let me write this topic which I belife with its importanc "by my hand ...."


I will introduce three representation for Multi Variable Control Systems "MVCS"
 a- Differential Operator Representation [DOR].
 b- State Space Representation [SSR]
 c- Transfer Matrix Representation [TMR].

Note: a and b are time-domain representation, but c is frequency-domain representation.



Summary for each representation:

a- [DOR]
 linear, time invariant and continuous dynamical MVCS can written as the following ODE.

P(D) z(t) = Q(D) u(t)
and
y(t)= R(D) z(t) +W(D) u(t)

P[qxq],Q[qxm],R [pxq]and W[pxm]: are matrix differential operator.
u and y : vectors with time functions represent inputs and outputs.

b- [SSR]
time domain more detailed description about the system based on converting the [DOR] from high order DE to n 1st ODE

x'=Ax+Bu
y=Cx+Eu

A: evolution matrix
B: control matrix
C: observation matrix
E: direct transmittion matrix 
u,y : input, output vectors
x: state vector


c- [TMR]
relation between Laplace transformed output and input vector.

y(s)=T(s) u(s)

y,u: output and input vectors
T: transfer matrix "in the SISO called Transfer Function" T is not scaler matrix .. it is Polynomial matrix.

Strategic Relations:

[DOR] --Laplace transformation --->[TMR]
[SSR] -- Rosenbrock --->[TMR]