Habermayer, M.

English

Periodica Polytechnica (Budapest University of Technology and Economics)

STABILITY DEGREE ANALYSIS OF LINEAR CONTROL SYSTEMS WITH DEAD TIME BY A DIGITAL COMPUTER By M. HABER:\IAYER Department of Automation. Technical "University, Budapest (Received }Iarch 18, 1972) Presented by Prof. Dr. F. CSAKI In previous papers [6, 7] 'we have studied the stability region variation permitting to reach arbitrary phase margins in the case of the linear control system with dead time shown in Fig. 1, compensated in series by proportional (P), integral (I) and proportional-integral (PI)-type compensation elements, respectively. In this paper diagrams showing the variation of the stability region versus the phase margin, the dead time and the time constants of the system compeilsated by a proportional-differential (PD) element are presented along with a procedure written in the ALGOL language for the determination of the stability region permitting to reach arbitrary phase margins of the con-trol system shown in Fig. 1, compensated by P, PI, PD and PID-type compen-sation elements, respecth-ely. Proportional-differential control The transfer function of the controller (Fig. 1) is as follows: The transcendent equation derived from the transfer function of the open loop used to determine the angular frequency permitting to reach ar-b i trary phase margins is: (1) 1 t is easily seen that the transcendent equation (1) may be obtained from the I elationship :r _ on: _ tan -1 _2_'_T_w_ 2 1-T~ 0)2 WTi tan -1 __ --'-__ 1-Ti Td w~ (2) 356 M. HABERMAYER inyolving a PID-type compensation, by substituting liTi = 0. Our statc-ment is fulfilled if: ;r T 1 - + tan-1 OJ d = -- tan-2 1 Xl w- Y7(S) ~(S) e-S '{;'- X2 1+2J"Ts+T2s2 x2 Fig. 1 10 0,1L-____________ ~ ____ ~ ______ ~ __ ~ ____ ~ __ _L __ ~~~ ____ ~ 0.01 D.I 10 100 Fig. 2 Taking the tangents of both sides and after arranging we have: 1 1 In accordance with our above statement. The above statement is useful in developing the program for the deter-mination of the stability region yariation for P, PI, PD and PID-type controls. By substituting l/T = 0, the limit position of the stability region may be calculated from the following equality: (3) LLYEAR COSTROL SYSTEMS K 0,1 QOI~----~------~----------------+----7--L--~--~------------0.01 0,1 K 0.1 -~---------- --,--~~----~----~--~ 0,01 0,01 Fig. 3 FiJi,4 10 10 J = 0.3 'i:/T=10 357 700 loa 358 .H. FIABER.UAYER involving a PID-type compensation, with the angular frequency value ohtain-ed hy the numerical evaluation of the relationship (1) with arbitrary accuracy. Diagrams 2 through 16 show the functionality K = K(Td/-r) determined with the help of a digital computer for the phase margin values r' = O~, 30~, 45°, 60'\ 75°, with the time constants of the plant as parameters. K 10 s= 05 'i:fr= 0,2 01L-____ ~ ______ ~ ____ ~~ ______ ~ ______ ~ ______ ~ __ ~ __ ~~ __ ~ 0,01 0,1 10 100 Fig. 5 K 0,1 001L-__ ~-L __ ~ __ -L __ ~ __ ~~ ____ ~ __ ~~ __ ~ __ -L __ ~ __ ~~ __ ~ 0,01 0,1 10 100 Fig. 6 LLYEAR CO,'TROL SYSTE-IIS 359 0,01 0,1 10 700 Fig. 7 J = 0,7 70 0,1L-__ ~~ ______ ~ ______ ~ __ ~ __ ~ ____ ~ ________ ~ ____________ ~ 0,01 0,1 10 100 Fig. 8 360 JI. HABERJIAYER K 0,1 QOIL-____________________________ ~ ____________ ~ ______ ~ ____ ~ QOI 10 Fig. 9 K O,OI~----~--------~--------------;----~----------~--~~~------0,01 0,1 10 TeI,:; :CD Fig. 10 LlSEAR CO.'TROLS YSTE.\[S 36J Fig. 11 From the diagrams the regularities described below follow: a) For TJr -r 0 and simultaneously TIT -r=, the controller may be substituted by a proportional element and the plant by a pure dead time ele-ment. Accordingly, the limit position of the stability region approaches 1. K , • f ( '/ • If': 0° 30° 45° 60° 75° 0,01 0. 