Online balanced truncation for linear time-varying systems using continuously differentiable interpolation on Grassmann manifold

—We consider model order reduction of linear timevarying systems on a ï¬nite time interval using balanced truncation. A standard way to perform MOR is to ï¬rst numerically integrate the associated pair of differential Lyapunov equations for the two gramians, then compute projection matrices using the square root method, and ï¬nally formulate the reduced systems at each time instant of a chosen grid. This approach is well-knownfordeliveringgoodapproximation,butrathercostly in computation and storage requirement. Furthermore, if one needs to compute the reduced system for any new time instant thatisnotincludedinthechosengrid,thementionedprocedure must be performed again without explicitly making use of the already computed data. For dealing with such a situation, we propose to store the projection matrices corresponding to a simpliï¬ed sparse time grid and to use them to recover the projection subspaces at any other time instant via curve interpolation on the Grassmann manifold. By doing this, we can avoid the repetition of solving the differential Lyapunov equations which is the most expensive step in the procedure and therefore, as shown in a numerical example, accelerate the online reduction process.