Keywords: Controllers with PI and PID dynamics, Takagi-Sugeno fuzzy models, Takagi- Sugeno fuzzy controllers, stability analysis, winding system.

Transcription

1 Ata Poytehnia Hungaria Vo. 2, No., 2005 Deveopment of Conventiona and Fuzzy Controers and Takagi-Sugeno Fuzzy Modes Dediated for Contro of Low Order Benhmarks with Time Variabe Parameters Stefan Preit *, Radu-Emi Preup *, Zsuzsa Preit *,** * Poitehnia University of Timisoara, Dept. of Automation and App. Inform. Bd. V. Parvan 2, RO Timisoara, Romania Phone: , -30, -24, -26, Fax: E-mai: spreit@aut.utt.ro, rpreup@aut.utt.ro ** Budapest University of Tehnoogy and Eonomis, Dept. of Automation and Appied Informatis, MTA-BME Contro Researh Group Godmann Gy. tér 3, H-, Budapest, Hungary; fax : E-mai : preit@aut.bme.hu Abstrat: The paper presents deveopment and tuning tehniques and soutions for PI and PID ontroers, and Takagi-Sugeno fuzzy ontroers with PI and PID type dynamis meant for appiations whih an be haraterised with ow order benhmark type modees (for exampe eetria and hydraui driving and positioning system.two type of pants and two ontro strutures with homogenous and with non-homogenous information proessing with respet to the inputs are presented, inuding tuning and optimization aspets. Then Takagi-Sugeno fuzzy modes dediated to a ass of pants haraterized by Two Input- Singe Output inear time-varying systems are presented. It is offered a stabiity test agorithm of the fuzzy ontro systems invoving Takagi-Sugeno fuzzy ontroers, to ontro the aepted ass of pants. The tuning methods are briefy presented in reation with a ontro soution for a drive system with a variabe inertia strip winding system. Keywords: Controers with PI and PID dynamis, Takagi-Sugeno fuzzy modes, Takagi- Sugeno fuzzy ontroers, stabiity anaysis, winding system. Introdution Take the ass of pants (P) having the transfer funtions expressed as: k P H P ( (a), s( + st ) Σ k P H P ( (b) (.) ( + st )( + st ) Σ 75

2 S. Preit et a. Deveopment of Conventiona and Fuzzy Controers and Takagi-Sugeno Fuzzy Modes Dediated for Contro of Low Order Benhmarks with Time Variabe Parameters kp H P ( (a), s( + st )( + st ) Σ kp H ( (b). (.2) P ( + st )( + st )( + st ) 2 Σ The parametres k P an be onstant or variabe, and it is asumed that T Σ < T 2 <(<<) T ; this ass of modes haraterize we enough eetria and hydraui driving and possitioning systems (ontro of possition and drive appiations as ontroed pant [], [2], [6]. The paper s aim is to deveop Takagi-Sugeno (TS) fuzzy ontroers (FC based on assia deveopment methods, meant for position ontro of eetria and hydraui drives with inear onstant or time-varying (LTV) parameters haraterized by benhmark type modes of form () and (2). If the parameters are ontinuousy varying, the LTV systems may resut as inearized noninear systems in the viinity of a set of operating points or of an operating trajetory. These features determine the wide appiation area of robust ontro, adaptive ontro and TS fuzzy modes. Regarded to the use of TS fuzzy modes, the appiation is based in spite of their drawbaks suh as: - The behavior of the goba TS fuzzy mode an signifianty divert from the expeted behavior obtained by the merge of the oa modes; - The stabiity anaysis and testing of fuzzy ontro systems based on TS fuzzy modes is reativey diffiut, beause of the ompex aggregation of the oa modes in the inferene engine. Firsty, the paper presents two assia deveopment proedures for ontinuousy and quasi-ontinuousy working PI and PID ontroers, based on extensions of the widey used Symmetria Optimum Method (Setion 2). Then, a ass of TS modes for Two Input-Singe Output (TISO) LTV pants is presented (Setion 3). In Setion 4 there are defined the TS fuzzy ontroers meant for ontroing the TS fuzzy modes. Based on these, a stabiity test agorithm is presented (based on Lyapunov s stabiity theory) for a ass of fuzzy systems with TS fuzzy ontroers ontroing the TISO LTV pants (Setion 5). Resuts onerning the deveopment of onventiona and fuzzy ontro soutions for a drive system with two output ouped motors, appiabe to the ros of a hot roing mi and to a variabe inertia strip winding system, are presented in Setion 6. Setion 7 is foused on the onuding part of the paper. 76

