TUNING CONTROLLERS TO T TE MODEL OBECTS WIT INERTIA TIME DELAY AND ASTATISM IN TE CASCADE ADE CONTROL SYSTEM Irina COUARI Technical University of Moldova Chişinău REZUMAT. În lucrare se roune un algoritm de acordare a regulatoarelor tiizate P PI PID în sisteme de reglare în cascadă cu două bucle de reglare. Procesul condus se rezintă rin două subrocese cu inerţie tim mort şi astatism. Pentru acordarea regulatoarelor în conturul interior şi conturul exterior se utilizează metoda gradului ui maximal de stabilitate. În conturul interior se roune de a acorda regulatoare P PI iar în conturul exterior se roune de a acorda regulatoarele P PI şi PID. Cuvinte cheie: metoda gradului maximal de stabilitate contur interior şi exterior reglare în cascadă acordarea regulatoarelor ABSTRACT. This aer rooses a tuning algorithm of linear controllers P PI PID in the multile-loo loo feedbac control systems. The control object consists of two subrocesses which are described by the dynamical models with astatism inertia and time delay. The controllers in the internal contour and in the external contour tuning use the maximal stability degree method. The P and d PI controllers are used in the internal contour and the P PI PID controllers are used in the external contour. Keywords: multile-loo feedbac control system internal contour external contour maximal stability degree method.. INTRODUCTION Many tuning methods of tyical controllers are used at the rojecting of multile-loo control systems: frequency method criteria (of modulus) method etc. [4]. The rocedure of tuning controllers in the multile-loo feedbac control system becomes difficult. In this aer it is roosed to use the maximal stability degree method (M.S.D) [5] for tuning of tyical controllers P PI PID for a class of control objects models with inertia time delay and astatism which are connected in cascade reresented by two subrocesses and as result with two regulating loos.. TE TUNING ALGORITM OF CONTROLLERS The multile-loo feedbac control system is reresented by two contours: internal contour with controller s transfer function R (s) and subrocess F (s) and external contour with controller s transfer function R (s) and subrocess F (s) (figure ). It is recommended to carry out the tuning of controllers first in the internal contour then in the external contour. r R(s) R(s) F(s) F(s) - - Fig.. The multile-loo feedbac control system. The control object consists of two subrocesses with transfer functions: () ( s) s T s T s c s c s c s F ( )( ) 0 where c 0 TT ; c T T ; c. τ s e F ( s) with T < T T. () ( Ts ) In exressions () and () we have the notations: are transfer coefficients of the subrocesses; T T T are time constants of resective subrocesses; τ - time delay. Buletinul AGIR nr. 4/0 octombrie-decembrie 87 F(s)
. TE TUNING CONTROLLERS IN TE INTERNAL CONTOUR The tuning of controller with transfer function (t.f.) R (s) from internal contour to the subrocess with t.f. F (s) is imlemented. We assume that P and PI controllers are used. P controller is tuned to the object with transfer function () alied M.S.D. method and tuning arameters of controller are determined from relation [5]: τ e ( T ). () In the relation () is the maximal stability degree which is chosen from the following condition >0. To determinate the t.f. of the internal contour in case of tuning P and PI controllers it is roosed to aroximate the value e τ s with Pade aroximant []: τ s e. (4) τ s The transfer function of the P controller is R ( s). (5) The t.f. of internal contour with P controller tuning is: ' R ( s) F ( s) ' F ( s) (6) R ( s) F ( s) l0s ls l where τt τ T ' ; l0 ; l ; l. PI controller is tuned to the object with the transfer function () alied the M.S.D. method and tuning arameters of controller are determined from relations τ e ( τt ( τ T ) ) (7) e ( T T ). (8) τ i τ τ We can obtain the values of arameters i changing the >0 value for that the erformances of control system are sated. The transfer function of the PI controller is ( ) s i R s s The t.f. of internal contour with PI controller tuning using exression (4) is: ' R ( s) F ( s) r0 s r F ( s) (0) s s l s l s l s l ( ) ( ). (9) R F 0 where τt ( τ T ) r0 ; r ; l0 ; l ; l ; l. i i i i 4. TE TUNING CONTROLLERS IN TE EXTERNAL CONTOUR The structure bloc scheme of external contour is reresented in the figure a b. Fig.. Structure bloc schema of external contour. For the tuning of P PI PID controllers in the external contour it is necessary to determine the equivalent transfer function of object (6) with P controller tuning in