A Novel Dual-Loop Disturbance Observer-Based Attitude Control for a Space-Based Observation Microsatellite

Abstract

Aiming at strong coupling and nonlinear dynamic equations for a space-based observation satellite, a sliding mode controller with feed-forward compensation is proposed in this paper. The theorem of moment of momentum is applied to formulate the exact nonlinear dynamic equations of a multibody satellite. On this basis, sliding-mode control-based, dual-loop forward-feed compensation control is used to control the attitude of the space-based observation microsatellite. By comparing with the conventional control method, simulation results demonstrate that the proposed control method has superior performance in terms of suppression from external disturbances and vibration. Better dynamic and static performance indices than the conventional control method are achieved.

1. Introduction

Space observation systems consist of two parts, that is, the space- and ground-based space observation systems. The latter is subject to atmospheric propagation jitter, astronomical refraction, ionosphere scintillation, and limits of national boundaries. Space-based observation techniques can effectively solve such problems and have the advantages of flexible maneuverability, wide monitoring ranges, long-distance observation, and so forth. Space-based observation is an important direction of future space target detection.

However, space-based observation satellites contain rotating rigid parts and flexible appendages. During the observation process, the rotation of rigid parts, elastic vibration of flexible appendages, and motion of satellite attitude influence each other, comprising a strongly coupling nonlinear system. Highly accurate and highly stable attitude control is difficult to realize, especially for micro-satellites used in space target observation, because of the small mass and moment inertias of micro-satellites, as well as the limited quantity and capacity of actuators. Therefore, some special control methods should be applied to the attitude control system of space-based observation micro-satellites.

Numerous efforts have been exerted to achieve high-accuracy satellite attitude control. Cheng et al. [1] have used the feedback linearization method to eliminate the coupling terms between the attitude angular velocity and reaction wheel angular momentum, as well as reduce the influences of the vibration of panels and turning of rigid parts on the satellite attitude. Li et al. [2] have proposed a method combining variable structure control with neural networks based on the Lyapunov stability theory, which can ensure that the attitude angular velocity exponentially converges to the given limited range in the case of uncertain disturbance. Chen et al. [3] has proposed an adaptive mixed 𝐻₂/𝐻_∞ attitude control for nonlinear spacecraft systems with unknown or uncertain inertia matrices and external disturbances. Experimental simulation results based on the Republic of China Satellite-1 spacecraft system are presented to demonstrate the effectiveness of the proposed design methods. Azadi et al. [4] have applied an adaptive-robust control scheme to control the three-axes maneuver of a flexible satellite. The controller is robust against disturbances and uncertainties, and adapted to the bounds of the unknown parameters of the system. Jan and Chiou [5] have applied sliding mode control to perform a rigid spacecraft large-angle maneuver in minimal time.

Modeling of nonlinear dynamic factors, parameter changes during movement, and external disturbances can be called disturbance terms. Disturbance observer (DOB) can suppress such disturbances. The design of observers has been investigated exhaustively in the theoretical framework and also in many related applications [6–10]. De La Sen and Luo [7] designs linear observers for a class of linear hybrid systems. Such systems are composed of continuous-time and digital substates and possess. Santhakumar [9] investigates the performance of a robust tracking control on the basis of proportional-derivative observer-based backstepping control applied on a three degrees of freedom underwater spatial manipulator. Yamashita et al. [6] have applied the disturbance observation-based PID control method to the Japanese satellite HALCA, whose mission is very long baseline interferometry observations in orbit.

However, all the above mentioned methods involve the linearization process or some simplification of the satellite dynamic model, which to some extent affects the final precision. In this paper, the nonlinear dynamic equations of multi-body satellites with flexible appendages and rigid turntables were explored based on the theorem of moment of momentum. On this basis, satellite attitude control with variable structure based on dual-loop feed-forward compensation of disturbance estimator was proposed. The mathematical simulation results showed that the proposed method is far superior to the traditional method of attitude control. The Lyapunov analysis and its treatment on stability and robustness of the proposed scheme under parameter uncertainties are not explicitly dealt with, and is left in this work for the scope of future work. The dynamic model of a space-based observation satellite derived in this paper has a high reference value for the design of attitude control systems of space-based observation satellites.

2. Dynamic Modeling of a Multi-Body Satellite

2.1. Kinematic Equation of a Space-Based Observation Satellite

The angular velocity of a camera coordinate system relevant to the satellite body frame is denoted as 𝜔^𝐵1𝐵 so that its projection under a camera coordinate system is written as 𝝎^{𝐵1𝐵𝐵1}=𝜔_𝑥1𝜔_𝑦1𝜔_𝑧1^𝑇. (2.1) As shown in Figure 1, the motion of the turntable of the camera along the z-axes is constrained relative to the satellite body. Therefore, the rotation angle turntable of the camera along the 𝑥₂- and 𝑦₂-axes in the camera coordination system is set as 𝛼 and 𝛽. The corresponding angular velocities are 𝛼̇ and 𝛽̇, respectively.

Figure 1 

Simplified structure of a space-based observation satellite.

