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M_1001

CH.2

E2.1, E2.12, E2.13, P2.25, P2.50, CP2.9

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A unity, negative feedback system has a nonlinear function , as

shown in Figure E2.1. For an input r in the range of 0 to 4. Calculate and plot the

open-loop and closed-loop output versus input and show that the feedback system

results in a more linear relationship.

E2.1

We have for the open-loop

y = 2

and for the closed-loopand y = 2

So, and

Ans :

For example, if r = 1,

then implies that e = 0.618.Thus,

y = 0.382. A plotversus r is shown in

Figure E2.1 FIGURE E2.1

Plot of open-loop versus closed-loop.

2 0e e r+ =2e r e=

e r y=

2 1 0e e+ =

2( )y f e e= =

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E2.12 Off-road vehicles experience many disturbance inputs as they traverse over

rough roads. An active suspension system can be controlled by a sensor that looks

"ahead" at the road conditions. An example of a simple suspension system that can

accommodate the bumps is shown in Figure E2.12. Find the appropriate gain K1so

that the vehicle does not bounce when the desired deflection is R(s) = 0 and thedisturbance is Td(s).

Ans :

FIGURE E2.12 Signal flow graph.

The transfer function from Td(s) to Y(s) is

If we set

then Y(s) = 0 for any Td

(s).

The signal flow graph is shown in Fig. E2.12. Find Y(s) whenR(s) = 0.

( ) ( )( )( )

( )1 2 1 22 2

( ) ( ) (1 ) ( )( )

1 1 ( )

d d dG s T s K K G s T s G s K K T sY s

K G s K G s

= =

+

1 2 1

2

11, ;K K K

K= =

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H. S. Black is noted for developing a negative feedback amplifier in 1927.

Often overlooked is the fact that three years earlier he had invented a circuit

design technique known as feedforward correction. Recent experiments have

shown that this technique offers the potential for yielding excellent amplifier

stabilization. Black's amplifier is shown in Figure P2.25(a) in the form recordedin 1924. The block diagram is shown in Figure P2.25(b). Determine the transfer

function between the output Y(s)and the input R(s)and between the output and

the disturbance Td(s). G(s) is used to denote the amplifier represented by in

Figure P2.25(a).

P2.25

Ans :

The transfer function fromR(s) and Td

(s) to Y(s) is given by

Thus,Y(s)

R(s)= G(s) ;

Also, we have that Y(s) = 0;

whenR(s) = 0. Therefore, the effect of the disturbance, Td(s), is eliminated.

1( ) ( )( ( ) ( ( ) ( ) ( ))) ( ) ( ) ( ) ( ) ( )

( )d d

Y s G s R s G s R s T s T s G s R s G s R sG s

= + + + =

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P2.50 A closed-loop control system is shown in Figure P2.50.

(a) Determine the transfer function T(s) =Y(s)

R(s).

(b) Determine the poles and zeros of T(s).

(c) Use a unit step input,R(s) = 1s, and obtain the partial fraction expansion for

Y(s) and the value of the residues.

(d) Ploty(t) and discuss the effect of the real and complex poles of T(s). Do the

complex poles or the real poles dominate the response?

(e) Predict the final value ofy(t) for the unit step input.

(a) The closed-loop transfer function is

(b) The poles of T(s) are s1

= - 5 and s2,3

= - 20

j50.(c) The partial fraction expansion (with a step input) is

(d) The step response is shown in Figure 2.50. The real and complex

(e) The final value of y(t) is

Ans:

0.9655 1.0275 0.0310 0.0390 0.0310 0.0390( )

5 20 50 20 50

j jY s

s s s j s j

+= + +

+ + + +

( )

( ) 3 2s 14000

( )1 s 45 3100 14500

GT s

G s s s= =

+ + + +

0lim ( ) 0.9655ss sy sY s= =

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CP 2.9 Consider the feedback control system in Figure CP2.9, where

G(s) =+1

+2 and H(s) =

1

+1

(a,b) Computing the closed-loop transfer function yields

The poles are s = - 3 ;-1 and the zeros are s = - 1; -1.(c) Yes, there is one pole-zero cancellation. The transfer function

(after pole-zero cancellation) is

(d) Only after all pole-zero cancellations have occurred is the transfer function

of minimal complexity obtained.

( )2

2

( ) 2 1

1 ( ) ( ) 4 3

G s s sT s

G s H s s s

+ += =

+ + +

1( )

3

sT s

s

+=

+

Ans:

(a) Using an m-file, determine the closed-loop tra

nsfer function.

(b) Obtain the pole-zero map using the pzmap function. Where

are the closed-loop system poles and zeros?

(c) Are there any pole-zero cancellations? If so, use the minreai function

to cancel common poles and zeros in the closed-loop transfer function.

(d) Why is it important to cancel common poles andzeros in the transfer function?