Sub-optimal discrete model reduction by multipoint Padé approximation
Let \(y(iT)\) be the impulse response of a linear discrete-time system and \(y_k(iT)\) the impulse response of a \(k\)th order reduced system. Let \(G(z)\) be the transfer function of the original system and \(G_k(z)\) the transfer function of the reduced system. If the poles of \(G_k\) are fixed to be \(p_1,\dots,p_k\), then minimizing the least squares error \(\sum^\infty_{i=0} [y(iT)-y_k(iT)]^2\) is equivalent to interpolating \(G(z)\) by \(G_k(z)\) in the reciprocals \(1/p_1,\dots,1/p_k\). Fixing the poles of the approximant has the advantage that the least squares problem is linear in the numerator coefficients and iteration can be avoided. The problem has become a multipoint Padé-type approximation problem. The problem can be solved by an efficient Routh-array type of algorithm. The method can easily adapted for the case when the \(y(iT)\) do not correspond to an impulse response but are the response to a step input or a ramp input. However, this method may give a reduced system that does not match the first Markov parameter of the given system. If one insists of matching at least this first Markov parameter, a least squares problem with a constraint should be solved. Using a Lagrange multiplier technique, this is elaborated in the paper under review and it results in a modified interpolation problem that can be solved by methods of the type given above.
scientific article; zbMATH DE number 902659