From Euler’s theorem, the rotation of a rigid body around a fixed point can be decomposed into a number of limited rotations around such point. As shown in Figure 2, the coordinate transformation matrix 𝐓_𝐵1𝐵 from the satellite body coordinate system to the camera coordinate system with a rotation sequence of 2-3-1 is as follows: 𝐓_𝐵1𝐵𝛼,𝛽,0=𝐓_𝑥(𝛼)𝐓_𝑧(0)𝐓_𝑦𝛽=c𝛽0−s𝛽s𝛼s𝛽c𝛼s𝛼c𝛽c𝛼s𝛽−s𝛼c𝛼c𝛽. (2.2) The angular velocity of the camera 𝝎^𝐵1𝐵 is connected to the transformation matrix 𝐓_𝐵1𝐵 by the following relationship: Ω_𝐵1𝐵=−𝐓^⋅_𝐵1𝐵𝛼,𝛽,0𝐓^𝑇_𝐵1𝐵𝛼,𝛽,0, (2.3) where Ω_𝐵1𝐵 is the antisymmetric matrix of 𝝎^𝐵1𝐵.

Figure 2 

Transformation between the satellite body frame and camera frame.

Based on the above equation, the following matrix can be obtained: 𝜔_𝑥1𝜔_𝑦1𝜔_𝑧1=𝛼̇𝛽̇𝛽̇c𝛼⋅−s𝛼⋅. (2.4)

2.2. Dynamic Equations of a Space-Based Observation Satellite Attitude

As shown in Figure 3, a space-based observation satellite can be simplified into three parts: camera turntable, satellite body, and solar panels. The camera turntable and satellite body can be approximated as rigid bodies, and the solar panels as flexible appendages. The following subsections describe the attitude dynamic modeling of a space-based observation satellite.

Figure 3 

A space-based observation satellite.

2.2.1. Attitude Dynamic Modeling of a Space-Based Observation Satellite

A space-based observation satellite can be simplified into a camera turntable and a satellite body when not considering the reaction wheel and flexible appendages. In this case, the attitude dynamic equation can be derived from the theorem of moment of momentum as follows: 𝐷^𝐼𝐋_𝐵^𝐵𝐼=𝐌_𝐵, (2.5) where 𝐋_𝐵^𝐵𝐼 is the total moment of momentum, and 𝐌_𝐵 is the total outside torque the satellite experiences relative to the centroid 𝐵.

The total moment of momentum of satellite consists of two parts, which can be written as 𝐋_𝐵^𝐵𝐼=𝐋^𝐵1𝐼_𝐵+𝐋^𝐵2𝐼_𝐵, (2.6) where 𝐋^𝐵1𝐼_𝐵,𝐋^𝐵2𝐼_𝐵 can be expressed as 𝐋^𝐵1𝐼_𝐵=𝐋^𝐵1𝐼_𝐵1+𝑚_𝐵1𝐫_𝐵1𝐵×𝐷^𝐼𝐫_𝐵1𝐵=𝐈^𝐵1_𝐵1𝝎^𝐵1𝐼+𝑚_𝐵1𝐑_𝐵1𝐵𝐷^𝐼𝐫_𝐵1𝐵,𝐋^𝐵2𝐼_𝐵=𝐈^𝐵2_𝐵2𝝎^𝐵2𝐼+𝑚_𝐵2𝐑_𝐵2𝐵𝐷^𝐼𝐫_𝐵2𝐵, (2.7) where 𝐑_𝐵1𝐵 represents the antisymmetric matrix of 𝐫_𝐵1𝐵.

Equations (2.5)–(2.7) can yield 𝐷^𝐼𝐈^𝐵1_𝐵1𝝎^𝐵1𝐼+𝑚_𝐵1𝐑_𝐵1𝐵𝐷^𝐼𝐫_𝐵1𝐵+𝐈^𝐵2_𝐵2𝝎^𝐵2𝐼+𝑚_𝐵2𝐑_𝐵2𝐵𝐷^𝐼𝐫_𝐵2𝐵=𝑀_𝐵. (2.8)

Given that 𝑚_𝐵1𝐫_𝐵1𝐵+𝑚_𝐵2𝐫_𝐵2𝐵=0, (2.8) can yield 𝐷^𝐼𝐈^𝐵1_𝐵1𝝎^𝐵1𝐼+𝐷^𝐼𝐈^𝐵2_𝐵2𝝎^𝐵2𝐼+𝐷^𝐼𝑚_𝐵2𝐑_𝐵2𝐵𝐷^𝐼𝐫_𝐵2𝐵1=𝑀_𝐵. (2.9) The transforms 𝐷^𝐼(𝐈^𝐵1_𝐵1𝝎^𝐵1𝐼), 𝐷^𝐼(𝐈^𝐵2_𝐵2𝝎^𝐵2𝐼), and 𝐷^𝐼(𝑚_𝐵2𝐑_𝐵2𝐵𝐷^𝐼𝐫_𝐵2𝐵1) are discussed below.

For 𝐷^𝐼(𝐈^𝐵1_𝐵1𝝎^𝐵1𝐼), based on the equation 𝐷^𝐼(𝑠)=𝐷^𝐵1(𝑠)+𝜔^𝐵1𝐼×𝑠, we can obtain 𝐷^𝐼𝐈^𝐵1_𝐵1𝝎^𝐵1𝐼=𝐷^𝐵1𝐈^𝐵1_𝐵1𝝎^𝐵1𝐼+𝝎^𝐵1𝐼×𝐈^𝐵1_𝐵1𝝎^𝐵1𝐼=𝐈^𝐵1_𝐵1𝐷^𝐵1𝝎^𝐵1𝐼+Ω^𝐵1𝐼𝐈^𝐵1_𝐵1𝝎^𝐵1𝐼, (2.10) where Ω^𝐵1𝐼 is the antisymmetric matrix of 𝝎^𝐵1𝐼.

