Root locus plots

Root locus

The root locus analysis is a graphical method for examining how the roots of a system change with variation of a certain system parameter. This is a technique used a stability criterion which can determine stability of the system. The root locus plots the poles of the frequency response function in the complex s-plane as a function of a parameter.

In addition to determining the stability of the system, the root locus can be used to design the damping ratio (ζ) and natural frequency (ω) of a system. Lines of constant damping ratio can be drawn radially from the origin and lines of constant natural frequency can be drawn as arccosine whose center points coincide with the origin. By selecting a point along the root locus that coincides with a desired damping ratio and natural frequency, a parameter can be calculated and implemented in the system.

To generate this plot which is obtained at a linearization time, select the Root locus item in the Linear Analysis menu.

Figure 54: Root locus item in the toolbar

It opens a dedicated window that helps to generate the root locus.

Figure 55: Root locus tool

The plot is generated according to the linearization time selected from the drop-down list.

The Update button allows you to refresh the drop-down list with the latest available simulation.

The plot appears with individual eigenvalues denoted by a symbol. The default symbol is ‘+’ but for the first in the sequence an circle is used and for the last a diamond.

Figure 56: Example of a root locus plot

In cases with extreme values of some eigenvalues, you can use the Custom scale option to limit the range of frequencies displayed. When you select the check box, the Maximum frequency (Hz) field is enabled. Here, you can enter the maximum frequency that is displayed in the plot so that extreme values are excluded.

Figure 57: Setting a custom scale

Figure 58: Root locus with maximum frequency option

Note

Depending on the location of the poles on the root locus, you can predict the dynamic properties of the system

Figure 59: Root locus for different locations of eigenvalues