1 10 100 Fig. 12 362 .11. HABERJIAYER <P= 75° 0.1 0,01~-----------------------0,01 0,1 Fig. 13 iD j=1 'C/I = 10 lOG b) When the dead time is lower than, or at the most equals the time cons-tant of the second order lag, then the loop gain shows a maximulll in the range of 0.1 ;§ Tdjr: ;§ 10. , , , ~---~~~.----10 G,D1 0,1 Fig. 14 10 K 0,1 K 0,01 60° 75 i 0,1 LISEAR COSTROL SYSTK\lS 10 Fig.l5 j = 2 Z/i:= 10 363 100 0,1 >-----------------~-----'---:::=_~__:c_~~----~----~- --------aOI~---------L----------0,01 0,1 10 100 Fig.ln 3 Peri",ii.-" Pui,-tedllli.-" EL xnL 364 J1. HABERMAYER c) For Td'T --+ :>0 the controller operates as a differential element. The stability region variation may be eharacterized in this case by the relationship 1 T ST 1 + 2 C(T /T) (ST) + (TjTF (ST)~ = K T f TT ; ~,J . exp(-sT) (4) For high Td!T values the functionality K = K(Td/T) with (r', TiT, ; as par-ameters is linear, in accordance with (4). The significance of thc above state-me.ilt is merely theoretical, as no pure differential elements are used as con-trollcrs in practice. Proportional-integral-differential control The transfer function of the controller in the case of a PID-type COIn-pensation IS as follows: }~(S) = K (1 (5) The limit of the stability region permitting to reach arbitrary phase mar-gins may be determined by rdationship (3) after thc cvaluation of (2) with any arhitrary accuracy. For the proportional-integral-differential control the number of the parameters is more by one than either for the PI-, or the PD-controls. There-fore the number of the diagrams representing the stability region variation for various time constant values increases considerably as compared to the PI, or PD-type compensations. So in the case of the PID control the loop gain values belonging to the arbitrary phase margins will not be plotted with the time constants of the system as parameters. However we specify the procedure written in the ALGOL programming language and supplied with a PID identifier suitable to determine the stabil-ity region of P, PI, PD and PID controls for any arbitrary time constant and phase margin. For selecting the series compensation element of the linear control with dead time shown in Fig. 1 the diagrams published in [5] may also proye satisfactory in many cases. These diagrams show the critical loop gain (r' = 0°) versus the time constants of the system. If the stability region is to he determined in the case of q:' > 0° as 'well, this can he easily done by activat-ing the PROCEDURE PID given in the ApPE:.\"DIx. By this means the varia-tion of the loop gain permitting to reach arbitrary phase margins of the COIl-LISEAR COSTROL SYSTEJIS 365 trol syst~m under investigation may be easily produced also in the case of a PID compensation for various T; and Td parameter values of the controller hesides of the fixed T, C, T parameters of the plant. Description of the PROCEDURE PID The PROCEDURE serHS for the determination of the stability region permitting to reach arhitrary phase margins of the linear, one-loop control system with dead time shown in Fig. 1. In the general case the controller muy be a PID element with the same transfer function as under (5). In the special case of the PID control the proportional, proportional-integral and proportion-er -OD W3 CJ9"Wt, CJ2= 70wI t + log W I I I I I I I I I I IBOo-rp' Fig. 1, ul-differential compensated loop gain values may be calculated according to Tahle I. For the sake of tractability the PROCEDURE PID contains