代做ECE5550: Applied Kalman Filtering THE LINEAR KALMAN FILTER代做Java程序

ECE5550: Applied Kalman Filtering

THE LINEAR KALMAN FILTER

4.1: Introduction

■ The principal goal of this course is to learn how to estimate the

present hidden state (vector) value of a dynamic system, using noisy measurements that are somehow related to that state (vector).

■ We assume a general, possibly nonlinear, model

xk = fk−1(xk−1, uk−1,wk−1)

z k = hk (xk, uk ,vk ),

where uk is a known (deterministic/measured) input signal, wk is a    process-noise random input, and vk is a sensor-noise random input.

SEQUENTIAL PROBABILISTIC INFERENCE: Estimate the present state xk of a dynamic system using all measurements Zk = {z0 , z 1 , ··· , z k } .

■ This notes chapter provides a unified theoretic framework to develop a family of estimators for this task: particle filters, Kalman filters, extended Kalman filters, sigma-point (unscented) Kalman filters. . .

A smattering of estimation theory

■ There are various approaches to “optimal estimation” of some unknown quantity x .

■ One says that we would like to minimize the expected magnitude

(length) of the error vector between x and the estimatex(ˆ) .

■ This turns out to be the median of the a posteriori pdf f (x | Z).

■ A similar result, but easier to derive analytically minimizes the expected length squared of that error vector.

■ This is the minimum mean square error (MMSE) estimator

■ We solve forx(ˆ) by differentiating the cost function and setting the

result to zero

■ Another approach to estimation is to optimize a likelihood function

■ Yet a fourth is the maximum a posteriori estimate

■ In general,x(^)MME  /=x(^)MMSE  /=x(^)ML  /=x(^)MAP , so which is “best”?

■ Answer: It probably depends on the application.

■ The text gives some metrics for comparison: bias, MSE, etc.

■ Here, we usex(^)MMSE  = E[x | Z] because it “makes sense” and works

well in a lot of applications and is mathematically tractable.

Some examples

■ In example 1, mean, median, and mode are identical. Any of these statistics would make a good estimator of x .

■ In example 2, mean, median, and mode are all different. Which to choose is not necessarily obvious.

■ In example 3, the distribution is multi-modal. None of the estimates is likely to be satisfactory!

4.2: Developing the framework

■ The Kalman filter applies the MMSE estimation criteria to a dynamic system. That is, our state estimate is the conditional mean

where Rxk is the set comprising the range of possible xk .

■ To make progress toward implementing this estimator, we must break f (xk | Zk ) into simpler pieces.

■ We first use Bayes’ rule to write:

■ We then break up Zk into smaller constituent parts within the joint probabilities as Zk−1  and z k

■ Thirdly, we use the joint probability rule f (a , b) = f (a | b)f (b) on the numerator and denominator terms

■ Next, we apply Bayes’ rule once again in the terms within the [   ]

■ We now cancel some terms from numerator and denominator

■ Finally, recognize that zk is conditionally independent of Zk−1  given xk

■ So, overall, we have shown that

KEY POINT #1: This shows that we can compute the desired density recursively with two steps per iteration:

■ The first step computes probability densities for predicting xk given all past observations

■ The second step updates the prediction via

■ Therefore, the general sequential inference solution breaks naturally into a prediction/update scenario.

■ To proceed further using this approach, the relevant probability densities may be computed as

KEY POINT #2: Closed-form. solutions to solving the multi-dimensional integrals is intractable for most real-world systems.

■ For applications that justify the computational expense, the integrals may be approximated using Monte Carlo methods (particle filters).

■ But, besides applications using particle filters, this approach appears to be a dead end.

KEY POINT #3: A simplified solution may be obtained if we are willing to

make the assumption that all probability densities are Gaussian.

■ This is the basis of the original Kalman filter, the extended Kalman

filter, and the sigma-point (unscented) Kalman filters to be discussed.