Balanced Truncation: Difference between revisions

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Revision as of 08:47, 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.

References

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-A stable system <math>/Sigma</math> , realized by (A,B,C,D) is called balanced, if the Gramians, i.e. the solutions P,Q  of the Lyapunov equations
+A stable system <math>\Sigma</math> , realized by (A,B,C,D) is called balanced, if the Gramians, i.e. the solutions P,Q  of the Lyapunov equations
 <math> AP+PA^T+BB^T=0,\quad A^TQ+QA+C^TC=0</math>
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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.
 ==References==