# How to convert linear inequality constraints into box constraints

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 $x'_2$.

• Thank you. Change of variables will really help. – Ali Khan Dec 27 '17 at 12:56