Modern Control Engineering

Problems 857

R(s) Observer U(s) Y(s)

+– controller

s^2 + 2 s+ 50

s(s+ 4) (s+ 6)

Figure 10–60
Control system with observer controller in the
feedforward path.

It is desired to design a regulator for this system. Using the
pole-placement-with-observer approach, design an observer
controller.
Choose the desired closed-loop poles for the pole-
placement part to be

s=–0.7071+j0.7071, s=–0.7071-j0.7071

and assuming that we use a minimum-order observer, choose
the desired observer pole at

s=–5

B–10–14.Consider the system

where

Design a regulator system by the pole-placement-with-
observer approach. Assume that the desired closed-loop
poles for pole placement are located at

s=–1+j, s=–1-j, s=–5

The desired observer poles are located at

s=–6, s=–6, s=–6

Also, obtain the transfer function of the observer controller.

B–10–15.Using the pole-placement-with-observer approach,
design observer controllers (one with a full-order observer and
the other with a minimum-order observer) for the system
shown in Figure 10–60. The desired closed-loop poles for the
pole-placement part are

s=–1+j2, s=–1-j2, s=–5

A=C

0

0

- 6

1

0

- 11

0

1

- 6

S, B= C

0

0

1

S, C=[ 1 0 0 ]

y =Cx

x# =Ax+Bu

The desired observer poles are

s=–10, s=–10, s=–10 for the full-order observer

s=–10, s=–10 for the minimum-order observer.

Compare the unit-step responses of the designed systems.

Compare also the bandwidths of both systems.

B–10–16.Using the pole-placement-with-observer approach,

design the control systems shown in Figures 10–61(a) and (b).

Assume that the desired closed-loop poles for the pole place-

ment are located at

s=–2+j2, s=–2-j2

and the desired observer poles are located at

s=–8, s=–8

Obtain the transfer function of the observer controller.

Compare the unit-step responses of both systems. [In System

(b), determine the constant Nso that the steady-state out-

puty(q) is unity when the input is a unit-step input.]

R(s) Observer Y(s)

+– controller

1

s(s+ 1)

1

s(s+ 1)

Y(s)

Observer

controller

+–

(b)

R(s)

N

(a)

Plant

Figure 10–61

Control systems with observer controller: (a) observer

controller in the feedforward path; (b) observer controller

in the feedback path.

B–10–17.Consider the system defined by

where

a=adjustable parameter 70

A=C

0

0

- 1

1

0

- 2

0

1

• a

S

x

=Ax