From 2d870d6e9a37155e7e80f595f7f3a7c7c4374d6e Mon Sep 17 00:00:00 2001 From: Jouke van Westrenen <148793668+jwestrenen3@users.noreply.github.com> Date: Wed, 1 Oct 2025 12:26:48 -0400 Subject: [PATCH] Create Example_LQR_vs_LQG Example describing the differences between lqr and lqg --- .../background_materials/Example_LQR_vs_LQG | 1369 +++++++++++++++++ 1 file changed, 1369 insertions(+) create mode 100644 class07/background_materials/Example_LQR_vs_LQG diff --git a/class07/background_materials/Example_LQR_vs_LQG b/class07/background_materials/Example_LQR_vs_LQG new file mode 100644 index 0000000..0ce6ea0 --- /dev/null +++ b/class07/background_materials/Example_LQR_vs_LQG @@ -0,0 +1,1369 @@ +### A Pluto.jl notebook ### +# v0.20.19 + +using Markdown +using InteractiveUtils + +# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error). +macro bind(def, element) + #! format: off + return quote + local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end + local el = $(esc(element)) + global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el) + el + end + #! format: on +end + +# ╔═╡ 126ab500-9ece-11f0-17b5-f97a5f8c9065 +begin + using Random, Plots, PlutoUI +end + +# ╔═╡ 94f46214-ea21-498b-8575-15f98487bb04 +# ------------------------------- +# 1. System definition +# ------------------------------- +begin + # Parameters + A = 0.9 + B = 0.5 + + # LQR weights + Q = 1.0 + R = 0.1 + + # Simulation length + N = 60 + x0 = 60.0 # initial temperature + + # Setpoint profile + setpoint = zeros(Float64, N) + setpoint[1:20] .= 60.0 + setpoint[21:40] .= 80.0 + setpoint[41:end] .= 70.0 + + setpoint +end + + +# ╔═╡ bdd67f65-4add-40a8-8eeb-c0c19807fa09 +# ------------------------------- +# 2. Compute LQR gain +# ------------------------------- +begin + P = Q + for _ in 1:200 + P = Q + A^2 * P - (A * P * B)^2 / (R + B^2 * P) + end + K = (B * P * A) / (R + B^2 * P) + K +end + +# ╔═╡ aa4c742e-3ad0-4cd2-8bc4-93048ce2ac5d +md""" +## Question 1 +We have computed the optimal **LQR gain** `K = K`. +What happens if we apply this controller in a perfect, noise-free world? +""" + +# ╔═╡ d7da5bb9-c95d-4412-86fa-1a5d311d5f8d +# ------------------------------- +# 3. Deterministic LQR (no noise) +# ------------------------------- +begin + x = zeros(N) + u = zeros(N) + x_true = x0 + + for k in 1:N + u[k] = -K * (x_true - setpoint[k]) + x_true = A * x_true + B * u[k] + x[k] = x_true + end + + plot(x, label="State (no noise)", lw=2, color=:blue) + plot!(setpoint, linestyle=:dash, label="Setpoint", color=:black) + xlabel!("Time step"); ylabel!("Temperature (°C)") + title!("LQR works perfectly without noise") +end + + +# ╔═╡ e5d8f40e-0812-4c9c-ba4d-e27935031b9a +md""" +## Question 2 +Why does it not reach the set point? +""" + +# ╔═╡ 26b6604e-c8be-4772-b334-a12be54f3cf2 +md""" +## Question 3 +Now let’s add **measurement noise**. +How do you expect the LQR controller will behave if it directly reacts to noisy measurements? +""" + +# ╔═╡ f4992a67-2f38-4af8-90c2-3a6f6afae217 +# ------------------------------- +# 4. Naive LQR with noise +# ------------------------------- +begin + @bind noise_slider Slider(0:0.5:10, default=5, show_value=true) +end + + +# ╔═╡ c4da12f1-4fc0-4a17-a844-621b00e0f197 +begin + measurement_noise_std = noise_slider + process_noise_std = 0.0 + + x_lqr = zeros(N) + u_lqr = zeros(N) + x_true_lqr = x0 + y_meas_lqr = zeros(N) + + for k in 1:N + y = x_true_lqr + randn() * measurement_noise_std + y_meas_lqr[k] = y + + u = -K * (y - setpoint[k]) + u = clamp(u, -50, 50) + u_lqr[k] = u + + x_true_lqr = A * x_true_lqr + B * u + randn() * process_noise_std + x_lqr[k] = x_true_lqr + end + + plot(x_lqr, label="Naive LQR (noisy)", color=:red, lw=2) + scatter!(y_meas_lqr, label="Measurements", color=:black, alpha=0.4, ms=3) + plot!(setpoint, linestyle=:dash, label="Setpoint", color=:black) + xlabel!("Time step"); ylabel!("Temperature (°C)") + title!