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diff --git a/class07/Kalman_Filtering.jl b/class07/Kalman_Filtering.jl
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--- a/class07/Kalman_Filtering.jl
+++ b/class07/Kalman_Filtering.jl
@@ -1,81 +1,1400 @@
-## Kalman Filtering
-# Goal: Estimate the true state of a system when measurements are noisy.
-#
-# In many practical systems, we cannot directly observe the full state x_t.
-# Instead, we have noisy measurements y_t. The Kalman filter provides the
-# optimal linear estimate of the hidden state, assuming linear dynamics
-# and Gaussian noise.
-
-using LinearAlgebra
-
-# The Kalman filter operates in two steps: prediction and update.
-
-## Prediction Step:
-# Forecast the next state based on the previous estimate and the system model.
-# x_pred = A * x_prev + B * u_prev
-# - This uses the known dynamics of the system (A, B) and the previous control input u_prev
-# P_pred = A * P_prev * A' + Q
-# - The uncertainty (covariance) is propagated forward
-# - Q represents process noise: uncertainty in how the system evolves
-# - Larger P_pred means we are less confident in the prediction
-
-## Update Step:
-# Correct the prediction using the latest measurement y
-# K = P_pred * C' * inv(C * P_pred * C' + R)
-# - The Kalman gain K determines how much we trust the measurement vs. prediction
-# x_hat = x_pred + K * (y - C * x_pred)
-# - Innovation (y - C*x_pred) adjusts our estimate based on new information
-# P = (I - K * C) * P_pred
-# - Covariance decreases after incorporating measurement, increasing confidence
-
-function kalman_filter(A, B, C, Q, R, y, u=nothing)
- n = size(A,1)
- N = length(y)
- x_hat = zeros(n, N) # estimated states
- P = zeros(n,n) # covariance
- I = Matrix(I, n, n)
-
- # Handle optional control input
- u = isnothing(u) ? zeros(size(B,2), N) : u
+### 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
+
+# ╔═╡ eb2ae410-c3bc-11f0-0c2e-ffdf8b66341d
+begin
+ using Markdown
+ using LinearAlgebra
+end
+
+# ╔═╡ 1d151eed-1e89-4b67-8365-cc5c1319cfd4
+begin
+ using PlutoUI
+ @bind noise_slider Slider(0:0.5:10, default=5, show_value=true)
+end
+
+# ╔═╡ 0037f4e3-d46a-481c-b1d6-7de99026e02e
+### Introduction
+md"""
+# Kalman Filtering
+### Estimating hidden states from noisy measurements
+
+Many real-world systems have internal states $x_t$ that **cannot be directly measured**:
+
+- A robot cannot directly observe the true position of all joints
+- A drone cannot perfectly measure its velocity
+- A temperature system may only measure surface temperature, not internal temperature
+- A navigation system only gets noisy GPS measurements
+
+To reconstruct these hidden states, we use the **Kalman filter** — the optimal estimator for:
+
+- **Linear** dynamical systems
+- **Gaussian** noise (process + measurement)
+
+It produces the best possible estimate (in a least-squares sense) of the true system state.
+"""
+
+
+# ╔═╡ 590597d0-9b09-450a-a72e-c1a57f2bb10e
+### System Model
+md"""
+## 1. System Model
+
+### **State Dynamics**
+$x_{t+1} = A x_t + B u_t + w_t$
+
+### **Measurement Model**
+$y_t = C x_t + v_t$
+
+Where:
+- \($x_t$\): true state (hidden)
+- \($u_t$\): control input
+- \($y_t$\): noisy measurement
+- \($A$\): state transition matrix
+- \($B$\): input matrix
+- \($C$\): measurement matrix
+- \($w_t \sim \mathcal{N}(0, Q)$\): process noise
+- \($v_t \sim \mathcal{N}(0, R)$\): measurement noise
+
+
+
+
+### Interpretation
+
+- **Q**: describes uncertainty in how the system evolves
+- **R**: describes uncertainty in the measurement
+
+The Kalman filter maintains not only a state estimate \($\hat{x}_t$\) but also its **uncertainty**, represented by a covariance matrix \($P_t$\).
+"""
+
+
+# ╔═╡ e6acf358-d786-4412-81e0-4c12c1e4bb50
+### Prediction & Update Steps
+md"""
+## 2. The Two-Stage Structure of the Kalman Filter
+
+### **(1) Prediction Step**
+
+$\hat{x}_{t|t-1} = A \hat{x}_{t-1|t-1} + B u_{t-1}$
+
+
+$P_{t|t-1} = A P_{t-1|t-1} A^\top + Q$
+
+
+### **(2) Update Step**
+$r_t = y_t - C \hat{x}_{t|t-1},\quad
+S_t = C P_{t|t-1} C^\top + R,\quad
+K_t = P_{t|t-1} C^\top S_t^{-1}$
+
+$\hat{x}_{t|t} = \hat{x}_{t|t-1} + K_t r_t,\quad
+P_{t|t} = (I - K_t C) P_{t|t-1}$
+"""
+
+
+# ╔═╡ 6c067283-b21f-4f5f-be31-fe74160af86a
+begin
+ 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
+
+# ╔═╡ bed5c65e-db0a-42cd-a21e-8595436cd929
+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
+
+# ╔═╡ 58cc53d8-092f-4790-8b3e-e4c283381a4e
+begin
+ x = zeros(N)
+ u = zeros(N)
+ x_true = copy(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
+end
+
+
+
+# ╔═╡ 107edffe-613a-4f99-bc52-a17ffccc425d
+begin
+ using Plots
+ 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
+
+# ╔═╡ 0ba5cfc8-9f65-4b02-924d-b17f7b723cdc
+
+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
+
+# ╔═╡ be08fffb-f837-4916-b853-0ba5898a7194
+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
- if k == 1
- x_pred = zeros(n) # initial guess for first step
- P_pred = P + Q
- else
- x_pred = A * x_hat[:,k-1] + B * u[:,k-1]
- P_pred = A * P * A' + Q
- end
+ x_pred = A * x_hat + B * u_prev
+ P_pred = A^2 * P_est + process_noise_std^2
# Update
- K = P_pred * C' * inv(C * P_pred * C' + R)
- x_hat[:,k] = x_pred + K * (y[k] - C * x_pred)
- P = (I - K * C) * P_pred
+ 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
- return x_hat
+ 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
-## Example usage:
-# A simple demonstration with a small system. This shows how the filter
-# reconstructs the true state from noisy measurements.
