Automatic Control System

V.

Compensation

CompensationIf the satisfactory design of the process dynamics can’t be obtained by a

gain adjustment alone, then some methods are indicated to modify the loop’s structure or compensate the loop’s dynamics.

While the variety of compensation schemes is great the classical three-term controller design has been found to be particularly simple and effective. This compensation scheme has historically implemented using analogue mechanical, pneumatically and electrical controllers.

Today the controllers are microprocessor based in which the compensation is implemented in a software. An extra steps is required to convert the analogue signal to digital information and the digital information to analogue signal. Nowadays in the industrial area the most popular A/D and D/A converter contain 12 bits.

The sample time

If the sampling time is much more less than the sum of time constants of the process than the theory and design methodology is the same in digital and analog technologies.

If one wants the digital PIDT1 compensator seems to be such as analogue must choose appropriate sample time period.

15050 0 jj T

TT

Grey model (Tj are the time constants.)

Black model (Ts is the settling time.)750250 0ss T

TT

Structure of classical three-term controller

1

eCK

IsT

1

D

D

sT

sT

1.01

u

My

r

}sT

A1

1

sT

sT

11{K)s(G

DD

D

icc

}dt

)t(dwT1.0

dt

)t(deTdt)t(e

T

1)t(e{K)t(u DD

0IC

!nm

w

T

TA D

D

10AD

t

( )h t

t

t

( )h t

( )h t

Structure of classical three-term controller in sampled controller

1

ecK

n

ii

ieT

T

0

0 )(

D0D T1.0T

)1n(w1.0T

u

My

r

}1.0

)(1.0)1()()()({)1(

00

0

DD

n

Ic TT

nyneneTie

T

TneKnu

w

Real combination of the three kinds

Ic sTsG

1)(

Proportional P

Cc KsG )( )()( teKtu C )()1( neKnu C

Proportional-Integral PI

}1.01

1{}

1.01

9.01{)(

D

DD

D

DCc sT

sTK

sT

TsKsG

}1

1{)(I

Cc sTKsG })(

1)({)(

0

dtteT

teKtuI

C })()({)1(0

0 n

IC iu

T

TnuKnu

Proportional-Derivative PD

Integral I

0

)(1

)( dtteT

tuI

n

I

ieT

Tnu

0

0 )()1(

Proportional-Integral-Derivative PID DI TT 4

}T1.0T

)n(w1.0)1n(u)n(uT)n(u{K)1n(u

D0DC

}dt

)t(dwT1.0

dt

)t(deT)t(e{K)t(u DDC

Tuning controllers

• Which type of PID is the best solution?

• Which parameters are optimal ?

The type of PID depend on transfer function of the plant (process model)

The optimal parameters depend on which is the required performance specification of the feedback control

If the simple feedback loop with compensation in cascade is unsatisfactory for the required performance specification than must modify the structure of control loop or exchange the classical PID scheme of control algorithm.

Position of the compensation

cascade

This is the most frequently when single feedback loop are used to compensate of the dynamical

parameters of process or machines.

( )e s ( )My sGC(s)

( )r s GE(s)

Cascade and feedback

( )e s ( )My sGC1(s)

( )r s GE(s)

GC2(s)

Position of the compensation

( )r s

parallel

feedback

( )e s

( )My s

GC(s) GP(s)

( )e s( )My s

GC(s)

GP(s)

( )r s

The parallel and the feedback arrangement are popular in the electronic circuit technique.

The PIPDT compensator

( )e s ( )u s

PPIPDT

I

I

1 sT

sT

D1 sT

1 sT

CK

t

( )h t

t

( )h t

I DPIPDT C

I

1 sT 1 sTG (s) K

sT 1 sT

Where the TI > 4TD ; TD >

5T

t

( )h t

It is popular in the electronic circuit technique, and the very similar lead-lag was used in the

pneumatic controller.1 D

LL CI D

1 s 1 sG (s) K

1 s 1 s

Compensation techniques

Depends on the identification of the process field transfer function

• Step response of the process field transfer function: Black-box model and recommendations for the controller type and value of the parameters.

• Frequency response of the process field transfer function: Black-box model, and fitting of PI, PDT1, or PIDT1 controller‘s frequency response to the process field transfer function.

• Pole replacement: Grey/box model. Poles of the process field transfer function is known.

Modelled the process from reaction curveby dead-time first order transfer function HPT1

t

uym ,

u

my

uT gT

1( )

1usTm

pg

yG s e

u sT

pK

Suggested parameters for HPT1minimum settling time and without overshoot

system performance with following trajectory of reference signal

Kc Ti Td

P 1

0.4 g

p u

T

K T

PI 1

0.36 g

p u

T

K T

0.85( )g uT T

PID 1

0.28 g

p u

T

K T

0.85( )g uT T

0.16( )g uT T

This parameters based on the absolute integral criterion of performance.

There are any other parameter commendation

Modelled the process from reaction curveby “n” order transfer function PTn

t

uym ,

uU

mm yY

30t

npp sTKsG

)1(

1)(

70t

%30

%70

%10

10t

Determination of system parameters

First the order “n” of the system is determined by the time percent co-efficient are best fittedThe time percent co-efficient n= 1 2 3 4 5 6

30

10

t

t0.30 0.48 0.58 0.63 0.87 0.70

70

10

t

t0.09 0.22 0.31 0.37 0.42 0.45

Dominant time 221TTT

1

30

T

t0.36 1.10 1.91 2.76 3.63 5.52

2

70

T

t1.20 2.44 3.62 4.76 5.89 7.01

Suggested parameters for PTnminimum settling time and maximum 20% overshoot

system performance with following trajectory of reference signal

Kc Ti Td P

n=1 pK

20

PI n=1

pp Kn

n

K

321

TT

n

n

2

1

3

2

PI n=2,3

pp Kn

n

K

1

1

11

T

n

n

2

2

PID n=4,5 2

1

nn

K p

Tn

n

1

2

5

T

I n=6

nT2

Modelled the process from reaction curveby dead-time integral first order transfer function HIT1

t

uym ,

u

gT

1 1( )

1

pI g

G ssT sT

IT

Suggested parameter for IT1 from Friedlich

Summary questions • Which transfer functions are used approaching of the real

reaction curve. • Interpret how the steady-state characteristics of blocks of a

closed loop are fitted each other?• Define the closed loop transfer functions and the opened loop

transfer function of a feedback control system. Highlight what this term “system type” means.

• How the steady-state remaining error depends on the system type?

• Draw the oscillatory and the monotonic response of unit step.Explain the terms of the performance specifications of response.

Summary questions

• What is the goal of the compensation?• What is the classical three-term controller? Outline the control

actions associated with each of its constituent parts.• Which transfer functions are used approaching of the real

reaction curve.