Numerical Methods For Periodic Time-Varying Control And Its Application In Satellites Formation Keeping

One of the main issues of satellites formation flying is the technique of satellites formation keeping control. The formation keeping control is necessary to suppress disturbance or to decrease its influence to satellites formation configuration, otherwise perturbative force would cause satellites to drift off objective orbit in the progress of formation flying.The satellites formation keeping control law designed according to C-W equation, which is a time-invariant system model and also called Hill equation, can not meet the requirement of high orbit and large eccentricity formation cases. The limitation of circle reference orbit relative dynamics model is now noticed and given up by researchers. Recently formation keeping control law based on Lawden equation, which is a time-varying system model, has been accepted by researchers gradually. But, general time-varying system controller without using the time-varying periodic characters of the Lawden equation may still have some problems in the implementation of real satellites formation keeping control. In this thesis, time-varying periodic control law which based on the numerical solution of periodic Riccati differential equation, periodic H_∞-Riccati differential equation, and periodic Lyapunov differential equation is designed using the periodic property of Lawden equation.Firstly, based on the property of periodic system, a Fourier series expanding method combined with precise integration method for solving periodic Riccati differential equation is proposed in this thesis. Comparing with the fourth-order Runge-Kutta method and symplectic conservative perturbation method, this method needs less computation work and has higher precision even in the case of large integration time step. And then, numerical methods for solving periodic Lyapunov differential equation are proposed in this thesis, which involves homogeneous Riccati equation method and dimension expanding method. Homogeneous Riccati equation method is better than dimension expanding method in precision and efficiency, while the dimension expanding method can substitute the homogeneous Riccati method when the solution of Riccati equation is singular. The proposed numerical methods for periodic Riccati differential equation and periodic Lyapunov differential equation are core technique in designing periodic time-varying control systems, which can be expanded to solve satellite formation keeping control problems also. For satellites formation keeping control problems, the optimal periodic control method and periodic H_∞robust control method only need one period feedback gain matrix, which decrease the computation cost in the control system design and implementation. In long time satellite formation keeping progress, the control output can be generated by invoking the stored periodic gain matrices repeatedly. Therefore, controller design with periodic characters can decrease computation and storage cost. Furthermore, periodic control methods provide more reasonable controller structure essentially.