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Numerics.DimRed.H2Model

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ℋ2-Optimal Model Reduction of Bilinear Control systems

adapted from the work of T. Breiten, P. Benner (MPI Magdeburg)

This approach to dimension reduction of linear/bilinear control systems is based on the idea of finding an optimal (low-dimensional) system that approximates as closely as possible the transfer function/matrix of the original (high-dimensional) system. As explained in detail in [1], the method builds on solving a generalized Sylvester equation by the Bilinear Iterative Rational Krylov Algorithm (BIRKA). It aims at the construction of bilinear reduced-order models that are (locally) optimal with respect to the bilinear H2-norm. Based on the idea of iterative correction, the algorithm can be seen as a suitable extension of its linear analogue (IRKA). Locally ℋ2-optimal reduced-order models fulfill an abstract Hermite-type interpolation condition for the Volterra series of the bilinear control system. Similar as in the linear case, these optimality conditions can also be stated in terms of (generalized) linear matrix equations. This in turn allows to construct the required projection subspaces as solutions to certain generalized Sylvester equations.

In practice, the algorithm works as follows. The original n-dimensional bilinear control system (A, N, B, C) is approximated by a d-dimensional reduced order model (denoted by hats). Initially chosen by random, the reduced system (A^, N^, B^, C^) is refined by the following iteration:

Two generalized Sylvester equations have to be solved

$\begin{matrix}AX+X\hat{A}^\ast +\sum_{k=1}^mN_kX\hat{N}_k^\ast+B\hat{B}^\ast&=&0klzzwxh:0021A^\ast Y+Y\hat{A}+\sum_{k=1}^m N_k^\ast Y\hat{N}_k-C^\ast\hat{C}&=&0\end{matrix}$

yielding matrices X,Y∈R^n×d. Note the formal similarity with the generalized Lyapunov equations for the calculations of Gramians; however, there is a sign change in front of C^*C. Direct methods for solving such equations have a numerical complexity of O(d³n³) where d stands for the diension of the reduced model. Instead, by mapping the matrices X, Y onto vectors, the generalized Sylvester equations can be understood as systems of coupled linear equations which can be solved, e. g., by means of the biconjugate gradient method, preferentially with pre-conditioning [2]. Alternatively, the iterative methods [3,4] can be used instead which requires the solution of a standard Sylvester equation in each step.

$\begin{array}{rcl}AX_0+X_0\hat{A}^\ast +B\hat{B}^\ast&=&0klzzwxh:0039AX_j+X_j\hat{A}^\ast+\sum_{k=1}^m N_kX_{j-1}\hat{N}^\ast_k+B\hat{B}^\ast&=&0,\,j.gt.0\end{array}$

and similarly for the second (dual) equation.
Then, a QR-decomposition of matrices X, Y is performed

$\begin{matrix} X&=&VRklzzwxh:0013Y&=&WZ\end{matrix}$

where V, W are orthogonal matrices and R, Z are upper triangular matrices (not needed here). Then V, W are used to construct a refined system in the following way

$\begin{matrix}\hat{x}&=&Sxklzzwxh:0015\hat{A}&=&SAVklzzwxh:0016\hat{N}_k&=&SN_kVklzzwxh:0017\hat{B}&=&SBklzzwxh:0018\hat{C}&=&CV\end{matrix}$

which corresponds to a Petrov-Galerkin type projection with

$S=\left(W^\ast V\right)^{-1}W^\ast$
This iteration is repeated until the change in the spectrum of the reduced order system matrix A^ falls below a user prescribed tolerance.

References

P. Benner, T. Breiten: SIAM Journal on Matrix Analysis and Application, 33, 859–885 (2012)
T. Breiten, private communications (2014).
E. Wachspress: Appl. Math. Lett. 1, 87 (1988)
T. Damm: Numer. Lin. Alg. Appl. 15, 853 (2008)