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
172
173
174
175
|
#
# MIT No Attribution
#
# Copyright (C) 2010-2023 Joel Andersson, Joris Gillis, Moritz Diehl, KU Leuven.
#
# Permission is hereby granted, free of charge, to any person obtaining a copy of this
# software and associated documentation files (the "Software"), to deal in the Software
# without restriction, including without limitation the rights to use, copy, modify,
# merge, publish, distribute, sublicense, and/or sell copies of the Software, and to
# permit persons to whom the Software is furnished to do so.
#
# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED,
# INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A
# PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT
# HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION
# OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE
# SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
#
from pylab import *
from scipy.linalg import sqrtm
from casadi import *
from casadi.tools import *
# System states
states = struct_symSX(["x","y","dx","dy"])
x,y,dx,dy = states[...]
# System controls
controls = struct_symSX(["u","v"])
u,v = controls[...]
# System parameters
parameters = struct_symSX(["k","c","beta"])
k,c,beta = parameters[...]
# Provide some numerical values
parameters_ = parameters()
parameters_["k"] = 10
parameters_["beta"] = 1
parameters_["c"] = 1
vel = vertcat(dx,dy)
p = vertcat(x,y)
q = vertcat(u,v)
# System dynamics
F = -k*(p-q) - beta*v*sqrt(sumsqr(vel) + c**2)
# System right hand side
rhs = struct_SX(states)
rhs["x"] = dx
rhs["y"] = dy
rhs["dx"] = F[0]
rhs["dy"] = F[1]
# f = SX.fun("f", controldaeIn(x=states,p=parameters,u=controls),daeOut(ode=rhs))
# # Simulation output grid
# N = 100
# tgrid = linspace(0,10.0,N)
# # ControlSimulator will output on each node of the timegrid
# opts = {}
# opts["integrator"] = "cvodes"
# opts["integrator_options"] = {"abstol":1e-10,"reltol":1e-10}
# csim = ControlSimulator("csim", f, tgrid, opts)
# x0 = states(0)
# # Create input profile
# controls_ = controls.repeated(csim.getInput("u"))
# controls_[0,"u"] = 1 # Kick the system with u=1 at the start
# controls_[N/2,"v"] = 2 # Kick the system with v=2 at half the simulation time
# # Pure simulation
# csim.setInput(x0,"x0")
# csim.setInput(parameters_,"p")
# csim.setInput(controls_,"u")
# csim.evaluate()
# output = states.repeated(csim.getOutput())
# # Plot all states
# for k in states.keys():
# plot(tgrid,output[vertcat,:,k])
# xlabel("t")
# legend(tuple(states.keys()))
# print "xf=", output[-1]
# # The remainder of this file deals with methods to calculate the state covariance matrix as it propagates through the system dynamics
# # === Method 1: integrator sensitivity ===
# # PF = d(I)/d(x0) P0 [d(I)/d(x0)]^T
# P0 = states.squared()
# P0[:,:] = 0.01*DM.eye(states.size)
# P0["x","dy"] = P0["dy","x"] = 0.002
# # Not supported in current revision, cf. #929
# # J = csim.jacobian_old(csim.index_in("x0"),csim.index_out("xf"))
# # J.setInput(x0,"x0")
# # J.setInput(parameters_,"p")
# # J.setInput(controls_,"u")
# # J.evaluate()
# # Jk = states.squared_repeated(J.getOutput())
# # F = Jk[-1]
# # PF_method1 = mtimes([F,P0,F.T])
# # print "State cov (method 1) = ", PF_method1
# # === Method 2: Lyapunov equations ===
# # P' = A.P + P.A^T
# states_aug = struct_symSX([
# entry("orig",sym=states),
# entry("P",shapestruct=(states,states))
# ])
# A = jacobian(rhs,states)
# rhs_aug = struct_SX(states_aug)
# rhs_aug["orig"] = rhs
# rhs_aug["P"] = mtimes(A,states_aug["P"]) + mtimes(states_aug["P"],A.T)
# f_aug = SX.fun("f_aug", controldaeIn(x=states_aug,p=parameters,u=controls),daeOut(ode=rhs_aug))
# csim_aug = ControlSimulator("csim_aug", f_aug, tgrid, {"integrator":"cvodes"})
# states_aug(csim_aug.getInput("x0"))["orig"] = x0
# states_aug(csim_aug.getInput("x0"))["P"] = P0
# csim_aug.setInput(parameters_,"p")
# csim_aug.setInput(controls_,"u")
# csim_aug.evaluate()
# output = states_aug.repeated(csim_aug.getOutput())
# PF_method2 = output[-1,"P"]
# print "State cov (method 2) = ", PF_method2
# # === Method 3: Unscented propagation ===
# # Sample and simulate 2n+1 initial points
# n = states.size
# W0 = 0
# x0 = DM(x0)
# W = DM([ W0 ] + [(1.0-W0)/2/n for j in range(2*n)])
# sqm = sqrtm(n/(1.0-W0)*DM(P0)).real
# sample_x = [ x0 ] + [x0+sqm[:,i] for i in range(n)] + [x0-sqm[:,i] for i in range(n)]
# csim.setInput(parameters_,"p")
# csim.setInput(controls_,"u")
# simulated_x = [] # This can be parallelised
# for x0_ in sample_x:
# csim.setInput(x0_,"x0")
# csim.evaluate()
# simulated_x.append(csim.getOutput()[-1,:])
# simulated_x = vertcat(simulated_x).T
# Xf_mean = mtimes(simulated_x,W)
# x_dev = simulated_x-mtimes(Xf_mean,DM.ones(1,2*n+1))
# PF_method3 = mtimes([x_dev,diag(W),x_dev.T])
# print "State cov (method 3) = ", PF_method3
# show()
|
