Balanced Truncation: Difference between revisions

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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 $Σ$ , realized by $(A, B, C, D)$

$\dot{x} = A x + B u$

$y = C x + D u$

is called balanced^[1], if the systems Controllability Gramian and Observability Gramian, the solutions $W_{C}$ and $W_{O}$ of the Lyapunov equations

$A W_{C} + W_{C} A^{T} = - B B^{T}$

$A^{T} W_{O} + W_{O} A = - C^{T} C$

respectively, satisfy $W_{C} = W_{O} = d i a g (σ_{1}, \dots, σ_{n})$ with $σ_{1} \geq σ_{2} \geq \dots \geq σ_{n} > 0$ . Since in general, the spectrum of $W_{C} W_{O}$ are the squared Hankel Singular Values for such a balanced system, they are given by: $\sqrt{λ (W_{C} W_{O})} = {σ_{1}, \dots, σ_{n}}$ .

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:

$(\tilde{A}, \tilde{B}, \tilde{C}, \tilde{D}) = (T A T^{- 1}, T B, C T^{- 1}, D) .$

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:

$(\tilde{A}, \tilde{B}, \tilde{C}, \tilde{D}) = ([\begin{matrix} {\tilde{A}}_{11} & {\tilde{A}}_{12} \\ {\tilde{A}}_{21} & {\tilde{A}}_{22} \end{matrix}], [\begin{matrix} {\tilde{B}}_{1} \\ {\tilde{B}}_{2} \end{matrix}], [\begin{matrix} {\tilde{C}}_{1} & {\tilde{C}}_{2} \end{matrix}], \tilde{D})$ .

By truncating the discardable states, the truncated reduced system is then given by $\hat{Σ} = ({\tilde{A}}_{11}, {\tilde{B}}_{1}, {\tilde{C}}_{1}, \tilde{D})$ .

Implementation: SR Method

The necessary balancing transformation can be computed by the SR Method^[2]. First, the Cholesky factors of the gramians $W_{C} = S^{T} S, W_{O} = R^{T} R$ are computed. Next, the Singular Value Decomposition of $S R^{T}$ is computed:

$S R^{T} = U Σ V^{T} .$

Now, partitioning $U, V$ , for example based on the Hankel singuar Values, gives

$S R^{T} = [\begin{matrix} U_{1} U_{2} \end{matrix}] [\begin{matrix} Σ_{1} \\ Σ_{2} \end{matrix}] [\begin{matrix} V_{1}^{T} \\ V_{2}^{T} \end{matrix}] .$

The truncation of discardable partitions $U_{2}, V_{2}^{T}, Σ_{2}$ results in the reduced order model $(P^{T} A Q, P^{T} B, C Q, D)$ where

$P = R^{T} V_{1} Σ_{1}^{- \frac{1}{2}},$

$Q = S^{T} U_{1} Σ_{1}^{- \frac{1}{2}} .$

$Q^{T} P = I_{r}$ makes $Q P^{T}$ an oblique projector and hence Balanced Trunctation a Petrov-Galerkin projection method. The reduced model is stable with Hankel Singular Values given by $σ_{1}, \dots, σ_{r}$ , where r is the order of the reduced system. It is possible to choose $r$ via the computable error bound^[3]:

$‖ Σ - \hat{Σ} ‖_{2} \leq 2 ‖ u ‖_{2} \sum_{k = r + 1}^{n} σ_{k} .$

Direct Truncation

A related truncation-based approach is Direct Truncation^[4]. Given a stable and symmetric system $(A, B, C, D)$ , such that there exists a transformation $J$

$A J = J A^{T}$

$B = J C^{T}$

then the solution of the Sylvester Equation

$A W_{X} + W_{X} A = - B C$

is the Cross Gramian, of which the absolute value of its spectrum equals the Hankel Singular Values:

$| λ (W_{X}) | = {σ_{1}, \dots, σ_{r}}$ .

Thus the Singular Value Decomposition of the Cross Gramian

$W_{X} = U Σ V^{T}$

also allows a partitioning

$W_{X} = [\begin{matrix} U_{1} U_{2} \end{matrix}] [\begin{matrix} Σ_{1} \\ Σ_{2} \end{matrix}] [\begin{matrix} V_{1}^{T} \\ V_{2}^{T} \end{matrix}] .$

and a subsequent truncation of the discardable states, to which the above error bound also applies.

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
↑ Antoulas, Athanasios C. "Approximation of large-scale dynamical systems". Vol. 6. Society for Industrial and Applied Mathematics, 2009; ISBN 978-0-89871-529-3

[1] 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

[2] 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

[3] 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

[4] Antoulas, Athanasios C. "Approximation of large-scale dynamical systems". Vol. 6. Society for Industrial and Applied Mathematics, 2009; ISBN 978-0-89871-529-3

[1]

[2]

[3]

[4]

@@ Line 38: / Line 38: @@
 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.
-Next, the Singular Value Decomposition of <math> SR^T\;</math> is computed:
+Next, the [[wikipedia:Singular_Value_Decomposition|Singular Value Decomposition]] of <math> SR^T\;</math> is computed:
 <math> SR^T= U\Sigma V^T.</math>
@@ Line 55: / Line 55: @@
 <math> \|\Sigma-\hat{\Sigma}\|_2 \leq 2 \|u\|_2 \sum_{k=r+1}^n\sigma_k. </math>
+== Direct Truncation ==
+A related truncation-based approach is '''Direct 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, 2009; ISBN 978-0-89871-529-3</ref>.
+Given a stable and symmetric system <math>(A,B,C,D)</math>, such that there exists a transformation <math>J</math>
+<math>AJ = JA^T</math>
+<math>B = JC^T</math>
+then the solution of the [[wikipedia:Sylvester_Equation|Sylvester Equation]]
+<math>AW_X+W_XA=-BC</math>
+is the [[Cross Gramian]], of which the absolute value of its spectrum equals the Hankel Singular Values:
+<math>|\lambda(W_X)| = \{\sigma_1,\dots,\sigma_r\}</math>.
+Thus the [[wikipedia:Singular_Value_Decomposition|Singular Value Decomposition]] of the Cross Gramian
+<math>W_X = U\Sigma V^T</math>
+also allows a partitioning
+<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.
 ==References==
 <references/>