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A partial Padé-via-Lanczos method for reduced-order modeling. (English) Zbl 0980.93012

Consider a linear SISO system \(C\dot{x}=Gx+bu\), \(y=c^Tx\). The transfer function \(H(s)=c^T(G+sC)^{-1}b\) can be rewritten in the form \(l^T(I-(s-s_0)A)^{-1}r\) if \(G+s_0C\) is nonsingular. A reduced model can be obtained as a Padé approximant of \(H\) in the point \(s_0\). It is essentially the (1,1) element of \((I-(s-s_0)T_n)^{-1}\), where \(T_n\) is the truncated tridiagonal Lanczos matrix obtained by applying a nonsymmetric Lanczos process to the matrix \(A\). However, the reduced model need not be stable or passive. Both these conditions are related to the positioning of poles and zeros of the approximant. By a postmodification of the matrix \(T_n\), one may modify the poles or zeros of the approximant and place them at a desired position. This gives a partial Padé approximant (all remaining degrees of freedom are used to match the maximum number of Taylor coefficients of \(H\) at \(s_0\)). An algorithm for this partial Padé-via-Lanczos method is given. Thus far, there is no systematic procedure to place the prefixed poles or zeros in an optimal way.

MSC:

93B11 System structure simplification
41A21 Padé approximation
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