Real-time optimal state feedback control for tethered subsatellite system

Abstract

Introduction A LTHOUGH many are devoted to the study of control problems of the tethered subsatellite systems, only a few papers treat the control problems as optimization problems. Bainum and Kumar introduced an optimal-control law based on an application of the linear quadratic regulator (LQR) to the tethered satellite system. They apply the tension control law, employing feedback of tether length, in-plane angle, their rates, and the commanded length for the deployment and retrieval and stationkeeping of the subsatellite; the performance index for the optimization problems is set to correct deviations from the instantaneous equilibrium state at each time. No and Cochran show that the LQR can be applied to control stationkeeping and maneuvering of the tethered subsatellite by utilizing the atmospheric aerodynamics, and the performance index for the optimization problem is set to correct deviations from the reference trajectory and to converge into the desired state at the final time. Netzer and Kane obtain an optimal-length law of the tethered subsatellite system with the performance index involving penalties on the terminal values of the states as well as on the states and controllers throughout the maneuver, and the optimal solution is used as a nominal path to follow. They show that the LQR can be applied to derive tracking-type feedback control law to decrease the deviation of the subsatellite from the nominal path by using the thrusters. On the assumption that the control force is only the tether tension and that no control force or energy dissipation exists for motion perpendicular to the tether line, Fujii and Anazawa obtain an optimal path in the sense that the time integral of squared tension plus squared in-plane angle is the performance index with inequality constraints on the control force. The Lyapunov approach is applied to derive the tracking-type feedback control law to follow the optimal path in Ref. 4. It is generally difficult to obtain the optimal path by solving the two-point boundary-value problem. To overcome this difficulty, Ohtsuka and Fujii adopted the stabilized continuation method, which converts the two-point boundary-value problem into the initial-value problem. Combining this method with the receding horizon control method, they have succeeded in developing the real-time optimal state feedback controller. This Note applies this real-time optimal state feedback control to the deployment and retrieval control problem of the tethered subsatellite and to determine the effective application of the control.