lag-leadCompensators.pdf

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Lag & Lead Compensator (PID-Controller) Design Techniques, Simulation &analysis methods with Bode Plot by MATLAB

Tariq MASOOD.CH Dr. Abdel-Aty Edris Prof. Dr. RK AggarwalQatar PetroleumDukhan Qatar

Manager Power Delivery (R&D)EPRI USA

University of BathBath _ UK

Prof. Dr. Suhail A. Qureshi Prof. Dr. Abdul Jabber Khan Yacob Y. Al-MullaUniversity of Engineering & Technology

Lahore PakistanRachna College of Engineering &Technology

Gujranwala PakistanIEEE Chair

Doha Qatar

Author contact Details : Email: maaat 2001@ ieee.org Ph:: 00974 560 75 72;; P.O Box 1000 52 Dukhan Qatar

Keywords: Compensator design structure Compensator gain attenuation zeros & poles gain crossover frequency

A. Introduction: -The purpose of lag/lead compensator design in thefrequency domain generally is to satisfy the specificationon steady state accuracy and phase margin. The overall

philosophy in the design procedure presented here is forthe compensator to adjust the phase curve to establish again-crossover frequency for the disturbing systemdetermining the phase of the loop transfer function L(s) atarbitrary points on the s-plane is an important skill forthose students who are in introductory control subject.Evans showed that the magnitude and phase of the transferfunction can be determined from the pole-zero map usingsimple vector geometry [1],[2]. For example themagnitude an phase of a transfer function of the from:

( ) s z

L ss p

=

Frequency (rad/sec)

P h a s e

( d e g

) ; M a g n

i t u

d e

( d B )

Bode Diagrams

-1

-0.5

0

0.5

1From: U(1)

10 -1 10 0 10 10

1

2

3x 10 -15

T o :

Y ( 1 )

rp rz

s1

Vector geometry for finding the magnitude

p z


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Figure 1. Vector geometry for finding the magnitude and phase. The quantity s1-p is presented by a vector whoselength is rp and magnitude is equal to rz/rp

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Real Axis

I m a g

A x

i s

Pole zero map

X

Figure 2. Pole-zero phase map of L(s) = 1/(s+1) thetransfer function pole is shown at s= -1. the blue arrows

point in the direction of the phase of L (s) for the value ofthe base of the arrow.

Characteristics of lag compensator:-

This relates to phase lag Low frequencies uses gain reduction to shift the zero-crossing

to the left introduces phase lag at frequencies below

wc

reduces bandwidth very large amount of compensation possible increase transient response time Suppress high frequencies noise effects Appreciable improvement in e ss steady state

accuracyCharacteristics of lead compensator:-

This relates to phase lead High frequencies uses phase advance at the zero-crossing increases high frequency gain and shifts wc to the

right increases bandwidth limited amount of compensation possible uses phase advance at the zero-crossing improve transient response Accentuates high frequencies noise Small effect e ss

B. Design procedure of LAG Compensator:-1. Compensator design structure

the basic lag compensator comprises with gain, one poleand one zero, therefore based on specific structure of thecompensator it will reflect as.

+

+=

++=

1

1)(1 _

pos

zos

pos Zos

KcG LagCom the final

control will be significant

++=

11

_ ss

KcG LagCom

e.g. typical bode plot lag compensator isKc = 1; P0=0.4; Z0= 2.5 so 25.64.0/5.2 ==

01

01

p Z == hereby changing the gain merely moves

the magnitude curve by 20log10|Kc|. The majorcontribution from the lag compensator is the constantattenuation in magnitude at high frequencies and zero

phase shifts at high frequencies. Bode plot transferfunction for the lag compensator is

125.214.0

++= ssKcG r compensatolag

Frequency (rad/sec)

P h a s e

( d e g

) ; M a g n

i t u

d e

( d B )

Bode Diagrams

-15

-10

-5

0Gm = Inf, Pm=180 deg. (at 5.2684e-009 rad/sec)

10 -1 10 0 10 1-50

-40

-30

-20

-10

2. How to Determine the Compensator Gain.In the design procedure, the first step is to deter mine thevalve of Kc. The gain is used to satisfy steady-state errorspecification. Therefore, the gain can be computed asgiven below.

