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

$AP+PA^T+BB^T=0,\quad A^TQ+QA+C^TC=0$

satisfy $P=Q=diag(\sigma_1,\dots,\sigma_n)$ with $\sigma_1\geq\sigma_2\geq \dots\geq\sigma_n\geq0$

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

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