Minimax Design of Prefilters for Maneuvering Flexible Structures

This paper focuses on the design of prefilters for maneuvering structures with the objective of desensitizing the controller to errors in the system model. Given information about the expected variation of the uncertain parameters, a minimax optimization problem is formulated to minimize the maximum value of the residual energy over the range of the uncertain parameter. The proposed technique is illustrated on a spring-mass-dashpot system with uncertainties in both the damping and stiffness constants and on a two-mass two-spring, two input system. INTRODUCTION Control of vibratory structures by filtering the reference input to the system has been addressed by numerous researchers [14], [7], [3], [8] etc. Singh and Singhose [2] present a tutorial related to the design of input shapers/time-delay filters and include a comprehensive list of relevant papers. Smith [14] proposed a wave cancellation technique to drive a second order system to its final position in finite time. However, this technique is sensitive to modeling errors. Singer and Seering [7] proposed a simple technique to desensitize the input shaper to modeling errors. This involved design of a sequence of impulses which forced the magnitude of the residual energy and its derivative with respect to damping or natural frequency, to zero. Singh and Vadali [8] arrived at the same results of Singer and Seering [7] by the design of a time-delay filter which cancelled the poles of the system. They also showed that by cascading the time-delay filter designed to cancel the poles of the system, the resulting filter was insensitive to errors in modeled damping and frequency. The idea of locating multiple zeros Associate Professor, Mechanical & Aerospace Engineering, Member AIAA Graduate Student, Mechanical & Aerospace Engineering of a time-delay filter at the estimated location of the poles of the system has been exploited to design robust time-optimal control [9], [5], robust fuel-time optimal control [11], fuel constrainted time-optimal control [12] etc. Liu and Singh [4] extended this idea to nonlinear systems undergoing rest-to-rest maneuvers, by requiring the sensitivity of the system states with respect to uncertain parameters be zero at the final time. Techniques to increase the range of uncertain parameters where the residual vibration is below a prespecified amount has been addressed by Singhose et al. [13]. This was referred to as the extra insensitive input shaper. Pao et al. [6] included the probability distribution of the uncertain parameters into the design process to arrive at input shapers which weighted the nominal value of the uncertain parameter the most. This paper proposes a technique to design timedelay filters which minimize the maximum magnitude of the residual vibration over the range of the uncertain parameter. Closed form expressions for the analytical gradients of the cost function and constraints have been derived by Singh [10], which can be used to expedite the convergence of optimization algorithms for the minimax time-delay filters, studied in this paper. The resulting controller will be referred to as the minimax time-delay controller. The first section will review the development of the time-delay control. This will be followed by the development of the minimax timedelay controller which is illustrated on single input numerical example. The penultimate section presents a multi-input example and final section summarizes results generated in this paper. TIME-DELAY CONTROL This section reviews the time-delay control technique. Figure 1 represents the time-delay control of a second order underdamped system. The parameters A0 and T need to be determined so that the poles of the system are cancelled by a pair of zeros of the timedelay filter. The location of the zeros of the time-delay