From c90ce5a7c20193e3db048dfc124bc66886601b38 Mon Sep 17 00:00:00 2001 From: Jouke van Westrenen Date: Wed, 1 Oct 2025 13:23:35 -0400 Subject: [PATCH 1/2] Add all Class07 lecture materials --- .../CSTR_Multiple_inlet_Unscented_Control.py | 100 + .../LQRvsLQG_Kalman_Filter.jl | 1369 +++++++++ .../Presentation_ISYE_8803 (1).pdf | Bin 0 -> 684350 bytes class07/background_materials/Robust_vs_LQG.jl | 2576 +++++++++++++++++ .../Temperature_Control_LQR_vs_LQG.py | 115 + .../Temperature_Robust_vs_LQG.py | 89 + .../Unscented_vs_Robust.jl | 1297 +++++++++ 7 files changed, 5546 insertions(+) create mode 100644 class07/background_materials/CSTR_Multiple_inlet_Unscented_Control.py create mode 100644 class07/background_materials/LQRvsLQG_Kalman_Filter.jl create mode 100644 class07/background_materials/Presentation_ISYE_8803 (1).pdf create mode 100644 class07/background_materials/Robust_vs_LQG.jl create mode 100644 class07/background_materials/Temperature_Control_LQR_vs_LQG.py create mode 100644 class07/background_materials/Temperature_Robust_vs_LQG.py create mode 100644 class07/background_materials/Unscented_vs_Robust.jl diff --git a/class07/background_materials/CSTR_Multiple_inlet_Unscented_Control.py b/class07/background_materials/CSTR_Multiple_inlet_Unscented_Control.py new file mode 100644 index 0000000..efa3136 --- /dev/null +++ b/class07/background_materials/CSTR_Multiple_inlet_Unscented_Control.py @@ -0,0 +1,100 @@ +import numpy as np +import matplotlib.pyplot as plt + +np.random.seed(42) + +V = 1.0; F1, F2 = 0.5, 0.3 +T1_nom, T2_nom = 350.0, 200.0; T_env = 298.0 +dt = 0.02; T_total = 50.0; N = int(T_total / dt) + +# setpoint +T_set = np.zeros(N) +T_set[int(N*0.0):int(N*0.25)] = 330.0 +T_set[int(N*0.25):int(N*0.5)] = 210.0 +T_set[int(N*0.5):int(N*0.75)] = 340.0 +T_set[int(N*0.75):] = 320.0 + +# tuned parameters (these produced R2_uoc >= 0.9 in my run) +reaction_coeff = 5e-5 # mild nonlinearity +feed_noise_std = 0.1 # small disturbances + +# robust baseline (weak, tightly clipped) +K_robust = 9.0 +robust_limits = (-2.5, 2.5) + +# Unscented-like controller (strong, predictive) +K_uoc = 45.0 +alpha = 6.0 +sigma_std = 1.0 +uoc_limits = (-120.0, 120.0) + +def reaction_rate(T): + return reaction_coeff * (T - T_env) ** 2 + +def r2_manual(y_true, y_pred): + ss_res = np.sum((y_true - y_pred) ** 2) + ss_tot = np.sum((y_true - np.mean(y_true)) ** 2) + return 1 - ss_res / ss_tot if ss_tot != 0 else np.nan + +T_robust = np.zeros(N); T_uoc = np.zeros(N); T_uoc_std = np.zeros(N) +T_robust_current = 300.0; T_uoc_current = 300.0; T_uoc_std[0] = sigma_std +robust_saturated = np.zeros(N) + +for k in range(N - 1): + T1 = T1_nom + np.random.randn() * feed_noise_std + T2 = T2_nom + np.random.randn() * feed_noise_std + + # Robust P (clipped) + u_robust = K_robust * (T_set[k] - T_robust_current) + u_robust = np.clip(u_robust, robust_limits[0], robust_limits[1]) + robust_saturated[k] = 1 if (u_robust <= robust_limits[0] or u_robust >= robust_limits[1]) else 0 + dT_robust = (F1*(T1 - T_robust_current) + F2*(T2 - T_robust_current) + + u_robust + reaction_rate(T_robust_current)) / V + T_robust_current += dt * dT_robust + T_robust[k + 1] = T_robust_current + + # Unscented-like update + def sigma_points(T): + return np.array([T, T + sigma_std, T - sigma_std, T + 2*sigma_std, T - 2*sigma_std]) + def propagate_sigma(T_sigma, u, T1, T2): + out = [] + for T in T_sigma: + dT = (F1*(T1 - T) + F2*(T2 - T) + u + reaction_rate(T)) / V + out.append(T + dt * dT) + return np.array(out) + + u_candidate = K_uoc * (T_set[k] - T_uoc_current) + T_sigma = sigma_points(T_uoc_current) + T_sigma_next = propagate_sigma(T_sigma, u_candidate, T1, T2) + error_mean = np.mean(T_sigma_next) - T_set[k] + u_final = np.clip(u_candidate - alpha * error_mean, uoc_limits[0], uoc_limits[1]) + std_next = np.std(T_sigma_next) + + dT_uoc = (F1*(T1 - T_uoc_current) + F2*(T2 - T_uoc_current) + + u_final + reaction_rate(T_uoc_current)) / V + T_uoc_current += dt * dT_uoc + T_uoc[k + 1] = T_uoc_current + T_uoc_std[k + 1] = std_next + +r2_robust = r2_manual(T_set, T_robust) +r2_uoc = r2_manual(T_set, T_uoc) +rmse_robust = np.sqrt(np.mean((T_set - T_robust) ** 2)) +rmse_uoc = np.sqrt(np.mean((T_set - T_uoc) ** 2)) + +print(f"R² Robust control: {r2_robust:.4f}") +print(f"R² Unscented control: {r2_uoc:.4f}") +print(f"RMSE Robust: {rmse_robust:.3f} K, RMSE UOC: {rmse_uoc:.3f} K") +print(f"Robust actuator saturation fraction: {robust_saturated.mean():.3f}") +print("Max |T_robust|:", np.nanmax(np.abs(T_robust))) +print("Max |T_uoc|:", np.nanmax(np.abs(T_uoc))) + +# Plots +plt.figure(figsize=(10,3)) +plt.plot(T_set, linestyle='--') +plt.plot(T_robust) +plt.plot(T_uoc) +plt.fill_between(np.arange(len(T_uoc)), T_uoc - T_uoc_std, T_uoc + T_uoc_std, alpha=0.25) +plt.legend(['Setpoint', 'Robust', 'UOC', 'UOC sigma']) +plt.grid(True) +plt.show() + diff --git a/class07/background_materials/LQRvsLQG_Kalman_Filter.jl b/class07/background_materials/LQRvsLQG_Kalman_Filter.jl new file mode 100644 index 0000000..0ce6ea0 --- /dev/null +++ b/class07/background_materials/LQRvsLQG_Kalman_Filter.jl @@ -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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zaU!SoF#S_-%ZM0TFOw}SgFOP*@UdAL&G^mfcG9a#;7v=nUt|*Y9p74|>&4!1?F5jZw9{!VAQD6t{>O zd3=-q` 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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╟─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 diff --git a/class07/background_materials/Temperature_Control_LQR_vs_LQG.py b/class07/background_materials/Temperature_Control_LQR_vs_LQG.py new file mode 100644 index 0000000..95a0f1e --- /dev/null +++ b/class07/background_materials/Temperature_Control_LQR_vs_LQG.py @@ -0,0 +1,115 @@ +import numpy as np +import matplotlib.pyplot as plt + +np.random.seed(0) # reproducible + +# ------------------------------- +# System Definition (1D) +# ------------------------------- +A = 0.9 # state transition +B = 0.5 # control input effect + +# LQR cost weights (tune Q,R to change aggressiveness) +Q = 1.0 +R = 0.1 + +# Simulation parameters +N = 60 +x0 = 60.0 # initial temperature in °C + +# Noise parameters +process_noise_std = 0 # small non-zero process noise +measurement_noise_std = 5.0 + +# Step changes in setpoint +setpoint = np.zeros(N) +setpoint[:20] = 60.0 # start at 60°C +setpoint[20:40] = 80.0 # step to 80°C +setpoint[40:] = 70.0 # step to 70°C + +# ------------------------------- +# Compute LQR gain (discrete-time) +# ------------------------------- +P = Q +for _ in range(200): + P = Q + A**2 * P - (A * P * B)**2 / (R + B**2 * P) +K = (B * P * A) / (R + B**2 * P) +print("LQR gain K =", K) + +# Actuator limits (example) +u_min, u_max = -50.0, 50.0 + +# ------------------------------- +# Naive LQR (reacts directly to noisy measurements) +# ------------------------------- +x_lqr = np.zeros(N) +u_lqr = np.zeros(N) +x_true_lqr = x0 +y_meas_lqr = np.zeros(N) + +for k in range(N): + # Measurement + y = x_true_lqr + np.random.randn() * measurement_noise_std + y_meas_lqr[k] = y + + # Control reacts directly to noisy measurement error (no estimator) + u = -K * (y - setpoint[k]) + # saturate actuator + u = np.clip(u, u_min, u_max) + u_lqr[k] = u + + # State evolves (process noise) + x_true_lqr = A * x_true_lqr + B * u + np.random.randn() * process_noise_std + x_lqr[k] = x_true_lqr + +# ------------------------------- +# LQG (with Kalman filter) -- FIXED: include control in prediction +# ------------------------------- +x_lqg = np.zeros(N) +x_hat = x0 # initial estimate = initial temperature +P_est = 1.0 +u_lqg = np.zeros(N) +x_true = x0 +y_meas_lqg = np.zeros(N) +u_prev = 0.0 + +for k in range(N): + # Measurement + y = x_true + np.random.randn() * measurement_noise_std + y_meas_lqg[k] = y + + # Prediction (include previous control!) + x_pred = A * x_hat + B * u_prev + P_pred = A**2 * P_est + process_noise_std**2 + + # Kalman 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 + + # LQR control based on estimated state + u = -K * (x_hat - setpoint[k]) + u = np.clip(u, u_min, u_max) + u_lqg[k] = u + + # Apply control to true system (with process noise) + x_true = A * x_true + B * u + np.random.randn() * process_noise_std + x_lqg[k] = x_true + + # Save u for next prediction + u_prev = u + +# ------------------------------- +# Plot results +# ------------------------------- +plt.figure(figsize=(12,6)) +plt.plot(x_lqr, label='Naive LQR (reacts to noisy measurements)', color='red') +plt.plot(x_lqg, label='LQG (state estimated via Kalman filter)', color='blue') +plt.plot(setpoint, 'k--', label='Setpoint', alpha=0.8) +plt.plot(y_meas_lqg, 'kx', alpha=0.4, label='Measurements') +plt.xlabel('Time step') +plt.ylabel('Temperature (°C)') +plt.title('LQR vs LQG: Tracking Step Changes in Temperature (fixed Kalman prediction)') +plt.legend() +plt.grid(True) +plt.show() diff --git a/class07/background_materials/Temperature_Robust_vs_LQG.py b/class07/background_materials/Temperature_Robust_vs_LQG.py new file mode 100644 index 0000000..b52f754 --- /dev/null +++ b/class07/background_materials/Temperature_Robust_vs_LQG.py @@ -0,0 +1,89 @@ +# -*- coding: utf-8 -*- +""" +LQG vs H∞ demonstration: H∞ clearly better under persistent disturbance +""" + +import numpy as np +import matplotlib.pyplot as plt +from scipy.linalg import solve_continuous_are + +np.random.seed(42) + +# --------------------------- +# System parameters +# --------------------------- +a = 0.3 +b = 1.0 +dt = 0.1 +T = 50 +N = int(T/dt) + +# --------------------------- +# Setpoint profile +# --------------------------- +x_set = np.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):] = 50.0 + +# --------------------------- +# Disturbance: structured + Gaussian +# --------------------------- +# Persistent, worst-case disturbance for H∞ to handle +w_structured = 2 * np.sin(0.5*np.arange(N)) # structured +w_noise = 2.0 * np.random.randn(N) # small Gaussian noise +w_process = w_structured + w_noise + +v_meas = 15 * np.random.randn(N) # measurement noise + +# --------------------------- +# LQR gain +# --------------------------- +Q = 1.0 +R = 0.1 +P = solve_continuous_are(np.array([[-a]]), np.array([[b]]), np.array([[Q]]), np.array([[R]])) +K_lqr = (b * P / R).item() + +# --------------------------- +# H∞ gain: scale up to attenuate structured disturbance +# --------------------------- +K_hinf = K_lqr * 2.0 # aggressive gain to fight worst-case disturbance + +# --------------------------- +# Simulation +# --------------------------- +x_lqg = np.zeros(N) +x_hat = np.zeros(N) +x_true = 0.0 +x_hinf = np.zeros(N) +x_true_hinf = 0.0 + +for k in range(N-1): + # --- LQG --- + y = x_true + v_meas[k] + # Update estimate + L = 1 # Kalman-like gain (tuned) + 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])) + # Predict next state + x_lqg[k+1] = x_hat[k] + dt*(-a*(x_hat[k]-x_set[k]) + b*(-K_lqr*(x_hat[k]-x_set[k]))) + # True system + 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 + +# --------------------------- +# Plot results +# --------------------------- +plt.figure(figsize=(12,6)) +plt.plot(x_set, 'k--', label='Setpoint') +plt.plot(x_lqg, label='LQG') +plt.plot(x_hinf, label='H∞ Controller') +plt.xlabel('Time step') +plt.ylabel('Temperature / deviation') +plt.title('LQG vs H∞: Persistent disturbance scenario') +plt.legend() +plt.grid(True) +plt.show() diff --git a/class07/background_materials/Unscented_vs_Robust.jl b/class07/background_materials/Unscented_vs_Robust.jl new file mode 100644 index 0000000..c414d5d --- /dev/null +++ b/class07/background_materials/Unscented_vs_Robust.jl @@ -0,0 +1,1297 @@ +### A Pluto.jl notebook ### +# v0.20.19 + +using Markdown +using InteractiveUtils + +# ╔═╡ 624a9e60-9ed7-11f0-2d92-5f53e7aefee8 +using Random, Plots, Statistics + +# ╔═╡ 62c1d144-1e66-46ee-8e58-d3905609e25a +begin + Random.seed!(42) +end + +# ╔═╡ 176ce1f2-aab0-4896-8ba1-484fdb6bfe7a +# --------------------------- +# System parameters +# --------------------------- +begin + V = 1.0 + F1, F2 = 0.5, 0.3 + T1_nom, T2_nom = 350.0, 200.0 + T_env = 298.0 + + dt = 0.02 + T_total = 50.0 + N = Int(T_total / dt) +end + +# ╔═╡ acb1e09d-bf4a-433b-bf22-f6ee01435742 +# --------------------------- +# Setpoint definition +# --------------------------- +begin + T_set = zeros(N) + T_set[1:Int(N*0.25)] .= 330.0 + T_set[Int(N*0.25)+1:Int(N*0.5)] .= 210.0 + T_set[Int(N*0.5)+1:Int(N*0.75)] .= 340.0 + T_set[Int(N*0.75)+1:end] .= 320.0 +end + + +# ╔═╡ 0de7fd3a-c00e-49a2-96f4-64c3b4b1256c +md""" +# Question 1: + +Why do we choose these setpoint steps? + +""" + + +# ╔═╡ a767ed8c-ce75-45f4-8112-49a58a06f84a +# --------------------------- +# Reaction & disturbance parameters +# --------------------------- +begin + reaction_coeff = 5e-5 + feed_noise_std = 0.1 +end + +# ╔═╡ 89d5e261-95db-48a9-a57d-89d90a1de712 +# --------------------------- +# Robust baseline controller parameters +# --------------------------- +begin + K_robust = 9.0 + robust_limits = (-2.5, 2.5) +end + + +# ╔═╡ eed0ea34-670b-4b3e-9228-eda8d08d53dd +# --------------------------- +# Unscented-like controller parameters +# --------------------------- +begin + K_uoc = 45.0 + alpha = 6.0 + sigma_std = 1.0 + uoc_limits = (-120.0, 120.0) +end + + +# ╔═╡ cb000516-425a-4ab1-83e3-4a32a509de6d +# --------------------------- +# Reaction rate function +# --------------------------- +begin + reaction_rate(T) = reaction_coeff * (T - T_env)^2 +end + + +# ╔═╡ 9864ce09-1a94-4eb2-9fb3-0ad1443d9995 +# --------------------------- +# R² metric function +# --------------------------- +begin + function r2_manual(y_true, y_pred) + ss_res = sum((y_true .- y_pred).^2) + ss_tot = sum((y_true .- mean(y_true)).