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 $\mathrm{\Sigma }$ , 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 ${\sigma }_1\ge {\sigma }_2\ge \dots \ge {\sigma }_n>0$ 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

$(A,B,C,D)=(T\tilde{A}{T}^{-1},T\tilde{B},\tilde{C}{T}^{-1},\tilde{D})=(\left[\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right],\left[\begin{array}{c}{B}_1\\ B_2\end{array}\right]\left[\begin{array}{cc}{C}_1& C_2\end{array}\right],D)$ 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 $P=S^{T}S,Q=R^{T}R$. Then we compute the singular value decomposition of $S{R}^T$

$S{R}^T=\left[\begin{array}{c}{U}_1{U}_2\end{array}\right]\left[\begin{array}{cc}{\mathrm{\Sigma }}_1& \\ & {\mathrm{\Sigma }}_2\end{array}\right]\left[\begin{array}{c}{V}_1^T\\ V_2^T\end{array}\right]$

Then the reduced order model is given by $(W^{T}AV,W^{T}B,CV,D)$ where

$W=R^T{V}_1{\mathrm{\Sigma }}_1^{-\frac{1}{2}},V=S^T{U}_1{\mathrm{\Sigma }}_1^{-\frac{1}{2}}.$

We get then that $V^{T}W=I_r$ which makes $V{W}^T$ 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

$\Vert y-\hat{y}{\Vert }_2\le (2\sum _{k=r+1}^n{\sigma }_k)\Vert u{\Vert }_2.$