Csáki, F.

Keviczky, L.

Kovács, T.

English

Periodica Polytechnica (Budapest University of Technology and Economics)

DIAGRAMS FOR THE OPTII.\IUM SETTING OF CONTROLS By F. CS..\KI, L. KEVICZKY and T. Kov_~cs Department of Automation, Technical University, Bndape~l (Received !lIay 'L 1971) The adjustment of the conventional (series connected) PI, PID, etc. controllcrs of linear, concentrated-parameter, single loop control circuits, i.e. the appropriate choice of the compensation parameters has been amply dealt with by the professional literature. Thr applied discussion methods relied in most cases on the frequency and the complex operator domains; so the utilized quality characteristics were also defined there. The illustrative power of the results defined as optimum obtained in this ·way is very poor-for lack of a simple connection with the time range-as the transition is simple only in principle. It can be stated generally that the designer with an -engineering attitude can hest appreciate the dynamic properties of a control process by the course of the time functions (e.g. the error signal) of a control circuit. This is why relationships between the frequency and the quality characteristics belonging to the time range were sought for. It must be admitted that these methods led also. even at best, to some thumb-rules only. It will be seen later that even one of the most accepted methods for setting the controller5. suggested by ZIEGLEH and ::\ICHOLS [1]. represents only a rathcr uncertain and 50metin1<'s inapplicable process. From alllong the methods used for setting the parameter" of the P. PI. PID. etc. controllers of conventional construction. the be:,t results \1"l'H' obtained by those adjusting the required cluality charaeteristies in the time range hy modelling a control circuit of some structure. Aeeolln to' of such investigations with modelling the control circuit hy analogue eOIllputers and finding the' optimum parameters hy adjusting thp miniIllum \'alue of some integral critf'rion are found among others in the literature [5,6]. The analogue computer is not suitahle for the great numher of measure-ments necessary for the t.'lahoration of a satisfactory setting diagram and for the required accuracy and so the published diagrams suppose a \'ery simple structure in the controlled section and are rather inaccurate. 242 F. CS.iKI " al. Because of the problems outlined above, it is advised to rely on a digital computer for the solution of the optimum dimensioning. At the Chair of Automation of the Technical University Budapest an ALGOL program family, suitable for the optimum dimensioning of the parameters of single-loop, linear, concentrated-parameter control circuits and of those with dead time based on the quadratic integral of some charac-teristic time function of the system has been developed. The description of the program package and the utilized computation process are found in [2, 3]; the latter are shortly summarized in the Appendix of this paper. Further experiences and the successful tests are described in [4, 9]. Xa Xc Ysz (s) Y (5) Xs ~ Fig. 1. Xa - reference signal: xe - error signal; Xs - controlled signal In this paper the setting diagrams 'worked out with the help of the abovc program package for PI and PID controllers are presented. In our investigations the simplified control circuit in Fig. 1 was used. A proportional element of threc time lags was chosen for the structure of the controlled section. This construction is satisfactory in the majority of the practical cases. It must also be noted that at present the identification of the three highest time constants of a controlled section may be regarded already to be a simple task [6]. So Y(s) had the form of Y(s) 1 The optimum adjustment of a PI and a PID control for the controlled section supposed to have this form has heen carried out with thc help of the program package for quite a wide range (0.01 -+ 1.0) of the parameters T?/TI and T3/T1, respectively. The transfer function of the applied PI control: Y 8Z(s) = YPI(s) = K li 1 + _1_) sTI while that of the PID control: STD " 1 0.2sTD , DIAGRAMS FOR THE OPTIMU.H SETTDiG OF CO-'iTROLS 2-43 (K being the loop amplification. T J the integration time and T D the differen-tiation time). With the latter control the differentiation effect is of an approximative character in order to have it nearer to the controls generally used in practice. (The time constant of the differentiation may be regarded to be a good average.) The results are summarized in the following. As the compensation depends on the relative values of the time constants, the section has been character-ized by two parameters: and The obtained optimum adjustment values ·were given also in relative values, accordingly in the form of and (Here K kr-critical circuit amplification; Tkr - critical oscillation period.) The starting points for the searching algorithms of the programs were chosen as T~ Tl and TJ; = max {T2 , T 3} (supposing the relation Tl > T 2, T3 to be valid). The initial point of the parameter K ·was chosen as half of the critical circuit amplification of the system modified in this way, i.e.: where Tt = max {Tz' T3} and T j = min {T~, T3}' The optimization was carried out on the basis of the quadratic integral criterion I 20 .