Hi, I have a question which relevant to controllability matrix. The question asks to determine the general form of sequence of control: $\displaystyle u_0, u_1$ and $\displaystyle u_2$,

if a system of difference equation of the form $\displaystyle X_{t+1}=AX_t+BU_t$, where $\displaystyle A=\begin{pmatrix}3 & 2 & 2 \\ -1 & 0 & -1 \\ 0 & 0 & 1 \end{pmatrix}$ and $\displaystyle B=\begin{pmatrix}0 & 0 \\ 0 & 1 \\ 1 & 0\end{pmatrix}$, is to be controlled from $\displaystyle X_0=0$ to $\displaystyle X_3=\begin{pmatrix}2 & 1 & 2\end{pmatrix}$,

then show that, in fact, this target could have been achieved at $\displaystyle X_2$.

What I did for this question is:

First, it's simple, just find the controollability matrix, set it's C, then

$\displaystyle C=\begin{pmatrix}A & AB & A^{2}B\end{pmatrix}=\begin{pmatrix} 0 & 0 &|& 2 & 2 & |& 6 & 6\\ 0 & 1 & |& -1 & 0 & | & -3 & -2 \\ 1 & 0 & | & 1 & 0 &| & 1 & 0\end{pmatrix}$

In order to solve the system of equation $\displaystyle CU=X_3-A^{3}X_0$, I have chosen the first three column from above matrix, then

the system has the matrix form:

$\displaystyle \begin{pmatrix}0 & 0 & 2\\ 0 & 1 & -1\\1 & 0 &1\end{pmatrix}\begin{pmatrix}u_2 \\ u_1 \\ u_0\end{pmatrix}=\begin{pmatrix}2\\1\\2\end{pmatri x}-0$

for which the argumented matrix is:

$\displaystyle \begin{pmatrix}0&0&2&|&2\\0&1&-1&|&1\\1&0&1&|&2\end{pmatrix}$ ---> $\displaystyle \begin{pmatrix}1&0&1&|&2\\0&1&-1&|&1\\0&0&2&|&2\end{pmatrix}$ ---> $\displaystyle \begin{pmatrix}1&0&1&|&2\\0&1&-1&|&1\\0&0&1&|&1\end{pmatrix}$

then I found that

$\displaystyle u_0=1$,

$\displaystyle u_1-u_0=1=>u_1=2$,

$\displaystyle u_2+u_0=2=>u_2=1$.

This is the control sequence in a particular case, I'm not sure that how to find it's general case.

And then substituting into the original system of difference equation to show that the target could have been achieved at $\displaystyle X_2$:

$\displaystyle X_{t+1}=AX_t+BU_t$

then

$\displaystyle X_1=AX_0+BU_0=0+\begin{pmatrix}0&0\\0&1\\1&0\end{p matrix}*1=\begin{pmatrix}0&0\\0&1\\1&0\end{pmatrix }$

$\displaystyle X_2=AX_1+BU_1=\begin{pmatrix}3 & 2 & 2 \\ -1 & 0 & -1 \\ 0 & 0 & 1 \end{pmatrix}\begin{pmatrix}0&0\\0&1\\1&0\end{pmat rix}+\begin{pmatrix}0&0\\0&1\\1&0\end{pmatrix}*2=\ begin{pmatrix}2&2\\-1&0\\1&0\end{pmatrix}$

which does not match the target $\displaystyle X_3=\begin{pmatrix}2 & 1 & 2\end{pmatrix}$, so it's absolutely a wrong result. I guess the problem is from finding the control sequence, it should be in general form.

I have the correct answer which shows

$\displaystyle u_0=\begin{pmatrix}M \\ R\end{pmatrix}$,

$\displaystyle u_1=\begin{pmatrix}1-S-3T-3R\\ S\end{pmatrix}$,

$\displaystyle u_2=\begin{pmatrix}1+S+2T+3R\\ 2-S-R\end{pmatrix}$.

But I really have no idea about how to obtain this results. Hope someone can help me this for a bit. Thanks a lot.