+# ╔═╡ efc27e04-3d85-4fdc-b2a6-7acc8a42b8dc
+###Explanation of Output
+md"""
+# Understanding the Output
-if abspath(PROGRAM_FILE) == @__FILE__
- # Define system matrices (example: 2-state thermal system)
- A = [1.0 1.0; 0.0 1.0] # state transition
- B = [0.0; 1.0] # control input
- C = [1.0 0.0] # measurement matrix
- Q = 0.01 * I(2) # process noise covariance
- R = 0.1 # measurement noise covariance
+$\hat{x}_t =
+\begin{bmatrix}
+\hat{T}_t \\
+\widehat{\dot{T}}_t
+\end{bmatrix}$
- # Simulated noisy measurements
- true_states = [20.0, 21.0, 23.0, 22.0]
- y = true_states .+ randn(length(true_states)) * sqrt(R)
+Even if we never measure \($\dot{T}$\), the Kalman filter reconstructs it because:
- # Run Kalman filter
- x_hat = kalman_filter(A, B, C, Q, R, y)
+- The model links temperature to its rate
+- The filter observes how temperature evolves
+- It infers unobserved states from dynamics
+"""
- println("Noisy measurements: ", y)
- println("Estimated states: ", x_hat)
-end
+
+# ╔═╡ 2ab7f491-0025-4161-8ab2-eb3401b0574f
+### Cell 9: Key Takeaways
+md"""
+## 6. Summary (Key Takeaways)
+
+- Optimal estimate for linear-Gaussian systems
+- Prediction + update steps
+- Kalman gain balances trust between model & measurement
+- Outputs state estimate + uncertainty
+- Can reconstruct hidden states like temperature rate
+- Widely used in robotics, navigation, finance, thermal control
+"""
+
+
+# ╔═╡ 00000000-0000-0000-0000-000000000001
+PLUTO_PROJECT_TOML_CONTENTS = """
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diff --git a/class07/LQG_Control.jl b/class07/LQG_Control.jl
index d3f0634..0da0483 100644
--- a/class07/LQG_Control.jl
+++ b/class07/LQG_Control.jl
@@ -1,84 +1,1241 @@
-# Linear Quadratic Gaussian (LQG) Control
-#
-# LQG control is about designing an optimal controller for linear systems
-# when there is uncertainty in both the system dynamics and the measurements.
-#
-# System setup:
-# x_{t+1} = A * x_t + B * u_t + w_t
-# y_t = C * x_t + v_t
-# Here, w_t and v_t are Gaussian process and measurement noise.
-#
-# The goal is to minimize a quadratic cost function:
-# J = E[ sum(x_t' * Q * x_t + u_t' * R * u_t) ]
-# which balances performance (state tracking) and control effort.
-
-using LinearAlgebra
-
-# Linear Quadratic Regulator (LQR)
-#
-# The LQR computes the optimal control law for a deterministic linear system
-# when the full state is known. For continuous-time systems:
-# dx/dt = A*x + B*u
-# The quadratic cost is:
-# J = ∫ (x'Qx + u'Ru) dt
-# The optimal control law is:
-# u = -K*x
-# where K is obtained by solving the continuous-time Riccati equation:
-# A'*P + P*A - P*B*R^-1*B'*P + Q = 0
-#
-# Here, we implement a discrete-time version.
-
-function dlqr(A, B, Q, R)
- P = copy(Q)
- maxiter = 1000
- tol = 1e-8
- for i in 1:maxiter
- P_new = A' * P * A - A' * P * B * inv(R + B' * P * B) * B' * P * A + Q
- if norm(P_new - P) < tol
- P = P_new
- break
- end
- P = P_new
- end
- K = inv(R + B' * P * B) * B' * P * A
- return K, P
-end
+### A Pluto.jl notebook ###
+# v0.20.19
+
+using Markdown
+using InteractiveUtils
-# Kalman Filter (State Estimation)
-#
-# In LQG, the controller does not know the true state x_t.
-# Instead, it receives noisy measurements y_t and uses a Kalman filter
-# to produce an optimal estimate x_hat_t.
-#
-# The filter updates the estimate according to:
-# x_hat_{t+1} = A*x_hat_t + B*u_t + L*(y_t - C*x_hat_t)
-# where L is the Kalman gain chosen to minimize estimation error variance.
-
-function kalman_filter(A, B, C, Q, R, y)
- n = size(A, 1)
- x_hat = zeros(n, length(y))
- P = zeros(n, n)
- for t in 1:length(y)
- # Prediction step
- if t == 1
- x_pred = zeros(n)
- P_pred = P + Q
- else
- x_pred = A * x_hat[:, t-1]
- P_pred = A * P * A' + Q
- end
- # Update step
- K = P_pred * C' * inv(C * P_pred * C' + R)
- x_hat[:, t] = x_pred + K * (y[t] - C * x_pred)
- P = (I - K * C) * P_pred
- end
- return x_hat
+# ╔═╡ a65b6bda-b377-4fce-8a20-9666f8c7ed02
+begin
+ using Random, Plots, PlutoUI, LinearAlgebra
end
-# Separation Principle
-#
-# A key property of LQG control is the separation principle:
-# The design of the controller (LQR) and the estimator (Kalman filter)
-# can be done independently. Once K and L are computed, the control input is:
-# u_t = -K * x_hat_t
-# This allows the system to act as if the true state were known, using only the estimated state.
+
+# ╔═╡ 62981e50-c3c7-11f0-274e-55979329fb62
+### A Pluto.jl notebook ###
+# v0.20.19
+# Chapter: Linear Quadratic Gaussian (LQG) Control — Theory, Implementation, and Examples
+
+
+# ╔═╡ 91e83c50-a183-4493-bb9f-4d00d74c66c9
+md"""
+# Chapter: Linear Quadratic Gaussian (LQG) Control
+
+**Learning objectives:**
+1. Understand the discrete-time LQR problem: objective, Riccati equation, and feedback law.
+2. Understand the discrete-time Kalman filter for linear-Gaussian systems: prediction, update, and gain selection.
+3. Learn the Separation Principle: when and why LQR and Kalman designs can be performed independently.
+4. Implement and simulate a scalar temperature-control example.
+5. Measure performance with quadratic cost and diagnostic metrics.
+
+**Prerequisites**
+- Basic linear algebra (matrices, eigenvalues).
+- Probability basics (Gaussian mean/covariance).
+- Concepts of stability, controllability, and observability.
+
+This is a self-contained teaching chapter. Each code cell is accompanied by a thorough explanation of variables and derivations.
+"""
+
+
+# ╔═╡ 39bd64fe-4442-457e-8dd7-9f299fdd3981
+md"""
+## 1 — System setup & notation
+
+We consider a discrete-time linear dynamical system with additive Gaussian noise:
+
+
+$x_{k+1} = A x_k + B u_k + w_k \qquad (1)$
+
+$y_k = C x_k + v_k \qquad (2)$
+
+- \($x_k\in\mathbb{R}^n$\) is the *state* at time step \(k\).
+- \($u_k\in\mathbb{R}^m$\) is the *control input*.
+- \($y_k\in\mathbb{R}^p$\) is the *measurement*.
+- \($w_k$\) is *process noise*, modeled as zero-mean Gaussian with covariance \($W$\): \($w_k \sim \mathcal{N}(0, W)$\).
+- \($v_k$\) is *measurement noise*, zero-mean Gaussian with covariance \($V$\): \($v_k \sim \mathcal{N}(0, V)$\).
+
+**Goal (LQG):** design a control law that minimizes the expected infinite-horizon quadratic cost
+
+$J = \mathbb{E}\!\left[\sum_{k=0}^{\infty} x_k^\top Q x_k + u_k^\top R u_k \right]$
+subject to the dynamics (1) and noisy measurements (2).
+- \($Q\succeq 0$\) (state penalty), \($R\succ 0$\) (control penalty).
+
+**Separation principle:**
+If \(A,B,C\) are such that the standard assumptions hold (controllable/observable, noise Gaussian), then the optimal LQG controller is the combination of:
+1. an LQR state-feedback controller,
+2. a Kalman filter producing state estimates from measurements.
+
+We will derive both parts and show the combined simulation in this chapter.
