Root locus analysis

In control theory, the root locus is the locus of the poles of a system's transfer function as the system gain K is varied on some interval. The root locus is a useful tools 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 or inside the unit circle for the z-plane.

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) = (CP)/(1+CP). Thus the poles (roots) 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 poles and zeros. 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 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 easy, but unnecessary with computer root locus solving tools (e.g. MATLAB's rltool). However, drawing them by hand it is a useful learning tool and can provide additional insight into a system.

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 to determine its behavior.

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.