Find the control function

Question

Determine the general form of $u_0, u_1 ~\text{and} ~ u_2$ if a system of difference equations of the form

$$x_{n+1} = Ax_n + Bu_n,$$

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 for $x_0 = 0 ~ to ~ x_3 = [2, 1, 2]^T$ .

Answer 1

Edit: The controllability matrix is $$ C=\begin{pmatrix} B,AB,A^2B \end{pmatrix} =\begin{pmatrix} 0&0&2&2&6&6\\ 0&1&-1&0&-3&-2\\ 1&0&1&0&1&0 \end{pmatrix}. $$ Clearly it has full rank. Thus the system is always controllable. In fact, if you take $u_0=0,\,u_1=(1,0)^T$ and $u_2=(1,2)^T$, you can get $x_3$ from $x_0$. For the general form of $u_0=(a,b)^T, u_1=(c,d)^T, u_2=(e,f)^T$, solve the system of linear equations $Cv=(2,1,2)^T$, where $v=(a,b,c,d,e,f)^T$.

Answer 2

@user1551 do you agree with the general form of $ u_0 u_1 $and $ u_2 $ is

$ u_0 =(1-3r_3 +2r_2 +r_1, 2-r_1 -r_3)^T u_1 = (1-3r_3 -3r_2 -r_1, r_1)^T u_2=(r_2, r_3)^T $