getUR

This example shows how to obtain the $\mathit{U}$ and $\mathit{R}$ factors of the snapshot collection $\mathit{X}$ processed by an incrementalPOD object. These factors provide a low-rank approximation of the data ${\mathit{XX}}^{\mathit{T}}=\left(\mathit{UR}\right){\left(\mathit{UR}\right)}^{\mathit{T}}$.

Create a random state-space model and run a simulation to collect state data.

rng(0)
sys = drss(50);
t = 0:1:100; 
u = zeros(size(t)); 
u(1) = 1;
[~,~,x] = lsim(sys,u,t);
X = x';

Now, perform an incremental POD for the same system using lsim.

xPOD = incrementalPOD;
[~,~,~,~,xPOD] = lsim(sys,u,t,xPOD);

Obtain the factors from the POD object.

[U,R] = getUR(xPOD);
size(U)

ans = 1×2

    50    22

This provides an approximation of rank 22 of ${\mathit{XX}}^{\mathit{T}}$ with size 50-by-50. You can verify that error percentage of the original state date with this approximation is less than the value specified in the RankTol property of xPOD.

sqrt(norm(X*X'-(U*R)*(U*R)',"fro"))/norm(X,"fro")

ans = 
5.7146e-07