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"""disk_margins.py
Demonstrate disk-based stability margin calculations.
References:
[1] Blight, James D., R. Lane Dailey, and Dagfinn Gangsaas. “Practical
Control Law Design for Aircraft Using Multivariable Techniques.”
International Journal of Control 59, no. 1 (January 1994): 93-137.
https://doi.org/10.1080/00207179408923071.
[2] Seiler, Peter, Andrew Packard, and Pascal Gahinet. “An Introduction
to Disk Margins [Lecture Notes].” IEEE Control Systems Magazine 40,
no. 5 (October 2020): 78-95.
[3] P. Benner, V. Mehrmann, V. Sima, S. Van Huffel, and A. Varga, "SLICOT
- A Subroutine Library in Systems and Control Theory", Applied and
Computational Control, Signals, and Circuits (Birkhauser), Vol. 1, Ch.
10, pp. 505-546, 1999.
[4] S. Van Huffel, V. Sima, A. Varga, S. Hammarling, and F. Delebecque,
"Development of High Performance Numerical Software for Control", IEEE
Control Systems Magazine, Vol. 24, Nr. 1, Feb., pp. 60-76, 2004.
[5] Deodhare, G., & Patel, V. (1998, August). A "Modern" Look at Gain
and Phase Margins: An H-Infinity/mu Approach. In Guidance, Navigation,
and Control Conference and Exhibit (p. 4134).
"""
import os
import control
import matplotlib.pyplot as plt
import numpy as np
def plot_allowable_region(alpha_max, skew, ax=None):
"""Plot region of allowable gain/phase variation, given worst-case disk margin.
Parameters
----------
alpha_max : float (scalar or list)
worst-case disk margin(s) across all frequencies. May be a scalar or list.
skew : float (scalar or list)
skew parameter(s) for disk margin calculation.
skew=0 uses the "balanced" sensitivity function 0.5*(S - T)
skew=1 uses the sensitivity function S
skew=-1 uses the complementary sensitivity function T
ax : axes to plot bounding curve(s) onto
Returns
-------
DM : ndarray
1D array of frequency-dependent disk margins. DM is the same
size as "omega" parameter.
GM : ndarray
1D array of frequency-dependent disk-based gain margins, in dB.
GM is the same size as "omega" parameter.
PM : ndarray
1D array of frequency-dependent disk-based phase margins, in deg.
PM is the same size as "omega" parameter.
"""
# Create axis if needed
if ax is None:
ax = plt.gca()
# Allow scalar or vector arguments (to overlay plots)
if np.isscalar(alpha_max):
alpha_max = np.asarray([alpha_max])
else:
alpha_max = np.asarray(alpha_max)
if np.isscalar(skew):
skew=np.asarray([skew])
else:
skew=np.asarray(skew)
# Add a plot for each (alpha, skew) pair present
theta = np.linspace(0, np.pi, 500)
legend_list = []
for ii in range(0, skew.shape[0]):
legend_str = "$\\sigma$ = %.1f, $\\alpha_{max}$ = %.2f" %(\
skew[ii], alpha_max[ii])
legend_list.append(legend_str)
# Complex bounding curve of stable gain/phase variations
f = (2 + alpha_max[ii] * (1 - skew[ii]) * np.exp(1j * theta))\
/(2 - alpha_max[ii] * (1 + skew[ii]) * np.exp(1j * theta))
# Allowable combined gain/phase variations
gamma_dB = control.ctrlutil.mag2db(np.abs(f)) # gain margin (dB)
phi_deg = np.rad2deg(np.angle(f)) # phase margin (deg)
# Plot the allowable combined gain/phase variations
out = ax.plot(gamma_dB, phi_deg, alpha=0.25, label='_nolegend_')
ax.fill_between(ax.lines[ii].get_xydata()[:,0],\
ax.lines[ii].get_xydata()[:,1], alpha=0.25)
plt.ylabel('Phase Variation (deg)')
plt.xlabel('Gain Variation (dB)')
plt.title('Range of Gain and Phase Variations')
plt.legend(legend_list)
plt.grid()
plt.tight_layout()
return out
def test_siso1():
#
# Disk-based stability margins for example
# SISO loop transfer function(s)
#
# Frequencies of interest
omega = np.logspace(-1, 2, 1001)
# Loop transfer gain
L = control.tf(25, [1, 10, 10, 10])
print("------------- Python control built-in (S) -------------")
GM_, PM_, SM_ = control.stability_margins(L)[:3] # python-control default (S-based...?)
