z-plane Versus s-plane
The root locus can also be computed in the z-plane, the discrete counterpart of the s-plane. An equation (z = esT) maps continuous s-plane poles (not zeros) into the z-domain, where T is the sampling period. The stable, left half s-plane maps into the interior of the unit circle of the z-plane, with the s-plane origin equating to |z| = 1 (because e0 = 1). A diagonal line of constant damping in the s-plane maps around a spiral from (1,0) in the z plane as it curves in toward the origin. Note also that the Nyquist aliasing criteria is expressed graphically in the z-plane by the x-axis, where (wnT = π). The line of constant damping just described spirals in indefinitely but in sampled data systems, frequency content is aliased down to lower frequencies by integral multiples of the Nyquist frequency. That is, the sampled response appears as a lower frequency and better damped as well since the root in the z-plane maps equally well to the first loop of a different, better damped spiral curve of constant damping. Many other interesting and relevant mapping properties can be described, not least that z-plane controllers, having the property that they may be directly implemented from the z-plane transfer function (zero/pole ratio of polynomials), can be imagined graphically on a z-plane plot of the open loop transfer function, and immediately analyzed utilizing root locus.
Since root locus is a graphical angle technique, root locus rules work the same in the z and s planes.
The idea of a root locus can be applied to many systems where a single parameter K is varied. For example, it is useful to sweep any system parameter for which the exact value is uncertain, in order to determine its behavior.
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