[SciPy-user] Impulse/Step response troubles

[Kumar Appaiah]

Reply below quote.

On Mon, Mar 31, 2008 at 08:52:57AM +0530, Kumar Appaiah wrote:
> The question is to determine (plot) the step and ramp responses of
> K / (s^2 + 8s + K) for K = 7, 16 and 80
> 
> So, the following code is used, and I'll keep changing a line:
> <code>
> from scipy import *
> from pylab import *
> 
> K = 16.0 # should be redone with 7.0, 16.0 and 80.0
> 
> r = signal.impulse(([K],[1.0,8.0,K]), T=r_[0:5:0.001])
> plot(r[0], r[1], linewidth=2)
> show()
> </code>
> 
> The above code plots the impulse response of the given function. Now,
> on to the action:
> 
> 1. Running the above code with K = 7.0 and K = 16.0 gives expected
> results. However, running the code with K = 16.0 doesn't work; if K =
> 16.0, the response should be 16 * t * exp(-4t) * u(t), which it
> isn't. Changing K to 16 + or - .00000000001 fixes it. There surely is
> some problem when the value of K is such that a double root is hit.

Well, I've zeroed in on the issue. Running the code of
signal.lti.impulse, we get here:

s,v = linalg.eig(sys.A)
vi = linalg.inv(v)

Now, v is a matrix which has the eigen vectors, which are EQUAL in
this case:

print sys.A
[[-0.9701425  -0.9701425 ]
 [ 0.24253563  0.24253563]]

which is singular. However, the code goes on to invert this happily,
resulting in a bad matrix and horrendous values. I would be surprised
if nobody has encountered this yet (didn't find this on the Tickets).

What would be the best way to handle repeated roots in the transfer
function's denominator?

Thanks.

Kumar
-- 
Kumar Appaiah,
458, Jamuna Hostel,
Indian Institute of Technology Madras,
Chennai - 600 036