Basic Control Engineering

Prof. Wonhee Kim

Ch.7. Steady-state Error

2

Steady-state Error: Frequency domain

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Steady-state Error: Frequency domain

Fig. Steady-state error: a. step input; b. ramp input

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Steady-state Error: Frequency domain

P(s)+

-

R(s) C(s)PC(s)

E(s)

1

1

C

C

E s R s C s

C s P s P s E s

E s R sP s P s

0lim

1sC

sR se

P s P s

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Steady-state Error: Frequency domain

Example 7.1) 2

5

7 5T s

s s

2

2

2

1

51

7 5

1 7 5

7 10

E s R s

s s

s s

s s s

0

1lim

1 2sC

sR se

P s P s

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Steady-state Error: Frequency domain

0lim

1sC

sR se

P s P s

0

1

1 lim Cs

eP s P s

1

R ss

0

lim Cs

P s P s

1 2

1 2

C n

s z s zP s P s

s s p s p

Integrator is required to eliminate the steady-state error

For

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Static Error Constants and System Type

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Static Error Constants and System Type

Example 7.4)

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Static Error Constants and System Type

Example 7.4)

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Static Error Constants and System Type

Example 7.4)

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Static Error Constants and System Type

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Steady-State Error Specifications

For example, if a control system has the specification Kv = 1000, we can draw several conclusions:

1. The system is stable.

2. The system is of Type 1, since only Type 1 systems have Kv’s that are finite constants. Recall that Kv = 0

for Type 0 systems, whereas Kv =∞ for Type 2 systems.

3. A ramp input is the test signal. Since Kv is specified as a finite constant, and the steady-state error for a

ramp input is inversely proportional to Kv, we know the test input is a ramp.

4. The steady-state error between the input ramp and the output ramp is 1/ Kv per unit of input slope.

13

Static Error Constants and System Type

Example 7.5) What information is contained in the specification Kp = 1000?

The system is stable. The system is Type 0, since only a Type 0 system has a finite

Kp. Type 1 and Type 2 systems have Kp = . The input test signal is a step, since

Kp is specified. Finally, the error per unit step is

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Static Error Constants and System Type

Example 7.6)

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Steady-state Error with Disturbance

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Steady-state Error with Disturbance

Increasing G1(s) can reduce both eR and eD

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Steady-state Error with Disturbance

Example 7.7)

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Steady-State Error for Nonunity Feedback Systems

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Steady-State Error for Nonunity Feedback Systems

Example 7.8)

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Steady-State Error for Nonunity Feedback Systems

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Steady-State Error for Nonunity Feedback Systems

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Steady-State Error for Nonunity Feedback Systems