plant x

required x

Specified ss

Plant ssc K

K

e

eK

== Where the K x is the

corresponding error constant of the system and e ss is thesteady state error for particular type of input, such as stepor ramp. The steady state error and error constant are(assuming that the closed-loop system is bounded-inputand bounded output.e.g. the steady state error which has been specified = 0.05when the reference input is a unit ramp function, thisrequires an error constant K x-required = 1/0.05 = 20. Assumethat plant transfer function

( )( )54200

)( ++=

ssG s p

which is type of "0" then the compensator must have a one pole at zero. The error constant is K=200/(4X5) = 10 sothe steady state error for a ramp input is 1/10=0.1resulting in compensator required gain has value of Kc=0.1/0.05 = = 20/10 =2


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Example # (1) PM (phase margin) 40; and ess (steady-state error) 0.05 ramp input .

)3)(2)(1(60

)( +++=

sssG s p

Frequency (rad/sec)

P h a s e

( d e g

) ; M a g n

i t u

d e

( d B )

Bode Diagrams

-100

-50

0

50From: U(1)

10 -1 10 0 10 1 10 2-300

-200

-100

0

T o :

Y ( 1 )

Unstable closed loop

Time (sec.)

A m p

l i t u

d e

Step Response

0 3 6 9 12 15 180

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8From: U(1)

T o :

Y ( 1 )

Kc=2

Frequency (rad/sec)

P h a s e

( d e g

) ; M a g n

i t u

d e

( d B )

Bode Diagrams

-100

-50

0

50From: U(1)

10 -1 10 0 10 1 10 2-300

-200

-100

0

T o :

Y ( 1 )

3. How to determine bode plot technique:-In this step plot the magnitude and phase as function offrequency omega for the series combination ofcompensator gain or any compensator poles at s=0. bode

plot transfer function will be used to determine the valuesof zeros and poles as well as if more then one stagecompensation is required. Magnitude (jw) vs. frequencyon a scale log and phase in degree vs. frequency on a logscale. The values of zeros and poles will be chosen tosatisfy the phase margin and crossover frequencyspecification. If the w =0 the magnitude |(jw/zc +1)/(jw/pc+1)|= 1 and this will be equal to 0 dB. Similar the phase

margin (jw/zc +1)/(jw/pc+1) = 0 degrees. Therefore thelow-frequency parts of the curves will be unchanged thesteady state error specification will be remain satisfied.

4. Gain crossover Frequency techniqueThe definition of gain crossover frequency is that where

omega is |G(jw)|=1 in absolute value or |G(jw)|=0 in dBvalue. The purpose of lag compensator's pole and zerocombination is to drop the magnitude of the transferfunction. The main goal of the lag compensator is to adjustthe magnitude curve as per requirement with shifting the

phase curve. Eventually, the phase margin is increased 5-10degree to account that the lag compensator is not ideal.Therefore the gain crossover frequency, which has beencompensated ( compensated) ) phase shift angle will be = -180+PM specified +10. E.g. if the phase margin specified 40degree then the phase shift angle will be -130 degree withabove equation. This ( compensated) ) is the frequency wherethe magnitude of the compensated system |Gc_lag(jw)Gp(jw)| =1 --> 0dB.

11

)( _ ++=

pc jw

zc jw

jwlagcG )2(1

) _( +=

jw jwG jw p

5. How to determine the attenuationOnce the compensated gain crossover frequency has beendetermined, then the amount of attenuation has been

provided by lag compensator must be evaluated as given below. Eventual, the all frequencies which are greater then4*zc, the higher frequencies (relative to zc) will be

attenuated by -20log10( )

( )

=20

10

dB jG d compensate

e.g. the PM 40 and e ss steady-state-error is 0.05 rampinput.

( )( )( )32160

)( +++=

sssG s p The value of required K

compensator = 1/0.05 = 20 and the plant gain is K plant =60/1.2.3 = 10 therefore the compensator gain will reflect:20/10 = 2dBHowever, the new equation will emerge as given below.The attenuation angle will be equal to 50 by 20log(10)^33.4/24 the answer is 46.6 attenuation of the givenangle.

