Balanced Truncation - Revision history

Kuerschner: Don't know how to cite the same paper again without increasing the counter.

2018-01-17T16:01:52Z

Don't know how to cite the same paper again without increasing the counter.

← Older revision		Revision as of 16:01, 17 January 2018
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	== Time- and frequency-limited BT ==		== Time- and frequency-limited BT ==

	'''Time- and frequency-limited BT''' <ref>Gawronski, Juang "<span class="plainlinks">[http://dx.doi.org/10.1080/00207729008910366 Model reduction in limited time and frequency intervals]</span>", Int. J. Syst. Sci. 21(2), pp.349–376, 1990;</ref> ~~is a modification~~ of BT targeted ~~to achieve~~ a high approximation quality within finite time <math>[0,T], T<\infty</math> or frequency regions <math>[\omega_1,\omega_2], 0\leq\omega_1<\omega_2<\infty</math>, while allowing large approximation errors outside these regions. Starting point of the derivation are the integral expressions of the Gramians		'''Time- and frequency-limited BT''' <ref>Gawronski, Juang "<span class="plainlinks">[http://dx.doi.org/10.1080/00207729008910366 Model reduction in limited time and frequency intervals]</span>", Int. J. Syst. Sci. 21(2), pp.349–376, 1990;</ref> are modifications of BT targeted at achieving a high approximation quality within finite time <math>[0,T], T<\infty</math> or frequency regions <math>[\omega_1,\omega_2], 0\leq\omega_1<\omega_2<\infty</math>, while allowing large approximation errors outside these regions. Starting point of the derivation are the integral expressions of the Gramians

	:<math> W_{C}=\int\limits_0^{\infty}\mathrm{e}^{At}BB^T \mathrm{e}^{A^Tt}\mathrm{d}t=\frac{1}{2\pi}\int\limits_{-\infty}^{\infty}(i\omega I-A)^{-1}BB^T (i\omega I-A)^{-H}\mathrm{d}\omega,\quad W_O=\int\limits_0^{\infty}\mathrm{e}^{A^Tt}C^TC \mathrm{e}^{At}\mathrm{d}t=\frac{1}{2\pi}\int\limits_{-\infty}^{\infty}(i\omega I-A)^{-H}C^TC (i\omega I-A)^{-1}\mathrm{d}\omega.</math>		:<math> W_{C}=\int\limits_0^{\infty}\mathrm{e}^{At}BB^T \mathrm{e}^{A^Tt}\mathrm{d}t=\frac{1}{2\pi}\int\limits_{-\infty}^{\infty}(i\omega I-A)^{-1}BB^T (i\omega I-A)^{-H}\mathrm{d}\omega,\quad W_O=\int\limits_0^{\infty}\mathrm{e}^{A^Tt}C^TC \mathrm{e}^{At}\mathrm{d}t=\frac{1}{2\pi}\int\limits_{-\infty}^{\infty}(i\omega I-A)^{-H}C^TC (i\omega I-A)^{-1}\mathrm{d}\omega.</math>