Given that 𝜔^𝐵1𝐼=𝜔^𝐵1𝐵+𝜔^𝐵𝐼, substitution into (2.10) yields 𝐷^𝐼𝐈^𝐵1_𝐵1𝝎^𝐵1𝐼=𝐈^𝐵1_𝐵1𝐷^𝐵1𝝎^𝐵1𝐵+𝐈^𝐵1_𝐵1𝐷^𝐵𝝎^𝐵𝐼+Ω^𝐵1𝐵𝐈^𝐵1_𝐵1+Ω^𝐵𝐼𝐈^𝐵1_𝐵1−𝐈^𝐵1_𝐵1Ω^𝐵1𝐵𝝎^𝐵𝐼+Ω^𝐵1𝐵+Ω^𝐵𝐼𝐈^𝐵1_𝐵1𝜔^𝐵1𝐵. (2.11)

For 𝐷^𝐼(𝐈^𝐵2_𝐵2𝝎^𝐵2𝐼), 𝐷^𝐼𝐈^𝐵2_𝐵2𝝎^𝐵1𝐼=𝐈^𝐵2_𝐵2𝐷^𝐵2𝝎^𝐵2𝐵+𝐈^𝐵2_𝐵2𝐷^𝐵𝝎^𝐵𝐼+Ω^𝐵2𝐵𝐈^𝐵2_𝐵2+Ω^𝐵𝐼𝐈^𝐵2_𝐵2−𝐈^𝐵2_𝐵2Ω^𝐵2𝐵𝝎^𝐵𝐼+Ω^𝐵2𝐵+Ω^𝐵𝐼𝐈^𝐵2_𝐵2𝜔^𝐵2𝐵. (2.12)

The following conversion relations exist between the angular velocities: 𝝎^𝐵2𝐼=𝝎^𝐵2𝐵+𝝎^𝐵𝐼,𝝎^𝐵2𝐵=𝟎, (2.13) where 𝝎^𝐵2𝐵=𝟎 because 𝐵₂ is fixed in the satellite body coordinate system.

Substituting (2.13) into (2.12) yields 𝐷^𝐼𝐈^𝐵2_𝐵2𝜔^𝐵2𝐼=𝐈^𝐵2_𝐵2𝐷^𝐵𝝎^𝐵𝐼+Ω^𝐵𝐼𝐈^𝐵2_𝐵2𝝎^𝐵𝐼. (2.14)

For 𝐷^𝐼(𝑚_𝐵2𝐑_𝐵2𝐵𝐷^𝐼𝐫_𝐵2𝐵1), 𝐷^𝐼𝑚_𝐵2𝐑_𝐵2𝐵𝐷^𝐼𝐫_𝐵2𝐵1=𝐷^𝐼𝑚_𝐵2𝐫_𝐵2𝐵×𝐷^𝐼𝐫_𝐵2𝐵1=𝑚_𝐵2𝐷^𝐼𝐫_𝐵2𝐵×𝐷^𝐼𝐫_𝐵2𝐵1+𝑚_𝐵2𝐑_𝐵2𝐵𝐷^𝐼𝐷^𝐼𝐫_𝐵2𝐵1, (2.15) where 𝐷^𝐼𝐫_𝐵2𝐵×𝐷^𝐼𝐫_𝐵2𝐵𝐷=0,_𝐼𝐫_𝐵2𝐵1=Ω^𝐵𝐼𝐫_𝐵2𝐵1+𝐷^𝐵𝐫_𝐵2𝐵1, (2.16)𝐷^𝐵𝐫_𝐵2𝐵1=𝟎 because the position vector 𝐫_𝐵2𝐵1 remains unchanged in the body coordinate system.

Substituting (2.16) into (2.15) yields 𝐷^𝐼𝑚_𝐵2𝐑_𝐵2𝐵𝐷^𝐼𝐫_𝐵1𝐵=𝑚_𝐵2𝐑_𝐵2𝐵𝐷^𝐼Ω^𝐵𝐼𝐫_𝐵2𝐵1. (2.17)

Based on 𝐷^𝐼(𝑠)=𝐷^𝐵1(𝑠)+𝜔^𝐵1𝐼×𝑠, (2.17) becomes 𝑚_𝐵2𝐑_𝐵2𝐵𝐷^𝐼Ω^𝐵𝐼𝐫_𝐵2𝐵1=𝑚_𝐵2𝐑_𝐵2𝐵Ω^𝐵𝐼Ω^𝐵𝐼𝐫_𝐵2𝐵1+𝑚_𝐵2𝐑_𝐵2𝐵𝐷^𝐵Ω^𝐵𝐼𝐫_𝐵2𝐵1. (2.18)

