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Where 

SysC=system(Ac,Bc,Cc,Dc)

, 

Sys=system(A,B,C,D)

, and 

nz 

is the dimension of 

z

 and 

nw

 is the dimension of 

w

:

Given the above, 

SysCL

 is calculated as shown in Figure 3-7.

Figure 3-7.  Calculation of the Closed Loop System (SysCL)

The closed-loop system is assumed to be well-posed—(I – D

) must 

be invertible). A well-posed closed-loop system assures that if two given 
systems, 

Sys

 and 

SysC

, are proper (only proper transfer functions can be 

represented in state space), then the resulting closed-loop system, 

SysCL

, 

also is proper and therefore realizable in state space.

Figure 3-8 is an example of an ill-posed feedback system, where the 
closed-loop transfer function is 

s+1

, which cannot be represented as 

a state-space system.

B is

C is C

,C

D is

A+B

–

(

)

1

–

–

(

)

1

–

–

(

)

1

–

+

–

(

)

1

–

+

=

–

(

)

1

–

+

–

(

)

1

–

+

=

–

(

)

1

–

+

=

–

(

)

1

–

+

–

(

)

1

–

=