Kuerschner: /* Time- and frequency-limited BT */

2018-01-17T15:59:38Z

Time- and frequency-limited BT

← Older revision		Revision as of 15:59, 17 January 2018
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	In general, neither time- nor frequency-limited BT guarantee that the stability of the original system is preserved and, hence, do also not provide an <math>H_{\infty}</math> error bound similar to the one of unrestricted BT.		In general, neither time- nor frequency-limited BT guarantee that the stability of the original system is preserved and, hence, do also not provide an <math>H_{\infty}</math> error bound similar to the one of unrestricted BT.
	Modified time- or frequency-limited BT <ref>Antoulas, A. C., Gugercin, S."<span class="plainlinks">[http://dx.doi.org/10.1080/00207170410001713448 A survey of model reduction by balanced truncation and some new results]</span>". Int. J. Control 77(8), pp. 748–766, 2004.</ref> fixes this, but has been found to be inferior compared to unrestricted BT and (unmodified) time-/frequency-limited BT in terms of computational efficiency. The achieved accuracy is also lower compared to time-/frequency-limited BT.		Modified time- or frequency-limited BT <ref>Antoulas, A. C., Gugercin, S."<span class="plainlinks">[http://dx.doi.org/10.1080/00207170410001713448 A survey of model reduction by balanced truncation and some new results]</span>". Int. J. Control 77(8), pp. 748–766, 2004.</ref> fixes this, but has been found to be inferior compared to unrestricted BT and (unmodified) time-/frequency-limited BT in terms of computational efficiency. The achieved accuracy is also lower compared to time-/frequency-limited BT.
	In <ref>Redmann, M., Kürschner, P. "<span class="plainlinks">[https://arxiv.org/abs/1710.07572 An H2-Type Error Bound for Time-Limited Balanced Truncation]</span>". arXiv e-print no. 1710.07572, 2017.</ref> H2-type error bounds for '''Time-limited BT''' are derived which do not rely on stability preservation. Moreover, some experiments in <ref></ref> indicate that '''Time-limited BT''' might be a potential candidate for reducing unstable systems.		In <ref>Redmann, M., Kürschner, P. "<span class="plainlinks">[https://arxiv.org/abs/1710.07572 An H2-Type Error Bound for Time-Limited Balanced Truncation]</span>". arXiv e-print no. 1710.07572, 2017.</ref> H2-type error bounds for '''Time-limited BT''' are derived which do not rely on stability preservation. Moreover, some experiments in <ref>Kürschner, P. "<span class="plainlinks">[https://arxiv.org/abs/1707.02839 Balanced truncation model order reduction in limited time intervals for large systems]</span>". arXiv e-print no. 1707.02839, 2017.</ref> indicate that '''Time-limited BT''' might be a potential candidate for reducing unstable systems.

	== Cross Gramian MOR ==		== Cross Gramian MOR ==

Kuerschner: /* Time- and frequency-limited BT */

2018-01-17T15:57:28Z

Time- and frequency-limited BT

← Older revision		Revision as of 15:57, 17 January 2018
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	:<math> AW_{C,T}+W_{C,T}A^T=f(A)BB^Tf(A)^T-BB^T,\quad A^TW_{O,T}+W_{O,T}A=f(A)^TC^TCf(A)-C^TC\quad\text{with}\quad f(A)= \mathrm{e}^{AT}.</math>		:<math> AW_{C,T}+W_{C,T}A^T=f(A)BB^Tf(A)^T-BB^T,\quad A^TW_{O,T}+W_{O,T}A=f(A)^TC^TCf(A)-C^TC\quad\text{with}\quad f(A)= \mathrm{e}^{AT}.</math>

	~~'''Time-limited BT''' is then obtained by using Cholesky- or low-rank factors of <math>W_{C,T},W_{0,T}</math> instead of <math>W_C,W_0</math>.~~

	Likewise, restricting the frequency-domain integral expressions of <math>W_C,W_0</math> to <math>\Omega:=[-\omega_2,-\omega_2]\cup[\omega_1,\omega_2]</math> leads to the frequency-limited Lyapunov equations		Likewise, restricting the frequency-domain integral expressions of <math>W_C,W_0</math> to <math>\Omega:=[-\omega_2,-\omega_2]\cup[\omega_1,\omega_2]</math> leads to the frequency-limited Lyapunov equations
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	:<math> AW_{C,\Omega}+W_{C,\Omega}A^T=-f(A)BB^T-BB^Tf(A)^T,\quad A^TW_{O,\Omega}+W_{O,\Omega}A=-f(A)^TC^TC-C^TCf(A)\quad\text{with}\quad f(A)= \frac{1}{\pi}\mathrm{Re}\left(i\log\left((A+i\omega_1I)^{-1}(A+i\omega_2I)\right)\right),</math>		:<math> AW_{C,\Omega}+W_{C,\Omega}A^T=-f(A)BB^T-BB^Tf(A)^T,\quad A^TW_{O,\Omega}+W_{O,\Omega}A=-f(A)^TC^TC-C^TCf(A)\quad\text{with}\quad f(A)= \frac{1}{\pi}\mathrm{Re}\left(i\log\left((A+i\omega_1I)^{-1}(A+i\omega_2I)\right)\right),</math>

