简介
This practical text demonstrates by example how to build Kalman filters and shows how the filtering equations can be applied to real-life problems. Computer code written in FORTRAN, MATLAB, and True BASIC accompanies all of the examples. Unlike other texts on Kalman filtering, the book does not devote any time to derivations; instead, Kalman filtering equations are explained in simple terms, and most of the text is dedicated to applying Kalman filtering to actual problems. All background and numerical techniques for understanding concepts is presented in the first chapter. Because real-life problems are seldom presented in the form of differential equations, a great deal of time is spent setting up a problem before the Kalman filter is actually formulated, to give the reader an intuitive feel for the problem being addressed. The authors illustrate several different filtering approaches for tackling a specific problem, so that readers can gain experience in software and performance tradeoffs for determining the best filtering approach. This third edition offers three new chapters on fixed- or finite-memory filters, use of the chain rule from calculus for filter initialization, and use of a bank of linear sine-wave Kalman filters for estimating the actual frequency of noisy sinusoidal measurements. An appendix serves as a central location for key concepts and formulas. Zarchan is affiliated with the technical staff at MIT Lincoln Laboratory. Musoff was affiliated with the Charles Stark Draper Laboratory Annotation c2009 Book News, Inc., Portland, OR (booknews.com)
目录
Preface p. xv
Introduction p. xvii
Acknowledgments p. xxv
Numerical Basics p. 1
Introduction p. 1
Simple Vector Operations p. 1
Simple Matrix Operations p. 3
Numerical Integration of Differential Equations p. 13
Noise and Random Variables p. 19
Gaussian Noise Example p. 23
Calculating Standard Deviation p. 26
White Noise p. 28
Simulating White Noise p. 30
State-Space Notation p. 33
Fundamental Matrix p. 34
Summary p. 38
References p. 38
Method of Least Squares p. 41
Introduction p. 41
Overview p. 41
Zeroth-Order or One-State Filter p. 42
First-Order or Two-State Filter p. 46
Second-Order or Three-State Least-Squares Filter p. 50
Third-Order System p. 56
Experiments with Zeroth-Order or One-State Filter p. 59
Experiments with First-Order or Two-State Filter p. 64
Experiments with Second-Order or Three-State Filter p. 71
Comparison of Filters p. 78
Accelerometer Testing Example p. 80
Summary p. 89
References p. 90
Recursive Least-Squares Filtering p. 91
Introduction p. 91
Making Zeroth-Order Least-Squares Filter Recursive p. 91
Properties of Zeroth-Order or One-State Filter p. 93
Properties of First-Order or Two-State Filter p. 103
Properties of Second-Order or Three-State Filter p. 112
Summary p. 124
References p. 128
Polynomial Kalman Filters p. 129
Introduction p. 129
General Equations p. 129
Derivation of Scalar Riccati Equations p. 131
Polynomial Kalman Filter (Zero Process Noise) p. 134
Comparing Zeroth-Order Recursive Least-Squares and Kalman Filters p. 136
Comparing First-Order Recursive Least-Squares and Kalman Filters p. 139
Comparing Second-Order Recursive Least-Squares and Kalman Filters p. 142
Comparing Different-Order Filters p. 148
Initial Covariance Matrix p. 151
Riccati Equations with Process Noise p. 155
Example of Kalman Filter Tracking a Falling Object p. 159
Revisiting Accelerometer Testing Example p. 171
Summary p. 179
References p. 182
Kalman Filters in a Nonpolynomial World p. 183
Introduction p. 183
Polynomial Kalman Filter and Sinusoidal Measurement p. 183