Given that 𝐷^𝐵Ω^𝐵𝐼𝐫_𝐵2𝐵1=𝐷^𝐵𝝎^𝐵𝐼×𝐫_𝐵2𝐵1=−𝐷^𝐵𝐫_𝐵2𝐵1×𝝎^𝐵𝐼=−𝐑_𝐵2𝐵1𝐷^𝐵𝝎^𝐵𝐼, (2.19) then 𝐷^𝐼𝑚_𝐵2𝐑_𝐵2𝐵𝐷^𝐼𝐫_𝐵1𝐵=𝑚_𝐵2𝐑_𝐵2𝐵Ω^𝐵𝐼Ω^𝐵𝐼𝐫_𝐵2𝐵1−𝑚_𝐵2𝐑_𝐵2𝐵𝐑_𝐵2𝐵1𝐷^𝐵𝝎^𝐵𝐼. (2.20) Based on (2.9), (2.11), (2.14), and (2.20), the attitude dynamic equation of a space-based observation satellite is as follows: 𝐈^𝐵1_𝐵1𝐷^𝐵1𝝎^𝐵1𝐵+𝐈^𝐵1_𝐵1𝐷^𝐵𝝎^𝐵𝐼+Ω^𝐵1𝐵𝐈^𝐵1_𝐵1+Ω^𝐵𝐼𝐈^𝐵1_𝐵1−𝐈^𝐵1_𝐵1Ω^𝐵1𝐵𝝎^𝐵𝐼+Ω^𝐵1𝐵+Ω^𝐵𝐼𝐈^𝐵1_𝐵1𝝎^𝐵1𝐵+𝐈^𝐵2_𝐵2𝐷^𝐵𝝎^𝐵𝐼+Ω^𝐵𝐼𝐈^𝐵2_𝐵2𝝎^𝐵𝐼+𝑚_𝐵2𝐑_𝐵2𝐵Ω^𝐵𝐼Ω^𝐵𝐼𝐫_𝐵2𝐵1−𝑚_𝐵2𝐑_𝐵2𝐵𝐑_𝐵2𝐵1𝐷^𝐵𝝎^𝐵𝐼=𝐌_𝐵, (2.21) where 𝐈^𝐵1_𝐵1^𝐵=𝐓_𝐵𝐵1𝐈^𝐵1_𝐵1^𝐵1𝐓_𝐵𝐵1^𝑇,Ω^{𝐵1𝐵𝐵}=𝐓_𝐵𝐵1Ω^{𝐵1𝐵𝐵1}𝐓_𝐵𝐵1^𝑇,𝝎^{𝐵1𝐵𝐵}=𝐓_𝐵𝐵1𝝎^{𝐵1𝐵𝐵1}. (2.22) Considering the orthogonality of [𝐓_𝐵𝐵1], the dynamic equation of attitude in the body coordinate system is as follows: 𝐓_𝐵𝐵1𝐈^𝐵1_𝐵1^𝐵1𝐷^𝐵1𝝎^{𝐵1𝐵𝐵1}+𝐓_𝐵𝐵1𝐈^𝐵1_𝐵1^𝐵1𝐓_𝐵𝐵1^𝑇𝐷^𝐵𝝎^𝐵𝐼+𝐓_𝐵𝐵1Ω^{𝐵1𝐵𝐵1}𝐈^𝐵1_𝐵1^𝐵1𝐓_𝐵𝐵1^𝑇+Ω^𝐵𝐼𝐓_𝐵𝐵1𝐈^𝐵1_𝐵1^𝐵1𝐓_𝐵𝐵1^𝑇−𝐓_𝐵𝐵1𝐈^𝐵1_𝐵1^𝐵1Ω^{𝐵1𝐵𝐵1}𝐓_𝐵𝐵1^𝑇𝝎^𝐵𝐼+𝐓_𝐵𝐵1Ω^{𝐵1𝐵𝐵1}𝐓_𝐵𝐵1^𝑇+Ω^𝐵𝐼𝐵𝐓_𝐵𝐵1𝐈^𝐵1_𝐵1^𝐵1𝝎^{𝐵1𝐵𝐵1}+𝐈^𝐵2_𝐵2𝐷^𝐵𝝎^𝐵𝐼+Ω^𝐵𝐼𝐈^𝐵2_𝐵2𝝎^𝐵𝐼+𝑚_𝐵2𝐑_𝐵2𝐵Ω^𝐵𝐼Ω^𝐵𝐼𝐫_𝐵2𝐵1−𝑚_𝐵2𝐑_𝐵2𝐵𝐑_𝐵2𝐵1𝐷^𝐵𝝎^𝐵𝐼=𝐌_𝐵. (2.23) Assuming the disturbance torque introduced by the camera turntable is 𝐌_rotate, then 𝐌_rotate=𝐓_𝐵𝐵1𝐈^𝐵1_𝐵1^𝐵1𝐷^𝐵1𝝎^{𝐵1𝐵𝐵1}+𝐓_𝐵𝐵1𝐈^𝐵1_𝐵1^𝐵1𝐓_𝐵𝐵1^𝑇𝐷^𝐵𝝎^𝐵𝐼+𝐓_𝐵𝐵1Ω^{𝐵1𝐵𝐵1}𝐈^𝐵1_𝐵1^𝐵1𝐓_𝐵𝐵1^𝑇+Ω^𝐵𝐼𝐓_𝐵𝐵1𝐈^𝐵1_𝐵1^𝐵1𝐓_𝐵𝐵1^𝑇−𝐓_𝐵𝐵1𝐈^𝐵1_𝐵1^𝐵1Ω^{𝐵1𝐵𝐵1}𝐓_𝐵𝐵1^𝑇𝝎^𝐵𝐼+𝐓_𝐵𝐵1Ω^{𝐵1𝐵𝐵1}𝐓_𝐵𝐵1^𝑇+Ω^𝐵𝐼𝐵𝐓_𝐵𝐵1𝐈^𝐵1_𝐵1^𝐵1𝝎^{𝐵1𝐵𝐵1}. (2.24)

