Balanced Truncation: Difference between revisions

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Revision as of 11:36, 25 March 2013

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

$(A, B, C, D) \Rightarrow (T A T^{- 1}, T B, C T^{- 1}, D)$

References

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 satisfy <math> P=Q=diag(\sigma_1,\dots,\sigma_n)</math> with <math> \sigma_1\geq\sigma_2\geq \dots\geq\sigma_n\geq0</math>
 The spectrum of <math> (PQ)^{\frac{1}{2}}</math> which is <math>\{\sigma_1,\dots,\sigma_n\}</math> are the Hankel singular values.
+In order to do balanced truncation one has to first compute a balanced realization via state-space transformation
+<math> (A,B,C,D)\Rightarrow (TAT^{-1},TB,CT^{-1},D)</math>
 ==References==