Add SOC materials for class07 (new folder soc-materials)

Author: jwestrenen3

class07/soc-materials/Kalman Filtering.html

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class07/soc-materials/Kalman_Filtering.jl

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+### 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  
+"""
+
+
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