O. Benzineb H. Mekki D. Boukhetala M. Tadjine M. S. Boucherit

Transcription

1 O. Benzineb H. Mekki D. Boukhetala M. Tajine M. S. Boucheit J. Electical Systems: special issue N (: 9- Regula pape Implicit Fault Toleant Contol Technique Base Backstepping: Application to Inuction Moto The aim of this pape is to apply a esent implicit fault toleant contol (FTC technique base backstepping to inuction moto (IM. Mathematical moels of the moto, the istubances as well as the faults signals have been evelope. Using these moels, the ynamic behavio of moto is stuie. A backstepping contol is then synthesize an applie to the system. Base on the simulation esults, it is evient that this contol law is obust to the unknown toque loa an paametic vaiations but is not to the ejection of the faults effect. In oe to etect an compensate this late, a ecent FTC technique is then stuie an applie to the moto moel. The pefomances of the applie technique ae highlighte by simulations. about the consequent benefits as fo EMC of this ive ae also pesente. Keywos: Fault Toleant Contol, Backstepping Contol, Mathematical Moel, Inuction Moto, Fault Detection an Isolation.. θ s NOMENCLATURE s, Stato, oto inex., q Synchonous efeence fame inexes. α, β Fixe stato efeence fame inexes. V, i, Φ Voltage, cuent, flux. L s, L Stato, oto inuctance. R s, R Stato, oto esistance. T s, T Stato, oto time constant. M Mutual inuctance. Ω Roto spee. Roto flux angula position. Coesponing autho : Pocess Contol Laboatoy, Ecole Nationale supéieu Polytechnique (ENP, BP 6, Elhaach, Algies, Algeia. Copyight JES on-line : jounals/esgoups.og/jes

2 O. Benzineb et al: Implicit Fault Toleant Contol Tech Base Backstepping: Application to I M. INTRODUCTION Fault toleance is an issue that has been aesse by many authos. Compehensive intouction to this aea can fo example be foun in [] an []. Reseach on fault toleant contol systems has eceive incease attention ecently ue to significantly inceasing eman fo eliability, maintainability an suvivability of safety-citical systems, such as automotive an aeospace systems, nuclea powe [3]. Many effots have ecently been evote to stuy fault toleant contol systems []. The inuction moto is efinitely one of the most use electic machines in the wol. When supplie by a symmetical an balance sinusoial theephase voltage an opeating accoing to manufactues instuctions namely egaing the envionment an loa type the inuction moto is a vey obust machine [5]. A beakown in the motos can cause the stop of the pouction facility o equie the use of eunant equipment to cicumvent the poblem. Seveal failues can affect electical moto ives [6] an can appea on the level of oto o stato of inuction moto [7]. They can be electic, mechanical o magnetic. Thei causes vey vaie. Inee, of the stuies showe that each faults eveale hamonics at specific fequencies in the cuents of the machine [8], [9] an []. These fequencies epen on the moto chaacteistics. The objective of this pape is to esign a obust contol (backstepping [], [] compae to the toque of loa an to the paametic istubances but on t pesent an insufficiency as fo the ejection of the faults effect. Into this case, we intouce the intenal moel which geneates an aitive tem that we as to the nominal contol to compensate the faults effect on the system an which epouces the unknown state exactly a pioi exogenos system simulating the faults [3].. INDUCTION MOTOR DISTURBANCE AND FAULTS MODELISATION.. Inuction Moto Moeling The setting in the state fom of the inuction moto moel allows the simulation of the latte, in a stationay efeence ( α β fame choosing us like states vaiables the stato cuents, oto flux an the mechanic spee Ω [], []. An like contol vecto the stato tensions. The moel of the IM is given by the state fom: = f ( x + Bu + TL ( The state vecto, the enty matix B an the vecto ae given by:

3 J. Electical Systems SI (: 9- T 3 5 sα sβ α β T x = [ x x x x x ] = [ i i Φ Φ Ω] b B = b [ ] T = With the following expession of fiel vecto f(: f( = ax + ax3 + a3xx5 f( = ax + a5x3x5 + a6x f3( = a7x + a8x3 + a9xx5 ( f = + + ( ax ax3x5 ax f5( = a3xx + axx3 + a5x5 The components of this vecto ae expesse accoing to the paametes of the IM as follows: σ σ n = = + p ( σ a a ; a = a6 = ; a3 = a5 = ; T s σ T σ T Mσ Mσ M a 7 = a = ; a8 = a = ; a9 = a = np ; a 3 T np M = a = ; JL M With: σ = L, L s.. Distubance Moeling a T f = J 5 ; L T = et R s T s =. Rs T b = b = ; = ; σ J An unknown tem Δ( x, Δai but all the time limite is ae in the moel ( which takes the fom: = f ( + Bu + TL +Δ( x, Δai (3 A way of simulating the paametic vaiations effect on the behavio of the moto is to cause at a given moment, a anom change in the system ( coefficients: ai a i + Δai, bi b i + Δbi an + Δ The istubances Δ x, Δa will take the following fom: ( i L L s

4 O. Benzineb et al: Implicit Fault Toleant Contol Tech Base Backstepping: Application to I M Δ ax +Δ ax 3+Δ ax 3 x5+δbu ax ax 5 3x5 ax 6 bu Δ +Δ +Δ +Δ Δ( x, Δ ai = Δ a7x+δ a8x3+δa9xx 5 Δ ax +Δ ax3x5 +Δax a3xx axx3 a5x5 T Δ +Δ +Δ +Δ L.3. Faults moeling In this section we uneline the change of the IM moel in the pesence of faults. A tem V epesenting an unknown istubance but all the time limite [3] -which esults fom the pesence of one o moe faults in the moto- is ae in moel ( which witten in the fom: = f ( + Bu + TL +σv (5 In absence of faults, V is ientically invali an we have: T V σ = an V = (6 V A way of intoucing the hamonics which affecting the stato cuents in the state system ( is to use a stable linea iffeential system epesente in this fom: z = S z (7 The ynamic matix S being the only known chaacteistic of the system, it is consiste of the pulsations ωi = π f i whee f i pesents the fequency chaacteistic of the faults: S = iag( Si ω i (8 Si = ωi We can wite the faults istubances V in this fom: aq + Q S V = z = Γz aqq + QqS (9 Γ [ aq + Q S] Γ = = [ ] Γ aqq + QqS Finally in the pesence of faults an paametic istubances the moel of the IM becomes: = f ( + Bu + T +Δ( x, Δ a +σv ( L i (

5 J. Electical Systems SI (: 9-3. BACKSTEPPING CONTROL BASEE ON THE PRINCIPLE OF FIELD-ORIENTED 3.. Pinciple of Fiel-Oiente Contol: The objective of fiel oiente contol is to have an electomagnetic toque popotional to the stato cuent of the moto (as in the case of a DC machine with an aim of contolling the electomagnetic toque an consequently the mechanical spee of the moto []. This metho consists in oienting oto flux accoing to the iection of the evolving axis (, which makes it possible to tansfom the moto moel given by ( in the efeence mak tuning ( q [3], [5]. In these new cooinates. The oto position is efine by the angle θ s as follows: Φβ θs = actan ( Φα The tansfomation αβ q is one as follows: x x s sq cos( θs = sin( θs sin( θs x cos( θs x sα sβ ( With ( can be use fo the cuent, flux an tension. The new moel of machine in ( q efeence is given by: = ax + θ s x + ax 3 + bu = θ s x + ax + a5x 3x 5 + bu 3 = a8x 3 + ax = 5 = ax x 3 + a5x 5 + TL (3 T With: x x x x3 x x5 is isq q = [ ] = [ Φ Φ Ω] Φ = Φ = Φα + Φβ ; θs x θ s = =Ω+ a7 t x Backstepping Contol The backstepping contol technique is a metho of synthesis into nonlinea when it is ifficult to apply the iect metho of Lyapunov. It is a question of choosing at the stat a Lyapunov function fo the fist subsystem an of inceasing it as we stability the vaious successive subsystems, to lea finally T 3