The satellite attitude dynamic equation can then be simplified as 𝐈^𝐵2_𝐵2𝐷^𝐵𝝎^𝐵𝐼+Ω^𝐵𝐼𝐈^𝐵2_𝐵2𝝎^𝐵𝐼+𝐌_rotate=𝐌_𝐵. (2.25)

2.2.2. Dynamic Model of a Space-Based Observation Satellite with Reaction Wheels

The reaction wheel, as the actuator of a satellite, is used to adjust the satellite attitude. When equipped with reaction wheels, the total moment of momentum of a satellite becomes 𝐷^𝐼𝐋_𝐵^𝐵𝐼=𝐷^𝐼𝐋^𝐵1𝐼_𝐵+𝐋^𝐵2𝐼_𝐵+𝐋_𝐵^𝑊𝐵=𝐷^𝐼𝐋^𝐵1𝐼_𝐵+𝐋^𝐵2𝐼_𝐵+𝐷^𝐵𝐋_𝐵^𝑊𝐵+Ω^𝐵𝐼𝐋_𝐵^𝑊𝐵. (2.26) From (2.25) and (2.26), the attitude dynamic equation of a space-based observation satellite with reaction wheels is as follows: 𝐈^𝐵2_𝐵2𝐷^𝐵𝝎^𝐵𝐼+Ω^𝐵𝐼𝐈^𝐵2_𝐵2𝝎^𝐵𝐼+Ω^𝐵𝐼𝐋_𝐵^𝑊𝐵+𝐌_rotate=𝐌_𝐵+𝐌_𝑓, (2.27) where 𝐌_𝑓=−𝐷^𝐵𝐋_𝐵^𝑊𝐵 is the control torque produced by a reaction wheel.

2.2.3. Dynamic Model of a Space-Based Observation Satellite with Flexible Appendages

The motion equation of solar panels is [6] 𝐪̈+2𝝃𝐪̇+Λ²𝐪+𝐂^𝑇𝝎̇=0, (2.28) where 𝐪 is the flexible modal coordinates, 𝐂^𝑇 is the coupling matrix between the solar panel and satellite body, 𝝃 is the structural damping of the solar panel, and Λ is the vibration frequency of the solar panel.

Then, the attitude dynamic model of the satellite with a reaction wheel and flexible appendages is 𝐈^𝐵2_𝐵2𝐷^𝐵𝝎^𝐵𝐼+Ω^𝐵𝐼𝐈^𝐵2_𝐵2𝝎^𝐵𝐼+Ω^𝐵𝐼𝐋_𝐵^𝑊𝐵+𝐂𝐪̈+M_rotate=𝐌_𝐵+𝐌_𝑓. (2.29)

3. Dual-Loop Compensation-Based High-Precision Attitude Control

A matter of primary concern in designing high-accuracy attitude control is how to eliminate the disturbance introduced by rotation of rigid parts and flexible solar panels. [6] have applied the attitude control system as shown in Figure 4(a) to the Japanese satellite HALCA for the very long baseline interferometry observations in orbit. F(s) is a third-order low-pass filter, called the Chebyshev-type filter. Therefore, 𝐹(𝑠)=10.686𝑠³+0.736𝑠²,+2.061𝜏𝑠+1 (3.1) where the cut-off frequency of F(s) is 0.286 Hz.

(a)

(b)

Figure 4 

Structure of the attitude control system. (a) The conventional method used in Japanese HALCA satellite. (b) The method proposed in this paper.

In this paper, a novel dual-loop DOB is proposed as shown in Figure 4(b), where the DOB includes two parts: loop 1 is used to compensate for the disturbance introduced by the motion of the turntable, while loop 2 is used to compensate for the disturbance made by solar panels. The numerical simulation results show that the proposed method is far superior to the conventional method.

