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Dynamic infinity‐norm constrained control allocation for attitude tracking control of overactuated combined spacecraft

2019; Institution of Engineering and Technology; Volume: 13; Issue: 11 Linguagem: Inglês

10.1049/iet-cta.2018.5707

1751-8652

Xiuwei Huang, Guang‐Ren Duan,

Stability and Control of Uncertain Systems

IET Control Theory & ApplicationsVolume 13, Issue 11 p. 1692-1703 Research ArticleFree Access Dynamic infinity-norm constrained control allocation for attitude tracking control of overactuated combined spacecraft Xiuwei Huang, Xiuwei Huang orcid.org/0000-0001-7107-6596 Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin, People's Republic of ChinaSearch for more papers by this authorGuang-Ren Duan, Corresponding Author Guang-Ren Duan g.r.duan@hit.edu.cn Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin, People's Republic of China State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin, People's Republic of ChinaSearch for more papers by this author Xiuwei Huang, Xiuwei Huang orcid.org/0000-0001-7107-6596 Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin, People's Republic of ChinaSearch for more papers by this authorGuang-Ren Duan, Corresponding Author Guang-Ren Duan g.r.duan@hit.edu.cn Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin, People's Republic of China State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin, People's Republic of ChinaSearch for more papers by this author First published: 23 May 2019 https://doi.org/10.1049/iet-cta.2018.5707Citations: 3AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract In this study, a dynamic (infinity-norm) constrained control allocation scheme is designed for attitude tracking control of combined spacecraft with inertia uncertainties and external disturbances. A disturbance-observer-based constrained backstepping control law is developed to generate the control command signals considering non-symmetric constraints on control input, where the lumped disturbance containing inertia uncertainties and external disturbances is compensated by the output of stable non-linear disturbance observer. The control scheme can guarantee that the attitude and angular velocity tracking error converge to small neighbourhood of zero by appropriately tuning the control parameters. With the consideration of physical amplitude and rate constraints on actuators, the dynamic constrained control allocation problem is solved by linear programming technique. Numerical examples demonstrate the effectiveness of the proposed disturbance-observer-based constrained backstepping control method and the dynamic constrained control allocation algorithm. 1 Introduction In recent years, there are increasing on-orbit failures since more spacecraft have been launched into space every year [1]. In order to extend the operational lifetime of non-functioning spacecraft, one promising way is to capture the non-functioning spacecraft by a service spacecraft with a healthy control system and these two spacecraft form a combined spacecraft [2]. Then the service spacecraft will take charge of the attitude and orbit control of the combined spacecraft. Furthermore, the attitude takeover control of the combined spacecraft after the non-functioning tumbling spacecraft or on-orbit service commercial satellite has been captured becomes the urgent and important problem needed to be settled. Some contributions have been achieved in this area. Bandyopadhyay et al. [3] developed a new non-linear tracking controller for a spacecraft carrying a large object, such as an asteroid or a boulder. Huang et al. [4] designed a reconfigurable attitude takeover control for combined spacecraft. A robust control law for a control moment gyroscope actuated space robot in the presence of system uncertainties and closed-chain constraints was investigated [5]. After the non-functioning spacecraft has been captured by space robot mounted on the service spacecraft, the mass centre will suffer a large shift, which will lead to the actuators' configuration change [6]. In order to maintain the control performance of the combined spacecraft, there is a need to perform the automatic distribution of the desired control command among available actuators for combined spacecraft under amplitude and rate constraints [7]. By control re-allocation, the desired control command can be redistributed among the actuators based on the new configuration matrix. To ensure the combined system's manoeuvrability, reliability, safety, and fault tolerance capability, there are usually more actuators than degrees of freedom for a spacecraft manoeuvre. Then infinite ways exist to reallocate control commands over the existing actuators. Thus, an optimal actuator allocation criterion must be selected. Most of present control allocation problems are addressed by minimising the sum of the squares of each actuator [8–11], i.e. based optimisation. However, the -based solution may lead to the actuators exceeding saturation by unnecessarily minimising the magnitude of each actuator [12]. To overcome the deficiency of based optimisation, some scholars turn to based optimisation which is defined as the absolute value of the largest component of the actuator manifold. In [13], the control allocation problem was cast as a piecewise linear programme and furthermore a mixed-integer linear programming, which could account for non-linearities in the moment/effector relationships and enforce position constraints on the effectors. A new approach for thruster force allocation based on