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 minimal (controllable and observable) 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 \( \sigma_1\geq\sigma_2\geq \dots\geq\sigma_n>0\) Since in general the spectrum of \( (PQ)^{\frac{1}{2}}\) are the Hankel singular values for such a balanced system they are given by\[\{\sigma_1,\dots,\sigma_n\}\]

Given an arbitrary system \((\tilde{A},\tilde{B},\tilde{C},\tilde{D})\) we transform into a balanced one via a state-space transformation

\( (A,B,C,D)= (T\tilde{A}T^{-1},T\tilde{B},\tilde{C}T^{-1},\tilde{D})=\left (\begin{bmatrix}A_{11} & A_{12}\\ A_{21} & A_{22}\end{bmatrix},\begin{bmatrix}B_1\\B_2\end{bmatrix}\begin{bmatrix} C_1 &C_2 \end{bmatrix},D\right)\) This transformed system has transformed Gramians \(P=T\tilde{P}T^T\) and \(Q=T^{-T}\tilde{Q}T^{-1}\) which are equal and diagonal. 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

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