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

$(A,B,C,D)\Rightarrow (TAT^{-1},TB,CT^{-1},D)=\left (\begin{bmatrix}A_{11} & A_{12}\\ A_{21} & A_{22}\end{bmatrix},\begin{bmatrix}B_1\\B_2\end{bmatrix}\begin{bmatrix} C_1 &C_2 \end{bmatrix},D\right)$

The truncated reduced system is then given by

$(\hat{A},\hat{B},\hat{C},\hat{D})=(A_{11},B_1,C_1,D)$

Implementation: SR Method

One computes it for example by the SR Method. First one computes the (Cholesky) factors of the gramians $P=S^TS,\; Q=R^TR$. Then we compute the singular value decomposition of $SR^T\;$

$SR^T=\begin{bmatrix} U_1 U_2 \end{bmatrix} \begin{bmatrix} \Sigma_1 & \\ & \Sigma_2\end{bmatrix} \begin{bmatrix} V_1^T\\V_2^T\end{bmatrix}$

Then the reduced order model is given by $(W^TAV,W^TB,CV,D)\;$ where

$W=R^T V_1\Sigma_1^{-\frac{1}{2}},\quad V= S^T U_1 \Sigma_1^{-\frac{1}{2}}.$

We get then that $V^TW=I_r$ which makes $VW^T$ an oblique projector and hence balanced trunctation a Petrov-Galerkin projection method. The reduced model is stable with Hankel Singular Values given by $\sigma_1,\dots,\sigma_r$, where r is the order of the reduced system. It is possible to choose r via the computable error bound

$\|y-\hat{y}\|_2\leq (2\sum_{k=r+1}^n\sigma_k)\|u\|_2.$