Lecture Control system design: Stability in the frequency domain - Nguyễn Công Phương

This chapter include all of the following content: Mapping contours in the s – plane, the nyquist criterio, relative stability and the nyquist criterion, time – domain performance criteria in the frequency domain, system bandwidth, the stability of control systems with time delays, pid controllers in the frequency domain, stability in the frequency domain using control design software.

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Nguyễn Công Phương

CONTROL SYSTEM DESIGN

Stability in the Frequency Domain

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I Introduction

II Mathematical Models of 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

sites.google.com/site/ncpdhbkhn 2

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1 Mapping Contours in the s – Plane

2 The Nyquist Criterion

3 Relative Stability and the Nyquist Criterion

4 Time – Domain Performance Criteria in the

Frequency Domain

5 System Bandwidth

6 The Stability of Control Systems with Time

Delays

7 PID Controllers in the Frequency Domain

8 Stability in the Frequency Domain Using

Control Design Software

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1 2 2

σ ω

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-1 -0.5 0 0.5 1 1.5

D

A B

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-2 -1 0 1 2 3 -2

-1 0 1 2

F s = s +

Cauchy’s theorem: If a contour Γ s in the s-plane encircles Z zeros and P poles of F(s) and does not pass through any poles or zeros of F(s) and the traversal is in the

encircles the origin of the F(s)-plane N = Z – P times in the clockwise direction

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-0.05 0 0.05 0.1 0.15

u

F(s)-plane

through any poles or zeros of F(s) and the traversal is in the clockwise direction along

F(s)-plane N = Z – P times in the clockwise direction

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through any poles or zeros of F(s) and the traversal is in the clockwise direction along

F(s)-plane N = Z – P times in the clockwise direction

-1 -0.5 0 0.5 1 1.5

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2 The Nyquist Criterion

Frequency Domain

5 System Bandwidth

Delays

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The Nyquist Criterion (1)

• F(s) = 1 + L(s) = 0

• A feedback system is stable if and only

if the contour ΓL in the L(s) – plane

does not encircle the (–1, 0) point when

the number of poles of L(s) in the right

– hand s – plane is zero (P = 0).

• (when the number of poles of L(s) in the

right – hand s – plane is other than

zero) A feedback system is stable if and

only if, for the contour ΓL , the number

of counterclockwise encirclements of

the (–1, 0) point is equal to the number

of poles of L(s) with positive real parts.

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1

1 1

A feedback system is stable if and only if the contour ΓL in the L(s) – plane does not encircle

the (–1, 0) point when the number of poles of L(s) in the right – hand s – plane is zero (P = 0).

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Cauchy’s theorem: If a contour Γ s in the s-plane encircles Z zeros and P poles of L(s)

and does not pass through any poles or zeros of L(s) and the traversal is in the

encircles the origin of the L(s)-plane N = Z – P times in the clockwise direction

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• It is sufficient to

construct the contour

ΓL for the frequency

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-6 -4 -2 0 2 4 -6

-4 -2 0 2 4 6

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u

K = 3, τ1 = 1, τ2 = 1

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u

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- 1.5 -1

- 0.5 0 0.5 1 1.5 2 2.5

-1.5 -1 -0.5 0 0.5 1 1.5 2

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-1.5 -1 -0.5 0 0.5 1 1.5 2

2 2

2

1 2

4 2 1/

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-1.5 -1 -0.5 0 0.5 1 1.5 2

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3 Relative Stability and the Nyquist Criterion

Frequency Domain

5 System Bandwidth

Delays

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Relative Stability and the Nyquist Criterion (1)

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1 Gain margin G M 20 log

α

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Ex 2

-1 -0.5 0 0.5

2

6 ( )

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Ex 2

-1 -0.5 0 0.5

2

6 ( )

2 2

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1 Gain margin G M 20 log

The gain margin is the increase in the

system gain when phase = –180° that

will result in a marginally stable

system with intersection of the

–1 + j0 point on the Nyquist diagram.

The phase margin is the amount of phase

shift of the L(jωt) at unity magnitude that

will result in a marginally stable system

with intersection of the – 1 + j0 point on

the Nyquist diagram.

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Ex 2

2

6 ( )

-20 -15 -10 -5 0 5

-180 -135 -90 -45

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-225 -180 -135 -90

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

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4 Time – Domain Performance Criteria in the

Frequency Domain

5 System Bandwidth

Delays

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Time – Domain Performance Criteria

in the Frequency Domain (1)

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0.5 1 1.5 2 2.5 3 3.5 4 4.5

K > K

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-1 -0.5 0 0.5 1 1.5

( )

( ) 1 ( ) ( )

j c

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-15 -10 -5 0 5 10 15 20

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Frequency Domain

5 System Bandwidth

Delays

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0.2 0.4 0.6 0.8 1

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Frequency Domain

5 System Bandwidth

6 The Stability of Control Systems with Time

Delays

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The Stability of Control Systems

with Time Delays (1)

of an even at one point in a system and its

resulting action at another point in the system.

magnitude of the transfer function.

system with a time delay.

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The Stability of Control Systems

with Time Delays (2)

Ex.

2

31.5 ( )

1 2

sT

T s e

T s

≈

+

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Frequency Domain

5 System Bandwidth

Delays

7 PID Controllers in the Frequency Domain

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K = –7000, τ = 5s Design the PID controller so that

2 2

( / ) ( / ) ( )

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PID Controllers

Ex.

K = –7000, τ = 5s Design the PID controller so that

2 2

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Frequency Domain

5 System Bandwidth

Delays

8 Stability in the Frequency Domain Using

Control Design Software

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Stability in the Frequency Domain Using Control Design Software

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