-
Notifications
You must be signed in to change notification settings - Fork 412
/
delay.py
125 lines (111 loc) · 3.95 KB
/
delay.py
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
# -*-coding: utf-8-*-
#! TODO: add module docstring
# delay.py - functions involving time delays
#
# Author: Sawyer Fuller
# Date: 26 Aug 2010
#
# This file contains functions for implementing time delays (currently
# only the pade() function).
#
# Copyright (c) 2010 by California Institute of Technology
# All rights reserved.
#
# Redistribution and use in source and binary forms, with or without
# modification, are permitted provided that the following conditions
# are met:
#
# 1. Redistributions of source code must retain the above copyright
# notice, this list of conditions and the following disclaimer.
#
# 2. Redistributions in binary form must reproduce the above copyright
# notice, this list of conditions and the following disclaimer in the
# documentation and/or other materials provided with the distribution.
#
# 3. Neither the name of the California Institute of Technology nor
# the names of its contributors may be used to endorse or promote
# products derived from this software without specific prior
# written permission.
#
# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
# "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
# LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
# FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL CALTECH
# OR THE CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
# SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
# LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF
# USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
# ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
# OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT
# OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
# SUCH DAMAGE.
#
# $Id$
__all__ = ['pade']
def pade(T, n=1, numdeg=None):
"""
Create a linear system that approximates a delay.
Return the numerator and denominator coefficients of the Pade approximation.
Parameters
----------
T : number
time delay
n : positive integer
degree of denominator of approximation
numdeg: integer, or None (the default)
If None, numerator degree equals denominator degree
If >= 0, specifies degree of numerator
If < 0, numerator degree is n+numdeg
Returns
-------
num, den : array
Polynomial coefficients of the delay model, in descending powers of s.
Notes
-----
Based on:
1. Algorithm 11.3.1 in Golub and van Loan, "Matrix Computation" 3rd.
Ed. pp. 572-574
2. M. Vajta, "Some remarks on Padé-approximations",
3rd TEMPUS-INTCOM Symposium
Examples
--------
>>> delay = 1
>>> num, den = ct.pade(delay, 3)
>>> num, den
([-1.0, 12.0, -60.0, 120.0], [1.0, 12.0, 60.0, 120.0])
>>> num, den = ct.pade(delay, 3, -2)
>>> num, den
([-6.0, 24.0], [1.0, 6.0, 18.0, 24.0])
"""
if numdeg is None:
numdeg = n
elif numdeg < 0:
numdeg += n
if not T >= 0:
raise ValueError("require T >= 0")
if not n >= 0:
raise ValueError("require n >= 0")
if not (0 <= numdeg <= n):
raise ValueError("require 0 <= numdeg <= n")
if T == 0:
num = [1,]
den = [1,]
else:
num = [0. for i in range(numdeg+1)]
num[-1] = 1.
cn = 1.
for k in range(1, numdeg+1):
# derived from Gloub and van Loan eq. for Dpq(z) on p. 572
# this accumulative style follows Alg 11.3.1
cn *= -T * (numdeg - k + 1)/(numdeg + n - k + 1)/k
num[numdeg-k] = cn
den = [0. for i in range(n+1)]
den[-1] = 1.
cd = 1.
for k in range(1, n+1):
# see cn above
cd *= T * (n - k + 1)/(numdeg + n - k + 1)/k
den[n-k] = cd
num = [coeff/den[0] for coeff in num]
den = [coeff/den[0] for coeff in den]
return num, den