Sparse Balanced Truncation of Thermal Model

Since R2023b

This example shows how to perform balanced truncation of a sparse state-space model obtained from linearizing a thermal model of heat distribution in a circular cylindrical rod.

Load the model data.

load cylindricalRod.mat
sys = sparss(A,B,C,D,E);
size(sys)

Sparse state-space model with 3 outputs, 1 inputs, and 7522 states.

The thermal model contains 7522 states.

Create a balanced truncation specification object for sys and run the algorithm.

R = reducespec(sys,"balanced");
R = process(R)

Initializing...
Running ADI with built-in shifts.........
Running ADI with adaptive shifts..
Solved Lyapunov equations to desired accuracy.

R = 
  SparseBalancedTruncation with properties:

        Sigma: [40x1 double]
       Energy: [40x1 double]
        Error: [40x1 double]
           Lr: [7522x40 double]
           Lo: [7522x123 double]
    Residuals: [6.2579e-09 8.9726e-09]
      Options: [1x1 mor.SparseBalancedTruncationOptions]

Use the view command to visualize the state contributions as Hankel singular values.

view(R,"sigma")

Obtain the reduced-order model with maximum error of 1e-6. This results in a model with order 8.

rsys = getrom(R,MaxError=1e-6,Method="truncate");

Compare the singular value response of the models.

w = logspace(-7,-3,20);
fsys = frd(sys,w);
sigma(fsys,fsys-rsys,'r--')

The reduced-order model is a good match for the full-order model.