| The popularity of magnetic attitude control system for earth observatorysatellite is due primarily to their evident advantages. On one hand, as a semipassiveactuator, the magnetorquer utilize the earth magnetic field which is anenvironmental force to generate control torque. The satellite could have a long(essentially unlimited) mission life since no expendables are involved. On the otherhand, the momentum bias design for a satellite obviates the requirement for a yawattitude sensor for its gryroscopic stiffness. This is a highly desirable result,significantly reducing control system cost and complexity. This paper mainlystudies the magnetic attitude control of the momentum-biased satellites.Firstly, according to the character that the earth magnetic field variesperiodically on the satellite orbit, a linear periodic model is established for themomentum-biased satellites with the consideration of gravity gradient. The systemmatrix is time-invariant while its control matrix is periodic. It could be verified thatthe roll/yaw dynamics is uniformly completely controllable, and all of theopen-loop poles are on the imaginary axis of complex plane.Then establishing design method based on preliminary Lyapunov differentialequation for the momentum-biased satellites controlled by magnetorquers. Thisdesign method can not only make the system semiglobal stability with guranteedconvergence rate in limited time, but also minimizes some index functions. Bytunning the free parameter in the feedback control law, for any fixed initialconditions, the magnitude of control can be made as small as desired. Applying tothe control of attitude, it means the control torques could be selected arbitrarilyunder the limitation which the magnetorquers can afford.To implement the controller designed by the preliminary Lyapunov differentialequation based stabilization approach, one only needs to solve a linear differentialequation. Two numerical approaches, namely, periodic generator method andmulti-shot method, are proposed. The basic idea of the periodic generator method isthat by specifically choosing some initial condition, the positive definite solutioncan be obtained by solving a differential equation with standard integration method,while the multi-shot method also rely on some discretization techniques, which ismore numerical reliable.Finally, for a giving initial condition deviated from the balance, simulationshows that the proposed control law can stabilize the system in limited time. Thesimulation also shows that the closed loop system has a strong inhibition to interference, and the inhibition effect heighten while a bigger parameter is selected.A critical value of the free parameter has been found for the same initial condition.When the free parameter is small than the critical value, the torques needed tostabilize the attitude will be no more than the maximum torque whichmagnetorquers could provide. |