3.1. Design of Loop 1

Based on (2.24), the estimation of disturbance introduced by the turntable is as follows: 𝐌_rotate=𝐓_𝐵𝐵1𝐈^𝐵1_𝐵1^𝐵1𝐷^𝐵1𝝎^{𝐵1𝐵𝐵1}+𝐓_𝐵𝐵1𝐈^𝐵1_𝐵1^𝐵1𝐓_𝐵𝐵1^𝑇𝐷^𝐵𝝎^𝐵𝐼+Ω^𝐵𝐼𝐓_𝐵𝐵1𝐈^𝐵1_𝐵1^𝐵1𝐓_𝐵𝐵1^𝑇𝝎^𝐵𝐼+𝐓_𝐵𝐵1Ω^{𝐵1𝐵𝐵1}𝐈^𝐵1_𝐵1^𝐵1𝝎^{𝐵1𝐵𝐵1}+𝑚_𝐵2𝐑_𝐵2𝐵Ω^𝐵𝐼Ω^𝐵𝐼𝐫_𝐵2𝐵1−𝑚_𝐵2𝐑_𝐵2𝐵𝐑_𝐵2𝐵1𝐷^𝐵𝝎^𝐵𝐼+𝐓^𝐵𝐵1Ω^{𝐵1𝐵𝐵1}𝐈^𝐵1_𝐵1^𝐵1𝐓^{𝐵𝐵1𝑇}−𝐓^𝐵𝐵1𝐈^𝐵1_𝐵1^𝐵1Ω^{𝐵1𝐵𝐵1}𝐓^{𝐵𝐵1𝑇}𝝎^𝐵𝐼+Ω^𝐵𝐼𝐵𝐓^𝐵𝐵1𝐈^𝐵1_𝐵1^𝐵1𝝎^{𝐵1𝐵𝐵1},𝐌_rotate=𝑀_𝐵𝑥𝑀_𝐵𝑦𝑀_𝐵𝑧. (3.2) Then the estimation of disturbance in x-axis is 𝑀_𝐵𝑥=𝐼_𝐵1𝑥𝜔̇_𝑥1c𝛽+𝜔̇_𝑦1s𝛼s𝛽+𝜔̇_𝑧1c𝛼s𝛽+𝜔̇_𝑥+𝜔_𝑦𝜔_𝑧1c𝛼c𝛽−𝜔_𝑦𝜔_𝑥1s𝛽+𝜔_𝑦𝜔_𝑦1s𝛼c𝛽−𝜔_𝑧𝜔_𝑦1c𝛼+𝜔_𝑧𝜔_𝑧1s𝛼+𝐼_𝐵1𝑦−𝜔_𝑦𝜔_𝑧1c𝛼c𝛽+𝜔_𝑦𝜔_𝑥1s𝛽−𝜔_𝑦𝜔_𝑦1s𝛼c𝛽−𝜔_𝑦𝜔_𝑧−𝜔_𝑦1𝜔_𝑧1c²𝛼c𝛽+𝜔_𝑥1𝜔_𝑦1c𝛼s𝛽−𝜔²_𝑦1s𝛼c𝛼c𝛽+𝜔²_𝑧1s𝛼c𝛼c𝛽−𝜔_𝑥1𝜔_𝑧1s𝛼s𝛽+𝜔_𝑦1𝜔_𝑧s²𝛼c𝛽−𝜔_𝑧𝜔_𝑦1c𝛼+𝜔_𝑧𝜔_𝑧1s𝛼+𝐼_𝐵1𝑧𝜔_𝑧𝜔_𝑦1c𝛼−𝜔_𝑧𝜔_𝑧1s𝛼+𝜔_𝑦𝜔_𝑧−𝜔_𝑥1𝜔_𝑦1c𝛼s𝛽+𝜔_𝑥1𝜔_𝑧1s𝛼s𝛽+𝜔²_𝑦1s𝛼c𝛼c𝛽−𝜔_𝑦1𝜔_𝑧1s²𝛼c𝛽+𝜔_𝑦1𝜔_𝑧1c²𝛼c𝛽−𝜔²_𝑧1s𝛼c𝛼c𝛽−𝜔_𝑦𝜔_𝑥1s𝛽+𝜔_𝑦𝜔_𝑦1s𝛼c𝛽+𝜔_𝑦𝜔_𝑧1c𝛼c𝛽+𝐼_𝐵2𝑥𝜔̇_𝑥−𝜔_𝑦𝜔_𝑧𝐼_𝐵2𝑦+𝜔_𝑦𝜔_𝑧𝐼_𝐵2𝑧, (3.3) where 𝛼, 𝛽, 𝜔_𝑥, 𝜔_𝑦, and 𝜔_𝑧 can be measured by sensors on the satellite. 𝜔_𝑥1, 𝜔_𝑦1, and 𝜔_𝑧1 can be yielded by (2.4).

The model of the reaction wheel is described as 𝐴(𝑠)=𝐾_𝑚𝑇_𝑚𝑠+1𝑠. (3.4)

Therefore, Z(s) is given by 𝑍(𝑠)=𝑇_𝑚𝑠+1𝐾_𝑚𝑠. (3.5)

The compensation of loop1 in x-axis is then described as 𝑈₂=𝑍(𝑠)𝑀_𝐵𝑥. (3.6)

3.2. Design of Loop 2

A description of the basic structure of disturbance observer is shown in Figure 5. The system output can be expressed in terms of the reference control input, the external disturbance, and the measurement noise. Thus, the behavior of the actual system is to be the same as the given nominal model.

Figure 5 

Block diagram of the disturbance observer.

From Figure 5 we can find 𝑢(𝑠)=𝑐(𝑠)−𝑑(𝑠)+𝑑(𝑠), (3.7) where 𝑐(𝑠) is the controller output and 𝑑(𝑠) is the disturbance estimation.

And we can also find 𝐺_𝐶𝑌(𝑠)=𝑌(𝑠)𝑈(𝑠)=𝐺_𝑝(𝑠)𝐺_𝑛(𝑠)𝐺_𝑛(𝑠)+𝑄(𝑠)𝐺_𝑝(𝑠)−𝐺_𝑛(𝑠),𝐺_𝐷𝑌(𝑠)=𝑌(𝑠)𝐷(𝑠)=𝐺_𝑝(𝑠)𝐺_𝑛(𝑠)(1−𝑄(𝑠))𝐺_𝑛(𝑠)+𝑄(𝑠)𝐺_𝑝(𝑠)−𝐺_𝑛(𝑠),𝐺_𝑁𝑌(𝑠)=𝑌(𝑠)𝑁(𝑠)=𝐺_𝑝(𝑠)𝑄(𝑠)𝐺_𝑛(𝑠)+𝑄(𝑠)𝐺_𝑝(𝑠)−𝐺_𝑛(𝑠). (3.8)

Assuming 𝑓_𝑞 is the cut-off frequency of low-pass filter 𝑄(𝑠) and f is the signal frequency. When 𝑓≤𝑓_𝑞, 𝑄≈1, 𝐺_𝐶𝑌(𝑠)≈𝐺_𝑛(𝑠), 𝐺_𝐷𝑌≈0, 𝐺_𝑁𝑌≈1.

