I am trying to solve an MPC type problem in which I have equality constraints of type
$x(k+1) = Ax(k) + Bu(k)$
which are because of system dynamics and I have inequality constraints of type
$h_{min} \le ax_1 + bx_2 \le h_{max}$
where $h_{min}$ and $h_{max}$ are constants, $x_1$ and $x_2$ are components of state vector.
I want to ask that is there any way I can convert these inequality constraints into box constraints on $x_1$ and $x_2$ i.e.
$c \le x_1 \le d$ and $e \le x_2 \le f$
where c, d, e and f are constants.
Just change a variables:
$$x'_1=x_1$$
$$x'_2=a x_1 + b x_2$$
Then apply
$$h_{\min}\le x'_2\le h_{\max}$$
you can simply retrieve back $x_2$ from $x'_1$ and $x'_2$.