CDS 110b: Kalman Filters

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CDS 110b

Schedule

Project

FAQ

Reading

In this lecture we introduce the optimal estimation problem and describe its solution, the Kalman (Bucy) filter. We discuss the extension of Kalman filters to nonlinear systems (the EKF) as well as the Linear Quadratic Guassian (LQG) problem.

Course Materials

• Lecture presentation (MP3)
• Lecture Notes on Kalman Filters
• HW #3 (due 24 Jan 07)

References and Further Reading

• Friedland, Chapter 11
• K. J. Åström and R. M. Murray, Feedback Systems: An Introduction for Scientists and Engineers, Preprint, 2006. Chapter 7 - Output Feedback. Section 7.4 covers the discrete time Kalman filter.

Frequently Asked Questions

Q: How do you determine the covariance and how does it relate to random processes

The covariance of two random variables $x$ and $y$ is given by

$E\{(x-\mu )(y-\mu )\}=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }(x-\mu )(y-\mu )p(x,y)dxdy$

For the case when $x=y$, the covariance $P(x,y)$ is called the variance, $\sigma ^{2}$.

For a random process, $x(t)$, with zero mean, we define the covariance as

$P(t)=E\{x(t)x^{T}(t)\}.$

If $x$ is a vector of length $n$, then the covariance matrix is an $n\times n$ matrix with entries

$E\{x_{i}(t)x_{j}(t)\}=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }x_{i}x_{j}p(x_{i},x_{j};t,t)dx_{i}dx_{j}$

where $p(x_{i},x_{j};t,t)$ is the joint distribution desity function between $x_{i}$ and $x_{j}$.

Intuitively, the covariance of a vector random process $x(t)$ describes how elements of the process vary together. If the covariance is zero, then the two elements are independent.

• Wikipedia entry on covariance

Q: you asked what the estimator for the ducted fan would show (compared to eigenvalue placement). What should we be looking at and how would we be making those guesses?

This was not such a great question because you didn't have enough information to really make an informed guess. The main feature that is surprising about the result is that the convergence rate is much slower than eigenvalue placement.