DETERMINATION OF PERFORMANCE INDICES OF LLNEAR CONTROL SYSTEMS WITH DEAD TIME

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DETERMINATION OF PERFORMANCE INDICES OF LLNEAR CONTROL SYSTEMS WITH DEAD TIME

By

M. RABER}IA YER

Department of Automation, Technical University, Budapest (Received March 24, 1974)

Presented by Prof. Dr. F. CS . .\.KI

The practical identification processes are often approximating some- times rather closely - the process to be controlled, by means of systems that ean be described by first-order or more frequently by second-order lag transfer functions with dead time. So in planning control processes, the task mostly consists in selecting and adjusting the parameters of the controller to be used in the systems with dead time.

To design the control systems with dead time by classic methods encoun- ters computation difficulties. But modern synthesis methods permit the synthesis of processes with dead time at the desired accuracy if a digital computer is available.

The analysis results are useful for designing control systems in case of less strict requirements. Numerical analyses in the frequency and the time domains permit quick selection of the compensating element meeting the prescribed performance indices at a satisfactory accuracy. Thus, numerical results of analyses in the frequency and the time domains of the control systems of various structures 'with dead time are rather useful.

The stability tests of linear control systems containing a first-order lag plant with dead time and the determination of their transfer processes have been dealt with by several papers [1, 2, 3,4, 5, 6].

The results of frequency domain analyses of control systems descrihed by linear, second-order lag transfer functions with dead time in the case of serial PID (and in the resultant specific cases of P, I, PI, PD) controllers in a single-loop control system (Fig. 1) have heen presented in [7, 8, 9, 10, 11, 12, 13].

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382 JI. HABER.UAYER

60

20

o

~ ^CC,, 1-+"'-"2)';:'.",1' i' '

j

0.2 03

o

Fig. 1

Fig. 2

75⁰

0.1 0.2 0.3 0.4 K 0.5 0.6 0.7 0.8

Fig. 3

·f'~75°

0.4 0.5 0.6 0.7 0.8 0.9 K

Fig. 4

1.0 1.1 1.2 1.3

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60.

20.

PERFORJIA.YCE ISDICES OF LISE.-lR COSTROL SYSTE.US

30.⁰ tIT, =10.

5=0.7 t IT=1

0.5 K 1

Fig. 5

of' =30.⁰

~

⁴⁵⁰ ^60.⁰

^{50

^{tIT; =1}

1.5

Q~~--~~ __ ~ __ ~ __ L -_ _ _ _ ~~ _ _ ~

DJ 0..2 0..3 0..4 0..5 0..6 0.7 0..8 0.9 K

Fig. 6

---~~~---

1'/T=0.1

0. 0..2 a.:. 0.5 _1.3

Fig. ;-

60. t IT=l

~ "ft''' ---- ^-l'~ -- :.~:f-'-O~ - ---2

201~_~-~~Z~.======~~~=====~==~~--? k ./.. . .... , 7~0.~, '''"''

^{. .}

0. 0..2 0..4 0..6 O.S K 1 1.2 1.4 1.6 1.S

Fig. 8

3:)3

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384

r7

~ 10

80

60 c

40

20

o

JI. HABER.UA YER

---

- - - -

---

r

IT. =10 __ - - - -

,

---

0.1

60

20

0.2 03 0.4 0.5 K 0.6 Fig. 9

Fig. 10 0.7

---

!=~~---

</"=60°

75"---7?--

000 2

K

Fig. 11

0.8 0.9 1.1

~ ^I

^j⁼² t I T = l O , [

f=300 1:' IT =1

I _ - - - ...

06 tIT =10 ---~ ~'-

I

l

'1"=45° '

. ---

~--."

2: 0.⁴

1 _ - - - - ---..--.. ' , - I

' " " 1I - -- _{i:.§Q;:. -} _- _- _{- -} " " _'- ',tiT·=O.1 _"- _I I _,

02, _ - - - - - - - _ "~ i

·1 __ -~ I

o 0.1 0.2 03 0.4

750, c 750 I

, I 0.5 0.6 0.7

1\

Fig. 12

0.8 0.9 1.0 1.1 1.2 1.3

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PERFOR.lIASCE L·WICES OF LiNEAR CO.'TROL S)STEMS 385

The results of tim e domain analyses for a linear control system with dead time shown in Fig. I, with selection of an I controller, were given in [15].

