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add tutorial analyzing hybrid systems #903

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merged 4 commits into from
Dec 4, 2023
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add tutorial analyzing hybrid systems #903

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3e653f2
add zoh example
baggepinnen Nov 14, 2023
e7a45f1
add exact translation to continuous time
baggepinnen Dec 4, 2023
a7def92
add tutorial analyzing hybrid systems
baggepinnen Dec 4, 2023
d52a74c
rm code file
baggepinnen Dec 4, 2023
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add exact translation to continuous time
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@baggepinnen
baggepinnen committed Dec 4, 2023
commit e7a45f1c53307fa789e3cd32e707294d7190853f
21 changes: 21 additions & 0 deletions example/zoh.jl
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Expand Up @@ -49,3 +49,24 @@ resP = lsim(P, ufun, Tf)
resPz = lsim(Pz, ufun, 0:0.01:Tf)
resPd = lsim(Pd, ufun, Tf)
plot([resP, resPz, resPd], lab=["P" "Pz" "Pd"])


##
# Discrete-time systems can be converted to continuous-time systems with delays in such a way that the frequency-response of the two systems match exactly.
# To this end, the function `d2d_exact` is used. The resulting system is useful for frrequency-domain simulations only and cannot be used for time-domain simulations.
# This is useful when analyzing hybrid systems with both continuous-time and
# discrete-time components and accurate frequency-domain calculations are required over the entire frequency range up to the Nyquist frequency.
# Below, we compare the frequency response of a discrete-time system with delays to
# two continuous-tiem systems, one translated using the exact method and one using the standard `d2c` method with inverse ZOH sampling.

w = exp10.(LinRange(-2, 1, 200))
Pdc = d2c_exact(ss(Pd))
Pc = d2c(Pd)

bodeplot(Pd, w, lab="Discrete")
bodeplot!(Pdc, w, lab="Continuous with delays", l=:dash)
bodeplot!(Pc, w, lab="Continuous d2c-ZoH")
vline!([0.5 0.5], l=(:black, :dash), lab="Nyquist freq.", legend=:bottomleft)