131 lines
3.3 KiB
Python
131 lines
3.3 KiB
Python
import sympy as sm
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import numpy as np
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import matplotlib.pyplot as plt
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sm.init_printing()
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print("PROBLEM 1:")
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print("part a:")
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s, z, t = sm.symbols("s z t")
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J, B, RHO, RADIUS, L, RESIST, T = sm.symbols(
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"J, B, rho, r, L, R, T", Real=True, Positive=True
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)
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K_T, K_B = sm.symbols("K_T, K_b")
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# loop gain
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Loop_Gain = K_T * 1 / (J * s + B)
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# transfer function from motor current command to linear displacement (ignoring feedback)
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X_over_I = Loop_Gain * RHO / s * RADIUS
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sm.pprint(X_over_I)
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X_over_I = X_over_I.expand().simplify()
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sm.pprint(X_over_I)
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print("part b:")
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# recall ZOH_eq
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# G(z) = (1-z**-1) Z{L**-1{G(s)/s}}
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print("Finding inverse laplace of G/s")
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G_s = X_over_I
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sm.pprint(G_s / s)
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g_t = sm.inverse_laplace_transform(G_s / s, s, t)
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sm.pprint(g_t.expand())
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# make z substitution subs
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G_z = (1 - z**-1) * (
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K_T * T * RADIUS * RHO / B * (z / (z - 1) ** 2)
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- J * K_T * RADIUS * RHO / B**2
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+ J * K_T * RADIUS * RHO / B**2 * (z / (z - sm.exp(-B / J * T)))
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)
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sm.pprint(G_z)
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# part c was on the board
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print("part c")
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F = sm.Matrix([[0, RHO], [0, -B / J]])
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G = sm.Matrix([0, K_T / J])
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# part d
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print("part d")
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A = sm.exp(F * T)
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B = sm.integrate(A, (T, 0, T))
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sm.pprint(F)
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sm.pprint(G)
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sm.pprint(A)
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sm.pprint(B)
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print(sm.latex(A))
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print(sm.latex(B))
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########################################################a
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print("PROBLEM 4:")
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print("bilinear transform")
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T = 0.2 # s
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G_s = 10 / (s + 10)
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G_z = G_s.subs(s, 2 * (z - 1) / T / (z + 1))
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sm.pprint(G_z)
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sm.pprint(G_z.simplify())
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print("bilinear with warping")
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omega_star = 10 # rad/s
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G_z_warped = G_s.subs(s, (omega_star / np.tan(omega_star * T / 2)) * (z - 1) / (z + 1))
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sm.pprint(G_z_warped)
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sm.pprint(G_z_warped.simplify())
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# Create a frequency vector from 0 to 10 rad/s (avoid exactly 0 to prevent log(0) issues)
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omega = np.linspace(0, 10, 500)
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# Continuous-time frequency response (evaluate at s = j*omega)
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G_s_response = 10 / (10 + 1j * omega)
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# For the discrete transfer functions, substitute z = exp(j*omega*T)
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z_vals = np.exp(1j * omega * T)
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# Convert symbolic expressions into functions that accept numpy arrays
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G_z_func = sm.lambdify(z, G_z, "numpy")
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G_z_warped_func = sm.lambdify(z, G_z_warped, "numpy")
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G_z_response = G_z_func(z_vals)
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G_z_warped_response = G_z_warped_func(z_vals)
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# Helper function to compute magnitude (in dB) and phase (in degrees)
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def compute_mag_phase(response):
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mag = 20 * np.log10(np.abs(response))
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phase = np.angle(response, deg=True)
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return mag, phase
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# Compute responses
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mag_s, phase_s = compute_mag_phase(G_s_response)
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mag_z, phase_z = compute_mag_phase(G_z_response)
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mag_z_warped, phase_z_warped = compute_mag_phase(G_z_warped_response)
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# Plot the Bode plots
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plt.figure(figsize=(10, 8))
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# Magnitude plot
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plt.subplot(2, 1, 1)
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plt.semilogx(omega, mag_s, label="Continuous: G(s)")
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plt.semilogx(omega, mag_z, label="Bilinear: G(z)")
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plt.semilogx(omega, mag_z_warped, label="Warped: G(z)_warped")
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plt.title("Bode Magnitude Plot")
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plt.ylabel("Magnitude (dB)")
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plt.legend()
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plt.grid(True)
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# Phase plot
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plt.subplot(2, 1, 2)
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plt.semilogx(omega, phase_s, label="Continuous: G(s)")
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plt.semilogx(omega, phase_z, label="Bilinear: G(z)")
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plt.semilogx(omega, phase_z_warped, label="Warped: G(z)_warped")
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plt.title("Bode Phase Plot")
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plt.xlabel("Frequency (rad/s)")
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plt.ylabel("Phase (degrees)")
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plt.legend()
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plt.grid(True)
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plt.tight_layout()
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plt.show()
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