print(f"SM_ = {SM_}")
print(f"GM_ = {GM_} dB")
print(f"PM_ = {PM_} deg\n")
print("------------- Sensitivity function (S) -------------")
DM, GM, PM = control.disk_margins(L, omega, skew=1.0, returnall=True) # S-based (S)
print(f"min(DM) = {min(DM)} (omega = {omega[np.argmin(DM)]})")
print(f"GM = {GM[np.argmin(DM)]} dB")
print(f"PM = {PM[np.argmin(DM)]} deg")
print(f"min(GM) = {min(GM)} dB")
print(f"min(PM) = {min(PM)} deg\n")
plt.figure(1)
plt.subplot(3, 3, 1)
plt.semilogx(omega, DM, label='$\\alpha$')
plt.ylabel('Disk Margin (abs)')
plt.legend()
plt.title('S-Based Margins')
plt.grid()
plt.xlim([omega[0], omega[-1]])
plt.ylim([0, 2])
plt.figure(1)
plt.subplot(3, 3, 4)
plt.semilogx(omega, GM, label='$\\gamma_{m}$')
plt.ylabel('Gain Margin (dB)')
plt.legend()
#plt.title('Gain-Only Margin')
plt.grid()
plt.xlim([omega[0], omega[-1]])
plt.ylim([0, 40])
plt.figure(1)
plt.subplot(3, 3, 7)
plt.semilogx(omega, PM, label='$\\phi_{m}$')
plt.ylabel('Phase Margin (deg)')
plt.legend()
#plt.title('Phase-Only Margin')
plt.grid()
plt.xlim([omega[0], omega[-1]])
plt.ylim([0, 90])
plt.xlabel('Frequency (rad/s)')
print("------------- Complementary sensitivity function (T) -------------")
DM, GM, PM = control.disk_margins(L, omega, skew=-1.0, returnall=True) # T-based (T)
print(f"min(DM) = {min(DM)} (omega = {omega[np.argmin(DM)]})")
print(f"GM = {GM[np.argmin(DM)]} dB")
print(f"PM = {PM[np.argmin(DM)]} deg")
print(f"min(GM) = {min(GM)} dB")
print(f"min(PM) = {min(PM)} deg\n")
plt.figure(1)
plt.subplot(3, 3, 2)
plt.semilogx(omega, DM, label='$\\alpha$')
plt.ylabel('Disk Margin (abs)')
plt.legend()
plt.title('T_Based Margins')
plt.grid()
plt.xlim([omega[0], omega[-1]])
plt.ylim([0, 2])
plt.figure(1)
plt.subplot(3, 3, 5)
plt.semilogx(omega, GM, label='$\\gamma_{m}$')
plt.ylabel('Gain Margin (dB)')
plt.legend()
#plt.title('Gain-Only Margin')
plt.grid()
plt.xlim([omega[0], omega[-1]])
plt.ylim([0, 40])
plt.figure(1)
plt.subplot(3, 3, 8)
plt.semilogx(omega, PM, label='$\\phi_{m}$')
plt.ylabel('Phase Margin (deg)')
plt.legend()
#plt.title('Phase-Only Margin')
plt.grid()
plt.xlim([omega[0], omega[-1]])
plt.ylim([0, 90])
plt.xlabel('Frequency (rad/s)')
print("------------- Balanced sensitivity function (S - T) -------------")
DM, GM, PM = control.disk_margins(L, omega, skew=0.0, returnall=True) # balanced (S - T)
print(f"min(DM) = {min(DM)} (omega = {omega[np.argmin(DM)]})")
print(f"GM = {GM[np.argmin(DM)]} dB")
print(f"PM = {PM[np.argmin(DM)]} deg")
print(f"min(GM) = {min(GM)} dB")
print(f"min(PM) = {min(PM)} deg\n")
plt.figure(1)
plt.subplot(3, 3, 3)
plt.semilogx(omega, DM, label='$\\alpha$')
plt.ylabel('Disk Margin (abs)')
plt.legend()
plt.title('Balanced Margins')
plt.grid()
plt.xlim([omega[0], omega[-1]])
plt.ylim([0, 2])
plt.figure(1)
plt.subplot(3, 3, 6)
plt.semilogx(omega, GM, label='$\\gamma_{m}$')
plt.ylabel('Gain Margin (dB)')
plt.legend()
#plt.title('Gain-Only Margin')
plt.grid()
plt.xlim([omega[0], omega[-1]])