3 Ata Poytehnia Hungaria Vo. 2, No., Deveopment of Continuousy and Quasi- Continuousy Operating PI and PID Contro Agorithms Many ontro appiations prefer strutures with typia ontro agorithms with homogenous or non-homogenous information proessing on the two input hannes [], [3], [4]. Suh strutures have the genera form given in Fig. 2.-a -b and - presents some partiuar ontro aws regarding the inputs. The boks ()... (5) an be desribed by its speifi transfer funtions (t.f.; w(t) or r(t) the referene signa (or the fitered referne signa), y(t) - the measured output, u(t) - the ontro signa (or its omponents, with index), e(t) - the ontro error. There an be estabished reations between suh ontroers and the 2-DOF ontroers [3]. For exampe, a bok diagram of a 2-DOF ontro struture is presented in figure 2.2. R(z), S(z) and T(z) are the harateristi boks of the 2- DOF ontroer, P is the pant; v (t), v 2 (t) pant disturbanes. Figure 2. Typia I-DOF ontroer strutures and partiuarization Figure 2.2 Struture of the 2-DOF ontroer. The 2-DOF ontroer an be restrutured in different ways; for the given ow order pants from a pratia point of view the presene of a onventiona 77

4 S. Preit et a. Deveopment of Conventiona and Fuzzy Controers and Takagi-Sugeno Fuzzy Modes Dediated for Contro of Low Order Benhmarks with Time Variabe Parameters ontroer (partiuary PI or a PID and signa fiter an be highighted. Two types of strutures are detaied in figure 2.3. These rearrangements aow: - to take over design experiene from ase of PI and PID ontroers; - an easy introdution of suppementary boks speifi to PI and PID ontroers (Anti Windup iruit, bumpess swithing and other; - the transformation of PI and PID ontroers into 2 DOF strutures and vie versa. The ontroers in figure 2.3 wi be haraterized for exampe by ontinuous t.f.s C(, C F (, C * (, C P (, in whih the parae reaisation tuning parameters are highighted{k C, T i, T d, T f }; disretizing, the digita ontro agorithm is obtained. Taking the basi ontroer C( of PID type, it an be written: - for the basi PID struture in Fig. 2. (b) (parae form): u( C k e( std ( + + st + st ( β ) TiTd s + ( α ) T * is + r ( ( + st f ) ), F( r( TiTd s + Ti s + ( + st ) ( C 2 i f f 2 ; (2.) Figure 2.3 Two aternatives for rearranging a 2 DOF ontroer. - for the struture (a) in figure 2.3: u( st u f ( d std C( kr ( + + ), CF ( kr ( α + β ) ; (2.2) e( st + st r( + st i - for the struture (b) in figure 2.3 (with the notation C(C * (): f f 78

