root locus

The idea of a root locus can be applied to many systems where a single parameter K is varied.

In combination with the angle condition, these two mathematical expressions fully determine the root locus.

The resulting root locus traces the unit circle.

Controllers can be designed through the polynomial design, root locus design methods to name just two of the more popular.

The root locus can also be computed in the z-plane, the discrete counterpart of the s-plane.

The root locus, Nyquist plot, and Nichols plot techniques all make use of the complex plane.

These include graphical systems like the root locus, Bode plots or the Nyquist plots.

The root locus plots the poles of the closed loop transfer function as a function of a gain parameter.

So to test a point for inclusion on the root locus, all you do is add the angles to all the open loop poles and zeros.

The plot of the root locus then gives an idea of the stability and dynamics of this feedback system for different values of k.

Frequency domain analysis (Bode plot, Root locus, Nyquist plot)

In addition to determining the stability of the system, the root locus can be used to design the damping ratio and natural frequency of a feedback system.

Since root locus is a graphical angle technique, root locus rules work the same in the z and s planes.

In particular, as a root locus analysis would show, increasing feedback gain will drive a closed-loop pole toward marginal stability at the DC zero introduced by the differentiator.

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.

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.

Bode plot, Nyquist stability criterion, Nichols plot, and root locus are the usual tools for SISO system analysis.

By selecting a point along the root locus that coincides with a desired damping ratio and natural frequency a gain, K, can be calculated and implemented in the controller.

Root locus analysis is a graphical method for examining how the roots of a system change with variation of a certain system parameter, commonly a gain within a feedback system.

In the root locus method, it is especially important whether the poles and zeros are in the left or right half planes, i.e. have real part greater than or less than zero.

More elaborate techniques of controller design using the root locus are available in most control textbooks: for instance, lag, lead, PI, PD and PID controllers can be designed approximately with this technique.

Gregory Walter Evans, "Bringing root locus to the classroom: the story of Walter R. Evans and his textbook Control System Dynamics", IEEE Control Magazine, pp.