ABSTRACT
This section develops further the concept of the system full transfer function matrix introduced in [81]. The general de…nitions presented in the sequel are equivalent to those in [148].
The full transfer function matrix FIO(s) of the IO system (2.9) (Section 2.1), repeated as
A()Y(t) = B()I(t); t 2 T; (7.1) describes in the complex domain C how the system temporally transfers a simultaneous in‡uence of an arbitrary input vector I(t); of any input initial conditions I0 , I
(1) 0 , ..., I
(1) 0 , and of arbitrary output initial conditions Y0 , Y
(1) 0 , ...,
Y (1) 0 on the system output response Y(t), Fig. 7.1 [148]. Int T0 is the interior of T0,
Int T0 = ft : t 2 T0; t > 0g =]0;1[; Int T0 T0; inf (Int T0) = 0 2 T0, sup (Int T0) =1. (7.2)
De…nition 104 a) The full (complete) (IO) matrix transfer function FIO(:), FIO(:) : C ! CNx[(+1)M+N ], of the IO system (7.1) is a matrix function of the complex variable s such that it determines uniquely (left, right) Laplace transform Y()(s) of the system output Y(t) as a homogenous linear function of (left, right) Laplace transform I()(s) of the system input vector I(t) for an arbitrary variation of I(t), of arbitrary initial vector values I1 0() and Y
1 0() of the extended input vector I
1(t) and of the extended output vector Y1(t) at t = 0(); respectively, and its matrix value is the system
Figure 7.1: The full block diagram of the IO system shows the system transfer function matrices relative to the input vector and relative to all initial conditions in which m = and v = .
