Projection based MOR

All the existing model order reduction (MOR) methods are 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

$E \frac{dx(t)}{dt}=A x(t)+B u(t),$ $y(t)=Cx(t)+Du(t),$

as an example to explain the basic idea. Assuming that an orthonormal basis $V=(v_1,v_2, \ldots, v_q)$ of the subspace $S_1$ has been found, then the approximation $\tilde x(t)$ in $S_1$ can be represented by the basis as $\tilde x(t)=V z(t)$. Therefore $x(t)$ can be approximated by $x(t) \approx V z(t)$. Here $z$ is a vector of length $q \ll n$.

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

Step 1. By replacing $x$ in (1) with $Vz$, we get

$E \frac{d{Vz}}{dt} \approx A Vz+Bu(t),$ $y(t) \approx CV z.$

Step 2. The residual is denoted as $e=AVz+Bu(t)-E \frac{d{Vz}}{dt}$. Force $e=0$ in a properly chosen subspace $S_2$ of $\mathbb {R}^n$ leads to the Petrov-Galerkin projection: $W^T e=0$, where the columns of $W$ are the basis of $S_2$. Finally, the reduced model is

$W^TE \frac{d{V z}}{t}=W^TA Vz +W^T B u(t),$ $\hat{y}(t)=CVz.$

\end{enumerate} % % By defining $\hat{E}=W^TEV$, $\hat {A}=W^TAV$, $\hat{B}=W^TB$, $\hat{C}=CV$, we get the final reduced model % % \begin{equation} \label{sys3} \begin{array}{rcl} \hat{E} \frac{d{\bf z}}{dt}&=&\hat{A}{\bf z}+\hat{B}{\bf u}(t), \\ \hatTemplate:\bf y(t)& = &\hat{C}{\bf z}. \end{array} \end{equation} % % Notice that the approximation $\tilde {\bf x}(t)=V{\bf z}(t)$ of ${\bf x}(t)$ can be obtained from ${\bf z}(t)$ by solving the system in~(\ref{sys3}). The system in~(\ref{sys3}) is much smaller than the system in~(\ref{sys1}) in the sense that there are many less equations in~(\ref{sys3}) than in~(\ref{sys1}). Therefore, the system in~(\ref{sys3}) is much easier to be solved, which is the so-called reduced model. In order to save the simulation time of solving~(\ref{sys1}), the system in~(\ref {sys3}) can be used to replace the original large system in~(\ref{sys1}) to attain a fast simulation. Furthermore, the error between the two systems should be within acceptable tolerance. The error can be measured through the error between the output responses or the transfer functions of the two systems.