This is too large for the single stage compensation in fact itrequired two compensators for this transfer function. If 10 then lag compensator needs to be used to prevent anyexcessively large component values and limit the undesired

phase shift. In other case if the > 10 then we need coupleof identical series compensator stages to accomplish toabove object.

stage stage totaln = We can consider with these n stages


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1

2 10 1003 100 1000

, 10 10

total

stage totaln n

total

n

n

< <

<

e.g.: if the value = 50 assumed to be too large, then twostages of compensation can be used. Then the each stage

would be given attenuation 50 7.1stage = = . The gain

can also be divided in two _ 2 1.414c stageK = = 6. How to determine the zeros and poles

This is last step to design the transfer function of the lagcompensator, whereas to determine the values of the poleand zero. Eventually in last step Ive already determine theratio of . Firstly, well choose to place the compensatorzero and then compute the pole location from pc = zc/

and10

compensated c

n stage

z

=

Model design Lag-lag compensators:-To meet the following specifications as listed below.

Closed loop bandwidth, 1 BW rad/sec(PM) Phase margin 30 degree(GM)Gain margin 10 dB(ess) Steady-state-error due to a unit step input 0.01.

Consider the system:-( )( )2

100.1 100

plant Gs s s

=+ + +

Step 1: first of all open loop bode plot has been plotted asgiven below.

Frequency (rad/sec)

P h a s e

( d e g

) ; M a g n

i t u

d e

( d B )

Bode Diagrams

-100

-80

-60

-40

-20

0Gm=20.01 dB (at 10.005 rad/sec), Pm=180 deg. (at 1.2702e-008 rad/sec)

10 -2 10 -1 10 0 10 1 10 2-300

-200

-100

0

Step 2: the gain k has been used to desired crossoverfrequency. PM phase margin 20 decibel scale B thereforethe change in to 20log |Hw| =H dB = 20

Log10 |Hw| =202010 10= therefore the K = 10

( )( )210

0.1 100 plant G k

s s s=

+ + +

Frequency (rad/sec)

P h a s e

( d e g

) ; M a g n

i t u

d e

( d B )

Bode Diagrams

-80

-60

-40

-20

0

20Gm=0.0095492 dB (at 10.005 rad/sec), Pm=0.60167 deg. (at 9.9997 rad/sec)

10 -2 10 -1 10 0 10 1 10 2-300

-200

-100

0

This is not very compatible result, since the resonant peakat = 10 rad/sec is a causing the bode plot to experiencecrossovers of the 0dB point and thus, w c, GM and pM arenot very much definable.Step 3: need to use the some sort of controller to pushthe resonant peak below the 0dB line. Therefore, welluse LAG controller.Objective : to push the lag in a position such thedownward sloping magnitude contributions of the lagsilky the peak. Therefore the peak sits in between the lagzero and pole.

Time (sec.)

A m p

l i t u d e

Step Response

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-3

-2

-1

0

1

2

3

4

5From: U(1)

T o :

Y ( 1 )


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Time (sec.)

A m p

l i t u

d e

Step Response

0 20 40 60 80 100 120 140 160 180 2000

1

2

3

4

5

6

7

8

9

10From: U(1)

T o :

Y ( 1 )

Step 4: herewith, well use other gain value to regain ouroriginal cross over of 1 rad/secFrom the above bode plot to obtain the crossover

frequency of 1 rad/sec, we need to subtract 18 dB fromthe magnitude plot.PM phase margin -18 decibel scale B therefore the changein to 20log |Hw| =H dB = -18

Log10 |Hw| =18

2010 0.126

= therefore the valve of K =0.126

Frequency (rad/sec)

P h a s e

( d e g

) ; M a g n

i t u

d e

( d B )

Bode Diagrams

-100

-50

0

50Gm=13.881 dB (at 9.3755 rad/sec), Pm=68.598 deg. (at 1.1122 rad/sec)

10 -2 10 -1 10 0 10 1 10 2-400

-300

-200

-100

0

Firstly, I'll calculate the steady state error to diminish

possible instability of the loop.