This shows that in low-frequency range the disturbance observer could make system characteristics approximately equal to the nominal model characteristics, and the disturbance observer can overcome the various disturbances effectively.

In this section, the nonlinear terms and small quantities in the satellite dynamics are treated as the external disturbances so that linear controller design tools (Laplace transform, disturbance observer) can be utilized. We take the x-axes as examples to design loop 2. Taking the Ω^𝐵𝐼𝐋_𝐵^𝑊𝐵,𝐂𝐪̈,𝐌_rotate in the satellite dynamics equation (2.29) as external disturbances, then the x-axis nominal modal could be written as 𝐺_𝑛(𝑠)=1𝐼_𝑥𝜔_𝑥𝑠+𝐼_𝑧−𝐼_𝑦𝜔_𝑧𝜔_𝑦. (3.9)

Then the inverse model is 𝐺_𝑛⁻¹(𝑠)=𝐼_𝑥𝜔_𝑥𝑠+𝐼_𝑧−𝐼_𝑦𝜔_𝑧𝜔_𝑦. (3.10)

The 𝑄(𝑠) is a key part in loop 2, which has the form [11]: 𝑄(𝑠)=1+^𝑁−𝑟_𝑘=1𝑎_𝑘(𝜏𝑠)^𝑘1+^𝑁_𝑘=1𝑎_𝑘(𝜏𝑠)^𝑘−1, (3.11) where 𝑁 is the order of 𝑄(𝑠), 𝜏 is the filter time constant, and 𝑟 is the relative degree of 𝑄(𝑠).

The main issue in the design of 𝑄(𝑠) is the tradeoff between making |𝑄(𝑗𝑤)| small and |1−𝑄(𝑗𝑤)| small [12]. One of the typical solutions is to choose 𝑄(𝑠) so that the right-hand-side slope of |𝑄(𝑗𝑤)| is the same as the left-hand-side slope of |1−𝑄(𝑗𝑤)|. In the frequency domain, the slope of |𝑄(𝑗𝑤)| in low frequency and the slope of |1−𝑄(𝑗𝑤)| in high frequency are, respectively, approximated as [12] Lm1−𝑄𝑗𝑤(𝑤)𝑄Lm≈𝑁−𝑟+1,Lm𝑗𝑤Lm(𝑤)≈−𝑟, (3.12) where Lm is log magnitude. Since the optimal 𝑄(𝑠) is obtained by letting the slopes be equivalent, the following relation is achieved: ()𝑟=𝑁+12. (3.13)

It is notable that the 𝑄₃₁-filter is obtained as the minimum-order 𝑄-filter for the second-order system [12] 𝑄₃₁(𝑠)=3𝜏𝑠+1𝑠³+3𝜏𝑠²,+3𝜏𝑠+1 (3.14) where 𝜏 is the time constant.

The compensation of loop 2 as shown in Figure 4(b) in x-axis is then described as 𝑈₃=𝑍(𝑠)𝑄(𝑠)𝑈_𝑀−𝐼_𝑥𝜔_𝑥𝑠−𝐼_𝑧−𝐼_𝑦𝜔_𝑦𝜔_𝑧. (3.15)

Other axes employ the same compensation structure as (3.15).

3.3. Design of the External Loop

There are many uncertain factors in the parameters and structure of an observation satellite. For example, the moment of inertia 𝐽 of the system is not quite accurate, and the space environmental disturbance torques, such as gravity gradient torque, aerodynamic torque and, solar radiation torque, are uncertainties. For a satellite in 1000 km orbit, a typical environmental disturbance torque equation is written as 𝑀_𝑥=10⁻⁵3cos𝜔₀,𝑀𝑡+1_𝑦=10⁻⁵1.5sin𝜔₀𝑡+3cos𝜔₀𝑡,𝑀_𝑧=10⁻⁵3sin𝜔₀𝑡+1. (3.16)

Consequently, the control scheme must robustly restrain all uncertainties, and the property of the actuators must be considered. The control law should be suitable for limited and discontinuous control torque and gas consumption.

The SMC method has some advantages, such as robustness to parameter uncertainty, insensitivity to bounded disturbances, fast dynamic response, a remarkable computational simplicity with respect to other robust control approaches, and easy implementation of the controller.

