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The relationship between transfer functions and the state-space representation

In the following, the Eqs. (12.4) and (12.5) will be transformed into the domain using the Laplace transform, which will be done analogously to the scalar case in section 2.5. For this, the operator notation

   

from section 2.1 is adopted and when applying it to Eq. (12.4), one obtains

   

or rearranged

   

The solution of the state equation in the domain is then given by

(12.6)

with

(12.7)

Similarly, for Eq. (12.5) yields

   

Substituting from Eq. (12.6), the system output in the domain is

   

To obtain the relationship with transfer functions, the initial condition has to be set to zero. For a single-input-single-output system according to Eqs. (12.2) and (12.3) the system output is

   

Comparing this equation with Eq. (3.3) the transfer function is given by

(12.8)

The matrix from Eq. (12.7) is a matrix of rational functions of , which can always be represented by

(12.9)

where is a matrix with polynomial elements in . From Eq. (3.2) it is obvious that

(12.10)

and

(12.11)

which is the characteristic polynomial of the system. The zeros of this polynomial are the poles of the transfer function and at the same time eigenvalues of the system matrix . If the system in the state-space representation is fully controllable and observable (see section 12.6), then the number of poles are equal to the number of eigenvalues.



Next: State-space vs transfer function Up: State-space representation Previous: State-space representation of multi-input-multi-output   Contents
Christian Schmid 2005-05-09