Determine the general form of control sequence

**jackw** · May 4th 2009, 08:14 PM

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.

**elldeegee** · March 21st 2012, 05:13 AM

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?

pretty please!

If anybody now has some hints they could give us, please reply.

To Moderators: I know i may get told off for posting on old threads, so i apologise in advance!

Thread: Determine the general form of control sequence

LinkBack

Thread Tools

Display

Determine the general form of control sequence

Re: Determine the general form of control sequence

Similar Math Help Forum Discussions

Writing transform. form of a quad. function in the general form?

quadratic function general form to standard form

Write an equation of the circle described in both the standard form and general form

general form to standard form

rewrite in slope-intercepr form and general form

Search Tags