《Iterative Learning Control: An Optimization Paradigm》
《Iterative Learning Control: An Optimization Paradigm》迭代学习控制：一种优化范例
作者：
David H. Owens
University of Sheffield
出版社：Springer
出版时间：2016年
目录
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Control Systems, Models and Algorithms . . . . . . . . . . . . . . . 2
1.2 Repetition and Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Periodic Demand Signals . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Repetitive Control and Multipass Systems . . . . . . . . . 4
1.2.3 Iterative Control Examples . . . . . . . . . . . . . . . . . . . . 6
1.3 Dynamical Properties of Iteration: A Review of Ideas . . . . . . . 9
1.4 So What Do We Need? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4.1 An Overview of Mathematical Techniques . . . . . . . . . 13
1.4.2 The Conceptual Basis for Algorithms . . . . . . . . . . . . 15
1.5 Discussion and Further Background Reading . . . . . . . . . . . . . 16
2 Mathematical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1 Elements of Matrix Theory . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Quadratic Optimization and Quadratic Forms . . . . . . . . . . . . . 27
2.2.1 Completing the Square . . . . . . . . . . . . . . . . . . . . . . 27
2.2.2 Singular Values, Lagrangians
and Matrix Norms. . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3 Banach Spaces, Operators, Norms and Convergent
Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3.1 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3.2 Normed Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3.3 Convergence, Closure, Completeness
and Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3.4 Linear Operators and Dense Subsets . . . . . . . . . . . . . 34
2.4 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4.1 Inner Products and Norms . . . . . . . . . . . . . . . . . . . . 37
2.4.2 Norm and Weak Convergence . . . . . . . . . . . . . . . . . 39
2.4.3 Adjoint and Self-adjoint Operators
in Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.5 Real Hilbert Spaces, Convex Sets and Projections. . . . . . . . . . 46
2.6 Optimal Control Problems in Hilbert Space . . . . . . . . . . . . . . 48
2.6.1 Proof by Completing the Square . . . . . . . . . . . . . . . . 50
2.6.2 Proof Using the Projection Theorem . . . . . . . . . . . . . 51
2.6.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.7 Further Discussion and Bibliography . . . . . . . . . . . . . . . . . . . 53
3 State Space Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.1 Models of Continuous State Space Systems . . . . . . . . . . . . . . 57
3.1.1 Solution of the State Equations. . . . . . . . . . . . . . . . . 58
3.1.2 The Convolution Operator and the Impulse
Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.1.3 The System as an Operator Between
Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2 Laplace Transforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3 Transfer Function Matrices, Poles, Zeros
and Relative Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.4 The System Frequency Response . . . . . . . . . . . . . . . . . . . . . 63
3.5 Discrete Time, Sampled Data State Space Models. . . . . . . . . . 64
3.5.1 State Space Models as Difference Equations. . . . . . . . 64
3.5.2 Solution of Linear, Discrete Time State Equations. . . . 65
3.5.3 The Discrete Convolution Operator
and the Discrete Impulse Response Sequence . . . . . . . 66
3.6 Z-Transforms and the Discrete Transfer Function Matrix . . . . 67
3.6.1 Discrete Transfer Function Matrices, Poles,
Zeros and the Relative Degree . . . . . . . . . . . . . . . . . 68
3.6.2 The Discrete System Frequency Response . . . . . . . . . 69
3.7 Multi-rate Discrete Time Systems . . . . . . . . . . . . . . . . . . . . . 70
3.8 Controllability, Observability, Minimal Realizations
and Pole Allocation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.9 Inverse Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.9.1 The Case of m ¼ ‘, Zeros and ν . . . . . . . . . . . . . . . 72