the internal contour with the t.f. of subrocess () ' ( s) ( s) ( s) () F F F 0 5 4 a0s as as as a4s τt T T ; a0 ; where r r - l s l s l s( T s )( T s ) TT ( τ T ) ( T T ) τt ( T T )( τ T ) τt a ; a TT ; τ T a T T ; a. 4 - R(s) F(s) F(s) R(s) a) b) F(s) For object with t.f. () P PI PID controllers can be tune alied the M.S.D. method using the relations from [5]. Control system with P controller: 5 4 ( a a a a a 0 4 ). () Control system with PI controller: 5 4 (6 a 0 5 a 4 a a a 4 ) () 6 5 4 ( a a a a a i 0 4 ). (4) Control system with PID controller: 5 4 (6 a 0 5 a 4 a a a 4 ) (5) d 6 5 4 ( a a a a a i 0 4 ) d (6) 4 ( 5 a 0 0 a d 6 a a a 4 ). (7) For the tuning of P PI PID controllers in the external contour for the case when in the internal contour was tuning PI controler it is necessary to y y 88 Buletinul AGIR nr. 4/0 octombrie-decembrie
determine the equivalent t.f of object (0) with PI controller in the internal contour with the t.f. of subrocess () r0 s r ( s) (8) F 0 l s l s l s l s( T s )( T s ) b s b a s a s a s a s a s a s 0 6 5 4 0 4 5 where τtt T b0 ; b ; a0 ; TT ( τ T ) ( T T ) τt a ; i i i i ( T T )( τ T ) τ T TT ( ) ; a i i i ( τ T ) ( T T )( ) a TT ; i i a T T ; a. 4 5 i For object with t.f. (8) P PI PID controllers can be tuned by alying the M.S.D. method using the relation from [5]. Control system with P controller: 6 5 4 - a0 a - a a - a4 a5 b - b 0 Control system with PI controller: d - d d - d d - d d 7 6 5 4 0 4 5 6 ( b - b0 ) where d0 6 a0b0 ; d 7a0b 5 ab 0; d 4a b 6 a b ; d 5a b a b ; 0 0 d a b 4 a b ; d a b a b ; d a b. 4 4 0 5 4 5 0 6 5 7 6 5 4 a0 - a a - a a4 - a5 i. b - b 0 Control system with PID controller: d d d d d d d d d 8 7 6 5 4 0 4 5 6 7 8 d 4 ( b b0 ) where d 0 a b ; d 00a b b 0a b ; 0 0 0 0 0 0 0 0 0 0 d a b b 68a b b a b ; d 4a b b 6a b 4a b 78a b b ; 0 0 0 0 0 0 4 0 ; d4 50a b b 0a b a b b a b d 8a b b 8a b b 0a b ; 5 0 4 0 6 a4b0b ab d7 a5 0 4 8 a5 d ; b b 6 a b ; d b. (9) (0) () () Values of i arameters are obtained varied d the value for that the erformances of control system are sated. The following rocedure is roosed to determine the otimal values of arameters from i d relations () (7) (8) ()-(7) (9) - (4) which reresent the deendences of the maximal stability degree and arameters T T T The values T T T consider are nown and constant and the variable is changing and the curves f ( ) f ( ) f ( ) for the resectively i d controller and object were obtained. After the sets of values of the arameters get the otimal and i d quasiotimal value of. For each set of values of the arameters it was maing the simulation and i d it was determined the transition rocess for that the obtained erformance corresonding with the sated erformance. 5. APLICATIONS AND COMPUTER SIMULATION To show the efficiency of the roosed algorithm of tuning the tyical controllers in the multile-loo feedbac control system with inertia using the resented relations an examle with the object model which has the following arameters: F (s)- T τ and F (s) 5 T 5 T 8 was examined. The P PI controllers were tuning in internal contour and P PI and PID controllers were tuned in external contour using the maximal stability degree method which ermitted to obtain the high erformance varied values of and choosing the resectively values of the P PI PID controllers. The obtained results were comared with results obtained tunig controllers using the Ziegler Nichols method. The comuter simulation has been made in MATLAB and the simulation diagram of multile-loo feedbac control system is resented in the figure. d - d d - d d - d d 7 6 5 4 0 4 5 6 (b - b0 ) where d0 6 a0b0 ; d 7a0b 5 ab 0; d 4a b 6 a b ; d 5a b a b ; 0 0 d a b 4 a b ; d a b a b ; d a b. 4 4 0 5 4 5 0 6 5 a - a a - a a - a. 7 6 5 4 0 4 5 i - d b - b0 d () (4) Fig.. Simulation diagrams of the control system. The transition rocesses of the multile-loo feedbac control system for external contour is resented in the figure 4 in case of tunnig controllers using the maximal stability degree method where: a) transition rocess in the external contour with P controller tuning in the internal contour and P PI PID Buletinul AGIR nr. 4/0 octombrie-decembrie 89
