Hello MIMO Control Friends,

This note demonstrates how one can obtain a state space realization
(A,B,C,D) for a general SISO transfer function. This may be useful when
using MATLAB, SIMULINK, and the other MATHWORKS Toolboxes.

AAR



PURPOSE OF NOTE: STATE SPACE REALIZATIONS FOR SISO TRANSFER FUNCTIONS
To obtain a state space realization (A,B,C,D) for a SISO transfer function.

Consider the SISO LTI system with transfer function given by:

     H(s) = { [b2 s^2 + b1 s + bo ]/[s^3 + a2 s^2 + a1 s + ao] } +  d

This system may be realized as follows:

                   xdot = Ax + Bu     (State Equation)     (1)
                      y = Cx + Du     (Output Equation)    (2)
where 
         A is a 3x3 matrix, 
         B is a 3x1 matrix (column vector), 
         C is a 1x3 matrix (row vector), 
         D is a 1x1 matrix (scalar), 

         u    is an 1x1 vector - called the control or input vector, 
         x    is an 3x1 vector - called the state vector,
         xdot is an 3x1 vector - denoting the time derivative of x, and
         y is    a  1x1 vector - called the output vector.


         A = [  0    1    0
                0    0    1
              -ao  -a1  -a2 ]

         B = [ 0
               0
               1 ]

         C = [ bo    b1    b2 ]

         D = [ d ]

That (A,B,C,D) constitute a valid state space quadruple for H(s) can be
verified by showing that C(sI - A)^{-1}B + D is precisely H(s). Please do
the linear algebra.

Extending the above realization method to more general higher order LTI
systems H(s) should be apparent.
Is it?

I hope this note is useful to you.

Thank you very much.
AAR