plt.ylim([0, 40])
plt.figure(1)
plt.subplot(3, 3, 9)
plt.semilogx(omega, PM, label='$\\phi_{m}$')
plt.ylabel('Phase Margin (deg)')
plt.legend()
#plt.title('Phase-Only Margin')
plt.grid()
plt.xlim([omega[0], omega[-1]])
plt.ylim([0, 90])
plt.xlabel('Frequency (rad/s)')
# Disk margin plot of admissible gain/phase variations for which
DM_plot = []
DM_plot.append(control.disk_margins(L, omega, skew=-2.0)[0])
DM_plot.append(control.disk_margins(L, omega, skew=0.0)[0])
DM_plot.append(control.disk_margins(L, omega, skew=2.0)[0])
plt.figure(10); plt.clf()
plot_allowable_region(DM_plot, skew=[-2.0, 0.0, 2.0])
return
def test_siso2():
#
# Disk-based stability margins for example
# SISO loop transfer function(s)
#
# Frequencies of interest
omega = np.logspace(-1, 2, 1001)
# Laplace variable
s = control.tf('s')
# Loop transfer gain
L = (6.25 * (s + 3) * (s + 5)) / (s * (s + 1)**2 * (s**2 + 0.18 * s + 100))
print("------------- Python control built-in (S) -------------")
GM_, PM_, SM_ = control.stability_margins(L)[:3] # python-control default (S-based...?)
print(f"SM_ = {SM_}")
print(f"GM_ = {GM_} dB")
print(f"PM_ = {PM_} deg\n")
print("------------- Sensitivity function (S) -------------")
DM, GM, PM = control.disk_margins(L, omega, skew=1.0, returnall=True) # S-based (S)
print(f"min(DM) = {min(DM)} (omega = {omega[np.argmin(DM)]})")
print(f"GM = {GM[np.argmin(DM)]} dB")
print(f"PM = {PM[np.argmin(DM)]} deg")
print(f"min(GM) = {min(GM)} dB")
print(f"min(PM) = {min(PM)} deg\n")
plt.figure(2)
plt.subplot(3, 3, 1)
plt.semilogx(omega, DM, label='$\\alpha$')
plt.ylabel('Disk Margin (abs)')
plt.legend()
plt.title('S-Based Margins')
plt.grid()
plt.xlim([omega[0], omega[-1]])
plt.ylim([0, 2])
plt.figure(2)
plt.subplot(3, 3, 4)
plt.semilogx(omega, GM, label='$\\gamma_{m}$')
plt.ylabel('Gain Margin (dB)')
plt.legend()
#plt.title('Gain-Only Margin')
plt.grid()
plt.xlim([omega[0], omega[-1]])
plt.ylim([0, 40])
plt.figure(2)
plt.subplot(3, 3, 7)
plt.semilogx(omega, PM, label='$\\phi_{m}$')
plt.ylabel('Phase Margin (deg)')
plt.legend()
#plt.title('Phase-Only Margin')
plt.grid()
plt.xlim([omega[0], omega[-1]])
plt.ylim([0, 90])
plt.xlabel('Frequency (rad/s)')
print("------------- Complementary sensitivity function (T) -------------")
DM, GM, PM = control.disk_margins(L, omega, skew=-1.0, returnall=True) # T-based (T)
print(f"min(DM) = {min(DM)} (omega = {omega[np.argmin(DM)]})")
print(f"GM = {GM[np.argmin(DM)]} dB")
print(f"PM = {PM[np.argmin(DM)]} deg")
print(f"min(GM) = {min(GM)} dB")
print(f"min(PM) = {min(PM)} deg\n")
plt.figure(2)
plt.subplot(3, 3, 2)
plt.semilogx(omega, DM, label='$\\alpha$')
plt.ylabel('Disk Margin (abs)')
plt.legend()
plt.title('T-Based Margins')
plt.grid()
plt.xlim([omega[0], omega[-1]])
plt.ylim([0, 2])
plt.figure(2)
plt.subplot(3, 3, 5)
plt.semilogx(omega, GM, label='$\\gamma_{m}$')
plt.ylabel('Gain Margin (dB)')
plt.legend()
#plt.title('Gain-Only Margin')
plt.grid()