+"""
+
+
+# ╔═╡ 63db473b-066c-4fa7-8a99-327efd224ea0
+md"""
+## 2 — LQR (Linear Quadratic Regulator)
+
+**Problem statement (deterministic state known):**
+
+Given the deterministic linear system \($x_{k+1} = A x_k + B u_k$\), choose a control law \($u_k = -K x_k$\) that minimizes
+
+
+$J = \sum_{k=0}^{\infty} x_k^\top Q x_k + u_k^\top R u_k.$
+
+**Discrete Algebraic Riccati Equation (DARE):** The (positive semidefinite) matrix \(P\) solving
+
+$P = A^\top P A - A^\top P B (R + B^\top P B)^{-1} B^\top P A + Q$
+
+yields the optimal feedback gain
+
+$K = (R + B^\top P B)^{-1} B^\top P A$
+"""
+
+
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--- a/class07/Robust_Control.jl
+++ b/class07/Robust_Control.jl
@@ -1,76 +1,364 @@
-## Robust Control
-# Goal: Ensure stability and performance under model uncertainty and disturbances.
-#
-# In real-world systems, models are never perfect. Uncertainties in parameters,
-# unmodeled dynamics, and unexpected disturbances can degrade performance.
-# Robust control explicitly accounts for these worst-case scenarios.
-
-using LinearAlgebra
-
-## Core Idea:
-# Robust control designs controllers to maintain stability and acceptable performance
-# even when the system deviates from the nominal model or experiences large disturbances.
-
-## Two main philosophies:
-# 1. Stochastic approaches (e.g., LQG) assume uncertainties are random with known statistics.
-# 2. Worst-case approaches (e.g., H-infinity) plan for the most adverse possible uncertainty
-# within a specified bound.
-
-# Example system with stochastic uncertainty (LQG view):
-# x_{k+1} = A*x_k + B*u_k + w_k, w_k ~ N(0, Q)
-# Measurement noise is handled via Kalman filtering.
-
-# Example system with worst-case uncertainty:
-# x_{k+1} = A*x_k + B*u_k + d_k, ||d_k|| <= delta
-# The controller is designed to handle the largest possible disturbance d_k.
-
-## Robust Trajectory Optimization:
-# Controller chooses inputs to minimize cost, while nature chooses worst-case
-# uncertainties and disturbances to maximize cost:
-# min_u max_{ΔA, ΔB, w} J = sum(x_t' Q x_t + u_t' R u_t)
-# subject to x_{t+1} = (A + ΔA)x_t + (B + ΔB)u_t + w_t
-# and ||ΔA|| <= ΔA_bar, ||ΔB|| <= ΔB_bar, ||w_t|| <= w_bar
-#
-# Interpretation:
-# - u_t: controller input minimizing cost
-# - ΔA, ΔB, w_t: nature’s adversarial choice
-# - Ensures robust performance under bounded uncertainties
-
-## H-infinity Control (Frequency-domain worst-case):
-# For infinite-horizon systems, worst-case amplification of disturbances is bounded:
-# max_{w != 0} ||z||_2 / ||w||_2 = ||T_{zw}||_∞ < γ
-# where T_{zw} is the transfer function from disturbance w to critical output z.
-
-# State-space H-infinity formulation:
-# dx/dt = A x + B_u u + B_w w
-# z = C_z x + D_zu u + D_zw w
-# y = C_y x + D_yu u + D_yw w
-# Goal: Design u(t) to minimize effect of worst-case w(t) on z(t).
-
-## Structured Uncertainty and μ-Synthesis:
-# H-infinity: optimizes against norm-bounded uncertainties.
-# μ-synthesis: extends to structured uncertainties (Δ), e.g., actuator or sensor errors.
-# μ < 1 ensures robustness against all allowed structured uncertainties.
-# D-K iteration procedure:
-# 1. D-step: compute scaling matrix D to bound effect of uncertainties
-# 2. K-step: design controller K to minimize μ under current scaling
-# Repeat until convergence.
-
-# Example comparison: LQG vs H-infinity under persistent disturbance
-#
-# System: single-state process with dynamics
-# dx/dt = -a*(x - x_set) + b*u + w_process
-# y = x + v_meas
-# where w_process = 2*sin(0.5*t) + Gaussian noise
-# and v_meas ~ N(0, 15^2)
-#
-# Setpoint: step changes over time
-# Controller design:
-# - LQG: optimal state feedback + Kalman filter
-# - H-infinity: gain to attenuate worst-case disturbances
-#
-# Outcome:
-# - LQG handles stochastic noise well
-# - H-infinity maintains performance under worst-case disturbances
-
-println("Robust control chapter loaded. Use this file to implement H∞ or μ-synthesis controllers.")
+### A Pluto.jl notebook ###
+# v0.20.19
+
+using Markdown
+using InteractiveUtils
+
+# ╔═╡ 6527df00-c3c9-11f0-13e2-7d05788e7333
+md"""
+# Robust Control
+### Ensuring stability and performance under **model uncertainty**, **unmodeled dynamics**, and **external disturbances**
+
+In real-world systems including temperature control systems, vehicles, chemical processes, and financial models, our mathematical models are *never* exact.
+Parameters drift, sensors degrade, actuators saturate, and disturbances appear in unexpected ways.
+
+**Robust control** is the branch of control theory that explicitly accounts for these uncertainties and guarantees acceptable performance *even under worst-case conditions*.
+
+This notebook walks through:
+
+1. What model uncertainty means and how it is represented
+2. Stochastic vs worst-case philosophies
+3. LQG and stochastic robustness
+4. H∞ (H-infinity) worst-case robustness
+5. μ-synthesis and structured uncertainty
+6. Robust trajectory optimization
+7. A temperature-control example contrasting LQG and H∞ controllers
+"""
+
+
+# ╔═╡ efd5146a-fe33-48fb-9048-e60b9373a461
+md"""
+# ## 1. What Is Model Uncertainty?
+
+Even if we write a model
+
+$x_{k+1} = A x_k + B u_k$
+
+we know that in reality:
+
+- The matrix **A** is not exact
+- The matrix **B** is not exact
+- There are disturbances, offsets, nonlinearities, bias, noise
+
+This leads to a more realistic model:
+
+$x_{k+1} = (A + \Delta A)x_k + (B + \Delta B)u_k + w_k$
+
+Where:
+
+- **$ΔA, ΔB$**: unknown deviations from the nominal model
+- **$w_k$**: disturbance or process noise
+- **$v_k$**: measurement noise (if sensors are used)
+
+The goal of robust control is to design a controller **u_k** that works for all allowed $ΔA, ΔB, w_k$.
+
+Different robust-control frameworks differ by **how uncertainty is assumed to behave**.
+"""
+
+
+# ╔═╡ 3fb8940e-3ef7-43b0-90bc-38bd320120e7
+md"""
+# 2. Two Philosophies of Robustness
+
+Robust control is centered around two fundamentally different interpretations of uncertainty:
+
+---
+
+## **2.1 Stochastic Approaches (example: LQG)**
+These assume that uncertainties behave like **random variables** with known distributions.
+
+Example assumptions:
+
+- Process noise \($w_k \sim \mathcal{N}(0, Q)$\)
+- Measurement noise \($v_k \sim \mathcal{N}(0, R)$\)
+
+You design:
+
+- A **Kalman filter** to optimally estimate the state from noisy measurements
+- An **LQR** controller to compute optimal control inputs
+
+Together they form the **LQG controller**.