5 Ata Poytehnia Hungaria Vo. 2, No., 2005 ( ) uf ( u s std std C ( kr[( α ) + + ( β) ], CP ( kr( α+ β ). (2.3) e( st + st r( + st i Tabe 2. Connetions between 2 DOF ontroer and extended DOF ontroer struture. Fig. 2. F( - F(C( C( Remarks Fig. 2.3-a - C F C( C F ( C( - Fig. 2.3-b - C P C*( C*(+C P ( - α β - - (ref. hanne) (feedbak) PID PID DOF 0 PDL2 DL PI PID 0 PD2L2 P PID-L PID PL2 PDL2 I PID α β PID ontroer with pre-fitering (2 DOF ontroer) f ontroer DOF with nonhomogenous behavior Remarks: P proportiona, D derivative, L ag, I integrator modues. Depending on the vaues of α and β parameters, for the presented boks the behaviors from in Tabe 2. are obtained. The hoie of a ertain representation of the ontroer depends on [3]: - the struture of the avaiabe ontroer; - the adopted agorithmi design method, and the resut of this design. In the presene of an integra (I) omponent and a imitation bok in the ontroer struture, figure 2.4 (a), the use of the AWR measure (Anti-Windup- Reset) is reommended. A assia struture for introduing the AWR measure on a ontroer struture with integra omponent is presented in figure 2.4 (b). The AWR measure an be gobay impemented with respet to ontroer output or oay, with respet to integra (I) omponent of the ontroer. f Figure 2.4 Cassia struture for introduing ARW measure on the ontroer struture The transfer funtions of the ontinuous PI (PID) ontroers are written reated to the design proedure and the impementation (disretization) proedure. For the seria form of PI and PID ontroers the t.f.s are: 79

6 S. Preit et a. Deveopment of Conventiona and Fuzzy Controers and Takagi-Sugeno Fuzzy Modes Dediated for Contro of Low Order Benhmarks with Time Variabe Parameters k PI: {k, T } C ( ( + st ) (2.4) s k s, PID: {k, T, T } C ( ( + st )( + st ) (2.5) The impementation of a quasi-ontinuousy (QC) operating PID digita ontro agorithm an be based on the informationa diagram presented in figure 2.5; the appearane of a suppementary state variabes x k, is assoiated to integra (I) omponent and the adding of the AWR measure. The parameter vaues {K pid, K i, K d,k arw } depends on the ontinuous parameters {k r, T r, T r } and on the samping time vaue, T e. Figure 2.5 Quasi-ontinuousy operating PID digita ontro agorithm impementation. The impementation of non-homogenous information proessing (Fig. 2.-) has two requirements [3], [4]: - an I or PI behavior with respet to the referene hanne; - a PI or PID behavior with respet to the feedbak hanne. The nonhomogenous information proessing struture respets the foowing informations (Tabe 2.2). Tabe 2.2 Transfer funtions of boks in Fig. 2.- (parae form) Case Channe Bok 3 Bok 4 Bok 5 Type () w, (r) I: (/st i ) ---- P: (k C ) I: (/st i ) y I: (/st i ) P: () P: (k C ) PI: (+/st i ) (2) w, (r) PI: (+/st i ) ---- P: (k C ) PI: (+/st i ) y PI: (+/st i ) D: (st d ) P: (k C ) PID 80

7 Ata Poytehnia Hungaria Vo. 2, No., 2005 In order to avoid diffiuties due to ontraditory resuts obtained from design aording to referene traking and disturbane rejetion, different optima - or in speia ases, optima-ike tuning tehniques an be adopted. Two of these methods are onsidered here to be representative: - the Moduus Optimum method, (MO-m), [], - the Symmetria Optimum method (SO-m) [] and two modified (extended versions, the Extended Symmetria Optimum method (ESO-m) [2] and the extended (modified version) of the extended Symmetria Optimum method (2E-SO-m)Criterion. The main advantage of this ESO-m and 2E-SOm onsist in the possibiity to inrease the ontro system phase reserve and for speifi ases in a better oad disturbane rejetion. The ontroer parameters in its seria form an be auated using the reations synthetised in Tabe 2.3. The design parameter β beongs usuay to the domain 4 β 6. In the ase of impementation, the probem of bump-ess transfer from one oa risp ontroer to another is soved in a risp manner, exempified here for two oa ontroers of digita PI-type, the od one with the parameters {q, q 0 }, and the new one with the parameters {q *, q * 0 }: u. (2.6) * * * k u k + qek + q0ek, u k u k + q ek + q0ek It is neessary to ompute previousy past vaues whih are neessary to the new ontroer. As it an be observed in (6), e k- * represent these new initia onditions (the past vaue. 3 A Cass of Takagi-Sugeno Fuzzy Modes The foowing Takagi-Sugeno fuzzy mode to represent a TISO LTV system wi be used that modes the ontroed pant [7]: s Tabe 2.3 Tuning reations after [2], [6] P( Contr. type Tuning reations 0 2 k p PI k 3/ 2 2 β k T (ESO-m.[2]) p Σ ( + stσ ) {k, T } T β T Σ 8

