ControlSystemsDesign Part: Optimisation

Slovak University of TechnologyFaculty of Material Science and Technology in Trnava

2007

Control Process Optimisation Optimisation Process

control loop structure design

optimum criteria selection

optimum control parameters computation

control process simulation

control parameters refinement

control process quality evaluation

documentation production

...SAT

Control Process Optimisation

control process qualitycontrol process stability

steady state - process variable deviation

dynamic control process overshooting

time of control process treg

integral criteria f(dev)

non oscillation control processes

Control Process Optimisation control process stability

characteristic polynomial

characteristic polynomial roots negative part of complex roots!

degree of the stability

critical parameters single control loop with P controller

critical GAIN

critical period Tkr

Control Process Optimisation

steady state - process variable deviation should be = 0; Deviation = Set Point - Process

the P controller problem: GAIN has to be as large as possible; (!) stability violation for higher order systems

else process deviation = 0 --- I part of controller; destabilisation of control loop

stability versus quality - solution is compromise

Control Process Optimisation

Dynamic control process optimisation standard forms of a characteristic polynomial

Ziegler Nichols method

method of optimum module

methods of integral criterions

Dynamic control process optimisationstandard forms of a characteristic polynomial

Naslin form of characteristic polynomial

0*0

n

i

ii sa

1 12

. . i i ia a a

is according the

i = 1,2, .... n-1

Dynamic control process optimisationstandard forms of a characteristic polynomial

Naslin form of characteristic polynomial

The parameter depends on the chosen overshooting

xmax according the table:

1.7 1.75 1.8 1.9 2.0

xmax 20 16 12 8 5

Dynamic control process optimisationstandard forms of a characteristic polynomial

Graham - Lathrop form

n Characteristic polynomial q

1 1q

2 1q.4,1q2

3 1q.15,2q.75,1q 23

4 1q.7,2q.4,3q.1,2q 234

5 1q.4,3q.5,5q.0,5q.8,2q 2345

0ω

s

Dynamic control process optimisationZiegler Nichols method

input data:

GAINcr - critical gain

Tcr - critical period

measured or computed at the stability boundary of the single control loop with P - controller

Dynamic control process optimisationZiegler Nichols method

Cont

roller

Para

metersValues

P Kr 0,5 . Kkr

Kr 0,45 . KkrPI

Ti 0,85 . Tkr

Kr 0,6 . Kkr

Ti 0,5 . TkrPID

Td 0,12 . Tkr

Dynamic control process optimisationmethod of optimum module

The transfer function of a controlled system is supposed in the form

The control parameters are for the ideal parallel PID algorithm r0, r-1 and r1:

i

n

ii

s

sa

KsF

.0

0r

KK sr

0

1

r

rTd

1

0

r

rTi

Dynamic control process optimisationmethod of optimum module

PI controller

2021

20

0

1

23

01

..2.5,0.

aaa

a

r

r

aa

aa

Dynamic control process optimisationmethod of optimum module

PID controller

401122

2021

20

1

0

1

345

123

01

..2..2

..2.5,0.

0

aaaaa

aaa

a

r

r

r

aaa

aaa

aa

Dynamic control process optimisationmethods of integral criterions

IAE Integral of Absolute Error

ITAE Integral of Absolute Error multiplied by Time

Dynamic system approximation

by K, T and D:

DsesT

K

.

1.

Dynamic control process optimisationmethods of integral criterions IAE:

A B

P 0,758 -0,861PI

I 1,020 -0,323

P 1,086 -0,869

I 0,740 -1,130

IAE

PID

D 0,348 0,914

Dynamic control process optimisationmethods of integral criterions ITAE:

A B

P 0,586 -0,916PI

I 1,030 -0,165

P 0,965 -0,865

I 0,796 -0,147

ITAE

PID

D 0,308 0,929

Dynamic control process optimisationmethods of integral criterions

B

T

DAY

.

rs KKY .

T

DBA

T

T

i

.

B

d

T

DA

T

T

.

For GAIN

For time constants

For Ti

For Td

Dynamic control process optimisationhalf dumping criterion

Dynamic control process optimisation half dumping criterion

Dynamic system approximation

by K, T and D:

DsesT

K

.

1.

Control Process Optimisation half dumping criterion

PI controller:

946,0

.928,0

T

DY

rs KKY .

583,0

.928,0

T

D

T

Ti

Auxiliary parameter

For GAIN

For integral time constant Ti

Control Process Optimisation half dumping criterion

PID controller:

Auxiliary parameter

95,0

.37,1

T

DY

rs KKY .738,0

.74,0

T

D

T

Ti

95,0

.365,0

T

D

T

Td

For GAIN

For integral time constant Ti

For derivative time constant Td