Balancing Transformation

Available identifiers

MaRDI QIDQ6674140

coordinate transformation T under which controllability and observability Gramians become equal and diagonal matrices comprising the Hankel singular values

Description

The transformation T is a contragredient transformation and exists whenever 𝑊𝑐 and 𝑊ₒ are symmetric and positive definite. Note that the squared HSVs are the eigenvalues of the product of 𝑊𝑐 and 𝑊ₒ.

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Defining Formula: $T^{-1}{W}_c{\left(T^{-1}\right)}^{*}=T^{*}{W}_{o}T=\left(\begin{array}{ccc}{\sigma }_1& & 0\\ & \ddots & \\ 0& & {\sigma }_n\end{array}\right)=\Sigma$

$T$ symbol represents MOR Transformation Matrix

$W_c$ symbol represents Controllability Gramian

$W_o$ symbol represents Observability Gramian

$\sigma$ symbol represents Hankel Singular Value

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