Lecture Fundamentals of control systems: Chapter 5

Lecture Notes Lecture Notes

Fundamentals of Control Systems Fundamentals of Control Systems

Instructor: Assoc. Prof. 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 5 Chapter 5

ANALYSIS OF CONTROL SYSTEM ANALYSIS OF CONTROL SYSTEM

PERFORMANCE PERFORMANCE

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Content Content

 Performance criteria  Steady state error  Transient response  Transient response  The optimal performance index  Relationship between frequency domain performances and  Relationship between frequency domain performances and

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time domain performances.

Performance criteria Performance criteria

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Performance criteria: Steady state error Performance criteria: Steady state error

yfb(t) (t)

ess

Y(s) R(s) E(s)

r(t)

G(s) +_

Yfb(s)

e(t)

ess

t

0 0

 Error: is the difference between the set point (input) and the  Error: is the difference between the set-point (input) and the

H(s)

feedback signal.

te )(

tr )(

t )(

sE )(

sR )(

sY )(



y fb

fb

 Steady-state error: is the error when time approaching infinity.

sE E

s )( )(



e ss



lim li s 0 

lim te li )( )( t 

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

Percent of Overshoot (POT) Performance criteria –– Percent of Overshoot (POT) Performance criteria

 Overshoot: refers to an output exceeding its steady-state value.  Overshoot: refers to an output exceeding its steady state value

y(t)

overshoot overshoot

ymax

ymax yss

yss

t

No overshoot

t

 Percentage of Overshoot (POT) is an index to quantify the

%100

POT





 max ssy y

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overshoot of a system, POT is calculated as: overshoot of a system, POT is calculated as:

 Settling time (t ): is the time required for the response of a  Settling time (ts): is the time required for the response of a system to reach and stay within a range about the steady- state value of size specified by absolute percentage of the steady-state value (usually 2% or 5%)

Settling time and rise time Performance criteria –– Settling time and rise time Performance criteria

 Rise time (tr):

is the time required for the response of a

y(t)

(1+)yss yss (1)yss

yss 0.9yss

0.1yss 0 0

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system to rise from 10% to 90% of its steady-state value. system to rise from 10% to 90% of its steady state value

state error Steady--state error Steady

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state error Steady--state error Steady

R(s) R( ) Y(s) Y( ) E(s) E( )

G(s) +_

Yfb(s) Y ( )

 Error expression:

sE )(



sR )( )( sHsG )(



 Steady-state error:

s )(



e ss

lim s 0 

sR s )( )( )( )( )( sHsG



H(s)

 Remark: Steady-state error not only depends on the structure p and parameters of the system but also depends on the input signal.

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p p y

state error to step input Steady--state error to step input Steady

 Step input:  Step input:

sR sR

s s

/1)(  /1)( 

 Steady-state error:



e ss

1 pK1 K 1 

sHsG )( )(



lim s 0 

yfb(t)

with (position constant)

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G(s)H(s) does not have G(s)H(s) does not have any deal integral factor G(s)H(s) has at least 1 G(s)H(s) has at least 1 ideal integral factor

state error to ramp input Steady--state error to ramp input Steady

 Ramp input:  Ramp input:

sR sR

2/1)( s /1)( s 



sHs )( )(



lim s s 0 0 

1 e  ss K vK

yfb(t)

r(t)

ess 0 e 0

ess = 0 0

e(t) 

G(s)H(s) has 1 ideal G(s)H(s) has 1 ideal

G(s)H(s) does not have G(s)H(s) does not have deal integral factor

integral factor

G(s)H(s) has at least G(s)H(s) has at least 2 ideal integral factors

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with (velocity constant)

state error to parabolic input Steady--state error to parabolic input Steady

 Parabolic input: t b li

3/1)(R 3/1)( sR s 

P i

 

sHsGs sHsGs )( )( )( )(

K K  a

lim 2 lim s 0 

with with

1 e e  ss K

yfb(t)

r(t)

ess0 ss

ess 0 ess= 0

e(t) 

G(s)H(s) has 2 ideal G(s)H(s) has 2 ideal

G(s)H(s) has less than G(s)H(s) has less than 2 ideal integral factors

G(s)H(s) has more than G(s)H(s) has more than 2 ideal integral factors

integral factors

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(acceleration constant )

Transient response Transient response

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order system First--order system First

K 1Ts

 Transfer function:

sG )( )( 

1 1

K Ts T

 First order system has 1 real pole:

p 1  1

 Transient response:

sY )(

sGsR )( )(



1 s s

1 T T K 1Ts 1 Ts 



R(s) Y(s)

/Tte





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ty )( K 1( )

order system (cont’) First--order system (cont’) First

y(t)

Im s

(1+).K K (1).K

Re s

0.63K

1/T

Pole – zero plot Pole zero plot of a first order system Transient response Transient response of the first order

/Tte  e

 



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ty )( )( ty K K 1( 1( ) )

First--order system First

Remarks order system –– Remarks

 First order system has only one real pole at (1/T),

its

 Time constant T: is the time required for the step response of f

transient response doesn’t have overshoot.

i d f th ti t T i th t

 The further  The further

t Ti the system to reach 63% its steady-state value.

the pole (1/T) of the pole ( 1/T) of

 Settling time of the first order system is:

lnT lT



ts t

 1     

    

the system is from the the system is from the imaginary axis, the smaller the time constant and the faster the time response of the system.