+
+### Strengths
+- Optimal when noise statistics are correct
+- Very strong performance in predictable environments
+
+### Weaknesses
+- Not robust to **worst-case disturbances**
+- Sensitive when the real noise distribution deviates from assumptions
+
+---
+
+## **2.2 Worst-Case Approaches (example: H∞)**
+Here uncertainty is treated as **an adversary** that tries to degrade performance.
+
+Instead of assuming a distribution, we assume a **bound**:
+
+
+$\|\Delta A\| \le \bar{\Delta A}, \quad \|\Delta B\| \le \bar{\Delta B}, \quad \|w_k\| \le w_{\max}$
+
+The controller is designed so that:
+
+- Even the **worst choice** of disturbances cannot destabilize the system
+- Performance measures remain within acceptable bounds
+
+### Strengths
+- Guaranteed performance under worst conditions
+- Robust to modeling errors
+
+### Weaknesses
+- More conservative
+- Can sacrifice performance during nominal conditions
+"""
+
+
+# ╔═╡ bb01c629-ffd9-4d40-b505-0a76f779f500
+md"""
+# 3. Detailed Stochastic Example: LQG Modeling
+
+We begin with a discrete-time stochastic system:
+
+$x_{k+1} = A x_k + B u_k + w_k,
+\quad w_k \sim \mathcal{N}(0, Q)$
+
+and noisy observations:
+
+$y_k = C x_k + v_k,
+\quad v_k \sim \mathcal{N}(0, R)$
+
+### Inside LQG:
+
+1. **Kalman filter** produces an optimal state estimate
+2. **LQR** computes
+ $u_k = -K \hat{x}_k$
+3. LQG = LQR + Kalman filter
+
+This is conceptually robust to *random* noise, but not robust to **bounded, structured uncertainties** in A or B.
+"""
+
+
+# ╔═╡ fd8746c9-42ca-4ecb-8040-17960982b533
+md"""
+# 4. Worst-Case Formulation: H∞ Control
+
+### Core idea:
+The controller competes against an adversarial disturbance that tries to maximize the output energy.
+
+We want the induced L2 gain from disturbance input \($w$\) to regulated output \($z$\) to satisfy:
+
+
+$\|T_{zw}\|_\infty < \gamma$
+
+Where:
+
+- \($T_{zw}$\) is the closed-loop transfer function
+- \($\gamma$\) is the worst-case amplification bound
+
+Smaller \($\gamma$\) = more robust to disturbances.
+
+---
+
+## State-space model:
+
+$\dot{x} = A x + B_u u + B_w w$
+
+$z = C_z x + D_{zu} u + D_{zw} w$
+
+$y = C_y x + D_{yu} u + D_{yw} w$
+
+The H∞ control problem finds a controller \($u(t)$\) that makes the system robust against the worst possible disturbance input \($w(t)$\).
+
+This yields:
+
+- Guaranteed stability under model uncertainty
+- Guaranteed bound on performance degradation
+"""
+
+
+# ╔═╡ 35ae220e-f0e8-407c-b957-5ebf869c0c47
+md"""
+# ## 5. μ-Synthesis: Handling Structured Uncertainty
+
+### Why H∞ is not enough
+H∞ is great for unknown but “norm-bounded” uncertainties (i.e., “anything with size < δ”).
+But many real systems have **structured uncertainty**, e.g.:
+
+- Sensor bias
+- Actuator saturation
+- Temperature-dependent parameters
+- Known block-diagonal uncertainty structures
+- Gain drift only in one channel
+
+H∞ cannot capture these structures.
+
+---
+
+## μ-synthesis solves:
+
+$\mu < 1$
+
+where **μ** (mu) is the structured singular value.
+
+### What μ measures:
+It measures whether structured uncertainty Δ can destabilize the system:
+
+- If μ < 1 → stable under all allowed Δ
+- If μ ≥ 1 → system might become unstable
+
+---
+
+## D–K Iteration (the μ-synthesis algorithm)
+
+### Step 1: **D-step**
+Compute a scaling matrix \($D$\) that upper-bounds the effect of structured uncertainty.
+
+### Step 2: **K-step**
+Design an H∞ controller using that scaling.
+
+### Repeat:
+D → K → D → K
+until convergence.
+
+μ-synthesis produces controllers that are:
+
+- More robust than H∞
+- Less conservative
+- Explicitly account for structured uncertainty
+"""
+
+
+# ╔═╡ 82075bd8-4908-4b4e-9f7b-b616386d59a5
+md"""
+# 6. Robust Trajectory Optimization
+
+Worst-case robust control can be formulated as a **min-max game**:
+
+$\min_{u_{0:T}} \max_{\Delta A, \Delta B, w_{0:T}}
+\sum_{t=0}^{T} x_t^\top Q x_t + u_t^\top R u_t$
+
+subject to:
+
+$x_{t+1} = (A + \Delta A)x_t + (B + \Delta B)u_t + w_t$
+
+with bounds:
+
+$\|\Delta A\| \le \bar{\Delta A}, \quad
+\|\Delta B\| \le \bar{\Delta B}, \quad
+\|w_t\| \le w_{\max}$
+
+Interpretation:
+
+- **Controller (minimizer)** chooses u(t) to keep performance good
+- **Nature (maximizer)** chooses worst possible disturbance
+
+This viewpoint unifies many robust control methods:
+
+- Online robust MPC
+- H∞ design
+- Adversarial learning
+- Distributionally-robust optimization
+"""
+
+
+# ╔═╡ 3e1b18e2-06b6-4299-8d58-202c99d399ef
+md"""
+# 7. Temperature Control Example: LQG vs H∞
+
+We consider a scalar thermal process:
+
+
+$\frac{dx}{dt} = -a(x - x_\text{set}) + b u + w_{\text{process}}$
+
+where:
+
+- \($x$\): system temperature
+- \($x_\text{set}$\): desired temperature
+- \($a > 0$\): thermal dissipation rate
+- \($b$\): actuator effectiveness (heater/cooler)
+- \($w_{\text{process}}$\): external heat disturbances
+
+### Disturbance model:
+
+$w_{\text{process}}(t) = 2\sin(0.5t) + \text{Gaussian noise}$
+
+### Measurement:
+$y = x + v_{\text{meas}}, \quad v_{\text{meas}} \sim \mathcal{N}(0, 15^2)$
+
+---
+
+## **LQG Controller**
+
+- Assumes noise is Gaussian with known covariance
+- Kalman filter estimates the temperature
+- LQR regulates temperature
+
+**Good performance under expected/stochastic conditions**.
+
+---
+
+## **H∞ Controller**
+
+- Assumes worst-case disturbance with bounded magnitude
+- Gains chosen to attenuate worst disturbances
+- No assumption of sinusoid or Gaussian behavior
+
+**More conservative, but guaranteed to perform well under extreme disturbance conditions.**
+
+---
+
+## Outcome Comparison:
+
+### LQG:
+- Excellent tracking when noise is reasonably small
+- Vulnerable to persistent sinusoidal disturbances
+- Not robust to model mismatch (e.g., actuator weakening)
+
+### H∞:
+- Very stable under persistent oscillations
+- Handles sudden changes and worst-case heat pulses
+- Slightly worse nominal tracking due to conservative nature
+"""
+
+
+# ╔═╡ bcc3e98a-d94d-4d1c-945c-11f770b12bae
+md"""
+---
+# Final Message
+This notebook provides an introduction to robust control, covering:
+
+- Stochastic vs worst-case uncertainty
+- LQG, H∞, μ-synthesis
+- Structured vs unstructured uncertainty
+- Min-max robust trajectory optimization
+- Realistic temperature control example
+"""
+
+
+# ╔═╡ 00000000-0000-0000-0000-000000000001
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+# ╟─00000000-0000-0000-0000-000000000002
diff --git a/class07/Robust_vs_LQG.jl b/class07/Robust_vs_LQG.jl
new file mode 100644
index 0000000..c0feeab
--- /dev/null
+++ b/class07/Robust_vs_LQG.jl
@@ -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?
