b_lecture16 bode compensation and pid controller automatic control system

The Design of Feedback Control System

Introduction

The performance of a feedback control system is primary importance.

What is a suitable control system?

-- It is stable. -- It results in an acceptable response to input commands.

-- It is less sensitive to system parameter changes. -- It results in a minimum steady-state error for input.

-- It is able to reduce the effect of undesirable disturbances.

)(ty

t0

overshoot

pt st

Performance specifications

performance specifications in the time domain

Overshoot

Setting time

Steady-state error

%

st

sse

c

h

Glg20

G

0

r b

rM

)0(707.0 M

performance specifications in the frequency domain

Closed-loop Open-loop

Resonant peak Gain-crossover frequency

Resonant frequency Gain margin

Bandwidth Phase margin

c

r

b

rM

/ hh L


o

j

n

n

T1

Typical complex domain indices are represented by the

location of the dominant poles

%100%21

e

n

st

5.3

or

Tts 3


What is compensation or correction of a control system ?

)1)(1()( :exampleFor

sTss

KsG

stable becan system loop-closed thismake

:getcan wecriterion, Hurwitz-Routh toAccording 0)T 0(K 1

11

TT

TK

.or ngonly varyi stable benot can system

loop-closed thisCriterion, Hurwitz-Routh toAccording )1(

)( :ifBut 2

TK

Tss

KsG

Solution:

Example

TTss

sKsG

)1(

)1()( :make weIf

2

This closed-loop system can be stable.

We make the system stable by increasing a component.

Compensation & Compensator

Increasing a component ,which makes the systems performance to be improved, other than only varying

the systems parameters, this procedure is called

the compensation or correction of the system.

The compensating device may be electric,

mechanical, hydraulic, pneumatic, or some other

type of device or network and is often called

a compensator.

A compensator is an additional component that is

inserted into a control system to compensate for a

deficient performance.


The compensator can be placed in a suitable location within the

structure of the system.

The compensator placed in forward path is called a cascade (or

series) compensator.

)(sGC )(sG)(sR )(sY

Controll

er

controlled

process

r.compensato a is 1)s( stable, becan system the

, 1)s( component increase to,)1(

)( :Example2

Tss

KsG


Similarly, the other compensation schemes are called feedback, output, input and disturbance compensation.

)(sGC

)(sG

)(sH

)(sR )(sY

)(sG

)(sH

)(sGC

)(sR )(sY

)(sG

)(sH

)(sGC

)(sR )(sY

)(sG)(sGC

)(sR)(sY

)(sGn)(sN

In the following sections, we will assume that the process has been

improved as much as possible and that the G(s) representing the

process is unalterable.

For frequency response methods, we are concerned with altering the system so that the frequency response of the compensated system will satisfy the system specifications.

Alternatively the design of a control system can be accomplished in the s-plane by root locus methods. For the case of the s-plane, the designer wishes to alter and reshape the root locus so that the roots of the system will lie in the desired position in the s-plane.

We shall consider the addition of so-called phase-lead , phase-lag and phase lag-lead compensation network ,and describe the design of the network by frequency response techniques.

Approaches to Compensation

Consider the first-order compensator with the transfer function

The design problem then becomes the selection of parameters ,

in order to provide a suitable performance.

T and

)1( 1

1)(

Ts

TssGc

TTjGc11 tantan)(

Tm

1

1

1arcsin

m

The maximum value of the phase lead occurs at frequency

The maximum phase lead is

Phase-lead Compensation Network

The frequency is the geometric mean of

and . Tp /1Tz /1m

lg10)(lg20 mjG

L

[+20]

T

1

m

0

0 T1

T1

cG

lg10

)1( 901

1arcsin0 00

m

The above equation is very useful for calculating a necessary ratio

between the pole and zero of a compensator in order to provide a required

maximum phase lead.

)1( 1

1)(

Ts

TssGc

1R

2R

C

1V 2V

1

11

1

1

)(

)()(

1

21

2

1

21

2

1

2

Ts

Ts

CsRRR

R

CsR

RR

R

sV

sVsGc

2

21

R

RR C

RR

RRT

21

21

Example : Phase-lead electric network compensation

Phase-lead Compensation Network

Summary of Effects of Phase-lead Compensation

Advantages and disadvantages of Phase-lead controller on performance are :

1. Improving damping and reducing maximum overshoot.

2. Improving h(Lh) and . 3. Increasing Wc.

4. Reducing setting time because of increasing Wc

5. Possibly accentuating noise at higher frequencies.

Black curve 1 controlled process G(s) Bode plot Red curve controller GC(s) Bode plot Green curve 2 Compensated system GC(s)G(s) Bode plot

Let us consider a single-loop feedback control system, where

)(sG)(sGC

)(sR )(sY

)11.0()(

ss

KsG

We want to have steady-state error ess=0.01 for an unit ramp input. Furthermore, we desire that the phase margin of the system be at least

450 and the gain crossover frequency be at least 40 rad/s.

100 01.01

KK

ess

Example: A phase-lead compensator design for

a second-order system using the Bode diagram

The first step is to plot the Bode diagram of the

uncompensated transfer function.

