Applied Interval Analysis: With Examples in Parameter and State Estimation, Robust Control and Robotics, Volume 1

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Springer Science & Business Media, Aug 30, 2001 - Computers - 379 pages
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At the core of many engineering problems is the solution of sets of equa tions and inequalities, and the optimization of cost functions. Unfortunately, except in special cases, such as when a set of equations is linear in its un knowns or when a convex cost function has to be minimized under convex constraints, the results obtained by conventional numerical methods are only local and cannot be guaranteed. This means, for example, that the actual global minimum of a cost function may not be reached, or that some global minimizers of this cost function may escape detection. By contrast, interval analysis makes it possible to obtain guaranteed approximations of the set of all the actual solutions of the problem being considered. This, together with the lack of books presenting interval techniques in such a way that they could become part of any engineering numerical tool kit, motivated the writing of this book. The adventure started in 1991 with the preparation by Luc Jaulin of his PhD thesis, under Eric Walter's supervision. It continued with their joint supervision of Olivier Didrit's and Michel Kieffer's PhD theses. More than two years ago, when we presented our book project to Springer, we naively thought that redaction would be a simple matter, given what had already been achieved . . .
 

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Contents

1 Introduction
3
11 What Are the Key Concepts?
4
13 What About Complexity?
5
14 How is the Book Organized?
6
Tools
9
2 Interval Analysis
11
222 Extended operations
12
223 Properties of set operators
13
721 Characteristic polynomial
189
723 Stability degree
190
73 Basic Tests for Robust Stability
193
731 Interval polynomials
195
732 Polytope polynomials
196
734 Conclusion
198
742 Stability degree
201
743 Valueset approach
205

224 Wrappers
15
23 Interval Analysis
17
231 Intervals
18
232 Interval computation
19
233 Closed intervals
20
234 Interval vectors
23
235 Interval matrices
25
24 Inclusion Functions
27
242 Natural inclusion functions
29
243 Centred inclusion functions
33
244 Mixed centred inclusion functions
34
245 Taylor inclusion functions
35
25 Inclusion Tests
38
252 Tests
40
253 Inclusion tests for sets
42
3 Subpavings
45
32 Set Topology
46
322 Enclosure of compact sets between subpavings
48
33 Regular Subpavings
49
331 Pavings and subpavings
50
332 Representing a regular subpaving as a binary tree
51
333 Basic operations on regular subpavings
52
34 Implementation of Set Computation
54
341 Set inversion
55
342 Image evaluation
59
35 Conclusions
63
4 Contractors
65
42 Basic Contractors
67
422 Intervalization of finite subsolvers
69
423 Fixedpoint methods
72
424 Forward backward propagation
77
425 Linear programming approach
81
43 External Approximation
82
431 Principle
83
432 Preconditioning
84
433 Newton contractor
86
434 Parallel linearization
87
435 Using formal transformations
88
44 Collaboration Between Contractors
90
442 Contractors and inclusion functions
95
45 Contractors for Sets
97
452 Sets defined by equality and inequality constraints
99
46 Conclusions
100
5 Solvers
103
52 Solving Square Systems of Nonlinear Equations
104
53 Characterizing Sets Defined by Inequalities
106
54 Interval Hull of a Set Defined by Inequalities
111
541 First approach
112
542 Second approach
113
55 Global Optimization
117
551 The MooreSkelboe algorithm
120
552 Hansens algorithm
121
553 Using interval constraint propagation
125
56 Minimax Optimization
126
561 Unconstrained case
127
562 Constrained case
131
563 Dealing with quantifiers
133
57 Cost Contours
135
58 Conclusions
136
Applications
139
6 Estimation
141
62 Parameter Estimation Via Optimization
144
621 Leastsquare parameter estimation in compartmental modelling
145
622 Minimax parameter estimation
148
63 Parameter Bounding
155
632 The values of the independent variables are known
158
633 Robustification against outliers
160
634 The values of the independent variables are uncertain
164
635 Computation of the interval hull of the posterior feasible set
167
64 State Bounding
168
642 Bounding the initial state
171
644 Bounding by constraint propagation
174
65 Conclusions
184
7 Robust Control
187
72 Stability of Deterministic Linear Systems
188
744 Robust stability margins
211
745 Stability radius
216
75 Controller Design
220
76 Conclusions
223
8 Robotics
225
82 Forward Kinematics Problem for StewartGough Platforms
226
822 From the frame of the mobile plate to that of the base
227
823 Equations to be solved
229
824 Solution
230
83 Path Planning
234
831 Graph discretization of configuration space
237
832 Algorithms for finding a feasible path
239
833 Test case
241
84 Localization and Tracking of a Mobile Robot
248
841 Formulation of the static localization problem
249
842 Model of the measurement process
253
843 Set inversion
257
844 Dealing with outliers
259
845 Static localization example
260
846 Tracking
263
847 Example
264
85 Conclusions
267
Implementation
269
9 Automatic Differentiation
271
921 Forward differentiation
272
922 Backward differentiation
273
93 Differentiation of Algorithms
275
932 Second assumption
278
933 Third assumption
279
94 Examples
281
942 Example 2
284
95 Conclusions
285
10 Guaranteed Computation with Floatingpoint Numbers
287
1021 Representation
288
1022 Rounding
289
1023 Special quantities
291
103 Intervals and IEEE 754
292
1031 Machine intervals
293
1032 Closed interval arithmetic
294
1033 Handling elementary functions
295
1034 Improvements
297
105 Conclusions
299
11 Do It Yourself
301
1121 Program structure
302
1122 Standard types
303
1123 Pointers
304
113 INTERVAL Class
305
1131 Constructors and destructor
307
1132 Other member functions
308
1133 Mathematical functions
313
114 Intervals with PROFILBIAS
315
1142 PROFIL
316
1143 Getting started
317
115 Exercises on Intervals
318
116 Interval Vectors
319
1161 INTERVAL_VECTOR class
320
1162 Constructors assignment and function call operators
321
1163 Friend functions
323
1164 Utilities
325
117 Vectors with PROFILBIAS
326
118 Exercises on Interval Vectors
327
119 Interval Matrices
331
1110 Matrices with PROFILBIAS
332
1111 Exercises on Interval Matrices
333
1112 Regular Subpavings with PROFILBIAS
336
11122 Set inversion with subpavings
339
11123 Image evaluation with subpavings
342
11124 System simulation and state estimation with subpavings
347
1113 Error Handling
349
11132 Exception handling
350
11133 Mathematical errors
351
References
353
Index
373
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