several PROCEDURE-s and REAL PROCEDURE-s with th~ additional advantage that when the plant contains more than two time' lags, then the procedure' can he easily adjusted, by a slight modification of the main bodies of the PROCE-Table I Controller lrr t Td P 0 0 PI yariable 0 PD 0 yariahle PID variable variable 3* 366 .If. HABER.UAYER DURE-s, FI, DFI and PID, to make it suitable for the determination of th·, stability of control systems with dead time containing more than two time lags. But as such systems are rarely met 'with in practice, or even if control systems of this type occur sometimes, a result of satisfactory accuracy can be obtained in most cases by substituting the plant by second order lag, the above adjustment is seldom needed. The calculation of the angular frequency giving the limit position of the stability region is hased on the considerations as fol-10wE. Let us trace the resultant phase-angular frequency diagram of the open loop of the control system sho'wn in Fig. 1. The character of the curve q; ((J)) is mostly as exhibited in Fig. 17. The angular frequency permitting to reach an arbitrary phase margin satisfying the equation (r ((J)c;') = 1800 -- Cf' is e,-aluated in the following steps: Starting from an arbitrarily low angular frequency of (1)0 V . Q and assumin:s that V 10, we find the frequency decade [(J)l' (!)~] within 'which the required (!)q;' drops (PROCEDURE FIND). As to the value of Q, it is advisable to choose it between .0001 and .001. In the second step wc reduce the frequency decadc by the method of hisection until the deviation of tlll' phase margin from the required value of the function is under a not too low error limit specified in ach-ance (PROCE-DURE HALF). ''''h(,l1 the conditions (180°--- (() ,.-/ EPS and q ') , / EPS respectively, are satisfied, the required angular frequency e,)'!· can he evaluated with arbitrary accuracy (DELT) by the :\"EWTO:> . RAPHso:> iteration method (REAL PROCEDURE .:\"£"\YTo:». In the knowledge of this angular frequmcy the PROCEDURE PID supplies the valuc of the loop gain helonging to the given phase margin. The REAL PROCEDURE FI and REAL PROCEDURE DFI produce thc phase angle and its derivative, respectively. In the interest of machine tim(' economy the program has heen developed in a '\-ay that first it should be run for the maximum phase margin, then the calculations arc to be iterated with eyer decreasing er' yalues. If the critical loop gain is to he determined for one phase margin only, then the COrres-ponding phase margin repres('nts at the ",ame time the maximum phase margin (FDIX). LD"E.·IR CO.v-fROL :; YSTE.\J S 367 The PROCEDURE PID is presented in the APPENDIX. For aetiyating the procedure the programmer must know the meanings of the formal pa-rameters appearing in it. These are: ZETA B TdiT A TiT TAU T FIPM FIMX er maXllllum log w 180°-rf'--t----------~"'_:___:;;IL-~I...:_----Fig. 18 The aboye parameters are input parameters. T~leir yalue must be specified hefore the PID procedure is actiyated. Also the yalu~s C = l!Ti' Q, EPS and DELT must be specified before running the program. The role of Q has been mentioned already. By slightly yarying EPS and DELT, the running time of the program may exhibit considerable deyiations. If yery strict EPS DELT error limits are imposed, then the solution will be eOllyergent, hut the rUIlning time may turn out to be yery long, whereas if the EPS is chosen too high, a diyergent solution might be obtained. The choice of the EPS yalue is decided by the time constants of the control system for which the course of the stability region is to be determined. The recommended yalues are: EPS ~ 0.1-0.2, DELT ~ 10- 4 • The course of the phase-angular frequency characteristic curyc is not monotonously decreasing in eyery case. For instance, the introduction of thc differentiation effect may result in the course of 'l.