Mariu M. B LA Aurel Vlaicu Univerity of Arad, Engineering Faculty Bd. Revolu iei nr. 77, 3030, Arad, Romania, E-mail: mariu.bala@ieee.org THE IDENTIFICATION OF THE OPERATING REGIMES OF THE CONTROLLERS BY THE HELP OF THE PHASE TRAJECTORY Note: The paper wa preented to the International Summer School SOFT COMPUTING Arad-Moneaa 6.08.08.004 89
Abtract The paper preent an on-line identification method of the operating regime of the cloed loop con-troller. Thi method ue a main tool the phae trajectory of the control error, which i analyed in a qualitative manner. The operating domain i divided into four regime: variable, teady, ocillating and untable. Thi operation may be performed by fuzzy or by interpolative controller, on the bae of general knowledge on the PID controller adjutment. A high quality elf-adaptation of the controller may thu be obtained, covering all the poible operation regime. Keyword: fuzzy interpolative controller, adaptive control, phae trajectory, heuritic control rule. 90
. INTRODUCTION The control of the non-linear and time variable plant demand a high quality elf-adaptation, covering a wide range of poible combination of the parameter of the ytem. A conventional control fail when it ha to deal with contradictory claue. For intance the integrative effect of the PID controller may produce notable overhoot and ocillation during the vari-able regime but in the ame time it help i welcome during the teady one. The adjutment of a dc drive controller for ordinary peed doe not matche at low peed, becaue of to the non-linearity of the friction load torque, which i growing when the peed i decreaing. However thee contradiction may be managed by the help of the fuion of everal controller each one deigned for a pecific operating regime. The fuzzy fuion of the controller became in the lat year a reliable olution to thi problem. The goal of thi leon i to offer a imple and reliable method for the on-line identification of each regime that could poible produce contradiction when adjuting a PID baic controller. The leon i baed on a previou paper [8].. THE BASIC ASSUMPTIONS The firt quetion that come in our mind when dealing with barely controllable plant (highly nonlinear, death time, mathematically unknown, etc.) i: how would a human operator control thi plant? Many pecialit may diregard thi heuritic approach, but till it ha fundamental advantage over the numerical algorithm: it can alway be applied! Of coure the performance may be not optimal, and a rigorou olution to the tability problem i not poible, but in turn the cot of the development of new product i getting lower and pecial applicative meaure againt intability can alway be 9
conidered. And neverthele, the accumulation of experimental fact about the heuritic controlled ytem i a baic condition for the developing of further numerical algorithm. The baic aumption that we will take in account are the next one: The PID i the fundamental control algorithm ince, in a general way of peaking, it can handle the preent (P), the hitory (I) and the future (D) of the evolution of the ytem. The expert knowledge of the PID adjutment offer reliable control rule for the mot part of the poible control application. The elf-adaptation i trictly neceary for highly nonlinear and/or time variable plant. A very attractive adaptation tool for the on-line control i the D phae trajectory of the control error = f( ) (the dependence between the change of error and the error ). The phae trajectory of the error i a fundamental tool, which ha a ignificant weight in the elaboration of the control deciion by the human operator. The adaptive trategy will be a heuritic one, the only one that i able to cope with the highly nonlinear ytem. The fuzzy logic i a baic tool that allow u to cope with the pecific incertitude of the complicated nonlinear ytem and with the qualitative or heuritic approache [6], [7]. The fuzzy fuion of the controller or only of the adaptive part of the controller i neceary for the management of the contradiction. The fuzzy controller may be developed by a linguitic method. In order to obtain reliable implementation we will ue only fuzzy controller that have a linear interpolation correpondent (o called fuzzy-interpolative controller). The prod-um Sugeno fuzzy inference and COG defuzzyfication i an obviou firt choice [3]. 9