plt.xlim([omega[0], omega[-1]])
plt.ylim([0, 40])
plt.figure(2)
plt.subplot(3, 3, 8)
plt.semilogx(omega, PM, label='$\\phi_{m}$')
plt.ylabel('Phase Margin (deg)')
plt.legend()
#plt.title('Phase-Only Margin')
plt.grid()
plt.xlim([omega[0], omega[-1]])
plt.ylim([0, 90])
plt.xlabel('Frequency (rad/s)')
print("------------- Balanced sensitivity function (S - T) -------------")
DM, GM, PM = control.disk_margins(L, omega, skew=0.0, returnall=True) # balanced (S - T)
print(f"min(DM) = {min(DM)} (omega = {omega[np.argmin(DM)]})")
print(f"GM = {GM[np.argmin(DM)]} dB")
print(f"PM = {PM[np.argmin(DM)]} deg")
print(f"min(GM) = {min(GM)} dB")
print(f"min(PM) = {min(PM)} deg\n")
plt.figure(2)
plt.subplot(3, 3, 3)
plt.semilogx(omega, DM, label='$\\alpha$')
plt.ylabel('Disk Margin (abs)')
plt.legend()
plt.title('Balanced Margins')
plt.grid()
plt.xlim([omega[0], omega[-1]])
plt.ylim([0, 2])
plt.figure(2)
plt.subplot(3, 3, 6)
plt.semilogx(omega, GM, label='$\\gamma_{m}$')
plt.ylabel('Gain Margin (dB)')
plt.legend()
#plt.title('Gain-Only Margin')
plt.grid()
plt.xlim([omega[0], omega[-1]])
plt.ylim([0, 40])
plt.figure(2)
plt.subplot(3, 3, 9)
plt.semilogx(omega, PM, label='$\\phi_{m}$')
plt.ylabel('Phase Margin (deg)')
plt.legend()
#plt.title('Phase-Only Margin')
plt.grid()
plt.xlim([omega[0], omega[-1]])
plt.ylim([0, 90])
plt.xlabel('Frequency (rad/s)')
# Disk margin plot of admissible gain/phase variations for which
# the feedback loop still remains stable, for each skew parameter
DM_plot = []
DM_plot.append(control.disk_margins(L, omega, skew=-1.0)[0]) # T-based (T)
DM_plot.append(control.disk_margins(L, omega, skew=0.0)[0]) # balanced (S - T)
DM_plot.append(control.disk_margins(L, omega, skew=1.0)[0]) # S-based (S)
plt.figure(20)
plot_allowable_region(DM_plot, skew=[-1.0, 0.0, 1.0])
return
def test_mimo():
#
# Disk-based stability margins for example
# MIMO loop transfer function(s)
#
# Frequencies of interest
omega = np.logspace(-1, 3, 1001)
# Loop transfer gain
P = control.ss([[0, 10],[-10, 0]], np.eye(2), [[1, 10], [-10, 1]], 0) # plant
K = control.ss([], [], [], [[1, -2], [0, 1]]) # controller
L = P * K # loop gain
print("------------- Sensitivity function (S) -------------")
DM, GM, PM = control.disk_margins(L, omega, skew=1.0, returnall=True) # S-based (S)
print(f"min(DM) = {min(DM)} (omega = {omega[np.argmin(DM)]})")
print(f"GM = {GM[np.argmin(DM)]} dB")
print(f"PM = {PM[np.argmin(DM)]} deg")
print(f"min(GM) = {min(GM)} dB")
print(f"min(PM) = {min(PM)} deg\n")
plt.figure(3)
plt.subplot(3, 3, 1)
plt.semilogx(omega, DM, label='$\\alpha$')
plt.ylabel('Disk Margin (abs)')
plt.legend()
plt.title('S-Based Margins')
plt.grid()
plt.xlim([omega[0], omega[-1]])
plt.ylim([0, 2])
plt.figure(3)
plt.subplot(3, 3, 4)
plt.semilogx(omega, GM, label='$\\gamma_{m}$')
plt.ylabel('Gain Margin (dB)')
plt.legend()
#plt.title('Gain-Only Margin')
plt.grid()
plt.xlim([omega[0], omega[-1]])
plt.ylim([0, 40])
plt.figure(3)