In the present pa per diagrams determined for various parameter values are presented, giving the maximum oyershoot a and the control time te of the examined control system with dead time.

Proportional-plus-integral control

The transfer function of the plant in the linear cOlitrol system with dead time to be analysed numerically is

10000

1000

i

100

10 0.001

I I

I

,~, ^,

/11- 1

0/

_I

_' ^, " ^, ^,

^I ⁵^{= 0.3}

' ','

^t'/T=O.1

" ^" " ,

"" I " , ,

'J _i"

"" ^,,'\

"

t'/T'=~~ , _I

I

r,"

^~'=300

I I _{" I} ,,'<

, ".'=45⁰

tt(" ^",x

^.,~~.j"=6QO

"

^'~,^\

I

"

^{\ \ \}

i

1.f'=75j ",

\

^\

I

I I ^I \ \ \ I

I

I I

IV; -

I I ^I

I

I ~!TI-O.11 ^{Fig. 13}

0.01

K 0.1

i

j I

!

I

I ,

i I ,

I

ⁱ

,

10

: I

_i

!

, I

I

I :

Fig. 14

K

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386 ^_1[.H.-JBER.IIA YER

I

I I I

^I ^I

I I ^I ^I

1000

I '

X'I _I

qk/Ti~10 _I , i" ,

~J.

',J

^''[{h30⁰ I ^,

"

" I

I "". '--4 ' " ;P'~45° " , i I ; 1

I I 1--~~6<{'

,

_r---- I I

f~T"~,,1 "V"

I

r,

1 , '

i ^I

IZf~'(l " ,

I TlTi~ll" I

"

I

100

i i ⁱⁱ

1''-( " "

^,

r--l--

l =07 : I I I ⁱ ~

'" " """ ", ^",

^'-,

10

1 0,001

I

:

,

't'/T~0.1

I

i,

i I !

:

, I

j ,

!,

!

i

0,01

1 I

,

I

,

K

Fig. 15

,,-"

_'-

'"

}, ~ I /

t/Ti~O.1-

i 1

I I

I _I

0.1

i

I tlTi=O.l -

I, I " I '

1 L-~L-~~ _ _ ~I ____ ~~~ __ ~ ____ L-~-J

0.01 Ql

K

Fig. 16

I

_I

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with T

T

PERFORJL-LYGE rIDIGES OF LL'iEAR GOSTROL SYSTE.US

exp (-s T) C(s)

=

1 2~ Ts

+

^T2^S2

dead time,

time constant of the second-order lag, damping factor.

I I

M

³⁰⁰

75° 45⁰

J 100

i

⁶⁰⁰

I I

10 _5=0.7

1

_t-"/T=10

I 1 I

1 001

I

ⁱ

I

I I

I

I I

I 0.1 K Fig. 17

9~0" I

1 I

I

I !

I

^irtT'~l'i⁴⁵^I^{0 -}

I ^~Oo

I

J

^7S

J I

I

1

i

L-Lll~l,-~+-~~~~-,

__ - , __

~

I i t/T=Ql'! , . , " , . " . , ,

10

ITI

1 0.001

i

1

I

!

I ;

I

i

,

0.01

i

^{' I"}

,I

^'!-i '--i. " ^,,, ^,,",

i

'r~

,

^~

i I

~~.: ^{I I}

I

. _i _!

1

,

I

^,

, i

!

0.1 10

K

Fig. 18 4 Periodica Polytechnica El 18/4

387

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388 .\J. H.1BER.\IA YER

The senes controller IS the proportional-plus-integral PI lag with ?

transfer function:

where K -- loop gain,

G(s)

=

K

(1 + _1_",) ,

sTi

Ti integral time constant.