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0.05 (5% criterion) where  = 0 02 (2% criterion) or  = 0 05 (5% criterion) where  0.02 (2% criterion) or 

order system First--order system First

 The further the pole of the system is from the imaginary axis, the time the time

The relationship between the pole and the time response The relationship between the pole and the time response The relationship between the pole and the time response The relationship between the pole and the time response

the time constant the time constant and the faster and the faster

Im s

y(t)

Re s

the smaller the smaller response of the system.

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Transient response Transient response of the first order Pole – zero plot Pole zero plot of a first order system

order oscillating system Second--order oscillating system Second

22 sT

K 2 



 The transfer function of the second-order oscillating system:

0 , 0

)1 )1

 

   

sG )( )( sG

 

( (   n

1 T

22 sT



K 2 



2 nK  2 ns 2  n

 The system has two complex conjugate poles:





2 j 1 



 n

p 2,1

sY sY )( )(

sGsR )( )( )( )( sGsR



 Transient response:  Transient response:

2 2

1 . . ss



2 nK  2 2 ns 2   n

tn  

)( ty ty )(

K K

sin sin

) )

t t





2 ( 1 ( 1  



 

 

(cos  (cos  ) )

 

R(s) Y(s)

 



2 

  1 1   

     

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order oscillating system (cont’) Second--order oscillating system (cont’) Second

y(t)

Im s

cos = 

2 j 1  

n

(1+).K K ) (1).K (

Re s R



n

2 j 1   1 n  j

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Pole – zero plot of a second order oscillating system Transient response of a second order oscillating system

Second--order oscillating system Second

Remark order oscillating system –– Remark

 A second order oscillation system has two conjugated  A second order oscillation system has two conjugated

= 0 = 0.2

= 0.4

 If = 0, transient response i If  0 t is a stable oscillation signal at the frequency n  n is called natural oscillation frequency.  ,

 If 0<<1, transient

= 0.6 = 0.6

response is a decaying oscillation signal  is called damping constant, , p g the larger the value , (the closer the poles are to the real axis) the faster the real axis) the faster the response decays.

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complex poles, its transient response is a oscillation signal.

 Transient response of the second order oscillating system  Transient response of the second order oscillating system

Second--order oscillating system Second Overshoot order oscillating system –– Overshoot

 

has overshoot.

POT

%100









   exp  

   . 2  

 The larger the value ,

)

The percentage of overshoot:

( ( T O P

 The smaller the value , (the closer the poles are (the closer the poles are to the imaginary axis) the larger the POT

 The relationship The relationship between POT and 

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(the closer the poles are to t e ea a s) t e to the real axis) the smaller the POT.

Second--order oscillating system Second

Settling time order oscillating system –– Settling time

 Settling time:

s

3 3  n

5% criterion:

s

4 n 

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2% criterion:

order oscillating system Second--order oscillating system Second

Th f th th th f t t

Im s Im s

y(t) y(t)

cos =  cos = 

Re s R



Relationship between pole location and transient response Relationship between pole location and transient response Relationship between pole location and transient response Relationship between pole location and transient response  The 2nd order systems that have the poles located in the same rays starting from the origin have the same damping constant, then the percentage of overshoots are the same. The further h th the poles from the origin, the shorter the settling time.

0 Transient response of a second Transient response of a second order oscillating system

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Pole – zero plot of a second Pole zero plot of a second order oscillating system

order oscillating system Second--order oscillating system Second

th i

Relationship between pole location and transient response (cont’) Relationship between pole location and transient response (cont ) Relationship between pole location and transient response (cont ) Relationship between pole location and transient response (cont’)  The 2nd order systems that have the poles located in the same distance from the origin have the same natural oscillation frequency. The closer the poles to the imaginary axis, the smaller t l th f the damping constant, then the higher the POT.

s Im s

y(t) y(t)

n 

Re s R

Pole zero plot of a second Pole – zero plot of a second order oscillating system

Transient response of a second Transient response of a second order oscillating system

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order oscillating system Second--order oscillating system Second

