Add all Class07 lecture materials

class07/Kalman_Filtering.jl

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+## 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
+
+    for k in 1:N
+        # 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
+
+        # 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
+    end
+
+    return x_hat
+end
+
+## Example usage:
+# A simple demonstration with a small system. This shows how the filter
+# reconstructs the true state from noisy measurements.
+
+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
+
+    # Simulated noisy measurements
+    true_states = [20.0, 21.0, 23.0, 22.0]
+    y = true_states .+ randn(length(true_states)) * sqrt(R)
+
+    # Run Kalman filter
+    x_hat = kalman_filter(A, B, C, Q, R, y)
+
+    println("Noisy measurements: ", y)
+    println("Estimated states: ", x_hat)
+end

class07/LQG_Control.jl

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+# 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
+
+# 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
+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.