controllers tuning in the external contour; b) transition rocess in the external contour with PI controller tuning in the internal contour and P PI PID controllers tuning in the external contour. The obtained values of resectively controllers are resented in the table where the number of rows corresond with number of curves from figure 4 a) b). a) Table. The values of tuning controllers by maximal stability degree method Item ot 0.9 0.004 ot 0.04 0.98 ot 0.0 0.49 i 0.00 ot 0.06 ot. iot 0.04 dot 8.794 4 0.0 0.68 i 0.006 d.9 5 0.08 0.768 i 0.00 d 6.464 6 0.768 i 0.00 d 6.464 ot 0. 0.9 i 0.04 ot 0.0 0.006 ot 0.0 0.00675 i 0.0000 ot 0.04 ot 0.0067 iot 0.000067 dot 0.068 0.0 0.004 i 0.000008 d 0.007 0.06 0.0054 i 0.0009 d 0.05 0.008 i 0.00005 d 0.044 The tuning values for curves 5 were obtained using the maximal stability degree method and curve 6 was obtained for the case of otimization the tuning value of the PID controller in the external contour using MATLAB. The obtained transition rocesses for the case of tuning controller using the Ziegler-Nichols method are resented in the figure 5 where: a) transition rocess in the external contour with P controller tuning in the internal contour and P controllers tuning in the external contour; b) transition rocess in the external contour with PI controller tuning in the internal contour and P controllers tuning in the external contour. Fig. 4. Transition rocesses of the multile-loo feedbac control system. For the cases of tuning PI and PID controllers in the external contour the control system was instable. The obtained values of resectively controllers are resented in the table. Table. The values of tuning controllers by Ziegler-Nichols method Item 0.775 0.6975 i 0.9 0.05 0.084 0.09 0.065 i 0.04 ot 0.048 iot 0.0 dot 5. i 0.0 ot 0.075 iot 0.009 dot 5.4 b) Fig. 5. Transition rocesses of the multile-loo feedbac control system. 90 Buletinul AGIR nr. 4/0 octombrie-decembrie
In the figure 6 a) is resented transition rocesses of the multile-loo feedbac control system in the figure 6 b) is resented the reartition of oles for the following cases: - control system with P controller tuning in the internal contour and P in the external contour using the Ziegler-Nichold method - control system with P controller tuning in the internal contour and P in the external contour using the maximal stability degree method - control system with PI controller tuning in the internal contour and P contour in the external contour using the the maximal stability degree method. 6. CONCLUSIONS As a result after tuning the P PI PID controllers in the multile-loo feedbac control system with object s models () () with nown arameters the following conclusions can be made:. The tuning of P PI controller in the internal contour in conformity with the maximal stability degree method ermitted to obtain the high results varying the value of the >0 and choosing the arameters of the resectively controllers for obtaining the sated erformance of the internal contour.. The tuning of P PI PID controllers in the external contour using the maximal stability degree method ermitted to obtain the high results varying the value of the >0 and obtained the otimal value and the subotimal values of the and choosing the sets of the values of controllers arameters for obtaining the sated erformances.. The robustness of control system with PID controller in the external contour and P controller in the internal contour tuning by M.S.D. is higher than the robustness of control system with PID controller in the external contour and PI controller in the internal contour tuning by M.S.D and using the otimization rocedure from MATLAB. REFERENCE Fig. 6. Transition rocesses of the multile-loo feedbac control system (a) and resentation of distribution of the systems oles (b). Analyzing the distribution of oles of characteristic equations of control system with controllers tuning by M.S.D. it can be observed that the relative stability of the control system with P controllers in the external contour and PI controller in the internal contour tuning by M.S.D. method has the highst reserve of stability. [] Dumitrache I. şi al. Automatizări electronice EDP Bucureşti 99 ISBN 97-0-96-X. [] Rotach V. Ia.: Teoria avtomatichesogo uravlenia termoenerghetichesimi rotzessami Energoatomizdat Mosva 985 9 s. [] Luas V. A. Teoria avtomatichesogo uravlenia Mosva Nedra 990 ISBN 5-47-007-. [4] Tan N. Atherton D. P. Design of stabilizing PI and PID controllers. International ournal of Systems Science Vol. 7. Issue 8 6/0/006.54-554 ISSN: 00077. [5] Izvoreanu B. Fiodorov I. Cojuhari I. Tonu S. Tuning of Controllers to the Fourth Order Advance Delay Objects with Nonminimal Phase and Astatism. In: Proceedings of the 4-rd International Conference on Microelectronics and Comuter Science ICMCS- 05 v.. U.T.M Chişinău 005. 08 ISBN 9975-66-040-. Buletinul AGIR nr. 4/0 octombrie-decembrie 9
9 Buletinul AGIR nr. 4/0 octombrie-decembrie