# CDS 110b: Kalman Filtering

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In this lecture we introduce the optimal estimation problem and describe its solution, the Kalman (Bucy) filter.

## Lecture Outline

1. State Space Computation for Stochastic Response
2. Optimal Estimation
3. Kalman Filter

## Lecture Materials

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.

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.