DEAD TIME COMPENSATED PI CONTROLLER STABILITY TEST OF LINEAR CONTROL SYSTEMS

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STABILITY TEST OF LINEAR CONTROL SYSTEMS WITH DEAD TIME COMPENSATED BY PI CONTROLLER

By

M. HABER:.\IA YEll

Department of :\.utomatiol1. Technical LlliYersity. Budapest (Received J line 30. 1969)

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

In a previous paper [1] ,,'e haye theoretically invcstigated the stability region variations of a linear control system of second order lag and dead time shown in Fig. 1 with a unite feedback, compensated, generally speaking, hy a PID controller. We have deriyed that the transcendental equation determin-

Control/er System

G(s - exp (-57;") ) - 1 + 2S Ts + T2 s2

Fig.

ing ^O)er.the angular frequency belonging to t 11(' critical loop gam

ICe·

^1;;:

( 1) 1

In the knowledge of ^C') ^='-=('Jer ohtained from (1) hy iteration the loop gain may he eyaluated:

(

^')\- ;

Control systems with second order lag and dead time compensated by P, I, PI, and PD type controllers may hc considered and calculated as special cases of the ahoye control system.

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54 M. HABER.UAYER

In a following paper [10] we presented graphs showing the stability region variations in function of the dead time and the system time constants for controllers of types

P

and

I

with values of

Kcr

obtained by digital computer.

Proportional-pins-integral control

The present paper investigates the stability region variations of the tiame cuntrol system, if

it

is compensated hy PI type controller.

The transcendental equation determining ^('Jcr, the angular frequency belonging to

Kcr

putting

Td

0 into (1) 'with the assumption of

Ti

1 is:

(3) Figs

2-6

show the values of

Kcr

for time-constant ratio T;

T

=

0.2, 0.6,

1, 4, 10 vs 0.05 TIT! .c' 10 with parameter values ~ = O.L 0.2, 0.3, 004, 0.5,

0.6, 0.7, 0.8, 0.9, 1, 1.25, 1.5, 1.75,

:2 in log-log scale for the sake of clcarnes~.

The figures demonstrate that

a)

'with increasing ~, thc stability region increases under a given value of TITi, over this value it diminishes. The value of TITi, ill the yicillity of which the critical loop gain depends hut slighthly on the value of ~, diminishes with increasing TIT;

b)

'with increasing TIT for very

la-w

values of TIT! the stability limit more and more approximates the loop gain value

Kcr

= 1 independently of tlw damping factor ~.

From this last hehayiour of the control "'ystem follows:

when TIT! and TIT simultaneously approach to 0 and to

=,

respectively, the control system 'with second order lag and dead time compen8ated by PI type controller may he suhstituted }JY a purE' dead time system, for wich the stability limit is

Kcr

l.

The corrcctness of thc last statements is easy to see. The stability limit from the oyer-all transfer function of the opeIl loop system is:

Y-( ,

s) = .1.\.cr

"[-

^lf

l · 1

-;- - -^j'

Tis

After somt' arrangements we have:

eXflJ_S_T)_

1

+

²^~T^S

+

^{T~ s~}

\Vhen TIT, -0 the critical loop gain approximately IS:

1+2 ~T!T(ST)--'c-(T;TnsT)

Ker

~

---'---'-,---'----'-

exp

(-ST)

-1.

(1)

(3)

STABILITY TEST OF LI.YEAH CO.YTHOL S,-STEJIS 55 '<er

1

¹

20

Proportionol- plus- integral control

10

5

3 2

0,021--,---+----+---,--

o,OII---:--T---+---,--~--L--_+--~-~~~--~~---~

O'03~~~ __ ^~____ ^~____ ^~__ ^~____ ^{L _}^{_ _ _ _}^~^{_ _ _ _}^~^{_ _}^~^{_ _}^~_ _ _ _ _ _ ~ _ _ ~

0,02 0,03 0,05 0,1 0,2 0,3 0,5 2 3 5 10 7;/0

Fig. 2

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56 JI. HABEIUIA YER

10

Kef Proportional-plus-integral control

5

3 2

05

0,3

02~---i--'--~---+

OUJL __ - L __ ~ ______ ~ ____ ~ __ J -__ ~ ____ ~~ ____ ~ __ L -__ ~ ____ - L _ _ _ _ ~

0,02 0,03 0,05 0,1 0,2 03 05 2 3 5 ¹⁰ 7;/li

Fig. 3

From the last equation 'we can see, in the caSE- of small value of TIT thEC

;;;econd order lag character dominates still strongly.

With increasing TIT the second order lag character may be more and more neglected in comparison to the dead time, as -we can see from Figs 2--6.

