Performance of FCS2018 mk

The Steady – State Error of Feedback Control Systems 7.. The Steady – State Error of Feedback Control Systems 7.. • Many control systems experience input signals that are very similar to

Nguyễn Công Phương

CONTROL SYSTEM DESIGN

The Performance

of Feedback Control Systems

III State Variable Models

IV Feedback Control System Characteristics

V The Performance of Feedback Control Systems

VI The Stability of Linear Feedback Systems

VII The Root Locus Method

VIII.Frequency Response Methods

IX Stability in the Frequency Domain

X The Design of Feedback Control Systems

XI The Design of State Variable Feedback Systems

XII Robust Control Systems

XIII.Digital Control Systems

The Performance

of Feedback Control Systems

1 Introduction

2 Test Input Signals

3 Performance of Second – Order Systems

4 Effects of a Third Pole & a Zero on the Second –

Order System Response

5 The s – Plane Root Location & the Transient

Response

6 The Steady – State Error of Feedback Control Systems

7 Performance Indices

8 The Simplification of Linear Systems

9 System Performance Using Control Design Software

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The Performance

of Feedback Control Systems

1 Introduction

2 Test Input Signals

3 Performance of Second – Order Systems

4 Effects of a Third Pole & a Zero on the Second –

Order System Response

5 The s – Plane Root Location & the Transient

Response

6 The Steady – State Error of Feedback Control Systems

7 Performance Indices

8 The Simplification of Linear Systems

9 System Performance Using Control Design Software

s i tes.google.com/site/ncpdhbkhn 5

Test Input Signals (1)

• If the system is stable, the response to a specific input

signal will provide several measures of the performance

• But because the actual input signal of a system is usually

unknown, a standard test input signal is normally chosen.

• Using a standard input allows the designer to compare several competing designs

• Many control systems experience input signals that are very similar to the standard test signals

The Performance

of Feedback Control Systems

1 Introduction

2 Test Input Signals

3 Performance of Second – Order Systems

4 Effects of a Third Pole & a Zero on the Second –

Order System Response

5 The s – Plane Root Location & the Transient

Response

6 The Steady – State Error of Feedback Control Systems

7 Performance Indices

8 The Simplification of Linear Systems

9 System Performance Using Control Design Software

G s

s s

ω ζω

Performance of Second – Order

G s

s s

ω ζω

G s

s s

ω ζω

1 1

sin( )

n

n

t n

Performance of Second – Order

G s

s s

ω ζω

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0

Performance of Second – Order

Performance of Second – Order

Find K & p so that the transient response to a step should be as

fast as attainable while retaining an overshoot of less than 5%,

and the settling time should be less than 4 seconds

n

ζ ω

p K

ζω ω

Find K & p so that the transient response to a step should be as

fast as attainable while retaining an overshoot of less than 5%,

and the settling time should be less than 4 seconds

2; 2

0 0.2 0.4 0.6 0.8 1 1.2 1.4

T ime (s)

The Performance

of Feedback Control Systems

1 Introduction

2 Test Input Signals

3 Performance of Second – Order Systems

4 Effects of a Third Pole & a Zero on the Second –

Order System Response

5 The s – Plane Root Location & the Transient

Response

6 The Steady – State Error of Feedback Control Systems

7 Performance Indices

8 The Simplification of Linear Systems

9 System Performance Using Control Design Software

Effects of a Third Pole & a Zero on the

Second – Order System Response (2)

2

1 ( )

Settling time

n = 0.5

Effects of a Third Pole & a Zero on the

Second – Order System Response (4)

2

( / )( ) ( )

Settling time

Peak time

-3 -2 -1 0 1 2 3 4

The Performance

of Feedback Control Systems

1 Introduction

2 Test Input Signals

3 Performance of Second – Order Systems

4 Effects of a Third Pole & a Zero on the Second –

Order System Response

5 The s – Plane Root Location & the Transient

Response

6 The Steady – State Error of Feedback Control Systems

7 Performance Indices

8 The Simplification of Linear Systems

9 System Performance Using Control Design Software

s i tes.google.com/site/ncpdhbkhn 25

The s – Plane Root Location

& the Transient Response

-1 0 1

0 10

0 5 10 -10

0 10

0 5 10 -1

0 1

0 5 10 -1

0 1

5 10

0 5 10 0

1 2

0 0.5 1

The Performance

of Feedback Control Systems

1 Introduction

2 Test Input Signals

3 Performance of Second – Order Systems

4 Effects of a Third Pole & a Zero on the Second –

Order System Response

5 The s – Plane Root Location & the Transient

Response

6 The Steady – State Error of Feedback Control

Systems

7 Performance Indices

8 The Simplification of Linear Systems

9 System Performance Using Control Design Software

s i tes.google.com/site/ncpdhbkhn 27

The Steady – State Error

of Feedback Control Systems (1)

ss s

K

+

The Steady – State Error

of Feedback Control Systems (2)

