Transfer-Function Dipoles

The problem of the complex zero in the bank angle-aileron transfer function happens to be one of a class of transfer-function dipole problems in stability-augmenter design. In many lightly damped airplane modes, the root in question is near a complex zero. The pole-zero pair is called a dipole. Root locus rules generally make sure that, for the pair, the locus that originates at the pole ends at the zero, forming a semicircle along the way.

When the dipole is close to the root locus imaginary axis, the semicircle can pass into the unstable, or right-half, plane. Conversely, by assuring that the semicircle forms to the left, closing the loop increases the stability of that lightly damped mode. This is called phase stabilization. By far the most important application of phase stabilization is to the bending and torsional modes of an elastic airplane with stability augmentation. As in the case of the bank angle transfer function, the modes are phase-stabilized when the dipole zero has a lower undamped natural frequency than the pole, or root.

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