CompensationCompensation

Using the process field Using the process field GGPFPF(s) (s) step responsestep response

Analysis of step responseAnalysis of step response

The following analysis technique are also The following analysis technique are also used using the original recorded values or used using the original recorded values or the identified model created from these the identified model created from these recorded values.recorded values.

First it must be concluded if the process First it must be concluded if the process field has or hasn't got integral effect. field has or hasn't got integral effect. It possible from the reaction curve of the process It possible from the reaction curve of the process field. It takes a new steady-state or uniformly field. It takes a new steady-state or uniformly changing the amplitude.changing the amplitude.

It should be noted the time constants of the It should be noted the time constants of the approximation model.approximation model. It is possible editing the reaction curve.It is possible editing the reaction curve.

j ( )G(j ) A( )e

Self-adjusting process Self-adjusting process fieldfield

PI or PIDT1 PI or PIDT1 compensationcompensation

European structureEuropean structure

Without integral effectWithout integral effectIn this case the most commonly used In this case the most commonly used controller type the PI or if the controller type the PI or if the reaction curve starts relatively slowly reaction curve starts relatively slowly PIDT1PIDT1The 1 can also be used if the process field has got large dead timeThe 1 can also be used if the process field has got large dead time..

The approximation models, whose parameters can be The approximation models, whose parameters can be determined without computers the next:determined without computers the next:

HPTHPT11

PTnPTn

usTPFa P

g

1G (s) K e

sT 1

PFa P n

g

1G (s) K

sT 1

The transfer functionsThe transfer functionsPI

C PI CI

1G (s) G (s) K 1

sT

In case of PIDT1 the transfer function has got four variables.

You must be determine the AD differential gain to define the

T time constants!

PIDT1

DC PIDT C

I

sT1G (s) G (s) K 1

sT sT 1

The principle of The principle of compensationcompensation

Be plotted the step response of process fieldBe plotted the step response of process field.. The ratio of the steady-state amplitude of the input The ratio of the steady-state amplitude of the input

(energizing) and output (response) signals is the(energizing) and output (response) signals is the K KPP. . You should look for the inflection point of the You should look for the inflection point of the

reaction curvereaction curve.. You need to edit the crossover points of beginning You need to edit the crossover points of beginning

and final values of reaction curve with the line which and final values of reaction curve with the line which overlaid on the inflection pointoverlaid on the inflection point..

The intersection can be defined the apparentThe intersection can be defined the apparent T Tuu dead dead

time and the apparent time and the apparent TTgg first order time constantfirst order time constant..

HPT1 model from the reaction curve of process field

uT gT

1( )

1

usTm

pg

yG s e

u sT

,My u

PK

My

u

t

%

55%

45%

Edited parametersEdited parameters

Determination ofDetermination of K KPP, T, Tuu andand TTgg

The above figures are made MATLAB software. The amplitude of step command of MATLAB is unit, and so the read final value equals the process field gain KP = 0.72.Determination with editing of Tg and Tu is quite inaccuracy.Recommendation of Piwinger:

g

u

T

T

0 3.3

7.8 50

I PID PI

Recommendation of Chien-Hrones-Reswick

KC TI TD

P

10.3 g

p u

T

K T

PI

10.3 g

p u

T

K T

1.2 gT

PID

10.6 g

p u

T

K T

gT

0.5 uT

The initial conditions for optimization parameters:The process field is an ideal HPT1; The objective function is the fastest aperiodic transient at setpoint tracking; The optimization is based on the square-integral criterion.

Determination ofDetermination of K KCC andand T TII

Defined values: KP = 0.72, Tg = 10.6 sec., and Tu = 0.9 sec. The ratio of the time constants 11.8, and so the recommended compensation is PI.Using the above table:

The PI compensation is:

1 1 10.60.3 0.3 4.9

0.72 0.9 g

Cp u

TK

K T

1.2 1.2*10.6 12.7sec. I gT T

1 62.2 4.9( ) 1

12.7

PI C

I

sG s K

sT s

Step response of closed Step response of closed looploop

Important: It is not an optimal parameter choice!

Chien-Hrones-Reswick recommendations

KC TI TD

P

10.7 g

p u

T

K T

PI

10.6 g

p u

T

K T

gT

PID

10.95 g

p u

T

K T

1.35 gT

0.47 uT

The initial conditions for optimization parameters:The process field is an ideal HPT1; The objective function is the fastest periodically transient with maximum 20% overshoot at setpoint tracking; The optimization is based on the square-integral criterion.

A PI kompenzáló tag:

1 1 10.60.6 0.6 9.8

0.72 0.9 g

Cp u

TK

K T

10.6sec. I gT T

1 103.9 9.8( ) 1

10.6

PI C

I

sG s K

sT s

Determination ofDetermination of K KCC andand T TII

Defined values: KP = 0.72, Tg = 10.6 sec., and Tu = 0.9 sec. The ratio of the time constants 11.8, and so the recommended compensation is PI.Using the above table:

Step response of closed Step response of closed looploop

It can be seen that the approximation of process field the objective function is not satisfied.

