Pii: S0893-9659(99)00063-4

-A model order reduction technique for systems depending on two parameters is developed. Given a large system model, the method generates the descriptor matrices of a system model of lower order that is a rational interpolant of the transfer function of the large system--the transfer functions have identical values and derivatives for a finite set of parameter values. The new technique is a generalization of recently developed algorithms for one-parameter systems that are based on projections onto Krylov subspaces defined by the descriptor matrices. (~) 1999 Elsevier Science Ltd. All rights reserved. Keywords--Generalized Krylov subspaces, Model order reduction, Rational interpolants, Padd approximation, Two-dimensional linear systems. 1. I N T R O D U C T I O N This study presents a moment matching, model order reduction method for two-parameter problems of the form ( 8 1 E l -~82E 2 A ) x (81, 82) -b u (81, 82) , y (Sl, s2) = c*x (Sl, s2), (1) This work was supported in part by a grant from AFOSR via the MURIF program under contract F49620-961-0025, the National Science Foundation under Grants ECS-9502138 and CCR-9796315, a grant from the IBM corporation, and an NSF graduate fellowship. 0893-9659/99/$ see front matter. (~) 1999 Elsevier Science Ltd. All rights reserved. Typeset by , 4 ~ T E X PII: S0893-9659(99)00063-4 94 D.S. WEILE et al. where A, El , and E2 are n x n system descriptor matrices, b and c are input and output coupling n-vectors, u is an input, y is an output, the asterisk denotes complex conjugation, and sl and s2 parameters upon which the system response depends. The theory presented here results in a characterization of a reduced order model of dimension m of the form (81]~1q-82E2-A) x(81,82)=[~u(81,82) , (2) (sl , s2) = e*x (sl , s2), where ~(sl, s2) matches y(sl , s2) and its derivatives at several points in the (Sl, s2) plane for a unit input. Recently, significant progress has been made in model reduction of systems linearly dependent on a single parameter, see [1,2] and their references. While systems of one parameter are remarkably useful in practice as they naturally occur as Laplace transforms of linear time-invariant systems, there are important problems which do not fit this form. Two directions of generalization are required. The first direction involves relaxing the restriction that the elements of the matrix defining the systems to be reduced are linear functions of the parameter of interest. Indeed, for many important problems, the system matrices are nonlinearly or even transcendentally dependent on their parameter. The second direction entails relaxing the restriction to a single parameter. A general p × p linear system nonlinearly dependent on two parameters takes the form G (81,82) Xoo (Sl, 82) = ]~u (81, 82), (3) y (81,82) = C*Xo0 (81, S2). To accomplish model reduction on this system, a method which incorporates both directions of generalization is required. To cope with the nonlinearity, the matrix G may be approximated at various points in the (Sl, s2) plane by a truncated Taylor series [3] I i G(s1,82) ~ Z ~ ~_..oJoi-j "~'3°1"2 " (4) i=0 j=0 Defining xij = s~-Ss~x00 for 0 < j < i < I 1, and substituting (4) into (3) results in a system in the form of (1) where n = p(I 2 I + 2)/2, and b* = [0 . . . 0 l~*], c* = [c* 0 . . . 0] , x* = [:x~ x; . . . :K~-I],