SI&DA - Lecture 26 'Applications for the Kalman Filter'

Summary

Recursive system identification, we considered the following Linear Regression Model:

$$y(t)=\phi (t{)}^T\theta +e(t)$$

where:

• $\phi (t)$ : regressor vector which usually contains past values of input and output signals like in ARX or FIR models
• $\theta$ : vector of parameters to be estimated
• $e(t)$ : a corruption process or noise

We have found the solution to this problem, if we minimize the sum of squares of the one step ahead output prediction error :

$$\underset{\theta }{\min }\sum _{t=1}^N{\left(y(t)-\hat{y}(t|t-1)\right)}^2$$

We get the Least Square Solution:

$${\hat{\theta }}_{LS}={\left[\sum _{t=1}^N\phi (t){\phi }^T(t)\right]}^{-1}\cdot \sum _{t=1}^N\phi (t)y(t)$$

But in case we want to calculate the estimate at time $t$ to deliver it, instead at $t=N$ (when the measurement ends) we just need to change the formula a bit, in principle the same formula:

$${\hat{\theta }}_t={\left[\sum _{t=1}^t\phi (k){\phi }^T(k)\right]}^{-1}\cdot \sum _{k=1}^t\phi (k)y(k)$$

But a more clever solution to recursively update the estimate could be to implement the following Recursive Algorithm :

$$\{\begin{array}{l}{\hat{\theta }}_t={\hat{\theta }}_{t-1}+R(t{)}^{-1}\phi (t)\left(y(t)-{\phi }^T(t){\hat{\theta }}_{t-1}\right)\\ R(t)=R(t-1)+\phi (t){\phi }^T(t)\end{array}$$

where: ${\phi }^T(t){\hat{\theta }}_{t-1}=\hat{y}(t|t-1)$

This algorithm can also be upgraded, removing the need to compute the inverse of $R(t)$ at each iteration: We call $P(t)=R(t{)}^{-1}$, and using the property:

$$(A+BCD{)}^{-1}=A^{-1}-A^{-1}B{\left(C^{-1}+D{A}^{-1}B\right)}^{-1}D{A}^{-1}$$

we obtain:

$$P(t)=P(t-1)-\frac{P(t-1)\phi (t){\phi }^T(t)P(t-1)}{1+{\phi }^T(t)P(t-1)\phi (t)}$$

So the complete algorithm is:

$$\{\begin{array}{l}{\hat{\theta }}_t={\hat{\theta }}_{t-1}+P(t)\phi (t)\left(y(t)-{\phi }^T(t){\hat{\theta }}_{t-1}\right)\\ P(t)=P(t-1)-\frac{P(t-1)\phi (t){\phi }^T(t)P(t-1)}{1+{\phi }^T(t)P(t-1)\phi (t)}\end{array}$$

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#TODO What does the purple line mean?? #TODO Understand the graph