PTn model

E P n

1G (s) K

sT 1

My ,u

My

u

30%

70%

10%

t

70t

10t

30t45%

55%

Determination of system parameters

The number of the first order time constant (n)

N 1 2 3 4 5 6

10

30

t

t

0.30

0.48

0.58

0.63

0.87

0.70

10

70

t

t

0.09

0.22

0.31

0.37

0.42

0.45

Time constant 1 2T TT

2

30

1

t

T

0.36

1.10

1.91

2.76

3.63

5.52

70

2

t

T

1.20

2.44

3.62

4.76

5.89

7.01

Step response of process field

Determination ofDetermination of n n andand T T

Defined values t10 = 1.95sec, t30 = 4 sec., és t70 = 10.1 sec. The process gain KP = 0.72

Based on the table above the PT2 is the closest approximation: n = 2.

10

30

1.950.49

4

t

t

301

43.64sec.

1.1 1.1 t

T

1 2 3.64 4.143.9sec.

2. 2

T T

T

10

70

1.950.19

10.1

t

t

702

10.14.14sec.

2.44 2.44 t

T

Proposed parameters for PTn model

The fastest periodically transient with maximum 20% overshoot at setpoint tracking

KC TI TD

P n=1 P

20

K

PI n=1 P

3

K T

2

PI n=2,3 P

1

K 2n

Tn 2

PID n=4,5 P

3 n

K n 2 2n

Tn 1

T

5

I n=6

2nT

Proposed parameters for PTn model

n = 2, and so you choose PI.

CP

1 1K 1.4

K 0.72 I

2n 4T T 3.9 3.9sec

n 2 4

In the industrial area you never use a pure P compensation to control a self-tuning process field!

Step response of the closed loop

Compare the two models the PTn is the better approximation, if the process field has not got a real dead time.

Process field with Process field with integral effectintegral effect

P P oror PDT1 PDT1 compensationcompensation

European structureEuropean structure

Process field with Process field with integral effectintegral effect

In this case the most popular compensation is theIn this case the most popular compensation is the P P or if the response signal without noise thanor if the response signal without noise than PDT1 PDT1,, but in the later case be applied the but in the later case be applied the PIDT1 PIDT1 tootoo..

The approximate models IT1 or HIT1The approximate models IT1 or HIT1

PI g

1 1G (s)

sT sT 1

usT

PI g

1 1G (s) e

sT sT 1

IT1 model from the reaction curve of process field

45%

65%

My ,u

u

gT IT

t

PI g

1 1G (s)

sT sT 1

Recommendation of Friedlich for IT1

Típus KC TI TD

P

PDT1 Tg

PIDT1 3.2Tg 0.8Tg

I

g

T0.5

T

I

g

T0.5

T

I

g

T0.4

T

The initial conditions for optimization parameters:The process field is an ideal IT1; The objective function is the fastest periodically transient with maximum 20% overshoot at setpoint tracking; The optimization is based on the square-integral criterion.

Step response of the Step response of the process fieldprocess field

The compensation type does not depend on the ratio of The compensation type does not depend on the ratio of thethe T TII andand T Tgg. .

Parameters of theParameters of theP, PDT1, P, PDT1, andand PIDT1 PIDT1

PP IC

g

T 9.9K 0.5 0.5 3.6

T 1.26

PDTPDT IC

g

TK 0.5 3.6

T D gT T 1.26sec

g1

T T 0.14sec9

PIDTPIDT IC

g

TK 0.4 3.15

T D gT 0.8T 1sec

g1

T T 0.11sec9

It is possible otherIt is possible other A ADD value too.value too.

I gT 3.2T 4sec

Step response of closed loop Step response of closed loop with P compensationwith P compensation

The steady-state error is 0; settling time is 11.4 sec.; overshoot is 6.1%

The steady-state error is 0; the settling time is 10.1 sec.; there is not overshoot.

Step response of closed loop Step response of closed loop with PDT1 compensationwith PDT1 compensation

Very bad! It is convenient the open-loop transfer function analysis.

Step response of closed loop Step response of closed loop with PIDT1 compensationwith PIDT1 compensation

The Bode plot of open-loopThe Bode plot of open-loop (G(G00(s))(s)) with PIDT1 with PIDT1

compensationcompensation

It can be seen that increasing the gain of the compensation up to 17.4 a better phase margin value is obtained.

The result of the PIDT1 The result of the PIDT1 compensation with the new compensation with the new

parametersparameters

Better, but it is not good!

Tuning the PDT1 Tuning the PDT1 compensationcompensation

Replace the phase margin value from 95° to 90° the KC increasing by 2.8-fold.

It is good enough!

The result of the PIDT1 The result of the PIDT1 compensation with the new compensation with the new

parametersparameters