Sinusoidal Kalman Filter and Sinusoidal Measurement p. 194
Suspension System Example p. 203
Kalman Filter for Suspension System p. 207
Summary p. 218
References p. 218
Continuous Polynomial Kalman Filter p. 219
Introduction p. 219
Theoretical Equations p. 219
Zeroth-Order or One-State Continuous Polynomial Kalman Filter p. 221
First-Order or Two-State Continuous Polynomial Kalman Filter p. 227
Second-Order or Three-State Continuous Polynomial Kalman Filter p. 232
Transfer Function for Zeroth-Order Filter p. 238
Transfer Function for First-Order Filter p. 243
Transfer Function for Second-Order Filter p. 248
Filter Comparison p. 251
Summary p. 255
References p. 255
Extended Kalman Filtering p. 257
Introduction p. 257
Theoretical Equations p. 257
Drag Acting on Falling Object p. 259
First Attempt at Extended Kalman Filter p. 261
Second Attempt at Extended Kalman Filter p. 274
Third Attempt at Extended Kalman Filter p. 284
Summary p. 291
References p. 291
Drag and Falling Object p. 293
Introduction p. 293
Problem Setup p. 293
Changing Filter States p. 309
Why Process Noise Is Required p. 311
Linear Polynomial Kalman Filter p. 320
Summary p. 329
References p. 329
Cannon-Launched Projectile Tracking Problem p. 331
Introduction p. 331
Problem Statement p. 331
Extended Cartesian Kalman Filter p. 334
Polar Coordinate System p. 349
Extended Polar Kalman Filter p. 354
Using Linear Decoupled Polynomial Kalman Filters p. 367
Using Linear Coupled Polynomial Kalman Filters p. 376
Robustness Comparison of Extended and Linear Coupled Kalman Filters p. 385
Summary p. 393
Reference p. 394
Tracking a Sine Wave p. 395
Introduction p. 395
Extended Kalman Filter p. 395
Two-State Extended Kalman Filter with a Priori Information p. 408
Alternate Extended Kalman Filter for Sinusoidal Signal p. 417
Another Extended Kalman Filter for Sinusoidal Model p. 431
Summary p. 441
References p. 441
Satellite Navigation p. 443
Introduction p. 443
Problem with Perfect Range Measurements p. 443
Estimation Without Filtering p. 447
Linear Filtering of Range p. 453
Using Extended Kalman Filtering p. 455
Using Extended Kalman Filtering with One Satellite p. 465
Using Extended Kalman Filtering with Constant Velocity Receiver p. 474
Single Satellite with Constant Velocity Receiver p. 479
Using Extended Kalman Filtering with Variable Velocity Receiver p. 493
Variable Velocity Receiver and Single Satellite p. 505
Summary p. 513
References p. 513
Biases p. 515
Introduction p. 515
Influence of Bias p. 515
Estimating Satellite Bias with Known Receiver Location p. 519
Estimating Receiver Bias with Unknown Receiver Location and Two Satellites p. 525
Estimating Receiver Bias with Unknown Receiver Location and Three Satellites p. 533
Summary p. 544
Reference p. 547
Linearized Kalman Filtering p. 549
Introduction p. 549
Theoretical Equations p. 549
Falling Object Revisited p. 552
Developing a Linearized Kalman Filter p. 556
Cannon-Launched Projectile Revisited p. 569
Linearized Cartesian Kalman Filter p. 570
Summary p. 583
References p. 585
Miscellaneous Topics p. 587
Introduction p. 587
Sinusoidal Kalman Filter and Signal-to-Noise Ratio p. 587
When Only a Few Measurements Are Available p. 595
Detecting Filter Divergence in the Real World p. 606
Observability Example p. 618
Aiding p. 629
Summary p. 642
References p. 646
Fading-Memory Filter p. 647
Introduction p. 647
Fading-Memory-Filter Structure and Properties p. 647
Radar Tracking Problem p. 662
Summary p. 673
References p. 675
Assorted Techniques for Improving Kalman-Filter Performance p. 677
Introduction p. 677
