Abstract
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 sliding mode 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.
1. 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 time-varying 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 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.
2. Mathematical Preliminaries
2.1. Inverse Optimal Control via Control Lyapunov Function
Basic concepts in the realm of nonlinear stabilization are given below. We consider the nonlinear dynamic systemwhere denotes the system state, is the control vector, and are continuous functions, respectively, with .
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 is called a control Lyapunov function (CLF) of system (29) if, for each ,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 lawwhere 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.
Lemma 2 (see [29]). The globally stabilizing control law solves system (1) with respect to the cost functionalwhere .
If is a positive definite, radially unbounded function such that is achieved with the control law , it follows that
When the function is set to be , one obtainsand becomes a solution of the Hamilton-Jacobi-Bellman (HJB) equationAccording 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 and 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.
3. 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 and , respectively. Then, the kinematics in terms of the quaternion is given by [27]where with the identity matrix. Here, notation denotes the skew-symmetric matrix which is defined asThe components of satisfy the conditionwhere is the Euclidean norm. A more detailed explanation of the quaternion and other attitude representations are presented in [27, 30].
3.2. Dynamics of Flexible Spacecraft
The dynamics of a flexible spacecraft is governed by [28]where is the symmetric inertia matrix of the whole spacecraft, is the modal displacement vector with being the mode number, and is the coupling matrix between the central rigid body and the flexible attachments. denotes the control torque vector, represents the external disturbance torque, and and denote the stiffness and damping matrices, respectively, which are defined aswith damping and natural frequency .
For simplicity, let where . The relative dynamic equations (11) can be written as [31] The matrices , , and are given as where is the zero matrix and the identity matrix. Clearly, is a Hurwitz matrix.
If we letthen, the kinematic equation in terms of can be expressed aswherewithDefine a new variable aswhere is a symmetric positive definite matrix. It should be noted thatTherefore, the variable in (21) becomesThe first time derivative of is obtained asConsider the flexible spacecraft systems in (18), (15), and (24). We can rewrite these equations in the state-space form aswherewith
3.3. 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 achievethat is, , , , , and as .
4. 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 is a positive definite matrix that is a solution of the Lyapunov equation with a positive definite matrix .
The following assumptions are required in the subsequent controller design.
Assumption 4. The known matrices and are symmetric positive definite and bounded.
Assumption 5. The control input torque of actuator satisfies , where is a known constant.
It should be noted that, in practice, 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 , , and satisfy the following inequality:where and denote the minimum and maximum singular values of the matrix .
Proof. Since is symmetric positive definite, can be written aswhereWith and , this ensures that is positive definite. Taking the first derivative of with respect to time, we havewhere by substituting (18), (15), and (24) into (33) one obtainsUsing the fact that , and , we obtainWe know that the first time derivative of can be expressed asComparing (35) and (36), it follows thatTherefore, if , then we haveNext, we show that if , then for all . Substituting in (34), can be expressed aswhere and . It follows that if condition (30) is satisfied, then is achieved. This guarantees that the candidate is a CLF for the spacecraft motion system (25). This completes the proof.
Next, the main results of our proposed inverse optimal control for the spacecraft model are presented. The presented dynamic feedback control law is designed aswherewith
It should be noted that the Lyapunov stability theory needs the existence and uniqueness of a solution, for an 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 and satisfies existence and uniqueness solution requirement.
Proof. We know that and and these properties will be used in this proof. Since (15) and (18) are continuously differentiable, the following inequalities can be obtained:where , . It should be noted that in (19) the term only provides the negative or positive value of and it has no effect on the differentiation. Thus, the term is continuously differentiable.
Now, we consider system (24) with control input and obtainSince the terms and are continuously differentiable, one obtainsNow, can be rearranged aswhere .
Recalling (43), the following inequalities can be obtained:which is equivalent towhere . The last inequality (48) states that the spacecraft attitude system consisting of (15), (18), and (24) is locally Lipschitz in . 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) if the parameters , , , and satisfy the following inequality: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).
For the case , the time derivative of a CLF isFor the definition of CLF, for we obtain for all . For , is achieved for all because is always satisfied. Therefore, is achieved for all .