3. THE FFSAIC ADAPTIVE CONTROLLERS Any fuzzy controller i an interpolative one a well, and may be implemented by mean of a look-up table with linear interpolation (imilar to the Matlab-Simulink look-up table). Such an implementation may be conidered a a fuzzy interpolative controller [3]. The main advantage of thi kind of tructure conit in the eaine of the implementation (both oftware or hardware). Interpolative controller are able to perform quite imilarly to any other kind of controller having in the mean time the advantage of a low amount of calculation and of the peed [4], [5]. The electronic implementation are feaible in any poible technology, even in the analogical one, ince the only important mathematic operation involved i the linear interpolation. Yet the look-up table are trictly numerical tool, their repreentation in the human mind being inadequate, epecially when uing large or multidimenional table. Thu the fuzzy feature become ueful mainly for methodological reaon. The linguitic repreentation of the knowledge i revelatory for human, catalying the developing tage of the application. The fuzzy controller ued in thi paper will be fuzzyinterpolative, and by conequence they will be implemented by look-up table. We will conider a a recommendable tool a pecific controller tructure which i operating by analying the phae trajectory of the error and which wa deigned having in mind the previou aumption. Thi tructure will be called FSAIC (Fuzzy Self- Adapted Interpolative Controller) [3] and it i hown in Fig.. 93
Gain P Input program du/dt Gain D 0. Direct controller 3-D T(u) Product Derivative Integrator 0. Gain I Sat Sat3 ++ Tranfer Fcn Sat Adaptive PD fuzzy-interpolative controller Fig.. A FSAIC controller The characteritic feature of thi tructure are the following: FSAIC ha a variable tructure. During tranient regime the main controller i a PD one. During the teady regime an integrative effect i gradually introduced, the tructure becoming a PID one. Thi functionality may be achieved with a 3D look-up table having a input, and, the integrative of the error. The different PD table correponding to the dimenion differ only at the central rule, that i activated when = zero and = zero [], [], [3]. Thu the integrative effect i gradually activated, through a linear interpolation, only when teady regime occur. The block that fulfil thi functionality in Fig. i Direct controller. A fuzzy-interpolative PD controller (corrector) induce the adaptive feature. The adaptive controller i generating a multiplying correction (Gain) over the output of the main controller; the multiplying correction i preferable to the additive one by allowing a direct fuzzy fuion. The PD tructure i choen becaue it can be matched with the phae trajectory of the error (ee Table, next page). The correponding block in Fig. i Adaptive PD fuzzy-interpolative controller. 94
In the following paragraph we will focu on the adaptive correction. The adaptive control rule could be grouped into four cluter according to the next claification of the main operating regime: variable (G), teady (G), ocillating (G3) and untable (G4). Thi point of view i not the only poible, other claification being a well productive. The cluter of rule mut repect in great hape the next linguitic commitment: G: < Gain i medium and Integrative i zero > G: < Gain i great and Integrative i great > () G3 & G4: < Gain i mall and Integrative i zero > The Integrative variable i generated by Direct controller. The difference between G3 and G4 are not fundamental. If neceary they may be eparated by the help of upplementary criteria, for example with the help of the product of the firt and econd derivative of the control error, that i poitive for untable ytem [3]. Anyway, Gain mut be reduced in order to reject the ocillation and/or to tabilize the ytem, in the ene of the Nyquit tability criterion. Due to the non-linearity of the plant, an optimal crip value of the Gain would have no ene. In the next table a poible tructure of adaptive PD fuzzyinterpolative corrector i preented. 95
Table : An adaptive PD fuzzy-interpolative controller change of error error negative zero poitive poitive big G G G poitive mall G3 G3 G3 zero G G G negative mall G3 G3 G3 negative big G G G A typical phae trajectory of the error i underlying it the correpondence with the table. 4. THE FUZZY FUSION OF THE ADAPTIVE CORRECTORS When the blending of the rule i not poible or atifactory, the fundamental olution i the fuzzy fuion of the individual controller. The implet fuzzy fuion operate according to the weighted um formula: i (t) u i (t) u(t) i () i (t) i 96
