Reachable frequency response

reach_freq.jl

### A Pluto.jl notebook ###
# v0.16.4

using Markdown
using InteractiveUtils

# ╔═╡ 15da0afe-3e5b-11ec-031f-d9903463dce1
begin
	using ControlSystems
	using IntervalArithmetic
	using LazySets
	using LinearAlgebra
	using Plots
	import Optim
	import PlotlyJS
	plotlyjs()
end

# ╔═╡ cf2212a1-5bb4-4773-a6c7-93997acdf3ef
md"Define the Laplace variable `s`"

# ╔═╡ 7e51891c-c143-4184-9d37-bc8b797e8277
s = tf("s")

# ╔═╡ 6417c4b4-1940-476d-a3b4-9630f09618b7
md"These will be our uncertain parameters"

# ╔═╡ 55af1bee-f985-4171-aae6-27047b319783
a = 20 ± 0.05;

# ╔═╡ b564bda3-311a-4904-9ba9-3a8c2fb08141
b = 1 ± 0.001;

# ╔═╡ bb9bf2d7-412c-419e-af71-d5b511c92f56
md"Define system transfer function using the uncertain parameters"

# ╔═╡ b7bfac81-371b-4d44-aae7-e3ca77be15b8
G = 7.6 * (s^2 + 0.1s + 0.55) / (s * (s+b) * (s+a) * (s^2 + 0.06s + 0.5));

# ╔═╡ d18cea5b-952e-4b39-ab8f-61c0c1c3f8cc
md"Vector of frequencies to calculate at"

# ╔═╡ 4519131d-8b89-429c-891f-3c5aadefe0aa
ω = 10 .^ (-0.5:0.001:0.1);

# ╔═╡ 90bac778-1d19-4068-80a4-291422425943
md"Calculate phase and gain of frequency response"

# ╔═╡ 878ccad2-f02a-47a7-a370-0f382fcf9920
freq = freqrespv(G, ω);

# ╔═╡ 11b248cb-fed4-48ae-b7c0-3e31156b84f4
gain = @. 20log10(abs(freq));

# ╔═╡ c5c1d64a-b6fc-4de7-bf05-7499ec368c5a
phase = @. atan(imag(freq)/real(freq))*180/π - 180;  # Manual hack to prevent wrapping

# ╔═╡ c75b10e0-6923-408d-8d0d-cb54442944a3
md"Create an interval box of phase and gain"

# ╔═╡ 144c0a8b-3291-4d8c-97d6-773a134a7da6
phase_gain = phase .× gain;

# ╔═╡ 691fc845-f5dd-430d-8db5-127ad9cf64cd
md"Define requirements keepout zone ellipsoid. This is defined as a distance from the 'critical point' of -180° phase and 0 dB gain. Any frequency response that crosses this critical point will be unstable when the loop is closed. To ensure healthy robustness to unmodelled uncertainty, we require that our frequency response be outside of this ellipsoid."

# ╔═╡ 6937594d-34ea-439c-a42f-a4afc47353bc
phase_req = 15.0;

# ╔═╡ da46ae74-c80f-4be0-a26a-db9a742f9ed4
gain_req = 3.0;

# ╔═╡ e9ee7817-7a93-4286-a485-6f39027b0b3a
requirements_keepout = Ellipsoid([-180.0, 0.0], Diagonal([phase_req, gain_req]).^2);

# ╔═╡ a18e4472-ac28-4389-93f2-0b021af75b64
violations = intersect.(Ref(overapproximate(requirements_keepout, HPolygon, 0.01)), convert.(Hyperrectangle, phase_gain))

# ╔═╡ de3a2cc3-5e91-4a1d-b0cb-167f6f3152e8
md"Plot frequency response reachable sets against requirements"

# ╔═╡ b6e8b24c-1144-479d-9d31-6730f27e0e16
begin
	plot(xlabel="phase (deg)", ylabel="gain (dB)")
	plot!(phase_gain; label="Frequency response")
	plot!(filter(!isempty, violations))
	plot!(requirements_keepout; label="Requirements keepout zone", c=:red, alpha=0.2, linealpha=1)
end

# ╔═╡ 00000000-0000-0000-0000-000000000001
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deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "ece2350174195bb31de1a63bea3a41ae1aa593b6"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "0.9.1+5"
"""

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