Inverse Optimal Attitude Stabilization of Flexible Spacecraft with Actuator Saturation

This paper presents a new robust inverse optimal control strategy for flexible spacecraft attitude maneuvers in the presence of external disturbances and actuator constraint. A new constrained attitude controller for flexible spacecraft is designed based on the Sontag-type formula and a control Lyapunov function. This control law optimizes a meaningful cost functional and the stability of the resulting closed-loop system is ensured by the Lyapunov framework. A slidingmode disturbance observer is used to compensate unknown bounded external disturbances. The ultimate boundedness of estimation error dynamics is guaranteed via a rigorous Lyapunov analysis. Simulation results are provided to demonstrate the performance of the proposed control law.


Introduction
Attitude control of flexible spacecraft has been an important problem in many developments of spacecraft technology.Recent spacecraft requires the structure of a rigid hub with flexible appendages, such as antenna and solar array.Because of the strong coupling between the hub and flexible appendages, the vibration of flexible appendages will be induced.This may lead to performance degradation or instability if the vibration is not suppressed as rapid as possible.In addition, owing to the environment and measurement factors, the external torque disturbance cannot be avoided.Inertia matrix uncertainty and external disturbance are required to be considered by researcher in attitude controller designs of a spacecraft.Recently, great attention has been paid to the vibration suppression and robust attitude controller designs for flexible spacecraft such as passive control [1], input shaping [2], sliding mode control [3,4], and active disturbance rejection control [5,6].
Optimal fuel consumption and time optimal control problems for spacecraft attitude maneuvers are very practical and important issues.The synthesis of optimal attitude control law for spacecraft attitude maneuvers has become increasingly popular among researchers.In [7] the timevarying linear quadratic regulator (LQR) method was applied to a nonlinear control problem but the approximation of spacecraft system was required to meet the optimality and stability conditions.In [8,9] state-dependent Riccati equation (SDRE) techniques were successfully used to develop optimal controller for spacecraft attitude maneuvers.In Xin and Balakrishnan [10], SDRE and neural network schemes were merged to develop a robust optimal attitude control law for spacecraft under an uncertain moment of inertia.However, the SDRE method requires to solve the Riccati equation repetitively at every integration step.It may be difficult to use the SDRE method if the system order is higher.Xin and Pan [11] have applied the - technique to design a nonlinear integrated position and attitude suboptimal control law.An inverse optimal control approach was presented in [12] to construct the optimal controller for regulation of the rigid body.Recently, attitude controller designs for rigid spacecraft using inverse optimal control schemes have been developed (see, e.g., [13,14]).The inverse optimal control approach avoids the task of solving a Hamilton-Jacobi-Bellman equation and offers a globally asymptotically stabilizing control law which is optimal with respect to 2 International Journal of Aerospace Engineering a meaningful cost functional.Using this method, a stabilizing control law can be designed using Sontag's formula [15] or Freeman's formula [16] with directional information supplied by control Lyapunov function (CLF) [17,18].
As extensions of the above studies, optimal control and robust control have been merged to obtain robust optimal control laws.Various methods for developing robust optimal controllers for the attitude control of a rigid spacecraft have been proposed in the literature.Nonlinear  ∞ control strategies were proposed in [19,20] to develop stabilizing feedback controllers for the spacecraft tracking motion.Later, the inverse optimal control method was used to develop attitude controllers for a rigid spacecraft in [21].Luo et al. [22] developed an  ∞ inverse optimal adaptive controller for attitude tracking of spacecraft.Adaptive control and nonlinear  ∞ control have also been merged to design robust optimal controllers.Zheng and Wu [23] proposed nonlinear  ∞ controller designs for spacecraft and a -level disturbance attenuation could be attained.Pukdeboon and Zinober [24] developed robust optimal control laws based on the optimal sliding mode control technique for attitude tracking of spacecraft.Optimal sliding mode control and inverse optimal control schemes for flexible spacecraft attitude maneuvers have been designed in [25] and [5], respectively.However, these control methods did not consider the effect of actuator saturation.In practical situation, due to the physical restriction and energy consumption, if actuator saturation is not handled effectively, a performance degradation or system instability may occur.Moreover, control methods mentioned above may lead to the unwinding phenomenon encountered in unit quaternion based attitude systems since these control laws consider only one of two equilibrium points of unit quaternion [26].
The main contributions of this paper are as follows: (I) A robust inverse optimal control method for flexible spacecraft attitude maneuvers in the presence of actuator constraint is proposed in this paper.This controller can prevent the unwinding phenomenon.
(II) A new inverse optimal control problem for flexible spacecraft attitude maneuvers with input saturation is studied.The proper control Lyapunov function is chosen and then used to solve this problem.Finding a control Lyapunov function and optimal feedback control to solve the inverse optimal control problem for the flexible spacecraft attitude control system is very difficult and has not been previously examined.
(III) A new sliding mode disturbance observer is developed and combined with the proposed attitude controller.The uniformly ultimate boundedness stability of the proposed disturbance observer is guaranteed via a rigorous Lyapunov analysis.
This paper is organized as follows.Section 2 introduces some preliminary results which are required for the following discussion in this paper.In Section 3 the dynamic equations and attitude kinematics of a flexible spacecraft [27,28] are described.Also, the control objective is provided.Section 4 proposes an inverse optimal control design with input saturation.The proposed CLF is selected to solve the inverse optimal control problem of flexible spacecraft and then an optimal stabilizing controller is designed.In Section 5, a sliding mode disturbance observer is designed and then used to develop a robust optimal attitude controller.The ultimate boundedness of estimation errors is guaranteed using the Lyapunov stability theory.In Section 6 an example of spacecraft attitude manoeuvres is presented to illustrate the performance of the developed control law.In Section 7 we present conclusions.