^2) + ss_tot == 0 ? NaN : 1 - ss_res / ss_tot + end +end + + +# ╔═╡ dcbbbdd2-5d96-45b3-9d68-e803c727f3bd +# --------------------------- +# Simulation loop +# --------------------------- +begin +T_robust = zeros(N) +T_uoc = zeros(N) +T_uoc_std = zeros(N) +robust_saturated = zeros(N) + +T_robust[1] = 300.0 +T_uoc[1] = 300.0 +T_uoc_std[1] = sigma_std + +for k in 1:N-1 + T1 = T1_nom + randn() * feed_noise_std + T2 = T2_nom + randn() * feed_noise_std + + # ---- Robust P controller ---- + u_robust = K_robust * (T_set[k] - T_robust[k]) + u_robust = clamp(u_robust, robust_limits[1], robust_limits[2]) + robust_saturated[k] = u_robust in (robust_limits[1], robust_limits[2]) ? 1 : 0 + dT_robust = (F1*(T1 - T_robust[k]) + F2*(T2 - T_robust[k]) + + u_robust + reaction_rate(T_robust[k])) / V + T_robust[k+1] = T_robust[k] + dt * dT_robust + + # ---- Unscented-like controller ---- + sigma_points(T) = [T, T + sigma_std, T - sigma_std, T + 2*sigma_std, T - 2*sigma_std] + propagate_sigma(T_sigma, u, T1, T2) = [T + dt * (F1*(T1 - T) + F2*(T2 - T) + u + reaction_rate(T)) for T in T_sigma] + + u_candidate = K_uoc * (T_set[k] - T_uoc[k]) + T_sigma = sigma_points(T_uoc[k]) + T_sigma_next = propagate_sigma(T_sigma, u_candidate, T1, T2) + error_mean = mean(T_sigma_next) - T_set[k] + u_final = clamp(u_candidate - alpha * error_mean, uoc_limits[1], uoc_limits[2]) + std_next = std(T_sigma_next) + + dT_uoc = (F1*(T1 - T_uoc[k]) + F2*(T2 - T_uoc[k]) + + u_final + reaction_rate(T_uoc[k])) / V + T_uoc[k+1] = T_uoc[k] + dt * dT_uoc + T_uoc_std[k+1] = std_next +end +end + + + +# ╔═╡ 88cdfd9f-1365-4a64-8a6e-82ea581a5820 +md""" +# Question 2: +What is the role of T_{uoc}_{std}? + +It tracks the predicted standard deviation of the Unscented-like controller, +""" + + +# ╔═╡ d833e119-53f6-40fa-9104-667dd9b676bf +# --------------------------- +# Compute performance metrics +# --------------------------- +begin + r2_robust = r2_manual(T_set, T_robust) + r2_uoc = r2_manual(T_set, T_uoc) + rmse_robust = sqrt(mean((T_set .- T_robust).^2)) + rmse_uoc = sqrt(mean((T_set .- T_uoc).^2)) + + println("R² Robust control: ", round(r2_robust, digits=4)) + println("R² Unscented control: ", round(r2_uoc, digits=4)) + println("RMSE Robust: ", round(rmse_robust, digits=3), " K, RMSE UOC: ", round(rmse_uoc, digits=3), " K") + println("Robust actuator saturation fraction: ", round(mean(robust_saturated), digits=3)) + println("Max |T_robust|: ", maximum(abs.(T_robust))) + println("Max |T_uoc|: ", maximum(abs.(T_uoc))) +end + + +# ╔═╡ fccfd8cd-0a5a-4638-a74f-1137446ff478 +# --------------------------- +# Plot results +# --------------------------- +begin + plot(1:N, T_set, linestyle=:dash, label="Setpoint") + plot!(1:N, T_robust, label="Robust") + plot!(1:N, T_uoc, label="UOC") + plot!(1:N, T_uoc .- T_uoc_std, alpha=0.25, label="") + plot!(1:N, T_uoc .+ T_uoc_std, alpha=0.25, label="UOC sigma") +end + + +# ╔═╡ 50733baf-1683-46fc-9490-0c46c5cee1e9 +md""" +# Question 3: + +Why is the Unscented controller better here than 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"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: +# ╠═624a9e60-9ed7-11f0-2d92-5f53e7aefee8 +# ╠═62c1d144-1e66-46ee-8e58-d3905609e25a +# ╠═176ce1f2-aab0-4896-8ba1-484fdb6bfe7a +# ╠═acb1e09d-bf4a-433b-bf22-f6ee01435742 +# ╠═0de7fd3a-c00e-49a2-96f4-64c3b4b1256c +# ╠═a767ed8c-ce75-45f4-8112-49a58a06f84a +# ╠═89d5e261-95db-48a9-a57d-89d90a1de712 +# ╠═eed0ea34-670b-4b3e-9228-eda8d08d53dd +# ╠═cb000516-425a-4ab1-83e3-4a32a509de6d +# ╠═9864ce09-1a94-4eb2-9fb3-0ad1443d9995 +# ╠═dcbbbdd2-5d96-45b3-9d68-e803c727f3bd +# ╠═88cdfd9f-1365-4a64-8a6e-82ea581a5820 +# ╠═d833e119-53f6-40fa-9104-667dd9b676bf +# ╠═fccfd8cd-0a5a-4638-a74f-1137446ff478 +# ╠═50733baf-1683-46fc-9490-0c46c5cee1e9 +# ╟─00000000-0000-0000-0000-000000000001 +# ╟─00000000-0000-0000-0000-000000000002 From c765b6649188bedd9475a2c1957a33b0bce87e19 Mon Sep 17 00:00:00 2001 From: Jouke van Westrenen Date: Thu, 2 Oct 2025 11:19:53 -0400 Subject: [PATCH 2/2] Addressed PR comments: updated Project.toml --- class07/background_materials/Manifest.toml | 2561 ++++++++++++++++++++ class07/background_materials/Project.toml | 12 + 2 files changed, 2573 insertions(+) create mode 100644 class07/background_materials/Manifest.toml create mode 100644 class07/background_materials/Project.toml diff --git a/class07/background_materials/Manifest.toml b/class07/background_materials/Manifest.toml new file mode 100644 index 0000000..a1cb71c --- /dev/null +++ b/class07/background_materials/Manifest.toml @@ -0,0 +1,2561 @@ +# This file is machine-generated - editing it directly is not advised + +julia_version = "1.11.6" +manifest_format = "2.0" +project_hash = "d124f974301d45d704feaa6c61b9016423d54622" + +[[deps.ADTypes]] +git-tree-sha1 = "27cecae79e5cc9935255f90c53bb831cc3c870d7" +uuid = "47edcb42-4c32-4615-8424-f2b9edc5f35b" +version = "1.18.0" + + [deps.ADTypes.extensions] + ADTypesChainRulesCoreExt = "ChainRulesCore" + ADTypesConstructionBaseExt = "ConstructionBase" + ADTypesEnzymeCoreExt = "EnzymeCore" + + [deps.ADTypes.weakdeps] + ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4" + ConstructionBase = "187b0558-2788-49d3-abe0-74a17ed4e7c9" + EnzymeCore = "f151be2c-9106-41f4-ab19-57ee4f262869" + +[[deps.ASL_jll]] +deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] +git-tree-sha1 = "6252039f98492252f9e47c312c8ffda0e3b9e78d" +uuid = "ae81ac8f-d209-56e5-92de-9978fef736f9" +version = "0.1.3+0" + +[[deps.AbstractFFTs]] +deps = ["LinearAlgebra"] +git-tree-sha1 = "d92ad398961a3ed262d8bf04a1a2b8340f915fef" +uuid = "621f4979-c628-5d54-868e-fcf4e3e8185c" +version = "1.5.0" +weakdeps = ["ChainRulesCore", "Test"] + + [deps.AbstractFFTs.extensions] + AbstractFFTsChainRulesCoreExt = "ChainRulesCore" + AbstractFFTsTestExt = "Test" + +[[deps.AbstractPlutoDingetjes]] +deps = ["Pkg"] +git-tree-sha1 = "6e1d2a35f2f90a4bc7c2ed98079b2ba09c35b83a" +uuid = "6e696c72-6542-2067-7265-42206c756150" +version = "1.3.2" + +[[deps.AbstractTrees]] +git-tree-sha1 = "2d9c9a55f9c93e8887ad391fbae72f8ef55e1177" +uuid = "1520ce14-60c1-5f80-bbc7-55ef81b5835c" +version = "0.4.5" + +[[deps.Adapt]] +deps = ["LinearAlgebra", "Requires"] +git-tree-sha1 = 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+[deps] +CairoMakie = "13f3f980-e62b-5c42-98c6-ff1f3baf88f0" +Ipopt = "b6b21f68-93f8-5de0-b562-5493be1d77c9" +JuMP = "4076af6c-e467-56ae-b986-b466b2749572" +MarkdownLiteral = "736d6165-7244-6769-4267-6b50796e6954" +MeshCat = "283c5d60-a78f-5afe-a0af-af636b173e11" +Pluto = "c3e4b0f8-55cb-11ea-2926-15256bba5781" +PlutoTeachingTools = "661c6b06-c737-4d37-b85c-46df65de6f69" +PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8" +Revise = "295af30f-e4ad-537b-8983-00126c2a3abe" +RigidBodyDynamics = "366cf18f-59d5-5db9-a4de-86a9f6786172" +TORA = "a6da6c0f-f153-4ec6-bf42-6dc0ab733f84"