\ x~dt o of the error signal Xc' The minimum function point ·was determined in the field of the variables at an accuracy of 2.5 ~~. The resulting optimum values are plotted in a diagram. Figs 2, 3, 4 and 5 refer to the PI control and Figs 6, 7, 8, 9,10 and II to the PID control. The diagrams show the optimum relative yalues of the control parameters versus Tz/T1 and parametered in TlT1• The values of K"r and Tier, respectively, may he determined by the build-up oscillation of the system. If the oscillations cannot be tolerated by the sys-tem, then K"r and Tkr may be calculated from the identified highest time 244 F. CS . .fKI el al. 1 0,1; .{Opt!Kkr DJ 3 5 Fig. :! 0,1; t i1opl/Kk._ QJ 0,2 Fig. 3 DIAGRAJIS FOR THE OPTI.U[JJI SETTLYG OF CO.YTROLS 245 , 5 YjoDI/Txr PI 3 2 5 3 2 2 3 5 2 3 5 Fig . .}. 5 ··O,i 3 1----2 5 Fig. 5 246 t 10' ifoPI/Kkf 10 1 ,i{,opt/Kxr 2 3 5 F. CS . .fKI ,t al. PID Fig. 6 PID 2 Fig. 7 constants 1\, 1'~ and 1'3 as follo"Ws: ab and 3 5 DIAGRAMS FOR THE OPTI.'tIUJI SETTI1YG OF CONTROLS 247 The diagrams corresponding to these relationships are seen in Figs 12, 13 and 14. Naturally only an approximative solution is given by these for-mulae, as the controlled sections have in reality several time lags. Let us consider an example for the utilization of the diagrams. Be the controlled section, indeed. a proportional element of three time lags. Be the timf' constants: Tl 100 sec: T~ = 8 sec: T:; = 10 sec, PID 0,01 ==-====:::==~'~:::~"~~0'02 ,-0,03 · ,04 5 J 2 Fig. 8 2 .-----, ---,---i-··-----'-"-TjoPI/~r PID 5 3 2 Fig. 9 "'0,05 :-0,06 -007 :-do8 --0,09 .":'0,1 0/7/ ·0,1 0,2 248 F. CS.·iKI ,,' af. 07 ~~~----~~----~-------~.----------,---:u-2 2 3 5 10-7 2 3 5 100 Fig. 10 o 2 3 5 2 5 Fig. 11 DIAGRAMS FOR THE OPTIMUM SETTI"G OF CO;\"TIWLS 249 the relative parameters: 0.08 and Then and T"r = 51.71 sec as calculated from the parameters of the controlled section. The optimum relative values as read off the diagrams are: PI: and l:'ID: 0.69: T[opt/T"r = 3.18 and Hence, the optimum values of the control parameters: l" I: K opt 8.87: T[ opt = 254·.6 sec l-'ID: K opt 18,4,7: T[opt 164.5 sec; TDopt = 11.43 sec. The optimum error unit step Tesponses Xa(t) = l(t) for both controls art' seen in Fig. 15, where the transient proccss which is optimum on the hasis of thc quadratic integral criterion is seen to permit quite a high overshot (of about 15 %,). This fact was demonstrated in OUT investigations [2] as well. This overshot can he prevcnted 'with the help of some other optimization criteria hut only at the cost of the control period. In connection 'with the prepared diagrams 'we note: the digital computl'r delivered great many points for the yarious comhinations of parameters a and b (for two decades in the yalues of hoth) and the diagrams "were plotted corresponding to tbe accuracy of graphical readahility by these points. In some cases the identification of the CUT"';es (e.g. in the PID cast') was disregarded as their cours\' veTSUS the parameters appeared to he e:stremely entangled. From thl' COllI'St' of the diagrams some yaluable relationships may he deduced. At the beginning of our il1yestigations 'we have chosen the. conditions of ZIEGLER and :\"ICHOLS [8] for thc initial points of the scarching algorithms. hut the results haye shown that in part of the cases, these conditions resulted instahle systems (e.g. Tl 100 see: T~ = 100 sec; t:) 1 sec). The use of the widely accepted Z N adjustment method is much reduced hy this ohse1'-yation. This result and the course of some of the diagrams (e.g. of K opt! K kr in the PID case) may he explained by the phase conditions of the system heing 250 F. CSAKI el al. Fig. 12 - 0,0,2 '"-1-o,o~ :-0,05 , -0,06 , -0,07 -608 :::0,09 0,7 fundamentally changed by the compensation. Therefore conclusions concerning the compensated system on the basis of the initial one may be drawn only with a rather limited safety. An unexpected result is also that in the case of the PID compensation a ...-alue higher than that of the initial circuit amplification Fig. 13 DIAGIUJIS FOR THE OPTIJIClf SETTL"G OF COSTROLS 251 2 3 5 Fi.f!. l! 20G I.;OG .' [sec! Fig. 1.5 7 Periodic. Polytechnic. 