RootLocusPlot

The root locus plot for various pole-zero configurations:

Root locus plot of a transfer-function object:

Root locus plot of a state-space model object:

Label the axes:

Specify the origin as the axes origin:

Color stable and unstable parts green and red, respectively:

Show lines corresponding to a damping ratio 0.4 on the plane:

Show the circle corresponding to the natural frequency 3 radians per time unit:

The loci of points with damping 0.4 in the plane for a system with sampling period 1:

Negative feedback is assumed by default:

Positive feedback:

Root loci of an open-loop system with positive feedback:

A closed-loop system:

The method can be faster than :

By default, closed-loop poles at the midpoint of the parameter range as well as the open-loop poles and zeros are shown:

Show no markers:

Show the closed-loop poles only:

Use text or typeset labels:

Use graphics primitives as the pole-zero markers:

Use any 2D or 3D graphics:

Use more points to get a smoother curve:

Show the loci only in the region where the closed-loop system is stable:

Explore and determine critical points such as break-away, break-in, and imaginary axis crossings:

Plot the roots of a polynomial as a parameter is varied:

Analyze the effect of the sensor gain on a system:

For negative feedback systems, the root loci are points on the complex plane where the phase of the transfer function is , and for positive feedback systems it is 0:

The root locus plot does not depend on the sampling period:

If the number of poles is greater than the number of zeros, the root loci go to infinity with straight-line asymptotes as the parameter is increased:

The slopes of the asymptotes for a negative feedback system:

Find where the asymptotes intercept the real axis:

Plot the root loci and the asymptotes:

The slopes of the asymptotes for a positive feedback system:

Plot the root loci and the asymptotes:

The break-away and break-in points on the real axis can be computed from the poles and zeros of the transfer function:

Select those points for which kInterval[{0, 5}]:

Show the points on the root locus plot:

The complex-valued transfer function is a surface with "peaks" at the poles and "valleys" at the zeros:

The root locus plot projected on the surface travels from the "peaks" to the "valleys" along the lines of steepest descent:

The Bode magnitude plot is the intersection of the surface and the - plane:

The system must be proper:

The root loci may not be symmetric with respect to the real axis (but the roots are):