+"""
+
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diff --git a/class07/StochasticOptimalControl_Overview.md b/class07/StochasticOptimalControl_Overview.md
deleted file mode 100644
index 77d3712..0000000
--- a/class07/StochasticOptimalControl_Overview.md
+++ /dev/null
@@ -1,40 +0,0 @@
-# StochasticOptimalControl.jl
-# Overview of Stochastic Optimal Control (SOC)
-# This file provides a narrative introduction to SOC, its motivation, methods,
-# and applications, with formulas explained in context.
-
-println("Stochastic Optimal Control chapter loaded. This file can be used to understand and implement LQG, Robust, and Unscented control methods.")
-
-println("""
-Stochastic Optimal Control (SOC) is concerned with choosing control actions in systems where both the dynamics and the observations are noisy.
-In real-world systems, uncertainties arise from sensor noise, model inaccuracies, and external disturbances.
-SOC explicitly accounts for these uncertainties while attempting to optimize a performance objective.
-
-Consider a discrete-time system with dynamics given by:
- x_{t+1} = A * x_t + B * u_t + w_t
-where x_t represents the state at time t, u_t is the control input, and w_t is a stochastic disturbance, typically modeled as a zero-mean Gaussian with covariance Q_w.
-
-Measurements are also noisy:
- y_t = C * x_t + v_t
-where v_t represents measurement noise with covariance R_v.
-
-The control objective is usually formulated as the minimization of an expected quadratic cost:
- J = E[ sum_{t=0}^{T-1} (x_t' * Q * x_t + u_t' * R * u_t) ]
-Here, Q and R are weighting matrices that balance the importance of state deviations versus control effort.
-
-This framework allows controllers to explicitly trade off performance and robustness, producing actions that are principled and reliable even under uncertainty.
-
-SOC has broad applications. In aerospace engineering, it is used to stabilize aircraft under turbulence and wind gusts. In robotics, SOC ensures reliable navigation when sensors are noisy or imperfect. In finance, it helps optimize portfolios under stochastic returns. In process engineering, SOC helps control chemical reactors, distillation columns, and energy systems where measurement noise and disturbances are significant.
-
-Traditional deterministic control assumes perfect knowledge of the system state and neglects uncertainty. Controllers designed under this assumption often perform poorly or fail when noise or model mismatch is present. SOC remedies this by explicitly modeling uncertainties and incorporating them into the decision-making process.
-
-Measurement choices directly affect the uncertainties the controller must handle. For example, temperature can be measured using thermocouples, resistance temperature detectors (RTDs), or infrared sensors. Thermocouples are inexpensive and cover a wide temperature range, but they are noisy and less accurate. RTDs are more precise and stable but slower and costlier. Infrared sensors are non-contact and fast, but their readings depend on surface emissivity and line-of-sight. The method chosen determines the type and magnitude of errors the controller will encounter, which influences the design of the control strategy.
-
-Several stochastic control methods exist, each suited to different types of systems and uncertainty. Linear Quadratic Gaussian (LQG) control is optimal for linear systems with Gaussian noise, providing a separation principle between state estimation and control. Robust control, including H-infinity methods, focuses on worst-case scenarios, ensuring stability and bounded performance under model mismatch or bounded disturbances. Unscented Optimal Control (UOC) and iterative Linear Quadratic Gaussian (iLQG) approaches handle nonlinear dynamics and non-Gaussian noise. Sigma-point propagation, used in UOC, accurately tracks the mean and covariance of the state through nonlinear transformations. iLQG iteratively linearizes the system around a nominal trajectory and computes locally optimal control laws.
-
-The choice of method depends on the system's linearity, noise characteristics, uncertainty magnitude, and performance versus safety requirements. There is no single stochastic control method that is universally optimal; the system context and design priorities must guide the selection.
-
-In this chapter, we will examine four major areas. First, LQG control is introduced to illustrate optimal control for linear systems under Gaussian noise and the separation between estimation and control. Second, Kalman filtering is described as a recursive technique for estimating system states from noisy measurements. Third, robust control methods are discussed, contrasting stochastic and worst-case approaches, and introducing H-infinity methods for handling uncertainties and disturbances. Finally, Unscented Optimal Control and iLQG are presented, showing how sigma-point propagation and iterative trajectory optimization allow SOC methods to handle nonlinear stochastic systems effectively.
-
-This narrative provides the foundation for understanding stochastic optimal control, highlighting the importance of explicitly handling uncertainty and the trade-offs between performance and robustness in practical systems.
-""")
diff --git a/class07/Unscented Control.html b/class07/Unscented Control.html
new file mode 100644
index 0000000..26de5fc
--- /dev/null
+++ b/class07/Unscented Control.html
@@ -0,0 +1,19 @@
+
+
+
+
+
+
+
+
\ No newline at end of file
diff --git a/class07/Unscented_Control.jl b/class07/Unscented_Control.jl
index dba0d60..d68b4c9 100644
--- a/class07/Unscented_Control.jl
+++ b/class07/Unscented_Control.jl
@@ -1,101 +1,1539 @@
-# Unscented Optimal Control (UOC) Chapter
-#
-# Nonlinear stochastic optimal control is challenging because standard
-# methods like LQG assume linear dynamics or rely on linearization,
-# which can introduce significant errors when the system is strongly nonlinear.
-# Unscented control (UOC) addresses this by accurately propagating state
-# distributions through nonlinear dynamics using sigma points.
-
-# Motivation:
-# - LQG assumes linear system dynamics or uses a first-order approximation.
-# - Robust control ensures stability under worst-case disturbances but
-# may be conservative for nonlinear stochastic systems.
-# - Unscented control provides a middle ground by capturing nonlinear
-# stochastic effects without linearization.
-
-# Problem formulation:
-# Consider a discrete-time nonlinear system with additive process noise:
-# x_{k+1} = f(x_k, u_k) + w_k
-# The goal is to find a control sequence u_0, ..., u_{T-1} that minimizes
-# E[ sum_{k=0}^{T-1} ℓ(x_k, u_k) + ℓ_T(x_T) ]
-# where ℓ is the stage cost and ℓ_T is the terminal cost.
-# The expectation accounts for the stochastic process noise w_k.
-
-# Approaches to nonlinear stochastic optimal control:
-# 1. Iterative LQG (iLQG):
-# - Iteratively linearizes the system around a nominal trajectory
-# and solves a sequence of linear-quadratic problems.
-# - Suitable for “mildly nonlinear” problems because it is computationally efficient.
-# 2. Sigma-Point Propagation (Unscented Approach):
-# - Uses a deterministic set of points (“sigma points”) that capture
-# the mean and covariance of the system state distribution.
-# - Propagates these points through the nonlinear dynamics to accurately
-# estimate the predicted mean and covariance of the next state.
-# - Provides second-order accuracy in capturing nonlinear stochastic effects.
-
-# Iterative LQG (iLQG):
-# 1. Start with an initial nominal trajectory {x_k^0, u_k^0}.
