GPS with High Rate Sensors

State Estimation: Review (GPS)

Optimal State Estimation The main topic of this topic is the derivation and study of the topic of optimal state estimation. Sections 3.6 and 4.8 have already introduced the concept of state estimation. In those sections the state estimation gain was represented by L. The gain L could, for example, be selected by pole placement […]

Minimum Variance Gain Derivation (GPS)

Eqn. (5.12) is useful for the selection of the estimator gainbecause it allows us to evaluate the covariance matricesthat would result from alternative choices ofIn fact, the diagonal of ^contains the variance of each element of the state vector. The scalar function is the sum of the variances of the individual states (i.e., the mean-squared-error); […]

From WLS to the Kalman Filter (GPS) Part 1

This section considers an alternative derivation of the Kalman filter in a series of steps. Subsection 5.3.1 considers the weighted least squares problem. The batch and recursive solutions to the weighted least squares problem are considered in Subsections 5.3.2 and 5.3.3, respectively. Using the solution to the recursive least squares problem and the state space […]

From WLS to the Kalman Filter (GPS) Part 2

Recursive Least Squares (RLS) For the linear measurement model of eqn. (5.17) with the set of measurementsusing equations (5.27-5.28), it is possible to calculate the estimate x that is the optimal in the WLS, maximum likelihood, and mean-squared senses. The question of this section is how to efficiently produce the new optimal estimate of x […]

Kalman Filter Derivation Summary (GPS)

For the system described by Section 5.3.4 derived the following version of the Kalman filter: Eqns. (5.54-5.56) are referred to as the Kalman filter time update equations. Eqns. (5.57-5.60) are referred to as the Kalman filter measurement update equations. Eqn. (5.59) defines the Kalman filter measurement residual. Analysis of the measurement residual can provide information […]

Kalman Filter Properties (GPS)

Section 5.2 derived the Kalman filter in the stochastic process framework as an unbiased, minimum variance, linear stochastic estimator. The results of that derivation approach can yield valuable insight into the Kalman filter and its performance. Section 5.3 derived the Kalman filter as an extension of the weighted least squares approach from linear sets of […]

Implementation Issues (GPS)

This section discusses several issues related to the real-time implementation of Kalman filters. Scalar Measurement Processing The Kalman filter algorithm as presented in Table 5.5 is formulated to process a vector of m simultaneous measurements. The portion of the measurement update that requires the most computing operations (i.e., FLOP’s) is the covariance update and gain […]

Implementation Sequence (GPS)

The discrete-time Kalman filter equations are summarized in Table 5.5. Because of the fact that the Kalman gain and covariance update equations are not affected by the measurement data, the sequence of implementation of the equations can be manipulated to decrease the latency between arrival of the measurement and computation of the corrected state. After […]

Asynchronous Measurements (GPS)

Asynchronous do not occur at a periodic rate. It is straightforward to handle asynchronous measurements within the Kalman filter framework. Note that the Kalman filter derivation did not assume the measurements to be periodic. The Kalman filter does require propagation of the state estimate and its covariance from one measurement time instant to the next […]

Numeric Issues (GPS)

One of the implicit assumptions in the derivation of the Kalman filter algorithm was that the gains and the arithmetic necessary to calculate them would take place using infinite precision real valued numbers. Computer implementations of the Kalman filter use a finite accuracy subset of the rational numbers. Although the precision of each computation may […]

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