Lecture introduction to control systems chapter 5 analysis of control system performance (dr huynh thai hoang)

First--order system order system The relationship between the pole and the time response æ The further the pole of the system is from the imaginary axis, thesmaller the time constant and

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/

Chapter 5

ANALYSIS OF CONTROL SYSTEM

PERFORMANCE

æ The optimal performance index

æ Relationship between frequency domain performances and time

domain performances

Performance criteria

Performance criteria: Steady state error

()

(t r t y t

e = − fb ⇔ E(s) = R(s) −Y fb(s)

æ Steady-state error: is the error when time approaching infinity

)(

æ Overshoot: refers to an output exceeding its steady state value

Performance criteria

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

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

POT

ss

y

æ S ttli ti (t ) i th ti i d f th f t t

Performance criteria

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

æ Settling time (t s): is the time required for the response of a system toreach and stay within a range about the steady-state value of sizespecified by absolute percentage of the steady-state value (usually2% or 5%)

æ Rise time (t r): is the time required for the response of a system torise from 10% to 90% of its steady-state value

rise from 10% to 90% of its steady state value

Steady Steady state error state error

()

(

s H s G

s

R s

)

(lim

)(

sE

e

s s

ss = → = → +

æ Steady-state error:

)()(1

G(s)H(s) does not have

any deal integral factor

G(s)H(s) has at least 1 ideal integral factor

G(s)H(s) does not have

deal integral factor

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

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

G(s)H(s) has less than 2

ideal integral factors

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

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

Transient response

1

)(

1)

()()

s G s R s

Y

1

+

Ts s

e K

t

Pole – zero plot

of a first order system

Transient response

of the first order

)1

()

(t K e t / T

y(t) = K(1 e− )

First order system order system – – Remarks Remarks

æ First order system has only one real pole at (−1/T), its transientresponse doesn’t have overshoot

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

system to reach 63% its steady-state value

æ The further the pole ( 1/T) of the system is from the imaginary

æ The further the pole (−1/T) of the system is from the imaginaryaxis, the smaller the time constant and the faster the time response

where ε = 0 02 (2% criterion) or ε = 0 05 (5% criterion)

where ε 0.02 (2% criterion) or ε 0.05 (5% criterion)

First order system order system

The relationship between the pole and the time response

æ The further the pole of the system is from the imaginary axis, thesmaller the time constant and the faster the time response of the

Pole – zero plot

of a first order system

Transient response

of the first order

Second order oscillating system order oscillating system

12

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

1 ( ω = < ξ <

æ The system has two complex conjugate poles:

2 2

2 2

2 1

2

) (

n

n s s

Ts s

T

s G

ω ξω

( ) ( )

2 2

( ) ( )

(

n

n s s

s

s G s R s

Y

ω

ξω + +

t

n

) 1

( sin 1

1

1 )

(

Pole – zero plot of a second

order oscillating system

Second order oscillating system order oscillating system – – Remark Remark

æ A second order oscillation system has two conjugated complex

æ A second order oscillation system has two conjugated complex

poles, its transient response is a oscillation signal

ξ , ( the closer the poles are to

the real axis ) the faster the

response decays.

ξ = 0.6

response decays.

Second order oscillating system order oscillating system – – Overshoot Overshoot

æ Transient response of the second order oscillating system has

% 100

The percentage of overshoot:

Ø The larger the value ξ, (the closer the poles are to the real axis) the smaller the

POT

Ø The smaller the value ξ,

(the closer the poles are to

(the closer the poles are to the imaginary axis) the larger the POT

ξ

The relationshipThe relationship

between POT and ξ

Second order oscillating system order oscillating system

Relationship between pole location and transient response

æ The second order systems that have the poles located in the same

rays starting from the origin have the same damping constant, then

cosθ = ξ

Pole – zero plot of a second Transient response of a second

t

0Pole zero plot of a second

order oscillating system

Transient response of a second order oscillating system

Second order oscillating system order oscillating system

Relationship between pole location and transient response (cont’)

æ The second order systems that have the poles located in the same

distance from the origin have the same natural oscillation

Pole zero plot of a second

order oscillating system

Transient response of a second order oscillating system

Second order oscillating system order oscillating system

Relationship between pole location and transient response (cont’)

Pole zero plot of a second

order oscillating system

Transient response of a second order oscillating system

Transient response of high order system

æ High order systems are the system that have more than 2 poles

æ High-order systems are the system that have more than 2 poles

æ If a high order system have a pair of poles located closer to the

imaginary axis than the others then the high order system can beapproximated to a second order system The pair of poles nearest tothe imaginary axis are called the dominant poles

A high order system can be approximated

by a dominated-pole second order systems

Performance indices

Integral performance indices

)

( dt t e

2

)

( dt t e

t

J ITAE

Optimum systems

æ A control system is optimum when the selected performance index is

æ A control system is optimum when the selected performance index is

ITAE optimal control

æ ITAE is usually used in design of control system

æ ITAE is usually used in design of control system

æ An n-order system is optimal according to ITAE criterion if thedenominator of its transfer function has the form:

ITAE optimal control (cont’)

æ Optimal response according to ITAE criterion

æ Optimal response according to ITAE criterion

Relationship between frequency domain p p q q y y

performances and time domain performances

Relationship between frequency response and steady state error

) (

lim )

( ) (

lim

0

ω G j H j s

H s G

K

s

p = → = →

) (

) (

lim )

( ) (

lim

0

ω j G j H j s

H s G s

K

s

v = → = →

) (

) (

) (

lim )

( ) (

K

s

a = → = →

Relationship between frequency response and steady state error

æ Steady state error of the closed loop system depends on the

æ Steady state error of the closed-loop system depends on the

magnitude response of the open-loop system at low frequencies butnot at high frequencies

æ The higher the magnitude response of the open-loop system at low

frequencies, the smaller the steady-state error of the closed-loop

s stem

system

æ In particular, if the magnitude response of the open-loop system isinfinity as frequency approaching zero, then the steady-state error ofy q y pp g , ythe closed-loop system to step input is zero

Relationship between frequency response and transient response

æ In the frequency range ω <ω because G( jω) >1 then:

æ In the frequency range ω <ωc, because G( jω) >1 then:

1)

(

)(

)(

1

)

()

ω ω

j G

j G j

G

j

G j

G cl

)(

)( jω G jω

G

æ In the frequency range ω >ωc, because G( jω) <1 then:

)(

)( jω G jω

G

⇒ Bandwidth of the closed loop system is approximate the gain

)

(1

)(

)(

1

)

()

j

G j

+

=

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

crossover frequency of the open-loop system

Relationship between frequency response and transient response

closed-loop system

Relationship between frequency response and transient response

æ The higher the gain crossover frequency of open loop system the

æ 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

qd c

t

ω

π ω

Example of relationship between gain crossover frequency and settling time

) 1 08

0 )(

1 1

0 (

10 )

(

+ +

=

s s

s

s G

0 )(

1 1

0

s

Example of relationship between gain crossover frequency and settling time

0 (

50 )

(

+

=

s s

s

G

) 1 1

0 ( s +

s

Example of relationship between phase margin and POT

) 1 08

0 )(

1 1

0 (

6 )

(

+ +

=

s s

s

s G

0 )(

1 1

0

s

Example of relationship between phase margin and POT (cont’)

) 1 1

0 (

6 )

(

+

=

s s

s G

0 ( s +

s