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