15/3. may result as optimum yalue (see Figs 6 and 7), although the phase-increasing influence of the D-effect is generally known. The hehayiour of the curves can he explained in nearly all the cases. Let us consider e.g. the case in Fig. 5. For Tl = T ~ and T3 q T 1, or T ~ T3 q T 1: the phase characteristic of the system intersects the -180° straight line at a very little slope, which means that the cutting frequency belonging to a given phase margin falls very far from the critical circuit fre-quency and accordingly the optimum TJ/T"r yalue will also he very high; "whereas for Tl ,~ T ~ r-...- Ta' the phase curve will intersect the -180° straight line at a great slope and so the cutting frequency belonging to thc same phase margin will he near to the critical yalue. So the optimum TIlT/er value is ahout 1 in this case. The analysis of the diagrams is likely to sho'w that the optimum param-eters of the controls vary within quite a wide range, so it is difficult to contract the relationships in some simple thumh-rule. On the hasis of the diagrams a nearly optimum setting may he suggested only for the P1D control, namely "hen T]: T ~ and T 3 are within one decade, i.e. 0.1 1 and then the following approximative relationships hold: and where x = min (T~, T 3)/T]. T J opt 1 lTlir ; IX l: This latter relationship is acceptable for other time constant ratios as well. Appendix Let us suppose that the L-transform of the error signal can be expressed in the form of Then the integral value I 20 = ,\' x;(t)dt o may be calculated hy the formula [1] I 20 = ( 1)"+1 ~­:2d"H" C(s) D(s) DIAGRAlIS FOR THE OPTI.\lC\I SETTISG OF COYHlOL.' 253 Here H" is the n-th Hurwitz-determinant belonging to the equation D(s) = 0, while B is also an n-order determinant produced by substituting the first line of the determinant Hc:: 0, ... 0 by the line The coefficients appearing here are those of the polynomial C(s)C( .) b "/l .• " ; b ., , b -" =-= :~ .. >-2.S- - .-- ••• - '2,s- I U· Thus, the described calculation method corresponds to the solution of a linear equation system. Also the yalue I:!(j is produced by our computf'r program as the solution of an equation system. The minimum of I:!o a:3 a multiyariahle function is determined by our extreme yalue searching algorithm operating according to the sequential sim plex method [7]. SUlllluary Our investigations proved again that the digital computers represent a very useful help in many fields of control technics. enm for the solution of problems considered today as classical ones. The analysis of the obtained results supplied many valuable experiences concerning the optimum setting of the PI and PID control parameters. The diagrams are of a great help in the design of single loop, linear, concentrated-parameter control circuits. Of course. no optimum results are given for the real systems by diagrams elaborated similarly. supposing the controlled section to have a preliminarily given structure. Yet digital computer outputs have shown that even so they are far better (apart from a few extreme cases) than those of the most widely accepted thumb rules for the setting of the controls. As to the future: we intend to elaborate similar diagrams for systems containing dead time as well. ~.,References 1. CSAKI. F.: Szabah;ozasok dinamikaja. Akadcmiai Kiado. Budapest, 1966. 2. CS<\KI. F.-KEnCZKY. L. KO·L\CS. T.: Optimal synthesis of linear control systems bv digital computers, Part I -H. Acta Technica. Acad. Sci. Hung. (In the press.) :3. Koy.-I.cs. T. KEVICZKY, L.: Algol procedures for the evaluation of quadratic integral-criteria. Periodica Polytechnica El. 14, 2 (1970). 4·. KOLl.CS. T.-KEYICZKY. L.: Program System for the Analysis and Synthesis of Automatic Control Systems, CO}fPCO:\"TROL 1970, }Iisko1c. Cl. KEYICZKY. L.; Szabalyozok optimalis beaIlitasa teryezett kiserletek segitsegevel. Elektro-technika 63, 1-3 (1970). 6. HARIIIOTT. P.: Process Control. McGraw-Hill Book Co .. :\"ew York-San Francisco, 1964. 7. W·ILDE. D . .T.: Optimum Seeking }Iethods. Prentice-Ha11. Inc. Englewood Cliffs, 1964. 8. ZIEGLEH. J. G.-:\"ICHOLS. :'-i. B.: Optimum setting for automatic controllers. Trans. A.S.}I.E .. 64 (1942). 9. KEYICZKY. L. KOyACS. T.: Diagramok szahalyozok optimalis heaIlitasara. VI. .'\-Iagyar Automatizalasi Konferencia, 'Budapest. 1970. Prof. Dr. Frigyes CS . .\KI Dr. Laszi6 KEVICZKY Tiyadar Koy . .\cs 7* Budapest XI.. Garami E. ter 3. Hungary 

DIAGRAMS FOR THE OPTIMUM SETTING OF CONTROLS

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DIAGRAMS FOR THE OPTIMUM SETTING OF CONTROLS

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Periodica Polytechnica (Budapest University of Technology and Economics)