Variable structure control (VSC) is recognized as a powerful theoretical technique for robust control, even under variations in system parameters or in the presence of external disturbances. A well-known property of a VSC system is that its sliding modes are invariant with respect to a class of disturbances and parameter perturbations. If the disturbances and perturbations are bounded, the so-called reaching process is also robust against them with the controller appropriately designed. Thus, in this paper, VSC was employed as the external loop controller, which is written as (𝑠𝑠|𝐔=−𝑇⋅sgn)|>𝛿₁−𝑇_𝑓⋅sgn(𝑠)𝛿<|𝑠|<𝛿₁−𝑇_𝑓⋅𝑠𝛿|𝑠|̇<𝛿,𝑠=𝑐𝑒+𝑒, (3.17) where 𝑠 is the sliding mode, 𝛿₁ is a bound value for adjusting the thrusters and wheels, 𝛿 is the cutting gene to avoid vibrations of the VSC on the switching surface, 𝑇_𝑝 is the thruster maximal output torque, 𝑇_𝑓 is the wheel maximal output torque, 𝑒 is the angle error (Figure 6), and 𝑒̇ is the angular velocity error.

Figure 6 

Track of the system in the phase plane.

Considering the effects produced by the motion of the two-dimensional turntable and elastic vibration of flexible panels, the disturbance estimation can be used as the forward to compensate for the disturbance from the camera turntable.

Then, the final control law is rewritten as follows: (𝑠𝑠|𝐔=−𝑇⋅sgn)|>𝛿₁−𝑇_𝑓⋅sgn(𝑠)𝛿<|𝑠|<𝛿₁−𝑇_𝑓⋅𝑠𝛿+𝐔_𝟐+𝐔_𝟑|𝑠|<𝛿, (3.18) where 𝐔₂, 𝐔₃ are the compensations of disturbance by the DOB.

Filter 𝐹_𝐻(s) is to prevent excitation of the ignored higher flexible-mode’s vibration, where we chose a common filter structure as 𝐹_𝐻=𝜔_ℎ𝑠+𝜔_ℎ, (3.19) where 𝜔_ℎ is the cut-off frequency. It is preferred that the cut-off frequency 𝜔_ℎ of 𝐹_𝐻 is designed equal to or lower than the bandwidth of the 𝑄(𝑠). In this paper, the cut-off frequency of 𝑄(𝑠) is 0.6, so 𝜔_ℎ is set as 0.6.

4. Numerical Simulation and Results

In this section, the results and analysis from the algorithm developed to control the space-based observation micro-satellite are presented. For some space observation tasks, the objective of the control system is to keep the satellite attitude angle error < 0.02° and the angular velocity error < 0.001°/s.

4.1. Initial Assumptions

See Tables 1 and 2.

Table 1 

Table 2 

Satellite parameters.

4.2. Results

See Figures 7, 8, and 9, and Table 3.

Table 3 

Comparison of the results of the two cases.

(a)

(b)

(c)

Figure 7 

Angle error results: (a) x-, (b) y-, and (c) z-axes. The red line represents proposed method. The blue line represents conventional method.

(a)

(b)

(c)

Figure 8 

Angular velocity error results: (a) x-, (b) y-, and (c) z-axes. The red line represents proposed method. The blue line represents conventional method.

(a)

(b)

(c)

Figure 9 

Estimation of the disturbance torques.

4.3. Conclusion

From Figures 7–9 and Table 3, we can conclude that by the dual-loop DOB, the control method proposed in this paper can significantly reduce the errors of attitude angle and angular velocity (as shown in Table 3), and the proposed control scheme can meet the requirements of the attitude control task. In addition, the derived-dynamic model of a space-based observation satellite has a high reference value to the design of attitude control systems for space-based observation satellites.

Nomenclature

𝐵∶	Centroid of satellite
𝐵1:	Centroid of payload
𝐵2:	Centroid of platform
𝑚𝐵1:	Mass of payload
𝑚𝐵2:	Mass of platform
𝐫𝐵2𝐵:	Distance vector from 𝐵 to 𝐵2, 𝐑𝐵2𝐵 is its antisymmetric matrix
𝐫𝐵1𝐵:	Distance vector from 𝐵to𝐵1,𝐑𝐵1𝐵 is its antisymmetric matrix
𝝎𝐵𝐼:	Angular velocity vector of the satellite body frame relative to the inertial frame
Ω𝐵𝐼:	Is its antisymmetric matrix
𝝎𝐵1𝐵:	Angular velocity vector of the camera frame relative to the satellite body frame
Ω𝐵1𝐵:	Is its antisymmetric matrix
𝐷𝐼=(𝑑/𝑑𝑡)|𝐼:	Differential in the inertial frame
𝐷𝐵=(𝑑/𝑑𝑡)|𝐵:	Differential in the satellite body frame
𝐷𝐵1=(𝑑/𝑑𝑡)|𝐵1:	Differential in the camera frame
𝐈𝐶𝐶𝐼:	Inertia matrix of rigid body𝐶relative to point𝐶
𝐋𝐵1𝐼𝐵:	Moment of momentum of the camera turntable relative to𝐵in the inertial frame
𝐋𝐵2𝐼𝐵:	Moment of momentum of the platform relativeto𝐵in the inertial frame
𝐋𝐵𝑊𝐵:	Moment of momentum of the reaction wheel
𝐓𝐵𝐼:	Transformation matrix from the inertial frame to the satellite body frame
𝐓𝐵1𝐵:	Transformation matrix from the satellite body frame to the payload frame
s𝛼:	sin 𝛼
c𝛼∶	cos 𝛼
s𝛽∶	sin 𝛽
c𝛽∶	cos 𝛽.

Acknowledgments

This paper was funded by the National Natural Science Foundation of China no. 60904089 and the Fundamental Research Funds for the Central Universities of China.