diff --git a/control/freqplot.py b/control/freqplot.py
index a63ef20d3..8917690ba 100644
--- a/control/freqplot.py
+++ b/control/freqplot.py
@@ -1060,7 +1060,9 @@ def gen_zero_centered_series(val_min, val_max, period):
'nyquist.max_curve_magnitude': 20, # clip large values
'nyquist.max_curve_offset': 0.02, # offset of primary/mirror
'nyquist.start_marker': 'o', # marker at start of curve
- 'nyquist.start_marker_size': 4, # size of the maker
+ 'nyquist.start_marker_size': 4, # size of the marker
+ 'nyquist.circle_style': # style for unit circles
+ {'color': 'black', 'linestyle': 'dashed', 'linewidth': 1}
}
@@ -1477,8 +1479,8 @@ def nyquist_response(
def nyquist_plot(
data, omega=None, plot=None, label_freq=0, color=None, label=None,
- return_contour=None, title=None, legend_loc='upper right',
- ax=None, **kwargs):
+ return_contour=None, title=None, legend_loc='upper right', ax=None,
+ unit_circle=False, mt_circles=None, ms_circles=None, **kwargs):
"""Nyquist plot for a system.
Generates a Nyquist plot for the system over a (optional) frequency
@@ -1501,7 +1503,13 @@ def nyquist_plot(
``omega_limits``.
color : string, optional
Used to specify the color of the line and arrowhead.
-
+ unit_circle : bool, optional
+ If ``True``, display the unit circle, to read gain crossover frequency.
+ mt_circles : array_like, optional
+ Draw circles corresponding to the given magnitudes of sensitivity.
+ ms_circles : array_like, optional
+ Draw circles corresponding to the given magnitudes of complementary
+ sensitivity.
**kwargs : :func:`matplotlib.pyplot.plot` keyword properties, optional
Additional keywords (passed to `matplotlib`)
@@ -1856,7 +1864,48 @@ def _parse_linestyle(style_name, allow_false=False):
# Mark the -1 point
plt.plot([-1], [0], 'r+')
- # Label the frequencies of the points
+ #
+ # Draw circles for gain crossover and sensitivity functions
+ #
+ theta = np.linspace(0, 2*np.pi, 100)
+ cos = np.cos(theta)
+ sin = np.sin(theta)
+ label_pos = 15
+
+ # Display the unit circle, to read gain crossover frequency
+ if unit_circle:
+ plt.plot(cos, sin, **config.defaults['nyquist.circle_style'])
+
+ # Draw circles for given magnitudes of sensitivity
+ if ms_circles is not None:
+ for ms in ms_circles:
+ pos_x = -1 + (1/ms)*cos
+ pos_y = (1/ms)*sin
+ plt.plot(
+ pos_x, pos_y, **config.defaults['nyquist.circle_style'])
+ plt.text(pos_x[label_pos], pos_y[label_pos], ms)
+
+ # Draw circles for given magnitudes of complementary sensitivity
+ if mt_circles is not None:
+ for mt in mt_circles:
+ if mt != 1:
+ ct = -mt**2/(mt**2-1) # Mt center
+ rt = mt/(mt**2-1) # Mt radius
+ pos_x = ct+rt*cos
+ pos_y = rt*sin
+ plt.plot(
+ pos_x, pos_y,
+ **config.defaults['nyquist.circle_style'])
+ plt.text(pos_x[label_pos], pos_y[label_pos], mt)
+ else:
+ _, _, ymin, ymax = plt.axis()
+ pos_y = np.linspace(ymin, ymax, 100)
+ plt.vlines(
+ -0.5, ymin=ymin, ymax=ymax,
+ **config.defaults['nyquist.circle_style'])
+ plt.text(-0.5, pos_y[label_pos], 1)
+
+ # Label the frequencies of the points on the Nyquist curve
if label_freq:
ind = slice(None, None, label_freq)
omega_sys = np.imag(splane_contour[np.real(splane_contour) == 0])
diff --git a/control/tests/nyquist_test.py b/control/tests/nyquist_test.py
index 8354932d7..af9505354 100644
--- a/control/tests/nyquist_test.py
+++ b/control/tests/nyquist_test.py
@@ -214,6 +214,22 @@ def test_nyquist_arrows(arrows):
assert _Z(sys) == response.count + _P(sys)
+def test_sensitivity_circles():
+ A = np.array([
+ [-3.56355873, -1.22980795, -1.5626527 , -0.4626829],
+ [-8.52361371, -3.60331459, -3.71574266, -0.43839201],
+ [-2.50458726, -0.72361335, -1.77795489, -0.4038419],
+ [-0.281183 , 0.23391825, 0.19096003, -0.9771515]])
+ B = np.array([[-0.], [-1.42827213], [ 0.76806551], [-1.07987454]])
+ C = np.array([[-0., 0.35557249, 0.35941791, -0.]])
+ D = np.array([[0]])
+ sys1 = ct.ss(A, B, C, D)
+ sys2 = ct.ss(A, B, C, D, dt=0.1)
+ plt.figure()
+ ct.nyquist_plot(sys1, unit_circle=True, mt_circles=[0.9,1,1.1,1.2], ms_circles=[0.9,1,1.1,1.2])
+ ct.nyquist_plot(sys2, unit_circle=True, mt_circles=[0.9,1,1.1,1.2], ms_circles=[0.9,1,1.1,1.2])
+
+
def test_nyquist_encirclements():
# Example 14.14: effect of friction in a cart-pendulum system
s = ct.tf('s')
@@ -518,6 +534,9 @@ def test_nyquist_frd():
test_nyquist_arrows(3)
test_nyquist_arrows([0.1, 0.5, 0.9])
+ print("Test sensitivity circles")
+ test_sensitivity_circles()
+
print("Stability checks")
test_nyquist_encirclements()
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