On the other hand, in case of

T/T;

~ 10, i.e. when

TITi ---

^{X l ,} the sta- hility limit from the resultant tra11"fer function of the open loop system (4) IS:

1 -'-- :2

;(TjT)

(ST)

+ (T/if

(sr)~

Ker

,'CoO T)T(Si) ---'-.:...--'---'--'--'---'---'--'--

exp (--Si)

Consequently, when TIT; --. "', the system may be substituted hy a system with second order lag and dead time compensated by I type controller.

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STABILITY TE."T UF LISEAli COSTlWL .'YSTE.IJS

Proportionaf-pfus-iniegraf contro!

5 3 2

os

o,lr---'---~---r---+---~~----~~~

0,D2 0,03 0,05 0,1 0,2 O,3 0,5 Fig . . J

2 3 5 10 ?;/T:

3 Kcr 2

Proporlionai- pfus -integra! controf

7:/T=4

O~r---~--~---~---.. ---~-"·-~---·-··-···~~···,, 0,2r-~----··~---··-+---·--~·-··-~--

o,lL-~ ____ ~ ____ - L ____ ~ __ -L ____ ~ ____ ~ ____ ~ ______ ~ ______ ~ __ ~

0,02 0,03 0,05 0,1 0,2 0,3 0,5 2 3 !i 10 1;'/T:

Fig. 5

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JI. HA HEIUU YER

Kef Proportion ai- plus - integra! contra!

7:/T=IO •

O'21---~---~---·~---~

o,l.~~~~~ __ ~ ____ ~~ __ ~ ____ -L ____ ~~ __ ~ ____ -L __ ~

0,02 0,03 0,05 0,1 0,2 0,3 0,5 2 3 5 10

(;-IT.

Fig. 6

According to the last statements in case of r}Ti ~ 10 the critical loop gain values extrapolated from Figs

2-6

agree ·with the values

Ka

obtain-

rd for I type compensation.

For instance let he:

TIT = 0.6

= 0.5

30,

On the hasis of Fig. 3 the extrapolated \-alue is K_crP1 1.4· 10-~. On the other hand using Fig. 7 of Ref. [10] the corresponding Yalw' is:

K ~

=0.42

er

T.

₁ ^'1_,

Hence KerJ

0.42/30

=

1.4 . 10-

² is identical with Kerp^!"

Conclusion

In the knowledge of the stability region belonging to a given linear continuous control system 'with second order lag and dead time compensated by P, I and PI controllers we are already able to choose the most advantageous type of compensation.

We found out [10] that for 10·\1" dead time values a proportional cQmpen-

",ation is preferahle. For high dead time values the integral type control proves to he best. On the hasis of [10] and this paper the most suitahle controller may

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:;TABILln- TEST OF LLYEAR CO,YTIWL SYSTEJI::'

Le calculated for the intermediate dead time values as welL with the help of the given system constants.

For the sake of completeness the investigation of

PID

type compensation will he dealt with in a coming paper.

Summary

The prcsent paper givc,; the stability region variations of a linear continuous control 'y~tem with second order lag and dead time' compensated by proportional-plus-integral action controller. The critical loop gain yalues were evaluated by digital computer. The diagram~

representing Ker in function of the dead time for 0.05 ;S; rjTi;S; 10 are plotted in log-log scale for the sake of clearrwss with the system time constants as parameter.

References

1. CS,\Kl F.--HABER}IAYEH }I.: Stability test of linear control systems with dead time.

Periodica Polytechnica, El. 12, 3fl-318 (1968).

2. CSol.KI F.: Control dynamics (In Hung.) Akadcmiai Kiad6, Budapest 1966.

3. CSAK1 F.-BARS R.-BARK1 K.: Automatika 1. Tankonyvkiad6, Budapest 1966.

4. CS,\.K1 F.-BARS R.: Automatika

n.

Tankonyvkiad6, Budapest 1966.

5. SOLDIA"< J. L-A.l-SUAIKU A.: A state-space approach to the stability of continuous systems with finite delay. Control, Part 1, pp. 554--556. October 1965.

6. SOLDIAl' J. I.-AL-Sn.UKU A.: A state-space approach to the stability of continuou,.

systems with finite delay. Control, Part 2, pp. 626-628. 1'\ovember 1965.

7. CUOKSY 1'\. H.: Time lag systems. Progress in control engineering. Volume 1. London 1962.

8. EISEl'BEIIG L.: Stability· of linear systems with tran;port lag. IEEE Transactions on Autom. Ac-ll, 247-254 (1966).

9. GRABBE ::\I.-RA}IO S.- \VOOLDRIDGE D. E.: Handbook of Automation Computation and Control, Vol. 3, 1961.

10. HABER'lIAYER ::\1.: Stability test of linear control systems with dead time by digital computer. Periodica Polytechnica, El. 12, ·1c13--152 (1969).

Dr. 1Iaria HABER:'IL\ YER, Budapest. XL Egry

J

ozsef u. 20. Hungary