Transcript Lecture 18
ASEN 5070: Statistical Orbit Determination I Fall 2015 Professor Brandon A. Jones Lecture 18: Minimum Variance Estimator and Sequential Processing University of Colorado Boulder Exam 1 – Friday, October 9 University of Colorado Boulder 2 Minimum Variance w/ A Priori Sequential Processing w/ Minimum Variance University of Colorado Boulder 3 Minimum Variance w/ A Priori University of Colorado Boulder 4 With the least squares solution, we minimized the square of the residuals Instead, what if we want the estimate that gives us the highest confidence in the solution: ◦ What is the linear, unbiased, minimum variance estimate of the state x? University of Colorado Boulder 5 What is the linear, unbiased, minimum variance estimate of the state x ? ◦ This encompasses three elements Linear Unbiased, and Minimum Variance We consider each of these to formulate a solution University of Colorado Boulder 6 Turns out, we get the weighted, statistical least squares! Hence, the linear least squares gives us the solution with the smallest variance, i.e., highest confidence ◦ Of course, this is predicated on all of our statistical/linearization assumptions University of Colorado Boulder 7 To add a priori in the least squares, we augment the cost function J(x) to include the minimization of the a priori error. How do we control the weighting of the a priori solution and the observations in the cost function? University of Colorado Boulder 8 This is analogous to treating the a priori information as an observation of the estimated state at the epoch time University of Colorado Boulder 9 University of Colorado Boulder 10 University of Colorado Boulder 11 Like the previous case, the statistical least squares w/ a priori is equivalent to the minimum variance estimator To use the least squares estimator, do I have to use a statistical description of the observation/state errors? University of Colorado Boulder 12 Least squares does not require a probabilistic definition of the weights/state The minimum variance estimator demonstrates that, for a Gaussian definition of the observation and state errors, the LS is the best solution Also know as the Best Linear Unbiased Estimator (BLUE) Now, we can use the minimum variance estimator as a sequential estimator… University of Colorado Boulder 13 Minimum Variance and Sequential Processing University of Colorado Boulder 14 X* Batch – process all observations in a single run of the filter Sequential – process each observation vector individually (usually as they become available over time) University of Colorado Boulder 15 How do we map our a priori forward in time? X* University of Colorado Boulder 16 Well, we’ve kind of covered this one before: Note: Yesterdays estimate can become today’s a priori… ◦ Not used much for the batch, but will be used for sequential processing University of Colorado Boulder 17 University of Colorado Boulder 18 Now, we may map the previous estimate in time via the STM Can we leverage this information to sequentially process measurements in the minimum variance / least squares algorithm? University of Colorado Boulder 19 Given from a previous filter run: We have new a observation and mapping matrix: We can update the solution via: University of Colorado Boulder 20 Two principle phases in any sequential estimator ◦ Time Update Map previous state deviation and covariance matrix to the current time of interest ◦ Measurement Update Update the state deviation and covariance matrix given the new observations at the time of interest Jargon can change with communities ◦ Forecast and analysis ◦ Prediction and fusion ◦ others… University of Colorado Boulder 21 To perform the measurement updated, we only require one observation at tk. Wait, but what if we have fewer observations than unknowns at tk? ◦ Do we have an underdetermined system? University of Colorado Boulder 22 The a priori x is based on independent analysis or a previous estimate ◦ Independent analysis could be a product of: Expected launch vehicle performance Previous analysis of system Initial orbit determination solution University of Colorado Boulder 23 We still have to invert an n × n matrix Can be computationally expensive for large n ◦ Gravity field estimation: ~n2+2n-3 coefficients! May become sensitive to numeric issues University of Colorado Boulder 24 Is there a better sequential processing algorithm? ◦ YES! – This equations above may be manipulated to yield the Kalman filter (next week) University of Colorado Boulder 25