CompensationCompensation

Using the process field Using the process field GGPFPF(s) (s) transfer functiontransfer function

Analysis of Bode diagramAnalysis of Bode diagramThe Bode plot is popular, because many The Bode plot is popular, because many

properties of process field are well read.properties of process field are well read.

Recorded the measured values or the identified Recorded the measured values or the identified model the following compensation technique model the following compensation technique are also used.are also used.

It must be concluded if the process field has It must be concluded if the process field has or hasn't got integral effect. or hasn't got integral effect. It possible from the phase plot.It possible from the phase plot.

It is necessary to determine how many time It is necessary to determine how many time constants belongs to the process field and constants belongs to the process field and how close to each other these time constants.how close to each other these time constants. It possible from the Bode plot.It possible from the Bode plot.

j ( )G(j ) A( )e

Without integral Without integral effecteffect

PI compensationPI compensationEuropean structureEuropean structure

Without integral effectWithout integral effectIt is possible to determine from the phase plot of It is possible to determine from the phase plot of process field transfer function.process field transfer function. (sufficiently low frequencies the phase shift is nearly zero.)(sufficiently low frequencies the phase shift is nearly zero.) In this case the most commonly used structure is In this case the most commonly used structure is the PI, if the process field model has got more than the PI, if the process field model has got more than three poles relatively close to each other, then three poles relatively close to each other, then offered to use the PIDT structure. offered to use the PIDT structure. The quality properties of the closed loop depends on the phase The quality properties of the closed loop depends on the phase margin of themargin of the G G00(s) (s) open loop transfer function.open loop transfer function.

Transfer function of PI Transfer function of PI compensationcompensation

There are two variables.First step we chose the KC = 1, and TI = 1 rad/sec. values!

IC PI C C

I I

sT 11G (s) G (s) K 1 K

sT sT

In case of PIDT must be: TI > 4TD and TD > 5T conditions.

PI

s 1G (s)

s

PI compensationPI compensationTenfold value ofTenfold value of ωωII the amplitude gain nearly 0. and the phase shiftthe amplitude gain nearly 0. and the phase shift -5,7 -5,7°°

The principle of The principle of compensation of PIcompensation of PI

First we chose a considers appropriate phase marginFirst we chose a considers appropriate phase margin!!Rule:Rule: In case of more than 3 and relatively close to each other time constantsIn case of more than 3 and relatively close to each other time constants 9090°>°> pmpm >> 7700°°;; In other case In other case pm pm >> 4545° pm° pm: : phase marginphase margin

On the phase plot of process field must be looking for the On the phase plot of process field must be looking for the chosen phase margin and the corresponding frequency will be chosen phase margin and the corresponding frequency will be the future the future ωωCC gain-crossover frequency. gain-crossover frequency.At the chosen phase margin (pm) the phase shiftAt the chosen phase margin (pm) the phase shift is is ps = pm + 5ps = pm + 5..77°° - 180 - 180°°..

The controller gain KThe controller gain KCC value to be chosen so, that at the future value to be chosen so, that at the future

ωωCC gain-crossover frequency will be unit the K gain-crossover frequency will be unit the K00 loop-gain. loop-gain.The reciprocal value of the amplitude gain at the future gain/crossover The reciprocal value of the amplitude gain at the future gain/crossover

frequencyfrequency on the amplitude plot of the gon the amplitude plot of the g00 (the g (the g00 is the G is the G00(s) open-loop transfer (s) open-loop transfer

function withfunction with KKCC = 1 value) will be the actual K = 1 value) will be the actual KCC..

The PI compensation processThe PI compensation process

Have to plotted the process fieldHave to plotted the process field Bode Bode plotplot.. On the phase plot must be looking for theOn the phase plot must be looking for the futurefuture gain-gain-

crossover frequency which is corresponding the following crossover frequency which is corresponding the following phase shift: phase shift: ps = pm + 5,7 - 180.ps = pm + 5,7 - 180.

