Transfer-Function Numerators

Airplane transfer-function denominator factors, or roots, govern airplane motions following initial disturbances. Stable roots, having negative real parts, lead to subsidence of oscillatory or aperiodic motions. The same is true for the denominators of closed-loop transfer functions, and early root locus work, such as the material in Dr. William Bollay’s 1950 Wright Brothers Lecture, dealt with roots, or poles, of the closed-loop denominator. Transfer function numerator factors are called zeros. A response survey to step inputs for a systematic variation in pole-zero combinations (Elgerd and Stephens, 1959) gives striking results, particularly for the case of two real poles and one real zero. Depending on whether the zero is between the poles or to the right, the step response appears either deadbeat or with a large overshoot.

Transfer-function zeros play an important role in the closed-loop responses of stability – augmentation systems. The details are too involved to go into here, but some examples can be touched upon. In the altitude or glide path loop in which errors are corrected by elevator or stabilizer control, a zero called 1 /Th1 can be in the right half of the root locus plane. This occurs on the back side of the power required curve, or at airspeeds below the minimum drag point. Loop closure drives a closed-loop real root into the right half-plane, with consequent divergence. An inner stability augmentation loop can correct this.

Another example is the complex zero associated with bank angle control by the ailerons. The Systems Technology, Inc., symbol for the undamped natural frequency associated with this zero is Шф. For values of Шф that exceed the Dutch roll undamped natural frequency rnd, loop closure excites the Dutch roll and closed-loop stability is degraded. A complete tutorial discussion of this problem, as well as the altitude control zero problem, is given by Duane T McRuer and Donald E. Johnston (1975).