Root locus analysis

In control theory, a root locus is the locus of the poles and zeros of a system's S-function as the system gain is varied from 0 to infinity. The root locus is one of the most useful tools for analyzing single input single output linear dynamic systems. A system is stable if all of its poles are in the left-hand side of the s-plane.

As an example, suppose there is in the forward path a motor with a transfer function expression 1/(s + 5) driving a roller with an transfer function expression 1/s. A controller with an adjustable gain K and a transfer function expression 1/(s + 10). A unity feedback loop is introduced to complete this feedback system. Root locus analysis is then carried out on this feedback system.

The location of the roots (poles and zeros) are found for each adjustment of value K. A few simple steps, the root locus method, can locate the overall shape of the path (locus) traversed by the roots and can give you an approximate location of the poles as K value increases from 0 to a large number, hence, give an idea of the stability of this unity feedback system for different values of K.

The method is due to W.R. Evans (AIEE Transactions, 1948).

Root loci can also be computed in the "z-plane" which is the same as the s-plane, but for discrete systems. A mapping (z = e^(sT)) transforms variables, and has numerous interesting properties. T is the sample time. The stable, left half s-plane maps as a unit circle in the z plane, with the s-origin equating to z=1 (because e^0=1). You can follow a diagonal line of constant damping in the s-plane around a spiral from (1,0) in the z plane as it curves forever in toward the origin.

Root locus "rules" work the same in the z and s planes, and learning to draw them is easy, but unnecessary with computer root solving tools. Z-plane equations translate easily to finite diference arithmetic (let 1/z be defined as a single frame's delay, as 1/s is defined as an integrator, and you can work it all out. Besides sweeping a single gain K, it's useful to sweep any system parameter for which the exact value is uncertain.

External links

• MIT - Lectures in Classic Control Theory; lecture describing the root-locus, its properties, and a step-by-step procedure for constructing a root-locus. Case studies are included which illustrate how to use the root-locus for designing a control system.