# Interdisciplinary Applied Mathematics

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Often, second-order systems are encountered in microsystems (Bai, 2002; Ramaswamy and White, 2001), of the form

Mq(t) + Dq(t) + Kq(t) = Pu(t),    y(t) = QT q + RT q(t),

where Q and R are chosen depending on the output variable of interest. The second-order system can be formulated into an equivalent linear system of the form given in equation (17.10) such that the symmetry of the original

FIGURE 17.10. Basic transformations in the Krylov subspace method for macromodeling.

system is preserved, i.e.,

 q(t) , c = D M q(t) F0
 G= K0 , B = p , l = Q 1 0 R fa 1 o _i

where F can be any N x N nonsingular matrix. Generally, F is chosen to be the identity matrix, I, while F = M is also a reasonable choice if M, D, and K are symmetric. The advantages of the Krylov subspace method are: (i) It is fairly accurate for linear systems and can be automated. (ii) It is computationally very effective. The disadvantages are: (i) It is not very accurate for highly nonlinear systems. (ii) It does not preserve the physical meaning of the original system.

Moment Matching Techniques

The main idea behind the moment matching technique (Ismail, 2002) is to construct the transfer function directly from the system equations using Laplace transformation, and then to approximate the transfer function by some rational function. Consider again equation (17.9). Taking the Laplace transform of this equation, the frequency domain formulation is given by

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