3.9.2 Left and Right Inverses When m 6¼ ‘. . . . . . . . . . . . . 74
3.10 Quadratic Optimal Control of Linear Continuous Systems . . . . 76
3.10.1 The Relevant Operators and Spaces. . . . . . . . . . . . . . 76
3.10.2 Computation of the Adjoint Operator . . . . . . . . . . . . 78
3.10.3 The Two Point Boundary Value Problem. . . . . . . . . . 81
3.10.4 The Riccati Equation and a State Feedback
Plus Feedforward Representation . . . . . . . . . . . . . . . 82
3.10.5 An Alternative Riccati Representation . . . . . . . . . . . . 84
3.11 Further Reading and Bibliography. . . . . . . . . . . . . . . . . . . . . 85
4 Matrix Models, Supervectors and Discrete Systems . . . . . . . . . . . 87
4.1 Supervectors and the Matrix Model. . . . . . . . . . . . . . . . . . . . 87
4.2 The Algebra of Series and Parallel Connections . . . . . . . . . . . 88
4.3 The Transpose System and Time Reversal . . . . . . . . . . . . . . . 89
4.4 Invertibility, Range and Relative Degrees. . . . . . . . . . . . . . . . 90
4.4.1 The Relative Degree and the Kernel
and Range of G . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.4.2 The Range of G and Decoupling Theory . . . . . . . . . . 93
4.5 The Range and Kernel and the Use of the Inverse System . . . . 96
4.5.1 A Partition of the Inverse. . . . . . . . . . . . . . . . . . . . . 96
4.5.2 Ensuring Stability of P1ðzÞ. . . . . . . . . . . . . . . . . . . 98
4.6 The Range, Kernel and the C Canonical Form . . . . . . . . . . . 99
4.6.1 Factorization Using State Feedback
and Output Injection . . . . . . . . . . . . . . . . . . . . . . . . 99
4.6.2 The C Canonical Form. . . . . . . . . . . . . . . . . . . . . . 100
4.6.3 The Special Case of Uniform Rank Systems . . . . . . . 102
4.7 Quadratic Optimal Control of Linear Discrete Systems . . . . . . 104
4.7.1 The Adjoint and the Discrete Two Point
Boundary Value Problem. . . . . . . . . . . . . . . . . . . . . 105
4.7.2 A State Feedback/Feedforward Solution. . . . . . . . . . . 106
4.8 Frequency Domain Relationships . . . . . . . . . . . . . . . . . . . . . 107
4.8.1 Bounding Norms on Finite Intervals . . . . . . . . . . . . . 108
4.8.2 Computing the Norm Using the Frequency
Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.8.3 Quadratic Forms and Positive Real Transfer
Function Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.8.4 Frequency Dependent Lower Bounds . . . . . . . . . . . . 112
4.9 Discussion and Further Reading . . . . . . . . . . . . . . . . . . . . . . 116
5 Iterative Learning Control: A Formulation . . . . . . . . . . . . . . . . . 119
5.1 Abstract Formulation of a Design Problem. . . . . . . . . . . . . . . 119
5.1.1 The Design Problem . . . . . . . . . . . . . . . . . . . . . . . . 120
5.1.2 Input and Error Update Equations:
The Linear Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.1.3 Robustness and Uncertainty Models . . . . . . . . . . . . . 124
5.2 General Conditions for Convergence of Linear Iterations . . . . . 128
5.2.1 Spectral Radius and Norm Conditions . . . . . . . . . . . . 129
5.2.2 Infinite Dimensions with rðLÞ ¼ kLk ¼ 1
and L ¼ L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.2.3 Relaxation, Convergence and Robustness. . . . . . . . . . 134
5.2.4 Eigenstructure Interpretation . . . . . . . . . . . . . . . . . . . 138
5.2.5 Formal Computation of the Eigenvalues
and Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.3 Robustness, Positivity and Inverse Systems . . . . . . . . . . . . . . 141
5.4 Discussion and Further Reading . . . . . . . . . . . . . . . . . . . . . . 143
6 Control Using Inverse Model Algorithms . . . . . . . . . . . . . . . . . . . 145
6.1 Inverse Model Control: A Benchmark Algorithm . . . . . . . . . . 145
6.1.1 Use of a Right Inverse of the Plant . . . . . . . . . . . . . . 145
6.1.2 Use of a Left Inverse of the Plant . . . . . . . . . . . . . . . 147
6.1.3 Why the Inverse Model Is Important . . . . . . . . . . . . . 149