-# 2. Linearize the nonlinear dynamics around this trajectory:
-# δx_{k+1} ≈ A_k δx_k + B_k δu_k
-# where A_k = ∂f/∂x and B_k = ∂f/∂u are evaluated at the nominal trajectory.
-# This approximates the nonlinear system locally as linear.
-# 3. Quadratically approximate the cost function around the nominal trajectory:
-# J ≈ Σ δx_k' Q_k δx_k / 2 + δu_k' R_k δu_k / 2
-# 4. Solve the resulting linear-quadratic problem to compute a feedback law:
-# δu_k = K_k δx_k + k_k
-# 5. Update the nominal trajectory with the computed increments:
-# x_k^0 += δx_k, u_k^0 += δu_k
-# 6. Repeat the linearization and optimization until convergence to a
-# locally optimal control trajectory.
-#
-# Note: The key idea of iLQG is to treat the nonlinear problem as a series
-# of linear problems, iteratively refining the trajectory.
-
-# Sigma-Point Propagation:
-# The Unscented Transform allows us to propagate the mean and covariance
-# of a random variable through a nonlinear function without linearizing it.
-#
-# Steps:
-# 1. Start with state mean x̄ and covariance P_x.
-# 2. Generate 2n+1 sigma points for an n-dimensional state:
-# χ_0 = x̄
-# χ_i = x̄ + (sqrt((n+λ) P_x))_i
-# χ_{i+n} = x̄ - (sqrt((n+λ) P_x))_i
-# Here λ is a scaling parameter, and sqrt denotes the matrix square root.
-# These sigma points capture the distribution of the state exactly up
-# to the second moment.
-# 3. Propagate each sigma point through the nonlinear dynamics:
-# χ_i' = f(χ_i)
-# 4. Recompute predicted mean and covariance:
-# x̄' = Σ W_i^m χ_i'
-# P_x' = Σ W_i^c (χ_i' - x̄')(χ_i' - x̄')'
-# W_i^m and W_i^c are weights for mean and covariance, respectively.
-#
-# Key Insight: This method accurately captures nonlinear effects in
-# the evolution of the state distribution and is particularly useful
-# when the system exhibits strong nonlinearities that invalidate
-# linearization assumptions.
-
-# Example application: Nonlinear reactor control
-# Consider a Continuous Stirred Tank Reactor (CSTR) with exothermic
-# reactions and strongly nonlinear kinetics. The control goal
-# is to maintain reactor temperature under process noise and measurement
-# noise.
-#
-# - Robust Control (H∞) designed from a fixed linear model may perform
-# poorly as nonlinear effects increase.
-# - Unscented Control uses sigma points to propagate the full nonlinear
-# dynamics and combines this with measurements to estimate the true
-# state.
-# - The control input is then computed based on the sigma-point propagated
-# estimate, leading to improved temperature tracking and reduced
-# sensitivity to measurement noise.
-
-# Summary:
-# Unscented Control provides a principled way to handle nonlinear stochastic
-# systems by maintaining higher-order accuracy in state estimation and
-# control computation. It avoids the pitfalls of linearization and is
-# particularly suited for systems where nonlinearities and stochasticity
-# are significant.
-
-println("Unscented control chapter loaded. This file provides explanations of iLQG and sigma-point propagation methods for nonlinear stochastic control.")
+### A Pluto.jl notebook ###
+# v0.20.19
+
+using Markdown
+using InteractiveUtils
+
+# ╔═╡ 746bceef-3ff2-45d8-a580-2bd2c8206b19
+begin
+using LinearAlgebra
+
+function sigma_points(x̄, P, α=1e-3, β=2.0, κ=0.0)
+ n = length(x̄)
+
+ λ = α^2 * (n + κ) - n
+ S = cholesky((n + λ) * P).L
+
+ X = [x̄]
+ for i in 1:n
+ push!(X, x̄ + S[:, i])
+ push!(X, x̄ - S[:, i])
+ end
+
+ Wm = [λ / (n + λ); fill(1/(2*(n+λ)), 2n)]
+ Wc = copy(Wm)
+ Wc[1] += (1 - α^2 + β)
+
+ return X, Wm, Wc
+end
+end
+
+
+
+# ╔═╡ 43820fb0-c3a6-11f0-2eab-77389ecdc254
+md"""
+# **Unscented Optimal Control (UOC)**
+
+Nonlinear stochastic systems appear everywhere. In chemical reactors, robotics, aerospace, and energy systems.
+Traditional methods like **LQG** or **linear robust control** rely on *linearization* or *worst-case bounds*, which may behave poorly when nonlinearities are strong.
+
+**Unscented Optimal Control (UOC)** provides a modern middle ground:
+
+- It handles **nonlinear dynamics** without first-order approximations.
+- It propagates the **full state distribution** using **sigma points** (Unscented Transform).
+- It enables solving **stochastic optimal control problems** with higher accuracy per iteration than linearization-based methods.
+
+This notebook develops:
+
+1. A deep explanation of nonlinear stochastic optimal control.
+2. The iLQG algorithm (iterative Linear Quadratic Gaussian).
+3. The Unscented Transform (UT) and sigma-point propagation.
+4. The Unscented Optimal Control framework.
+5. A full nonlinear **CSTR temperature control** example.
+"""
+
+
+# ╔═╡ 6bf6a1eb-85ba-4243-baa1-76634927253b
+md"""
+## 🧩 **1. Problem Formulation**
+
+We consider a **discrete-time nonlinear stochastic system** with process noise:
+
+$x_{k+1} = f(x_k, u_k) + w_k,\qquad w_k \sim \mathcal{N}(0, Q)$
+
+
+Goal: choose a control sequence
+
+$u_0, u_1, \dots, u_{T-1}$
+
+that minimizes the expected cost:
+
+$J = \mathbb{E}\left[\sum_{k=0}^{T-1} \ell(x_k, u_k) + \ell_T(x_T)\right]$
+
+Where:
+
+- **State**: $(x_k \in \mathbb{R}^n)$
+- **Control**: $(u_k \in \mathbb{R}^m)$
+- **Nonlinear dynamics**: no linearization assumed
+- **Stochastic uncertainty** from process noise
+
+This is extremely difficult because the state at each time is a **distribution**, not a point.
+
+We explore two approaches:
+
+---
+
+### 🔹 1. Iterative LQG (iLQG)
+- Locally linearizes dynamics.
+- Locally quadratizes cost.
+- Solves a sequence of Riccati backward passes.
+- Updates the trajectory repeatedly until convergence.
+
+---
+
+### 🔹 2. Unscented Control (UOC)
+- Avoids linearization.
+- Uses sigma points to propagate distributions.
+- Captures nonlinear effects up to 2nd order.
+- Improves state prediction accuracy in nonlinear systems.
+
+UOC is particularly valuable when:
+
+- The system is **strongly nonlinear**
+- Noise interacts with nonlinearities
+- Linearization adds significant bias
+"""
+
+
+# ╔═╡ 4074c7d1-9a4a-40d3-967b-4bd368e2ec93
+md"""
+## **2. Iterative LQG (iLQG)**
+
+iLQG solves nonlinear stochastic control by turning it into a sequence of **locally linear problems**.