minimising the was proposed [14], which could be cast as a linear problem then easily incorporated the thruster saturation limits. Doman et al. [15] proposed a mixed-integer linear programming approach mixing continuous and pulsed control effectors. By applying the optimisation technique, a new iterative algorithm was developed by using a control theoretic approach to finding the minimum norm solution of undetermined problems [16]. To improve the robustness of the operation that converted virtual into real control inputs, the robust set of attainable moments was defined, which accounted for several uncertainties in the vehicle control effectiveness [17]. Tohidi et al. [18] developed an adaptive control allocation method based on the pseudo inverse along the null space of the control matrix to adaptively tolerate actuator faults, where the control allocation problem was solved with an exact solution and optimised of the control signal. In summary, as stated in [12], the -based optimisation problem minimises the maximum component of the actuators manifold, it allows the actuators to run within a safer range than the -based optimisation and provides more manoeuvrability for a subsequent controller action. Different from other methods where control distribution depends only on the current control demand, dynamic control allocation also depends on the distribution in a previous sampling instant [19], which can allow different actuators to produce control efforts at different frequency ranges. Guo and Liang [20] proposed a new dynamic control allocation algorithm to distribute the virtual control command among the redundant actuators with actuator's amplitude and rate constraints. A constrained quadratic programming-based robust dynamic control allocation was implemented to manage the redundant actuators for attitude stabilisation problem of a rigid spacecraft [21]. Furthermore, the dynamic control allocation problem was posed as a sequential quadratic programming problem [22]. In this study, the dynamic control allocation cooperating method with the consideration of amplitude and rate constraints is firstly investigated to address the control reallocation problem of the combined spacecraft. Owing to the limitation of actuators, the desired control input must be saturated, if not, the control allocation problem might not have a solution under actuator saturation. Attitude tracking control with input saturation has been quite extensively investigated and various approaches have been developed. Hu et al. [23] used the adaptive method to estimate the resulting approximation error between constrained control input and smooth function. An input saturated model was built and considered as the part of the control law [24, 25]. In [26–28], the effect of input saturation was compensated by the disturbance observer. Also, an auxiliary system was used to tackle the saturated control input problem [29]. Besides input saturation, angular velocity constraint has also been considered in some literature. Under angular velocity constraints, a virtual angular velocity command used to stabilise the attitude was designed by a hyperbolic tangent function [30]. Based on a two-loop structure, Li et al. [31] stabilised the attitude under angular velocity constraints by designing an outer-loop control law. A constrained input-to-stable controller was presented by a non-linear disturbance observer considering assigned angular velocity and control constraints [32]. A constrained backstepping control scheme [33] used command filters to implement the constraints on the control surfaces and virtual control states, in which the effect of saturation would be filtered out by using a modified tracking error definition. The control method proposed in this study is inspired by the constrained backstepping control method, where the effect of the constraints on the control input and virtual angular velocity can be filtered and removed from the parameter update laws to ensure a stable parameter estimation process even when these limitations are in effect. Another advantage of this approach is that the tedious analytic computation of the virtual control signal derivatives can be eliminated compared with the basic backstepping method. Though it has been point out in [33] that simulations of the controller would take up a lot of computational time in MATLAB/Simulink for a complex high-order system, the computational effect on the second-order attitude tracking system can be neglected. Furthermore, after the target has been captured, the inertia matrix of the combined spacecraft is hard to be obtained due to the non-cooperation of the target. Thus, inertia uncertainties need to be considered. In some studies, inertia uncertainties and external disturbances have been treated as a lumped disturbance after model transformation. Disturbance observer technique [34–36] has been applied to deal with the lumped disturbances in the system model. A stable non-linear disturbance observer [37, 38] was designed to estimate the lumped disturbance for control input with a simple form. Comparing with the disturbance observers in other studies, the proposed disturbance observer in [37] is simpler and more effective in control designing, while the performance of the observer could be tuned easily by only one parameter in the observer. Thus, it is will be used to estimate the lumped disturbance containing inertia uncertainties and external disturbances in this study. In this study, the desired control input is obtained by a disturbance-observer-based constrained backstepping control