	see, e.g., <ref>Petersson, D."<span class="plainlinks">[http://www.diva-portal.org/smash/get/diva2:647068/FULLTEXT01.pdf A Nonlinear Optimization Approach to H2-Optimal Modeling and Control]</span>", PhD thesis, Linköping University, 2013.</ref>. '''Frequency-limited BT''' is then ~~carried out~~ by using Cholesky- or low-rank factors of <math>W_{C,\Omega},W_{0,\Omega}</math> instead of <math>W_C,W_0</math>.		see, e.g., <ref>Petersson, D."<span class="plainlinks">[http://www.diva-portal.org/smash/get/diva2:647068/FULLTEXT01.pdf A Nonlinear Optimization Approach to H2-Optimal Modeling and Control]</span>", PhD thesis, Linköping University, 2013.</ref>. '''Time-limited''' and '''Frequency-limited BT''' are then obtained by using Cholesky- or low-rank factors of the restricted Gramians <math>W_{C,T},W_{O,T}</math> and, respectively,<math>W_{C,\Omega},W_{O,\Omega}</math> instead of the infinite Gramians <math>W_C,W_0</math>.

			Computational strategies for large-scale systems for dealing with the matrix functions <math>f(A)</math> and for computing the required low-rank factors of Gramians <math>W_{C,\Omega},W_{O,\Omega}</math> and <math>W_{C,T},W_{O,T}</math> are proposed in <ref>Benner, P., Kürschner, P., Saak, J. "<span class="plainlinks">[http://dx.doi.org/10.1137/15M1030911 Frequency-Limited Balanced Truncation with Low-Rank Approximations]</span>". SIAM J. Sci. Comp. 38(1), pp. A471-–A499, 2016.</ref>, <ref>Kürschner, P. "<span class="plainlinks">[https://arxiv.org/abs/1707.02839 Balanced truncation model order reduction in limited time intervals for large systems]</span>". arXiv e-print no. 1707.02839, 2017.</ref>.

	A computational framework for large-scale systems for dealing with the matrix functions <math>f(A)</math> and computing the required low-rank factors of <math>W_{C,\Omega},W_{0,\Omega}</math> is proposed in <ref>Benner, P., Kürschner, P., Saak, J. "<span class="plainlinks">[http://dx.doi.org/10.1137/15M1030911 Frequency-Limited Balanced Truncation with Low-Rank Approximations]</span>". SIAM J. Sci. Comp. 38(1), pp. A471-–A499, 2016.</ref>. The handling of generalized state-space systems can be adopted right away from the unrestricted BT case.		The handling of generalized state-space systems can be adopted right away from the unrestricted BT case.

	In general, neither time- nor frequency-limited BT guarantee that the stability of the original system is preserved and, hence, do also not provide an error bound similar to the one of unrestricted BT.		In general, neither time- nor frequency-limited BT guarantee that the stability of the original system is preserved and, hence, do also not provide an <math>H_{\infty}</math> error bound similar to the one of unrestricted BT.
	Modified time- or frequency-limited BT <ref>Antoulas, A. C., Gugercin, S."<span class="plainlinks">[http://dx.doi.org/10.1080/00207170410001713448 A survey of model reduction by balanced truncation and some new results]</span>". Int. J. Control 77(8), pp. 748–766, 2004.</ref> fixes this, but has been found to be inferior compared to unrestricted BT and (unmodified) time-/frequency-limited BT in terms of computational efficiency. The achieved accuracy is also lower compared to time-/frequency-limited BT.		Modified time- or frequency-limited BT <ref>Antoulas, A. C., Gugercin, S."<span class="plainlinks">[http://dx.doi.org/10.1080/00207170410001713448 A survey of model reduction by balanced truncation and some new results]</span>". Int. J. Control 77(8), pp. 748–766, 2004.</ref> fixes this, but has been found to be inferior compared to unrestricted BT and (unmodified) time-/frequency-limited BT in terms of computational efficiency. The achieved accuracy is also lower compared to time-/frequency-limited BT.
			In <ref>Redmann, M., Kürschner, P. "<span class="plainlinks">[https://arxiv.org/abs/1710.07572 An H2-Type Error Bound for Time-Limited Balanced Truncation]</span>". arXiv e-print no. 1710.07572, 2017.</ref> H2-type error bounds for '''Time-limited BT''' are derived which do not rely on stability preservation. Moreover, some experiments in <ref></ref> indicate that '''Time-limited BT''' might be a potential candidate for reducing unstable systems.

