STABILITY ANALYSIS OF FUZZY CONTROLLERS USING THE MODIFIED POPOV CRITERION

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1 SABILIY ANALYSIS OF FUZZY CONROLLERS USING HE MODIFIED POPOV CRIERION Mauricio Gonçalvs Santana Junior Instituto cnológico d Aronáutica Pça Mal Eduardo Goms, 50 Vila das Acácias - CEP São José dos Campos - SP mauricio@l.ita.cta.br Karl Hinz Kinitz Instituto cnológico d Aronáutica kinitz@i.org Abstract. his papr dals with th stability analysis of fuzzy logic control (FLC) systms. h output charactristic of this class of controllrs is nonlinar, thus it is ncssary to us stability analysis mthods for nonlinar systms. Hr w us th Popov critrion. h systms analysd hav a FLC with two input variabls and on output variabl, rsulting in a surfac as th nonlinar output charactristic. Using algbraic and graphic manipulation of th control systm stat quations and of th nonlinar output charactristic of th FLC, w can apply th Popov critrion to systms with this configuration. Kywords. Fuzzy logic, stability, Popov critrion.. Introduction h concpt of a fuzzy st was introducd by Lotfi A. Zadh (Univrsity of Califórnia, Brkly), in 965. H obsrvd that th availabl tchnological rsourcs wr insufficint to automat th activitis rlatd to th problms of industrial, biological or chmical natur, that includs ambiguous situations. Fuzzy st thory provids a mthod of translating vrbal, imprcis and qualitativ prssions, common in human communication, in numrical valus. hus, w can prss human knowldg in a way that th computrs can procss. hrfor, systms that us fuzzy st thory ar calld by intllignt, bcaus thy mulat som aspcts of human intllignc. Whn fuzzy st thory is usd in th logic contt, for ampl in knowldg basd systms, it is known as fuzzy logic. h fuzzy logic controllr dsignr nds a larg knowldg of th imprcision and th uncrtaintis in industrial procsss and plants and how thy affct th usual applications of modrn control thory (Shaw t all, 999). h tchniqus of fuzzy control originatd with th rsarch of E. H. Mamdani (974), of Qun Mary Collg, London Univrsity. In 974 h controlld a stam machin using fuzzy rasoning. o nhanc th powr of th framwork of fuzzy control, it is dsirabl to dmonstrat th stability of fuzzy systms, but this may not b straightforward. In this papr w ar concrnd with a stability analysis tool for a particular class of fuzzy systms. Whn w hav th mathmatical modl of th control systm, w can mak th stability analysis. h output charactristic of fuzzy logic controllrs is nonlinar. In vry spcific cass it can b linar. hrfor, w nd to us stability analysis mthods for nonlinar systms. h stability analysis tool discussd hr is basd on Popov critrion. It is applid to fuzzy logic controllrs (FLC) with two inputs and Mamdani implication. hus w prsnt and w discuss a modification of th Popov critrion introducd by Bühlr (994). 2. Popov critrion Considr a closd loop systm composd by a nonlinarity (NL) without mmory in cascad with a linar plant (L). h stability of this systm can b dtrmind basd on its frquncy rspons, using th Popov critrion (omovic, 966): r=0 NL u L y G(s) Figur. Basic structur for th application of th Popov critrion. & = A + b u ; y = C ; u = f ( t, y) ()

2 whr A is th stat matri, b is th input vctor, C is th output matri, u is th controllr s output, is th stat vctor and y is th systm s output. Suppos that A is Hurwitz, (A, b ) is controllabl, (A, C) is obsrvabl and f ( ) is a timinvariant nonlinarity that satisfis th condition (Santana Jr, 2003): [ f ( y) K y] [ f ( y) K y] 0 y Γ R and Kmin = 0 (2) min ma whr K is a positiv dfinit symmtric matri. Considr a Lyapunov function candidat of th Lur-typ: ( ) = P + 2 η f ( σ) y V K dσ (3) 0 whr η 0. Using th Kalman-Yakubovitch lmma (Yonyama t al, 2002), w can show that an nough condition for stability is th istnc of q > 0 such that: R [ q jω] G( jω) + > 0 ω R p + (4) k whr G(jω) is th frquncy rspons of th systm. his condition can b rprsntd graphically. G * ( ω) X + j Y whr X = R[G * ( jω) ] = R[G( jω) ] Y = Im[G * ( jω) ] = ω Im[G( jω)] = (5) hrfor, for absolut stability, th Eq. (4) can b prssd by: X q Y + k = 0 (6) h Eq. (6) is a straight lin quation, with slop /q and passing through th point /k. his straight lin is calld Popov s lin. h Eq. (4) is satisfid if th Popov plot G * (jω) lis ntirly to th right of this lin. h slop q can tak any ral, positiv valu. Y -/k X G* (jω ) ω Figur 2. Gomtrical intrprtation of th Popov critrion. hus, it is possibl to find th factor k, which dfins, according to th Fig. (3), th sctor of absolut stability: u k. u=f() () Figur 3. Zon 0 < f < k containing th nonlinar charactristic u = f().

