Hankel-Norm Approximation - Revision history

Werner at 09:04, 3 May 2018

2018-05-03T09:04:08Z

← Older revision		Revision as of 09:04, 3 May 2018
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	<math>\tilde{G}(s) = \tilde{C}(sI_{n-k} - \tilde{A})^{-1}\tilde{B} + \tilde{D} = G_{r}(s) + F(s),</math>		<math>\tilde{G}(s) = \tilde{C}(sI_{n-k} - \tilde{A})^{-1}\tilde{B} + \tilde{D} = G_{r}(s) + F(s),</math>
	where <math>F</math> is anti-stable and <math>G_{r}</math> is the <math>r</math>-th order stable Hankel-norm approximation.		where <math>F</math> is anti-stable and <math>G_{r}</math> is the <math>r</math>-th order stable Hankel-norm approximation.


	== References ==		== References ==

Werner at 09:03, 3 May 2018

2018-05-03T09:03:56Z

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	The '''Hankel-norm approximation''' method is a model reduction approach that solves the best-approximation problem in the Hankel semi-norm<ref name="morGlo84"></ref>.		The '''Hankel-norm approximation''' method is a model reduction approach that solves the best-approximation problem in the Hankel semi-norm<ref name="morGlo84"></ref>.


	== Description ==		== Description ==
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	For such error systems, the Hankel semi-norm is known to be <math>\lVert \mathcal{E} \rVert_{H} = \varsigma_{r + 1}.</math>		For such error systems, the Hankel semi-norm is known to be <math>\lVert \mathcal{E} \rVert_{H} = \varsigma_{r + 1}.</math>


	== Algorithm ==		== Algorithm ==

Werner: Added curly bracket with system name.

2018-01-05T15:37:31Z

Added curly bracket with system name.

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	Consider the standard linear-time invariant system		Consider the standard linear-time invariant system

	::<math>\begin{align} \dot{x}(t) & = Ax(t) + Bu(t),\\ y(t) & = Cx(t) + Du(t), \end{align}</math>		::<math>G:\left\{ \begin{align} \dot{x}(t) & = Ax(t) + Bu(t),\\ y(t) & = Cx(t) + Du(t), \end{align} \right.</math>

	with the matrices <math style="vertical-align: top;">A \in \mathbb{R}^{n \times n}</math>, <math style="vertical-align: top;">B \in \mathbb{R}^{n \times m}</math>, <math style="vertical-align: top;">C \in \mathbb{R}^{p \times n}</math> and <math style="vertical-align: top;">D \in \mathbb{R}^{p \times m}</math>.		with the matrices <math style="vertical-align: top;">A \in \mathbb{R}^{n \times n}</math>, <math style="vertical-align: top;">B \in \mathbb{R}^{n \times m}</math>, <math style="vertical-align: top;">C \in \mathbb{R}^{p \times n}</math> and <math style="vertical-align: top;">D \in \mathbb{R}^{p \times m}</math>.

Saak at 14:56, 5 January 2018

2018-01-05T14:56:21Z

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	~~{{preliminary}} <!-- Do not remove -->~~

	[[Category:method]]		[[Category:method]]
	[[Category:linear]]		[[Category:linear]]

Werner: Initial page for Hankel-norm approximation of standard LTI systems.

2017-12-21T15:54:07Z

Initial page for Hankel-norm approximation of standard LTI systems.

New page

{{preliminary}} 

[[Category:method]]
[[Category:linear]]
[[Category:time invariant]]
[[Category:first differential order]]
[[Category:linear algebra]]

The '''Hankel-norm approximation''' method is a model reduction approach that solves the best-approximation problem in the Hankel semi-norm<ref name="morGlo84"></ref>.

== Description ==
Consider the standard linear-time invariant system

::<math>\begin{align} \dot{x}(t) & = Ax(t) + Bu(t),\\ y(t) & = Cx(t) + Du(t), \end{align}</math>

with the matrices <math style="vertical-align: top;">A \in \mathbb{R}^{n \times n}</math>, <math style="vertical-align: top;">B \in \mathbb{R}^{n \times m}</math>, <math style="vertical-align: top;">C \in \mathbb{R}^{p \times n}</math> and <math style="vertical-align: top;">D \in \mathbb{R}^{p \times m}</math>.
For a system <math style="vertical-align: top;">G</math>, the Hankel operator <math style="vertical-align: top;">\mathcal{H}</math> maps past inputs <math>u_{-}</math> to future outputs <math>y_{+}</math> of the system, i.e., <math>y_{+} = \mathcal{H}u_{-}</math>.
Then, the Hankel semi-norm of the system <math>G</math> is defined as the <math>\mathcal{L}_{2}</math>-induced norm of the Hankel opertor

::<math>\lVert G \rVert_{H} := \sup\limits_{u_{-} \in \mathcal{L}_{2}\left(-\infty, 0\right]}\frac{\lVert y_{+} \rVert_{\mathcal{L}_{2}}}{\lVert u_{-} \rVert_{\mathcal{L}_{2}}}.</math>

