In this dissertation, the three-axis attitude stabilization for inertial or nadir pointing using only magnetic torquers has been investigated. Unlike a reaction wheel and control momentum gyro, which have high speed rotating parts, the magnetic torquers have been used to control the attitude of small satellites in low earth orbits (LEOs) when precise attitude control is not needed due to their simplicity, reliability, low cost, and light weight. The control torques are generated by the interaction (cross product) between the earth's magnetic field and the three orthogonal current-driven coils according to the calculated control input. Since the control torque of the magnetic torquers generated by the cross product is structurally singular, it is impossible to produce three independent control torques at each time instant. Thus, a magnetically actuated satellite constitutes a remarkable example of an underactuated system. Moreover, it is considered to be a more difficult problem compared to conventional control problem due to the earth's time-varying magnetic field. Until now, the attitude control problem using only magnetic torques has been limited to the nadir pointing of the circular orbit satellites. Therefore, in this paper, the attitude control in elliptic orbit for inertial and nadir pointing and in circular orbit for inertial pointing has been studied.
First of all, a linear time-varying model predictive control approach is applied to magnetically actuated satellites in elliptic low earth orbits for nadir and inertial pointing. The state-space model based on small angle approximations for both nadir- and inertial-pointing nonlinear dynamics models with linearized control torque and gravity gradient torque equations is obtained, and it is confirmed that the disturbing pitch torque acts as a disturbance in state-space model for nadir pointing. The state-space model is transformed into an augmented state-space model to apply a linear time-varying model predictive control, and the constraint on the magnitude of the control input is formulated during the control horizon to consider the saturation of the actuator. The Laguerre function is one of a set of discrete-time orthonormal basis functions, and the MPC formulated using Laguerre functions can be optimized for the cost function in real-time on the condition of linear inequality constraints. Simulation results of state-space model for nadir pointing in elliptic orbit show that the attitude stabilization is achieved without steady state error in the presence of the disturbances. Furthermore, exponential data weighting in the model predictive control design for inertial pointing is proposed to improve the numerically ill-conditioned problem and closed-loop stability. Nonlinear simulation results are used to demonstrate the effectiveness of the proposed methods.
Moreover, within the MPC scheme, the linearization control torque equation derived when ignoring the high-order terms uses the local magnetic field in the orbit frame for nadir pointing or in the inertial frame for inertial pointing. However, one problem is that the control torque derived from existing linearization methods cannot take into account the attitude of the satellite. Therefore, in this study, an improved linearization technique for the control torque including the first derivative terms of a Taylor expansion series is proposed to consider the satellite attitude at each time step. Numerical simulation results demonstrate the effectiveness of the proposed LTV-MPC formulation.
Up to now, the magnetically actuated satellite attitude control for inertial pointing was conducted by limited research using proportional derivative (PD) controller in circular orbit, and it was carried out by concentrating on the proof of stability by assuming no disturbance conditions. However, a satellite system in real-world situations is an uncertain multiple-input multiple-output (MIMO) nonlinear system with uncertainties and external disturbances. So, disturbances are an important factor to be considered in the problem of attitude control of a satellite. There are two main weaknesses in PD control: (1) the controller requires measurements of position and velocity. (2) the PD control cannot guarantee that the steady-state error becomes zero due to the existence of disturbances and uncertainties. The angular velocity is usually measured by rate gyros, which is expensive and often contaminated by noise. So, it is important to achieve stability without attitude rate measurements from a practical point of view. To improve the problems of existing PD controller, output feedback is used for inertial-pointing satellite control without measuring angular velocity. Moreover, the disturbance observer is used to estimate unknown disturbances in real-time, and the estimated disturbance can be compensated in the form of a feedforward control. The numerical simulation results demonstrate that the proposed new control approach allows faster convergence of the closed-loop system to the desired equilibrium and better control efficiency under the lumped uncertainty.