-
Notifications
You must be signed in to change notification settings - Fork 453
Expand file tree
/
Copy pathpvtol-nested.py
More file actions
171 lines (139 loc) · 4.46 KB
/
pvtol-nested.py
File metadata and controls
171 lines (139 loc) · 4.46 KB
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
# pvtol-nested.py - inner/outer design for vectored thrust aircraft
# RMM, 5 Sep 09
#
# This file works through a fairly complicated control design and
# analysis, corresponding to the planar vertical takeoff and landing
# (PVTOL) aircraft in Astrom and Murray, Chapter 11. It is intended
# to demonstrate the basic functionality of the python-control
# package.
#
import os
import matplotlib.pyplot as plt # MATLAB-like plotting functions
import control as ct
import numpy as np
# System parameters
m = 4 # mass of aircraft
J = 0.0475 # inertia around pitch axis
r = 0.25 # distance to center of force
g = 9.8 # gravitational constant
c = 0.05 # damping factor (estimated)
# Transfer functions for dynamics
Pi = ct.tf([r], [J, 0, 0]) # inner loop (roll)
Po = ct.tf([1], [m, c, 0]) # outer loop (position)
#
# Inner loop control design
#
# This is the controller for the pitch dynamics. Goal is to have
# fast response for the pitch dynamics so that we can use this as a
# control for the lateral dynamics
#
# Design a simple lead controller for the system
k, a, b = 200, 2, 50
Ci = k * ct.tf([1, a], [1, b]) # lead compensator
Li = Pi * Ci
# Bode plot for the open loop process
plt.figure(1)
ct.bode_plot(Pi)
# Bode plot for the loop transfer function, with margins
plt.figure(2)
ct.bode_plot(Li)
# Compute out the gain and phase margins
gm, pm, wcg, wcp = ct.margin(Li)
# Compute the sensitivity and complementary sensitivity functions
Si = ct.feedback(1, Li)
Ti = Li * Si
# Check to make sure that the specification is met
plt.figure(3)
ct.gangof4(Pi, Ci)
# Compute out the actual transfer function from u1 to v1 (see L8.2 notes)
# Hi = Ci*(1-m*g*Pi)/(1+Ci*Pi)
Hi = ct.parallel(ct.feedback(Ci, Pi), -m * g *ct.feedback(Ci * Pi, 1))
plt.figure(4)
ct.bode_plot(Hi)
# Now design the lateral control system
a, b, K = 0.02, 5, 2
Co = -K * ct.tf([1, 0.3], [1, 10]) # another lead compensator
Lo = -m*g*Po*Co
plt.figure(5)
ct.bode_plot(Lo) # margin(Lo)
# Finally compute the real outer-loop loop gain + responses
L = Co * Hi * Po
S = ct.feedback(1, L)
T = ct.feedback(L, 1)
# Compute stability margins
gm, pm, wgc, wpc = ct.margin(L)
print("Gain margin: %g at %g" % (gm, wgc))
print("Phase margin: %g at %g" % (pm, wpc))
plt.figure(6)
plt.clf()
ct.bode_plot(L, np.logspace(-4, 3))
# Add crossover line to the magnitude plot
#
# Note: in matplotlib before v2.1, the following code worked:
#
# plt.subplot(211); hold(True);
# loglog([1e-4, 1e3], [1, 1], 'k-')
#
# In later versions of matplotlib the call to plt.subplot will clear the
# axes and so we have to extract the axes that we want to use by hand.
# In addition, hold() is deprecated so we no longer require it.
#
for ax in plt.gcf().axes:
if ax.get_label() == 'control-bode-magnitude':
break
ax.semilogx([1e-4, 1e3], 20*np.log10([1, 1]), 'k-')
#
# Replot phase starting at -90 degrees
#
# Get the phase plot axes
for ax in plt.gcf().axes:
if ax.get_label() == 'control-bode-phase':
break
# Recreate the frequency response and shift the phase
mag, phase, w = ct.freqresp(L, np.logspace(-4, 3))
phase = phase - 360
# Replot the phase by hand
ax.semilogx([1e-4, 1e3], [-180, -180], 'k-')
ax.semilogx(w, np.squeeze(phase), 'b-')
ax.axis([1e-4, 1e3, -360, 0])
plt.xlabel('Frequency [deg]')
plt.ylabel('Phase [deg]')
# plt.set(gca, 'YTick', [-360, -270, -180, -90, 0])
# plt.set(gca, 'XTick', [10^-4, 10^-2, 1, 100])
#
# Nyquist plot for complete design
#
plt.figure(7)
plt.clf()
ct.nyquist_plot(L)
# Add a box in the region we are going to expand
plt.plot([-2, -2, 1, 1, -2], [-4, 4, 4, -4, -4], 'r-')
# Expanded region
plt.figure(8)
plt.clf()
ct.nyquist_plot(L)
plt.axis([-2, 1, -4, 4])
# set up the color
color = 'b'
# Add arrows to the plot
# H1 = L.evalfr(0.4); H2 = L.evalfr(0.41);
# arrow([real(H1), imag(H1)], [real(H2), imag(H2)], AM_normal_arrowsize, \
# 'EdgeColor', color, 'FaceColor', color);
# H1 = freqresp(L, 0.35); H2 = freqresp(L, 0.36);
# arrow([real(H2), -imag(H2)], [real(H1), -imag(H1)], AM_normal_arrowsize, \
# 'EdgeColor', color, 'FaceColor', color);
plt.figure(9)
Tvec, Yvec = ct.step_response(T, np.linspace(0, 20))
plt.plot(Tvec.T, Yvec.T)
Tvec, Yvec = ct.step_response(Co*S, np.linspace(0, 20))
plt.plot(Tvec.T, Yvec.T)
plt.figure(10)
plt.clf()
P, Z = ct.pzmap(T, plot=True, grid=True)
print("Closed loop poles and zeros: ", P, Z)
# Gang of Four
plt.figure(11)
plt.clf()
ct.gangof4_plot(Hi * Po, Co)
if 'PYCONTROL_TEST_EXAMPLES' not in os.environ:
plt.show()