On the Selection of Optimal Bandwidths for LSS Controllers

It has been established that any linear dynamical feedback controller (complete with model errors) will not destabilize a stable physical plant if the controller gain is "small" enough. It has also been established that any linear dynamical feedback controller will always destabilize the physical plant if the gains are "large" enough. The first result follows naturally from the laws of continuity. The latter result can be explained as the consequence of "modeling errors" That is, no mathematical model (upon which the controller is based) exactly describes any physical process. Hence, there are always modeling errors, and their significance depends upon the magnitude of the controller gains. The larger gains eventually cause the controller spectrum to exceed that spectrum for which the model is credible. In general terms then, the obvious conclusion is that there is a "best" gain in the sense that a smaller gain would yield degraded performance due to failure to exploit known model structure, and a larger gain would yield degraded performance due to the effects of modeling errors. In application to Large Space Structures this means that there exists a "best" set of gains associated with a controller of specified dimension. Furthermore, these gains cannot be determined from knowlege of the reduced model alone.