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Filtering and System Identification

Book Description

Filtering and system identification are powerful techniques for building models of complex systems. This 2007 book discusses the design of reliable numerical methods to retrieve missing information in models derived using these techniques. Emphasis is on the least squares approach as applied to the linear state-space model, and problems of increasing complexity are analyzed and solved within this framework, starting with the Kalman filter and concluding with the estimation of a full model, noise statistics and state estimator directly from the data. Key background topics, including linear matrix algebra and linear system theory, are covered, followed by different estimation and identification methods in the state-space model. With end-of-chapter exercises, MATLAB simulations and numerous illustrations, this book will appeal to graduate students and researchers in electrical, mechanical and aerospace engineering. It is also useful for practitioners. Additional resources for this title, including solutions for instructors, are available online at www.cambridge.org/9780521875127.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Contents
  6. Preface
  7. Notation and symbols
  8. List of abbreviations
  9. 1. Introduction
  10. 2. Linear algebra
    1. 2.1 Introduction
    2. 2.2 Vectors
    3. 2.3 Matrices
    4. 2.4 Square matrices
    5. 2.5 Matrix decompositions
    6. 2.6 Linear least-squares problems
      1. 2.6.1 Solution if the matrix F has full column rank
      2. 2.6.2 Solutions if the matrix F does not have full column rank
    7. 2.7 Weighted linear least-squares problems
    8. 2.8 Summary
  11. 3. Discrete-time signals and systems
    1. 3.1 Introduction
    2. 3.2 Signals
    3. 3.3 Signal transforms
      1. 3.3.1 The z-transform
      2. 3.3.2 The discrete-time Fourier transform
    4. 3.4 Linear systems
      1. 3.4.1 Linearization
      2. 3.4.2 System response and stability
      3. 3.4.3 Controllability and observability
      4. 3.4.4 Input–output descriptions
    5. 3.5 Interaction between systems
    6. 3.6 Summary
  12. 4. Random variables and signals
    1. 4.1 Introduction
    2. 4.2 Description of a random variable
      1. 4.2.1 Experiments and events
      2. 4.2.2 The probability model
      3. 4.2.3 Linear functions of a random variable
      4. 4.2.4 The expected value of a random variable
      5. 4.2.5 Gaussian random variables
      6. 4.2.6 Multiple random variables
    3. 4.3 Random signals
      1. 4.3.1 Expectations of random signals
      2. 4.3.2 Important classes of random signals
      3. 4.3.3 Stationary random signals
      4. 4.3.4 Ergodicity and time averages of random signals
    4. 4.4 Power spectra
    5. 4.5 Properties of least-squares estimates
      1. 4.5.1 The linear least-squares problem
      2. 4.5.2 The weighted linear least-squares problem
      3. 4.5.3 The stochastic linear least-squares problem
      4. 4.5.4 A square-root solution to the stochastic linear least-squares problem
      5. 4.5.5 Maximum-likelihood interpretation of the weighted linear least-squares problem
    6. 4.6 Summary
  13. 5. Kalman filtering
    1. 5.1 Introduction
    2. 5.2 The asymptotic observer
    3. 5.3 The Kalman-filter problem
    4. 5.4 The Kalman filter and stochastic least squares
    5. 5.5 The Kalman filter and weighted least squares
      1. 5.5.1 A weighted least-squares problem formulation
      2. 5.5.2 The measurement update
      3. 5.5.3 The time update
      4. 5.5.4 The combined measurement–time update
      5. 5.5.5 The innovation form representation
    6. 5.6 Fixed-interval smoothing
    7. 5.7 The Kalman filter for LTI systems
    8. 5.8 The Kalman filter for estimating unknown inputs
    9. 5.9 Summary
  14. 6. Estimation of spectra and frequency-response functions
    1. 6.1 Introduction
    2. 6.2 The discrete Fourier transform
    3. 6.3 Spectral leakage
    4. 6.4 The FFT algorithm
    5. 6.5 Estimation of signal spectra
    6. 6.6 Estimation of FRFs and disturbance spectra
      1. 6.6.1 Periodic input sequences
      2. 6.6.2 General input sequences
      3. 6.6.3 Estimating the disturbance spectrum
    7. 6.7 Summary
  15. 7. Output-error parametric model estimation
    1. 7.1 Introduction
    2. 7.2 Problems in estimating parameters of an LTI state-space model
    3. 7.3 Parameterizing a MIMO LTI state-space model
      1. 7.3.1 The output normal form
      2. 7.3.2 The tridiagonal form
    4. 7.4 The output-error cost function
    5. 7.5 Numerical parameter estimation
      1. 7.5.1 The Gauss–Newton method
      2. 7.5.2 Regularization in the Gauss–Newton method
      3. 7.5.3 The steepest descent method
      4. 7.5.4 Gradient projection
    6. 7.6 Analyzing the accuracy of the estimates
    7. 7.7 Dealing with colored measurement noise
      1. 7.7.1 Weighted least squares
      2. 7.7.2 Prediction-error methods
    8. 7.8 Summary
  16. 8. Prediction-error parametric model estimation
    1. 8.1 Introduction
    2. 8.2 Prediction-error methods for estimating state-space models
      1. 8.2.1 Parameterizing an innovation state-space model
      2. 8.2.2 The prediction-error cost function
      3. 8.2.3 Numerical parameter estimation
      4. 8.2.4 Analyzing the accuracy of the estimates
    3. 8.3 Specific model parameterizations for SISO systems
      1. 8.3.1 The ARMAX and ARX model structures
      2. 8.3.2 The Box–Jenkins and output-error model structures
    4. 8.4 Qualitative analysis of the model bias for SISO systems
    5. 8.5 Estimation problems in closed-loop systems
    6. 8.6 Summary
  17. 9. Subspace model identification
    1. 9.1 Introduction
    2. 9.2 Subspace model identification for deterministic systems
      1. 9.2.1 The data equation
      2. 9.2.2 Identification for autonomous systems
      3. 9.2.3 Identification using impulse input sequences
      4. 9.2.4 Identification using general input sequences
    3. 9.3 Subspace identification with white measurement noise
    4. 9.4 The use of instrumental variables
    5. 9.5 Subspace identification with colored measurement noise
    6. 9.6 Subspace identification with process and measurement noise
      1. 9.6.1 The PO-MOESP method
      2. 9.6.2 Subspace identification as a least-squares problem
      3. 9.6.3 Estimating the Kalman gain K[sub(T)]
      4. 9.6.4 Relations among different subspace identification methods
    7. 9.7 Using subspace identification with closed-loop data
    8. 9.8 Summary
  18. 10. The system-identification cycle
    1. 10.1 Introduction
    2. 10.2 Experiment design
      1. 10.2.1 Choice of sampling frequency
      2. 10.2.2 Transient-response analysis
      3. 10.2.3 Experiment duration
      4. 10.2.4 Persistency of excitation of the input sequence
      5. 10.2.5 Types of input sequence
    3. 10.3 Data pre-processing
      1. 10.3.1 Decimation
      2. 10.3.2 Detrending the data
      3. 10.3.3 Pre-filtering the data
      4. 10.3.4 Concatenating data sequences
    4. 10.4 Selection of the model structure
      1. 10.4.1 Delay estimation
      2. 10.4.2 Model-structure selection in ARMAX model estimation
      3. 10.4.3 Model-structure selection in subspace identification
    5. 10.5 Model validation
      1. 10.5.1 The auto-correlation test
      2. 10.5.2 The cross-correlation test
      3. 10.5.3 The cross-validation test
    6. 10.6 Summary
  19. References
  20. Index