CDS 110b: Kalman Filtering

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Course Home

L7-2: Sensitivity

L8-1: Robust Stability

L9-1: Robust Perf

Schedule

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

• Lecture presentation (MP3)
• Lecture Notes on Kalman Filters
• Reading: Friedland, Chapter 11

Frequently Asked Questions

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

The covariance of two random variables Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E\{(x - \mu) (y - \mu)\} = \int_{-\infty}^\infty \int_{-\infty}^\infty (x - \mu) (y - \mu) p(x, y) dx dy }

For the case when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = y} , the covariance Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(x, y)} is called the variance, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma^2} .

For a random process, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x(t)} , with zero mean, we define the covariance as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(t) = E\{x(t) x^T(t)\}. }

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is a vector of length Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} , then the covariance matrix is an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \times n} matrix with entries

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x_i, x_j; t, t)} is the joint distribution desity function between Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_j} .

Intuitively, the covariance of a vector random process Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.