Text: Brown and Hwang, "Introduction to Random Signals and  Applied Kalman Filtering" , Wiley  1992.

Course purpose:  To provide the student with a working, practical knowledge of modern techniques in stochastic processes and optimal estimation based on Kalman Filtering.

Method of Instruction: Overhead Projections with Notes,Numerical Software Provided

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Course summary:

   First a bridge between conventional linear systems, random process theory (Laplace transforms, autocorrelation function/ spectral density) and state space descriptions of linear systems and stochastic systems  for both continuous and discrete time formulations, is presented. Numerical methods are included. Then the discrete Kalman Filter is presented along with examples, modeling techniques, methods for controlling numerical error, real time error control, and suboptimal filter  evaluation techniques.

  This is followed by development of the continuous Kalman filter and its applications. Additional topics include optimal smoothing techniques and their application , the extended Kalman filter, and presentation of case studies in communication, control and navigation systems. MATLAB examples worked out, problems  given.

 Direct applications

  Optimum navigation systems                 Parameter estimation
  Optimal filters for control of noise       Fault Detection
  Estimation of complex systems             Evaluation of noisy test data

Ancillary benefits:

· Comfort with both stationary and non stationary random processes.
· Evaluation techniques for handling all sorts of stochastic processes affecting complex systems.
· State space representation of (stochastic) systems.
· Comfort of with systems from either "conventional" approach or from state space approach
· Ability to work with stochastic processes in either discrete time or continuous time.
  As ACF, power spectral density; or state space, covariance.
 
         Course requirements:
       Mid Term                   8 Homework Sets
       Final  Project              Computer assignments (Counts for 1 Exam)
 

 WEEK         TOPICS COVERED                             TEXT
                                                                      ASSIGNMENTS
1.    Review of key elements of probability theory:   1.7 1.16
      Fundamental Distributions
      Implications of the Central Limit theorem
      Correlation, Covariance  and  Orthogonality
      Multi variable Gaussian Distribution
      Linear transformations and the Gaussian random  variable
      Definitions of a Random Process                          2.1 2.4
 
2.  Random Processes Continued                            2.5 2.14
    Ensemble and time averages
    Autocorrelation, Cross Correlation , Power spectral
    Density, White noise
    Examples

3.  Random Processes though Linear Systems:       3.1 3.10

    The shaping filter, and use of White Noise          4.1  4.3 (p180)
    Conventions
    Equivalent White Noise
    Discrete Random Processes
    Derivation of the  (non causal) Weiner Filter
 
4. State Space Description of Random Processes:      5.2 5.3
       Factorization of Spectral Density                              Notes
       Role of the Shaping Filter in State Space Modeling
       Non Stationary and Deterministic Processes.
       Other Models
       Some Reverse Examples: State Space to  S Domain
       Covariance Propagation of Continuous Time Systems
 
5. Continuous and Discrete Propagation of State Covariance
      Non Stationary Examples                                           5.2 5.3
      Modeling considerations
      Steady State Techniques                                            Notes
      Numerical Methods for Time Discretization of the
      Stochastic System.

6.  Derivation of the Discrete  Time Kalman Filter          5.4 5.6
 
     Examples                                                                    Notes
     Observability
 

7.   Further Topics in Estimation Theory                 Parts Ch. 6

     The Steady State Filter, Z Transform Representation.
     Augmentation of the state variables for:                    Notes
         Accurate Error Modeling
         Performing Sensitivity Analysis/Evaluation
         Desensitizing Filters to Uncertainties in the Model
         Truth Model vs. Filter Model
         An Extreme: Least Squares Estimation and Curve Fitting  vs. Kalman Filtering

8.  Further Examples                                                  Notes

9.  The Continuous Time Kalman Filter                       Ch. 7
       Derivation.                                                             Notes
       Non White Measurement Noise
       Matrix Riccati Equation
       Steady State Solutions

10. Topics with, and Applications of Continuous Time
       Kalman  Filters                                                        Notes
 
       S Domain Representations of Steady State filters (Causal  Weiner Filter)
       Combining Continuous and Discrete Measurements

  11.  Discrete Time Optimal Smoothing                        Ch. 8
        Development
        Evaluation
        Applications

 12.  Additional Methods of for Real Time Implementation
       Sensitivity Analysis and Off Line Evaluation              Notes, Papers
       Use of the Innovations for Editing Data , on the Fly  Evaluation.
       Case studies

 13.  Linearized/ Extended Kalman Filter                      9.1 9.2
      Examples:  Trajectory Determination,  Navigation Systems
 
 14.  Further Applications/ Case Studies in:               Select. Ch. 9,10.
         Navigation Systems, System Identification  and  Fault Detection  .

 15.  Review, Catch-up