f}_{i2}(k),\hat{g}_{i0}(k)]^{\rm T}$ and $\boldsymbol{\phi}_i(k-1) = [e_i(k-1),e_i(k-2),u_i(k-1)]^{\rm T}$. The positive constants $\lambda_{e1}$ and $\lambda_{e2}$ satisfy $0 < \lambda_{e1} < 1$ and $\lambda_{e2}>0$,respectively. ${\rm Pro}(x)$ represents the orthogonal projection from $x$ to the bounded closed convex set $\Xi$. According to Table Ⅱ,we have

Usually,the projection operator plays a crucial role in the projection-type adaptive mapping scheme^[21]. However it is noticeable that the orthogonal projection algorithm only works when the estimated values leave the given intervals which have been tested.

By (13) and (14),we obtain the equivalent control of DTSMC law as follows:

The variable structure control of DTSMC law is designed as

where $p_i$ and$q_i$ are both odd positive integers with $p_i < q_i$ and $\rho_i$ and $\vartheta_i$ are positive constants. In order to avoid the singularity,the parameters $p_i$ and $q_i$ should be chosen carefully^[22]. $\textrm{sgn}(\cdot)$denotes the sign function,that is

The control input for the spacecraft is proposed as

Then we will analyze the behavior of sliding mode surface and the output tracking precision using the characteristic model-based DTSMC law. Before moving on,we give the following lemma and definition.

Definition 1^[23]. The Lypaunov function is chosen to be $V(k)$ $=$ $s^2(k)/2$. The sliding manifold $s(k)=0$ is attractive if $V(k$ $+$ $1)-V(k) < 0$. This condition is satisfied if

Lemma 2^[24]. Consider the scalar dynamical system

where $l>0$ and $0 < \alpha < 1$ is a ratio of odd integers. If $| j(k)|$ $\leq$ $\gamma$,$\gamma>0$,there is a finite number $K^*>0$ such that

where function $\psi(\alpha)$ is defined as

Based on Definition 1 and Lemma 2,we have the following theorem.

Theorem 1. If the error characteristic model of system (10) can be expressed with (12),the sliding mode surface is defined as (14),and the control law is designed as (19),then

a) The overall system (13) is stable with

b) There is a finite number $K^*_1>0$ such that

c) There is a finite number $K^*_2>0$ such that

where

$f_{i1}(\infty)$ and $f_{i2}(\infty)$ can be found in Table Ⅱ, and $e_i(0)$ is the initial error.

Proof. Choosing Lypapunov function $V_i(k)=s_i^2(k)/2$,the behavior of $s_i(k)$ can be prescribed as

Substituting (13) and (19) into (26) yields

and then

Hence the control law (19) makes (13) move to the predefined sliding mode surface. Observing $s_i(k)=0$ in sliding phase,it follows that

Substituting (16) into (29),we have $0.7258\leq L_1\hat{f}_{i1}\leq 0.764$ and $-0.618\leq L_2\hat{f}_{i2}\leq -0.5562$,which results in $| (\frac{cs_{i2}(k-1)}{cs_{i1}(k-1)})^\tau |$ $ < $ $1$. Thus the overall system is stable.

Then it follows from (27),considering $|-\vartheta_i\textrm{sgn}(s_i(k))|=\vartheta_i$,based on Lemma 2, there exists a finite number $K^*_1>0$ such that when $\forall k\geq K^*_1$ there is

Substituting (19) into (12) yields

Noticing that $| (L_2\hat{f}_{i2}(k))^\tau | < 1$ which results in $| (L_2\hat{f}_{i2}(k))^\tau |$ $\cdot$ $| e_i(k)|$ $ < $ $| (L_2f_{i2}{(\infty)})^\tau| \cdot| e_i(0)|$,it follows from Lemma 2 that there is a finite number $K^*_2>0$ such that when $\forall k\geq K^*_2$ there is

Remark 1. According to (29),when the error reaches its sliding surface,it will converge to zero with a speed determined by chosen $\tau$,which lies in $\left[0.728^\tau,0.85^\tau \right]$. Usually,we choose $2\leq \tau \leq 10$.

Remark 2. Actually,in practice,according to Table Ⅱ and (16),we can obtain $f_{i1}(\infty)\approx\lim\nolimits_{k \rightarrow K^*_2}\hat{f}_{i1}(k)\approx2$ and $f_{i2}(\infty)$ $\approx$ $\lim\nolimits_{k \rightarrow K^*_2}\hat{f}_{i2}(k)\approx-1$,which lead to

Theorem 1 demonstrates that for the closed system (10) with the characteristic model (12),under the DTSMC law (19),the sliding mode surface and the error will converge to its relevant bound in finite time and the predefined bound mainly depends on the adjustable parameters of the DTSMC law.