(w) as traced in Fig. 18. In such cases it is advi8ed to choose a lower EPS yalue. 368 J1. HABERJIA }"ER The PROCEDURE PID does not cOlltain any part-program for the determination of the loop gains belonging to each of the angular frequencies (!Jr,?'" CO~,q;' and (!J3,,/,' respectively, in Fig. 18. The introduction of such a part-program would multiply the running time. But this deficiency has no specific significance, as the angular frequencies w2 ,rp' and w3 ,rp' refer to the conditionally stable region permitting to reach the phase margin --q' [1]. For dimensioning purposes it is advisable to use only the K K(wr,'P') value in the calculation. N either does the procedure supplied with a PID identifier guarantee the production of the loop gain K = K(cor,'P.') in all cases. But this is of no specific significance either, as this occurs only in a short i/Ti interval for the increasing values of the differential time constant, as seen by the diagrams Kcr KcAi/Ti) traced for (p' = 0° [5]. If the variation of the angular frequency in this short interval if Ti is not univocal, then it is advised to repeat the calculation for the involved time constants with reduced V and EPS values in PROCEDURE FIND and PROCED"CRE HALF respectively, as needed. Conclusions The frequency region analysis of linear control systems with dead time imposed itself as little consideration of this problem is found in the literature so far, due to the difficulties in the calculation, Yet systems with dead time occur often in practice and they were dimensioned up to now by empiric thumb rules or by simplifying the control systems. ·With the advent of the digital computers the possibility of more accurate numerical investigations permitting to speed up considerably the synthesis of the linear controls with dead time was opened up. Investigation of the critical loop gain of the one-loop control system compensated in general by a PID controller versus the time constants of the system has bcen described in [8, 9]. [2, ... ,7] and the present paper deal with the determination of the stability region variation of a one-loop control system for the plant with dead time and second order lag, for various phase margin values when P, I, PI, PD and PID controllers are chosen respectively. Considering that systems with plant containing more than two time lags are rarely met with in control technics, the results in [2, ... , 9] and the present paper are suitable for investigating the stability of any linear control system with dead time, where the transfer function of the open loop can be written in the form Y(s) = Y1(s)exp( Si), with Y1(s) not containing the factor exp(--si) anymore. If the plant contains more than two time lags, then the substitution of the plant by two time lags proves to be a satisfactory approximation in the majority of the practical cases. LISEAR GOSTROL SYSTE.