4. Review of Classical Control

7/28/2019 4. Review of Classical Control

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1

Control of Smart Structures 4. Review of Classical Control 1

Department of Mechanical Engineering

Dr. G. Song, Associate Professor

4. Review of Classical Controls

A Continuation of

Dynamics and Controls Related Knowledge

in Intelligent Structural Systems


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2




Classification of Control Systems

Type Based on their ability to follow step, ramp and parabolic inputs.

The magnitude of steady state errors due to these individual inputs are

indicative of the goodness of the system.

Consider the unity feedback system as shown

The Open Loop Transfer function is ( note the pole of Multiplicity

N or the N INTEGRATORS )

1 2

( 1)( 1)...( 1)( )

( 1)( 1)...( 1)

a b m

N

p

K T s T s T sG s

s T s T s T s

+ + +=

+ + +

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

+-

E(s)


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3




A system is called Type 0, Type 1, Type 2 if N=0, N=1,

N=2,

This is DIFFERENT from order of the system.

For the closed loop system shown in previous slide, the steady state

error is given by (if the system is stable)

Now, we will define various error constants. Names of the errors doesnot implies, position or velocity in literal terms. However, they imply

output, rate of change of output and so on.

Classification of Control Systems

0 0

( )lim ( ) lim ( ) lim

1 ( )ss

t s s

sR se e t sE s

G s = = =

+


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Static Position Error Constant Kp: The steady state error of the systems

for a unit STEP input is

is defined as

Thus the steady state error is

Steady State Errors

0

1 1lim1 ( ) 1 (0)

sss

seG s s G

= =+ +

pK

0

lim ( ) (0)ps

K G s G

= =

1

1ss pe K=

+

(if the system is stable)


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For Type 0 system,

For Type 1 , system


Steady State Errors

00 01 2

( 1)( 1)...( 1)

lim ( ) lim ( 1)( 1)...( 1)

a b m

P s sp

K T s T s T s

G s Ks T s T s T s

+ + +

= = =+ + +

10 01 2

( 1)( 1)...( 1)lim ( ) lim 1( 1)( 1)...( 1)

a b mP

s sp

K T s T s T sK G s for Ns T s T s T s + + += = = + + +

10

1

0 1

ss

ss

e for TypeK

e for Type or higher systems

=+

=



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Static Velocity Error Constant Kv: The steady state error of the

systems for a unit RAMP input is

is defined as


Steady State Errors

20 0

1 1lim lim1 ( ) ( )

sss s

seG s s sG s

= =+

vK

0

lim ( )vs

K sG s

=

1

ssve K

=



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For Type 0 system,

For Type 1 , system

For Type 2 or higher systems

Steady State Errors

00 0 1 2

( 1)( 1)...( 1)lim ( ) lim 0

( 1)( 1)...( 1)

a b mv

s s p

sK T s T s T sK sG s

s T s T s T s

+ + += = =

+ + +

10 01 2

( 1)( 1)...( 1)lim ( ) lim

( 1)( 1)...( 1)a b mv

s sp

s K T s T s T ss G s K

s T s T s T s

+ + += = =+ + +

20 01 2

( 1)( 1)...( 1)lim ( ) lim 2

( 1)( 1)...( 1)

a b mv

s sp

sK T s T s T sK sG s for N

s T s T s T s

+ + += = =

+ + +


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Thus the steady state error for unit ramp input is

Thus, Type 0, system cannot follow ramp input. Type 2 system can

follow ramp input with zero error.

Steady State Errors

10

1 11

1 0 2

ss

v

ss

v

ss

v

e for Type

K

e for TypeK K

e for Type or higher systemsK

= =

= =

= =



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For Type 0 system,

For Type 1 , system

For Type 2

For Type 3 or higher system

Steady State Errors

22

0

0 0 1 2

( 1)( 1)...( 1)lim ( ) lim 0

( 1)( 1)...( 1)

a b ma

s s p

s K T s T s T sK s G s

s T s T s T s

+ + += = =

+ + +2

2

10 01 2

( 1)( 1)...( 1)lim ( ) lim 0

( 1)( 1)...( 1)

a b ma

s sp

s K T s T s T sK s G s

s T s T s T s

+ + += = =

+ + +

22

20 01 2

( 1)( 1)...( 1)lim ( ) lim

( 1)( 1)...( 1)

a b ma

s sp

s K T s T s T ss G s K

s T s T s T s

+ + += = =

+ + +

22

20 01 2

( 1)( 1)...( 1)lim ( ) lim 3

( 1)( 1)...( 1)

a b ma

s sp

s K T s T s T sK s G s for N

s T s T s T s

+ + += = =

+ + +


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Thus the steady state error for unit Parabolic input is

Thus, Type 0 and Type 1, system cannot follow parabolic input. Type

2 system can follow ramp input with finite error. Type 3 and highersystem can follow a parabolic input with zero error.