scheme for an attitude tracking control system under input saturation, inertia uncertainties and external disturbance. Furthermore, a dynamic control allocation scheme is designed with the consideration of amplitude and rate constraints. The main contribution of this study is that a disturbance-observer-based constrained backstepping approach is developed to generate the command control input for attitude tracking of combined spacecraft with input saturation, and then a dynamic constrained control allocation scheme is firstly proposed with actuator saturation. This paper is organised as follows: Section 2 describes the dynamic constrained control allocation problem of the attitude tracking system of combined spacecraft with inertia uncertainties and external disturbance. In Section 3, a disturbance-observer-based constrained backstepping control scheme is developed and tuning conditions of controller parameters are derived based on the Lyapunov analysis. A linear programming technique is employed to solve the dynamic constrained control allocation problem in Section 4. Numerical simulations are provided in Section 5 to illustrate the effectiveness of the proposed control law and control allocation approach. Notation. The notations used throughout the paper are fairly standard. All the vectors and matrices are written in bold italic font form, and denote the n -dimensional Euclidean space and the set of -dimensional real matrices, respectively. The superscript 'T' stands for matrix transposition. refers to the Euclidean vector norm of a vector or the induced matrix 2-norm of a matrix. and represent the minimal and maximal eigenvalue of a matrix, respectively. 2 Problem description 2.1 Kinematics and dynamics of attitude tracking system for combined spacecraft In this study, in order to form the attitude stabilisation control system of the combined spacecraft, several corresponding frames are presented as follows: the orbital frame defines the centroid of the combined spacecraft as its origin, and the axis is along the local horizontal direction in the orbital plane, the axis is along the orbital normal and the axis is collinear with a line that extends from the centre of the earth to the centroid of the combined spacecraft and completes a right-handed triad; the body frame of service spacecraft defines the mass centre of service spacecraft as its origin, and three mutually perpendicular axes , and coincident with the principle axis of inertia [4]; the body frame of target spacecraft defines the mass centre of target spacecraft as its origin, and three mutually perpendicular axes , and coincident with the principle axis of inertia; the body frame of combined spacecraft defines the mass centre of combined spacecraft as its origin, and three mutually perpendicular axes , and coincident with the principle axis of inertia. Furthermore, the Modified Rodrigues Parameters (MRPs) vector with Euler's principal rotation axis and angle is used to represent the attitude of the combined spacecraft. The combined spacecraft system consists of a rigid service spacecraft, a rigid target spacecraft and one rigid space manipulator, which is shown in Fig. 1. Then, the attitude motion of combined spacecraft can be described by the following kinematics and dynamics [6]: (1)where , is the angular velocity of the combined spacecraft expressed in , is the control torque, is the gravity gradient torque and is the inertia matrix of combined spacecraft expressed in , is the identity matrix, is the external disturbance and assumed to be bounded. For any vector , is defined as (2) Remark 1.If , the corresponding MRPs will go singular. It is possible to map the MRPs vector to its shadow counterpart through [39] (3)by switching the MRPs to when , the MRPs vector remains bounded within a unit sphere, global rotation representation can thus be ensured. Fig. 1Open in figure viewerPowerPoint Model of space manipulator robot on service spacecraft The gravity gradient torque can be computed by [40] (4)where is the orbit angular rate value; is the third column-vector of the direction cosine matrix expressed as (5) Furthermore, let represents the desired attitude frame with its attitude and angular velocity denoted by and , the rotation matrix from to is defined as (6)and can be expressed as (7) Define the MRPs tracking error and angular tracking error as [41] (8)Then, the attitude tracking motion of combined spacecraft can be described by the following kinematics and dynamics: (9)where . To signify the error between the true inertia matrix and the estimated one of combined spacecraft, we set (10)where denotes the estimated value of the inertia tensor of the combined spacecraft, denotes the estimated error of the inertia tensor of the combined spacecraft and is assumed to be bounded. The inverse matrix of is written as (11)where . Then (9) can be rewritten as (12)where , and the lumped disturbance is defined as (13)with (14) 2.2 Dynamic constrained control allocation problem In this paper, the dynamic constrained control allocation scheme will be proposed for the combined spacecraft. The control sequence is shown in Fig. 2, where the control allocation result will act on the combined spacecraft to achieve the attitude control goal via actuators. We thus expect the difference between and to be as small as possible. Fig. 2Open in figure viewerPowerPoint Block diagram of the dynamic constrained control allocation system Let us assume that the service spacecraft has a configuration of m thrusters. The mapping relationship between the actual control input and