Closed-form root loci for quasi-dipoles with control design applications

A quasi-dipole is defined to be a pole-zero pair embedded in a suitable approximation to a linear phase field. When complex conjugate plant poles occur near the stability boundary, quasi-dipoles often result from controller designs having compensating zeros. Closed-form root loci expressions are developed for quasi-dipoles; these expressions are shown to provide excellent root locus approximations for practical problems.