6.1.4 Inverse Model Algorithms for State
Space Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6.1.5 Robustness Tests and Multiplicative
Error Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.2 Frequency Domain Robustness Criteria . . . . . . . . . . . . . . . . . 156
6.2.1 Discrete System Robust Monotonicity Tests . . . . . . . . 156
6.2.2 Improving Robustness Using Relaxation . . . . . . . . . . 158
6.2.3 Discrete Systems: Robustness
and Non-monotonic Convergence . . . . . . . . . . . . . . . 159
6.3 Discussion and Further Reading . . . . . . . . . . . . . . . . . . . . . . 161
7 Monotonicity and Gradient Algorithms . . . . . . . . . . . . . . . . . . . . 165
7.1 Steepest Descent: Achieving Minimum Energy Solutions . . . . . 166
7.2 Application to Discrete Time State Space Systems . . . . . . . . . 168
7.2.1 Algorithm Construction . . . . . . . . . . . . . . . . . . . . . . 169
7.2.2 Eigenstructure Interpretation: Convergence
in Finite Iterations. . . . . . . . . . . . . . . . . . . . . . . . . . 171
7.2.3 Frequency Domain Attenuation. . . . . . . . . . . . . . . . . 174
7.3 Steepest Descent for Continuous Time State
Space Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
7.4 Monotonic Evolution Using General Gradients . . . . . . . . . . . . 180
7.5 Discrete State Space Models Revisited . . . . . . . . . . . . . . . . . 183
7.5.1 Gradients Using the Adjoint of a State
Space System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
7.5.2 Why the Case of m ¼ ‘ May Be Important
in Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
7.5.3 Robustness Tests in the Frequency Domain . . . . . . . . 194
7.5.4 Robustness and Relaxation. . . . . . . . . . . . . . . . . . . . 197
7.5.5 Non-monotonic Gradient-Based Control
and ε-Weighted Norms . . . . . . . . . . . . . . . . . . . . . . 198
7.5.6 A Steepest Descent Algorithm Using ε-Norms . . . . . . 203
7.6 Discussion, Comments and Further Generalizations . . . . . . . . . 203
7.6.1 Bringing the Ideas Together? . . . . . . . . . . . . . . . . . . 204
7.6.2 Factors Influencing Achievable Performance . . . . . . . 206
7.6.3 Notes on Continuous State Space Systems . . . . . . . . . 207
8 Combined Inverse and Gradient Based Design. . . . . . . . . . . . . . . 209
8.1 Inverse Algorithms: Robustness
and Bi-directional Filtering. . . . . . . . . . . . . . . . . . . . . . . . . . 209
8.2 General Issues in Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
8.2.1 Pre-conditioning Control Loops . . . . . . . . . . . . . . . . 214
8.2.2 Compensator Structures . . . . . . . . . . . . . . . . . . . . . . 216
8.2.3 Stable Inversion Algorithms . . . . . . . . . . . . . . . . . . . 218
8.2.4 All-Pass Networks and Non-minimum-phase
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
8.3 Gradients, Compensation and Feedback Design Methods . . . . . 226
8.3.1 Feedback Design: The Discrete Time Case. . . . . . . . . 227
8.3.2 Feedback Design: The Continuous Time Case . . . . . . 229
8.4 Discussion and Further Reading . . . . . . . . . . . . . . . . . . . . . . 229
9 Norm Optimal Iterative Learning Control . . . . . . . . . . . . . . . . . . 233
9.1 Problem Formulation and Formal Algorithm . . . . . . . . . . . . . 234
9.1.1 The Choice of Objective Function. . . . . . . . . . . . . . . 234
9.1.2 Relaxed Versions of NOILC . . . . . . . . . . . . . . . . . . 236
9.1.3 NOILC for Discrete-Time State Space Systems . . . . . 238
9.1.4 Relaxed NOILC for Discrete-Time
State Space Systems . . . . . . . . . . . . . . . . . . . . . . . . 240
9.1.5 A Note on Frequency Attenuation:
The Discrete Time Case . . . . . . . . . . . . . . . . . . . . . 241
9.1.6 NOILC: The Case of Continuous-Time State
Space Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
9.1.7 Convergence, Eigenstructure, ε2
and Spectral Bandwidth . . . . . . . . . . . . . . . . . . . . . . 244
9.1.8 Convergence: General Properties
of NOILC Algorithms . . . . . . . . . . . . . . . . . . . . . . . 248
9.2 Robustness of NOILC: Feedforward Implementation . . . . . . . . 252
