Lecture introduction to control systems chapter 3 system dynamics (dr huynh thai hoang)

Graphical representation of frequency responseæ Bode diagram: is a graph of the frequency response of a linear æ Bode diagram: is a graph of the frequency response of a linearsystem vers

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Lecture Notes

Introduction to Control Systems

Instructor: Dr Huynh Thai Hoang Department of Automatic Control Faculty of Electrical & Electronics Engineering

Ho Chi Minh City University of Technology

Email: hthoang@hcmut.edu.vn

huynhthaihoang@yahoo.com Homepage: www4.hcmut.edu.vn/~hthoang/

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Chapter 3

SYSTEM DYNAMICS

24 September 2011 © H T Hoàng - www4.hcmut.edu.vn/~hthoang/ 2

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æ The concept of system dynamics

Content

æ The concept of system dynamics

Ø Time response

Ø Frequency responseq y p

æ Dynamics of typical components

æ Dynamics of control systems

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The concept of system dynamics

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The concept of system dynamics

æ System dynamics is the study to understanding the behaviour of

complex systems over time

æ Systems described by similar mathematical model will expose

similar dynamic responses

æ To study the dynamic responses, input signals are usually chosen to

be basic signals such as Dirac impulse signal, step signal, or

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()

(s U s G s G s

{ }( ) { ( )} ( ))

(t 1 Y s 1 G s g t

⇒ Impulse response is the inverse Laplace transform of the transfer function

æ Impulse response is also referred as weighting function

æ It is possible to calculate the response of a system to a arbitrarily

input by taking convolution of the weighting function and the input

t t

t) ( )* ( ) ( ) ( )(

=

t y

0

)(

)()

(

*)()

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G s U s

s

G s

Y t

y

0

1 1

)(

)

()

⇒ The step response is the integral of the impulse response

æ The step response is also referred as the transient function of the

system

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Impulse and step response example

æ Calculate the impulse response and step response of the system

G(s)

U (s) Y (s)

æ Calculate the impulse response and step response of the system

described by the transfer function:

)5(

1)

)(

+

s s

)()

55

)5(

)()

(

s s

s G t

t

e t

5

45

1)

⇒

55

41

25

45

125

4)

5(

1)

()

+

−+

s s

s

s s

s

G t

44

1)

h

2525

5

)( = − − t5 +

e t

t h

⇒

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Frequency response

æ Observe the response of a linear system at steady state when theinput is a sinusoidal signal

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Frequency response definition

æ It can be observed that for linear system if the input is a sinusoidal

æ It can be observed that, for linear system, if the input is a sinusoidalsignal then the output signal at steady-state is also a sinusoidalsignal with the same frequency as the input, but different amplitude

d h

and phase

HT

u (t)=U m sin (jω ) y (t)=Y m sin (jω + ϕ)

æ Definition: Frequency response of a system is the ratio between the

U (jω ) Y (jω )

æ Definition: Frequency response of a system is the ratio between thesteady-state output and the sinusoidal input

)( jω

Y

F

)(

)

(ω

j U

j

=responseFrequency

It is proven that: Frequency response = G(s) = G( jω)

It is proven that: Frequency response G(s) ω G( jω)

j

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Magnitude response and phase response

æ In general G(jω) is a complex function and it can be represented in

æ In general, G(jω) is a complex function and it can be represented inalgebraic form or polar form

) ()

()

( ω ω ω ω jϕ ω

e M

jQ P

j

where:

)()

()

)

()

ω

ω ω

ω

ϕ

P

Q tg

æ Physical meaning of frequency response:

Ø The magnitude response provides information about the gain ofthe system with respect to frequency

Ø The phase response provides information about the phase shiftp p p p

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Graphical representation of frequency response

æ Bode diagram: is a graph of the frequency response of a linear

æ Bode diagram: is a graph of the frequency response of a linearsystem versus frequency plotted with a log-frequency axis Bodediagram consists of two plots:

L(ω) versus frequency ω

)(l

20)

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Graphical representation of frequency response (cont’)

Bode diagram Nyquist plot

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Crossover frequency

æ Gain crossover frequency(ω ): is the frequency where the amplitude

æ Gain crossover frequency(ωc): is the frequency where the amplitude

of the frequency response is 1 (or 0 dB)