("Naive LQR fails with measurement noise") +end + +# ╔═╡ 67b34377-863c-4230-8702-e9ac58af3744 +md""" +## Question 4 +Clearly, the naive LQR reacts to noise and performs poorly. +How can we **filter the measurements** to get a better estimate of the true state? +""" + +# ╔═╡ be56a6d2-cb1a-4886-b0e6-611b864f71f4 +# ------------------------------- +# 5. LQG with Kalman filter +# ------------------------------- +begin + x_true_lqg = x0 # <- different variable + x_lqg = zeros(N) + x_hat = x0 + P_est = 1.0 + u_lqg = zeros(N) + y_meas_lqg = zeros(N) + u_prev = 0.0 + + for k in 1:N + y = x_true_lqg + randn() * measurement_noise_std + y_meas_lqg[k] = y + + # Prediction + x_pred = A * x_hat + B * u_prev + P_pred = A^2 * P_est + process_noise_std^2 + + # Update + K_kalman = P_pred / (P_pred + measurement_noise_std^2) + x_hat = x_pred + K_kalman * (y - x_pred) + P_est = (1 - K_kalman) * P_pred + + # Control + u = -K * (x_hat - setpoint[k]) + u = clamp(u, -50, 50) + u_lqg[k] = u + + # True system + x_true_lqg = A * x_true_lqg + B * u + randn() * process_noise_std + x_lqg[k] = x_true_lqg + + u_prev = u + end + + plot(x_lqr, label="Naive LQR (noisy)", color=:red, lw=2) + plot!(x_lqg, label="LQG (Kalman filtered)", color=:blue, lw=2) + plot!(setpoint, linestyle=:dash, label="Setpoint", color=:black) + scatter!(y_meas_lqg, label="Measurements", color=:black, alpha=0.4, ms=3) + xlabel!("Time step"); ylabel!("Temperature (°C)") + title!("LQR vs LQG: Kalman filter restores stability") +end + + +# ╔═╡ aded6b23-985a-448c-8899-d752fb66aeac +md""" +## Question 4 +With the Kalman filter in place, the LQG controller tracks the setpoint much more smoothly. +- How does increasing the **measurement noise** affect the difference between LQR and LQG? +- Try adjusting the noise slider above! +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000001 +PLUTO_PROJECT_TOML_CONTENTS = """ +[deps] +Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" +PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8" +Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c" + +[compat] +Plots = "~1.40.18" +PlutoUI = "~0.7.69" +""" + +# ╔═╡ 00000000-0000-0000-0000-000000000002 +PLUTO_MANIFEST_TOML_CONTENTS = """ +# This file is machine-generated - editing it directly is not advised + +julia_version = "1.11.6" +manifest_format = "2.0" +project_hash = 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+git-tree-sha1 = "125eedcb0a4a0bba65b657251ce1d27c8714e9d6" +uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0" +version = "0.17.4+0" + +[[deps.libblastrampoline_jll]] +deps = ["Artifacts", "Libdl"] +uuid = "8e850b90-86db-534c-a0d3-1478176c7d93" +version = "5.11.0+0" + +[[deps.libdecor_jll]] +deps = ["Artifacts", "Dbus_jll", "JLLWrappers", "Libdl", "Libglvnd_jll", "Pango_jll", "Wayland_jll", "xkbcommon_jll"] +git-tree-sha1 = "9bf7903af251d2050b467f76bdbe57ce541f7f4f" +uuid = "1183f4f0-6f2a-5f1a-908b-139f9cdfea6f" +version = "0.2.2+0" + +[[deps.libevdev_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl"] +git-tree-sha1 = "56d643b57b188d30cccc25e331d416d3d358e557" +uuid = "2db6ffa8-e38f-5e21-84af-90c45d0032cc" +version = "1.13.4+0" + +[[deps.libfdk_aac_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl"] +git-tree-sha1 = "646634dd19587a56ee2f1199563ec056c5f228df" +uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280" +version = "2.0.4+0" + +[[deps.libinput_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "eudev_jll", "libevdev_jll", "mtdev_jll"] +git-tree-sha1 = "91d05d7f4a9f67205bd6cf395e488009fe85b499" +uuid = "36db933b-70db-51c0-b978-0f229ee0e533" +version = "1.28.1+0" + +[[deps.libpng_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Zlib_jll"] +git-tree-sha1 = "07b6a107d926093898e82b3b1db657ebe33134ec" +uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f" +version = "1.6.50+0" + +[[deps.libvorbis_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll"] +git-tree-sha1 = "11e1772e7f3cc987e9d3de991dd4f6b2602663a5" +uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a" +version = "1.3.8+0" + +[[deps.mtdev_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl"] +git-tree-sha1 = "b4d631fd51f2e9cdd93724ae25b2efc198b059b1" +uuid = "009596ad-96f7-51b1-9f1b-5ce2d5e8a71e" +version = "1.1.7+0" + +[[deps.nghttp2_jll]] +deps = ["Artifacts", "Libdl"] +uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d" +version = "1.59.0+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: +# ╠═126ab500-9ece-11f0-17b5-f97a5f8c9065 +# ╟─94f46214-ea21-498b-8575-15f98487bb04 +# ╟─bdd67f65-4add-40a8-8eeb-c0c19807fa09 +# ╟─aa4c742e-3ad0-4cd2-8bc4-93048ce2ac5d +# ╠═d7da5bb9-c95d-4412-86fa-1a5d311d5f8d +# ╟─e5d8f40e-0812-4c9c-ba4d-e27935031b9a +# ╟─26b6604e-c8be-4772-b334-a12be54f3cf2 +# ╟─f4992a67-2f38-4af8-90c2-3a6f6afae217 +# ╠═c4da12f1-4fc0-4a17-a844-621b00e0f197 +# ╟─67b34377-863c-4230-8702-e9ac58af3744 +# ╠═be56a6d2-cb1a-4886-b0e6-611b864f71f4 +# ╟─aded6b23-985a-448c-8899-d752fb66aeac +# ╟─00000000-0000-0000-0000-000000000001 +# ╟─00000000-0000-0000-0000-000000000002