3 3. Modification of th Popov critrion o us th Popov critrion in th stability analysis of a fuzzy control systm, it is of advantag to modify this critrion slightly: L r=0 u L y NL u G (s) Figur 4. Modifid structur for application of th Popov critrion. W chang th position of th linar (L) and nonlinar (NL) blocks of Fig. (), and w obtain th outlin shown in Fig. (4). hus, w analys th control rror = r y blonging to block L, and w dtrmin a nw block, which is calld L. hrfor, its transfr function bcoms: G () s G() s = (7) Bcaus of th signal invrsion, th Popov critrion givn by Eq. (6) is thn dfind by: ( jω) qy ( jω) k 0 X < (8) h Popov s lin is givn by: X q Y k = 0 (9) hus, for stability assuranc, th Popov plot Fig. (5): G * ( jω) must b placd to th lft of Popov s lin, as it is shown in Y G *(jω) ω /k X Figur 5. Gomtric intrprtation of th modifid Popov critrion. 4. Fuzzy logic control systm and th Popov critrion application 4.. Gnral rlationships h Fig. (6) prsnts th fuzzy control systm bas structur with th linar systm. his linar systm is calld S, and th fuzzy logic controllr is rprsntd by th block FLC:

4 r u S y FLC * R Figur 6. Fuzzy systm bas structur. h linar systm can b dscribd by th following stat quation: & = A + b u (0) whr th stat vctor has th dimnsion n. h output variabls ar in th vctor y and thy ar givn by: y = C () hy ar applid to th FLC inputs with th rfrnc r. h stat vctor is givn by: s = s2 (2) s3 h stats s and s2 ar usd to compos th FLC inputs. h vctor s3 contains all th othr stats. h stat vctor composs th input variabl of th FLC and th stat quation of th systm to control is givn by: = A + b R & (3) whr u is changd by th FLC output R *. So, w nd to apply th following linar transformations: - = ; A = A ; b = b (4) h matri is dtrmind so that th tangnt plan in th origin of th charactristic of th FLC attributs th sam output signal of R * in th domains of and. his tangnt plan corrsponds to th output charactristic of th fdback stat controllr (Bühlr, 994). ( r s) ks2 s2 u = ks (5) whr r - s corrsponds to th rror of th stat fdback control. h lin vctor k s is dfind from (5): k s = [ ks ks2] (6) In th FLC bas systm, th control rror = r y is formd from y and togthr with th othr lmnt y 2, it bcoms th vctor, with dimnsion n = 2. So, w us th following linar transformation (Bühlr, 994): = y (7) For th matri, thr is th following condition (Bühlr, 994): ks y = k = k y = k y (8) hr is also th introduction of th vctor, with dimnsion n = n s = 2: = [ ] (9)

5 W can s that th first lmnt has a ngativ signal, bcaus is proportional to th control rror. So, th matri bcoms: ks k 0 = (20) 0 ks2 k With th vctor, w can modify th control systm bas structur, as shown in Fig. (7). h rfrnc r dos not appar clarly. hrfor, th control systm is autonomous. u S FLC * R Figur 7. Modifid structur of th FLC systm (first modification) Nonlinar function of FLC h FLC output R * is a nonlinar function of th input vctor, givn by: ( ) R = f (2) h nonlinar function f(.) must rspct som conditions: R R R = 0 para > 0 para < 0 para = 0 < 0 > 0 (22) Finally, for small valus of th vctor, it is ncssary that th nonlinar function approachs of a linar rlationship dfind as (Bühlr, 994): R = k = - ks y para 0 (23) hus, at th limit of th origin, th nonlinar charactristic of th FLC bhavs as a fdback control. h nonlinar functions of th FLC rspct thos conditions, as wll as it happns with th fdback control (Bühlr, 994) Nonlinar transformation of th input variabls of th FLC o us th Popov critrion with a FLC with two inputs (n = 2), w nd to introduc a linar transformation to th vctor. W promot a rotation of 45º in th as of th output charactristic of th FLC. hus w obtain nw as t and t2 and w chang th thir scals. h mthod is illustratd by th Fig. (8):