If the system <math>G</math> is stable, the controllability and observability Gramians <math>\mathcal{G}_{c}</math> and <math>\mathcal{G}_{o}</math> of the system above are given as the unique positive semidefinite solutions of the two Lyapunov equations

::<math>\begin{align} A\mathcal{G}_{c} + \mathcal{G}_{c}A^{T} + BB^{T} & = 0,\\ A^{T}\mathcal{G}_{o} + \mathcal{G}_{o}A + C^{T}C & = 0. \end{align}</math>

The Hankel singular values of the system <math>G</math> are then defined as the square-roots of the eigenvalues of the multiplied system Gramians, i.e., <math>\sqrt{\Lambda(\mathcal{G}_{c}\mathcal{G}_{o})} = \{ \varsigma_{1}, \ldots, \varsigma_{n} \}</math>.
It can be shown, that the Hankel semi-norm of a system is given by the largest Hankel singular value <math>\lVert G \rVert_{H} = \varsigma_{\text{max}}</math>.

The idea of the Hankel-norm approximation method is, to construct a reduced-order model <math>G_{r}</math> of order <math>r</math> such that the error system <math>\mathcal{E} = G - G_{r}</math> has a scaled all-pass transfer function

::<math>\mathcal{E}(s)\mathcal{E}^{T}(-s) = \varsigma_{r + 1}^{2} I_{p},</math>

with <math>\varsigma_{r + 1}</math> the <math>(r + 1)</math>-st Hankel singular value of the system <math>G</math>.

For such error systems, the Hankel semi-norm is known to be <math>\lVert \mathcal{E} \rVert_{H} = \varsigma_{r + 1}.</math>

== Algorithm ==
Here, the algorithm of the Hankel-norm approximation method is shortly described <ref name="morBenQQ04"></ref>:

1. Compute a minimal balanced realization <math>(\check{A}, \check{B}, \check{C}, D)</math> using the [[Balanced Truncation#Balancing and Truncation|balanced truncation square-root method]].
2. Choose the Hankel singular value <math>\varsigma_{r + 1}</math>.
3. Permute the balanced realization such that the Gramians have the form
<math>\begin{align}\check{\mathcal{G}}_{c} = \check{\mathcal{G}}_{o} & = \mathrm{diag}(\varsigma_{1}, \ldots, \varsigma_{r}, \varsigma_{r + k + 1}, \ldots, \varsigma_{n}, \varsigma_{r + 1}I_{k})\\ & = \mathrm{diag}(\Sigma, \varsigma_{r + 1}I_{k}).\end{align}</math>
4. Partition the resulting permuted system according to the Gramians
<math>\check{A} = \begin{bmatrix} A_{11} & A_{12}\\ A_{21} & A_{22}\end{bmatrix}, ~~~ \check{B} = \begin{bmatrix} B_{1}\\ B_{2}\end{bmatrix}, \check{C} = \begin{bmatrix} C_{1} & C_{2}\end{bmatrix},</math>
where <math>A_{22} \in \mathbb{R}^{k \times k}</math>, <math>B_{2} \in \mathbb{R}^{k \times m}</math> and <math>C_{2} \in \mathbb{R}^{p \times k}</math>.
5. Compute the transformation
<math>\begin{align}\tilde{A} & = \Gamma^{-1}(\varsigma_{r+1}^{2}A_{11}^{T} + \Sigma A_{11} \Sigma + \varsigma_{r+1}C_{1}^{T}UB_{1}^{T}),\\ \tilde{B} & = \Gamma^{-1}(\Sigma B_{1} - \varsigma_{r+1}C_{1}^{T}U),\\ \tilde{C} & = C_{1}\Sigma - \varsigma_{r+1}UB_{1}^{T},\\ \tilde{D} & = D + \varsigma_{r+1}U,\end{align}</math>
with <math>U = \left(C_{2}^{T}\right)^{\dagger}B_{2}</math> and <math>\Gamma = \Sigma^{2} - \varsigma_{r+1}^{2}I_{n-k}</math>.
6. Compute the additive decomposition
<math>\tilde{G}(s) = \tilde{C}(sI_{n-k} - \tilde{A})^{-1}\tilde{B} + \tilde{D} = G_{r}(s) + F(s),</math>
where <math>F</math> is anti-stable and <math>G_{r}</math> is the <math>r</math>-th order stable Hankel-norm approximation.

== References ==

<references>
<ref name="morBenQQ04">P. Benner, E. S. Quintana-Ortí, and G. Quintana-Ortí. Computing optimal Hankel norm approximations of large-scale systems. In 2004 43rd IEEE Conference on Decision and Control (CDC), volume 3, pages 3078-3083, Atlantis, Paradise Island, Bahamas, December 2004. Institute of Electrical and Electronics Engineers.</ref>
<ref name="morGlo84">K. Glover. All optimal Hankel-norm approximations of linear multivariable systems and their <math style="vertical-align: top;">L^{\infty}</math>-error norms. Internat. J. Control, 39(6):1115-1193, 1984.</ref>
</references>