	== Cross Gramian MOR ==		== Cross Gramian MOR ==

Kuerschner at 09:26, 17 January 2017

2017-01-17T09:26:39Z

← Older revision		Revision as of 09:26, 17 January 2017
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	:<math> \\|\Sigma-\hat{\Sigma}\\|_2 \leq 2 \\|u\\|_2 \sum_{k=r+1}^n\sigma_k. </math>		:<math> \\|\Sigma-\hat{\Sigma}\\|_2 \leq 2 \\|u\\|_2 \sum_{k=r+1}^n\sigma_k. </math>

			== Time- and frequency-limited BT ==

			'''Time- and frequency-limited BT''' <ref>Gawronski, Juang "<span class="plainlinks">[http://dx.doi.org/10.1080/00207729008910366 Model reduction in limited time and frequency intervals]</span>", Int. J. Syst. Sci. 21(2), pp.349–376, 1990;</ref> is a modification of BT targeted to achieve a high approximation quality within finite time <math>[0,T], T<\infty</math> or frequency regions <math>[\omega_1,\omega_2], 0\leq\omega_1<\omega_2<\infty</math>, while allowing large approximation errors outside these regions. Starting point of the derivation are the integral expressions of the Gramians

			:<math> W_{C}=\int\limits_0^{\infty}\mathrm{e}^{At}BB^T \mathrm{e}^{A^Tt}\mathrm{d}t=\frac{1}{2\pi}\int\limits_{-\infty}^{\infty}(i\omega I-A)^{-1}BB^T (i\omega I-A)^{-H}\mathrm{d}\omega,\quad W_O=\int\limits_0^{\infty}\mathrm{e}^{A^Tt}C^TC \mathrm{e}^{At}\mathrm{d}t=\frac{1}{2\pi}\int\limits_{-\infty}^{\infty}(i\omega I-A)^{-H}C^TC (i\omega I-A)^{-1}\mathrm{d}\omega.</math>

			Restricting the integration domain of the time-domain integrals to <math>[0,T]</math> leads to the time-limited Gramians

			:<math> W_{C,T}=\int\limits_0^{T}\mathrm{e}^{At}BB^T \mathrm{e}^{A^Tt}\mathrm{d}t,\quad W_{O,T}=\int\limits_0^{T}\mathrm{e}^{A^Tt}C^TC \mathrm{e}^{At}\mathrm{d}t.</math>

			which are the solutions of the time-limited Lyapunov equations

			:<math> AW_{C,T}+W_{C,T}A^T=f(A)BB^Tf(A)^T-BB^T,\quad A^TW_{O,T}+W_{O,T}A=f(A)^TC^TCf(A)-C^TC\quad\text{with}\quad f(A)= \mathrm{e}^{AT}.</math>

			'''Time-limited BT''' is then obtained by using Cholesky- or low-rank factors of <math>W_{C,T},W_{0,T}</math> instead of <math>W_C,W_0</math>.

			Likewise, restricting the frequency-domain integral expressions of <math>W_C,W_0</math> to <math>\Omega:=[-\omega_2,-\omega_2]\cup[\omega_1,\omega_2]</math> leads to the frequency-limited Lyapunov equations

			:<math> AW_{C,\Omega}+W_{C,\Omega}A^T=-f(A)BB^T-BB^Tf(A)^T,\quad A^TW_{O,\Omega}+W_{O,\Omega}A=-f(A)^TC^TC-C^TCf(A)\quad\text{with}\quad f(A)= \frac{1}{\pi}\mathrm{Re}\left(i\log\left((A+i\omega_1I)^{-1}(A+i\omega_2I)\right)\right),</math>

			see, e.g., <ref>Petersson, D."<span class="plainlinks">[http://www.diva-portal.org/smash/get/diva2:647068/FULLTEXT01.pdf A Nonlinear Optimization Approach to H2-Optimal Modeling and Control]</span>", PhD thesis, Linköping University, 2013.</ref>. '''Frequency-limited BT''' is then carried out by using Cholesky- or low-rank factors of <math>W_{C,\Omega},W_{0,\Omega}</math> instead of <math>W_C,W_0</math>.

