Hello Control Friends,

PURPOSE OF NOTE
The purpose of this note is to 

(1) define transmission zeros and directions, and 
(2) demonstrate how they are found for MIMO LTI system 
    described by a state space quadruple (A,B,C,D).

Illustrative examples are also given.

___________________________________________________________________________

We begin by defining transmission zeros and directions.

DEFINITION: TRANSMISSION ZERO AND DIRECTIONS
A system has a transmission zero as s_o if their exists an input direction
u_o and an initial condition x_o (not both zero), such that if the input is
given by

       u(t) = u_o exp(s_o t)

and   
      x(o-) = x_o, 
then
      x(t) = x_o  exp(s_o t)
and 
      y(t) = 0

for all t greater than or equal to zero. The column vector [ x_o
                                                             u_o ]

is called the transmission zero direction associated with the transmission
zero s_o.

___________________________________________________________________________

The following can be used to determine the transmission zeros and
associated directions 
of a MIMO LTI system described by a state space quadruple (A,B,C,D).

CHARACTERIZATION OF TRANSMISSION ZEROS AN DIRECTIONS
The transmission zeros of a MIMO LTI system described by the state space
quadruple (A,B,C,D) are the roots of the following polynomial:


       det [ s_oI-A    -B
             C          D ] = 0       (1)

The corresponding zero directions (x_o, u_o) are found by solving:

| s_oI-A    -B || x_o |
| C          D || u_o | = 0           (2)

The following example illustrate how (1) and (2) are applied.

___________________________________________________________________________

EXAMPLE: COMPUTATION OF TRANSMISSION ZEROS AND DIRECTIONS FOR A SISO SYSTEM
Consider the SISO LTI system described by A = 0, B=C=1, D=0.
This system has transfer function given by

              H(s) = C(sI-A)^{-1}B + D = (1/s) + 1 = (s+1)/s 

This system clearly has a transmission zero at s_o = -1.

>From (1), one can verify this as follows:

     det [ s_oI-A    -B
             C          D ] =  det [s_o    -1
                                    1       1] = s_o + 1 = 0

This, however,  yields s_o = -1!  JOY!


>From (2), the corresponding zero direction may be found as follows:

| s_oI-A    -B || x_o |
| C          D || u_o | =  | -1   -1 || x_o |
                           |  1    1 || u_o |  = 0           
This, however, yields

                 | x_o | = k |  1 |
                 | u_o |     | -1 |

where k is an arbitrary constant.

___________________________________________________________________________

IMPORTANT NOTE: It is important to note that for MIMO LTI systems, the
transmission zeros are in general NOT the zeros of the individual transfer
functions within the transfer function matrix!

___________________________________________________________________________


The following MIMO example illustrates the above point.

EXAMPLE: COMPUTATION OF TRANSMISSION ZEROS AND DIRECTIONS FOR A MIMO SYSTEM
Consider the MIMO LTI system described by the following state space quadruple:

A = | -1   0   0 |     B = | 1  0 |
    |  1  -3   0 |         | 0  0 |
    |  0   0  -2 |         | 0  1 |

C = | 1   -3   1 |     D = | 0  0 |
    | 5    0   k |         | 0  0 |

Show that this system has a

(i) transfer function matrix given by

     H(s) = | s/(s+1)(s+3)    1/(s+2) |
            | 5/(s+1)         k/(s+2) |


(ii) transmission zero at s_o = 3 and a corresponding zero direction (x_o,
u_o) given by

          x_o = [ 6  1  -3}^T 

          u_o = [ 24  -15]^T

     for k = 10.

Note that for k = 10, the transmission zero at s_o = 3 is NOT a zero of any
of the individual transfer function entries within the transfer function
matrix H(s).
   
___________________________________________________________________________

I hope that the above note is helpful.

Thank you.
AAR