# calculating estimate of a state for a system with two observations of the state from different times

I'm struggling with a problem that I just can't seem to get a grasp of. I'm supposed to calculate an estimate for state $x(k)$ at times $k=1,2,3$ from the state-space

\begin{align} x(k+1) &= A x(k) + B u(k) + w(k) \\ y_1(k) &= x(k) + v1_(k) \\ y_2(k) &= x(k-1) + v_2(k) \\ \end{align}

where the covariances of the noises are given as

$E[w(k)w(k)']$ , $E[v_1(k)v_1(k)']$, $E[v_2(k)v_2(k)']$

and

$u(k)$ is given for $k=1,2,3$

$y_1(k)$ is given for $k=1,2,3$

$y_2(k)$ is given for $k=2,3$

What kind of technique am I supposed to use? Information Kalman filter?

• you can rewrite the equations so they are in the canonical form. Play with some substations. Don’t be too literal about what is a state. I vaguely recall that this a problem from Kailath and Sayeed . Please be upfront about homework if this is the case. – Stanley Pawlukiewicz Oct 6 '17 at 15:44
• This is indeed a homework problem, I am not asking for a solution, just a hint as to where to begin working on this problem. We've been studying linear estimation and Kalman Filter recently, but the problem does not seem to fit on anything. – Yeviipew Oct 6 '17 at 17:27
• rewrite the state transition equations so that they are the standard form and then form the Kalman Filter. Do I need to actually say that this is a hint? Stack y1 over y2 so it’s a vector. Then think about an H matrix that makes sense. What would a state vector look like? – Stanley Pawlukiewicz Oct 6 '17 at 21:02

$$\left| \begin{array}{cc} A & 0 \\ 0 & 1 \end{array} \right|$$