8 S. Preit et a. Deveopment of Conventiona and Fuzzy Controers and Takagi-Sugeno Fuzzy Modes Dediated for Contro of Low Order Benhmarks with Time Variabe Parameters k p s( + st )( + stσ ) k P, + st )( + st ) m T Σ /T ( Σ k P ( 2 + st )( + st )( + stσ ) m T Σ /T R : IF z( t) is F AND z 2 ( t) is F2 THEN y( P u ( u( + P v PID {k, T, T } PI {k, T } PID {k, T, T } AND ( v(, K m, kr 3/ 2 2 β k T (ESO-m[2]) p Σ T β T, T T Σ, k 2k T (MO-m) p Σ T T (+m) 2 k, β 3/2 k P T Σ ' m β T Σ ' m (m) (2E-SO-m) T [6] ( + m) 2 m (m)+(2-β /2 )m+m 2 k 2k T (MO-m) p, Σ ' T T T T2 (+m) 2 k, T T 2 β 3/2 k P T Σ ' m β T Σ ' m (m) T [6] ( + m) 2 m (m)+(2-β /2 )m+m 2 K AND z ( t) is n (2E-SO-m) F n (3.) where: u(, v(, y( the Lapae transform of the pant input (the ontro signa) u(t), of the disturbane input v(t) and of the ontroed output y(t); R the th inferene rue, m; m the number of inferene rues; z i (t) the measurabe pant variabes, i n, and: T z t ) [ z ( t) z ( t) z n ( t)] ; (3.2) ( 2 K n the number of measurabe pant (system) variabes pointing out the timevariation of the pant; F the inguisti terms assoiated to the measurabe variabe z i (t) and to the rue R ; P u ( and P v ( the oa t.f.s of the pant. 82

9 Ata Poytehnia Hungaria Vo. 2, No., 2005 The TS fuzzy mode (3.) inudes both the inferene rues as part of the rue base and the oa anayti modes of the TISO LTV system. The ontroed output is inferred by taking the weighted average of a oa modes appearing in (3.), whih haraterizes the properties of the ontroed pant in a oa region of the input spae; so it is referred to as fuzzy dynami oa mode [8], [9]. The foowing notation is introdued: µ ( t) µ ( z ( t)), Km, (3.3) for the membership degrees of the normaized membership funtions µ of the inferred fuzzy set F, where: F n I Fi, Km, (a) and µ ( t ). (b) (3.4) i m By using the produt inferene method in (3.4) (b) and the weighted average method for defuzzifiation, the TS fuzzy mode (3.) an be expressed in terms of the foowing fuzzy dynami goba mode that an be onsidered as TS fuzzy mode of the pant: y( P u P ( u ( u( + P m µ ( t) P (, P u v ( v(, v m µ ( t) P (. v (3.5) The mode (3.5) is LTV system beause the inferred transfer funtions, P u ( and P v (, have time-varying oeffiients regarded to oa inearised modes. 4 Takagi-Sugeno Fuzzy Controers. Cosed-Loop System Modes The TS fuzzy modes (3.) or (3.5) oud be very usefu in omparison with other onventiona tehniques in noninear ontro. This is the ase of pieewise inearization [8], where the pant mode is inearized around a nomina operating point, and there are appied inear ontro tehniques to the ontroer deveopment. This approah divides the input spae into risp subspaes, and the resut is in a non-smooth onnetion of the inear subsystems to buid the osedoop system mode. These modes are based on the division of the input spae into fuzzy subspaes and use inear oa modes in eah subspae. Furthermore, the fuzzy sets F i and the inferene method permit the smooth onnetion of the oa modes to buid the fuzzy dynami goba mode of the osed-oop system. 83

