Add the ability to define custom Nichols charts by passing magnitude and/or phase arrays to nichols_grid. Also some tweaks to the chart grid generation and axis configuration, and further clean-up of documentation and naming.

Allan McInnes · Allan McInnes · commit 6338d2be0fa9 · 2011-02-14T07:49:05.000Z
diff --git a/src/freqplot.py b/src/freqplot.py
@@ -181,7 +181,7 @@ def nyquist(syslist, omega=None):
 # Nichols plot
 # Contributed by Allan McInnes <Allan.McInnes@canterbury.ac.nz>
 #! TODO: need unit test code
-def nichols(syslist, omega=None):
+def nichols(syslist, omega=None, grid=True):
     """Nichols plot for a system
 
     Usage
@@ -196,6 +196,8 @@ def nichols(syslist, omega=None):
         List of linear input/output systems (single system is OK)
     omega : freq_range
         Range of frequencies (list or bounds) in rad/sec
+    grid : boolean
+        True if the plot should include a Nichols-chart grid
 
     Return values
     -------------
@@ -228,88 +230,114 @@ def nichols(syslist, omega=None):
 
     # Mark the -180 point
     plt.plot([-180], [0], 'r+')
+
+    # Add grid
+    if grid:
+        nichols_grid()
     
 # Nichols grid
-def nichols_grid():
+def nichols_grid(**kwargs):
     """Nichols chart grid
     
     Usage
     =====
     nichols_grid()
 
-    Plots a Nichols chart grid on the current axis.
+    Plots a Nichols chart grid on the current axis, or creates a new chart
+    if no plot already exists.
 
     Parameters
     ----------
-    None
+    cl_mags : array-like (dB)
+        Array of closed-loop magnitudes defining the iso-gain lines on a
+        custom Nichols chart.
+    cl_phases : array-like (degrees)
+        Array of closed-loop phases defining the iso-phase lines on a custom
+        Nichols chart. Must be in the range -360 < cl_phases < 0
 
     Return values
     -------------
     None    
     """
-    mag_min_default = -40.0 # dB
-    mag_step = 20.0         # dB
-
-    # Chart defaults
-    phase_min, phase_max, mag_min, mag_max = -360.0, 0.0, mag_min_default, 40.0
-
-    # Set actual chart bounds based on current plot
+    # Default chart size
+    ol_phase_min = -359.99
+    ol_phase_max = 0.0
+    ol_mag_min = -40.0
+    ol_mag_max = default_ol_mag_max = 50.0
+    
+    # Find bounds of the current dataset, if there is one. 
     if plt.gcf().gca().has_data():
-        phase_min, phase_max, mag_min, mag_max = plt.axis()
-        
+        ol_phase_min, ol_phase_max, ol_mag_min, ol_mag_max = plt.axis()
+
     # M-circle magnitudes.
-    # The "fixed" set are always generated, since this guarantees a recognizable
-    # Nichols chart grid.
-    mags_fixed = np.array([-40.0, -20.0, -12.0, -6.0, -3.0, -1.0, -0.5, 0.0,
-                                 0.25, 0.5, 1.0, 3.0, 6.0, 12.0])
-
-    if mag_min < mag_min_default:
-        # Outside the "fixed" set of magnitudes, the generated M-circles
-        # are extended in steps of 'mag_step' dB to cover anything made
-        # visible by the range of the existing plot
-        mags_adjust = np.arange(mag_step*np.floor(mag_min/mag_step),
-                                mag_min_default, mag_step)
-        mags = np.concatenate((mags_adjust, mags_fixed))
+    if kwargs.has_key('cl_mags'):
+        # Custom chart
+        cl_mags = kwargs['cl_mags']
     else:
-        mags = mags_fixed
+        # Default chart magnitudes
+        # The key set of magnitudes are always generated, since this
+        # guarantees a recognizable Nichols chart grid.
+        key_cl_mags = np.array([-40.0, -20.0, -12.0, -6.0, -3.0, -1.0, -0.5, 0.0,
+                                     0.25, 0.5, 1.0, 3.0, 6.0, 12.0])
+        # Extend the range of magnitudes if necessary. The extended arange
+        # will end up empty if no extension is required. Assumes that closed-loop
+        # magnitudes are approximately aligned with open-loop magnitudes beyond
+        # the value of np.min(key_cl_mags)
+        cl_mag_step = -20.0 # dB
+        extended_cl_mags = np.arange(np.min(key_cl_mags),
+                                     ol_mag_min + cl_mag_step, cl_mag_step)
+        cl_mags = np.concatenate((extended_cl_mags, key_cl_mags))
                      
