Transmission zeros and high-authority/low-authority control of flexible space structures

Flexible space structures generally have very low levels of inherent damping, which makes the problem of controlling such systems quite challenging. For instance, unmodeled modes can easily be made unstable by spillover effects. One way to avoid these difficulties is to use a low-authority control loop to simply increase structural damping, typically to 5-10% of critical, and then use a more sophisticated high-authority control loop to achieve the desired control objectives. A very important design question that then arises is: what are the best choices of locations for the dampers used to implement the low-authority controller? This question is studied here, and it is shown that it is more important to increase the damping of the zeros of the damping-augmented structure than that of the poles. A simple algorithm is then derived for determining the damper locations that make the zeros as heavily damped as possible. Finally, the operation of this algorithm is illustrated by application to a simple cantilever beam example. In this approach, a LAC loop, acting through dampers placed on the structure, is used to augment the inherent damping of the vehicle, typically raising it to 5-10% of critical. The intention is to make it easier for the main HAC loop, acting through its own actuators, to give good closed-loop system performance as measured at the HAC sensor stations. The increase in damping provided by the LAC loop also helps to prevent any unmodeled modes being destabilized by spillover from the main control loop. The high-authority controller is usually chosen to be an optimal regulator, as many FSS control prob- lems are readily expressible in an optimization form; an exam- ple is the problem of minimizing the rms surface deflection of a flexible antenna without using excessive amounts of propel- lant. The LAC design problem considered in this paper is as follows: Given an optimal regulator HAC loop with specified sensor and actuator locations and a fixed number of dampers of given strengths, where should these dampers be placed on the structure to give the best overall system performance? Central to this analysis are the results recently derived in Ref. 11 concerning the optimal root loci8 of FSS, which in turn are closely related to the generic properties19 of the transmission zeros6 of these systems. In particular, if the HAC loop is permitted to use fairly high gains (as is necessary if reasonably good performance is to be obtained), then the finite poles of the closed-loop system (FSS + LAC + HAC) will approach the zeros of (FSS + LAC). Typical LAC analyses in the past1-22 have concentrated on positioning dampers so as to maximize the damping of the poles of (FSS + LAC), with no attention being paid to the zeros of this damping-augmented system. It can therefore be seen that placing dampers in this traditional way will not necessarily improve the performance of the overall closed-loop system (FSS + LAC + HAC) signif- icantly. This question will be studied in the paper and a simple algorithm given, based on orthogonality to the HAC modal influence matrix, for determining damper locations that pro- duce as much zeros damping as possible. The poles will always also be damped by this scheme. The converse, however, would not be true: a method designed to maximize the damping of the poles may leave the zeros totally undamped. Numerical results for a simple cantilever beam will be used to illustrate these results.