ABSTRACT
There exist two approaches to de…ne the system transfer function G(s) of the SISO system and its generalization, the transfer function matrix G(s) of the MIMO system. Laplace transform with its properties is the basis for both. Its application to derivatives of a (scalar or vector, either input or output) variable introduces the initial conditions (of the variable and of its derivatives) in the complex domain. Since system (scalar or matrix) parameters multiply the variable and its derivatives, therefore, the same parameters multiply the corresponding Laplace transforms of the variable and of its derivatives. The result is a sum of Laplace transform of the variable and of the (second) sum of the products of the parameters and initial values. The second sum introduces (altogether) the double sum of the products of the parameters and initial values as soon as the system order is higher than one. This holds for both SISO and MIMO systems. The double sum in the initial conditions of the input variable and its deriva-
tives together with the double sum of the initial conditions of the output variable and its derivatives appeared as a mathematical obstacle to determine the linear homogeneous relationship between Laplace transform of the output variable and Laplace transform of the input variable together with all initial conditions. The accepted exit from this mathematical complication was to accept the unjusti…able assumption that all initial conditions are equal to (scalar or vector) zero. This is common to both approaches to de…ne the system transfer function (matrix) G(s). The older approach de…nes ([21], [46], [62], [214]- [216], [234]-[269], [270],
[282], [300]) the transfer function G(s) as the ratio of left Laplace transform Y (s) of the output variable Y (t) and of left Laplace transform I(s) of the input variable I(t) under all (input and output) initial conditions (at t = 0)
equal to zero:
G(s) = Y (s) I(s)
; I1(0) = 0 ; Y1(0) = 0; (4.1)
for the -th order SISO system. It enables the linear homogenous relationship between Y (s) and I(s),
Y (s) = G(s)I(s); I1(0) = 0 ; Y1(0) = 0: (4.2)
This takes the vector-matrix form for the N dimensional -th order MIMO system,
Y(s) = G(s)I(s); I1(0) = 0 ; Y1(0) = 0; (4.3)
where G(s) is the matrix composed of all transfer functions of the system, which is the system transfer function matrix. It is the matrix value of the complex matrix function G(:) that relates in the linear homogeneous form Y(s) to I(s) under all zero initial conditions. If all input variables are Dirac impulse, of which Laplace transform is one, then (4.2) becomes
Y (s) = G(s); I(s) = 1, I1(0) = 0 ; Y1(0) = 0; (4.4)
and (4.3) takes the following form:
Y(s) = G(s)1M ; I(s) = 1M ; I1(0) = 0 ; Y1(0) = 0: (4.5)
Equations (4.4) and (4.5) provide the explanation of the physical meaning of G(s) that it is left Laplace transform of the output unit impulse response of the system under all zero initial conditions. The latter approach ([8], [23], [29], [198], [318]-[320]) de…nes the transfer
function G(s) as left Laplace transform of the SISO system output response to the unit impulse input under all zero initial conditions. The transfer function matrix G(s) of the MIMO system is then left Laplace transform of the system output vector response to the unit impulse action of all input variables under all zero initial conditions.
