ABSTRACT
What follows in general is the basis for the next subsections. The mathematical modeling of many physical systems results (after a pos-
sible linearization) in the time-invariant linear vector Input-Output (IO) di¤erential equation (2.1) to be called the IO system,
AkY (k)(t) =
BkI (k)(t); detA 6= 0, 8t 2 T, 1; 0 ;
Y(k)(t) = dkY(t)
dtk ; Ak2RNxN , Bk2RNxM , k = 0; 1; ::; ;
< =) Bi = O, i = + 1, + 2; :::; . (2.1)
This mathematical description can be the general IO mathematical description of an object/plant, of a controller and of a whole control system. Let Ck be the k-dimensional complex vector space, Rk be the k-dimensional
real vector space, OMxN be the zero matrix in theMxN -dimensional real matrix space RMxN , and ON be the zero matrix in RNxN , ON = ONxN . Analogously, let 0k 2 Rk be the zero vector in Rk. Let I = [I1 I2 ::: IM ]T 2RM be the input vector, and Y = [Y1 Y2 ::: YN ]
T 2RN be the output vector. The values Ik and Ym are measured with respect to the total zeros of these variables if they have total zeros. If a variable does not have a total zero, then some its value is accepted to play the role of its total zero. Temperature has the total zero that is Kelvin zero. Position does not have the total zero.