1)

( c =

M (ωc) ⇔ L(ωc c) = 0

æ Phase crossover frequency (ω−π): is the frequency where phase shift

of the frequency response is equal to −1800 (or equal to −π radian)

0180)

(ω−π = −

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Stability margin

æ Gain margin (GM):

)(

M

æ Phase margin (ΦM)

)(

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Dynamics of basic factors

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The proportional gain

g = δ

) ( 1 )

(ω =

ϕ

⇒

Ø Phase response:

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The proportional gain – – Time response Time response

(a) Weighting function (b) Transient function

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The proportional gain – – Frequency response Frequency response

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g =

) ( 1 )

h =

æ Frequency response:

ωω

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Time response of integral factor

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Frequency response of integral factor

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) ( )

(

Ø Phase response:

⇒

090)

(ω =

ϕ

Ø Phase response:

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Time response of derivative factor

Transient function

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Frequency response of derivative factor

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æ Time response:

1

1)

T Ts

1)

Ts s

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Time response of first

Time response of first order lag factor order lag factor

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Frequency response of first

Frequency response of first order lag factor order lag factor

1)

lg20)

)(

(

æ Approximation of the Bode diagram by asymptotes:

Øω <1/T : the asymptote lies on the horizontal axis: the asymptote lies on the horizontal axis

Ø : the asymptote has the slope of −20dB/dec

T

/1

<

ω

T

/1

>

ω

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Frequency response of first order lag factor (cont’) order lag factor (cont’)

corner frequency

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()

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Time response of first

Time response of first order lead factor order lead factor

Transient function

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Frequency response of first order lead factor order lead factor

Ø Magnitude response:

1 )

( jω = Tjω +

G

2 21

lg20)

)(

)

ϕ = tg− T

Ø Phase response:

Øω <1/T : the asymptote lies on the horizontal axis

Øω >1/T : the asymptote has the slope of +20dB/dec

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Frequency response of first order lead factor (cont’) order lead factor (cont’)

corner frequency

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++

=

Ts s

T

s G

ξ (0 < ξ <1)

Ø Impulse response: g t e [ n t]

t n

n

)1

(

sin1

(

sin1

1)

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Time response of second

Time response of second order oscillating factor order oscillating factor

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Frequency response of second

Frequency response of second order oscillating factor order oscillating factor

1

12

1)

++

−

=

ωξ

ω

Tj T

j G

1)

2 2

4)

1(

)

(

ωξ

ω

T T

2 2

4)

1(l

20)

1(lg20)

11

2)

ϕ

T

T tg

⎠

⎝ − 2 2

1 T ω

Ø : the asymptote lies on the horizontal axis

Ø : the asymptote has the slope of −40dB/dec

T

/1

<

ω

T

/1

>

ω

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Frequency response of second

Frequency response of second order oscillating factor order oscillating factor

Corner frequency

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Time delay factor

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Time response of time delay factor

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Frequency response of time delay factor

1)(

Ø Magnitude response:

ωω

ϕ( ) = −T

Ø Phase response:

⇒

1)(ω =

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Frequency response of time delay factor (cont’)

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Dynamics of control systems

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Time response of control systems

æ Consider a control system which has the transfer function G (s):

n n

m m

a s a s

a s

a

b s b s

b s

G

+ +

1

1 1 0

s a s

a0 + 1 + + −1 +

æ Laplace transform of the transient function:

1

s a s a s

a s

b s b s

b s

s G s

H

n n

m m

) (

) ( )

(

1

1 1 0

1

1 1 0

+ +

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Remarks on the time response of control systems

æ If G(s) does not contain a ideal integral or derivative factor then:

Ü Weighting function decays to 0

b s b s

b s

b

0 lim

) ( lim

) (

1

1 1 0

1 1

0 0

+ +

n n

m m

s

s a s a s a s a

b s b s

b s

b s s

sG

g

L L

Ü Transient function approaches to non-zero value at steady state:

0

1 lim

) ( lim

)

1 1

b b

s b s

b s

s sH

0

lim )

( lim

) (

1

1 1 0

n

n n

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Remarks on the time response of control systems (cont’)

æ If G( ) t i id l i t l f t ( 0) th

æ If G(s) contain a ideal integral factor (a n = 0) then:

Ü Weighting function has non-zero steady-state:

b s b s

b s

b m m

0 lim

) ( lim

) (

1

1 1 0

1 1

0 0

+

+ +

b s

b s s

sG g

n

n n

m m

sH

m m

1

1 1 0

1 lim

) ( lim

s s

1

1 1 0

0

0 ( ) lim lim

) (

L

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æ If G( ) t i id l d i ti f t (b 0) th

æ If G(s) contain a ideal derivative factor (b m = 0) then:

Ü Weighting function approaches zero at steady-state

s b s

b s

b

0 lim

) ( lim

) (

1

1 1 0

1 1

0 0

+ +

n n

m s

s a s a s a s a

s b s

b s

s sG

g

L L

Ü Transient function approaches zero at steady-state

0

1 lim

) ( lim

)

1 1

s b s

b s

s sH

0

lim )

( lim

) (

1

1 1 0

n n

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æ If G( ) i ( ≤ ) th h(0) 0

æ If G(s) is proper (m ≤ n) then h(0) = 0

0

1 lim )

( lim )

0

1 1

m

m m

s b s

b s

H

1

1 1

n n

s

s s a s a s L a s a

æ If G(s)( ) is strictly proper (y p p (m < n) then) g(0) = 0 g( )

0 lim

) ( lim )

0 (

1

1 1 0

1

1 1

+ +

n n

m

m m

s

s a s a s a s a

s b s

b s

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Frequency response of control system

æ Consider a control system which has the transfer function G(s)

æ Consider a control system which has the transfer function G(s).Suppose that G(s) consists of basis factors in series:

G

1

)()

G

1

)(

ϕ

⇒ The Bode diagram of a system consisting of basic factors in series

equals to the summation of the Bode diagram of the basic factors

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Approximation of Bode diagram

æ Suppose that the transfer function of the system is of the form:

K)()

()

(s Ks G1 s G2 s G3 s

(α>0: the system has ideal derivative factor(s)

æ Step 1: Determine all the corner frequency ω =1/T and sort them

(α>0: the system has ideal derivative factor(s)

α<0: the system has ideal integral factor(s))

æ Step 1: Determine all the corner frequency ωi =1/T i , and sort them

in ascending order ω1 <ω2 < ω3 …

æ Step 2: The approximated Bode diagram passes the point A having

20)

L

ω0 is a frequency satisfying ω0 < ω1 If ω1 > 1 then it is possible to

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Approximation of Bode diagram (cont’)

æ Step 3: Through point A draw an asymptote with the slope:

æ Step 3: Through point A, draw an asymptote with the slope:

Ø (− 20 dB/dec × α) if G(s) has α ideal integral factors

Ø (+ 20 dB/dec × α) if G(s) has α ideal derivative factors

The asymptote extends to the next corner frequency

æ Step 4: At the corner frequency ωS ep : e co e eque cy ωi i =1/T / i i , the slope of the asymptote is , e s ope o e asy p o e sadded with:

Ø (−20dB/dec × β i) if G i (s) is a first-order lag factor (multiple βi)

Ø (−40dB/dec × β i) if G i (s) is a second-order oscillating factor (multiple βi)

The asymptote extends to the next corner frequency

æ Step 5: Repeat the step 4 until the asymptote at the last corner

frequency is plotted

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Approximation of Bode diagram – – Example 1 Example 1

æ Plot the Bode diagram using asymptotes:

) 1 01

, 0 (

) 1 1

, 0 (

100 )

s s

, 0

1 1

, 0

1 1

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Approximation of Bode diagram – – Example 1 (cont’) Example 1 (cont’)

10 2

3

æ In the Bode diagram, the gain crossover frequency is 103 rad/sec

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26

54 = +

−

æ The slope of segment CD:

æ The corner frequencies:

(rad/sec)5

020

26

400

301

2 2

1

)1(

)1)(

1

()

s T s

T

K s

40lg

Tiêu đề	System Dynamics
Người hướng dẫn	Dr. Huynh Thai Hoang
Trường học	Ho Chi Minh City University of Technology
Chuyên ngành	Automatic Control
Thể loại	Lecture Notes
Năm xuất bản	2011
Thành phố	Ho Chi Minh City

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