In the last post, we present a well-known observer for deterministic systems. In this post, I will present the estimators for noisy systems.

Noisy system:

$x_{i+1}=Ax_i+Bu_i+Bw_i$
$y_i=Cx_i+v_i$

Observer:

$\hat{x}_{i+1}=A\hat{x}_i+Bu_i+K(y_i-\hat{y}_i)$
$\hat{y}_i=C\hat{x}_i$

It should be noted that in the noisy system, there are $w_i, v_i$. But in the estimator there is no these terms because we don’t know $w_i, v_i$ and they can’t be used in our estimator!!

The state error is $\tilde{x}_i=x_i-\hat{x}_i$. We have the estimation error model:

$\tilde{x}_{i+1}=(A-KC)\tilde{x}_i+Bw_i-Kv_i$

There are random terms in the error model. Unlike the deterministic cases, we should no longer expect $\tilde{x}_{i+1}\rightarrow 0$!!! Instead, taking expectation of the above equation gives

$\mathbf{E}\left(\tilde{x}_{i+1}\right)=(A-KC)\mathbf{E}\left(\tilde{x}_{i}\right)$

So the point in this case is to make $\mathbf{E}\left(\tilde{x}_{i+1}\right)\rightarrow 0$. This is not enough because the expectation may be zero but errors can fluctuate about the mean value dramatically. So we also need to minimize the variance of the error. Usually we need to minimize $\mathbf{E}\left(\tilde{x}_{i+1}\tilde{x}_{i+1}^T\right)$. Kalman did nothing but proposed an estimator minimizing the error variance!!

PS: it is assumed $\mathbf{E}(w_i)=0, \mathbf{E}(v_i)=0, \mathbf{E}(x_0)=0$. In practice, this may be not true. But we can always turn them into the zero-mean cases. Besides, the states here is no longer deterministic, they are random variables and have expectation and covariance. I still don’t understand why $x_0$ is a random variable and we require $\mathbf{E}(x_0)=0$.