Interdisciplinary Applied Mathematics

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Krylov Subspace Technique Based on the Lanczos Method


The governing equation for a continuous time-invariant multi-input multioutput (MIMO) system (e.g., a comb-drive microresonator) is of the form (Srinivasan et al., 2001; Bai, 2002)


Cx(t) + Gx(t) = Bu(t),    y(t) = LT x(t),    (17.9)


where t is the time variable, x(t) G is a state vector, u(t) G is the input excitation vector, and y(t) G Жр is the output vector. Here C, G G NxN are system matrices, B G Nxm and L G Nxp are the input and output distribution arrays, respectively, N is the state space dimension, m and p are much smaller than N, and m > p.


A variety of analyses can be performed for the linear dynamical system given in equation (17.9). For example:


1. A static analysis to compute the equilibrium condition.


2.    A steady-state analysis, also called the frequency response analysis, to determine the frequency responses of the system to external steady-state oscillatory (e.g., sinusoidal) excitation.


3. A transient analysis to compute the output behavior y(t) subject to time varying excitation u(t).


4. A sensitivity analysis to determine the proportional changes in the time response y(t) and/or steady-state response to a proportional change in system parameters.


Some of these analyses can be very expensive, especially if performed using equation (17.9). If a reduced-order system to equation (17.9) can be developed, the analysis can be performed quickly. The reduced-order system should have the following desirable attributes:


1. The reduced system should have a much smaller state-space dimension compared to the state-space dimension of the full-order system.

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