10 31

44

dB6

decdB /20

decdB /40

Glg20

022

88

sradc / 3109.17

ionsspecificat hesatisfy tt don' and c



)11.0(

100)(

sssG

101136.0

104544.0)(

s

ssGc

Black line : 20log ( )G j

Green line : 20log ( ) ( )cG j G j

10 31

44

dB6

decdB /20

decdB /40

Glg20

022

88

)1( 1

1)(

Ts

TssGc

88T

1T ,22

T

1T 21

here

08.49

/44

101136.0)11.0(

)104544.0(100)()(

srad

sss

ssGsG

c

c



Blue line : controlled process Bode plot 20log ( ) and ( )G j G j

Green line : Compensated system Bode plot 20log ( ) ( ) and ( ) ( )c cG j G j G j G j



Consider the first-order compensator with the transfer function

The design problem then becomes the selection of parameters and

T, in order to provide a suitable performance.

TTjGc11 tantan)(

The phase of this compensator is always negative. Thus it is called a

phase-lag compensator.

Phase-lag Compensation Network

)(sG)(sGC

)(sR )(sY

1

11

c

TsG s

Ts

)( lg20

)( lg20

0

)(

1

11

1

T

TTT

T

L

L

Phase-lag Compensation Network

lg20

L

Summary of Effects of Phase-lag Compensation

1 T

1

T

Black curve 1 controlled process G(s) Bode plot Red curve controller GC(s) Bode plot Green curve 2 Compensated system GC(s)G(s) Bode plot

Advantages and disadvantages of Phase-lag controller on performance are :


2. Improving h(Lh) and . 3. Filtering out high-frequency noise (lessening noise at higher frequencies).

4. Decreasing Wc.

5. Increasing settling time because of decreasing Wc .

The phase lag-lead compensator

The phase-lead compensator improves settling time,

phase margin and increase the bandwidth.

However, phase-lag compensator when applied properly

improves phase margin but usually results in a longer

settling time.

Therefore, each of these control schemes has its advantages, disadvantages,and limitations, and there are many systems that cannot be satisfactorily compensated by either scheme acting alone.

It is natural, therefore, whenever necessary, to consider

using a combination of the lead-lag compensator, so

that the advantages of both schemes are utilized.

The transfer function of a lag-lead compensator can be written as

sT

saT

sT

sTsGsGsG ccc

2

2

1

121

1

1

1

1)()()(

)10 ,1(

lead lag

It is usually assumed that the two break frequencies of the lag portion are

lower than the two break frequencies of the lead portion.

1

1

T1

1

T 2

1

T 2

1

T

||lg20)( GL

0


G)(


1R

2R

2Cie

i

0e1C

Phase lag-lead electric network compensation

PID controllers in the frequency domain

The PID controller provides a proportional term, an integral term,

and a derivative term.

We have the PID controller transfer function as

sKs

KKsG D

IPc )(

If we set , we have the PI controller 0DK

s

KKsG IPc )(

If we set , we have the PD controller 0IK

sKKsG DPc )(

Effects are similar to phase lag-lead compensation.

Effects are similar to phase-lag compensation.

Effects are similar to phase-lead compensation.

PD controller

Effects are similar to phase-lead compensation

)( :function transfer sKKsG Dpc

C(s) G(s)

R(s)

pK

sKD

)(sGC)2(

)( :Assuming

2

n

n

sssG

)2(

)()()( :is system dcompensate theoffunction transfer loop-open The

2

n

DPnc

ss

sKKsGsG

D

P

K

Ks :at zero loopopen a adding toequivalent is controller PD that theshowsIt

Advantages and disadvantages of PD controller on the performance are :


2. Improving h(Lh) and . 3. Increasing Wc.

4. Reducing setting time because of increasing Wc .

5. Possibly accentuating noise at higher frequencies.

s

KKsG Ipc1

)( :functionTransfer

)2()( :Assuming

2

n

n

sssG

C(s) G(s)

R(s)

pK

sK I

1

)(sGC

)2(

)(

)2(

)1

(

)()(

:is system dcompensate theoffunction transfer loopopen The

2

22

n

IPn

n

IPn

css

KsK

ss

sKK

sGsG

0s :at pole a and :at zero loopopen a adding toequivalent is controller PI P

I

K

Ks

PI controller

Effects are similar to phase-lag compensation

Advantages and disadvantages of PI controller on the performance are :


2. Improving h(Lh) and . 3. Filtering out high-frequency noise (lessening noise at higher frequencies).

4. Decreasing Wc.

5. Increasing settling time because of decreasing Wc .

Transfer function: sKs

KK(s)G DIpc 1

PID controller have advantages both of PI and PD.

G(s) R(s) C(s)

- + pK

sKI

1

sKD)(sGC

PID controller

Effects are similar to phase lag-lead compensation

Circuits of PI , PD and PID

_

+

C

R1 u

r u0

PI controller

R2

)1

1(21

2

)(

)(0

CsRR

R

UU

s

s

R

_

+ C

R1 u

r u0

PD controller

R2 )1( 11

1

2

)(

)(0 sCRR

R

UU

s

s

R

_

+ C

1

R1 ur

u0

PID controller

C2 R2

?)(

)(0 s

s

RUU

b_lecture16 bode compensation and pid controller automatic control system

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