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

$A P + P A^{T} + B B^{T} = 0, A^{T} Q + Q A + C^{T} C = 0$

satisfy $P = Q = d i a g (σ_{1}, \dots, σ_{n})$ with $σ_{1} \geq σ_{2} \geq \dots \geq σ_{n} \geq 0$

The spectrum of $(P Q)^{\frac{1}{2}}$ which is ${σ_{1}, \dots, σ_{n}}$ are the Hankel singular values.

In order to do balanced truncation one has to first compute a balanced realization via state-space transformation

$(A, B, C, D) \Rightarrow (T A T^{- 1}, T B, C T^{- 1}, D) = ([\begin{matrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{matrix}], [\begin{matrix} B_{1} \\ B_{2} \end{matrix}] [\begin{matrix} C_{1} & C_{2} \end{matrix}], D)$

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 $S R^{T}$

$S R^{T} = [\begin{matrix} U_{1} U_{2} \end{matrix}] [\begin{matrix} Σ_{1} \\ Σ_{2} \end{matrix}] [\begin{matrix} V_{1}^{T} \\ V_{2}^{T} \end{matrix}]$

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

$W = R^{T} V_{1} Σ_{1}^{- \frac{1}{2}}, V = S^{T} U_{1} Σ_{1}^{- \frac{1}{2}} .$

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 $σ_{1}, \dots, σ_{r}$ , where r is the order of the reduced system. It is possible to choose r via the computable error bound

$‖ y - \hat{y} ‖_{2} \leq (2 \sum_{k = r + 1}^{n} σ_{k}) ‖ u ‖_{2} .$

References