The unit step responses for the various parameters of the control system

U

⁼ 0.3, 0.7, 1, 2; TIT = 0.1, 1, 10 and TiTi = 0.1, 1, 10) were determined by the digital simulation method using a computer [12]. The maximum overshoot and the control time were plotted in diagrams versus the loop gain, by connecting the points belonging to identical phase margin values (( (Figs 2 to 12 and 13 to 23, respectinly).

In some cases the a-values dttermined by the digital computer werp compared with the approximative values obtained from the Jl-x curves [1.:1:].

The best approximation was obtained, as it has been established also ^III the case of the integral control, in the 45° (F 60° range [13].

From 0' 0' (K) diagrams (Figs 2 to 12) the value of the maximum overshoot is seen to greatly depend on the time constants of the system.

By varying the parameters of the controller in the range 0.1 TITi

<

10, in the case of a given phase margin, the value of 0' greatly depends on the timc constants of the controller.

In plotting the control time diagrams, the time needed for the deviation of the unit step response from the final value to keep below --'-- 5 per cent was regarded as control time.

From the feiTi

=

f(K) diagrams (Figs 13 to 22) it may he read off that a) for a given phase margin the value of f,iT! is decreasing with increasing loop gain,

b) for a given plant and fixed Ti , the maximum control time greatly varies as a function of the phase margin.

An important characteristic of the unit step response of a control· beside the maximum overshoot and the control time - is also the number N of deflections occurring up to steady state. For meeting the performance indices N_max 3 is generally demanded. The 0' and fe diagrams detcrmined for the actually studied control system do not yield, however, the N_maxvalue. But in the knowledge of the unit step responses [14.] it can he stated that with decreasing damping factor values the deflection tendency is increasing, as expected. When T ~ Ti , then for low ~ values the number of the deflections N}> 3.

The ahoye considerations show that if a PI controller is chosen for the compensation of the tested control system, no far-reaching conclusions can he drawn on the characteristics of the unit step response. But utilizing the

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PERFOR.1fA.VCE LVDICES OF LLVEAR COSTROL STSTE.1fS 389

frequency domain analysis results the presented diagrams may be of help in establishing the course of the unit step response of the control system of the given structure at a fair accuracy without calculations in the time domain.

1 GO i---r-"tr"':;C"

10

4*

Fig. 19

0.1 K

Fig. 20

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390

Conclusions

The time domain analysis of the linear control system with dead time given in Fig. 1 suggests that in case of less strict requirements, the performance indices of control systems may be estimated from empirical relationships with a sufficient accuracy in the environment of the phase margin of 45⁰ [14].

But for synthetizing control systems, subject to stricter requirements, correct results can only be obtained by computer analysis. If the behaviour of the plant can be characterized by a linear, second-order lag transfer function with

I I IJ

_I

-t±

^I ^I

~Ti~10. ^I ^I

I I ¹ ¹

1-- I I

!

"~1 ~ r".,.L ³⁰

^o

^I

^! ^I

I • r", i

I

: _{I 'L}I i'.. _'l

I'

_'1.fh45° _"~~~ I I

: _{I I}

1000

I I !

h .

.p'~o~,

"

^'t'/Ti~li I

I !

i 1 1 I

J. ¹ ^I i

I

^!^,

" 't '''1

. --"';

i

¹^I ¹ⁱ

^I

I

i ¹^,

^!

I f'~75°"i ^I"i~' \

l

^,,,

~

ⁱⁱ

100

I I

h-.i h '

^I

10

1 I I

1 I

'9-.:i

I'"

. )=2

I ^1t'~I''''

It'/Ti~·l ^II 1 I

I

^I ^I ^j

I

^~,

i I ! I I

i'"

1 1 1 I 1 1

i I

^I

1

Î ^, ÎÎ Î

I

1 ^I

0001 0.01 0.1

K 10

Fig. 21

1 ! i ⁱ I I i 1 ~J

0.001 001 0.1

K Fig. 22

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PERFORJIA:VCE LYDICES OF LISEAR COSTROL SYSTE.US

0.1 K

Fig. 23

391

dead time and the practical identification processes are aimed in many- cases at the estimation of the parameters of such structures then the maximum overshoot and the control time values given in [15] and those determined in the present paper are useful in designing the control system witll dead time corresponding to Fig. 1.