6 0 t ,5-0,75-0,25 2 0,25 0,5 0, ,5-0, t 0,5 0, Figur 8. Linar transformation of th FLC input. hr ar th following gnral rlationships btwn th vctors = [ 2] t = [ t t2] (Bühlr, 994): t = t ; = t t (24) hrfor, th transformation matri must b: t = (25) hus, th bhavior at th limit 0 (origin) of th nonlinar charactristic, according to Eq. (23), it bcoms: R = k t t = k [ ] t = k t (26) that is, th slop of th tangnt in th origin of th charactristic, in rlation to th ais t, is givn by: k R t = (27) W can obsrv that th bhavior at th limit ( t 0) dos not dpnd of th variabl t. Continuing th computation of th nonlinar charactristic R * = f( ) = f( t), w introduc th following rlationship (Bühlr, 994): R = kr t (28) whr k R is a nonlinar function of t, givn by: kr = f ( ) R t = (29) t h valu k R dos not dpnd only of t, but it also dpnds on th othr lmnts of vctor t. According to th nonlinar charactristic givn this factor changs among rights limits which dtrmin a sctor whr this charactristic must b locatd. h valu k R indicats th slop of th hyprplan, which cross th origin of th spac ( t, R * ). his hyprplan has th sam coordinats that th nonlinar function at th oprational point. hus, w nd to calculat th valus k R for all th points, using th Eq. (29).

7 4.4. Fuzzy logic control systm modifid structur Now w can modify th fuzzy logic control systm structur. hus, w considr that th control systm (calld now by S t ) has th variabl t as th only output variabl. S Fig. (9). Howvr, th variabl t2 also acts on th FLC. * R S t t FLC * R t2 Figur 9. Modifid structur of th FLC systm (scond modification). hus, w analys th FLC with two inputs as a FLC with on input. hrfor, his output charactristic is studid as a function of th spac ( t, R * ). h stat-spac quation and th output quation of th ar givn by: S t = A + b R & (30) t = c t (3) k First of all, w gnraliz th Eq. (26) to dtrminat th lin vctor c t (Bühlr, 994): k t t = k t = k thus w can conclud that: [ 0] t = [ 0] t (32) (33) Considring Eq. (24),Eq. (7), Eq. (8) and Eq. (), w can mak succssivly th following transformations: t= [ 0] t = [ 0] t = y= ks C (34) k A comparison with Eq. (3) shows that: c t = ks C (35) 4.5. ransfr function of th modifid control systm and th Popov critrion By comparison of th Fig. (9) and Fig. (4), w can s that modifid systm structur of th FLC, according to th sction IV.4.3, it corrsponds to th structur modifid by th application of th Popov critrion, according to th sction IV.3. hus, w can us th Popov critrion givn by Eq. (8), to analyz th stability of th FLC. According to th structur of th Fig. (9), th control systm S t, is dscribd by Eq. (30) and Eq. (3). By th istnt rlationships btwn th stat quations and th transfr function, w can find th following transfr function for th modifid systm: Gt () s t = R () s () s = c t k ( s I A) b= k.c ( s I A) b k s (36) For a concrt cas, w can comput th frquncy rspons G t (jω), ( s = jω ), and thn, w can comput th Popov plot G t * (jω) of th modifid systm. hus, w can dtrminat th Popov s lin, as in th Fig. (5), and w can obtain th factor k, which dfins th sction whr th nonlinar charactristic must b locatd to nsur th absolut stability of th nonlinar closd loop systm. h nonlinar charactristic of th FLC is givn by:

8 R = f ( t ) = kr t (37) As w alrady plaind, th factor k R is a nonlinar function of t t2. hus, thr is th situation prsntd by th Fig. (0). h straight lin k R. t cross th opration point of th nonlinar function R * = f( t). h nonlinar control stability is guarantd if Eq. (36) is satisfid. hn, w hav th condition: 0 < kr < k (38) ( t) t Figur 0. Nonlinar charactristic of FLC insid a sctor 0 k Discussion on th Popov critrion prsntd by Bühlr (994) In th prvious sctions w showd th dscription of th Popov critrion, and w saw that it can b applid only to a tim-invariant nonlinar function. Howvr, whn w apply th Popov critrion to a modifid fuzzy systm, th input 2 is a drivativ of th position y and so th output of th CN is a nonlinar tim-variant function. hn th modification prsntd by Bühlr is not appropriat. 5. Rsults Hr w study th modifid Popov critrion application for a linar plant that satisfis all th conditions introducd prviously. h stat-spac modl of th systm is givn by: 0 0 A =, b, C = 0. = [ 0] h pols of this systm ar: s = -0,05 + 0,9987*I s = -0,05-0,9987*i. Evry pols hav ngativ ral parts and th systm is controllabl and obsrvabl. hrfor, th Popov critrion can b applid to this systm. o control this systm, w chosn th function shown blow: u = K(, )y Whr: s s2 R K( s, s2 s s2 ) = 0 K( s, s2 ) s, s2 R + + s s2 So this nonlinar function rspcts all th conditions (22). h nonlinar charactristic and th tangnt plan at th origin of this charactristic ar shown in th Fig. (2):

9 Figur 2. h controllr s nonlinar charactristic and his tangnt plan. his plan is obtaind from th istnt points about th origin of th charactristic. From this plan quation w obtain th stat lin vctor k s and th nonlinarity paramtr: k = 0,909 nonlinarity paramtr (tangnt plan s slop); k s = [0,909 0,909] lin vctor of th fdback stat controllr. W comput now, point to point, th slop of all th points of th nonlinar charactristic. hus, w can obtain th maimum valu of th slop. In this cas, th maimum slop is k Rma = 0,909. h modifid output quations ar givn by (thy ar th FLC inputs): 0 s 0 y = C = C t = 0 s2 0 From th matri C t, from th vctor k s and from th paramtr k, togthr with th matri A and b of th systm to control, w obtain th transfr function of th modifid systm G t (s): G t () s = s 2 s + 0, s + From th G t (s) w can draw th Popov plot and thn w can obtain th Popov s lin. h Fig. (3) (a) shows th Nyquist plot of th linar systm and (b) shows th Popov plot. Figur 3. (a) Nyquist plot of th linar systm.(b) Popov plot of th modifid systm. h invrs of th intrsction of Popov s lin with th ral as supplis th valu of th slop k of th ara of stability, in which th nonlinar charactristic must b locatd.

10 k = 0,0327 k = 30,557 By comparison w s that th valu of k is largr than th valu of th slop k Rma, bcaus k Rma = 0,909< k = 30,557, so w can conclud that th systm is stabl. Howvr, whn this systm is simulatd w can obsrv that this systm is unstabl. h Fig. (4) (a) shows th simulation diagram (b) shows th position plot. Figur 4. (a) h blocks diagram.(b) Position plot. 6. Conclusions In this papr w discuss a mthodology to analysis for th Popov critrion application to a FLC with two inputs and on output. W prsntd th modifid Popov Critrion dvlopd by Bühlr. W obsrv that th dvlopd linar transformation is corrct and w can apply th analysis ovr this transformation. Howvr, th stability critrion chosn by Bühlr is not appropriat. As w shown prviously, th Popov critrion can b applid only on nonlinar timinvariant functions, and th nonlinar charactristic of FLC with two inputs and on output is a tim-variant function. W usd a function that rspcts th conditions (22) to show that th Popov critrion can not b applid to functions lik that. So, w nd to find othr mthod to analysis to apply ovr th linar transformations prsntd abov. 7. Rfrncs Bühlr, H.,994, Réglag par Logiqu Flou, Prsss Polytchniqus t Univrsitairs Romands, Lausann. Khalil, H. K.,996, Nonlinar Systms, Ed. Prntic Hall, Scond Edition, U.S.A. Santana Jr, M. G., 2003, Anális d Estabilidad d Sistmas Nbulosos Via Critério d Popov, s d Mstrado, Instituto cnológico d Aronáutica, São José dos Campos, Brazil. Shaw, I. S.; Simõs, M. G.,999, Control Modlagm Fuzzy, Ed. Edgard Blüchr, First Edition, São Paulo, Brazil. omovic, R., 966, Introduction to Nonlinar Systms, Ed. John Wily and Sons LD, Iugoslávia. Faliros, A. C.; Yonyama,., 2002, oria Matmática d Sistmas, Ed. Art & Ciência, São José dos Campos, Brazil. Zadh, L. A., 965, Fuzzy Sts, Information and Control 8, Univrsity of California, Brkly, U.S.A.. Mamdani, E. H.,974, Application of Fuzzy Algorithms for Control of Simpl Dynamic Plant, Proc. IEE, v. 2, n 2, pp