Optimal measurement methods for distributed parameter system identification

Title:

Personal Author:

UcinskiI, Dariusz

Publication Information:

Boca Raton, FL : CRC Press, 2005

ISBN:

9780849323133

Subject Term:

Distributed parameter systems

Mathematical optimization

System analysis

Control theory

Available:*

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010082565	QA402 U24 2005	Open Access Book	Book	Searching... Unknown

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. With the expense of scanning and moving sensors, optimal placement presents a critical problem.

Optimal Measurement Methods for Distributed Parameter System Identification discusses the characteristic features of the sensor placement problem, analyzes classical and recent approaches, and proposes a wide range of original solutions, culminating in the most comprehensive and timely treatment of the issue available. By presenting a step-by-step guide to theoretical aspects and to practical design methods, this book provides a sound understanding of sensor location techniques.

Both researchers and practitioners will find the case studies, the proposed algorithms, and the numerical examples to be invaluable. This text also offers results that translate easily to MATLAB and to Maple. Assuming only a basic familiarity with partial differential equations, vector spaces, and probability and statistics, and avoiding too many technicalities, this is a superb resource for researchers and practitioners in the fields of applied mathematics, electrical, civil, geotechnical, mechanical, chemical, and environmental engineering.

Author Notes

Ucinski, Dariusz

Preface	p. xv
1 Introduction	p. 1
1.1 The optimum experimental design problem in context	p. 1
1.2 A general review of the literature	p. 3
2 Key ideas of identification and experimental design	p. 9
2.1 System description	p. 9
2.2 Parameter identification	p. 13
2.3 Measurement-location problem	p. 14
2.4 Main impediments	p. 19
2.4.1 High dimensionality of the multimodal optimization problem	p. 19
2.4.2 Loss of the underlying properties of the estimator for finite horizons of observation	p. 20
2.4.3 Sensor clusterization	p. 20
2.4.4 Dependence of the solution on the parameters to be identified	p. 22
2.5 Deterministic interpretation of the FIM	p. 24
2.6 Calculation of sensitivity coefficients	p. 27
2.6.1 Finite-difference method	p. 27
2.6.2 Direct-differentiation method	p. 28
2.6.3 Adjoint method	p. 29
2.7 A final introductory note	p. 31
3 Locally optimal designs for stationary sensors	p. 33
3.1 Linear-in-parameters lumped models	p. 33
3.1.1 Problem statement	p. 34
3.1.2 Characterization of the solutions	p. 38
3.1.3 Algorithms	p. 49
3.2 Construction of minimax designs	p. 68
3.3 Continuous designs in measurement optimization	p. 74
3.4 Clusterization-free designs	p. 83
3.5 Nonlinear programming approach	p. 88
3.6 A critical note on a deterministic approach	p. 92
3.7 Modifications required by other settings	p. 95
3.7.1 Discrete-time measurements	p. 95
3.7.2 Multiresponse systems and inaccessibility of state measurements	p. 95
3.7.3 Simplifications for static DPSs	p. 96
3.8 Summary	p. 100
4 Locally optimal strategies for scanning and moving observations	p. 103
4.1 Optimal activation policies for scanning sensors	p. 103
4.1.1 Exchange scheme based on clusterization-free designs	p. 105
4.1.2 Scanning sensor scheduling as a constrained optimal control problem	p. 118
4.1.3 Equivalent Mayer formulation	p. 120
4.1.4 Computational procedure based on the control parameterization-enhancing technique	p. 121
4.2 Adapting the idea of continuous designs for moving sensors	p. 125
4.2.1 Optimal time-dependent measures	p. 125
4.2.2 Parameterization of sensor trajectories	p. 129
4.3 Optimization of sensor trajectories based on optimal-control techniques	p. 131
4.3.1 Statement of the problem and notation	p. 131
4.3.2 Equivalent Mayer problem and existence results	p. 134
4.3.3 Linearization of the optimal-control problem	p. 136
4.3.4 A numerical technique for solving the optimal measurement problem	p. 137
4.3.5 Special cases	p. 142
4.4 Concluding remarks	p. 149
5 Measurement strategies with alternative design objectives	p. 153
5.1 Optimal sensor location for prediction	p. 153
5.1.1 Problem formulation	p. 153
5.1.2 Optimal-control formulation	p. 155
5.1.3 Minimization algorithm	p. 156
5.2 Sensor location for model discrimination	p. 159