A tenth of this frequency is A tenth of this frequency is ωωII, , and and TTII is the reciprocal of is the reciprocal of ωωII.. Must plot the Bode diagram of Must plot the Bode diagram of ..

On thisOn this g0 g0 Bode plot have to be looking forBode plot have to be looking for the frequency the frequency which is corresponding the chosen phase margin, next have to which is corresponding the chosen phase margin, next have to read the gain at this frequency.read the gain at this frequency. TheThe K KCC controller gain is equal controller gain is equal

the reciprocal value of the readings gain. the reciprocal value of the readings gain. (In dB the readings value changes the sign)(In dB the readings value changes the sign)

PFG (j )

IE

I

sT 1g0 G ( j )

sT

The identified LTI The identified LTI model from the model from the

measured valuesmeasured valuesThe model seems to be self-adjusting The model seems to be self-adjusting nature. (Without integral effect)nature. (Without integral effect)

If the equitation is known, you can determine If the equitation is known, you can determine the time cons-tants, but they have generally the time cons-tants, but they have generally become during the identification. become during the identification. When the number of the roots 4, and they are When the number of the roots 4, and they are relatively close to each other is the search relatively close to each other is the search phase shift value: phase shift value: psps°° ≈ pm+5.7-180 ≈ -104.3 ≈ pm+5.7-180 ≈ -104.3

E 4 3 2

1.3G (s)

s 42s 198s 154s 18

pm 70o;

Bode plot ofBode plot of G GPFPF(s) (s)

It can be seen the rounding is permitted, but must be documented!

Determination of theDetermination of the T TII and and the the gg00

0 4 3 2

25s 1 1.3g (s)

25s s 42s 198s 154s 18

Based on the above figure 10wI = 0.4 rad/sec., and so wI = 0.04 rad/sec. Creating the reciprocal value: TI = 25 sec.The g0 open/loop transfer function with KC = 1:

On the Bode plot of g0 should look for the kC gain which corresponding the phase margin = 70°.

Bode plot of Bode plot of gg00(s)(s)

The gain of the controller converted from dB: KC = 47.9

To check this, the Bode plot To check this, the Bode plot of of GG00(s)(s)

With the rounding inaccuracy is the required phase margin.

Step response of the Step response of the closed-loopclosed-loop

It needs tuning. The integral time constant is big.

There aren’t overshoot,

and steady-state error

With integral effectWith integral effectPDT1 PDT1

compensationcompensationEuropean stuctureEuropean stucture

Analysis of Bode diagramAnalysis of Bode diagramThe Bode plot is popular, because many The Bode plot is popular, because many

properties of process field are well read.properties of process field are well read.

Recorded the measured values or the identified Recorded the measured values or the identified model the following compensation technique model the following compensation technique are also used.are also used.

It must be concluded if the process field has It must be concluded if the process field has or hasn't got integral effect. or hasn't got integral effect. It possible from the phase plot.It possible from the phase plot.

It is necessary to determine how many time It is necessary to determine how many time constants belongs to the process field and constants belongs to the process field and how close to each other these time constants.how close to each other these time constants. It possible from the Bode plot.It possible from the Bode plot.

j ( )G(j ) A( )e

Transfer function of theTransfer function of the PDT1PDT1 compensation compensation

There are three variables. The first step you choose KC = 1, TD = 0.9 rad/sec, and T = 0.1 rad/sec.

D DC PDT C C

sT s(T T) 1G (s) G (s) K 1 K

sT 1 sT 1

Case PDT1 compensation must be TD > 5T.

PDT

s 1G (s)

0.1s 1

Bode plot of Bode plot of PDT1PDT1TheThe φφmaxmax phase shift depends on thephase shift depends on the A ADD differential gaindifferential gain..

Present examplePresent example A ADD = 9, = 9, and soand so φφmaxmax = 54.9 = 54.9°°..

The principle of The principle of compensation of PDTcompensation of PDTFirst we chose a considers appropriate phase marginFirst we chose a considers appropriate phase margin!!