9.2.1 Computational Aspects of Feedforward NOILC . . . . . 253
9.2.2 The Case of Right Multiplicative
Modelling Errors . . . . . . . . . . . . . . . . . . . . . . . . . . 254
9.2.3 Discrete State Space Systems with Right
Multiplicative Errors . . . . . . . . . . . . . . . . . . . . . . . . 259
9.2.4 The Case of Left Multiplicative Modelling Errors . . . . 262
9.2.5 Discrete Systems with Left Multiplicative
Modelling Errors . . . . . . . . . . . . . . . . . . . . . . . . . . 267
9.2.6 Monotonicity in Y with Respect to the Norm k kY . . . 268
9.3 Non-minimum-phase Properties and Flat-Lining . . . . . . . . . . . 269
9.4 Discussion and Further Reading . . . . . . . . . . . . . . . . . . . . . . 272
9.4.1 Background Comments . . . . . . . . . . . . . . . . . . . . . . 272
9.4.2 Practical Observations . . . . . . . . . . . . . . . . . . . . . . . 273
9.4.3 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
9.4.4 Robustness and the Inverse Algorithm . . . . . . . . . . . . 274
9.4.5 Alternatives? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
9.4.6 Q, R and Dyadic Expansions . . . . . . . . . . . . . . . . . . 276
10 NOILC: Natural Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
10.1 Filtering Using Input and Error Weighting . . . . . . . . . . . . . . . 277
10.2 Multi-rate Sampled Discrete Time Systems . . . . . . . . . . . . . . 279
10.3 Initial Conditions as Control Signals . . . . . . . . . . . . . . . . . . . 280
10.4 Problems with Several Objectives . . . . . . . . . . . . . . . . . . . . . 284
10.5 Intermediate Point Problems. . . . . . . . . . . . . . . . . . . . . . . . . 286
10.5.1 Continuous Time Systems:
An Intermediate Point Problem. . . . . . . . . . . . . . . . . 286
10.5.2 Discrete Time Systems: An Intermediate
Point Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
10.5.3 IPNOILC: Additional Issues and Robustness . . . . . . . 290
10.6 Multi-task NOILC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
10.6.1 Continuous State Space Systems. . . . . . . . . . . . . . . . 294
10.6.2 Adding Initial Conditions as Controls . . . . . . . . . . . . 299
10.6.3 Discrete State Space Systems . . . . . . . . . . . . . . . . . . 300
10.7 Multi-models and Predictive NOILC . . . . . . . . . . . . . . . . . . . 301
10.7.1 Predictive NOILC—General Theory
and a Link to Inversion . . . . . . . . . . . . . . . . . . . . . . 301
10.7.2 A Multi-model Representation . . . . . . . . . . . . . . . . . 304
10.7.3 The Case of Linear, State Space Models . . . . . . . . . . 305
10.7.4 Convergence and Other Algorithm Properties . . . . . . . 308
10.7.5 The Special Cases of M ¼ 2 and M ¼1 . . . . . . . . . 313
10.7.6 A Note on Robustness of Feedforward
Predictive NOILC. . . . . . . . . . . . . . . . . . . . . . . . . . 315
10.8 Discussion and Further Reading . . . . . . . . . . . . . . . . . . . . . . 319
11 Iteration and Auxiliary Optimization. . . . . . . . . . . . . . . . . . . . . . 323
11.1 Models with Auxiliary Variables and Problem
Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
11.2 A Right Inverse Model Solution . . . . . . . . . . . . . . . . . . . . . . 325
11.3 Solutions Using Switching Algorithms. . . . . . . . . . . . . . . . . . 327
11.3.1 Switching Algorithm Construction . . . . . . . . . . . . . . 327
11.3.2 Properties of the Switching Algorithm . . . . . . . . . . . . 328
11.3.3 Characterization of Convergence Rates . . . . . . . . . . . 331
11.3.4 Decoupling Minimum Energy Representations
from NOILC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
11.3.5 Intermediate Point Tracking
and the Choice G1 ¼ G . . . . . . . . . . . . . . . . . . . . . 334
11.3.6 Restructuring the NOILC Spectrum
by Choosing G1 ¼ Ge . . . . . . . . . . . . . . . . . . . . . . 335
11.4 A Note on Robustness of Switching Algorithms . . . . . . . . . . . 338
11.5 The Switching Algorithm When GeGe Is Invertible . . . . . . . . . 341
11.6 Discussion and Further Reading . . . . . . . . . . . . . . . . . . . . . . 344
