Edit Figure: Figure 7.6: State feedback control of a steering system

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[[Image:figure-7.6-steering_place.png]] '''Figure 7.6:''' State feedback control of a steering system. Unit step responses (from zero initial condition) obtained with controllers designed with <math>\zeta_c = 0.7</math> and <math>\omega_c = 0.5</math>, <math>0.7</math>, and <math>1</math> [rad/s] are shown in (a). The dashed lines indicate <math>\pm 5</math>% deviations from the setpoint. Notice that response speed increases with increasing <math>\omega_c</math>, but that large <math>\omega_c</math> also give large initial control actions. Unit step responses obtained with a controller designed with <math>\omega_c = 0.7</math> and <math>\zeta_c = 0.5</math>, <math>0.7</math>, and <math>1</math> are shown in (b). <nowiki> # example-7.4-steering_place.py - <Short description> # RMM, 28 Nov 2024 # # Figure 7.6: State feedback control of a steering system. Unit step # responses (from zero initial condition) obtained with controllers # designed with zeta_c = 0.7 and omega_c = 0.5, 0.7, and 1 [rad/s] are # shown in (a). The dashed lines indicate ±5% deviations from the # setpoint. Notice that response speed increases with increasing omega_c, # but that large omega_c also give large initial control actions. Unit step # responses obtained with a controller designed with omega_c = 0.7 and # zeta_c = 0.5, 0.7, and 1 are shown 2in (b). import control as ct import numpy as np import matplotlib.pyplot as plt ct.use_fbs_defaults() # # System dynamics # # Get the normalized linear dynamics from steering import linearize_lateral sys = linearize_lateral(normalize=True, output_full_state=True) # Function to place the poles at desired values def steering_place(sys, omega, zeta): # Get the pole locations based on omega and zeta desired_poly = np.polynomial.Polynomial([omega**2, 2 * zeta * omega, 1]) return ct.place(sys.A, sys.B, desired_poly.roots()) # Set up the plotting grid to match the layout in the book fig = plt.figure(constrained_layout=True) gs = fig.add_gridspec(3, 14) # allow some space in the middle left = slice(0, 6) right = slice(7, 13) # # (a) Unit step response for varying omega_c # ax_pos = fig.add_subplot(gs[0, left]) ax_delta = fig.add_subplot(gs[1, left]) ax_pos.set_title( r"(a) Unit step response for varying $\omega_c$", size='medium') timepts = np.linspace(0, 20) zeta_c = 0.7 for omega_c in [0.5, 0.7, 1]: # Compute the gains for the controller K = steering_place(sys, omega_c, zeta_c) kf = omega_c**2 # Compute the closed loop system clsys = ct.feedback(sys, K) * kf # Simulate the closed loop dynamics response = ct.forced_response(clsys, timepts, 1, X0=0) ax_pos.plot(response.time, response.states[0], 'b') ax_delta.plot(response.time, (kf - K @ response.states)[0], 'b') # Label the plot ax_pos.set_ylabel(r"Lateral position $y/b$") ax_delta.set_xlabel(r"Normalized time $v_0 t/b$") ax_delta.set_ylabel(r"Steering angle $\delta$ [rad]") ax_pos.axhline(0.95, color='k', linestyle='--', linewidth=0.5) ax_pos.axhline(1.05, color='k', linestyle='--', linewidth=0.5) ax_pos.annotate( "", xy=(3.5, 0.3), xytext=[0, 0.8], arrowprops={'arrowstyle': '<-'}) ax_pos.text(3.6, 0.25, r"$\omega_c$") ax_delta.annotate( "", xy=(4, 0.1), xytext=(0, -0.2), arrowprops={'arrowstyle': '<-'}) ax_delta.text(3.5, 0.15, r"$\omega_c$") # # (b) Unit step response for varying zeta_c # ax_pos = fig.add_subplot(gs[0, right], sharey=ax_pos) ax_delta = fig.add_subplot(gs[1, right], sharey=ax_delta) ax_pos.set_title( r"(b) Unit step response for varying $\zeta_c$", size='medium') timepts = np.linspace(0, 20) omega_c = 0.7 for zeta_c in [0.5, 0.7, 1]: # Compute the gains for the controller K = steering_place(sys, omega_c, zeta_c) kf = omega_c**2 # Compute the closed loop system clsys = ct.feedback(sys, K) * kf # Simulate the closed loop dynamics response = ct.forced_response(clsys, timepts, 1, X0=0) ax_pos.plot(response.time, response.states[0], 'b') ax_delta.plot(response.time, (kf - K @ response.states)[0], 'b') # Label the plot ax_pos.set_ylabel(r"Lateral position $y/b$") ax_delta.set_xlabel(r"Normalized time $v_0 t/b$") ax_delta.set_ylabel(r"Steering angle $\delta$ [rad]") ax_pos.axhline(0.95, color='k', linestyle='--', linewidth=0.5) ax_pos.axhline(1.05, color='k', linestyle='--', linewidth=0.5) ax_pos.annotate( "", xy=(1.7, 1.15), xytext=(5.2, 0.6), arrowprops={'arrowstyle': '<-'}) ax_pos.text(5.5, 0.5, r"$\zeta_c$") ax_delta.annotate( "", xy=(5.5, -0.2), xytext=(3.5, 0.15), arrowprops={'arrowstyle': '<-'}) ax_delta.text(3, 0.2, r"$\zeta_c$") # Save the figure fig.align_ylabels() plt.savefig("figure-7.4-steering_place.png", bbox_inches='tight') </nowiki>

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