Incorrect rootlocus plot

I’m using the control package in a modified jupyter/scipy docker container. The modification to the container were just to install the controls package (RUN pip3 install control). When i try to plot the root locus of a inverted pendulum with 2 roots at 0, it outputs the wrong rootlocus.

Minimal working example:

import numpy as np
import control

A = np.array([[0,1,0,0],[0,0,-1,0],[0,0,0,1],[0,0,5,0]])
B = np.array([[0],[1],[0],[-2]])
C = np.array([1,0,0,0])
D = 0
sys = control.ss(A,B,C,D)
r = control.root_locus(sys, xlim=[-2,2],ylim=[-7,7])

The above code results in the following plot When the same system in plotted in octave, it gives the following rlocus

As can be seen there should be no branch immediately to the left of 0

[murrayrm]

I chased this down a bit more. The difference seems to come from the way that slycot creates the transfer function representation of the system versus the way that SciPy carries out the calculation (via scipy.signal.ss2tf).

Slycot:

• Poles: array([ 2.23606798e+00, -2.23606798e+00, 1.26650220e-17, 1.26650216e-17])
• Zeros: array([ 1.73205081, -1.73205081])

SciPy:

• Poles: array([-2.23606798, 2.23606798, 0. , 0. ])
• Zeros: array([-2.68871620e+14, 1.73205081e+00, -1.73205081e+00])

Interestingly, SciPy throws a warning when computing the poles:

/Users/murray/anaconda3/envs/python3.9-slycot/lib/python3.9/site-
packages/scipy/signal/filter_design.py:1630: BadCoefficients: Badly
conditioned filter coefficients (numerator): the results may be meaningless

So it looks like the problem is the (fast, stable) zero at -2.689e+14, which completely throws off the root locus plot. Not sure how to fix without implementing our own transfer function calculation rather than rely on SciPy.

I looked at the source code for the SciPy function and the basic calculation is:

den = poly(A)
num[k] = poly(A - dot(B, C[k:k+1, :])) + (D[k] - 1) * den

The Slycot computation for tb04ad (the function that gets calls) says:

The method for transfer matrices factorized by rows will be described here: T(s) factorized by columns is dealt with by operating on the dual of the original system. Each row of T(s) is simply a single-output relatively left prime polynomial matrix representation, so can be calculated by applying a simplified version of the Orthogonal Structure Theorem to a minimal state-space representation for the corresponding row of the given system. A minimal state-space representation is obtained using the Orthogonal Canonical Form to first separate out a completely controllable one for the overall system and then, for each row in turn, applying it again to the resulting dual SIMO (single-input multi-output) system. Note that the elements of the transformed matrix A so calculated are individually scaled in a way which guarantees a monic denominator polynomial.

[warlock31415]

Doesn’t work without slycot with the pacakage versions you mentioned. Works with the mentioned packages with slycot