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

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

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 $S_2$. The the reduced model is

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

By defining $\hat{E}=W^TEV$, $\hat {A}=W^TAV, \hat{B}=W^TB$, $\hat{C}=CV$, we get the final reduced model

$\hat{E} \frac{dz(t)}{dt}=\hat{A}z(t)+\hat{B}u(t),$ $\hat{y}(t)=\hat{C}z(t).$

Notice that the approximation $\hat x(t)=Vz(t)$ of $x(t)$ can be obtained from $z(t)$ by solving the system in (3). The system in~(3) is much smaller than the system in~(1) in the sense that there are many less equations in (3) than in (1). Therefore, the system in (3) is much easier to be solved, which is the so-called reduced model. In order to save the simulation time of solving (1), reduced model can be used to replace (1) 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 the state vectors $x(t), \hat x(t)$, or between the output responses $y(t), \hat y(t)$, or between the transfer functions of the two systems.