10 S. Preit et a. Deveopment of Conventiona and Fuzzy Controers and Takagi-Sugeno Fuzzy Modes Dediated for Contro of Low Order Benhmarks with Time Variabe Parameters To ontro the TISO LTV pant (3.5) there is proposed a TS fuzzy ontroer with the foowing mode: R : 2 IF z( t) is F AND z2( t) is F AND K AND THEN u( C ( e(, Km, z ( t) is F n n, (4.) where e( is the Lapae transform of the ontro error e( t) r( t) y( t) ; r(t) is the referene input; C ( the t.f. of the oa ontroers, m. The oa ontroers in (4.) are deveoped for the oa anayti (inear) modes in (3.) by parae distributed ompensation [0]. By the feedbak onnetion of the pant (3.) and of the fuzzy ontroer (4.) in terms of the onventiona ontro struture presented in figure 4., the osed-oop system an be desribed by the foowing fuzzy dynami oa mode: R : IF z( t) is F AND z2 ( t) is F2 AND K AND THEN y( H ( r( + H ( v(, Km, r, v, z ( t) n is F n, (4.2) where H r, ( and H v, ( - the oa t.f.s of the osed-oop system, m. Figure 4. Contro system struture. In the onditions (3.3) (3.5), by aepting the same inferene method and defuzzifiation method as in the previous Setion, the fuzzy dynami goba mode of the osed-oop system an be expressed in terms of (4.3): y( H r r ( r( + H m H ( µ ( t) H ( v(, m, (, H ( µ r v v ( t) H v, (, (4.3) where the inferred t.f.s H r ( and H v ( have time-varying oeffiients. It is justified to onsider the TS fuzzy mode (4.3) as TISO LTV system; for its anaysis there an be appied methods speifi to LTV systems [8] [0] whih require numeria tehniques for the auation of H r ( and H v (. For the deveopment of the fuzzy ontroers it is neessary to perform the stabiity anaysis and testing; a stabiity anaysis test agorithm for the osed-oop system (4.3) are presented in the next Setion. 84

11 Ata Poytehnia Hungaria Vo. 2, No., Stabiity Test Agorithm To perform the stabiity anaysis of the fuzzy ontro systems two approahes an be empoyed: - the first one, based on the use of the fuzzy dynami goba mode (4.3) and, - the seond one an be deveoped by starting with the definition of a pieewise smooth quadrati Lyapunov funtion [0], [] based on the fuzzy dynami oa mode (4.2). In the ase of the system (4.2) there an be used severa approahes based on either transferring the ideas from hybrid systems [0] or by using, sine this system an be onsidered as a variabe struture one with possibe disontinuous right-hand side, stabiity anaysis methods dediated to variabe struture systems []. For the stabiity anaysis and testing of the fuzzy ontro system modeed by the fuzzy dynami goba mode (4.3) it wi be presented as foows the first approah, based on the Lyapunov stabiity theory in terms of the definition of a pieewise smooth quadrati Lyapunov funtion V: m V q V, V x P x, (5.) T where x the state vetor, dim x (, n S ), P positive definite symmetri matries, dim P (n S, n S ), q weighting oeffiients ensuring the smoothness of the funtion V, m, n S system order. The matries P are obtained by ensuring the negative definiteness of the derivative of the Lyapunov funtion. This an be ensured by soving the agebrai Riati equations (5.2): T A P + P A Q, Km, (5.2) with Q positive definite symmetri matries, dim Q (n S, n S ), and A the system matries in the systemi reaizations orresponding to the osed-oop transfer funtions H r, ( and H v, (, dim A (n S, n S ). The stabiity anaysis test agorithm onsists in four steps, detaied in [3]. Resuming: - Step : based on the knowedge and experiene onerning the ontroed pant operation, determine the number of inferene rues m for ontroing the pant, the partition of the input spae in fuzzy regions, assign the inguisti terms F i to the measurabe pant variabes z i (t), i n, and define the membership funtions orresponding to F i, m; 85

