Modern Control Engineering

718 Chapter 9 / Control Systems Analysis in State Space

If a pole-zero cancellation occurs in the transfer function, then the controllability and observability

vary, depending on how the state variables are chosen. Remember that, to be completely state con-

trollable and observable, the transfer function must not have any pole-zero cancellations.

A–9–18. Prove that the system defined by

where

is completely observable if and only if the composite mn*nmatrixP, where

is of rank n.

Solution.We shall first obtain the necessary condition. Suppose that

Then there exists x(0)such that

or

Hence, we obtain, for a certain x(0),

Notice that from Equation (9–48) or (9–50), we have

wherem(mn)is the degree of the minimal polynomial for A. Hence, for a certain x(0), we have

CeAt x(0)=CCa 0 (t) I+a 1 (t) A+a 2 (t) A^2 +p+am- 1 (t) Am-^1 D x(0)= 0

eAt=a 0 (t) I+a 1 (t) A+a 2 (t) A^2 +p+am- 1 (t) Am-^1

CAi x( 0 )= 0 , fori=0, 1, 2,p,n- 1

Px(0)= F

C

CA







CAn-^1

Vx(0)= F

Cx(0)

CAx(0)







CAn-^1 x(0)

V = 0

Px(0)= 0

rank P 6 n

P= F

C

CA







CAn-^1

V

C=m*n matrix

A=n*n matrix

y=output vector (m-vector) (mn)

x=state vector (n-vector)

y=Cx

x# =Ax

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