Steady State Errors

10 1

1 12

1 0 3

ss

v

ss

a

ss

a

e for Type and Type

K

e for TypeK K

e for Type or higher systemsK

= =

= =

= =


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12




Unit Step Input Unit Ramp InputUnit Accleration

Input

Type 01

1 pK+

Type 1 0 1

vK

Type 2 0 0 1

a

Steady State Errors

Steady State Error in Terms of Error Constants


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14


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14




Resonant Frequency and Resonant Peak Value

The magnitude of

This magnitude is maximum when the denominator is minimum. The

minimum of denominator occurs when

This is called the resonant frequency

2

2 22

2

1( )

1 2

1( )

1 2

n n

n n

G j

j j

is

G j

=

+ +

=

+

21 2 0 0.707r n for =

21 2n =

C l f S S 4 R i f Cl i l C l 15


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For the resonant frequency is less than the damped

natural frequency which is exhibited in transient

response.

The magnitude of the resonant peak can be found as

As approaches zero approaches infinity. This means that for an

undamped system excited at its natural frequency the magnitude

becomes infinite.

Resonant Frequency and Resonant Peak Value

max 2

1( ) ( ) 0 0.707

2 1r rM G j G j for

= = =

r

0 0.707

21d n =

C t l f S t St t 4 R i f Cl i l C t l 16


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Relationship between System Type and Magnitude Curve

Consider a unity feedback control system.

The static position, velocity and acceleration constant describe the low

frequency behavior of the Type 0, Type 1 and Type 2 systems

respectively.

For a given system only one of the static error constant is finite and

significant. (refer table)

The larger the value of the finite static error constant, the higher theloop gain is as approaches zero.

The type of the system determines the slope of the log-magnitude

curve at low frequencies.

Thus information about the steady state error of a control system to a

given input can be determined from the low frequency region of the

log magnitude curve.

C t l f S t St t 4 R i f Cl i l C t l 17


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For a unity feedback Open Loop Transfer Function is

For a Type 0 system (Magnitude plot

shown)

It follows that the low frequency asymptote is a horizontal line at


1 2

( 1)( 1)...( 1)( )

( 1)( 1)...( 1)

a b m

N

p

K T s T s T sG s

s T s T s T s

+ + +=

+ + +

0lim ( ) pG j K K

= =

20log .pK dB

Control of Smart Str ct res 4 Re ie of Classical Control 18


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In a Type 1 system (Magnitude plot

shown)

The intersection of the initial -20

dB/decade segment or its extension with

the 0 dB line has a frequency numericallyequal to Kv.


( ) 1vK

G j for w

j

=

1

1

1

2

1 2 3

( ) 1v

v

KG j

jK

= =

=

=

Control of Smart Structures 4 Review of Classical Control 19


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In a Type 2 system (Magnitude plot

shown)

The intersection of the initial -40

dB/decade segment or its extension with

the 0 dB line has a frequency numericallyequal to the square root of Ka.

( )

2( ) 1v

KG j for w

j

=

( )220log 20log1 0a

a

a a

K

j

K

= =

=



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Gain and Phase Margin

Phase Margin: The phase margin is that amount of additional phase lag

at the gain cross over frequency required to bring the system to the

verge of instability.

The gain cross over frequency is the frequency at which the magnitude

of the open loop transfer function is unity.

The phase margin is plus the phase angle of the open loop

transfer function at the gain cross over frequency or,

180

180 = +



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Gain Margin: The gain margin is the reciprocal of the magnitude at the

frequency at which the phase angle is .

Defining the phase cross over frequency to be the frequency at

which the phase angle of the open loop transfer function equals

gives the gain margin

In term of dB


180

180

1

gK

1

( )gK

G j=

120log 20log ( )g gK dB K G j= =



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The values of and are almost the same for small values of . Thus for small values of , the value of is indicative of the speed

of transient response of the system.