\lS 369 Summary Diagrams produced by a digital computer, and representing the stability regions permitting to reach arbitrary phase margins of a linear, one-loop control system with dead time and second order lag and compensated in series by a proportional-differential element are presented. The PROCED1:RE PID of the APPE);DIX written in the ALGOL language permits to determine the stability region variation versns the phase margin and the time constants of the system for the above control compensated by proportional, proportional-integral, proportional-differential and proportional-integral-differential elements, respectively. The diagrams in [6, 7] and in the present paper. along with the PROCEDURE PID permit the easy choice of the P, I, PI, PD and PID compensations, the most advantageous in the case of the actually involved linear, one-loop control system with dead time and second order lag. APPENDIX PROCEDURE PID (ZETA,B,A,TAU,FIPM,FIMX,K); VALUE ZETA,RA,TAU,FIPl\I,FIMX; REAL ZETA,B,A,TAU,FIPM,FIMX,K: REAL PROCEDURE FI(OMEG); VALUE Ol\IEG: REAL OMEG; BEGI1V REAL M8,:M9; M2:=Ol\IEG ~ 2; M3:=M2 XTD; M4:=T ~ 2xM2; MS:=C--M3; M6:=1--M4; l\17:=Ml X OMEG; IF ABS(MS) < 10 -30 THE1V BEGI1"- M8:= 1.S70796: GO TO Ll Ei'fD ELSE l\I8:=ARCTAX(OMEGjl\IS): IF M8<0 THEN M8:=3.141S93+M8; Ll:IF ABS(M6)<lO--30 THKY BEGIN M9: 1.S70796: GO TO L2 E1YD ELSE M9:=ARCTAN(l\17/M6): IF M9<0 THEN M9: 3.141S93+M9; L2:FI:= 1.S70796+0MEGxTAU +M9 --·M8; END FI: REAL PROCEDURE DFI; BEGIlV l\IS:=yIS X MS; M6:=M6 xM6; M7 :=1\17 X lV17: DFI:=TAL --,-::m X (1+M4)/(M6-t-M7) -(C+M3)!(::VIS...LM2) END DFI: 370 .H. HABER.IIA 1"ER PROCEDURE FIND(V,Q,OlVIl,OM2); VALUE V,Q; REAL V,Q,OMl,OM2; BEGnV OlVIEG:=Q; OlVIl:=O; FOR OlVIEG:=VxOlVIEG WHILE FI(OMEG)< F DO OlVIl:= OlVIEG; OM2:=VxOlVIl END FIND; PROCED URE HALF(OlVIl ,OM2,EPS,OMEG); VALUE OMl,OlVI2,EPS; REAL OlVIl,OM2,EPS,OlVIEG; BEGnV FOR OlVIEG:=(OMl+0M2)xO.5 WHILE ABS(FI(OlHEG) -F»EPS DO BEGIN IF FI(OMEG)<F THE1V OlVIl:=OlVIEG ELSE OM2:=OMEG END; OMEG:=0.5 X (OMl OlVI2) END HALF; REAL PROCEDURE NEWT(OMEG,DELT); VALUE DELT; REAL OlVIEG,DELT; BEGIN REAL Z; Z:=OMEG; FOR Z:=Z -(FI(OlVIEG) -F)/DFI WHILE ABS(Z -OlVIEG»DELT DO OMEG:=Z; NEWT:=OMEG END NEWT; BEGIN T:=TAU/A; TD:=BXTAU; lVIl:=2xZETAxT; F:=3.141593 X (1-FIPlVIjlS0); IF FIlVIX --0.1 >FIFM THE1Y FIND(2,Q,OlVIl,O:;YI2) ELSE FIND(10,Q,OlVIl,OlVI2); HALF(OlVIl,OM2,EPS,OlHEG); K:=NEWT(OMEG,DELT) x SQRT( (lVI6+lVI7)j(M5+M2) ) E;'\'D PID; References 1. CSAKI. F. BAHS. R: Automatics. In Hungarian. Tankonyvkiad6, Budapest ) 969. 2. CS"tKI. F.-H"\BEH)IAYEH, )1.: Stability test of linear control systems with dead time. Periodica Polytechnica, EL 12, 311-318 (1968). 3. HABERHAYEH. :\1.: Stahility test oflincar control systems with dead time by digital computer. Periodica Polyteclmica, El. 12, 443-452 (1969). 4. HABEH}IAYEH. M.: Stability test of linear control systems with dead time compensated by PI controller. Periodica Polytechnica, El. H., 53-59 (1970). LL\EAR COSTROL SYSTE.llS 371 5. HABER~IAYER. M.: Stability test of linear control systems with dead time compensated by PID controller. Periodica Polytechnica. El. 14, 181-194 (1970). 6. HABER)IAYER. M.: Stability degree analysis of linear control systems with dead time. Periodica Polytechnica. El. (in press). 7. HABERJIlAYER. M.: Stability degree analysis of linear control systems with dead time compen-sated by a proportional-plus-integral controller. Periodica Polytechnica. El. (in press). 8. SOLIMAN. J. L-AL-SHAIKH. A.: A state-space approach to the stability of continuous systems with finite delay. Control. Part 1. 554-556. October (1965). 9. SOLBIAN. J. L-AL-SHAIKH. A.: A state-space approach to the stability of continuous systems with fiuite delay. ControL Part 2. 626-628. :;';o\"ember (1965). Maria HABERyIAYER, 1052 Budapest, XI., Garami E. ter 3. Hungary 

STABILITY DEGREE ANALYSIS OF LINEAR CONTROL SYSTEMS WITH DEAD TIME BY A DIGITAL COMPUTER

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STABILITY DEGREE ANALYSIS OF LINEAR CONTROL SYSTEMS WITH DEAD TIME BY A DIGITAL COMPUTER

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