On Thursday, September 3, 2015 at 7:19:38 PM UTC+3, Carl Sandrock wrote:
>
> 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
>
>
This error message does not appear in the current master. However, the
result is very complicated and not necessarily correct for all values of
the parameters.
>
> 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}
> ]
>
>
The results will be better if exact (rational) values are substituted. The
integrator can not deal with floats in many cases.
> 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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