Relationship between pole location and transient response (cont ) Relationship between pole location and transient response (cont’) Relationship between pole location and transient response (cont ) Relationship between pole location and transient response (cont’)

ttli

 The 2nd order systems that have the poles located in the same distance from the imaginary axis have the same n, then the settling time are the same. The further the poles from the real l Th f axis, the smaller the damping constant, then the higher the POT

s Im s

y(t) y(t)

Re s R

n

Pole zero plot of a second Pole – zero plot of a second order oscillating system

Transient response of a second Transient response of a second order oscillating system

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Transient response of high order system Transient response of high order system

 High-order systems are the system that have more than 2 poles  High-order systems are the system that have more than 2 poles  If a high order system have a pair of poles located closer to the

t d t

imaginary axis than the others then the high order system can be approximated to a second order system. The pair of poles d nearest to the imaginary axis are called the dominant poles.

s Im s

y(t) y(t)

R Response of high f hi h order system

Re s Re s

Response of second order system with the dominant poles the dominant poles

High order systems High order systems have more than 2 poles

A high order system can be A high order system can be approximated by a 2nd order system

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Performance indices Performance indices

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Integral performance indices Integral performance indices

 IAE criterion  IAE criterion



te )( dt

J IAE

(Integral of the Absolute Magnitude of the Error )

 

 ISE criterion  ISE criterion



2 2 te

)( dt

J ISE

 

 ITAE criterion

(Integral of the Square of the Error)

tet tet

)( dt )( dt

J ITAE J

   0

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(Integral of Time multiplied by the Absolute Value of the Error)

 A control system is optimal when the selected performance  A control system is optimal when the selected performance

707

systems Optimal systems Optimal

index is minimized  Second order system:

707

.0 5.0 50 .0

minIAEJ minISEJ i J minITAE J

=0.3

y(t)

=0.5

=0.707

=0.9

0 0

when when when

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Transient response of second order systems

 ITAE is usually used in design of control system  ITAE is usually used in design of control system

 An n-order system is optimal according to ITAE criterion if the

ITAE optimal control ITAE optimal control

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denominator of its transfer function has the form:

 Optimal response according to ITAE criterion  Optimal response according to ITAE criterion

y(t)

1st order system 1st

2nd order system

3rd order system

4th order system

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ITAE optimal control (cont’) ITAE optimal control (cont’)

Relationship between frequency domain Relationship between frequency domain

time domain performances performances & time domain performances performances &

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Relationship between frequency response & Relationship between frequency

steady state error response & steady state error

Y(s) Y(s) R(s) R(s)

sHsG )(

(



jHjG ) ( ) 

lim s 0 

lim)(  0 



(



jHjGj ) ) ( 

lim)(  0 



sHsGs )( lim s 0 

(



j ) 

jHjG ) ( ) 

lim)(  0 



2 sHsGs lim )( 0 s 

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G(s) +

Relationship between frequency response & Relationship between frequency

steady state error response & steady state error

Y(s) Y( ) R(s) R( )

 Steady state error of the closed-loop system depends on the low

G(s) + 

the open-loop system at

magnitude response of frequencies but not at high frequencies.

 The higher the magnitude response of the open-loop system at low frequencies, the smaller the steady-state error of the closed-loop system.

 In particular, if the magnitude response of the open-loop system is infinity as frequency approaching zero, then the steady-state error of the closed-loop system to step input is zero.

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p y

Relationship between frequency response & Relationship between frequency

transient response response & transient response

Y(s) Y(s) R(s) R(s)

G(s) +

(

)

jG

 In the frequency range <c , because

) 







jGcl (

jG ( jG (

)  ) 

jG ( )  jG ( )  

)

then:

 In the frequency range >c , because

jG jG ( (

) ) 







) ) 

jGcl jG ( (

jG ( j ( ( ) ) jG  1

j ( ( ) ) jG  jG ( )  

 Bandwidth of the closed loop system is approximate the gain  Bandwidth of the closed-loop system is approximate the gain

then:

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crossover frequency of the open-loop system.

Relationship between frequency response & Relationship between frequency

transient response response & transient response

Bode plot of a open-loop system

Bode plot of the corresponding closed-loop system

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Relationship between frequency response & Relationship between frequency

transient response response & transient response

Y(s) Y(s) R(s) R(s)

 The higher the gain crossover frequency of open-loop system, the wider the bandwidth of closed-loop system  the faster the response of close-loop system, the shorter the settling time.



   c

4 4    c

 The higher the phase margin of the open-loop system, the

G(s) +

p y g g p p

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f smaller the POT of closed-loop system. In most of the cases, if the phase margin of the open-loop system is larger than 600 then the POT of the closed-loop system is smaller than 10%.