Rule:Rule: In case of more than 3 and relatively close to each other time constantsIn case of more than 3 and relatively close to each other time constants 9090°>°> pmpm >> 7700°°;; In other case In other case pm pm >> 4545°. pm°. pm: : phase marginphase margin

On the phase plot of process field must look for the chosen On the phase plot of process field must look for the chosen phase margin and the corresponding frequency will be the phase margin and the corresponding frequency will be the future future ωωCC gain-crossover frequency. gain-crossover frequency.At the chosen phase margin (pm) the phase shiftAt the chosen phase margin (pm) the phase shift is is ps = pm ps = pm –– 5 54.94.9°° - 180 - 180°°..

The controller gain KThe controller gain KCC value to be chosen so, that at the future value to be chosen so, that at the future

ωωCC gain-crossover frequency will be unit the K gain-crossover frequency will be unit the K00 loop-gain. loop-gain.The reciprocal value of the amplitude gain at the future gain/crossover The reciprocal value of the amplitude gain at the future gain/crossover

frequencyfrequency on the amplitude plot of the gon the amplitude plot of the g00 (the g (the g00 is the G is the G00(s) open-loop transfer (s) open-loop transfer

function withfunction with KKCC = 1 value) will be the actual K = 1 value) will be the actual KCC..

The PDT1 compensation processThe PDT1 compensation process Have to plotted the process fieldHave to plotted the process field Bode Bode plotplot.. On the phase plot have to look for theOn the phase plot have to look for the futurefuture gain-crossover gain-crossover

frequency which is corresponding the following phase shift: frequency which is corresponding the following phase shift: ps = pm ps = pm –– 5 54.94.9 - 180. - 180. (Assuming A (Assuming ADD == 9) 9)

IfIf A ADD = 9 = 9 then this frequency-thirdthen this frequency-third equalsequals ωωDD, , and this and this frequency of three-timesfrequency of three-times isis ωωT T . . The reciprocal values of the The reciprocal values of the counted frequencies arecounted frequencies are T TDD andand T. T.

Must plot the Bode diagram of Must plot the Bode diagram of ..

On thisOn this g0 g0 Bode plot have to be looking forBode plot have to be looking for the frequency the frequency which is corresponding the chosen phase margin, next have to which is corresponding the chosen phase margin, next have to read the gain at this frequency.read the gain at this frequency. TheThe K KCC controller gain is equal controller gain is equal the reciprocal value of the readings gain.the reciprocal value of the readings gain.

PFG (j )

DE

s(T T) 1g0 G ( j )

sT 1

The identifiedThe identified LTI LTI modelmodel

The model has got integral effect, because has The model has got integral effect, because has a zero root in the denominator.a zero root in the denominator.

If the equitation is known, you can determine If the equitation is known, you can determine the time constants, but they have generally the time constants, but they have generally become during the identification. become during the identification. When the number of the roots 4, and they are When the number of the roots 4, and they are relatively close to each other the phase margin relatively close to each other the phase margin be:be:

The search phase shift value is: The search phase shift value is: psps°° ≈ 65 – 54.9 ≈ 65 – 54.9 -180 = -169.9 -180 = -169.9

E 4 3 2

1.3G (s)

s 42s 190s 127s

pm 65o;

A GA GEE(s) Bode diagramja(s) Bode diagramja

Assuming that AD = 9.

max D T( )

Determination ofDetermination of T TII andand g g00

0 4 3 2

2.1s 1 1.3g (s)

0.24s 1 s 42s 190s 127s

Based on above figurewD = 0.47 rad/sec. és wT = 4.2 rad/sec. and so TD = 2.1 sec., és T = 0.24 sec. The open-loop transfer function with KC = 1 g0 is :

On the Bode plot of g0 should look for the kC gain which corresponding the phase margin = 65°.

Bode plot ofBode plot of g g00(s)(s)

A kompenzáló tag erősítése átváltva KC = 92.3

Step response of feedback Step response of feedback systemsystem

Possible, but not necessary additional tuning.