12 S. Preit et a. Deveopment of Conventiona and Fuzzy Controers and Takagi-Sugeno Fuzzy Modes Dediated for Contro of Low Order Benhmarks with Time Variabe Parameters - Step 2: for eah inferene rue R, m, derive the inear oa modes of the pant, haraterized by the transfer funtions P u ( and P v ( ; - Step 3: deveop a onventiona ontroer with the transfer funtion H C. ( for eah of the oa modes of the pant by a inear ontro deveopment tehnique suh that the m osed-oop oa systems, with the transfer funtions H r, ( and H v, (, m, have the required ontro system performane; - Step 4: set the vaues of the positive definite symmetri matries Q and sove the agebrai Riati equations (5.2); if the soutions P of (5.2) prove to be not positive definite, then jump to the step 3; otherwise, the system is stabe. The soving of the agebrai Riati equations (5.2) and the required anaysis requires the argest omputationa effort. 6 Appiation: Winding System Contro Soution A typia appiation for eetria drives with variabe inertia (Variabe Inertia Drive System, VIDS) is in the fied of roing mis and of winding mehanisms. The ontro of suh systems represents a diffiut task due to: - the existene of output ouping between severa subsystems requires the deveopment of ontro systems with referene input ompensation; - the presene of possibe osiations due to the eastiity of the shaft; - the noninearities of the ontroed pant inuding bakash and stik-sip; - the modifiation of the inertia during the pant operation determine timevarying parameters of the ontroed pant. The simpified funtiona diagram and the informationa diagram of an eetria drive with DC motor variabe inertia appearing in appiations where a strip is winded on a drum are shown in Fig. 6.-a and -b [2]. In the winding proess, the referene input must be orreated with the modifiation of work ro radius. In this ontext, two basi aspets our at the deveopment of the ontro struture: the modifiation of the referene input (ω 0 (t)), and tuning the ontroer parameters. For the first one, the ondition (6.) must be fufied by the ontro soution: v(t) onst ω o (t) k/r(t), (6.) where by the measurement of r(t) there an be ensured the ontinuous modifiation of the referene input ω o (t). 86

13 Ata Poytehnia Hungaria Vo. 2, No., 2005 (a) (b) Figure 6. Funtiona diagram of VIDS and referene input orretion system. The probem of ontroing the speed of the winding system an be soved in various ways: by the use of a asade ontro struture with two, urrent and speed, ontroers, or by the use of a state feedbak ontro struture. For both versions, the variane of the moment of inertia, aording to (6.2): J(t) (/2) ρ π R 4 (t), (6.2) requires muh attention in the ontroer design. In this paper wi be presented a soutions based on ontro oops with inear PI and PI-fuzzy ontroers with parameter adaptation. The state-spae mathematia mode of VIDS has the state variabes {x i a, x 2 ω m, x 3 f t }, and a orresponding informationa bok diagram given in figure 6.2 [2]. x (t) (R a /L a )x (t) (k e /L a )x 2 (t)+(k h /L a )u (t), x 2 (t)(k m /J e (t))x (t) (/J e (t))m f (x 2 (t)) (r t (t)/j e (t))x 3 (t)) (/J e (t))(j e (t))r t (t), x 3 (t) b r t (t)x 2 (t) b v s (t). (6.3) Linearizing the modes in some representive funtiona points, mathematia modes (benhmark t.f. in form of (.) and (.2) an be obtained. Conerning the oa inearized pant modes (.), (.2), the speed ontroer design is based on the tuning methods desribed in hapter 2, appied in its various, dediated versions [2], [4], [6]. An atrative tuning version, regarding TS fuzzy modes, TS fuzzy ontroers and TS fuzzy osed-oop system modes (Setions 3 and 4), by aepting that the ontroer parameters ensure a maximum phase reserve for eah oa inearised pant mode and the orresponding TISO LTV systems are handed as in Setions 3-5. It an be onsidered equivaent with a re-tuning of the ontroer parameters as funtion of radius modifiation. This version permits the obtaining of better ontro system performane. 87

