When the practical demands of World War II required a body of theoretical knowledge to facilitate the design of automatic weapons systems, modern control-system theory was born. Its early development was nearly explosive, for, in the beginning, a considerable stock of nourishing food was readily available.
Thus mathematical machinery defining the dynamic behavior of linear electrical and mechanical networks was already highly developed. Once the basic principle of negative feedback was recognized, the synthesis of this knowledge into a form descriptive of closed-loop control systems was readily accomplished. In fact, such a synthesis had already been made for feedback amplifiers by Nyquist and Bode. The linear theory so developed was successfully applied to many practical control problems which arose during and after the war.
It is only in recent years, with the development of interest in adaptive control systems, that the limitations of the earlier theory have become increasingly serious. Such limitations are at once apparent to anyone attempting to describe a biological control system. These systems almost always contain essential nonlinearities arising from individual system components, from parametric feedback loops, or both. In addition, they are frequently characterized by multiple feedback loops arranged in some sort of hierarchy. To a considerable extent, the applicability of simple linear control-system theory to a biological system will depend upon the degree of detail in which we are interested and whether our goal is a special empirical or a general theoretical description. For example, over a restricted amplitude range, the frequency response of a biological "black box" may be indistinguishable from that of a simple linear first- or second-order system.
This may be all of the information required for a particular application. On the contrary, it may not satisfy us at all. We may wish to examine the contents of the box in detail and to derive equations for its over-all behavior on theoretical grounds. (Brashers-Krug, Shadmehr & Bizzi, 1996) When we do so we may obtain expressions which are very complex in form but which behave rather simply under certain special conditions, just as relativistic mechanics reduces to Newtonian at "ordinary" velocities. Hence we may approach the analysis of a biological control system in more than one way, and each approach will have its own special advantages and limitations.
(Rosenbaum, 1991) Since our main concern is with control systems and since feedback is an essential feature of such systems, let us take an introductory look at the general nature and usefulness of feedback. If one knows the frequency response of a system, what can be said about its transient response? For first- and second-order systems, the answer is both simple and complete. Here it is obvious that the Bode frequency diagram corresponds exactly to a particular transient response. But what about higher-order systems? We have indicated that frequency analysis was of particular value here because of its relative simplicity.
(Alexander, 1991) The question now arises as to whether this simplicity extends to the prediction of the exact transient response once the frequency response is known. Unfortunately, the answer is that there is no general method of doing this which is both simple and rigorous. An approximate method, often used in engineering applications and based largely upon empirical experience, assumes that the higher-order system may be treated as if it were second order. Thus it is assumed that a design based on the frequency method which holds the maximum gain below that of the peak transient overshoot of a second-order system will have a transient response which does not differ significantly from the corresponding second-order transient.
(Guenther & Barreca, (1997) This approximation will come close to the truth if the higher-order system has one pair of conjugate complex roots which lie much closer to the imaginary axis than any other roots, for then the transient response. (Dean & Bruwer, 1997) ...