			A computational framework for large-scale systems for dealing with the matrix functions <math>f(A)</math> and computing the required low-rank factors of <math>W_{C,\Omega},W_{0,\Omega}</math> is proposed in <ref>Benner, P., Kürschner, P., Saak, J. "<span class="plainlinks">[http://dx.doi.org/10.1137/15M1030911 Frequency-Limited Balanced Truncation with Low-Rank Approximations]</span>". SIAM J. Sci. Comp. 38(1), pp. A471-–A499, 2016.</ref>. The handling of generalized state-space systems can be adopted right away from the unrestricted BT case.

			In general, neither time- nor frequency-limited BT guarantee that the stability of the original system is preserved and, hence, do also not provide an error bound similar to the one of unrestricted BT.
			Modified time- or frequency-limited BT <ref>Antoulas, A. C., Gugercin, S."<span class="plainlinks">[http://dx.doi.org/10.1080/00207170410001713448 A survey of model reduction by balanced truncation and some new results]</span>". Int. J. Control 77(8), pp. 748–766, 2004.</ref> fixes this, but has been found to be inferior compared to unrestricted BT and (unmodified) time-/frequency-limited BT in terms of computational efficiency. The achieved accuracy is also lower compared to time-/frequency-limited BT.

	== Cross Gramian MOR ==		== Cross Gramian MOR ==

Himpe: control and observe refs

2014-08-02T16:36:04Z

control and observe refs

← Older revision		Revision as of 16:36, 2 August 2014
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	[[Category:linear algebra]]		[[Category:linear algebra]]

	'''Balanced Truncation''' is an important [[Projection based MOR\|projection]] method which delivers high quality reduced models by making an extra effort in choosing the projection subspaces.		'''Balanced Truncation''' is an important [[Projection based MOR\|projection]] method which delivers high quality reduced models by making an extra effort in choosing the projection subspaces based on the [[wikipedia:Controllability\|Controllability]] and [[wikipedia:Observability\|Observability]] of the underlying control system.




	==Derivation ==		==Derivation ==
	A stable minimal (controllable and observable) system <math>\Sigma</math>, realized by <math>(A,B,C)</math>		A stable minimal (controllable and observable) linear system <math>\Sigma</math>, realized by <math>(A,B,C)</math>

	:<math> \dot{x} = Ax + Bu</math>		:<math> \dot{x} = Ax + Bu</math>

Himpe: /* Cross Gramian MOR */

2013-06-13T13:00:55Z

Cross Gramian MOR

← Older revision		Revision as of 13:00, 13 June 2013
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	:<math>W_X = \begin{bmatrix} U_1 U_2 \end{bmatrix} \begin{bmatrix} \Sigma_1 & \\ & \Sigma_2\end{bmatrix} \begin{bmatrix} V_1^T\\V_2^T\end{bmatrix}.</math>		:<math>W_X = \begin{bmatrix} U_1 U_2 \end{bmatrix} \begin{bmatrix} \Sigma_1 & \\ & \Sigma_2\end{bmatrix} \begin{bmatrix} V_1^T\\V_2^T\end{bmatrix}.</math>

	and a subsequent truncation of the discardable states, to which the above error bound also applies. Note that, although the similarities to the standard balanced truncation approach, the reduced order model obtained with this method is not balanced.		and a subsequent truncation of the discardable states, to which the above error bound also applies. Note that, although the similarities to the standard balanced truncation approach, the reduced order model obtained with this method is not balanced for non-symmetric systems.

	==References==		==References==

	<references/>		<references/>

Kuerschner at 10:16, 3 June 2013

2013-06-03T10:16:35Z

← Older revision		Revision as of 10:16, 3 June 2013
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	:<math>W_X = \begin{bmatrix} U_1 U_2 \end{bmatrix} \begin{bmatrix} \Sigma_1 & \\ & \Sigma_2\end{bmatrix} \begin{bmatrix} V_1^T\\V_2^T\end{bmatrix}.</math>		:<math>W_X = \begin{bmatrix} U_1 U_2 \end{bmatrix} \begin{bmatrix} \Sigma_1 & \\ & \Sigma_2\end{bmatrix} \begin{bmatrix} V_1^T\\V_2^T\end{bmatrix}.</math>

	and a subsequent truncation of the discardable states, to which the above error bound also applies. Note although the ~~similarity~~ to the standard balanced truncation approach, the reduced order model obtained with this method is not balanced.		and a subsequent truncation of the discardable states, to which the above error bound also applies. Note that, although the similarities to the standard balanced truncation approach, the reduced order model obtained with this method is not balanced.

