Modern Control Engineering

Problems 855

x.=Ax+Bu y=Cx

x 2

x 3

k 2

k 1

k 3

r u

x

y=x 1

+– +–

Figure 10–58
Type 1 servo system.

Problems

B–10–1.Consider the system defined by

where

Transform the system equations into (a) controllable canon-
ical form and (b) observable canonical form.

B–10–2.Consider the system defined by

where

Transform the system equations into the observable canon-
ical form.

B–10–3.Consider the system defined by

where

By using the state-feedback control it is desired to
have the closed-loop poles at Deter-
mine the state-feedback gain matrix K.

B–10–4.Solve Problem B–10–3 with MATLAB.

s=- 2 ;j4,s=-10.

u=-Kx,

A= C

0

0

- 1

1

0

- 5

0

1

- 6

S, B= C

0

1

1

S

x# =Ax+Bu

A= C

- 1

1

0

0

- 2

0

1

0

- 3

S, B= C

0

1

1

S, C=[1 1 1]

y =Cx

x# =Ax+Bu

A=C C=[1 1 0]

- 1

1

0

0

- 2

0

1

0

- 3

S, B= C

0

0

1

S,

y =Cx

x# =Ax+Bu

B–10–5.Consider the system defined by

Show that this system cannot be stabilized by the state-

feedback control whatever matrix Kis chosen.

B–10–6.A regulator system has a plant

Define state variables as

By use of the state-feedback control it is desired

to place the closed-loop poles at

Determine the necessary state-feedback gain matrix K.

B–10–7.Solve Problem B–10–6 with MATLAB.

B–10–8.Consider the type 1 servo system shown in Figure

10–58. Matrices A,B, and Cin Figure 10–58 are given by

Determine the feedback gain constants k 1 , k 2 ,andk 3 such

that the closed-loop poles are located at

Obtain the unit-step response and plot the output

y(t)-versus-tcurve.

s=- 2 +j4, s=- 2 - j4, s=- 10

A= C

0

0

0

1

0

- 5

0

1

- 6

S, B= C

0

0

1

S, C=[1 0 0]

s=- 2 +j2 13 , s=- 2 - j2 13 , s=- 10

u=-Kx,

x 3 =x# 2

x 2 =x

1

x 1 =y

Y(s)

U(s)

=

10

(s+1)(s+2)(s+3)

u=-Kx,

B

x# 1

x# 2

R = B

0

0

1

2

RB

x 1

x 2

R + B

1

0

Ru