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Control Systems (Transfer Function; Open / Closed Loop System; Observable / Unobservable)

A system described in the attachment is under feedback control of the form u = Kx + r where r is the reference input.
(i) Show that (A,C) is observable.
(ii) Compute a K of the form {see attachment} so that (A - BK, C) is unobservable. (I.e., the closed loop system is unobservable)
(iii) Find the transfer function of the open loop system as well as the transfer funtion of the closed loop system, and compare the transfer functions to determine what the unobservability is due to.


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I will use "^T" for the transpose (Superscipt T) of a matrix. So, A^T is the transpose of matrix A.

Please check to make sure all the matrix calculations and algebra are correct.

1. Show that (A,C) is observable.
It is observable if the matrix P = [ C^T (A^T)*(C^T) ] is of rank 2; that is, determinant[P] is NOT zero.

Vector C^T

Matrix A^T
-3 1
+1 0

Vector (A^T)*(C^T)

Matrix P:
1 -1
2 +1

det[P] = 1+2 = 3
Hence, P ...

Solution Summary

The solution involves calculations and explanations for control systems. It covers the concepts of observable/unobservable systems and transfer functions for both open and closed loop systems.