Root locus analysis

In control theory, the root locus is the locus of the poles of a transfer function as the system gain K is varied on some interval. The root locus is a useful tool for analyzing single input single output (SISO) linear dynamic systems. A system is stable if all of its poles are in the left-hand side of the s-plane (for continuous systems) or inside the unit circle of the z-plane (for discrete systems).

As an example, suppose there is a motor with a transfer function expression P(s), a controller with an adjustable gain K and a transfer function expression C(s). A unity feedback loop is constructed to complete this feedback system. For this system, the overall transfer function is given by T(s) = (KCP)/(1+KCP). Thus the closed-loop poles (roots) of the transfer function are the solutions to the equation 1+ KC(s)P(s) = 0. From this function T(s), we can also see that the zeros of the open loop system (CP) are also the zeros of the closed loop system. It is important to note that the root locus only gives the location of closed loop poles as the gain K is varied, given the open loop transfer function. The zeros of a system can not be moved.

Using a few basic rules, the root locus method can plot the overall shape of the path (locus) traversed by the roots as the value of K varies. The plot of the root locus then gives an idea of the stability and dynamics of this 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, the discrete counterpart of the s-plane. An equation ($z=e^{sT}$) maps continuous s-plane poles (not zeros) into the z-domain, where T is the sample period. The stable, left half s-plane maps as the unit circle into the z-plane, with the s-plane origin equating to z=1 (because $e^{0}=1$). 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.

Root locus rules work the same in the z and s planes, and learning to draw them is relatively straight-forward, but unnecessary with computer root locus solving tools (e.g., MATLAB's rltool, Scilab's evans). However, drawing them by hand is a useful learning tool.

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.

External links

[[1]] Tutorial