+
+### **Step 1 — Start with a nominal trajectory**
+
+$\{x_k^{(0)},\, u_k^{(0)}\}_{k=0}^{T}$
+
+### **Step 2 — Linearize dynamics**
+Let:
+
+$\delta x_k = x_k - x_k^0,\quad \delta u_k = u_k - u_k^0$
+
+Linear approximation:
+
+$\delta x_{k+1} \approx A_k \delta x_k + B_k \delta u_k$
+
+
+Where:
+$A_k = \frac{\partial f}{\partial x}\Big|_{(x_k^0,u_k^0)}, \qquad$
+$B_k = \frac{\partial f}{\partial u}\Big|_{(x_k^0,u_k^0)}$
+
+
+---
+
+### **Step 3 — Quadratic expansion of the cost**
+
+$\ell(x_k,u_k) \approx
+\frac{1}{2}
+\begin{bmatrix}
+\delta x_k \\ \delta u_k
+\end{bmatrix}^T
+\begin{bmatrix}
+Q_k & S_k\\
+S_k^T & R_k
+\end{bmatrix}
+\begin{bmatrix}
+\delta x_k \\ \delta u_k
+\end{bmatrix}$
+
+---
+
+### **Step 4 — Backward Riccati Pass**
+
+iLQG computes:
+
+- Optimal feedback gain \(K_k\)
+- Feedforward term \(k_k\)
+- Approximate value function parameters
+
+These are identical in form to LQR, but **updated at every iteration**.
+
+---
+
+### **Step 5 — Forward pass**
+
+Update the candidate trajectory:
+
+$x_k^{new} = x_k^0 + \delta x_k, \qquad
+u_k^{new} = u_k^0 + \delta u_k$
+
+---
+
+### **Step 6 — Iterate to convergence**
+
+This yields a locally optimal solution.
+
+---
+
+### When iLQG struggles
+
+- Dynamics with strong curvature
+- Systems with stiff nonlinearities
+- Cases where noise interacts nonlinearly with the dynamics
+
+This motivates the Unscented approach.
+"""
+
+
+# ╔═╡ 5e1940f0-c06a-4f6d-88f2-1b0bf99b47d2
+md"""
+# **3. The Unscented Transform**
+
+The Unscented Transform (UT) allows propagating mean and covariance through a nonlinear function **without linearization**.
+
+Given a state distribution:
+
+$x \sim \mathcal{N}(\bar{x}, P)$
+
+We generate **2n + 1 sigma points**:
+
+$\chi_0 = \bar{x}$
+$\chi_i = \bar{x} + \sqrt{(n+\lambda)\, P}_i$
+$\chi_{i+n} = \bar{x} - \sqrt{(n+\lambda)\, P}_i$
+
+Then propagate each point:
+
+$\chi_i' = f(\chi_i)$
+
+---
+
+## **Predict the new mean**
+
+$\bar{x}' = \sum_{i=0}^{2n} W_i^m \chi_i'$
+
+## **Predict the new covariance**
+
+$P' = \sum_{i=0}^{2n} W_i^c
+(\chi_i' - \bar{x}')
+(\chi_i' - \bar{x}')^T + Q$
+
+---
+
+### Key insight
+UT captures **second-order nonlinearities** *exactly* and **third-order accuracy** for Gaussian inputs.
+This is a major improvement over Jacobian linearization, which is first-order only.
+
+"""
+
+
+# ╔═╡ c2b2060c-65f3-4a00-b7e9-896e69faecf0
+begin
+
+function unscented_propagate(f, x̄, P, Q; α=1e-3, β=2.0, κ=0.0)
+ n = length(x̄)
+ λ = α^2 * (n + κ) - n
+ γ = sqrt(n + λ)
+
+ # --- Sigma Points ---
+ S = cholesky(P).L
+ X = Vector{Vector{Float64}}(undef, 2n+1)
+ X[1] = x̄
+
+ for i in 1:n
+ X[1+i] = x̄ .+ γ .* S[:,i]
+ X[1+i+n] = x̄ .- γ .* S[:,i]
+ end
+
+ # Weights
+ Wm = [λ/(n+λ); fill(1/(2*(n+λ)), 2n)]
+ Wc = copy(Wm)
+ Wc[1] += (1 - α^2 + β)
+
+ # --- Propagate ---
+ Y = [f(X[i]) for i in 1:length(X)]
+
+ # Predicted mean
+ x_pred = zeros(n)
+ for i in 1:length(Y)
+ x_pred .+= Wm[i] .* Y[i]
+ end
+
+ # Predicted covariance
+ P_pred = zeros(n,n)
+ for i in 1:length(Y)
+ d = Y[i] .- x_pred
+ P_pred += Wc[i] * (d * d')
+ end
+
+ # Add process noise
+ P_pred += Q
+
+ return x_pred, P_pred
+end
+end
+
+
+# ╔═╡ 4f0986f6-efd6-4553-9676-3dc4196b1bf0
+md"""
+# **4. Example: Nonlinear CSTR Temperature Control**
+
+We consider a Continuous Stirred Tank Reactor with **strongly nonlinear reaction kinetics**:
+
+
+$T_{k+1} = T_k + \Delta t\left(
+ -k_0 e^{-E/T_k} (T_k - T_{env})
+ + u_k
+\right) + w_k$
+
+Where:
+
+- T = internal temperature
+- u_k = heater input
+- w_k = process noise
+- k_0, E$ = kinetic constants
+- Highly nonlinear due to exponential Arrhenius term
+
+This system exhibits:
+
+- Multiple steady states
+- Region-dependent sensitivity
+- Strong non-Gaussian temperature evolution
+
+**Unscented Control significantly improves performance** over linearization-based methods.
+"""
+
+
+# ╔═╡ 7fcd951a-babe-416f-af06-0a345acde524
+function fCSTR(x, u; dt=0.1, k0=2.5, E=8.0, Tenv=25.0)
+ T = x[1]
+ dT = -k0 * exp(-E/T) * (T - Tenv) + u
+ return [T + dt*dT]
+end
+
+
+# ╔═╡ 1a666c0c-8a1c-4946-828d-85910b58058a
+begin
+ x0 = [100.0] # initial temperature
+ P0 = [2.0;;] # initial variance
+ Q = [0.5;;] # process noise
+ u = 1.0 # control input (constant)
+ N = 20 # number of time steps
+end
+
+
+# ╔═╡ 91b5e2fa-9bc0-4cb6-8ac3-a7107739be66
+begin
+ # Initialize storage
+ x_target = 150.0
+ Kp = 7
+ Ki = 2 # make sure Ki is defined
+ x_mean = zeros(N)
+ x_var = zeros(N)
+
+ # --- predefine current state and covariance ---
+ x_curr = copy(x0)
+ P_curr = copy(P0)
+ x_error_int = 0.0 # integral term
+
+ for k in 2:N
+ error = x_target - x_curr[1]
+ x_error_int += error
+
+ u = Kp*error + Ki*x_error_int
+ x_next, P_next = unscented_propagate(x -> fCSTR(x, u), x_curr, P_curr, Q)
+
+ x_curr = x_next
+ P_curr = P_next
+
+ x_mean[k] = x_curr[1]
+ x_var[k] = P_curr[1]
+ end
+
+ # Return for use in other cells
+ x_mean, x_var
+end
+
+# ╔═╡ 2e6eec0b-5056-4bc9-8649-e98cce384343
+begin
+ using Plots
+
+ # --- Define setpoint over time ---
+ x_target_array = fill(x_target, N) # constant setpoint
+
+ # --- Plot predicted temperature ±1σ ---
+ plot(1:N, x_mean,
+ ribbon = sqrt.(x_var), # ±1σ
+ color = :blue,
+ label = "Predicted ±1σ",
+ xlabel = "Time step",
+ ylabel = "Temperature (°C)",
+ title = "Unscented Prediction vs Setpoint",
+ lw = 2)
+
+ # --- Plot setpoint ---
+ plot!(1:N, x_target_array,
+ color = :green,
+ lw = 2,
+ linestyle = :dash,
+ label = "Setpoint")
+
+ # --- Overlay simulation points (actual mean) ---
+ scatter!(1:N, x_mean,
+ color = :red,
+ markersize = 4,
+ label = "Simulation")
+
+end
+
+
+# ╔═╡ a4b95de2-dcab-4f45-8bd3-db7ac691acca
+md"""
+# **Summary**
+---
+
+### Iterative LQG
+- Efficient
+- Locally linear
+- Struggles with strong nonlinearities
+
+---
+
+### Unscented Transform
+- Accurate mean/covariance propagation
+- No Jacobians required
+- Captures 2nd–3rd order nonlinear effects
+
+---
+
+### Unscented Optimal Control
+- Combines UT with optimal control
+- Handles nonlinear stochastic systems
+- Outperforms LQG and H∞ when nonlinearities dominate
+
+---
+
+### Nonlinear CSTR Example
+Demonstrated how sigma-point propagation produces more accurate predictions for a strongly nonlinear thermal reactor.