	==References==		==References==

	<references/>		<references/>

Kuerschner at 10:16, 3 June 2013

2013-06-03T10:16:04Z

← Older revision		Revision as of 10:16, 3 June 2013
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	== Cross Gramian MOR ==		== Cross Gramian MOR ==

	A related Gramian-based approach is '''Cross Gramian ~~Balanced Truncation~~'''<ref>Antoulas, A. C. "<span class="plainlinks">[http://dx.doi.org/10.1137/1.9780898718713 Approximation of large-scale dynamical systems]</span>". Vol. 6. Society for Industrial and Applied Mathematics, pp.376--377, 2009; ISBN 978-0-89871-529-3</ref>,<ref>D.C. Sorensen and A.C. Antoulas "<span class="plainlinks">[http://10.1016/S0024-3795(02)00283-5 The Sylvester equation and approximate balanced reduction]</span>", Linear Algebra and its Applications, vol. 351-352(0), pp. 671-700, 2002,		A related Gramian-based approach is '''Cross Gramian Model Order Reduction'''<ref>Antoulas, A. C. "<span class="plainlinks">[http://dx.doi.org/10.1137/1.9780898718713 Approximation of large-scale dynamical systems]</span>". Vol. 6. Society for Industrial and Applied Mathematics, pp.376--377, 2009; ISBN 978-0-89871-529-3</ref>,<ref>D.C. Sorensen and A.C. Antoulas "<span class="plainlinks">[http://10.1016/S0024-3795(02)00283-5 The Sylvester equation and approximate balanced reduction]</span>", Linear Algebra and its Applications, vol. 351-352(0), pp. 671-700, 2002,
	</ref>.		</ref>.
	Given a stable and symmetric system <math>(A,B,C,D)</math>, such that there exists a transformation <math>J</math>		Given a stable and symmetric system <math>(A,B,C,D)</math>, such that there exists a transformation <math>J</math>
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	:<math>W_X = \begin{bmatrix} U_1 U_2 \end{bmatrix} \begin{bmatrix} \Sigma_1 & \\ & \Sigma_2\end{bmatrix} \begin{bmatrix} V_1^T\\V_2^T\end{bmatrix}.</math>		:<math>W_X = \begin{bmatrix} U_1 U_2 \end{bmatrix} \begin{bmatrix} \Sigma_1 & \\ & \Sigma_2\end{bmatrix} \begin{bmatrix} V_1^T\\V_2^T\end{bmatrix}.</math>

	and a subsequent truncation of the discardable states, to which the above error bound also applies.		and a subsequent truncation of the discardable states, to which the above error bound also applies. Note although the similarity to the standard balanced truncation approach, the reduced order model obtained with this method is not balanced.

	==References==		==References==

	<references/>		<references/>

Himpe: /* Implementation: SR Method */ svd for gramian decomposiition

2013-05-19T20:29:47Z

Implementation: SR Method: svd for gramian decomposiition

← Older revision		Revision as of 20:29, 19 May 2013
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	By truncating the discardable states, the truncated reduced system is then given by <math> \hat{\Sigma}=(\tilde{E}_{11},\tilde{A}_{11},\tilde{B}_1,\tilde{C}_1,\tilde{D}) </math>.		By truncating the discardable states, the truncated reduced system is then given by <math> \hat{\Sigma}=(\tilde{E}_{11},\tilde{A}_{11},\tilde{B}_1,\tilde{C}_1,\tilde{D}) </math>.