plt.subplot(3, 3, 7)
plt.semilogx(omega, PM, label='$\\phi_{m}$')
plt.ylabel('Phase Margin (deg)')
plt.legend()
#plt.title('Phase-Only Margin')
plt.grid()
plt.xlim([omega[0], omega[-1]])
plt.ylim([0, 90])
plt.xlabel('Frequency (rad/s)')
print("------------- Complementary sensitivity function (T) -------------")
DM, GM, PM = control.disk_margins(L, omega, skew=-1.0, returnall=True) # T-based (T)
print(f"min(DM) = {min(DM)} (omega = {omega[np.argmin(DM)]})")
print(f"GM = {GM[np.argmin(DM)]} dB")
print(f"PM = {PM[np.argmin(DM)]} deg")
print(f"min(GM) = {min(GM)} dB")
print(f"min(PM) = {min(PM)} deg\n")
plt.figure(3)
plt.subplot(3, 3, 2)
plt.semilogx(omega, DM, label='$\\alpha$')
plt.ylabel('Disk Margin (abs)')
plt.legend()
plt.title('T-Based Margins')
plt.grid()
plt.xlim([omega[0], omega[-1]])
plt.ylim([0, 2])
plt.figure(3)
plt.subplot(3, 3, 5)
plt.semilogx(omega, GM, label='$\\gamma_{m}$')
plt.ylabel('Gain Margin (dB)')
plt.legend()
#plt.title('Gain-Only Margin')
plt.grid()
plt.xlim([omega[0], omega[-1]])
plt.ylim([0, 40])
plt.figure(3)
plt.subplot(3, 3, 8)
plt.semilogx(omega, PM, label='$\\phi_{m}$')
plt.ylabel('Phase Margin (deg)')
plt.legend()
#plt.title('Phase-Only Margin')
plt.grid()
plt.xlim([omega[0], omega[-1]])
plt.ylim([0, 90])
plt.xlabel('Frequency (rad/s)')
print("------------- Balanced sensitivity function (S - T) -------------")
DM, GM, PM = control.disk_margins(L, omega, skew=0.0, returnall=True) # balanced (S - T)
print(f"min(DM) = {min(DM)} (omega = {omega[np.argmin(DM)]})")
print(f"GM = {GM[np.argmin(DM)]} dB")
print(f"PM = {PM[np.argmin(DM)]} deg")
print(f"min(GM) = {min(GM)} dB")
print(f"min(PM) = {min(PM)} deg\n")
plt.figure(3)
plt.subplot(3, 3, 3)
plt.semilogx(omega, DM, label='$\\alpha$')
plt.ylabel('Disk Margin (abs)')
plt.legend()
plt.title('Balanced Margins')
plt.grid()
plt.xlim([omega[0], omega[-1]])
plt.ylim([0, 2])
plt.figure(3)
plt.subplot(3, 3, 6)
plt.semilogx(omega, GM, label='$\\gamma_{m}$')
plt.ylabel('Gain Margin (dB)')
plt.legend()
#plt.title('Gain-Only Margin')
plt.grid()
plt.xlim([omega[0], omega[-1]])
plt.ylim([0, 40])
plt.figure(3)
plt.subplot(3, 3, 9)
plt.semilogx(omega, PM, label='$\\phi_{m}$')
plt.ylabel('Phase Margin (deg)')
plt.legend()
#plt.title('Phase-Only Margin')
plt.grid()
plt.xlim([omega[0], omega[-1]])
plt.ylim([0, 90])
plt.xlabel('Frequency (rad/s)')
# Disk margin plot of admissible gain/phase variations for which
# the feedback loop still remains stable, for each skew parameter
DM_plot = []
DM_plot.append(control.disk_margins(L, omega, skew=-1.0)[0]) # T-based (T)
DM_plot.append(control.disk_margins(L, omega, skew=0.0)[0]) # balanced (S - T)
DM_plot.append(control.disk_margins(L, omega, skew=1.0)[0]) # S-based (S)
plt.figure(30)
plot_allowable_region(DM_plot, skew=[-1.0, 0.0, 1.0])
return
if __name__ == '__main__':
#test_siso1()
#test_siso2()
test_mimo()
if 'PYCONTROL_TEST_EXAMPLES' not in os.environ:
#plt.tight_layout()
plt.show()