6 O. Benzineb et al: Implicit Fault Toleant Contol Tech Base Backstepping: Application to I M to a total Lyapunov function which stabilizes the total system. The backstepping esign poceue consists of the following two steps [], []. Step : This fist step consists to ientify the eos ε an ε which espectively epesent the eo between eal spee Ω an efeence Ω as well as the moule of flux Φ an that of efeence Φ. ε = x ε = x x3 The eo eivative is given by: ε = 5 5 = 5 axx3 a5x5 T ε = = a x a x 5 3 x L ( (5 The fist Lyapunov function is efine by: v = ( ε + ε (6 We selecte stabilizing functions as follows: ( x ( k x a x T ef = ε L ax3 ( x ( k x a x ef = ε a Then the eos ynamics is given by: ε = kε ε = k ε The eivative of (6 compae to time is: (7 (8 = kε kε < v (9 With k an k ae positive esign constants. So the contol in (7 is asymptotically stabilizing. Step : In this step, we efinite two new eos of the components of the stato cuent given by: ( ε ε = ( x x = k + x a x T x 3 ef L ax3 ( ε ε = ( x x = k + x a x x ef a (5 will be: Then the equation

7 J. Electical Systems SI (: 9- ε = kε + aε3 ε = k ε + a ε The eivative of (9 gives us: ε = ( = ( 3 ε = ( ef ef = ( ef ef f ( + bu f ( + bu ( ( Whee: f ( = ax + θs x + ax3 f ( x = θs x + a x + a x x To efine the contol laws, we aopt a new Lyapunov function escibe by the following expession: v = ( ε + ε + ε 3 + ε ( Thus the eivative of the final Lyapunov function is: v = ε ε + ε ε + ε ε + ε Its eivative is given by: v 3 3 ε = k + ε + ε ε 3 k ( k3ε 3 + ( ef f( bu ( k ε + ( f ( bu ε k 3ε 3 ef k ε < We choice k 3 > an k >. Finely to make ( v the eivative of the complete Lyapunov function be negative efinite, the voltage contol input is chosen as follows: u = Vs = ( ef + kε f( b (3 u = Vsq = ( ef + k3ε 3 f( b The eo eivativeε 3 an ε will be as follows: ε 3 = k3ε 3 ax3ε ( ε = a ε k ε. A RECENT IMPLICIT FAULTS TOLERANT CONTROL The iea behin implicit FTC is that of esigning a contol unit able to automatically offset the faults effect, without nee of an explicit FDI pocess an consequent explicit econfiguation [3]. This objective will be pusue fo the IM by means of the contol scheme sketche in Fig.. 5

8 O. Benzineb et al: Implicit Fault Toleant Contol Tech Base Backstepping: Application to I M By supposing that the faults effects on the system can be suitably moele by an exogenos signal esulting fom a stable autonomous system calle "exosystem ". An aitive contol is ae to the nominal contol an setting to compensate the faults effect (aspect FTC. This aitive contol esults fom the intenal moel whose ole is to epouce the signal epesenting the faults effect (aspect FDI. Exosystem Refeences Faults Nominal contol FDI ξ u nom + + u u a Intenal moel V System Output Fig. Stuctue of an implicit faults toleant contol. The toque of loa an the istubances paametic ae compensate by the nominal contol. Fo this ( becomes: = f ( + Bu + σv (5 The new contol is expesse by: u = u + u nom a u unom = + ua (6 u = u unom ua On the basis of which we calculate the unknown tem u a with the expession which we etaine fom the nominal contol (7 an (3. The instantaneous iffeence between the state eivative of the system an the efeence becomes: + Γ kx bu a z x + Γ k3x bu a z = 3 = 3 = ax k 3 x3 (7 x x 5 5 axx3 + k 5 x5 Let us notice that the fist two equations o not epen on the vaiables x 3, xan x 5. 6

