Lecture 13

The name "Kalman filter" refers to the estimation equation (13.1) and takes its name from Rudolf Kalman (1930 –), who developed it in the years 1958-64. He also coined the terms controllable and observable, and gave the criteria that we have seen in previous lectures. The fact that a system is controllable iff the matrix $[B\ AB\ \cdots\ A^{n-1}B]$ is of full rank is sometimes called Kalman's criteria. In the IEEE biography of Kalman it is stated

The Kalman filter, and its later extensions to nonlinear problems, represents perhaps the most widely applied by-product of modern control theory. It has been used in space vehicle navigation and control (e.g. the Apollo vehicle), radar tracking algorithms for ABM applications, process control, and socioeconomic systems.

The theory in this lecture is admittedly quite tricky - partly because the notation. As a test of memory, can you say what roles in the theory are taken by each of these?

 $x_t$,  $u_t$, $A$, $B$, $\epsilon_t$, $y_t$, $C$, $\eta_t$, $\hat x_t$, $\Delta_t$, $\xi_t$, $\zeta_t$, $R$, $S$, $Q$, $K_t$, $\Pi_t$, $N$, $L$, $M$, $H_t$,  $V_t$. 

 You will understand the ideas better once you have worked through the details of a scalar example (in which $n=m=p=1$). You do this in Example Sheet 3 Question 2. When you do this question, start by supposing that $\hat x_t=\hat x_{t-1}+u_{t-1}+h_t(y_t-\hat x_{t-1})$, and then find the value of $h_t$ that minimizes the variance of $\hat x_t$. You can start by subtracting $x_t=x_{t-1}+u_{t-1}+3\epsilon_t$ and using $y_t=x_{t-1}+2\eta_t$. You get,

$\hat{x}_{t+1}-{x}_{t+1}=\Delta_{t+1}=\Delta_t+2\epsilon_t-h_t\Delta_t+h_t2\eta_t.$

Then square, take the expected value, and minimize with respect to $h_t$ to find a formula for $V_{t+1}$ in terms of $V_t$.

You will not be asked to reproduce the statement or proofs of Theorem 13.1 or 13.2 in examinations. You should simply know that $\hat{x}_t$ is computed from $\hat{x}_{t-1}$ and $y_t$ in the linear manner specified by (13.1), and that the covariance matrix $V_t$ satisfies a Riccati equation. You are not expected to memorize Riccati equations.

Notice that the Riccati equation for $V_t$, i.e. $V_t = g\, V_{t-1}$ runs in the opposite time direction to the one we had for $\Pi_t$ in lecture 10, where $\Pi_{t-1} = f\, \Pi_t$. We are given $V_0$ and $\Pi_h$.