Modern Control Engineering

684 Chapter 9 / Control Systems Analysis in State Space

EXAMPLE 9–14 Consider the system described by

Is this system controllable and observable?

Since the rank of the matrix

is 2, the system is completely state controllable.

For output controllability, let us find the rank of the matrix Since

the rank of this matrix is 1. Hence, the system is completely output controllable.

To test the observability condition, examine the rank of Since

the rank of is 2. Hence, the system is completely observable.

Conditions for Complete Observability in the s Plane. The conditions for com-

plete observability can also be stated in terms of transfer functions or transfer matrices.

The necessary and sufficient conditions for complete observability is that no cancella-

tion occur in the transfer function or transfer matrix. If cancellation occurs, the canceled

mode cannot be observed in the output.

EXAMPLE 9–15 Show that the following system is not completely observable:

where

Note that the control function udoes not affect the complete observability of the system. To

examine complete observability, we may simply set u=0.For this system, we have

CCAC(A)^2 C*D= C

4

5

1

- 6

- 7

- 1

6

5

- 1

S

x= C

x 1

x 2

x 3

S, A= C

0

0

- 6

1

0

- 11

0

1

- 6

S, B= C

0

0

1

S, C=[4 5 1]

y =Cx

x# =Ax+Bu

CCAC*D

CCAC*D=B

1

0

1

1

R

[CAC*].

CCB  CABD=[0 1]

CCB  CABD.

CBABD= B

0

1

1

- 1

R

y =[1 0]B

x 1

x 2

R

B

x# 1

x# 2

R= B

1

- 2

1

- 1

RB

x 1

x 2

R + B

0

1

Ru

Openmirrors.com