9 J. Electical Systems SI (: 9- in the thi equation if x x3 in the fouth equation we have x = = in the fifth equation if x x5 In the continuation, fo the etemination of u a let us consie the subsystem: x x = (8 x Whose ynamics esults fom the system (7. z = S z kx + bu Γ z a (9 = = k3x + bu Γ a z Fom system (9 we can wite it in a matix fom: x = H( + Bua Γz (3 = k b H ( Ax et A = B = k (3 3 b In this case fo the etemination of intenal moel we intouce a esent implicit fault toleant contol appoach which oes not est on the esolution of the Sylveste equation. The intenal moel takes then this fom: ξ = Sξ + N( (3 im( ξ = im( z = n f Fo the calculation of the aitive contol we intouce a new vaiable. χ = M ( ξ z Gx χ = M ( ξ z Gx (33 Afte eplacement we fin: χ = MS ( ξ z + M N( G ( H( ua + Γz (3 Then ua is chosen like: u = a B Γξ (35 Consie the systems (3 an the aitive tem given by (35 in this case we have: = H ( + Γ ( ξ z (36 The new vaiable of eo is consiee: e = ( ξ z (37 Its eivative compae to time: e = ξ z = S ξ + N( + S z e = S e + N( (38 7

10 O. Benzineb et al: Implicit Fault Toleant Contol Tech Base Backstepping: Application to I M The equations escibing the ynamics of the eos in close loop ae thus: = A x + Γ e (39 e = S e + N( It is necessay to fin the expession of N (x which cancels the eo of obsevation of the faults e an makes it possible at the same time to eject thei effect fo it cancels also x. That is to say the Lyapunov function of the system (39: T T V = x x + e e Afte evelop of calculatesv becomes: V T T T T = x A x + e Γ x + e N( x ( In this case the choice of N(x is given by: T N( = Γ x ( Finally V is witten: V T = x A x (3 The system (39 becomes: Γ e = e = S e ( The objective of the contol is achieve by aopting the poceue suggeste an we able to compensate the faults effect on the system ( x an to epouce ( e thanks to the intenal moel. ( 5. SIMULATION RESULTS In the Fig. we intouce at t=. sec a loa toque equal to the nominal toque then at t =. sec a vaiation of 8% in the electical an mechanical paametes of machine, afte that the effect of only one fault in the stato geneating one hamonic of fequency 5Hz, amplitue 8 an null phase at t =.6 sec. Fo Fig.3 we consie the same situation (Fig. but in this case we intouce at t=.6 sec the effect of two faults one in the stato an anothe one in the oto geneating thee hamonics of null phases, fequency, 5, Hz an of amplitue 8,, 5 espectively. In Fig. an Fig.5, we simulate the close loop system with the ecent implicit FTC appoach. The FTC appoach which we synthesize ejects the effect of the loa toque, the paametic istubances an also the effect of one fault o two faults (in the stato an oto. 8

11 J. Electical Systems SI (: 9- Ωef (e an Ω (blue ( a s.5 Φ ef (e an Φ (blue (Wb 5 Fault I s (A I sq (A Fig. Simulations of backstepping contol in the pesence of one fault in the stato. 9

12 O. Benzineb et al: Implicit Fault Toleant Contol Tech Base Backstepping: Application to I M Ωef (e an Ω (blue ( a s Φ ef (e an Φ (blue (Wb Faults I s (A (A (A (A I sq (A (A (A Fig.3 Simulations of backstepping contol in the pesence of two faults (stato an oto.

13 J. Electical Systems SI (: 9- Ωef (e an Ω (blue ( a.5 Φ ef (e an Φ (blue (W 5 Fault (A I s (A (A I sq (A (A V s (e an V sq (blue (V (A z Obsevation eo Fig. Simulations of the ecent FTC appoach in the pesence of one fault.

14 O. Benzineb et al: Implicit Fault Toleant Contol Tech Base Backstepping: Application to I M Ωef (e an Ω (blue ( a s.5 Φ ef (e an Φ (blue (Wb 5 Faults I s (A I sq (A V s (e an V sq (blue (V. z Obsevation eo z Obsevation eo z 3 Obsevation eo Fig. 5 Simulations of the ecent FTC appoach in the pesence of two faults.

