I am trying to build a workbook to illustrate the effect of various
parameters of second order transfer functions. The full workbook is on
GitHub
<http://nbviewer.ipython.org/github/alchemyst/Dynamics-and-Control/blob/master/Second%20order%20systems.ipynb>,
but here is a minimal example of the problem:
import sympy
tau, zeta, t, w, K = sympy.symbols('tau, zeta, t, w, K', real=True,
positive=True)s = sympy.Symbol('s')
G = K/(tau**2*s**2 + 2*tau*zeta*s + 1)
The impulse response of the second order system is simply the inverse Laplace
of G. However, the nature of the inverse depends on the parameter ζ.
TAttempting directly to calculate the inverse results in
TypeError: cannot determine truth value of
-zeta/tau - sqrt(zeta + 1)*cos(atan2(0, zeta - 1)/2)*sqrt(Abs(zeta - 1))/tau <
oo
There are three cases of interest, ζ>1, ζ=1 and 0<ζ<1
In Sage, I would be able to use assume(zeta > 1) before calculating the inverse
to obtain the correct version of the inverse, but I have not found a way to
impose such constraints in SymPy. So, first question is whether I can find nice
solutions for these cases to the inverse.
Failing that, I want at least to be able to calculate the inverse with known
values of all the parameters so that I can animate the response using IPython
notebook widgets. Here I have also been out of luck, as some cases result in
special values which are not cleanly evaluated to
knownbadvalues = [{K: 5.05, tau: 5., zeta: 1.},
{K: 5.05, tau: 5.05, zeta: 1.05}
]for values in knownbadvalues:
print sympy.inverse_laplace_transform(G.subs(values), s, t, noconds=True)
(0.202*t - 0.404*EulerGamma - 0.404*polygamma(0, 1.0))*exp(-0.2*t)
0.198019801980198*meijerg(((0.728681938243239, 0.855476477598345), ()), ((),
(-0.271318061756761, -0.144523522401655)), exp(t))
ts = numpy.linspace(0, tmax, 100)
sympy.lambdify(t, invL(G.subs(values)), ['numpy', 'sympy'])(ts).n()
fails with "ValueError: sequence too large; must be smaller than 32"
Any advice on getting either getting the closed forms or just finding a
version which can be evaluated cleanly?
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