me 2046 hw4 complete!

ME_2046/HW4/hw4.py

@@ -33,13 +33,13 @@ sm.pprint(g_t.expand())
 
 # make z substitution subs
 
-G_z = (
+G_z = (1 - z**-1) * (
     K_T * T * RADIUS * RHO / B * (z / (z - 1) ** 2)
     - J * K_T * RADIUS * RHO / B**2
     + J * K_T * RADIUS * RHO / B**2 * (z / (z - sm.exp(-B / J * T)))
 )
 
-sm.pprint(G_z.simplify())
+sm.pprint(G_z)
 
 # part c was on the board
 print("part c")
@@ -54,3 +54,75 @@ sm.pprint(F)
 sm.pprint(G)
 sm.pprint(A)
 sm.pprint(B)
+
+########################################################a
+print("PROBLEM 4:")
+print("bilinear transform")
+T = 0.2  # s
+
+G_s = 10 / (s + 10)
+G_z = G_s.subs(s, 2 * (z - 1) / T * (z + 1))
+sm.pprint(G_z)
+sm.pprint(G_z.simplify())
+
+print("bilinear with warping")
+omega_star = 10  # rad/s
+G_z_warped = G_s.subs(s, (omega_star / np.tan(omega_star * T / 2)) * (z - 1) / (z + 1))
+sm.pprint(G_z_warped)
+sm.pprint(G_z_warped.simplify())
+
+# Create a frequency vector from 0 to 10 rad/s (avoid exactly 0 to prevent log(0) issues)
+omega = np.linspace(5, 15, 500)
+
+# Continuous-time frequency response (evaluate at s = j*omega)
+G_s_response = 10 / (10 + 1j * omega)
+
+# For the discrete transfer functions, substitute z = exp(j*omega*T)
+z_vals = np.exp(1j * omega * T)
+
+# Convert symbolic expressions into functions that accept numpy arrays
+G_z_func = sm.lambdify(z, G_z, "numpy")
+G_z_warped_func = sm.lambdify(z, G_z_warped, "numpy")
+
+G_z_response = G_z_func(z_vals)
+G_z_warped_response = G_z_warped_func(z_vals)
+
+
+# Helper function to compute magnitude (in dB) and phase (in degrees)
+def compute_mag_phase(response):
+    mag = 20 * np.log10(np.abs(response))
+    phase = np.angle(response, deg=True)
+    return mag, phase
+
+
+# Compute responses
+mag_s, phase_s = compute_mag_phase(G_s_response)
+mag_z, phase_z = compute_mag_phase(G_z_response)
+mag_z_warped, phase_z_warped = compute_mag_phase(G_z_warped_response)
+
+# Plot the Bode plots
+plt.figure(figsize=(10, 8))
+
+# Magnitude plot
+plt.subplot(2, 1, 1)
+plt.semilogx(omega, mag_s, label="Continuous: G(s)")
+plt.semilogx(omega, mag_z, label="Bilinear: G(z)")
+plt.semilogx(omega, mag_z_warped, label="Warped: G(z)_warped")
+plt.title("Bode Magnitude Plot")
+plt.ylabel("Magnitude (dB)")
+plt.legend()
+plt.grid(True)
+
+# Phase plot
+plt.subplot(2, 1, 2)
+plt.semilogx(omega, phase_s, label="Continuous: G(s)")
+plt.semilogx(omega, phase_z, label="Bilinear: G(z)")
+plt.semilogx(omega, phase_z_warped, label="Warped: G(z)_warped")
+plt.title("Bode Phase Plot")
+plt.xlabel("Frequency (rad/s)")
+plt.ylabel("Phase (degrees)")
+plt.legend()
+plt.grid(True)
+
+plt.tight_layout()
+plt.show()