Question

(Observable canonical form). Given a transfer function G^(s) ) let (Aˉ,Bˉ,Cˉ,Dˉ) be a realization for its transpose Gˉ(s):=G^(s)′. Show that (A,B,C,D) with

A:=Aˉ′,
B:=Cˉ′,
C:=Bˉ′,
D:=Dˉ′ is a realization for G(s).

Answer 1

To show that (A,B,C,D) is a realization for G(s) given a transpose realization (ABCD) for G(s), we must take the transpose of each matrix in the realization of G(s).

Step-by-step explanation:

The question is related to systems theory in the field of control engineering and concerns the properties of the transfer function and its realization. A realization of a transfer function is a state space representation, consisting of matrices A, B, C, D, that exhibits the same input-output behavior as the transfer function.

For a clearer understanding, here's a step-by-step explanation: If you have a transfer function (s) represented in terms of a state space model with matrices to find the realization of the original transfer function G(s), you can simply take the transpose of each matrix in that realization. Hence, A becomes the transpose of , B becomes the transpose of the transpose of and D the transpose of . This process will give you a new state space model that represents the original transfer function G(s).