Speed up forced_response/lsim #51
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Fixes python-control#48 The old algorithm called a general-purpose ODE integrator at each timestep, and this could be extremely slow (hundreds of times slower than scipy.signal.lsim). This new algorithm evaluates the matrix exponential once, and uses it to advance the simulation at each step. Importantly, the old algorithm would work for variable timesteps, but the new version assumes a fixed timestep (as does Matlab's lsim). This is the usual case, and it is the price one pays for increased speed. Note that this algorithm is the same idea as the algorithm used in scipy.signal.lsim, but the scipy version requires inverting the `A` matrix, and thus does not work if the system has a pole at the origin. The algorithm used here works even if there is a pole at the origin. This commit also adds a test for this case (the double integrator).
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Speed up forced_response/lsim
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Fixes #48.
This version of lsim assumes a constant timestep, as in Matlab. This algorithm is much faster than the previous implementation, which used a generic ODE integrator: the example in issue #48 now runs over 400x faster, and is now a little faster than scipy.signal.lsim. This algorithm is more general than the scipy version, since it works even if the system has a pole at the origin.
If we want a version that works with variable timesteps, I suggest adding an
lsim2function, as inscipy.signal, which could call the generic ODE integrator as before. I don't think this is necessary (for instance, there is nothing like that in Matlab, as far as I know), but if anybody needs it, we can always add it.