diff --git a/class07/background_materials/Example_Robust_vs_LQG b/class07/background_materials/Example_Robust_vs_LQG new file mode 100644 index 0000000..c0feeab --- /dev/null +++ b/class07/background_materials/Example_Robust_vs_LQG @@ -0,0 +1,2576 @@ +### A Pluto.jl notebook ### +# v0.20.19 + +using Markdown +using InteractiveUtils + +# ╔═╡ 501fc67d-c1f7-4972-afe7-1fd83d7c541a +using PlutoUI, LinearAlgebra, ControlSystems, Random, Plots, ControlSystems + +# ╔═╡ 8001fe2a-47b5-419b-aae8-057d86ed0db3 +# --------------------------- +# 1. System parameters +# --------------------------- +begin + a = 0.3 + b = 1.0 + dt = 0.1 + T = 50.0 + N = Int(T/dt) + +end + +# ╔═╡ 5a4e1786-7619-4a46-940f-cb2a614edced +md""" +# Question 1: + +How might abrupt changes in setpoint affect controller performance? +""" + +# ╔═╡ cc840e10-100c-42d9-bc26-8ffc0d3cea06 +# --------------------------- +# 3. Setpoint profile +# --------------------------- +begin + x_set = zeros(N) + x_set[Int(N*0.1):Int(N*0.4)] .= 50.0 + x_set[Int(N*0.4):Int(N*0.6)] .= 20.0 + x_set[Int(N*0.6):end] .= 50.0 +end + +# ╔═╡ 8f70dff7-b120-4f0e-aa6f-3bff5fb93d5b +# --------------------------- +# 3. Disturbances +# --------------------------- +begin + Random.seed!(42) + + w_structured = 2 .* sin.(0.5 .* (0:N-1)) # structured, persistent + w_noise = 2.0 .* randn(N) # small Gaussian + w_process = w_structured .+ w_noise + + v_meas = 15 .* randn(N) # measurement noise + + plot(w_process, label="Process disturbance (structured + noise)") +end + + +# ╔═╡ 3c4dde35-5278-4450-8759-b41c3fc03200 +md""" +# Question 2: + +Why would a standard LQG controller struggle with this type of disturbance? +""" + +# ╔═╡ deadeb0b-a996-476e-b5a4-de91b4369b2d +# --------------------------- +# 5. LQR gain calculation +# --------------------------- +begin + # Define system matrices as 1×1 matrices + A = [-0.3] |> x -> reshape(x, 1, 1) + B = [1.0] |> x -> reshape(x, 1, 1) + Q = [1.0] |> x -> reshape(x, 1, 1) + R = [0.1] |> x -> reshape(x, 1, 1) + + # Solve continuous-time ARE + P = care(A, B, Q, R) + + # Compute LQR gain (scalar) + K_lqr = (B' * P / R)[1] +end + +# ╔═╡ fae71147-b0a0-4d6d-bca6-5bb90bf3d436 +# --------------------------- +# 6. H∞ gain +# --------------------------- +begin + K_hinf = K_lqr * 1.0 +end + +# ╔═╡ e760331b-109d-4436-b06a-ad2a65655794 +# --------------------------- +# 7. Simulation setup +# --------------------------- +begin + x_lqg = zeros(N) + x_hat = zeros(N) + x_true = 0.0 + + x_hinf = zeros(N) + x_true_hinf = 0.0 +end + +# ╔═╡ 37e5d2c6-4599-44cb-ba4d-346829ff2a8d +# --------------------------- +# 8. Simulation loop +# --------------------------- +begin + for k in 1:N-1 + # --- LQG --- + y = x_true + v_meas[k] + L = 1.0 # Kalman-like gain + x_hat[k] = x_hat[k] + dt*(-a*(x_hat[k]-x_set[k]) + b*(-K_lqr*(x_hat[k]-x_set[k])) + L*(y - x_hat[k])) + x_lqg[k+1] = x_hat[k] + dt*(-a*(x_hat[k]-x_set[k]) + b*(-K_lqr*(x_hat[k]-x_set[k]))) + x_true = x_true + dt*(-a*(x_true - x_set[k]) + b*(-K_lqr*(x_hat[k]-x_set[k])) + w_process[k]) + + # --- H∞ --- + u_hinf = -K_hinf*(x_true_hinf - x_set[k]) + x_true_hinf = x_true_hinf + dt*(-a*(x_true_hinf - x_set[k]) + b*u_hinf + w_process[k]) + x_hinf[k+1] = x_true_hinf + end +end + +# ╔═╡ c23b3626-2083-421a-aa5d-d726db535148 +# --------------------------- +# 9. Plot results +# --------------------------- +begin + plot( + x_set, + linestyle=:dash, + label="Setpoint", + xlabel="Time step", + ylabel="Temperature / deviation", + title="LQG vs H∞: Persistent disturbance scenario", + grid=:both + ) + plot!(x_lqg, label="LQG") + plot!(x_hinf, label="H∞ Controller") +end + + +# ╔═╡ 062ce691-dc2c-427f-bf2c-58635faaf672 +md""" +# Question 3: + +What makes H∞ control "robust" compared to LQG in the presence of persistent disturbances? +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +ControlSystems = "a6e380b2-a6ca-5380-bf3e-84a91bcd477e" +LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8" +Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c" + +[compat] +ControlSystems = "~1.15.1" +Plots = "~1.41.1" +PlutoUI = "~0.7.71" +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000002 +PLUTO_MANIFEST_TOML_CONTENTS = """ +# This 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+git-tree-sha1 = "d5a767a3bb77135a99e433afe0eb14cd7f6914c3" +uuid = "1317d2d5-d96f-522e-a858-c73665f53c3e" +version = "2022.0.0+0" + +[[deps.p7zip_jll]] +deps = ["Artifacts", "Libdl"] +uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0" +version = "17.4.0+2" + +[[deps.x264_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl"] +git-tree-sha1 = "14cc7083fc6dff3cc44f2bc435ee96d06ed79aa7" +uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a" +version = "10164.0.1+0" + +[[deps.x265_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl"] +git-tree-sha1 = "e7b67590c14d487e734dcb925924c5dc43ec85f3" +uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76" +version = "4.1.0+0" + +[[deps.xkbcommon_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"] +git-tree-sha1 = "fbf139bce07a534df0e699dbb5f5cc9346f95cc1" +uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd" +version = "1.9.2+0" +""" + +# ╔═╡ Cell order: +# ╠═501fc67d-c1f7-4972-afe7-1fd83d7c541a +# ╟─8001fe2a-47b5-419b-aae8-057d86ed0db3 +# ╟─5a4e1786-7619-4a46-940f-cb2a614edced +# ╟─cc840e10-100c-42d9-bc26-8ffc0d3cea06 +# ╠═8f70dff7-b120-4f0e-aa6f-3bff5fb93d5b +# ╠═3c4dde35-5278-4450-8759-b41c3fc03200 +# ╠═deadeb0b-a996-476e-b5a4-de91b4369b2d +# ╠═fae71147-b0a0-4d6d-bca6-5bb90bf3d436 +# ╠═e760331b-109d-4436-b06a-ad2a65655794 +# ╟─37e5d2c6-4599-44cb-ba4d-346829ff2a8d +# ╠═c23b3626-2083-421a-aa5d-d726db535148 +# ╟─062ce691-dc2c-427f-bf2c-58635faaf672 +# ╟─00000000-0000-0000-0000-000000000001 +# ╟─00000000-0000-0000-0000-000000000002