# Math Help - Determine the general form of control sequence

1. ## Determine the general form of control sequence

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

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

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

What I did for this question is:

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

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

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

the system has the matrix form:

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

for which the argumented matrix is:

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

then I found that
$u_0=1$,
$u_1-u_0=1=>u_1=2$,
$u_2+u_0=2=>u_2=1$.

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 $X_2$:

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

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

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

which does not match the target $X_3=\begin{pmatrix}2 & 1 & 2\end{pmatrix}$, 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
$u_0=\begin{pmatrix}M \\ R\end{pmatrix}$,
$u_1=\begin{pmatrix}1-S-3T-3R\\ S\end{pmatrix}$,
$u_2=\begin{pmatrix}1+S+2T+3R\\ 2-S-R\end{pmatrix}$.

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

2. ## Re: Determine the general form of control sequence

Originally Posted by jackw
Hi, I .

Did you ever figure out how to do this question?

We'vee got this EXACT same question for our courseowrk and we have the same problems as you, so we were wondering if you could throw any hints our way?