the differential equation $\dot{x}(t) = ax(t) + bu(t)$ coupled with the algebraic equation $y(t) = cx(t) + du(t)$ where $a\in\mathbb{c}^{n\times n}$, $b\in\mathbb{c}^{n\times m}$, $c\in\mathbb{c}^{p\times n}$ is called a state space system and commonly employed to represent a linear operator from an input space to an output space in control theory. one major challenge with such a representation is that typically $n$, the dimension of the intermediate state function $x(t)$, is much larger than $m$ and $p$, the dimensions of the input function $u(t)$ and the output function $y(t)$. to reduce the order of such a system (dimension of the state space) the traditional approaches are based on minimizing the $h_{\infty}$ norm of the difference between the transfer functions of the original system and the reduced-order system. we pose a backward error minimization problem for model reduction in terms of the norms of the perturbations to the coefficients $a$, $b$ and $c$ such that the perturbed systems are equivalent to systems of order $r