Optimal Measurement Methods for Distributed Parameter System Identification / Edition 1

by Dariusz Ucinski

Overview

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.

Product Details

ISBN-13:	9780367393984
Publisher:	Taylor & Francis
Publication date:	09/19/2019
Pages:	392
Product dimensions:	6.12(w) x 9.19(h) x (d)

About the Author

Ucinski, Dariusz

Table of Contents

Preface xv

1 Introduction 1

1.1 The optimum experimental design problem in context 1

1.2 A general review of the literature 3

2 Key ideas of identification and experimental design 9

2.1 System description 9

2.2 Parameter identification 13

2.3 Measurement-location problem 14

2.4 Main impediments 19

2.4.1 High dimensionality of the multimodal optimization problem 19

2.4.2 Loss of the underlying properties of the estimator for finite horizons of observation 20

2.4.3 Sensor clusterization 20

2.4.4 Dependence of the solution on the parameters to be identified 22

2.5 Deterministic interpretation of the FIM 24

2.6 Calculation of sensitivity coefficients 27

2.6.1 Finite-difference method 27

2.6.2 Direct-differentiation method 28

2.6.3 Adjoint method 29

2.7 A final introductory note 31

3 Locally optimal designs for stationary sensors 33

3.1 Linear-in-parameters lumped models 33

3.1.1 Problem statement 34

3.1.2 Characterization of the solutions 38

3.1.3 Algorithms 49

3.2 Construction of minimax designs 68

3.3 Continuous designs in measurement optimization 74

3.4 Clusterization-free designs 83

3.5 Nonlinear programming approach 88

3.6 A critical note on a deterministic approach 92

3.7 Modifications required by other settings 95

3.7.1 Discrete-time measurements 95

3.7.2 Multiresponse systems and inaccessibility of state measurements 95

3.7.3 Simplifications for static DPSs 96

3.8 Summary 100

4 Locally optimal strategies for scanning and moving observations 103

4.1 Optimal activation policies for scanning sensors 103

4.1.1 Exchange scheme based on clusterization-free designs 105

4.1.2 Scanning sensor scheduling as a constrained optimal control problem 118

4.1.3 Equivalent Mayer formulation 120

4.1.4 Computational procedure based on the control parameterization-enhancing technique 121

4.2 Adapting the idea of continuous designs for moving sensors 125

4.2.1 Optimal time-dependent measures 125

4.2.2 Parameterization of sensor trajectories 129

4.3 Optimization of sensor trajectories based on optimal-control techniques 131

4.3.1 Statement of the problem and notation 131

4.3.2 Equivalent Mayer problem and existence results 134

4.3.3 Linearization of the optimal-control problem 136

4.3.4 A numerical technique for solving the optimal measurement problem 137

4.3.5 Special cases 142

4.4 Concluding remarks 149

5 Measurement strategies with alternative design objectives 153

5.1 Optimal sensor location for prediction 153

5.1.1 Problem formulation 153

5.1.2 Optimal-control formulation 155

5.1.3 Minimization algorithm 156

5.2 Sensor location for model discrimination 159

5.2.1 Competing models of a given distributed system 160

5.2.2 Theoretical problem setup 161

5.2.3 T₁₂-optimality conditions 164

5.2.4 Numerical construction of T₁₂-optimum designs 167

5.3 Conclusions 171

6 Robust designs for sensor location 173

6.1 Sequential designs 173

6.2 Optimal designs in the average sense 175

6.2.1 Problem statement 175

6.2.2 Stochastic-approximation algorithms 177

6.3 Optimal designs in the minimax sense 181

6.3.1 Problem statement and characterization 181

6.3.2 Numerical techniques for exact designs 182

6.4 Robust sensor location using randomized algorithms 187

6.4.1 A glance at complexity theory 188

6.4.2 NP-hard problems in control-system design 190

6.4.3 Weakened definitions of minima 191

6.4.4 Randomized algorithm for sensor placement 193

6.5 Concluding remarks 198

7 Towards even more challenging problems 201

7.1 Measurement strategies in the presence of correlated observations 201

7.1.1 Exchange algorithm for Ψ-optimum designs 203

7.2 Maximization of an observability measure 209

7.2.1 Observability in a quantitative sense 210

7.2.2 Scanning problem for optimal observability 211

7.2.3 Conversion to finding optimal sensor densities 212

7.3 Summary 216

8 Applications from engineering 217

8.1 Electrolytic reactor 217

8.1.1 Optimization of experimental effort 219

8.1.2 Clusterization-free designs 220

8.2 Calibration of smog-prediction models 221

8.3 Monitoring of groundwater resources quality 225

8.4 Diffusion process with correlated observational errors 230

8.5 Vibrating H-shaped membrane 232

9 Conclusions and future research directions 237

Appendices 245

A List of symbols 247

B Mathematical background 251

B.1 Matrix algebra 251

B.2 Symmetric, nonnegative definite and positive-definite matrices 255

B.3 Vector and matrix differentiation 260

B.4 Convex sets and convex functions 264

B.5 Convexity and differentiability of common optimality criteria 267

B.6 Differentiability of spectral functions 268

B.7 Monotonicity of common design criteria 271

B.8 Integration with respect to probability measures 272

B.9 Projection onto the canonical simplex 274

B.10 Conditional probability and conditional expectation 275

B.11 Some accessory inequalities from statistical learning theory 277

B.11.1 Hoeffding's inequality 277

B.11.2 Estimating the minima of functions 278

C Statistical properties of estimators 279

C.1 Best linear unbiased estimators in a stochastic-process setting 279

C.2 Best linear unbiased estimators in a partially uncorrelated framework 284

D Analysis of the largest eigenvalue 289

D.1 Directional differentiability 289

D.1.1 Case of the single largest eigenvalue 289

D.1.2 Case of the repeated largest eigenvalue 292

D.1.3 Smooth approximation to the largest eigenvalue 293

E Differentiation of nonlinear operators 297

E.1 Gateaux and Fréchet derivatives 297

E.2 Chain rule of differentiation 298

E.3 Partial derivatives 298

E.4 One-dimensional domains 299

E.5 Second derivatives 299

E.6 Functional on Hilbert spaces 300

E.7 Directional derivatives 301

E.8 Differentiability of max functions 301

F Accessory results for PDEs 303

F.1 Green formulae 303

F.2 Differentiability w.r.t. parameters 304

G Interpolation of tabulated sensitivity coefficients 313

G.1 Cubic spline interpolation for functions of one variable 313

G.2 Tricubic spline interpolation 314

H Differentials of Section 4.3.3 321

H.1 Derivation of formula (4.126) 321

H.2 Derivation of formula (4.129) 322

I Solving sensor-location problems using Maple & Matlab 323

I.1 Optimum experimental effort for a 1D problem 323

I.1.1 Forming the system of sensitivity equations 324

I.1.2 Solving the sensitivity equations 325

I.1.3 Determining the FIMs associated with individual spatial points 327

I.1.4 Optimizing the design weights 327

I.1.5 Plotting the results 328

I.2 Clusterization-free design for a 2D problem 329

I.2.1 Solving sensitivity equations using the PDE Toolbox 329

I.2.2 Determining potential contributions to the FIM from individual spatial points 335

I.2.3 Iterative optimization of sensor locations 336

References 339

Index 367