     # N-circle phases (should be in the range -360 to 0)
-    phases = np.array([-0.25, -10.0, -20.0, -30.0, -45.0, -60.0, -90.0,
-                       -120.0, -150.0, -180.0, -210.0, -240.0, -270.0,
-                       -310.0, -325.0, -340.0, -350.0, -359.75])
+    if kwargs.has_key('cl_phases'):
+        # Custom chart
+        cl_phases = kwargs['cl_phases']
+        assert ((-360.0 < np.min(cl_phases)) and (np.max(cl_phases) < 0.0)) 
+    else:
+        # Choose a reasonable set of default phases (denser if the open-loop
+        # data is restricted to a relatively small range of phases).
+        key_cl_phases = np.array([-0.25, -45.0, -90.0, -180.0, -270.0, -325.0, -359.75])
+        if np.abs(ol_phase_max - ol_phase_min) < 90.0:
+            other_cl_phases = np.arange(-10.0, -360.0, -10.0)
+        else:
+            other_cl_phases = np.arange(-10.0, -360.0, -20.0)
+        cl_phases = np.concatenate((key_cl_phases, other_cl_phases))
 
     # Find the M-contours
-    m = m_circles(mags, phase_min=np.min(phases), phase_max=np.max(phases))
+    m = m_circles(cl_mags, phase_min=np.min(cl_phases), phase_max=np.max(cl_phases))
     m_mag = 20*sp.log10(np.abs(m))
     m_phase = sp.mod(sp.degrees(sp.angle(m)), -360.0) # Unwrap
 
     # Find the N-contours
-    n = n_circles(phases, mag_min=np.min(mags), mag_max=np.max(mags))
+    n = n_circles(cl_phases, mag_min=np.min(cl_mags), mag_max=np.max(cl_mags))
     n_mag = 20*sp.log10(np.abs(n))
     n_phase = sp.mod(sp.degrees(sp.angle(n)), -360.0) # Unwrap
 
     # Plot the contours behind other plot elements.
     # The "phase offset" is used to produce copies of the chart that cover
     # the entire range of the plotted data, starting from a base chart computed
-    # over the range -360 < phase < 0 (see above). Given the range 
+    # over the range -360 < phase < 0. Given the range 
     # the base chart is computed over, the phase offset should be 0
-    # for -360 < phase_min < 0.
-    phase_offset_min = 360.0*np.ceil(phase_min/360.0)
-    phase_offset_max = 360.0*np.ceil(phase_max/360.0) + 360.0
+    # for -360 < ol_phase_min < 0.
+    phase_offset_min = 360.0*np.ceil(ol_phase_min/360.0)
+    phase_offset_max = 360.0*np.ceil(ol_phase_max/360.0) + 360.0
     phase_offsets = np.arange(phase_offset_min, phase_offset_max, 360.0)
+
     for phase_offset in phase_offsets:
+        # Draw M and N contours
         plt.plot(m_phase + phase_offset, m_mag, color='gray',
                  linestyle='dashed', zorder=0)
         plt.plot(n_phase + phase_offset, n_mag, color='gray',
                  linestyle='dashed', zorder=0)
 
-    # Add magnitude labels
-    for x, y, m in zip(m_phase[:][-1], m_mag[:][-1], mags):
-        align = 'right' if m < 0.0 else 'left'
-        plt.text(x, y, str(m) + ' dB', size='small', ha=align)
+        # Add magnitude labels
+        for x, y, m in zip(m_phase[:][-1] + phase_offset, m_mag[:][-1], cl_mags):
+            align = 'right' if m < 0.0 else 'left'
+            plt.text(x, y, str(m) + ' dB', size='small', ha=align)
 
