Projection based MOR

The basic idea of almost all the model order reduction (MOR) methods is based on projection. That is to find a subspace $S_1$ which approximates the manifold where the state vector $x(t)$ resides. Afterwards, $x(t)$ is approximated by a vector $\tilde{x}(t)$ 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 $E\frac{dx(t)}{dt}=Ax(t)+Bu(t),$ $y(t)=Cx(t)+Du(t).$ as an example to explain the basic idea. Assuming that an orthonormal basis $V=(v_1,v_2,\dots ,v_q)$ of the subspace $S_1$ has been found, then the approximation $\tilde{x}(t)$ in $S_1$ can be represented by the basis as $\tilde{x}(t)=Vz(t)$. Therefore $x(t)$ can be approximated by $x(t)\approx Vz(t)$. Here $z$ is a vector of length $q \ll n$.

Once $z(t)$ is computed, we can get an approximate solution $\tilde{x}(t)=Vz(t)$ for $x(t)$. The vector $z(t)$ can be computed from the reduced model which is derived by the following two steps.