Suppose that the SSOS is required to rapidly maneuver considering the tilt of the cantilevered appendage. Simulations are conducted for demonstrating the performance of the proposed DTSMC law. The closed system logic structure is shown in Fig.1.

Fig. 1

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Fig. 1. Control logic block diagram for the spacecraft.

Four different sinusoidal-wave disturbances as in (33) are added to the spacecraft randomly. The external disturbances used in the simulation are far worse than those typically observed in practice^[25].

The desired attitude velocity command is described by using the bang-coast-bang method and is chosen as

For the sliding mode,the parameter is $m=2$. The parameters of the controller are tuned as $p_i=1$,$q_i=5$,$\rho_i=1$,and $\vartheta_i =1\times 10^{-4}$. The simulation results are shown in Figs. 2-7. The simulation results reveal that converging times of three-axis are very close. Maneuvering angles are $43^\circ$, $20^\circ$,and $43^\circ$,respectively. The rapid maneuvering performance and point accuracy shown in Figs. 3 and 4 are significantly improved compared to the robust fuzzy $H_{\infty}$ controller^[1]. Table III shows the simulation results. The interference of the flexible appendage vibration is suppressed. It follows from Figs. 5 and 6 that tracking effects are achieved in finite time. The control input is shown in Fig.7. It illustrates that chattering is avoided because of the bounded DTSMC implementation.

Fig. 2

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Fig. 2. Quaternion responses.

Fig. 3

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Fig. 3. Attitude angles.

Fig. 4

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Fig. 4. Angular rates.

Fig. 5

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Fig. 5. Tracking errors.

Fig. 6

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Fig. 6. Partial enlargement of tracking errors.

Fig. 7

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Fig. 7. Control inputs.

Table Ⅲ

Table Ⅲ THE RESULTS OF COMPARISON WITH [1] Methods [1] DTSMC Time of maneuver process $({\rm s})$ 50 40 Angle of rotation $(^\circ)$ $(1,0.4,0.5)$ $(43,20,43)$ Time of transition process $({\rm s})$ 30 10 Attitude stability accuracy $(^\circ/{\rm s})$ $1\times 10^{-4}$ $1\times 10^{-5}$ Attitude pointing accuracy $(^\circ)$ $2.8\times 10^{-3}$ $1\times 10^{-3}$ Control input ${\rm (N\cdot m)}$ $\leq 1$ $\leq 50$ Flexible variables $\leq 0.003$ $\leq 0.003$

Table Ⅲ THE RESULTS OF COMPARISON WITH [1]

As can be seen from the phase plane in Figs. 8-10,the characteristic model-based sliding mode surface and the controller demonstrate a fast convergence and strong robustness.

Fig. 8

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Fig. 8. Phase plane of the first channel.

Fig. 9

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Fig. 9. Phase plane of the second channel.

Fig. 10

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Fig. 10. Phase plane of the third channel.

Then the rotational inertia of satellite is considered as $1.5{\pmb I}_s$ and $0.5{\pmb I}_s$,respectively. The simulation results in Table Ⅳ illuminate the proposed controller can provide a satisfactory tracking result.

Table Ⅳ

Table Ⅳ SIMULATION RESULTS Rotational inertia $1.5\boldsymbol{I}_s$ $\boldsymbol{I}_s$ $0.5\boldsymbol{I}_s$ Pointing accuracy $(^\circ)$ $0.005$ $0.005$ $0.005$ Time of stabilization$({\rm s})$ $8$ $10$ $13$ Attitude stability $(^\circ/{\rm s})$ $1\times 10^{-5}$ $1\times 10^{-5}$ $1\times 10^{-5}$

Table Ⅳ SIMULATION RESULTS

The requirement of attitude tracking in AOCSs is a challenging and important objective for many practical spacecraft missions. In this paper,a finite-time DTSMC law has been proposed to achieve this objective for the spacecraft with variable tilt of flexible structures using the characteristic modeling method. The designed methodology has used the characteristic model,which includes all the information of the spacecraft,instead of the dynamic model to avoid the enormously complex cantilever hybrid model. Based on such characteristic model, an adaptive sliding mode surface,which was composed by the parameters of the characteristic model and the relative angular velocity errors of the spacecraft,has been designed to construct the finite-time DTSMC law. Numerical simulations of a SSOS has been performed to validate the effectiveness of the proposed control law in the presence of severe model uncertainties and disturbances. Simulation results of the control performance have demonstrated that the spacecraft converges to the desired attitude and angular velocity in a short time. Furthermore,the vibration caused by the maneuver of the appendage has been suppressed as well as the chattering phenomenon caused by the switching control of the SMC has been reduced.