-    # Make sure axes conform to any pre-existing plot.
-    plt.axis([phase_min, phase_max, mag_min, mag_max])
+    # Fit axes to generated chart
+    plt.axis([phase_offset_min - 360.0, phase_offset_max - 360.0,
+              np.min(cl_mags), np.max([ol_mag_max, default_ol_mag_max])])
 
 # Gang of Four
 #! TODO: think about how (and whether) to handle lists of systems
@@ -417,38 +445,44 @@ def default_frequency_range(syslist):
     return omega
 
 # Compute contours of a closed-loop transfer function
-def closed_loop_contours(Hmag, Hphase):
-    """Contours of the function H = G/(1+G).
+def closed_loop_contours(Gcl_mags, Gcl_phases):
+    """Contours of the function Gcl = Gol/(1+Gol), where
+    Gol is an open-loop transfer function, and Gcl is a corresponding
+    closed-loop transfer function.
 
     Usage
     =====
-    contours = closed_loop_contours(mags, phases)
+    contours = closed_loop_contours(Gcl_mags, Gcl_phases)
 
     Parameters
     ----------
-    mags : array-like
-        Meshgrid array of magnitudes of the contours
-    phases : array-like
-        Meshgrid array of phases in radians of the contours
+    Gcl_mags : array-like
+        Array of magnitudes of the contours
+    Gcl_phases : array-like
+        Array of phases in radians of the contours
 
     Return values
     -------------
     contours : complex array
         Array of complex numbers corresponding to the contours.
     """
-    # Compute the contours in H-space
-    H = Hmag*sp.exp(1.j*Hphase)
+    # Compute the contours in Gcl-space. Since we're given closed-loop
+    # magnitudes and phases, this is just a case of converting them into
+    # a complex number.
+    Gcl = Gcl_mags*sp.exp(1.j*Gcl_phases)
 
-    # Invert H = G/(1+G) to get an expression for the contours in G-space
-    return H/(1.0 - H)
+    # Invert Gcl = Gol/(1+Gol) to map the contours into the open-loop space
+    return Gcl/(1.0 - Gcl)
 
 # M-circle
 def m_circles(mags, phase_min=-359.75, phase_max=-0.25):
-    """Constant-magnitude contours of the function H = G/(1+G).
+    """Constant-magnitude contours of the function Gcl = Gol/(1+Gol), where
+    Gol is an open-loop transfer function, and Gcl is a corresponding
+    closed-loop transfer function.
 
     Usage
     =====
-    contours = m_circles(mags)
+    contours = m_circles(mags, phase_min, phase_max)
 
     Parameters
     ----------
@@ -467,16 +501,18 @@ def m_circles(mags, phase_min=-359.75, phase_max=-0.25):
     # Convert magnitudes and phase range into a grid suitable for
     # building contours
     phases = sp.radians(sp.linspace(phase_min, phase_max, 500))
-    Hmag, Hphase = sp.meshgrid(10.0**(mags/20.0), phases)
-    return closed_loop_contours(Hmag, Hphase)
+    Gcl_mags, Gcl_phases = sp.meshgrid(10.0**(mags/20.0), phases)
+    return closed_loop_contours(Gcl_mags, Gcl_phases)
 
 # N-circle
 def n_circles(phases, mag_min=-40.0, mag_max=12.0):
-    """Constant-phase contours of the function H = G/(1+G).
+    """Constant-phase contours of the function Gcl = Gol/(1+Gol), where
+    Gol is an open-loop transfer function, and Gcl is a corresponding
+    closed-loop transfer function.
 
     Usage
     ===== 
-    contour = n_circles(angles)
+    contours = n_circles(phases, mag_min, mag_max)
 
     Parameters
     ----------
@@ -495,5 +531,5 @@ def n_circles(phases, mag_min=-40.0, mag_max=12.0):
     # Convert phases and magnitude range into a grid suitable for
     # building contours    
     mags = sp.linspace(10**(mag_min/20.0), 10**(mag_max/20.0), 2000)
-    Hphase, Hmag = sp.meshgrid(sp.radians(phases), mags)
-    return closed_loop_contours(Hmag, Hphase)
+    Gcl_phases, Gcl_mags = sp.meshgrid(sp.radians(phases), mags)
+    return closed_loop_contours(Gcl_mags, Gcl_phases)