5.2.1 Competing models of a given distributed system	p. 160
5.2.2 Theoretical problem setup	p. 161
5.2.3 T[subscript 12]-optimality conditions	p. 164
5.2.4 Numerical construction of T[subscript 12]-optimum designs	p. 167
5.3 Conclusions	p. 171
6 Robust designs for sensor location	p. 173
6.1 Sequential designs	p. 173
6.2 Optimal designs in the average sense	p. 175
6.2.1 Problem statement	p. 175
6.2.2 Stochastic-approximation algorithms	p. 177
6.3 Optimal designs in the minimax sense	p. 181
6.3.1 Problem statement and characterization	p. 181
6.3.2 Numerical techniques for exact designs	p. 182
6.4 Robust sensor location using randomized algorithms	p. 187
6.4.1 A glance at complexity theory	p. 188
6.4.2 NP-hard problems in control-system design	p. 190
6.4.3 Weakened definitions of minima	p. 191
6.4.4 Randomized algorithm for sensor placement	p. 193
6.5 Concluding remarks	p. 198
7 Towards even more challenging problems	p. 201
7.1 Measurement strategies in the presence of correlated observations	p. 201
7.1.1 Exchange algorithm for [psi]-optimum designs	p. 203
7.2 Maximization of an observability measure	p. 209
7.2.1 Observability in a quantitative sense	p. 210
7.2.2 Scanning problem for optimal observability	p. 211
7.2.3 Conversion to finding optimal sensor densities	p. 212
7.3 Summary	p. 216
8 Applications from engineering	p. 217
8.1 Electrolytic reactor	p. 217
8.1.1 Optimization of experimental effort	p. 219
8.1.2 Clusterization-free designs	p. 220
8.2 Calibration of smog-prediction models	p. 221
8.3 Monitoring of groundwater resources quality	p. 225
8.4 Diffusion process with correlated observational errors	p. 230
8.5 Vibrating H-shaped membrane	p. 232
9 Conclusions and future research directions	p. 237
Appendices	p. 245
A List of symbols	p. 247
B Mathematical background	p. 251
B.1 Matrix algebra	p. 251
B.2 Symmetric, nonnegative definite and positive-definite matrices	p. 255
B.3 Vector and matrix differentiation	p. 260
B.4 Convex sets and convex functions	p. 264
B.5 Convexity and differentiability of common optimality criteria	p. 267
B.6 Differentiability of spectral functions	p. 268
B.7 Monotonicity of common design criteria	p. 271
B.8 Integration with respect to probability measures	p. 272
B.9 Projection onto the canonical simplex	p. 274
B.10 Conditional probability and conditional expectation	p. 275
B.11 Some accessory inequalities from statistical learning theory	p. 277
B.11.1 Hoeffding's inequality	p. 277
B.11.2 Estimating the minima of functions	p. 278
C Statistical properties of estimators	p. 279
C.1 Best linear unbiased estimators in a stochastic-process setting	p. 279
C.2 Best linear unbiased estimators in a partially uncorrelated framework	p. 284
D Analysis of the largest eigenvalue	p. 289
D.1 Directional differentiability	p. 289
D.1.1 Case of the single largest eigenvalue	p. 289
D.1.2 Case of the repeated largest eigenvalue	p. 292
D.1.3 Smooth approximation to the largest eigenvalue	p. 293
E Differentiation of nonlinear operators	p. 297
E.1 Gateaux and Frechet derivatives	p. 297
E.2 Chain rule of differentiation	p. 298
E.3 Partial derivatives	p. 298
E.4 One-dimensional domains	p. 299
E.5 Second derivatives	p. 299
E.6 Functionals on Hilbert spaces	p. 300
E.7 Directional derivatives	p. 301
E.8 Differentiability of max functions	p. 301
F Accessory results for PDEs	p. 303
F.1 Green formulae	p. 303
F.2 Differentiability w.r.t. parameters	p. 304
G Interpolation of tabulated sensitivity coefficients	p. 313
G.1 Cubic spline interpolation for functions of one variable	p. 313
G.2 Tricubic spline interpolation	p. 314
H Differentials of Section 4.3.3	p. 321
H.1 Derivation of formula (4.126)	p. 321
H.2 Derivation of formula (4.129)	p. 322
I Solving sensor-location problems using Maple & Matlab	p. 323
I.1 Optimum experimental effort for a 1D problem	p. 323
I.1.1 Forming the system of sensitivity equations	p. 324
I.1.2 Solving the sensitivity equations	p. 325
I.1.3 Determining the FIMs associated with individual spatial points	p. 327
I.1.4 Optimizing the design weights	p. 327
I.1.5 Plotting the results	p. 328
I.2 Clusterization-free design for a 2D problem	p. 329
I.2.1 Solving sensitivity equations using the PDE Toolbox	p. 329
I.2.2 Determining potential contributions to the FIM from individual spatial points	p. 335
I.2.3 Iterative optimization of sensor locations	p. 336
References	p. 339
Index	p. 367

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Table of Contents