Increasing Data Rate p. 677
Adding a Second Measurement p. 682
Batch Processing p. 690
Adaptive Filtering-Multiple Filters p. 701
Adaptive Filtering-Single Filter with Variable Process Noise p. 710
Summary p. 720
Fundamentals of Kalman-Filtering Software p. 723
Software Details p. 723
MATLAB p. 724
True BASIC p. 730
Reference p. 738
Key Formula and Concept Summary p. 739
Overview of Kalman-Filter Operation Principles p. 739
Kalman-Filter Gains and the Riccati Equations p. 739
Kalman-Filter Gain Logic p. 740
Matrix Inverse p. 740
Numerical Integration p. 741
Postprocessing Formulas p. 741
Simulating Pseudo White Noise p. 742
Fundamental Matrix p. 742
Method of Least-Squares Summary p. 742
Fading-Memory Filter Summary p. 745
Index p. 747
Supporting Materials p. 765
Introduction p. xvii
Acknowledgments p. xxv
Numerical Basics p. 1
Introduction p. 1
Simple Vector Operations p. 1
Simple Matrix Operations p. 3
Numerical Integration of Differential Equations p. 13
Noise and Random Variables p. 19
Gaussian Noise Example p. 23
Calculating Standard Deviation p. 26
White Noise p. 28
Simulating White Noise p. 30
State-Space Notation p. 33
Fundamental Matrix p. 34
Summary p. 38
References p. 38
Method of Least Squares p. 41
Introduction p. 41
Overview p. 41
Zeroth-Order or One-State Filter p. 42
First-Order or Two-State Filter p. 46
Second-Order or Three-State Least-Squares Filter p. 50
Third-Order System p. 56
Experiments with Zeroth-Order or One-State Filter p. 59
Experiments with First-Order or Two-State Filter p. 64
Experiments with Second-Order or Three-State Filter p. 71
Comparison of Filters p. 78
Accelerometer Testing Example p. 80
Summary p. 89
References p. 90
Recursive Least-Squares Filtering p. 91
Introduction p. 91
Making Zeroth-Order Least-Squares Filter Recursive p. 91
Properties of Zeroth-Order or One-State Filter p. 93
Properties of First-Order or Two-State Filter p. 103
Properties of Second-Order or Three-State Filter p. 112
Summary p. 124
References p. 128
Polynomial Kalman Filters p. 129
Introduction p. 129
General Equations p. 129
Derivation of Scalar Riccati Equations p. 131
Polynomial Kalman Filter (Zero Process Noise) p. 134
Comparing Zeroth-Order Recursive Least-Squares and Kalman Filters p. 136
Comparing First-Order Recursive Least-Squares and Kalman Filters p. 139
Comparing Second-Order Recursive Least-Squares and Kalman Filters p. 142
Comparing Different-Order Filters p. 148
Initial Covariance Matrix p. 151
Riccati Equations with Process Noise p. 155
Example of Kalman Filter Tracking a Falling Object p. 159
Revisiting Accelerometer Testing Example p. 171
Summary p. 179
References p. 182
Kalman Filters in a Nonpolynomial World p. 183
Introduction p. 183
Polynomial Kalman Filter and Sinusoidal Measurement p. 183
Sinusoidal Kalman Filter and Sinusoidal Measurement p. 194
Suspension System Example p. 203
Kalman Filter for Suspension System p. 207
Summary p. 218
References p. 218
Continuous Polynomial Kalman Filter p. 219
Introduction p. 219
Theoretical Equations p. 219
Zeroth-Order or One-State Continuous Polynomial Kalman Filter p. 221
First-Order or Two-State Continuous Polynomial Kalman Filter p. 227
Second-Order or Three-State Continuous Polynomial Kalman Filter p. 232
Transfer Function for Zeroth-Order Filter p. 238
Transfer Function for First-Order Filter p. 243
Transfer Function for Second-Order Filter p. 248
Filter Comparison p. 251
Summary p. 255
References p. 255
Extended Kalman Filtering p. 257
Introduction p. 257
Theoretical Equations p. 257
Drag Acting on Falling Object p. 259