the force vector of the thrusters can be stated as [6] (15)where is the actual output torque of the actuators and (16)with the position matrix and the orientation matrix of the thrusters before the service spacecraft captures target spacecraft and the change of position matrix of thrusters after the target spacecraft is captured. The object for the dynamic control allocation is to make the actual control input track the desired control command . This object can be expressed as (17) In this study, we are going to minimise the of the thrust manifold rather than the conventional method to minimise the . In general, since the minimises the maximum component of the thrust manifold, it allows the thrusters to run within a safer range than the where more manoeuvrability for subsequent controller action is provided [12]. Owing to the physical limitations on thruster forces such as amplitude and rate constraints, thruster force satisfies (18)where and are the upper and lower amplitudes of thruster forces, and are the upper and lower bounds of thruster force rates. Since the control allocation is part of a digital control allocation, it is reasonable to appropriate the time derivative as (19)where T is the sampling time and is the thruster force in the previous sampling instant. Then one can get the overall thruster force constraints as (20)with (21) (22) As demonstrated in [12], can always find a feasible solution as long as a solution exists within the thruster forces' limits, whereas cannot. Since minimises the maximum component of the thruster force, it allows the thrusters to run within a safer range than providing more manoeuvrability for subsequent action. To ensure the stability and smoothness during the path following process, the error between the thruster force and the previous step thruster force is weighted. Thus, the dynamic constrained control allocation problem is established and formulated as the following optimisation problem: (23)with (24)where for any vector , , are positive-defined weighting matrices. To achieve the object that the attitude tracking errors and converge to zero, a dynamic constrained control allocation scheme is developed based on the attitude tracking model of combined spacecraft (12) considering input saturation. In this study, the solution we proposed for the constrained control allocation involves two aspects (i) a disturbance-observer-based constrained backstepping control scheme is proposed to provide the desired control command of the attitude tracking system for combined spacecraft with the consideration of non-symmetric input saturation, inertia uncertainties, and external disturbances; (ii) a dynamic constrained control allocation scheme is developed to distribute the amplitude constrained control command to each thruster and achieve the desired tracking control performance with the coupling efforts of all thrusters. 3 Disturbance-observer-based constrained backstepping control law 3.1 Control law design procedure Since there exist constraints on thruster forces of the combined spacecraft, the control input must be saturated by a certain value. To provide the limited control command , a disturbance-observer-based constrained backstepping control scheme will be designed to generate the limited total control command with input saturation. The control input with constraint is defined as (25)where is the nominal control input, and are the known saturation levels of the control input. Remark 2.Although it is easy to obtain the saturation levels and by linear optimal algorithm according to the actuators boundaries , and the control mapping relationship (15), there is a problem that the upper bound cannot be reached simultaneously, the same problem to the lower bound . In order to get smaller in (24), the saturation levels and must be cut down reasonably. Obviously, there exists a difference between the desired control input u and the nominal control input , which is defined as (26)To guarantee the controllability of the studied attitude system (12), is assumed to be bounded, i.e. with a constant without loss of generality [42]. Following the backstepping procedure, firstly define the error surface as (27) (28)where is the virtual control to be designed later. In order to compensate the effect the constraint on the virtual angular velocity and control input, we define the following modified tracking errors: (29) (30)with (31) (32)where and are arbitrary positive constants, and are nominal virtual control law and control input, respectively, and are the limited virtual control signals of and , respectively. In order to obtain the limited virtual signal and the derivative of the limited virtual control signal , here, we select the second-order model shown in Fig. 3 as the command filters [34] (33)where and are state vectors of the filters, and are positive constants representing the limitation of the bandwidths and damping, respectively, and denotes (34) Remark 3.It can be seen from (29) and (31) that when the limitation on the virtual angular velocity is not in effect, the modified tracking error converges to the tracking error . Also, the effect of implementing the limited control law instead of the desired one can be estimated with (32). Fig. 3Open in figure viewerPowerPoint Filter that generates the command and command derivative while enforcing magnitude Based on the defined vectors and filters, the controller design procedure carries on with the following two steps: (1) Step 1: Take the derivative of , we have (35) Design the virtual control input as (36)Substituting (36) into (35) yields (37) Define a Lyapunov function as (38)Then, based on (37), the time derivative of is (39) (2) Step 2: Proceeding to the second equation of (12), we design the control law in this step. The derivative of is (40) Design the control input command for (12) as (41)where , and is the output of the following non-linear disturbance observer [38] (42)Substituting (41) into (40) yields (43)where the estimate error . Now consider the following Lyapunov function candidate: (44)then the derivative of along the system trajectories satisfies (45) 3.2 Stability analysis Based on the constrained backstepping control theory, the following theorem is proposed. Theorem 1.Consider the attitude tracking model of combined spacecraft (12). If the designing of controller (41) augmented by non-linear disturbance observer (42) under with and , then attitude error and angular velocity error converge to smaller neighbourhood of zero by choosing suitable controller parameters. Proof.The Lyapunov function candidate of the whole attitude control system is taken as (46)With the property , the derivative of V along the system trajectories is given by (47)with then we can have (48)where Then according to the comparison principle, we have .It is assumed that is small varying signals if is viewed as slowly varying signals with respect to the fast dynamics of disturbance observer under large gains [7], can be derived with an unknown scalar , so that lim . According to the definition of , we know that and converge to smaller neighbourhoods of zero. Furthermore, we have and . Now consider (31) and (32), without loss of generality, set and , then we have (49) (50)Also, set , it is reasonable to assume that is bounded, set as , we can derive that (51)which leads to (52)thus can be made arbitrarily small by appropriately selecting and when .As for the virtual control , we have (53)thus, the size of can also be arbitrarily small by appropriating selection of when . Since the boundary of is a fixed constant, the constraint of will no longer take effect in finite time, i.e. . Then the angular tracking error can be rewritten as (54)then we can get (55)Thus, the tracking errors and will made arbitrarily small by appropriating selecting the parameters and as . □ Remark 4.Compared to the general backstepping method, the effect of the constraints on the control input and virtual angular velocity can be filtered by (31) and (32). Also, command filter (33) is used to eliminate the analytic computation of the time derivatives of the virtual control in the constrained backstepping, and moreover there is no need to introduce other state variables compared to the dynamic surface method. Using command filters to calculate the virtual control derivatives , it is still possible to prove stability in the sense of Lyapunov in the absence of constraints on the control input and virtual control variable [33]. Remark 5.In order to compensate the unknown lumped disturbance , the disturbance observer (42) is designed in the control command design. Comparing the proposed disturbance observer with the ones in previous studies [43, 44], the proposed disturbance observer is simple and effective in control designing, while the performance of the observer can be tuned easily by only one parameter in the observer. Remark 6.The convergence rate of the system states is mainly determined by controller, saturation filter and observer parameters and , the larger and lead to larger and faster convergence rate of attitude. However, the larger will lead to larger convergence domain of according to (55), and larger will lead to larger command input according to (41). Thus those control performance must be balanced when choosing parameters and . 4 Dynamic constrained control allocation The focus in this section is to develop the dynamic constrained control allocation scheme, which enables the actual control input to track the desired control command . From (23), we know that the problem of the dynamic constrained control allocation can be posed as a convex optimisation problem. We will use the linear programming technique to solve this constrained control allocation problem. Problem (23) is equal to (56)s.t. (57)with suitable positive definite diagonal matrix Now redefining (58) (59) (60)which leads to the following linear programming problem: (61)s.t. (62) (63) (64) (65)where denotes the j th column vector of identity matrix . The linear problem of (61) can be written in the following compact matrix form: (66)s.t. (67)where (68) with and . If a thruster breaks down or partly loses its driving capacity, the control input must be reallocated among the functioning thrusters. The diagonal elements of the weighting matrix penalise the upper and lower saturation limits of thrusters such that demanded faulty thruster does not exceed the available capacity of the faulty thrusters. The diagonal weighting matrix can be defined as (69)with . Finally, the fault-tolerant dynamic constrained control allocation problem is formulated as (70)s.t. (71)with (72) (73) If , i.e. is satisfied, the optimisation problem (61) can be transformed into (74)subject to (75) (76)and (77) (78) Furthermore, (70) can be rewritten as (79)subject to (80) (81)where (82) (83) (84) Finally, the fault-tolerant dynamic constrained control allocation problem can also be formulated as (85)s.t. (86) (87)with (88) Based on (70), the dynamic constrained control allocation result is obtained for the thrusters that control the combined spacecraft to achieve attitude tracking. The control sequence of this disturbance-observer-based const