Optimal Measurement Methods for Distributed Parameter System Identification

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ISBN: 9780849323133 | 0849323134
Cover: Hardcover
Copyright: 8/27/2004

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For dynamic distributed systems modeled by partial differential equations, existing methods of sensor location in parameter estimation experiments are either limited to one-dimensional spatial domains or require large investments in software systems. This book discusses the characteristic features of the sensor placement problem, analyzes classical and recent approaches, and proposes a wide range of original solutions. It also presents results that translate easily to MATLAB and to MAPLE. This is a superb resource for researchers and practitioners in the fields of applied mathematics and electrical, civil, geotechnical, mechanical, chemical, and environmental engineering.

Preface	p. xv
Introduction	p. 1
The optimum experimental design problem in context	p. 1
A general review of the literature	p. 3
Key ideas of identification and experimental design	p. 9
System description	p. 9
Parameter identification	p. 13
Measurement-location problem	p. 14
Main impediments	p. 19
High dimensionality of the multimodal optimization problem	p. 19
Loss of the underlying properties of the estimator for finite horizons of observation	p. 20
Sensor clusterization	p. 20
Dependence of the solution on the parameters to be identified	p. 22
Deterministic interpretation of the FIM	p. 24
Calculation of sensitivity coefficients	p. 27
Finite-difference method	p. 27
Direct-differentiation method	p. 28
Adjoint method	p. 29
A final introductory note	p. 31
Locally optimal designs for stationary sensors	p. 33
Linear-in-parameters lumped models	p. 33
Problem statement	p. 34
Characterization of the solutions	p. 38
Algorithms	p. 49
Construction of minimax designs	p. 68
Continuous designs in measurement optimization	p. 74
Clusterization-free designs	p. 83
Nonlinear programming approach	p. 88
A critical note on a deterministic approach	p. 92
Modifications required by other settings	p. 95
Discrete-time measurements	p. 95
Multiresponse systems and inaccessibility of state measurements	p. 95
Simplifications for static DPSs	p. 96
Summary	p. 100
Locally optimal strategies for scanning and moving observations	p. 103
Optimal activation policies for scanning sensors	p. 103
Exchange scheme based on clusterization-free designs	p. 105
Scanning sensor scheduling as a constrained optimal control problem	p. 118
Equivalent Mayer formulation	p. 120
Computational procedure based on the control parameterization-enhancing technique	p. 121
Adapting the idea of continuous designs for moving sensors	p. 125
Optimal time-dependent measures	p. 125
Parameterization of sensor trajectories	p. 129
Optimization of sensor trajectories based on optimal-control techniques	p. 131
Statement of the problem and notation	p. 131
Equivalent Mayer problem and existence results	p. 134
Linearization of the optimal-control problem	p. 136
A numerical technique for solving the optimal measurement problem	p. 137
Special cases	p. 142
Concluding remarks	p. 149
Measurement strategies with alternative design objectives	p. 153
Optimal sensor location for prediction	p. 153
Problem formulation	p. 153
Optimal-control formulation	p. 155
Minimization algorithm	p. 156
Sensor location for model discrimination	p. 159
Competing models of a given distributed system	p. 160
Theoretical problem setup	p. 161
T[subscript 12]-optimality conditions	p. 164
Numerical construction of T[subscript 12]-optimum designs	p. 167
Conclusions	p. 171
Robust designs for sensor location	p. 173
Sequential designs	p. 173
Optimal designs in the average sense	p. 175
Problem statement	p. 175
Stochastic-approximation algorithms	p. 177
Optimal designs in the minimax sense	p. 181
Problem statement and characterization	p. 181
Numerical techniques for exact designs	p. 182
Robust sensor location using randomized algorithms	p. 187
A glance at complexity theory	p. 188
NP-hard problems in control-system design	p. 190
Weakened definitions of minima	p. 191
Randomized algorithm for sensor placement	p. 193
Concluding remarks	p. 198
Towards even more challenging problems	p. 201
Measurement strategies in the presence of correlated observations	p. 201
Exchange algorithm for [psi]-optimum designs	p. 203
Maximization of an observability measure	p. 209
Observability in a quantitative sense	p. 210
Scanning problem for optimal observability	p. 211
Conversion to finding optimal sensor densities	p. 212
Summary	p. 216
Applications from engineering	p. 217
Electrolytic reactor	p. 217
Optimization of experimental effort	p. 219
Clusterization-free designs	p. 220
Calibration of smog-prediction models	p. 221
Monitoring of groundwater resources quality	p. 225
Diffusion process with correlated observational errors	p. 230
Vibrating H-shaped membrane	p. 232
Conclusions and future research directions	p. 237
Appendices	p. 245
List of symbols	p. 247
Mathematical background	p. 251
Matrix algebra	p. 251
Symmetric, nonnegative definite and positive-definite matrices	p. 255
Vector and matrix differentiation	p. 260
Convex sets and convex functions	p. 264
Convexity and differentiability of common optimality criteria	p. 267
Differentiability of spectral functions	p. 268
Monotonicity of common design criteria	p. 271
Integration with respect to probability measures	p. 272
Projection onto the canonical simplex	p. 274
Conditional probability and conditional expectation	p. 275
Some accessory inequalities from statistical learning theory	p. 277
Hoeffding's inequality	p. 277
Estimating the minima of functions	p. 278
Statistical properties of estimators	p. 279
Best linear unbiased estimators in a stochastic-process setting	p. 279
Best linear unbiased estimators in a partially uncorrelated framework	p. 284
Analysis of the largest eigenvalue	p. 289
Directional differentiability	p. 289
Case of the single largest eigenvalue	p. 289
Case of the repeated largest eigenvalue	p. 292
Smooth approximation to the largest eigenvalue	p. 293
Differentiation of nonlinear operators	p. 297
Gateaux and Frechet derivatives	p. 297
Chain rule of differentiation	p. 298
Partial derivatives	p. 298
One-dimensional domains	p. 299
Second derivatives	p. 299
Functionals on Hilbert spaces	p. 300
Directional derivatives	p. 301
Differentiability of max functions	p. 301
Accessory results for PDEs	p. 303
Green formulae	p. 303
Differentiability w.r.t. parameters	p. 304
Interpolation of tabulated sensitivity coefficients	p. 313
Cubic spline interpolation for functions of one variable	p. 313
Tricubic spline interpolation	p. 314
Differentials of Section 4.3.3	p. 321
Derivation of formula (4.126)	p. 321
Derivation of formula (4.129)	p. 322
Solving sensor-location problems using Maple & Matlab	p. 323
Optimum experimental effort for a 1D problem	p. 323
Forming the system of sensitivity equations	p. 324
Solving the sensitivity equations	p. 325
Determining the FIMs associated with individual spatial points	p. 327
Optimizing the design weights	p. 327
Plotting the results	p. 328
Clusterization-free design for a 2D problem	p. 329
Solving sensitivity equations using the PDE Toolbox	p. 329
Determining potential contributions to the FIM from individual spatial points	p. 335
Iterative optimization of sensor locations	p. 336
References	p. 339
Index	p. 367
Table of Contents provided by Rittenhouse. All Rights Reserved.

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