First Attempt at Extended Kalman Filter p. 261
Second Attempt at Extended Kalman Filter p. 274
Third Attempt at Extended Kalman Filter p. 284
Summary p. 291
References p. 291
Drag and Falling Object p. 293
Introduction p. 293
Problem Setup p. 293
Changing Filter States p. 309
Why Process Noise Is Required p. 311
Linear Polynomial Kalman Filter p. 320
Summary p. 329
References p. 329
Cannon-Launched Projectile Tracking Problem p. 331
Introduction p. 331
Problem Statement p. 331
Extended Cartesian Kalman Filter p. 334
Polar Coordinate System p. 349
Extended Polar Kalman Filter p. 354
Using Linear Decoupled Polynomial Kalman Filters p. 367
Using Linear Coupled Polynomial Kalman Filters p. 376
Robustness Comparison of Extended and Linear Coupled Kalman Filters p. 385
Summary p. 393
Reference p. 394
Tracking a Sine Wave p. 395
Introduction p. 395
Extended Kalman Filter p. 395
Two-State Extended Kalman Filter with a Priori Information p. 408
Alternate Extended Kalman Filter for Sinusoidal Signal p. 417
Another Extended Kalman Filter for Sinusoidal Model p. 431
Summary p. 441
References p. 441
Satellite Navigation p. 443
Introduction p. 443
Problem with Perfect Range Measurements p. 443
Estimation Without Filtering p. 447
Linear Filtering of Range p. 453
Using Extended Kalman Filtering p. 455
Using Extended Kalman Filtering with One Satellite p. 465
Using Extended Kalman Filtering with Constant Velocity Receiver p. 474
Single Satellite with Constant Velocity Receiver p. 479
Using Extended Kalman Filtering with Variable Velocity Receiver p. 493
Variable Velocity Receiver and Single Satellite p. 505
Summary p. 513
References p. 513
Biases p. 515
Introduction p. 515
Influence of Bias p. 515
Estimating Satellite Bias with Known Receiver Location p. 519
Estimating Receiver Bias with Unknown Receiver Location and Two Satellites p. 525
Estimating Receiver Bias with Unknown Receiver Location and Three Satellites p. 533
Summary p. 544
Reference p. 547
Linearized Kalman Filtering p. 549
Introduction p. 549
Theoretical Equations p. 549
Falling Object Revisited p. 552
Developing a Linearized Kalman Filter p. 556
Cannon-Launched Projectile Revisited p. 569
Linearized Cartesian Kalman Filter p. 570
Summary p. 583
References p. 585
Miscellaneous Topics p. 587
Introduction p. 587
Sinusoidal Kalman Filter and Signal-to-Noise Ratio p. 587
When Only a Few Measurements Are Available p. 595
Detecting Filter Divergence in the Real World p. 606
Observability Example p. 618
Aiding p. 629
Summary p. 642
References p. 646
Fading-Memory Filter p. 647
Introduction p. 647
Fading-Memory-Filter Structure and Properties p. 647
Radar Tracking Problem p. 662
Summary p. 673
References p. 675
Assorted Techniques for Improving Kalman-Filter Performance p. 677
Introduction p. 677
Increasing Data Rate p. 677
Adding a Second Measurement p. 682
Batch Processing p. 690
Adaptive Filtering-Multiple Filters p. 701
Adaptive Filtering-Single Filter with Variable Process Noise p. 710
Summary p. 720
Fundamentals of Kalman-Filtering Software p. 723
Software Details p. 723
MATLAB p. 724
True BASIC p. 730
Reference p. 738
Key Formula and Concept Summary p. 739
Overview of Kalman-Filter Operation Principles p. 739
Kalman-Filter Gains and the Riccati Equations p. 739
Kalman-Filter Gain Logic p. 740
Matrix Inverse p. 740
Numerical Integration p. 741
Postprocessing Formulas p. 741
Simulating Pseudo White Noise p. 742
Fundamental Matrix p. 742
Method of Least-Squares Summary p. 742
Fading-Memory Filter Summary p. 745
Index p. 747
Supporting Materials p. 765
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