Inverse Optimal Control via Control Lyapunov Function.
Basic concepts in the realm of nonlinear stabilization are given below.We consider the nonlinear dynamic system where  ∈ R  denotes the system state,  ∈ R  is the control vector,  : R  → R  and  : R  → R × are continuous functions, respectively, with (0) = 0.
Based on the definition of control Lyapunov function (CLF) in [17,29], we provide the following definition.
Definition 1 (see [17,29]).A differentiable, positive definite and radially unbounded function () : R  →  + is called a control Lyapunov function (CLF) of system (29) if, for each  ̸ = 0, where the functions    and    denote the Lie derivatives of the Lyapunov function () with respect to the vector fields defining the system.These functions are defined as    = [/]  () and    = [/]  ().
We next consider a globally stabilizing control law where () is a symmetric positive definite matrix.When a CLF is known for system (1), an inverse optimal controller can be designed using the following lemma.
If () is a positive definite, radially unbounded function such that V() ≤ 0 is achieved with the control law  = (1/2) * = −(1/2) −1 (  ())  , it follows that When the function () is set to be − V, one obtains and () becomes a solution of the Hamilton-Jacobi-Bellman (HJB) equation According to the standard result of optimal control theory, the control law (3) is optimal among all () that globally asymptotically stabilize system (1).
Remark 3. In the inverse optimal approach, a globally stabilizing feedback control law () is designed first and then it is required to find () ≥ 0 and () > 0 such that  optimizes (4).The problem is inverse because the functions () and () are a posteriori found by the stabilizing feedback law, rather than a priori selected by the designer.

Flexible Spacecraft Attitude Model
3.1.Kinematics of Flexible Spacecraft.In this paper the unit quaternion is used to represent the attitude system of a flexible spacecraft.We define the rigid body attitude and angular velocity of the rigid spacecraft body frame with respect to the inertia axis frame as  = [ 0   ]  ∈ R × R 3 and  ∈ R 3 , respectively.Then, the kinematics in terms of the quaternion is given by [27] q where () =  × +  0  3 with  3 the 3 × 3 identity matrix.
Here, notation  × denotes the skew-symmetric matrix which is defined as The components of  satisfy the condition where ‖⋅‖ is the Euclidean norm.A more detailed explanation of the quaternion and other attitude representations are presented in [27,30].

Dynamics of Flexible Spacecraft.
The dynamics of a flexible spacecraft is governed by [28]  ω +   η = − × ( +   η ) +  + where  ∈ R 3×3 is the symmetric inertia matrix of the whole spacecraft,  ∈ R  is the modal displacement vector with  being the mode number, and  ∈ R ×3 is the coupling matrix between the central rigid body and the flexible attachments. ∈ R 3 denotes the control torque vector,  ∈ R 3 represents the external disturbance torque, and  and  denote the stiffness and damping matrices, respectively, which are defined as with damping   and natural frequency   .
For simplicity, let where  = η + .The relative dynamic equations ( 11) can be written as [31] The matrices   , , and  are given as where 0 × is the  ×  zero matrix and  4 the 4 × 4 identity matrix.Clearly,  is a Hurwitz matrix.
If we let then, the kinematic equation in terms of ( 0 , ) can be expressed as where with International Journal of Aerospace Engineering Define a new variable (,  0 , ) as where  1 ∈ R 3×3 is a symmetric positive definite matrix.It should be noted that Therefore, the variable (,  0 , ) in ( 21) becomes The first time derivative of  is obtained as Consider the flexible spacecraft systems in ( 18), (15), and (24).We can rewrite these equations in the state-space form as where with

Control Objective.
In this work we consider rest-to-rest maneuvers of a flexible spacecraft.The control objective is to design a controller  to achieve the desired rotations of flexible spacecraft in the presence of external disturbances and actuator saturation.In other words, we will find a controller  to achieve lim that is,  → 0,  0 → ±1,  → 0,  → 0, and η → 0 as  → ∞.

