Inner/outer control design for vertical takeoff and landing aircraft

This script demonstrates the use of the python-control package for analysis and design of a controller for a vectored thrust aircraft model that is used as a running example through the text Feedback Systems by Astrom and Murray. This example makes use of MATLAB compatible commands.

Code

  1# pvtol-nested.py - inner/outer design for vectored thrust aircraft
  2# RMM, 5 Sep 09
  3#
  4# This file works through a fairly complicated control design and
  5# analysis, corresponding to the planar vertical takeoff and landing
  6# (PVTOL) aircraft in Astrom and Murray, Chapter 11.  It is intended
  7# to demonstrate the basic functionality of the python-control
  8# package.
  9#
 10
 11import os
 12import matplotlib.pyplot as plt  # MATLAB plotting functions
 13from control.matlab import *    # MATLAB-like functions
 14import numpy as np
 15
 16# System parameters
 17m = 4               # mass of aircraft
 18J = 0.0475          # inertia around pitch axis
 19r = 0.25            # distance to center of force
 20g = 9.8             # gravitational constant
 21c = 0.05            # damping factor (estimated)
 22
 23# Transfer functions for dynamics
 24Pi = tf([r], [J, 0, 0])  # inner loop (roll)
 25Po = tf([1], [m, c, 0])  # outer loop (position)
 26
 27#
 28# Inner loop control design
 29#
 30# This is the controller for the pitch dynamics.  Goal is to have
 31# fast response for the pitch dynamics so that we can use this as a
 32# control for the lateral dynamics
 33#
 34
 35# Design a simple lead controller for the system
 36k, a, b = 200, 2, 50
 37Ci = k*tf([1, a], [1, b])  # lead compensator
 38Li = Pi*Ci
 39
 40# Bode plot for the open loop process
 41plt.figure(1)
 42bode(Pi)
 43
 44# Bode plot for the loop transfer function, with margins
 45plt.figure(2)
 46bode(Li)
 47
 48# Compute out the gain and phase margins
 49#! Not implemented
 50# gm, pm, wcg, wcp = margin(Li)
 51
 52# Compute the sensitivity and complementary sensitivity functions
 53Si = feedback(1, Li)
 54Ti = Li*Si
 55
 56# Check to make sure that the specification is met
 57plt.figure(3)
 58gangof4(Pi, Ci)
 59
 60# Compute out the actual transfer function from u1 to v1 (see L8.2 notes)
 61# Hi = Ci*(1-m*g*Pi)/(1+Ci*Pi)
 62Hi = parallel(feedback(Ci, Pi), -m*g*feedback(Ci*Pi, 1))
 63
 64plt.figure(4)
 65plt.clf()
 66plt.subplot(221)
 67bode(Hi)
 68
 69# Now design the lateral control system
 70a, b, K = 0.02, 5, 2
 71Co = -K*tf([1, 0.3], [1, 10])  # another lead compensator
 72Lo = -m*g*Po*Co
 73
 74plt.figure(5)
 75bode(Lo)  # margin(Lo)
 76
 77# Finally compute the real outer-loop loop gain + responses
 78L = Co*Hi*Po
 79S = feedback(1, L)
 80T = feedback(L, 1)
 81
 82# Compute stability margins
 83gm, pm, wgc, wpc = margin(L)
 84print("Gain margin: %g at %g" % (gm, wgc))
 85print("Phase margin: %g at %g" % (pm, wpc))
 86
 87plt.figure(6)
 88plt.clf()
 89bode(L, np.logspace(-4, 3))
 90
 91# Add crossover line to the magnitude plot
 92#
 93# Note: in matplotlib before v2.1, the following code worked:
 94#
 95#   plt.subplot(211); hold(True);
 96#   loglog([1e-4, 1e3], [1, 1], 'k-')
 97#
 98# In later versions of matplotlib the call to plt.subplot will clear the
 99# axes and so we have to extract the axes that we want to use by hand.
100# In addition, hold() is deprecated so we no longer require it.
101#
102for ax in plt.gcf().axes:
103    if ax.get_label() == 'control-bode-magnitude':
104        break
105ax.semilogx([1e-4, 1e3], 20*np.log10([1, 1]), 'k-')
106
107#
108# Replot phase starting at -90 degrees
109#
110# Get the phase plot axes
111for ax in plt.gcf().axes:
112    if ax.get_label() == 'control-bode-phase':
113        break
114
115# Recreate the frequency response and shift the phase
116mag, phase, w = freqresp(L, np.logspace(-4, 3))
117phase = phase - 360
118
119# Replot the phase by hand
120ax.semilogx([1e-4, 1e3], [-180, -180], 'k-')
121ax.semilogx(w, np.squeeze(phase), 'b-')
122ax.axis([1e-4, 1e3, -360, 0])
123plt.xlabel('Frequency [deg]')
124plt.ylabel('Phase [deg]')
125# plt.set(gca, 'YTick', [-360, -270, -180, -90, 0])
126# plt.set(gca, 'XTick', [10^-4, 10^-2, 1, 100])
127
128#
129# Nyquist plot for complete design
130#
131plt.figure(7)
132plt.clf()
133nyquist(L, (0.0001, 1000))
134
135# Add a box in the region we are going to expand
136plt.plot([-2, -2, 1, 1, -2], [-4, 4, 4, -4, -4], 'r-')
137
138# Expanded region
139plt.figure(8)
140plt.clf()
141nyquist(L)
142plt.axis([-2, 1, -4, 4])
143
144# set up the color
145color = 'b'
146
147# Add arrows to the plot
148# H1 = L.evalfr(0.4); H2 = L.evalfr(0.41);
149# arrow([real(H1), imag(H1)], [real(H2), imag(H2)], AM_normal_arrowsize, \
150#  'EdgeColor', color, 'FaceColor', color);
151
152# H1 = freqresp(L, 0.35); H2 = freqresp(L, 0.36);
153# arrow([real(H2), -imag(H2)], [real(H1), -imag(H1)], AM_normal_arrowsize, \
154#  'EdgeColor', color, 'FaceColor', color);
155
156plt.figure(9)
157Yvec, Tvec = step(T, np.linspace(0, 20))
158plt.plot(Tvec.T, Yvec.T)
159
160Yvec, Tvec = step(Co*S, np.linspace(0, 20))
161plt.plot(Tvec.T, Yvec.T)
162
163plt.figure(10)
164plt.clf()
165P, Z = pzmap(T, plot=True, grid=True)
166print("Closed loop poles and zeros: ", P, Z)
167
168# Gang of Four
169plt.figure(11)
170plt.clf()
171gangof4(Hi*Po, Co)
172
173if 'PYCONTROL_TEST_EXAMPLES' not in os.environ:
174    plt.show()

Notes

1. The environment variable PYCONTROL_TEST_EXAMPLES is used for testing to turn off plotting of the outputs.