( )( ) ( )

+=

sGslagkk ssse ss

21

1)

1(0lim

( ) ( )( )( )( )( )( )

++++++

++++=)20)(126.0)(10(101001.02

1001.010lim 2

2

sssss

sssss

=0.074This does not satisfy the steady state error requirement, soadd another LAG at low frequencies to compensate this.Insofar the steady state error equation will become as

below.

( )( ) ( )

+

=sGs Lagslagkk s

sse ss )(11

)1

(0lim22

( ) ( )( )( )( )( )( )( )( ) ( )

+++++++++++++=

zss psssss pssssss

)20)(126.0)(10(101001.021001.010lim 2

2

01.025220

20 45 degree(ess) Steady-state-error


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The controller gain is:-K = 40

ss

ss

ssG lead 2.01

4.012.01

2.0211

1++=+ +=++=

ss

ss

ss

G lag 44.11186.21

86.24186.21

11

++=

++=

++=

Fi

nal compensator is:

lead lead r compensato final GKGG = _

ss

ss

K G r compensato final 44.11186.21

2.014.01

_ ++

++=

Frequency (rad/sec)

P h a s e

( d e g

) ; M a g n

i t u

d e

( d B )

Bode Diagrams

-100

-50

0

50

100Gm = Inf, Pm=44.889 deg. (at 3.5089 rad/sec)

10 -3 10 -2 10 -1 10 0 10 1 10 2-180

-160

-140

-120

-100

-80

Closed loop feedback control

Frequency (rad/sec)

P h a s e

( d e g

) ; M a g n

i t u

d e

( d B )

Bode Diagrams

-60

-40

-20

0

20Gm = Inf, Pm=76.801 deg. (at 4.7474 rad/sec)

10 -2 10 -1 10 0 10 1 10 2-200

-150

-100

-50

0

Time (sec.)

A m p

l i t u

d e

Step Response

0 1.4 2.8 4.2 5.6 70

0.2

0.4

0.6

0.8

1

1.2

1.4From: U(1)

T o :

Y ( 1 )

Conclusion:-Using pole-zero phase maps help to determine the

phase of the transfer function from a plot of the polesand zeros. In this paper we have described someholistic techniques which are dealing with design lag-lead compensators (PID Control). We have extendedsome useful classical control design techniquesdeveloped for a fixed system to the case wherevectors of uncertain parameters are present in thecontrol system. We have supposed that the

parameters appear linearly in the characteristic polynomial. This allows us to exactly construct theBode plot envelopes using the most compositesegments. We have also shown how bounds for theextremal gain and phase margins for such systems

may be found easily from these segments. Based onthese calculations suitable compensation can bedesigned to improve the worst case values of thesemargins. The results given here should be viewed asa start towards a more complete and complicatedcontrol theory of variant design doctrines which can

be dealt with lead-lag, lag-lag compensatorsdesigning core perceptions.

REFERENCES:-1) Frankline, Powell and Emami Feedback

Control of dynamics control system 4 th edition.2) G.J Thaler, automatic control systems 19893) J.J Dazzo and C.H Houpis, Linear control

system analysis and design McGraw-Hill 4 th edition, 1995.

4) I.H.Keel and S.P Bhattacharyya RobustParametric Classical Control design, Trans IEEEon automatic control, VOL. 39,No.& July, 1994.

5) [4] Kent H.Lundberg Pole Zeros phase mapfeature IEEE Control systems magazineFerruary-2005


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6) Kai Shing Yeung, King W. Wong and Kan-LinChen A Non-Trial-and-Errot method for lag-lead compensator design, IEEE Trans on

education Vol.41, No.1 February 1998.7) Thomas J.Cavicchi Phase Margin Revisited:Phase-Root Locus, Bode Plots and PhaseShifters, IEEE Tran on education, Vol. 46, No. 1February 2003.

8) Raymond C. Garcia and Bonnie S.HeckEnhancing Classical Controls Education viaInteractive GUI Design IEEE Control System,June 1999.

9) W.R. Evans, control system synthesis by rootlocus method, Trans AIEE, vol 69, pp, 66-69,1950.

10) W.R. Evans, control system dynamic Newyork:McGraw-Hill,1954, pp,99-100.

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