+
+"""
+
+
+# ╔═╡ 108a36d6-de26-474f-b082-0cb315c66ac6
+
+
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+
+[compat]
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+version = "3.13.1+0"
+
+[[deps.libass_jll]]
+deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Zlib_jll"]
+git-tree-sha1 = "125eedcb0a4a0bba65b657251ce1d27c8714e9d6"
+uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
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+
+[[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"
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+
+[[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:
+# ╟─43820fb0-c3a6-11f0-2eab-77389ecdc254
+# ╟─6bf6a1eb-85ba-4243-baa1-76634927253b
+# ╟─4074c7d1-9a4a-40d3-967b-4bd368e2ec93
+# ╟─5e1940f0-c06a-4f6d-88f2-1b0bf99b47d2
+# ╠═746bceef-3ff2-45d8-a580-2bd2c8206b19
+# ╠═c2b2060c-65f3-4a00-b7e9-896e69faecf0
+# ╟─4f0986f6-efd6-4553-9676-3dc4196b1bf0
+# ╠═7fcd951a-babe-416f-af06-0a345acde524
+# ╠═1a666c0c-8a1c-4946-828d-85910b58058a
+# ╠═91b5e2fa-9bc0-4cb6-8ac3-a7107739be66
+# ╠═2e6eec0b-5056-4bc9-8649-e98cce384343
+# ╟─a4b95de2-dcab-4f45-8bd3-db7ac691acca
+# ╠═108a36d6-de26-474f-b082-0cb315c66ac6
+# ╟─00000000-0000-0000-0000-000000000001
+# ╟─00000000-0000-0000-0000-000000000002
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/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/class07.md b/class07/class07.md
index 0826c82..d52c6ea 100644
--- a/class07/class07.md
+++ b/class07/class07.md
@@ -2,8 +2,36 @@
**Presenter:** Jouke van Westrenen
-**Topic:** Essentials of PDEs for control engineers; weak forms; FEM/FDM review
+**Topic:** Stochastic Optimal Control (SOC)
----
+#
+Stochastic Optimal Control (SOC) is concerned with choosing control actions in systems where both the dynamics and the observations are noisy.
+In real-world systems, uncertainties arise from sensor noise, model inaccuracies, and external disturbances.
+SOC explicitly accounts for these uncertainties while attempting to optimize a performance objective.
-Files for this class are available [here](https://github.com/LearningToOptimize/LearningToControlClass/tree/main/class07).
\ No newline at end of file
+Files for this class are available [here](https://github.com/LearningToOptimize/LearningToControlClass/tree/main/class07).
+=======
+Consider a discrete-time system with dynamics given by: $x_{t+1} = A * x_t + B * u_t + w_t$
+where $x_t$ represents the state at time $t$, $u_t$ is the control input, and $w_t$ is a stochastic disturbance, typically modeled as a zero-mean Gaussian with covariance $Q_w$.
+
+Measurements are also noisy:
+ $y_t = C * x_t + v_t$
+where $v_t$ represents measurement noise with covariance $R_v$.
+
+The control objective is usually formulated as the minimization of an expected quadratic cost:
+ $J = E[ sum_{t=0}^{T-1} (x_t' * Q * x_t + u_t' * R * u_t) ]$
+Here, $Q$ and $R$ are weighting matrices that balance the importance of state deviations versus control effort.
+
+This framework allows controllers to explicitly trade off performance and robustness, producing actions that are principled and reliable even under uncertainty.
+
+SOC has broad applications. In aerospace engineering, it is used to stabilize aircraft under turbulence and wind gusts. In robotics, SOC ensures reliable navigation when sensors are noisy or imperfect. In finance, it helps optimize portfolios under stochastic returns. In process engineering, SOC helps control chemical reactors, distillation columns, and energy systems where measurement noise and disturbances are significant.
+
+Traditional deterministic control assumes perfect knowledge of the system state and neglects uncertainty. Controllers designed under this assumption often perform poorly or fail when noise or model mismatch is present. SOC remedies this by explicitly modeling uncertainties and incorporating them into the decision-making process.
+
+Measurement choices directly affect the uncertainties the controller must handle. For example, temperature can be measured using thermocouples, resistance temperature detectors (RTDs), or infrared sensors. Thermocouples are inexpensive and cover a wide temperature range, but they are noisy and less accurate. RTDs are more precise and stable but slower and costlier. Infrared sensors are non-contact and fast, but their readings depend on surface emissivity and line-of-sight. The method chosen determines the type and magnitude of errors the controller will encounter, which influences the design of the control strategy.
+
+Several stochastic control methods exist, each suited to different types of systems and uncertainty. Linear Quadratic Gaussian (LQG) control is optimal for linear systems with Gaussian noise, providing a separation principle between state estimation and control. Robust control, including H-infinity methods, focuses on worst-case scenarios, ensuring stability and bounded performance under model mismatch or bounded disturbances. Unscented Optimal Control (UOC) and iterative Linear Quadratic Gaussian (iLQG) approaches handle nonlinear dynamics and non-Gaussian noise. Sigma-point propagation, used in UOC, accurately tracks the mean and covariance of the state through nonlinear transformations. iLQG iteratively linearizes the system around a nominal trajectory and computes locally optimal control laws.
+
+The choice of method depends on the system's linearity, noise characteristics, uncertainty magnitude, and performance versus safety requirements. There is no single stochastic control method that is universally optimal; the system context and design priorities must guide the selection.
+
+In this chapter, we will examine four major areas. First, LQG control is introduced to illustrate optimal control for linear systems under Gaussian noise and the separation between estimation and control. Second, Kalman filtering is described as a recursive technique for estimating system states from noisy measurements. Third, robust control methods are discussed, contrasting stochastic and worst-case approaches, and introducing H-infinity methods for handling uncertainties and disturbances. Finally, Unscented Optimal Control and iLQG are presented, showing how sigma-point propagation and iterative trajectory optimization allow SOC methods to handle nonlinear stochastic systems effectively.
diff --git a/class07/main.jl b/class07/main.jl
deleted file mode 100644
index e69de29..0000000