Interdisciplinary Applied Mathematics

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sCX (s) + GX(s) = BU(s),    Y(s) = LT X(s).    (17.11)


Eliminating the variable X(s) from equation (17.11), the input and output are related by a p x m matrix-valued rational function


H(s) = LT (G + sC)^1B,


where H(s) is known as the transfer function of the linear system, and the state-space dimension of the system is N. The Taylor series expansion of the scalar transfer function H(s) about so is given by


H (s) = lT (I-(s-so)A) 1 r = lT r+(lT Ar)(s-so ) + (lT A2r)(s-so )2 +—-


= mo + mi(s — so) + m2(s — so)2 + ••• ,


where mj are the moments about so. The objective is to approximate H(s) by a rational function Hq(s) G Rq-1,q over the range of frequencies of interest, where q < N. One choice is the Pade approximation (Bultheel and Barvel, 1986). A function Hq(s) G Rq-1,q is said to be a qth Pade approx-imant of H(s) about an expansion point so if it matches with moments of H(s) as far as possible. It is required that (Bai, 2002)


H(s)= Hq(s) + O((s — so)2q).    (17.12)


Note that we have 2q conditions on the 2q degrees of freedom that describe the approximation function. Specifically, let


Hq (s)



Pq-lOO


Qq(s)



ao + ais + a2s2 + ••• + aq-i sq 1 1 + bs + 62s2—-h bqsq


The coefficients [a*] and [bi] of the polynomials Pq-1(s) and Qq(s), and also the moments can be computed by multiplying both sides of equation (17.12) by Qq(s) and comparing the first q(s — so)k terms for к = 0,1, 2,…,n- 1. A system of 2q nonlinear equations is solved to find the 2q unknowns. This takes into    account    the dominant    q    poles,    while    the    poles    larger than    this

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