12 Iteration as Successive Projection . . . . . . . . . . . . . . . . . . . . . . . . 347
12.1 Convergence Versus Proximity . . . . . . . . . . . . . . . . . . . . . . . 347
12.2 Successive Projection and Proximity Algorithms . . . . . . . . . . . 349
12.3 Iterative Control with Constraints . . . . . . . . . . . . . . . . . . . . . 354
12.3.1 NOILC with Input Constraints . . . . . . . . . . . . . . . . . 355
12.3.2 General Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 358
12.3.3 Intermediate Point Control with Input
and Output Constraints . . . . . . . . . . . . . . . . . . . . . . 362
12.3.4 Iterative Control to Satisfy Auxiliary
Variable Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
12.3.5 An Overview and Summary . . . . . . . . . . . . . . . . . . . 366
12.4 “Iteration Management” by Operator Intervention . . . . . . . . . . 367
12.5 What Happens If S1 and S2 Do Not Intersect? . . . . . . . . . . . . 370
12.6 Discussion and Further Reading . . . . . . . . . . . . . . . . . . . . . . 373
13 Acceleration and Successive Projection . . . . . . . . . . . . . . . . . . . . 377
13.1 Replacing Plant Iterations by Off-Line Iterations . . . . . . . . . . . 378
13.2 Accelerating Algorithms Using Extrapolation . . . . . . . . . . . . . 378
13.2.1 Successive Projection and Extrapolation
Algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379
13.2.2 NOILC: Acceleration Using Extrapolation . . . . . . . . . 381
13.3 A Notch Algorithm Using Parameterized Sets . . . . . . . . . . . . 383
13.3.1 Creating a Spectral Notch: Computation
and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
13.3.2 The Notch Algorithm and Iterative Control
Using Successive Projection . . . . . . . . . . . . . . . . . . . 389
13.3.3 A Notch Algorithm for Discrete State
Space Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
13.3.4 Robustness of the Notch Algorithm
in Feedforward Form. . . . . . . . . . . . . . . . . . . . . . . . 396
13.4 Discussion and Further Reading . . . . . . . . . . . . . . . . . . . . . . 401
14 Parameter Optimal Iterative Control . . . . . . . . . . . . . . . . . . . . . . 403
14.1 Parameterizations and Norm Optimal Iteration . . . . . . . . . . . . 403
14.2 Parameter Optimal Control: The Single Parameter Case . . . . . . 408
14.2.1 Alternative Objective Functions . . . . . . . . . . . . . . . . 408
14.2.2 Problem Definition and Convergence
Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . 410
14.2.3 Convergence Properties: Dependence
on Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413
14.2.4 Choosing the Compensator. . . . . . . . . . . . . . . . . . . . 415
14.2.5 Computing tr½Γ0Γ0: Discrete State
Space Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
14.2.6 Choosing Parameters in JðβÞ . . . . . . . . . . . . . . . . . . 418
14.2.7 Iteration Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 420
14.2.8 Plateauing/Flatlining Phenomena. . . . . . . . . . . . . . . . 420
14.2.9 Switching Algorithms . . . . . . . . . . . . . . . . . . . . . . . 425
14.3 Robustness of POILC: The Single Parameter Case . . . . . . . . . 429
14.3.1 Robustness Using the Right Inverse . . . . . . . . . . . . . 429
14.3.2 Robustness: A More General Case . . . . . . . . . . . . . . 431
14.4 Multi-Parameter Learning Control . . . . . . . . . . . . . . . . . . . . . 433
14.4.1 The Form of the Parameterization . . . . . . . . . . . . . . . 433
14.4.2 Alternative Forms for ΩΓ
and the Objective Function . . . . . . . . . . . . . . . . . . . 434
14.4.3 The Multi-parameter POILC Algorithm . . . . . . . . . . . 437
14.4.4 Choice of Multi-parameter Parameterization . . . . . . . . 439
14.5 Discussion and Further Reading . . . . . . . . . . . . . . . . . . . . . . 441
14.5.1 Chapter Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 441
14.5.2 High Order POILC: A Brief Summary . . . . . . . . . . . 443
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451
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