1. Colorado Center for Astrodynamics Research The University of Colorado 1 STATISTICAL ORBIT DETERMINATION Kalman Filter with Process Noise Gauss- Markov Process Examples of SNC and DMC ASEN 5070 LECTURE 28,29 11/04,06/09

2. Colorado Center for Astrodynamics Research The University of Colorado 2 Exam #2 I. Probability & Statistics Review 1. Axioms of Probability 2. Conditional Probability 3. Independence 4. Density and Distribution Functions 5. Expected Values 6. Univariate & Bivariate Density Functions 7. Properties of Covariance & Correlations 8. Central Limit Theorem II. Statistical Interpretation of Least Squares 1. Observation Error Covariance 2. Estimation Error Covariance 3. Biased Estimators III. Minimum Variance Estimator IV. Kalman (Sequential) Filter V. Observability VI. Review Homework, Assignments 4-8

3. Colorado Center for Astrodynamics Research The University of Colorado 3 Estimation in the Presence of Process Noise State Noise Compensation Algorithm (section 4.9) The state dynamics of a linear system under the influence of process noise are described by The state deviation vector is propagated according to Where and are known functions and is a (white, i.e., uncorrelated in time) random noise process with (4.9.1)

4. Colorado Center for Astrodynamics Research The University of Colorado 4 Estimation in the Presence of Process Noise The Kalman Filter with process noise (also known as state noise compensation, SNC) is as follows Time update:

5. Colorado Center for Astrodynamics Research The University of Colorado 5 Estimation in the Presence of Process Noise Measurement update: is generally determined by trial and error and is usually a diagonal matrix. This formulation of the Kalman Filter in known as state noise compensation (SNC). See appendix F and web handout.

10. Numerous functions can be approximated by varying  and  For example, we can take the limit of eq (4.9.60) using L’Hospital’s Rule to show that: Also,

16. 16 Copyright 2006 Example of State Noise & Dynamic Model Compensation Consider a target particle that moves in one dimension along the x axis in the positive direction. Nominally, the particle’s velocity is a constant 10 m/sec. This constant velocity is perturbed by a sinusoidal acceleration in the x direction, which is unknown and is described by: Example Taken from Appendix F of the Text

17. 17 Copyright 2006 Example of State Noise & Dynamic Model Compensation Figure F.1.1: Perturbed particle position, velocity and acceleration

18. 18 Copyright 2006 Example of State Noise & Dynamic Model Compensation A simple estimator for this problem incorporates a two-parameter state vector consisting of position and velocity: The dynamic model assumes constant velocity for the particle. The state transition matrix for this estimator is: The observation/state mapping matrix is a two-by-two identity matrix: The filter was first run with no process noise. Observations are range and range rate at 10 HZ With Gaussian noise and  1m and 0.1m/s.

19. 19 Copyright 2006 Example of State Noise & Dynamic Model Compensation The filter quickly saturates and its estimation performance is poor.

20. 20 Copyright 2006 Example of State Noise & Dynamic Model Compensation The State Noise Compensation (SNC) algorithm improves estimation performance through partial compensation for the unknown acceleration. A simple SNC model uses a two-state filter but assumes that particle dynamics are perturbed by an acceleration that is characterized as simple white noise: where u(t) is a stationary Gaussian process with a mean of zero and a variance of Application of the equation results in

21. 21 Copyright 2006 State Noise Compensation Where the state propagation matrix A is identified as: and the process noise mapping matrix is: The state transition matrix is the same:

22. 22 Copyright 2006 State Noise Compensation The process noise covariance integral (see Eq. (4.9.44)) needed for the time update of the estimation error covariance matrix at time t is expressed as: The process noise covariance matrix is given by where t – t 0 is the measurement update interval The implication of this is that the original deterministic constant velocity model of particle motion is modified to include a random component σ u is a tuning parameter whose value can be optimized to improve performance

23. 23 Copyright 2006 State Noise Compensation The increase in the velocity variances’ prevents the components of the Kalman gain matrix from going to zero with the attendant saturation of the filter.

24. 24 Copyright 2006 Dynamic Model Compensation A more sophisticated process noise model is provided by the Dynamic Model Compensation (DMC) formulation. DMC assumes that the unknown acceleration can be characterized as a first-order linear stochastic differential equation where u(t) is a white zero-mean Gaussian process as described earlier and β is the inverse of the correlation time: The solution to this equation is a Gauss-Markov process

25. 25 Copyright 2006 Dynamic Model Compensation DMC yields a deterministic acceleration term as well as a purely random term. The deterministic acceleration can be added to the state vector and estimated along with the velocity and position The augmented state vector becomes a three-state filter with where η D (t) is the deterministic part The dynamic model of the particle’s motion becomes

26. 26 Copyright 2006 Dynamic Model Compensation The correlation time, τ, can also be added to the estimated state, resulting in a four- parameter state vector. Generally τ is set to a near-optimal value and held constant The observation/state mapping matrix is a simple extension of the two-state case: The state transition matrix is found to be

27. 27 Copyright 2006 Dynamic Model Compensation The state propagation matrix for this case is the process noise mapping matrix is the process noise covariance integral becomes Note that here τ designates the integration variable, not the correlation time. Elements of Q η (t) are given in Appendix F of the text

28. 28 Copyright 2006 Dynamic Model Compensation With τ fixed at 200.0 sec (or β = 0.005 sec -1 ), the optimal value of σ u is 0.26 m/sec 5/2.

29. 29 Copyright 2006 Dynamic Model Compensation Figure F.3.2 is a plot of the RMS of the position error as a function of σ u for both the two-state SNC filter and the three-state DMC filter. Although the optimally tuned SNC filter approaches the position error RMS performance of the DMC filter, it is much more sensitive to tuning.

30. 30 Copyright 2006 Dynamic Model Compensation The DMC filter provides an estimate of the unknown, unmodeled acceleration.