Correlation between Step Transient and

Frequency response for second order system The phase margin and damping ratio are directly related. Figure shows

a plot of the phase margin as a function of the damping ratio .

For a second order system the phase margin and damping ratio are

related by a straight line for as follows

0 0.6

100

=

rr

d



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Summary: Frequency Response

Information Obtained from Open Loop Frequency Response

- The low frequency region (far below cross over frequency) of the

locus indicates the steady state behaviorof the closed loop system.

- The medium frequency region indicates the relative stability

- The high frequency region (far above cross over frequency) of the

locus indicates the complexity of the closed loop system.



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Nyquist Plots: Stability and Relative Stability

Nyquist plot is the locus of vectors as is varied fromzero to infinity.

Nyquist stability criterion determines the stability of a closed loop

system from its open loop frequency response. ( Remember RootLocus also gives stability of closed loop system from open looptransfer function root locus plot)

The Nyquist stability criterion relates the open loop frequency

response to the number of zeros and poles of thecharacteristic equations that lie in the right halfs-plane.

Absolute stability of the closed loop system can be determined fromgraphically from the open loop frequency response and thus there is no

need for actually determining closed loop poles. This is important because in practical systems mathematical

expressions are often not available, only the frequency response data isavailable.

( ) ( )G j H j 1 ( ) ( )G s H s+

( ) ( )G j G j



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Control of Smart Structures 4. Review of Classical Control



TASK:

Self Study: Plotting of Nyquist Plots, Mapping Theorem (s-plane to

F(s)plane whereF(s)=1+G(s)H(s) )

Self study: Nyquist Stability Analysis and Relative Stability Analysis.

Nyquist Plots: Stability and Relative Stability



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27Department of Mechanical Engineering


Characteristics of Lead, Lag and Lag-Lead Compensators

Lead Compensatoressentially yields an appreciable improvement in

transient response and a small change in steady state accuracy. This

implies it can be used for vibration suppression.

Lag Compensatoryields an appreciable improvement in steady state

accuracy at the expense of increasing the transient response time.

Lag-Lead Compensatorcombines both the characteristics of Lead and

Lag compensators.



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Lead Control Design

Two approaches can be used for Lead Controller Design

1. Root Locus Approach

2. Bode Plot (Frequency Response Approach)

We will deal with both the approaches.



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+_

Lead comp. plant

PZT

sensor

C(s)

Gc(s) G(s)

R(s)

command

1. Phase-lead compensator

)10(1

1

1

1

)(

1

1

2

1


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System

5.0,2

31:..

424

)()(

1 2

==

=

++=

+=

n

jspolesLC

sssRsC

GGCL

)2(

4)(

+=

sssG

Close-loop uncompensated

5.0,4: == nObjectiveDesign

Steps: 1) Determine the desired location for the dominant C.L. poles(new) from

performance spaces.

jjs nn

n

o

3221

4

605.0cos

2

2,1 ==

=

===

Lead compensator (Root Locus)



31/52



2) Draw the root-locus plot of the uncompensated system, see if gain adjustment alonecan achieve the desired C.L. poles. If not (this case), calculate the angle deficiency ,

which will be contributed by the lead compensator.(we need to modify a little to satisfy

the angle condition)

new

4j

3j

2j

j120

old

60

60

r = 4

=+=+

=+

=

=

18030210)2(

4

210)2(

4

1

1

ss

ss

ssGc

ss




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The lead compensator will have a phase lead of 30.

)10(1

1

1

1

)(


33/52



68.4

1

)2(

4

4.5

9.2

1

4.59.2)(

537.0

345.04.5

1

9.21

1

=

=

++

+

=

++=

=

==

=

=

c

ss

c

c

cc

K

sss

sK

GGNow

ssKsG

T

T

T

15o

Design finished, verify the result




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m

mm

mzpm

pz

p

zc

ccc

TT

TTs

s

K

Ts

Ts

KTs

TsKsG

sin1

sin1

1

1sin

11

)1

,1

(

1

1

)10(1

1

1

1)(

2

+

=

+

=

===

==+

+=


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Lead compensator (Frequency Response)

Design example and procedures

Gc+- 1010.24ss

5.122 ++

Design requirement:

