TITLE: Linear Control Systems
QUESTION [0 upvotes]: The question reads: Determine the general form of $u_0$, $u_1$ and $u_2$ if a system of difference equations of the form
$$x_{n+1} = Ax_n + Bu_n.$$
$A$ is a $3\times3$ matrix and $B$ is a $3\times2$ matrix.
We were given information that $x_0 = 0$ to $x_3 = [2, 1, 2]^T$.
What confuses me is I know how to solve controllability but the fact that matrix B was $3\times2$ and not $3\times1$ completely threw me off.
I have spent hours trying to look for examples and cannot find any.

REPLY [0 votes]: Assume we have two matrices $M, N$, with sizes $m_1 \times m_2, n_1 \times n_2$ respectively. Then matrix multiplication of $MN$ is only defined if $m_2 = n_1$, and has resulting size $m_1 \times n_2$.
By looking at this argument, we know the dimensions. We know that $B u_n$ must be $3 \times 1$ matrix to add with $Ax_n$ another $3 \times 1$ matrix to get $x_{n+1}$ another $3 \times 1 $ matrix.
So  $u_n$ must be $2 \times 1$ matrices. 
Now, 
\begin{align}
x_3 &= Ax_2 + Bu_2\\
x_3 &= A^2x_1 + ABu_1 + Bu_2\\
x_3 &= A^3x_0 + A^2Bu_0 + ABu_1 + Bu_2
\end{align}
but $x_0 = 0$, so we have $$x_3 = + A^2Bu_0 + ABu_1 + Bu_2.$$
So if you know $u_0, u_1$, we know what $u_2$ looks like. Similarly, if we know $A,B$ better, or we know $x_n$, we can work out $u_n$.
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