Frequency Methods of Analysis

Frequency methods of analysis ushered in the age of modern airplane stability augmentation and autopilot analysis. In his 1950 Wright Brothers Lecture, Dr. William Bollay reminded us that the application of frequency-response methods to the airplane case came a full ten years after their use in the development of anti-aircraft gun directors. The footprints of the electrical engineering community in this field are still evident in the use of terms such as decibels and octaves in some airplane frequency-response studies.

Frequency response is the steady-state sinusoidal airplane motion perturbation in re­sponse to steady-state sinusoidal control surface input perturbation. Only the amplitude ratio and phase relationship of the two sinusoids are of interest. Frequency response of mechanical and electrical devices is readily found from the parameters of the linearized differential equations that describe the device’s motion or electrical properties. The formal mathematics that do this rest on the Laplace transformation, as explained in Chapters 3 and 4 of the classic 1948 text Principles of Servomechanisms, by Gordon S. Brown and Donald P Campbell.

Frequency-response analysis led stability and control engineers to an entirely new way to describe airplane dynamics, the transfer function. The transfer function is the mathematical operator by which any input function is multiplied to obtain the output function for that element. Transfer functions are numerical or literal expressions in the Laplace variable s. Transfer-function denominators are nothing but the characteristic equation, with Routh’s and Bryan’soperator X replaced by the complex Laplace variable s. Transfer-function numerators are governed by the input. Thus, the classical transfer function that converts elevator angle disturbances to pitch attitude disturbances is a second-degree polynomial in s divided by a fourth-degree polynomial in the variable s.

One of the first known applications of frequency response in airplane stability augmenter design was made by Roland J. White for the XB-47 yaw damper. White used the inverse frequency-response diagrams described by H. T Marcy in 1946, in an electrical engineering context. Over the years, frequency-response analysis has never gone out of fashion. For example, the demanding X-29A flight control system was designed using Bode frequency plot techniques by Grumman engineers led by Arnold Whitaker, James Chin, Howard L. Berman, and Robert Klein.

Frequency-response methods are used in some of the latest airplane flight control design methods, giving frequency response another lease on life. As detailed in a later section, singular value methods, associated with robust control theory, use frequency response.