Inverse Optimal Controller Design for Flexible Spacecraft
In this section, we now propose an inverse optimal controller for stabilizing the complete attitude motion of flexible spacecraft in the presence of external disturbances.
In order to design a stabilizing controller for solving the inverse optimal problem with an input constraint we first choose a CLF for system (25) as the following candidate positive definite function: where  is a positive constant and  ∈ R 8×8 is a positive definite matrix that is a solution of the Lyapunov equation    +  = − with a positive definite matrix  ∈ R 8×8 .
The following assumptions are required in the subsequent controller design.It should be noted that, in practice,  max is the maximum allowable torque input which actuators can produce.
We next show that the function () defined in ( 29) is a CLF for system (25) by using the following theorem.Theorem 6.In the absence of disturbance vector  in (25), under Assumptions 4 and 5, the positive definite function  defined in ( 29) is a CLF for the spacecraft motion equation (25), if ,  1 , and  satisfy the following inequality: where  min ( 1 ) and  max ( 1 ) denote the minimum and maximum singular values of the matrix  1 .
Next, the main results of our proposed inverse optimal control for the spacecraft model are presented.The presented dynamic feedback control law is designed as where with It should be noted that the Lyapunov stability theory needs the existence and uniqueness of a solution, for an International Journal of Aerospace Engineering initial condition.This is an important basis for the subsequent analysis of this paper.The following theorem shows that the system consisting of ( 15), (18), and (24) satisfies existence and uniqueness solution requirement since the system is locally Lipschitz [32].Theorem 7. In the absence of disturbance, the spacecraft attitude system consisting of ( 15), (18), and ( 24) is locally Lipschitz in (, , ) ∈ R 4 × R 3 × R 8 and satisfies existence and uniqueness solution requirement.
Proof.We know that ‖‖ = ‖ +  1 ‖ ≥ ‖‖ and ‖()‖ = 1 and these properties will be used in this proof.Since ( 15) and ( 18) are continuously differentiable, the following inequalities can be obtained: where  1 ,  2 ∈ R + .It should be noted that in (19) the term sign( 0 ) only provides the negative or positive value of  and it has no effect on the differentiation.Thus, the term sign( 0 )  is continuously differentiable.Now, we consider system (24) with control input  =  −1  and obtain Since the terms − Now, ‖  ż 1 −   ż 2 ‖ can be rearranged as where Recalling (43), the following inequalities can be obtained: which is equivalent to where The last inequality (48) states that the spacecraft attitude system consisting of ( 15), (18), and ( 24) is locally Lipschitz in (, , ) ∈ R 4 × R 3 × R 8 .Hence, the existence of a unique solution is guaranteed.
Remark 8.In this paper, the disturbance  in (24) will be estimated by the proposed observer in Section 5.With the estimation result, the disturbance  will be canceled out by the estimated one.Thus, it makes sense to ignore the disturbance  in Theorem 7 and subsequent analysis.Theorem 9.The control law (40) stabilizes the spacecraft system (25) where Proof.It is required to show that the control law  in (40) is a stabilizing controller for attitude control system (25).Consider the smooth positive definite radially unbounded Lyapunov function as (29).
Remark 10.Inequality ( 30) is considered as the special case of inequality (49).When  = 0 is set, inequality (49) is reduced to a more simple form as presented in (30).Thus, in the controller design, it is sufficient to select the parameters ,  1 ,  max , and  such that the inequalities (49) and (50) are satisfied.This automatically satisfies condition (30).
Next, when  in ( 25) is ignored, it can be shown that the feedback stabilizing controller  defined in (40) solves the inverse optimal control problem.

Theorem 12. The following dynamic feedback control law
solves the inverse optimal assignment problem for the attitude control system equation ( 25) by minimizing the cost functional where () is defined by and  is a final time.
In practice, the time instant  in (55) is the time period of a complete rotation of the spacecraft.Also, the proof of Theorem 12 is given.
Proof.With the derivations in Theorem 9 and the definitions of () and (), we obtain that () is positive definite and which shows that () is positive semidefinite.Although we have two cases, ‖‖ >  max and ‖‖ ≤  max , both cases obtain V < 0, for all  ̸ = 0.This means () is positive semidefinite for ‖‖ >  max and ‖‖ ≤  max .Therefore,   is a meaningful cost functional for the attitude control problem of flexible spacecraft, penalizing on  as well as the control effort .Hence, the control input (54) minimizes the cost functional   .One has the optimal cost  *  = 4(0) which is obtained by substituting () into (55).This completes the proof.