Summary

Diagrams of the maximum overshoot and the control time of a linear, one-loop control system with dead time yersus the time constants of the system. determined by means of a digital computer are presented. The plant was regarded, _. as it ·often OCCurs in' control proc-

~sses - as a second-order la!! element wilh dead time and a PI element was chosen as a series connected controller. ~

References

1. SOLDIAN, J. I.-AL-SHAIKH, A.: A State-space Approach to the Stability of Continuous Systems with Finite Delay. Control. October, 1965.

2. SOLDIA", J. I.-AL-SHAIKH, A.: A State-space Approach to the Stability of Continuous Systems with Finite Delay. Part 2. Control. ::'\oyember, 1965.

3. SCH:mDT, G.: Vergleich yerschiedener Totzeitregelsysteme. :JIessen, Steuern, Regeln. 10, 1967. H. 2.

4. GOSCIl'iSKY, A.: Przeregulowania w "Ckladach z Opoznieniem. Automatyki i Telemecha- niki. TO:JI-XV ZESZYT 1-1970.

5. CEPEJ3PrIHC~I1YI, A. 51.: l\1eTO,l onpe,le.leHH5I Beml'lllHbl :llaKcmla.lbHoro OTK,10HeHH5I B CHC- Te:llaX aBTO:IlaTlI'leCIWro perynllpOBaHlI5I, BK.l10^tJa101..Q1IX OObelZT C 3ana3,lblBaHHe:ll npll- OOPbI H ClICTe:llbl ynpaBjleHH5I. 1968, 11.

6. KEVICZKY, L.-CS . .\.KI, F.: Design of Control Systems with Dead Time in the Time Domain.

Acta Technica, 74, 1973 (1-2).

7. CS . .\.KI, F.-HABER)IAYER. :JI.: Stability Test of Linear Control Systems with Dead Time.

Pe~iodica Polytechnic~, El.Eng. 12~ 1968, 3. •

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392 ^.IT.H.·IBER.ITAYER

8. HABER)IAYER, }I.: Stability Test of Linear Control Systems with Dead Time by Digital Computer. Periodica Polytechnica, EI.Eng. 12, 1968, .1.

9. HABER)IAYER, }I.: Stability Test of Linear Control Systems with Dead Time Compensated by PI Controller. Periodica Polytechnica, EI.Eng. 14, 1970, 1.

10. }L~BER)lA YER, }I.: Stability Test of Linear Control Systems with Dead Time Compensated by PID ,-\ction Controller. Periodica Polytechnica, EI.Eng. 14, 1970, 2.

11. HABEK\IAYER. }I.: Stability Degree ,-\nalvsis of Linear Control SYstems with Dead Time.

Periodica Polvtechnica. 'El.E'!tg. 15. 19'71. 3. .

12. HABER)IAYER. }I.: Stability Deg;ee ,.\nalysi·s of Linear Control Systems with Dead Time Compensat'ed by a P~opo;tional'ph;s-integral Controller. Periodica Polytechnica, EI.Eng. 15, 1971. 3.

13. HABER)IAYER. }I.: Stability Degree Anah'si, of Linear Control SYstems with Dead Time by a Digital Computer: PerIodica Poiytechnica, El.Eng. 16, i972, .1.

U. HABER)IAYER, }I.: ;\.nalysis of Linear, One-loop Control Systems with Dead Time in the Frequency and the Time Domain. Ph. D. Thesis. Budapest, 1972. (In Hung.)

15. HABER)lAYER. ^}I.:Analysis of Linear Control SYStems with Dead Time in the Time Domaiu.

Periodica Polytechnica. EI.Eng. (in press):

Dr. llaria HABER:IIAYER, H-15:21 Budapes1

DETERMINATION OF PERFORMANCE INDICES OF LLNEAR CONTROL SYSTEMS WITH DEAD TIME