14 S. Preit et a. Deveopment of Conventiona and Fuzzy Controers and Takagi-Sugeno Fuzzy Modes Dediated for Contro of Low Order Benhmarks with Time Variabe Parameters Figure 6.2 Informationa bok diagram of VIDS. The envisaged ontro struture ontains two oops, the inner regarded to the urent and the externa regarded to the speed. The inner oop ontroer is auated based on the Moduus Optimum method with fixed parameter vaues. For the deveopment of the speed ontro oop, in the deveopment step it an be used ony one pant variabe, r t. For obtaining a bump-ess transfer from one oa ontroer to another at the same time with ensuring a reativey simpe impementation, the authors reommend that the membership funtions of the inguisti terms F i shoud have trianguar type and m shoud not exeed 3 or 5. Some detais regarding the deveopment step 3 dediated to the deveopment of the oa ontroers are presented as foows. For the onsidered ontroed pant, after inearization in the viinity of some signifiant operating points the mode an be brought to a simpified form haraterized by the foowing oa transfer funtions: P u k P (, Km, (6.4) s( + st ) Σ where the parameters: - T Σ (the sma time onstant oresponding to the sum of parasiti time onstant is a onstant vaue, and - k P (the ontroed pant gain) is time-varying due to the time-varying J e ; note that for the sake of simpiity the index was omitted. There an be used severa versions of oa transfer funtions ( depending on the types of disturbane inputs v(t) appied to the ontroed pant. For the oa pants (6.4) the use of PI ontroers having the transfer funtions: P v 88

15 Ata Poytehnia Hungaria Vo. 2, No., 2005 k C ( ( + st ), Km, (6.5) s an ensure very good ontro system performane when the ontroers are tuned in terms of the ESO-m [2]; the ontroer parameters are k (the ontroer gain) and T (the integra time onstant) (see tabe 2.3). By the hoie of the parameter β 9 4 the ontro struture ensures a good maximum phase reserve (55 0 <φ r,m < 60 0 ); this phse reserve is hanging during the pant operating. The hoie of suh a vaue ensures good robustnes, so that the stabiity anaysis was not performed (a) (b) () Figure 6.3 Simuation resuts The oa ontro system performane an be improved by adding a first- or seond-order referene fiter [2], [6]. This is the way the ontro struture obtains the features speifi to ontro strutures with 2 DOF ontroers. For to test the method, a theoretia appiation osey onneted to pratie was onsidered. The non-inear Simuink mode of the pant is presented in Appendix. Three PI ontroers were auated and the hange of the ontroers during the pant operating was based on a very simpe fuzzy seetion rue. The probem of bump-ess transfer from one oa risp ontroer to another is soved in a risp manner. 89

16 S. Preit et a. Deveopment of Conventiona and Fuzzy Controers and Takagi-Sugeno Fuzzy Modes Dediated for Contro of Low Order Benhmarks with Time Variabe Parameters Some simuation resuts are presented in figure 6.3: (a) the hange of the inear speed, v(t); (b) the hange of the radius hanging r(t); () the hange of the anguar speed, ω(t). The resuts an be aepted as good and onfirm the possibiitie of use Takagi- Sugeno fuzzy modes to represent a TISO LTV as modes of the ontroed pant. Conusions The paper presents ontinuous-time deveopment soutions for eetria drives with variabe inertia. The tuning reations are dedued for assia but generay aepted benhmark type pant modes. The presented TS fuzzy modes dediated to TISO LTV systems are suitabe for ontro strutures where the pant mathematia mode inearization offers oa inear modes. A stabiity test agorithm for the fuzzy ontro systems modeed by TS fuzzy modes based on Lyapunov stabiity theory is presented. The main imitation of the stabiity anaysis agorithm onerns its omputationa ompexity. The modes and the stabiity anaysis agorithm an be used in the deveopment of onventiona but aso of TS fuzzy ontroers based on the parae distributed ompensation with severa appiations. One rea-word appiation an be in the area of eetria drives with variabe inertia, where the deveopment of the oa ontroers an be performed in terms of the ESO-m or 2E-SO-m. The simuated appiation is regarded to a VIDS where the referene input must be orreated with the modifiation of working ro radius. Referenes [] Åström, K. J. and T. Häggund: PID Controers Theory: Design and Tuning, Instrument Soiety of Ameria, Researh Triange Park, 995 [2] Preit, St. and R.-E. Preup, An extension of tuning Reations after symmetria optimum method for PI and PID ontroers, Automatia, Esevier Siene, vo. 35, pp , 999 [3] Preit, Zsuzsa, Controer deveopment by agebrai methods. Anaysis and Matab-Simuink programs (in romanian). Master thesis, Poitehnia University of Timişoara, Romania, 2003 [4] Preup, R.-E. and St. Preit, Deveopment of Some Fuzzy Controers with Non-homogenous Dynamis with Respet to the Input Channes Meant for a Cass of Systems, Proeedings of ECCC-999, (e-format) Karsruhe, Germany [5] Preit, St., R.-E. Preup, Zsuzsa Preit, Two Degree of Freedom Fuzzy Controers: Struture and Deveopment, Proeedings of the In 90

