Fundamental approach to equivalent systems analysis

Abstract

CRITICAL review is presented, and an alternate extended procedure is offered, for the equivalent system technique1 for evaluating aircraft handling qualities. In the equivalent systems approach, a numerical search algorithm has been employed to find the reduced-order model, of classical aircraft form, such that the frequency response of the higher-order system (aircraft) is well approximated over a specified frequency range. Questions have been raised, however especially when a good approximation is not obtained with the method, and are related to the following2: 1) the nonuniqueness of solutions, 2) the interpretation of the matching cost, 3) the goodness of fit required, 4) the uncertainty as to whether to fix or free parameters in the lower order transfer functions, 5) the appropriate treatment of the multi-input/multi-output case, and 6) the concept of effective system dynamic order. The origins of some of these problems may be generated by the reduction procedure, while others may result from not considering certain aspects of closed-loop system analysis or from the inability to accurately define the task. And still others may arise because each system possesses an effective order, often times higher than that desired. Because of some of these fundamental difficulties, the reduced-order modeling objective of approximating the aircraft's frequency response is re-examined, and when and how to match multiple frequency responses will be reviewed. An alternate state-space model-reduction approach will be proposed. The original transfer function (matrix) G(s) of dynamic order n is to be reduced, via a state-space transformation T. The construction of T involves no numerical search algorithm. Furthermore, the resulting model Gr is unique for the selected dynamic order >. The least effective dynamic order is determined a priori by evaluating a set of frequency-domain matching error bounds. These error bounds, furthermore, are applicable to each i-j element of [G(s) — Gr(s)]s=J(0 over all CD. These error bounds may naturally be interpreted on a Bode plot. Finally, the approach is applicable to multi-input/multioutput systems.