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<math> y = Cx + Du</math> |
<math> y = Cx + Du</math> |
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| − | is called balanced<ref>B.C. Moore, "<span class=" |
+ | is called balanced<ref>B.C. Moore, "<span class="plainlinks">[http://dx.doi.org/10.1109/TAC.1981.1102568 Principal component analysis in linear systems: Controllability, observability, and model reduction]</span>", IEEE Transactions on Automatic Control , vol.26, no.1, pp.17,32, Feb 1981</ref>, if the systems [[wikipedia:Controllability_Gramian|Controllability Gramian]] and [[wikipedia:Observability_Gramian|Observability Gramian]], the solutions <math>W_C</math> and <math>W_O</math> of the Lyapunov equations |
<math> AW_C+W_CA^T=-BB^T </math> |
<math> AW_C+W_CA^T=-BB^T </math> |
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== Implementation: SR Method== |
== Implementation: SR Method== |
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| − | The necessary balancing transformation can be computed by the SR Method<ref>A.J. Laub; M.T. Heath; C. Paige; R. Ward, "<span class=" |
+ | 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>. |
First, the Cholesky factors of the gramians <math>W_C=S^TS,\; W_O=R^TR</math> are computed. |
First, the Cholesky factors of the gramians <math>W_C=S^TS,\; W_O=R^TR</math> are computed. |
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Next, the Singular Value Decomposition of <math> SR^T\;</math> is computed: |
Next, the Singular Value Decomposition of <math> SR^T\;</math> is computed: |
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<math> Q= S^T U_1 \Sigma_1^{-\frac{1}{2}}.</math> |
<math> Q= S^T U_1 \Sigma_1^{-\frac{1}{2}}.</math> |
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| − | <math>Q^TP=I_r</math> makes <math> QP^T</math> an oblique projector and hence '''Balanced Trunctation''' a Petrov-Galerkin projection method. The reduced model is stable with Hankel Singular Values given by <math>\sigma_1,\dots,\sigma_r</math>, where r is the order of the reduced system. It is possible to choose <math>r</math> via the computable error bound<ref>D.F. Enns, "<span class=" |
+ | <math>Q^TP=I_r</math> makes <math> QP^T</math> an oblique projector and hence '''Balanced Trunctation''' a Petrov-Galerkin projection method. The reduced model is stable with Hankel Singular Values given by <math>\sigma_1,\dots,\sigma_r</math>, where r is the order of the reduced system. It is possible to choose <math>r</math> via the computable error bound<ref>D.F. Enns, "<span class="plainlinks">[http://dx.doi.org/10.1109/CDC.1984.272286 Model reduction with balanced realizations: An error bound and a frequency weighted generalization]</span>," The 23rd IEEE Conference on Decision and Control, vol.23, pp.127,132, Dec. 1984</ref>: |
<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> |
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Revision as of 11:05, 23 April 2013
Balanced Truncation is an important projection model reduction method which delivers high quality reduced models by making an extra effort in choosing the projection subspaces.
Derivation
A stable minimal (controllable and observable) system \(\Sigma\), realized by \((A,B,C,D)\)
\( \dot{x} = Ax + Bu\)
\( y = Cx + Du\)
is called balanced[1], if the systems Controllability Gramian and Observability Gramian, the solutions \(W_C\) and \(W_O\) of the Lyapunov equations
\( AW_C+W_CA^T=-BB^T \)
\( A^TW_O+W_OA=-C^TC \)
respectively, satisfy \( W_C=W_O=diag(\sigma_1,\dots,\sigma_n)\) with \( \sigma_1\geq\sigma_2\geq \dots\geq\sigma_n>0\). Since in general, the spectrum of \(W_CW_O\) are the squared Hankel Singular Values for such a balanced system, they are given by\[\sqrt{\lambda(W_CW_O)} = \{\sigma_1,\dots,\sigma_n\}\].
An arbitrary system \((A,B,C,D)\) can be transformed into a balanced system \((\tilde{A},\tilde{B},\tilde{C},\tilde{D})\) via a state-space transformation\[ (\tilde{A},\tilde{B},\tilde{C},\tilde{D})= (TAT^{-1},TB,CT^{-1},D).\]
This transformed system has balanced Gramians \(W_C=T\tilde{W_C}T^T\) and \(W_O=T^{-T}\tilde{W_O}T^{-1}\) which are equal and diagonal. The balanced systems states are ordered (descendingly) by how controllable and observable they are, thus allowing a partion of the form\[ (\tilde{A},\tilde{B},\tilde{C},\tilde{D})= \left (\begin{bmatrix}\tilde{A}_{11} & \tilde{A}_{12}\\ \tilde{A}_{21} & \tilde{A}_{22}\end{bmatrix},\begin{bmatrix}\tilde{B}_1\\\tilde{B}_2\end{bmatrix},\begin{bmatrix} \tilde{C}_1 &\tilde{C}_2 \end{bmatrix},\tilde{D}\right)\].
By truncating the discardable states, the truncated reduced system is then given by \( \hat{\Sigma}=(\tilde{A}_{11},\tilde{B}_1,\tilde{C}_1,\tilde{D}) \).
Implementation: SR Method
The necessary balancing transformation can be computed by the SR Method[2]. First, the Cholesky factors of the gramians \(W_C=S^TS,\; W_O=R^TR\) are computed. Next, the Singular Value Decomposition of \( SR^T\;\) is computed\[ SR^T= U\Sigma V^T.\]
Now, partitioning \(U,V\), for example based on the Hankel singuar Values, gives
\(SR^T= \begin{bmatrix} U_1 U_2 \end{bmatrix} \begin{bmatrix} \Sigma_1 & \\ & \Sigma_2\end{bmatrix} \begin{bmatrix} V_1^T\\V_2^T\end{bmatrix}.\)
The truncation of discardable partitions \(U_2,V^T_2,\Sigma_2\) results in the reduced order model \((P^TAQ,P^TB,CQ,D)\;\) where
\( P=R^T V_1\Sigma_1^{-\frac{1}{2}},\)
\( Q= S^T U_1 \Sigma_1^{-\frac{1}{2}}.\)
\(Q^TP=I_r\) makes \( QP^T\) an oblique projector and hence Balanced Trunctation a Petrov-Galerkin projection method. The reduced model is stable with Hankel Singular Values given by \(\sigma_1,\dots,\sigma_r\), where r is the order of the reduced system. It is possible to choose \(r\) via the computable error bound[3]\[ \|\Sigma-\hat{\Sigma}\|_2 \leq 2 \|u\|_2 \sum_{k=r+1}^n\sigma_k. \]
References
- ↑ B.C. Moore, "Principal component analysis in linear systems: Controllability, observability, and model reduction", IEEE Transactions on Automatic Control , vol.26, no.1, pp.17,32, Feb 1981
- ↑ A.J. Laub; M.T. Heath; C. Paige; R. Ward, "Computation of system balancing transformations and other applications of simultaneous diagonalization algorithms," IEEE Transactions on Automatic Control, vol.32, no.2, pp.115,122, Feb 1987
- ↑ D.F. Enns, "Model reduction with balanced realizations: An error bound and a frequency weighted generalization," The 23rd IEEE Conference on Decision and Control, vol.23, pp.127,132, Dec. 1984