15 6. CONCLUSION J. Electical Systems SI (: 9- In this pape, a ecent implicit FTC technique is stuie an applie to inuction moto. The backstepping contol (nominal contol which we have synthesizes pesent obustness compae to the paametic istubances an the loa toque. But, pesent an insufficiency as fo the ejection of the effect of one fault (in the stato o two faults (in the stato an oto. We have shown how a esent implicit FTC appoach can be esigne in oe to compensate the faults effect stating with geneating fom the intenal moel state, an aitive tem wish we a to the nominal contol. The appoach that we ew fom the liteatue ests on the esolution of Sylveste equation which is at the oigin of its isavantages. These isavantages wee eliminate by the intoucing of a esent implicit fault toleant contol appoach fo the calculation of the intenal moel. Simulation esults have been pesente in oe to show the effectiveness of this ecent appoach. REFERENCES [] R. Patton, P. Fank an R. Clak, Issues of Fault Diagnosis fo Dynamic Systems. Spinge-Velag, Belin,. [] R. Isemann, Supevision, Fault-Detection an Fault Diagnosis Methos- an Intouction. Contol Engineeing Pactice, Vol. 5, pp , 997. [3] F. Tahami, A. Shojaei an D. Ahmai Khati, A Divesity Base Reconfiguable Metho fo Fault Toleant Contol of Inuction Motos SPEEDAM 6, S [] M. E. H. Benbouzi, D. Diallo an M. Zeaoulia, Avance Fault-Toleant Contol of Inuction-Moto Dives fo EV/HEV Taction Applications: Fom Conventional to Moen an Intelligent Contol Techniques. IEEE Tansaction on vehicula technology, vol. 56, no., Ma. 7. [5] A. S. M. Menes an A. J. Maques Caoso, A Fault-Toleant Opeating stategies appelie to Tee-phase Inuction Moto Dives IEEE tansactions on inustial electonics, vol. 53, no. 6, Dec. 6. [6] D. Diallo, M. E. H. Benbouzi an A. Makouf, A Fault-Toleant Contol Achitectue fo Inuction Moto Dives in Automotive Applications IEEE tansactions on vehicula technology, vol. 53, no. 6, Nov.. [7] S. Moeau, J.C. Tigeassou, G. Champenois, J.P. Gaubet, Diagnosis of Inuction Machines: a poceue fo electical fault etection an localization. SDEMPED 99, Gijon, Spain, pp. 5-9, Sep [8] M. E. H. Benbouzi, M. Vieia an C. Theys, Inuction motos faults etection an localization using stato cuent avance signal pocessing techniques. IEEE Tansaction on Powe Electonics, Vol., N, pp, Jan [9] S. Nani, A. Toliyat Hami, an L. Xiaoong, Conition Monitoing an Fault Diagnosis of Electical Motos IEEE Tansactions on enegy convesion, ( 5. pp [] D. Diallo, M. E. H. Benbouzi, D. Hama an X. Piee, Fault Detection an Diagnosis in an Inuction Machine Dive: A Patten Recognition Appoach Base 3

16 O. Benzineb et al: Implicit Fault Toleant Contol Tech Base Backstepping: Application to I M on Concoia Stato Mean Cuent Vecto. IEEE Tansactions on Enegy Convesion, vol., no. 3, Sep. 5. [] M. Belkheii an F. Boujema, Backstepping contol of an Inuction machine augmente by a Neual netwok. th Intenational Multi-Confeence (SSD 7, Volume I: Confeence on Systems Analysis & Automatic Contol, Tunisia, Ma. 7. [] S. Chaouch, A. Heizi, H. Seai an M. S. Naït-Saï, Lyapunov an Backstepping Contol Design of Inuction Moto System. th Intenational Multi-Confeence (SSD 7, Volume II: Confeence on Powe Electical Systems, Tunisia, Ma. 7. [3] C. Bonivento, A. Isioi, I. Maconi an A. Paoli, Implicit Fault Toleant Contol: Application to Inuction Motos. Automatica,, Rate Values Powe.8 KW Voltage / 38 V Fequency 5 Hz n p Rate Paametes R s Ω R 6.3 Ω L.6 H s L.6 H M. H Kg.m J. f.5 IS