17 Ata Poytehnia Hungaria Vo. 2, No., 2005 Memoriam John von Neumann Symposium, Deember, 2003, ISBN , pp , Budapest, Hungary [6] Preit, Zsuzsa, PI and PID Controer Tuning Method for a Cass of Systems, SACCS 2007 th Internationa Symposium on Automati Contro and Computer Siene, Otober 200, Iasi, Romania (e-format) [7] Preit St. and R.-E. Preup, On the Fuzzy Contro of a Cass of Linear Time-Varying Systems, A&QT-R 2004, IEEE-TTC- Conferene on Automation, Quaity and Testing, Robotis, May 3 5, 2004, Cuj- Napoa, Romania [8] Kózy, L. T., Fuzzy If-Then Rue Modes and Their Transformation into One Another. IEEE Trans. on SMC part A, 26, (996) [9] H. O. Wang, K. Tanaka and M. F. Griffin, An Approah to Fuzzy Contro of Noninear Systems: Stabiity and Design Issues, IEEE Transations on Fuzzy Systems, (996), vo. 4, pp [0] M. Johansson and A. Rantzer, Computation of Pieewise Quadrati Lyapunov Funtions for Hybrid Systems, IEEE Transations on Automati Contro, (998), vo. 43, pp [] M. Johansson, A. Rantzer and K. -E. Arzen, Pieewise Quadrati Stabiity of Fuzzy Systems, IEEE Transations on Fuzzy Systems, (999), vo. 7, pp [2] St. Preit and R.-E. Preup. PI ontroer design for speed ontro of DC drives with variabe moment of inertia. Bu.St. U.P.T., Trans. AC&CS. Timisoara, Vo. 42(56), pp [3] St. Preit, R.-E. Preup, On the Fuzzy Contro of a Cass of Linear Time- Varying Systems, AQTR 2004 (THETA 4), IEEE-TTTC- Conferene on Automation, Quaity and Testing, Robotis May, 2004, Cuj-Napoa, Romania 9

18 S. Preit et a. Deveopment of Conventiona and Fuzzy Controers and Takagi-Sugeno Fuzzy Modes Dediated for Contro of Low Order Benhmarks with Time Variabe Parameters Appendix. The non-inear Simuink mode of the pant Ua Ua U -K - Kee Sum ia La.s Integra ia -K - Ra -K - Sope Jsist(t) Jpart Jtgo+Jtransf+Jm r(t) * r(t) * w (t) Jinfas(t) Sum 4 -K - r(t) Gain Produt3 Ke Ke -K - Ke Km s Produt4 Integra w om ega ( w(t) ) w(t) Sope w om ega ( w(t) ) Sum To W orkspae om ega ( w(t) ) Vs V_iniar C C s Sum 2 Integra fh a a a*fh(t) r(t) Prod Sope3 Sope2 t -K - C*a r(t) * w (t) -K - Gain r(t) r(t) * w (t) r(t) * r(t) * w (t) r(t) Produt5 Produt2 Prod w(t) r(t) Sum 3 r0 r0 -K - s a*h/(2*pi) Integra w Cok To W orkspae 92