Distributed system transfer functions of exponential order

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Abstract
σo is a real number. We construct a transfer function algebra of fractions, viz. F˚(σo), for modelling possibly unstable distributed systems such that (i) [fcirc] in F˚(σo) is holomorphic in Re s ≧ σo, (i.e. is σo-stable), iff [fcirc] is σo-exponentially stable, and (ii) we allow delay in the direct input-output transmission of the system. This algebra is (a) a restriction of the algebra B˚(σo) developed by Callier and Desoer (1978, 1980 a), (b) an extension of the algebra of proper rational functions such that the exponential order properties of the latter transfer functions of lumped systems are maintained. The algebra F˚ (σo) can be used for modelling and feedback system design. It is shown that standard semigroup systems are better modelled by a transfer function in F˚(σo) rather than B˚(σo).