K

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The Steady – State Error

of Feedback Control Systems (3)

K

The Steady – State Error

of Feedback Control Systems (4)

K

= +

v

A K

A K

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The Steady – State Error

of Feedback Control Systems (5)

K K

The Steady – State Error

of Feedback Control Systems (6)

Step input Output Error

-1 -0.5 0 0.5

1

Ramp input Output Error

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The Steady – State Error

of Feedback Control Systems (7)

0 0.5

The Steady – State Error

of Feedback Control Systems (8)

c c

s K G s G s

s K G s G s

τ τ

1 lim ( )

1 lim ( ) ( )

ss

s

c s

s i tes.google.com/site/ncpdhbkhn 35

The Steady – State Error

of Feedback Control Systems (9)

1

K

Ex 2

1 0

1

ss

c s

e

→

= +

s→ s

= +

The Steady – State Error

of Feedback Control Systems (10)

The Performance

of Feedback Control Systems

1 Introduction

2 Test Input Signals

3 Performance of Second – Order Systems

4 Effects of a Third Pole & a Zero on the Second –

Order System Response

5 The s – Plane Root Location & the Transient

Response

6 The Steady – State Error of Feedback Control Systems

7 Performance Indices

8 The Simplification of Linear Systems

9 System Performance Using Control Design Software

s i tes.google.com/site/ncpdhbkhn 37

Performance Indices (1)

• A performance index is a quantitative measure of the performance of a system and is chosen so that emphasis is given to the important system

specifications.

• A system is considered an optimum control system

when the system parameters are adjusted so that the index reaches an extremum, commonly a

minimum value.

• A performance index must be a number that is

always positive or zero.

• Then the best system is defined as the system that minimizes this index.

-1 -0.5 0

ISE

s i tes.google.com/site/ncpdhbkhn 39

Performance Indices (2)

2 0

ISE ( )

T

e t dt

= The Integral of the Square of the Error

Integral of Absolute magnitude of Error, IAE

0 1000 2000

3000

Integral of T ime multiplied by Absolut e Error, ITAE

50 100 150 200 250

Integral of Time multiplied by Squared Error, ITSE

T

te t dt

= 

1 2 3 4 5 6 7 8

= 1 – (sum of all different loop gains)

+ (sum of the gain products of all combinations of two nontouching loops) – (sum of the gain products of all combinations of three nontouching loops) +

P k : gain of kth path from input to output

Δk (cofactor): the determinant Δ with the loop(s) touching the kth path removed.

= − + = → K3 = 10 = 3.2

Performance Indices (9)

0 1

( ) ( )

The Optimum Coefficients of T(s)

Based on the ITAE Criterion for a Step Input

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Performance Indices (10)

1 0 1

( ) ( )

The Optimum Coefficients of T(s)

Based on the ITAE Criterion for a Ramp Input

( )

Y s

2 0

ω ω ω

( )

Y s

2 0

1.2

Step Response

Time (seconds)

The Performance

of Feedback Control Systems

1 Introduction

2 Test Input Signals

3 Performance of Second – Order Systems

4 Effects of a Third Pole & a Zero on the Second –

Order System Response

5 The s – Plane Root Location & the Transient

Response

6 The Steady – State Error of Feedback Control Systems

7 Performance Indices

8 The Simplification of Linear Systems

9 System Performance Using Control Design Software

s i tes.google.com/site/ncpdhbkhn 51

The Simplification of Linear Systems (1)

T

1

0 0.2 0.4 0.6 0.8 1

T2

The Simplification of Linear Systems (2)

H L

k

k k

k

d

ds d

q k

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The Simplification of Linear Systems (3)

3 2

1 ( )

The Simplification of Linear Systems (4)

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The Simplification of Linear Systems (5)

The Simplification of Linear Systems (6)

0 1 2 3 4 5 6 7 8 9 10 0

The Simplification of Linear Systems (8)

The Performance

of Feedback Control Systems

1 Introduction

2 Test Input Signals

3 Performance of Second – Order Systems

4 Effects of a Third Pole & a Zero on the Second –

Order System Response

5 The s – Plane Root Location & the Transient

Response

6 The Steady – State Error of Feedback Control Systems

7 Performance Indices

8 The Simplification of Linear Systems

9 System Performance Using Control Design

Software

s i tes.google.com/site/ncpdhbkhn 59

System Performance Using Control Design Software