Add all Class07 lecture materials

class07/background_materials/CSTR_Multiple_inlet_Unscented_Control.py

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+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()
+