Kp (WHY Kp ? ) to be 10 times of the original one

phase margin at least 33

gain margin at least 10 dB



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Procedures:

1. Determine gainKto satisfy the requirement on the given static error constant

12

0

1 12.5 1( ) * ( )1 0.24 101 1

lim ( )

10*101/12.5 80.8

c

p cs

Ts TsG G s K G sTs s s Ts

K G G s

K

+ += =+ + + +

=

= =

2. Draw bode diagram of

G1 : gain adjusted O.L. transfer function (not include the compensator, but

include the compensators gain)

Evaluate the phase margin from bode plot

1( ) ( )G s KG s=




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3. Determine the necessary phaselead angle to be added to thesystem, allowing for a smallamount of angle addition to

compensate for the shift in gaincross over frequency.

33 - 0.195 = 32.805

32.80 + 5.20 = 38

Addition of a lead compensatormodifies the magnitude curveand shifts the gain cross overfrequency to the right. To

offset the increase phase lag ofdue to the

increase in the gain cross overfrequency, we provide someadditional angle, as above.


Bode Plot of Gain adjusted Open Loop system

KGjG =)(1


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6. Determine the corner frequency of the lead compensator

7. Determine Kc

1

122.87

1 122.87 95.32

0.24

m

m

p

T

T

T

=

= =

= = =

Zero:

Pole:


c

c

K K/ 80.8/0.24 336.66

s 22.87G 336.66s 95.32

= = =

+= +



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8. Check the gain margin and phase margin of the Open Loop compensated system




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Check the transient response


High Overshoots:

Characteristics oflead compensators.



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Basics: Zero is always located to the left of pole in s-plane for lag compensator. It

reduces steady state error at the expense of increasing the transient

response time. Given by:

Lag compensator

11 ( ) ( 1)

11c c c

sTs TG s K K Ts

TsT

++

= = >+ +

Figure shows a typical lag compensator.

Magnitude of the lag compensator (for this

case only) is 40 dB for low frequencies and

20 dB at high frequencies. Thus, the lag

compensator is essentially a low pass filter

-1/T -1/(T)



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In Lag compensators, pole and zero are chosen very

close to each other. By the use of lag compensator we

are intending to reduce the steady state error. To do

so, we do not want to alter the transient response

much. However, the addition of poles and zeros

change the root locus. To avoid major changes in the

root locus, the poles and zeros are chosen very close

to each other as well as very close to the origin.

The design will be dealt in detail, when we design a

lag lead compensator.

If we use large , Gc(s) will change the root loci of the system, therefore we hope to

use large to increase the steady state error Kp & KvcK

-1/T -1/(T)

s

Lag compensator



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Basics: Used to improve both transient response and steady state

response. The lag-lead compensator is given by

Lag-Lead Compensator (Root-Locus)

( )1 2

1 2

1 1

( ) ( ) ( ) 1, 11c c

s sT T

G s K

s sT T

+ +

= > >+ +

Lead Network Lag Network



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Problem

Problem Statement: For the Block diagram shown below design acompensator such that the dominant closed loop poles are located at

and the static velocity error constant is 50 sec-1.

Q. What kind of Compensator ?

A. Lag-Lead, because we need to reshape the root-locus and decrease the steadystate error by increasing .

Q. Which Compensator will be designed first?A. Lead, because lead compensator will change the root locus. After reshaping theroot locus, we can design lag . Lag compensator do not appreciably change theroot locus.

js 3222,1 =

vK

Gc+

-

10

s(s 2)(s 5)+ +

v



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Step1: Find angle deficiency, which will be compensated by the lead portion.

1 1 1

10100.89

( 2)( 5)s s s

=

+ +

Assume is not equal to for design simplicity

1 = 120

2 = 90

3 = 50

123

O.L.poles0-2-5

s1

Lag-Lead Compensator (Root Locus)

180 100.89 79.11angle deficiency = =



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Not to increase the system order, we choose 1/T1 =2, then (s+ 1/T1 ) will cancel the plant

pole at (s+2)

1

( 2) 101 33.6

( 20) ( 2)( 5)c c

s s

sK K

s s s s=

+= =

+ + +

Use Magnitude Condition

-21.6 -2

79.11

s1

1

1

20.00, 1/ 2 0.5, 10.0T

T

= = = =

01

1

5&11

1

2

1

2

1

2

1

2

1

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4. Review of Classical Control