For the case , substituting into (35), the time derivative of a CLF becomeswhich can be further written aswhere . Evidently, if (49) and (50) are satisfied, one obtains for all . Therefore, controller (40) asymptotically stabilizes the spacecraft system (25).
Remark 10. Inequality (30) is considered as the special case of inequality (49). When 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 , , , and such that the inequalities (49) and (50) are satisfied. This automatically satisfies condition (30).
Remark 11. When the disturbance vector is ignored, the proposed control law (40) can stabilize the two equilibrium points and . If the initial is chosen, the proposed law will force to . On the other hand, if the initial is selected, the proposed law will drive to . This means that the proposed law can avoid an unwinding phenomenon. However, most existing inverse optimal control methods (see, e.g., [13, 22, 33]) could only stabilize one equilibrium point and another equilibrium point is unstable. If the initial is chosen, these controllers would force to instead of , so an unwinding phenomenon may occur.
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 lawsolves the inverse optimal assignment problem for the attitude control system equation (25) by minimizing the cost functionalwhere is defined byand 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 andwhich shows that is positive semidefinite. Although we have two cases, and , both cases obtain , for all . This means is positive semidefinite for and . 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 which is obtained by substituting into (55). This completes the proof.
5. 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.
5.1. Sliding Mode Disturbance Observer
We now consider the following plant system aswhere , , and 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 as the total disturbance to system (58). The new SMDO for system (58) is constructed aswhere and , , are the observer gain parameters to be selected. is the observer variable used to estimate the total disturbance, denotes the first time derivative of the total disturbance , and is the estimation error of the SMDO. The following assumption must be included to ensure the convergence of the proposed SMDO.
Assumption 13. The total disturbance in (58) and its first time derivative are unknown but bounded; that is, and , where and denote positive constants.
The convergence of the states of proposed SMDO to the states of plant (58) is investigated in the following theorem.
Theorem 14. Let Assumptions 4, 5, and 13 hold. Consider system (58) with the SMDO (59). Then, there exist positive observer gains such that the ultimate boundedness of the observed errors is ensured.
Proof. To investigate the stability of the SMDO system, one must consider an expression for the observer error dynamics. We first define the observer errors , , , and . The observer error dynamics can be expressed asWe now show that the ultimate boundedness of observed states , , is guaranteed. The sliding surface is defined as . Consider the following Lyapunov function:The first time derivative of isWe could select , where . Then, . This would guarantee that , and the finite time convergence of to the sliding surface is also ensured. When the sliding mode occurs (i.e., and ), the resulting error system on the sliding mode becomesLetting , then (63) can be written aswhere ,If the matrix is Hurwitz, there exists a unique positive definite matrix such that , where is a positive definite matrix. Select the Lyapunov function asIts first time derivative isUsing (66) we know that and it follows thatSetting , one obtainswhere and . This ensures that the observed errors , , converge to a small bound of zero.
5.2. Robust Inverse Optimal Control Design
We now consider system (24) and define the auxiliary dynamicswhere , , , and .
Applying the SMDO (59) with , we have
Using the results from the SMDO, the estimated disturbance is determined by . Thus, the proposed robust inverse optimal control with an input constraint can be obtained asWith suitable control gains defined by the inverse optimal control approach based on CLF concept, the optimal feedback controller (72) contains both optimality and robustness performance to attenuate external disturbances.
6. 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 [28] and coupling matrices respectively. The natural frequencies and damping ration are provided by
The maximum torque input is restricted by N-m. The control parameters of the proposed controller (72) are set as , , and . Only the first four elastic modes have been considered in the controller design. The initial states of the rotation motion are given by [21] , , and . For the minimax IOAC in [21], the control parameters and are chosen. The periodic disturbances are provided by
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 is set as , the state should converge to the equilibrium point . Figure 4 shows that only the proposed controller (72) can force the state to the equilibrium point . 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.
7. 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.
Competing Interests
The author declares that there are no competing interests.
Acknowledgments
The research was supported by King Mongkut’s University of Technology North Bangkok, Contract no. KMUTNB-GOV-58-13.