Ang Li, Alessandro Astolfi,, and Ming Liu

Abstract—The attitude regulation problem with bounded control for a class of satellites in the presence of large disturbances, with bounded moving average, is solved using a Lyapunov-like design. The analysis and design approaches are introduced in the case in which the underlying system is an integrator and are then applied to the satellite attitude regulation problem. The performance of the resulting closed-loop systems are studied in detail and it is shown that trajectories are ultimately bounded despite the effect of the persistent disturbance. Simulation results on a model of a small satellite subject to large, but bounded in moving average, disturbances are presented.

I. INTRODUCTION

IN the past few decades satellites have been widely developed for commercial, communication, military and scientific purposes. These include tasks which play important roles in modern society, such as long-distance signal transmission [1], [2], navigation [3], [4], Earth observation [5]and weather and climate monitoring [6]. To guarantee reliable and repeatable operations, high-accuracy attitude control algorithms have to be implemented and various control methods have been utilized. Traditional control architectures exploit PD-like controllers [7], PID controllers [8], LQRbased designs [9], [10], sliding mode controllers [11] and back-stepping-based designs [12]. Of these, PD-like, PID and LQR controllers are most widely used in applications because of their simple structure and design process.

A class of satellites, which is becoming very popular because of the modest construction and deployment costs, is that of the so-called CubeSats [13]–[18]. Unfortunately,CubeSats have a limited power budget and, because of their small dimension, are very sensitive to environmental disturbances, such as the Earth gravity gradient, aerodynamic drag, solar radiation pressure [19], [20], and to disturbances resulting from the coupling of the on-board electronics with the Earth magnetic field (often known as the residual dipole torque) [21]. In addition, the size and weight limitations impose strict constraints on the attitude control torque, which is often smaller, in magnitude, than the combined torque generated by the disturbances [22].

CubeSats attitude regulation in the presence of disturbances has been studied in [23] (in which the gravity gradient is regarded as the main external disturbance) and in [24] (in which multiple external disturbances are considered). In both papers, however, no consideration has been given to control bounds or to the fact that the instantaneous amplitude of the external disturbance may exceed the available control torque.

Such considerations are the point of departure for this paper.In particular, we study control systems which are perturbed by(additive) external disturbances, the instantaneous amplitude of which is not constrained to be smaller than the amplitude of the control signal. To illustrate this class of control problems,which to the best of our knowledge has not received attention in the control literature, we study initially the case of a scalar linear system (an integrator) with matched additive disturbance and with bounded input. For thistoyexample we design a state feedback control law which, on the basis of an integral bound on the disturbance, yields ultimately bounded trajectories and stability of a set containing the origin.

This design idea is then extended to solve the attitude control problem for a satellite subject to external torque disturbances, which satisfy an integral bound, in the presence of bounded control. We consider a saturated PD-like control structure, the gains of which are tuned as a function of the available bound on the disturbance, the available control torque, and the desired attitude accuracy. Similarly to the case of thetoyexample, we show that the trajectories of the closedloop system are ultimately bounded and a residual set around the origin, the size of which depends upon the bound on the disturbances, is stable.

The paper is organised as follows. In Section II we discuss the class of disturbances considered and we present integral bounds which are used to characterize their long term properties, or their properties in a given time window.Relations among these bounds and implications in terms of the instantaneous amplitude of the disturbance are also discussed. In Section III a scalar linear system with matched additive disturbance and bounded input is studied as a motivating example. Two state feedback controllers, designed on the basis of a given integral bound on the external disturbance, are presented and the resulting properties of the closed-loop system are discussed. By exploiting the ideas of Section III, Section IV provides a solution to the bounded input attitude regulation problem for a satellite subject to disturbances satisfying an integral bound. In Section V simulation results for the systems studied in Sections III and IV driven by randomly generated disturbances satisfying the considered integral bounds are presented. Section VI contains conclusions and outlooks.

II. THE CLASS OF ADMISSIBLE DISTURBANCES

Fig. 3. L Tp,a norms (left), and ( right), for p=1,2.

Fig. 4. L Tp,a norms (left), and( right), for p =1,2.

V. SIMULATION RESULTS

A. Attitude Regulation

Consider now the attitude regulation problem described in Section IV. Let

and

Fig. 5. Realization of the disturbance w, with T =1.

Fig. 6. Time history of , with T =1.

Fig. 7. Time histories of the states of the system (2), with the disturbance w for T =1 (top) and T =0.05 (bellow), and for various initial conditions.

Fig. 8. Realization of the disturbance w for T = 1 s.

Fig. 9. Time history owith T =1 s.

Fig. 10. The quaternion energy for different control parameters and a fixed disturbance.

Fig. 13. Time histories of the quaternion q.

Fig. 11. Mean quaternion energy for different disturbances as a function of k1 and k2.

Fig. 14. Time histories of the control input u.

Fig. 12. Time histories of the angular velocity ω.

B. COMPASS-1 Based Simulation

Fig. 15. Realization of the disturbance w, with T = 1 s.

Fig. 16. Time history of

VI. CONCLUSION

This paper has studied the attitude regulation problem for a class of satellites with bounded control input in the presence of persistent disturbances with bounded windowed norms. A Lyapunov-like analysis, which is firstly introduced in the case in which the underlying system is a disturbed integrator and then extended to the satellite attitude regulation problem, is developed. A detailed analysis of the performance of the resulting closed-loop systems is given and it is shown that the trajectories are ultimately bounded even in the presence of a persistent disturbance. Simulation results on the model of a small satellite subject to large, but bounded in moving average, disturbances are presented.

Fig. 17. Time histories of the control input u, with k1=1.2×10?5 and k2=0.4×10?5.

Fig. 18. Time histories of the quaternion q and of the angular velocity ω.

Fig. 19. Time histories of quaternion q and of the angular velocity ω.

Future work will extend the proposed analysis tools to a more general classes system.

Fig. 20. Time histories of the control input u, with k1=1.2×10?7 and k2=0.4×10?7.

Fig. 21. Time histories of the control input u, with k1=1.2×10?3 and k2=0.4×10?3.

Fig. 22. Time histories of the quaternion q and of the angular velocity ω.