where u i i the output of the controller i and i i the memberhip value of the ame controller. More complicated hape for the memberhip value function may be ued when impoed performance mut be reached. A Fuioned Fuzzy Self-Adapted Interpolative Controller (FFSAIC) i preented in Fig.. P Error D Direct controller 3-D T (u) 0 Saturation Plant Reference I Evaluation/ Scaling Saturation Saturation Output caling factor Product3 ++ Death tim e Adaptation Product Fuzzy fuion -u Fcn Adaptation3 Product Fig.. A FFSAIC controller FFSAIC may include everal different PD adaptive controller (corrector), each one dedica-ted to a pecific operating regime. In Fig. a minimum FFSAIC variant i preented, having only two corrector, correponding to the regime G&G (which may be covered by the ame adaptive corrector) and G3. The Table controller may be applied in order to control the fuzzy fuion of three corrector. The heuritic meaning of the identification of G and G3 are obviou, while the identification of G3 (ocillation) i linked to the point = 0: if the error i zero (in the linguitic ene) the regime i teady, if not, the regime i ocillatory. 97
Table : A regime identification fuzzy-interpolative controller change of error negative zero poitive error poitive big G G G poitive mall G G3 G zero G G G negative mall G G3 G negative big G G G 5 COMPARING FSAIC TO LINEAR PID. SIMULATIONS RESULTS The next example have only an illutrative goal. Some detailed reult may be found in [3]. Fig. 3 preent a comparion of FSAIC and linear PID performance for the cae of an ocillatory plant, with the tranfer function P hae tra jecto rie fo r the pla nt /( 3 +3 * +3 *+) 6 P ID 4 F S AIC change of error 0 - -4-6 - -0.5 0 error 0.5.5 F ig. 3.T he performance of F S AIC comparing to P ID 98
H() (3) 3 3 3 Fig. 4 preent the time performance of FFSAIC when controlling four different plant: H () (4) H () H () 3 H () 4 e -0.05 0. (5) e -0.05 0 (6) e -0.05 3 3 3 (7) If linear PID controller would control them, the teted plant would produce extremely different repone, one of them being fully untable. 6 CONCLUSIONS The identification of the operating regime of the cloed loop control ytem may be obtained by the mean of the qualitative analye of the phae trajectory of the control error. Adaptive action baed on thi approach are able to improve the control of highly nonlinear and/or important dead time plant. Thi analyi may be achieved with the help of fuzzyinterpolative controller. A family of fuzzy elf-adaptive interpolative controller FSAIC i deigned to implement thi kind of operation. The adaptive part of FSAIC i a fuzzyinterpolative PD corrector. 99
The regime identification may enure a high quality adaptation even in the cae of contradictory claue. The tool that can cope with the contradictory poible regime (FFSAIC) i obtained by the fuzzy fuion of everal adapting FSAIC correcting controller. The on-line control of the fuzzy fuion i achieved a well by a fuzzy-interpolative controller. T h e F F S A IC c o n t ro lle r o p e ra t in g o ve r 4 d iffe re n t p la n t, d e m a n d in g c o n t ra d ic t o ry c o n t ro l. 6 e x p (-0. 0 5 )/ ( + 0. * + ). 4 / ( + * + ). 0. 8 0. 6 e x p (-0. 0 5 )/ ( 3 + 3 * + 3 * + ) 0. 4 0. e x p (-0. 0 5 )/ ( + 0 * + ) 0 0 0 0 3 0 4 0 5 0 6 0 t im e [ ] F ig. 4. T n e F F S A IC w it h t h e o n -lin e id e n t ific a t io n o f t h e o p e ra t in g re g im e FFSAIC can alo control ytem that are uffering untabiliing influence. Only two poible application of FSAIC are revealed o far: the air-conditioning [] and the ABS braking [] of the railway coache. Further reearch could produce important applicative achievement, ince the method i a veratile and eay to implement one. 00
REFERENCES: [] B la M., The temperature control by variable tructure interpolative PID controller. Proceeding of the 6 th International Conference on Engineering of Modern Electric Sytem. Univerity of Oradea, May, 00. [] B la M., B la V., Foulloy L., Galichet S., A model of the liding wheel during braking. Proceeding of the 5 th International Conference on Railway Bogie and Running Gear BOGIE'0, Budapet, Sept., 00. [3] B la M., Regulatoare fuzzy interpolative, Politehnica Timi- oara, 00. [4] Dragomir T.L., Interpolative Controller a Correctional and Improvement Alternative for Deign and Implementation. Proceeding of the 6 th International Conference on Engineering of Modern Electric Sytem EMES'0, Oradea, May, 00. [5] Dragomir T.L., Dale S., B la M., Some apect regarding interpolative control. The 3 th International Conference on Control Sytem, CSCS3, Bucure ti, Nov. 00. [6] Foulloy L., Qualitative Control and Fuzzy Control: Toward a Writing Methodology, AICOM Vol. 6, Nr. 3/4 Sept./Dec., pag. 47-54, 993. [7] Stoll K.E., Ralton P.A.S., Ward T.L., Linguitic Deign of Nonlinear Controller. Journal of Intelligent and Fuzzy Sytem, vol. 4, nr., 9-3, 996. [8] M. B la, V. B la, T.L. Dragomir, A Preamble on the Identification of the Operating Regime of the Controller by the Help of the Phae Trajectory, Electronic BUSEFAL nr. 87, June 00, http://www.univ-avoie.fr/ Portail/ Groupe/ LISTIC/ buefal/ Paper/ 87.zip/ 87_0 0