Moment-matching PMOR method

Description

The method introduced here is described in and , which is an extension of the moment-matching method for nonparametric systems (see , for moment-matching MOR). The method is applicable for linear parametrized systems, either in frequency domain or in time domain. For example, the parametric system in frequency domain:

$(E_0+s_1 E+s_2E_2+\ldots +s_pE_p)x=Bu(s_1,\ldots,s_p), \quad y=Cx, \quad \quad \quad \quad (1)$

where $s_1=j2 \pi f$ is the frequency domain variable, $f$ is the frequency. $s_2, s_3, \ldots, s_{p}$ are the parameters of the system. They can be any scalar functions of some source parameters, like $s_2=e^t$, where $t$ is time, or combinations of several physical (geometrical) parameters like $s_2=\rho v$, where $\rho$ and $v$ are two independent physical (geometrical) parameters. $x(t)\in \mathbb{R}^n$ is the state vector, $u \in \mathbb{R}^{d_I}$ and $y \in \mathbb{R}^{d_O}$ are the inputs and outputs of the system, respectively.

To obtain the reduced model in (2), a projection matrix $V \in \mathbb{R}^{n \times r}, r\ll n$ has to be computed.

$V^T(E_0+s_1E_1+s_2E_2+\ldots +s_pE_p)Vx=V^TBu(s_1,\ldots,s_p),$

$y=CVx. \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad(2)$

The matrix $V$ is derived by orthogonalizing a number of moment matrices of the system in (1) as follows, see or .

By defining $\tilde{E}=E_0+s_1^0E_1+s_2^0E_2+\cdots+s_p^0E_p,$ and $B_M=\tilde{E}^{-1}B, M_i=-\tilde{E}^{-1}E_i,i=1,2,\ldots,p$, we can expand $x$ in (1) at $s_1, s_2, \ldots, s_p$ around $p_0=[s_1^0,s_2^0,\cdots,s_p^0]$ as below,

$x=[I-(\sigma_1M_1+\ldots +\sigma_pM_p)]^{-1}B_Mu(s_1,\ldots,s_p) =\sum\limits_{i=0}^{\infty}(\sigma_1M_1+\ldots+\sigma_pM_p)^iB_Mu(s_1,\ldots,s_p).$

Here $\sigma_i=s_i-s_i^0, i=1,2,\ldots,p$. We call the coefficients in the above series expansion moment matrices of the parametrized system, i.e. $B_M, M_1B_M, \ldots, M_pB_M, M_1^2B_M, (M_1M_2+M_2M_1)B_M, \ldots, (M_1M_p+M_pM_1)B_M, M_p^2B_M, M_1^3B_M, \ldots$. The corresponding moments of the transfer function are those moment matrices multiplied by $C$ from the left. The matrix $V$ can be generated by first explicitly computing some of the moment matrices and then orthogonalizing them as suggested in . The resulting $V$ is desired to expand the subspace:

$\mathop{\mathrm{range}}\{V\}=\mathop{\mathrm{span}}\{B_M, \ M_1B_M,\ldots, M_pB_M,\ M_1^2B_M, \, (M_1M_2+M_2M_1)B_M, \ldots, (M_1M_p+M_pM_1)B_M,$

$M_p^2B_M, M_1^3B_M,\ldots, M_1^rB_M, \ldots,M_p^rB_M \}. \quad \quad \quad \quad (3)$

However, $V$ does not really span the whole subspace, because the latterly computed vectors in the subspace become linearly dependent due to numerical instability. Therefore, with this matrix $V$ one cannot get an accurate reduced model which matches all the moments algebraically included in the subspace.

Instead of directly computing the moment matrices in (3), a numerically robust method is proposed in ( the detailed algorithm is described in ), which combines the recursion in (5) with the modified Gram-Schmidt process to implicitly compute the moment matrices. The computed $V$ is actually an orthonormal basis of the subspace as below,

$\mathop{\mathrm{range}}\{V\}=\mathop{\mathrm{span}}\{R_0, R_1,\ldots, R_r \}. \quad \quad \quad \quad (4)$

$R_0 =[B_M],$

$R_1=[M_1R_0,\ldots, M_pR_0],$

$R_2=[M_1R_1,\ldots, M_pR_1], \quad \quad \quad \quad (5)$

$\vdots,$

$R_r=[M_1R_{r-1},\ldots, M_pR_{r-1}]$

$\vdots.$

Due to the numerical stability properties of the repeated modified Gram-Schmidt process employed in and , the reduced model derived from $V$ in (4) is computed in a numerically stable and accurate way. Applications of the method in , to the parametric models Gyroscope, Silicon nitride membrane, and Microthruster Unit, can be found in .