Balanced Truncation

An important projection model reduction method which delivers high quality reduced models by making an extra effort in choosing the projection subspaces.

A stable system $\mathrm{\Sigma }$ , realized by (A,B,C,D) is called balanced, if the Gramians, i.e. the solutions P,Q of the Lyapunov equations

$AP+P{A}^T+B{B}^T=0,A^{T}Q+QA+C^{T}C=0$

satisfy $P=Q=diag({\sigma }_1,\dots ,{\sigma }_n)$ with ${\sigma }_1\ge {\sigma }_2\ge \dots \ge {\sigma }_n\ge 0$

The spectrum of $(PQ{)}^{\frac{1}{2}}$ which is $\{{\sigma }_1,\dots ,{\sigma }_n\}$ are the Hankel singular values.

In order to do balanced truncation one has to first compute a balanced realization via state-space transformation

$(A,B,C,D)\Rightarrow (TA{T}^{-1},TB,C{T}^{-1},D)$