The basic idea of almost all the model order reduction (MOR) methods is to find a subspace $S_1$ which approximates the manifold where the state vector ${\bf x}(t)$ resides. Afterwards, ${\bf x}(t)$ is approximated by a vector $\tilde{\bf x}(t)$ in $S_1$. The reduced model is produced by Petrov-Galerkin projection onto a subspace $S_2$, or by Galerkin projection onto the same subspace $S_1$.
We use the system Failed to parse (unknown function "\begin"): \begin{equation} \label{sys1} \begin{array}{rcl} E \frac{d{\bf x}}{dt}&=&A {\bf x}+B {\bf u}(t), \\ {\bf y}(t)&=&C{\bf x}+D{\bf u}(t). \end{array} \end{equation}
as an example to explain the basic idea. Assuming that an orthonormal basis $V=(v_1,v_2, \ldots, v_q)$ of the subspace $S_1$ has been found, then the approximation $\tilde{\bf x}(t)$ in $S_1$ can be represented by the basis as $\tilde{\bf x}(t)=V{\bf z}(t)$. Therefore ${\bf x}(t)$ can be approximated by ${\bf x}(t) \approx V{\bf z}(t)$. Here ${\bf z}$ is a vector of length $q \ll n$.
Once ${\bf z}(t)$ is computed, we can get an approximate solution $\tilde{\bf x}(t)=V{\bf z}(t)$ for ${\bf x}(t)$. The vector ${\bf z}(t)$ can be computed from the reduced model which is derived by the following two steps.