Modern Control Engineering

Problems 721

B–9–11.Obtain a transfer-function representation of the
following system with MATLAB:

B–9–12.Obtain a transfer-function representation of the
following system with MATLAB:

B–9–13.Consider the system defined by

Is the system completely state controllable and completely
observable?

B–9–14.Consider the system given by

Is the system completely state controllable and completely
observable? Is the system completely output controllable?

B

y 1

y 2

R =B

1

0

0

1

0

0

RC

x 1

x 2

x 3

S

C

x# 1

x# 2

x# 3

S =C

2

0

0

0

2

3

0

0

1

SC

x 1

x 2

x 3

S + C

0

1

0

1

0

1

SB

u 1

u 2

R

y =[1 1 0]C

x 1

x 2

x 3

S

C

x# 1

x# 2

x# 3

S =C

- 1

0

1

- 2

- 1

0

- 2

1

- 1

SC

x 1

x 2

x 3

S +C

2

0

1

Su

y =[1 0 0]C

x 1

x 2

x 3

S

C

x# 1

x# 2

x# 3

S =C

2

0

0

1

2

1

0

0

3

SC

x 1

x 2

x 3

S + C

0

1

0

1

0

1

SB

u 1

u 2

R

y =[0 0 1]C

x 1

x 2

x 3

S

C

x# 1

x# 2

x# 3

S =C

0

- 1

1

1

- 1

0

0

0

0

SC

x 1

x 2

x 3

S +C

0

1

0

Su

B–9–15.Is the following system completely state control-

lable and completely observable?

B–9–16.Consider the system defined by

Except for an obvious choice of c 1 =c 2 =c 3 =0, find an

example of a set of c 1 , c 2 , c 3 that will make the system

unobservable.

B–9–17.Consider the system

The output is given by

(a)Show that the system is not completely observable.

(b)Show that the system is completely observable if the

output is given by

B

y 1

y 2

R = B

1

1

1

2

1

3

RC

x 1

x 2

x 3

S

y=[1 1 1]C

x 1

x 2

x 3

S

C

x# 1

x# 2

x# 3

S =C

2

0

0

0

2

3

0

0

1

SC

x 1

x 2

x 3

S

y = Cc 1 c 2 c 3 DC

x 1

x 2

x 3

S

C

x# 1

x# 2

x# 3

S= C

0

0

- 6

1

0

- 11

0

1

- 6

SC

x 1

x 2

x 3

S +C

0

0

1

Su

y =[20 9 1]C

x 1

x 2

x 3

S

C

x# 1

x# 2

x# 3

S= C

0

0

- 6

1

0

- 11

0

1

- 6

SC

x 1

x 2

x 3

S +C

0

0

1

Su