Robust Inverse Optimal Control with Sliding Mode Disturbance Observer
Recently, because of the great advances in nonlinear control theory, the extended state observer (ESO) [32,34] has high efficiency in accomplishing nonlinear dynamic estimation.
The main idea of ESO is the total disturbance vector representing system uncertainties and disturbances are considered as an added state of the system; then all states of the system including the added one will be observed accurately and rapidly.However, few rigorous proofs of ESO convergence have been studied.In this section a sliding mode disturbance observer (SMDO) which uses similar structure to the traditional ESO is presented and the ultimate boundedness of the proposed disturbance observer is ensured using the Lyapunov technique.

Sliding Mode Disturbance
Observer.We now consider the following plant system as where  = [ 1  2 ⋅ ⋅ ⋅   ]  ∈ R  ,  ∈ R, and  ∈ R are the states, input, and output of the system, respectively.() is the unknown total disturbance including disturbances and unknown part of the system, () denotes the known part of the plant model.We add an extended state  +1 as the total disturbance () to system (58).The new SMDO for system (58) is constructed as where Ẑ = [ Ẑ1 Ẑ2 ⋅ ⋅ ⋅ Ẑ+1 ]  ∈ R +1 and   ,  = 1, 2, . . ., +1, are the observer gain parameters to be selected.Ẑ+1 is the observer variable used to estimate the total disturbance, ℎ() denotes the first time derivative of the total disturbance ( Ẑ, ), and  is the estimation error of the SMDO.The following assumption must be included to ensure the convergence of the proposed SMDO.

Simulation Results
The rest-to-rest maneuver of flexible spacecraft is considered with numerical simulations to compare the performance of the proposed inverse optimal attitude control (proposed IOAC) and minimax inverse optimal attitude control (minimax IOAC) in [21].The spacecraft is assumed to have the nominal inertia matrix ] Simulation results of the proposed IOAC and minimax IOAC in [21] are presented in Figures 1-14.From Figures 1-3 we can see that the proposed controller (72) offers smoother attitude responses.For the simulation, since the initial condition of  0 is set as (0) = −0.3320< 0, the state  0 () should converge to the equilibrium point  0 = −1.Figure 4 shows that only the proposed controller (72) can force the state  0 () to the equilibrium point  0 = −1.Thus, the proposed controller (72) can avoid an unwinding phenomenon, while for the minimax IOAC in [21], an unwinding phenomenon occurs.In Figures 5-7, responses of angular velocities obtained by the proposed controller (72) are smoother.Figures 8-10 show that the minimax IOAC in [21] requires larger magnitude control torques.As shown in Figures 11-14 for minimax IOAC in [21], modal displacements converge to a small region around the zero.However, these good responses of modal displacements are obtained since the minimax IOAC takes large control magnitudes.As shown in Figures 15-17, the proposed SMDO provides fast and accurate estimations of the disturbance vector.
We have made comparisons between the simulation results obtained by minimax IOAC in [21] and proposed IOAC (72).Because both controllers have different cost functionals, they have different magnitudes of control forces and torques and take different convergence speeds to reach the origin.These are not valid enough reasons to find out which one of the two controllers is better.It is found that the proposed control law (72) offers smooth attitude and angular velocity responses and requires smaller magnitudes of control inputs.In view of these simulation results, the proposed IOAC (72) seems to be a more useful control approach for general cases of flexible spacecraft attitude maneuvers.It should be noted that the method of [21] did not consider the optimal control problem of flexible spacecraft, so it is unfair to directly compare this method with our proposed control scheme since our proposed method is designed for the flexible spacecraft system.Moreover, the control performance of the attitude controller proposed in [21] is determined by the control gain parameters, so the control performance of the attitude controller proposed in [21] could give better simulation results by choosing other control gain parameters.

Conclusion
A robust inverse optimal control scheme of attitude stabilization of a flexible spacecraft in the presence of external disturbances and actuator constraint has been developed.The concepts of inverse optimal control and CLF have been employed to design an inverse optimal attitude control law with actuator saturation.A new SMDO has been designed by using the similar structure to the traditional ESO.The ultimate boundedness of estimation errors has been proven using the Lyapunov technique.It has shown that the developed controller solves the inverse optimal control problem with input saturation and asymptotically converges to the origin.An example of a rest-to-rest maneuver is presented and simulation results are provided to verify the usefulness of the presented controller.

Assumption 4 .Assumption 5 .
The known matrices  and   are symmetric positive definite and bounded.The control input torque of actuator  ∈ R 3 satisfies ‖‖ ≤  max , where  max > 0 is a known constant.

Figure 12 :Figure 13 :
Figure 12: Time responses of the 2nd component of modal displacements.

Figure 14 :Figure 15 :
Figure 14: Time responses of the 4th component of modal displacements.

Figure 16 :Figure 17 :
Figure 16: Estimation of the 2nd component of the disturbance vector.