# 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.
– user28715
Oct 6, 2017 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. Oct 6, 2017 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?
– user28715
Oct 6, 2017 at 21:02

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