Robust time-optimal control of flexible spacecraft with structured uncertainties

A technique to design robust time-optimal controllers for nonlinear flexible structures is proposed in this paper. A parameter optimization problem is solved using the Switch Time Optimization algorithm to determine a bang-bang control profile that minimizes the maneuver time subject to the constraint that the sensitivity of the final states with respect to system parameters are zero. The proposed approach is illustrated on the benchmark floating oscillator problem followed by two nonlinear examples. Simulation results illustrate the reduction of residual vibrations of the system subject to the robust control profile, compared to the time-optimal control profile, in the presence of errors in system parameters. 1.0 Introduction The problem of design of time-optimal controllers for flexible spacecraft is a topic of current interest which is reflected by numerous papers that address this problem for linear and nonlinear systems." Typical time-optimal control solutions for rest-torest and spin-up maneuvers of flexible spacecraft can be found in [2] and [3], Time-optimal controllers are very sensitive to errors in system parameters. Therefore, development of techniques for robust time-optimal control intended to reduce residual oscillation have received increased attention in recent years." Byers et al., applied a smooth bang-bang control profile to deal with the rigid body mode, in conjunction with a variable structure controller to reduce the residual oscillations. Singer and Seering introduced a preshaping technique which superposed two impulse Graduate Student, Mechanical & Aerospace Engineering | Assistant Professor, Mechanical & Aerospace Engineering, Member of AIAA. Copyright © 1996 American Institute of Aeronautics and Astronautics, Inc. All rights reserved. 1 American Institute of Aeronautics and Astronautics commands to eliminate residual vibration. Upon using a series of properly arranged impulse inputs, robustness to system parameter uncertainty can be achieved. The authors also comment that as long as the system is varying slowly, the shaping technique tends to work for nonlinear systems. Liu and Wie presented a parameter optimization approach to minimize the final maneuver time while providing robustness with respect to structural parameter uncertainty by incorporating additional constraints for a time-optimal solutioa Another technique concerning structural parameter uncertainty for timeoptimal control problem was proposed by Singh and Vadali. They placed multiple zeros of a timedelay filter to cancel the poles of system and applied a parameter optimization method to solve the minimum time problem for both rest-to-rest and spin-up maneuvers. It should be noted that the last two approaches can only be applied to linear systems. In this paper, a new technique which can generate robust solutions with respect to structural uncertainty for time-optimal control of flexible nonlinear structures is proposed. The new approach considers the sensitivity of states with respect to the structural uncertain parameter as additional states. These sensitivity states with their boundary conditions are combined with the original state equations to form a higher order system. Now, all the traditional twopoint boundary value problem solving techniques can be applied to arrive at the optimal solution. The Switching Time Optimization (STO) algorithm which was developed by Meier and Bryson will be used in this work. We first design a robust controller for a spring-mass system with stiffness uncertainty, undergoing a rest-to-rest maneuver, and compare the solution to those available in literature. The applicability of the proposed approach to nonlinear systems is illustrated on two examples. The first, is a modification of the floating oscillator which includes a hardening spring and the second is a flexible slewing structure. The statement of the tune-optimal and the desensitized time-optimal control problem is included in section 2. Section 3.1 expounds the proposed technique which can be applied to solve robust time-optimal control problems with structural uncertainty. Section 3.2 illustrates that the optimal control for the two-mass-spring system determined by the proposed technique results in a control profile that coincides with results presented in the literature.' Investigations of the two-mass-spring system that contains hard spring characteristics is discussed in Section 3.3. The final nonlinear system of interest in this study is a flexible spacecraft which contain a rigid hub and two flexible appendages. An on-off torquer located at the center of the hub provides the required control input for attitude control and vibration attenuation. A set of equation of motion containing the nonlinearities due to link foreshortening as well as frame rotation coupling between body axis and inertia! axis is derived in Section 4.1. Numerical analysis of non-robust and robust time-optimal control for the flexible spacecraft undergoing rest-to-rest maneuvers is presented in Section 4.2. The final section summarizes results of this study. 2.0 Problem Statement The problem statement for the design of robust timeoptimal controllers for linear and nonlinear structures is: Determine the control that drives the system t = f(x, u) (1) where x e 9\" and u are the system states and the control input respectively, from specified initial conditions .x0 to final conditions xf while minimizing the cost function