	== ~~Implementation: SR Method~~==		== Balancing and Truncation ==

	The necessary balancing transformation can be computed by the ~~SR Method~~<ref>A.J. Laub; M.T. Heath; C. Paige; R. Ward, "<span class="plainlinks">[http://dx.doi.org/10.1109/TAC.1987.1104549 Computation of system balancing transformations and other applications of simultaneous diagonalization algorithms]</span>," IEEE Transactions on Automatic Control, vol.32, no.2, pp.115,122, Feb 1987</ref>.		The necessary balancing transformation can be computed by the Square-Root method<ref>A.J. Laub; M.T. Heath; C. Paige; R. Ward, "<span class="plainlinks">[http://dx.doi.org/10.1109/TAC.1987.1104549 Computation of system balancing transformations and other applications of simultaneous diagonalization algorithms]</span>," IEEE Transactions on Automatic Control, vol.32, no.2, pp.115,122, Feb 1987</ref>.
	First, the Cholesky factors of the Gramians <math>W_C=S^TS,\; W_O=R^TR</math> are computed.		First, the [[wikipedia:Cholesky_decomposition\|Cholesky factors]] of the Gramians <math>W_C=S^TS,\; W_O=R^TR</math> are computed.
	~~Next~~, the [[wikipedia:Singular_Value_Decomposition\|Singular Value Decomposition]] of <math> SR^T\;</math> is computed:		Alternatively to the Cholesky factorization, the [[wikipedia:Singular_Value_Decomposition\|Singular Value Decomposition]] can be employed: <math>W_O = U_O \Sigma_O U_O^T \Rightarrow S = (U_O \Sigma_O^\frac{1}{2})^T</math> and <math>W_C = U_C \Sigma_C U_C^T \Rightarrow R = (U_C \Sigma_C^\frac{1}{2})^T</math>.
			Next, the Singular Value Decomposition of <math> SR^T\;</math> is computed:

	:<math> SR^T= U\Sigma V^T.</math>		:<math> SR^T= U\Sigma V^T.</math>

	Now, partitioning <math>U,V</math>, for example based on the Hankel ~~singuar~~ Values, gives		Now, partitioning <math>U,V</math>, for example based on the [[wikipedia:Hankel_singular_value\|Hankel Singular Values]], gives

	:<math>SR^T= \begin{bmatrix} U_1 U_2 \end{bmatrix} \begin{bmatrix} \Sigma_1 & \\ & \Sigma_2\end{bmatrix} \begin{bmatrix} V_1^T\\V_2^T\end{bmatrix}.</math>		:<math>SR^T= \begin{bmatrix} U_1 U_2 \end{bmatrix} \begin{bmatrix} \Sigma_1 & \\ & \Sigma_2\end{bmatrix} \begin{bmatrix} V_1^T\\V_2^T\end{bmatrix}.</math>

Kuerschner at 06:38, 17 May 2013

2013-05-17T06:38:08Z

← Older revision		Revision as of 06:38, 17 May 2013
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	:<math> \\|\Sigma-\hat{\Sigma}\\|_2 \leq 2 \\|u\\|_2 \sum_{k=r+1}^n\sigma_k. </math>		:<math> \\|\Sigma-\hat{\Sigma}\\|_2 \leq 2 \\|u\\|_2 \sum_{k=r+1}^n\sigma_k. </math>

	== Cross Gramian ~~Balancing~~ ==		== Cross Gramian MOR ==

	A related ~~truncation~~-based approach is '''Cross Gramian Balanced Truncation'''<ref>Antoulas, ~~Athanasios~~ C. "<span class="plainlinks">[http://dx.doi.org/10.1137/1.9780898718713 Approximation of large-scale dynamical systems]</span>". Vol. 6. Society for Industrial and Applied Mathematics, pp.376--377, 2009; ISBN 978-0-89871-529-3</ref>.		A related Gramian-based approach is '''Cross Gramian Balanced Truncation'''<ref>Antoulas, A. C. "<span class="plainlinks">[http://dx.doi.org/10.1137/1.9780898718713 Approximation of large-scale dynamical systems]</span>". Vol. 6. Society for Industrial and Applied Mathematics, pp.376--377, 2009; ISBN 978-0-89871-529-3</ref>,<ref>D.C. Sorensen and A.C. Antoulas "<span class="plainlinks">[http://10.1016/S0024-3795(02)00283-5 The Sylvester equation and approximate balanced reduction]</span>", Linear Algebra and its Applications, vol. 351-352(0), pp. 671-700, 2002,
			</ref>.
	Given a stable and symmetric system <math>(A,B,C,D)</math>, such that there exists a transformation <math>J</math>		Given a stable and symmetric system <math>(A,B,C,D)</math>, such that there exists a transformation <math>J</math>