IMPLICIT FAULT TOLERANT CONTROL: APPLICATION TO INDUCTION MOTORS 1. DEIS, University of Bologna (Italy) viale Risorgimento 2, Bologna

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1 IMPLICIT FAULT TOLERANT CONTROL: APPLICATION TO INDUCTION MOTORS 1 Clauio Bonivento Alberto Isiori Lorenzo Marconi Anrea Paoli DEIS, University of Bologna Italy viale Risorgimento 2, 4136 Bologna {cbonivento, lmarconi, apaoli}@eis.unibo.it DIS, University of Roma La Sapienza Italy Via Euossiana 18, 184 Roma isiori@is.uniroma1.it Abstract: In this paper we propose an innovative way of ealing with the esign of fault tolerant control systems. We show how the nonlinear output regulation theory can be successfully aopte in orer to esign a regulator able to offset the effect of all the possible faults which can occur an, so oing, also to etect an isolate the occurre fault. The regulator is esigne by embeing the possible nonlinear internal moel of the fault. This iea is then applie to the esign of a fault tolerant controller for inuction motors in presence of both electrical an mechanical faults. Copyright IFAC 22. Keywors: Output regulation, fault tolerant control systems, fault etection an isolation, internal moel-base control, inuction motor 1. INTRODUCTION Many efforts in the control community have been recently evote to stuy fault tolerant control FTC systems, namely control systems able on one han to etect incipient faults in sensors an/or actuators an, on the other, to promptly aapt the control law in such a way to preserve pre-specifie performances in terms of quality of the prouction, safety, etc. The most common approach of ealing with such a problem see Frank, 199 an the reference therein is to ivie the overall esign in two istinct phases. The first phase regars the so-calle Fault Detection an Isolation FDI problem which consists of esigning a ynamical system filter which, processing input/output ata, is able to etect the presence of an incipient fault an eventually to precisely isolate it. Once the FDI unit has been esigne, the secon 1 This work was supporte by MURST. Corresponing author: Dr. Lorenzo Marconi, DEIS-University of Bologna, Via Risorgimento 2, 4136 Bologna, Italy. phase usually amounts in esigning a supervisory unit which, on the basis of the information provie by the FDI filter, achieves control reconfiguration to compensate for the effect of the fault an fulfill performances constraints. In general the latter phase is carrie out looking for a parameterize controller which is suitable upate by the supervisor accoring to the information yiele by FDI unit. It is clear from this brief iscussion that the classical approach to FDI an FTC relies upon a certainty equivalence iea extensively use in the context of aaptive control, since it involves the explicit estimation of unknown time varying signals/parameters in the specific case the faults by the FDI unit an then the explicit reconfiguration of the controller in presence of faults. The aim of this paper is to follow a ifferent perspective to fault tolerance control. In particular, restricting the analysis to that faults whose sie-effects in the system operation are suitably moele, we look for

2 a controller which embes an internal moel of the fault an hence is able to intrinsically compensate for its effect, regarless its entity an without its explicit estimation. In other wors the control reconfiguration oes not pass through an explicit FDI esign but, inee, is achieve by a proper esign of a ynamic controller which is implicitly fault tolerant to all the possible faults whose moel is embee in the regulator. This iea will be pursue using the theoretical machinery of the nonlinear output regulation theory see Byrnes et al., 1997 uner the assumption that the sie-effect generate by the occurrence of the fault can be moelle as an exogenous signal given by an autonomous neutrally stable system exosystem. As opposite to certainty equivalence-base aaptive control, the istinctive feature of the output regulation theory relies in the esign of a regulator which, embeing an internal moel of exosystem, is able to offset the effect of any exosystem-generate signal without explicitly estimating it. In view of this it is possible to say that the main iea pursue in this paper is exactly that of proposing a fault tolerant control esign which is to the classical FDI/FTC approach as the output regulation theory is to certainty equivalence-base aaptive control. It is interesting to see that, in this framework, the Fault Detection an Isolation phase which is usually the starting point in the esign of a Fault Tolerant Control systems, is postpone to that of control reconfiguration since it can be carrie out by testing the state of the internal moel unit which automatically activates to offset the presence of the fault. In this paper the approach above outline is specialize to the esign of a fault tolerant control system for Inuction Motors IM. As all the magnetic rotating machines the IM is subject to rotor an stator failures cause by a combination of thermal, electrical, mechanical, magnetic an environmental stresses. Due to these stresses the IM can operate into a failure conition whose effects show with spurious harmonic currents arising in the stator circuit see Bellini et al., 2, Vas, 1994 with frequencies which are irectly relate to the kin of the fault in general stator or rotor fault an amplitue an phase which epen on the gravity of the fault. This allows to see the fault tolerant problem in the perspective above outline an to cast the problem as an output regulation problem. More in etail we show how an inirect fiel oriente IFO controller see Peresaa an Tonielli, 2, Ortega et al., 1996 an the reference therein can be augmente with a ynamic unit esigne in orer to compensate the unknown spurious harmonic currents arising in the stator circuit in presence of rotor or stator faults. In this way a controller which is implicitly fault tolerant to all the faults belonging to the moel embee in the regulator is obtaine. With respect to the work presente in Paoli et al., 21, just a semiglobal result was reporte, this work presents the more interesting case of the global stabilizer. The paper is structure as follow. The next section briefly presents the IM moel an the global IFO controller able to enforce esire flux an spee profiles when no faults enter into the picture. Section 3 shows how the IM moel moifies in presence of stator an rotor faults while section 4 presents the main result of the paper namely the esign of a Global Fault Tolerant IFO controller. 2. THE IM MODEL AND THE GLOBAL IFO CONTROLLER Uner assumptions of linear magnetic circuits an balance operating conitions, the equivalent twophase moel of the symmetrical IM, represente in an arbitrary rotating two-phase reference frame q, is see Marino et al., 1993 ẋ = fx, ω + Bu + T L + bv y = Cx ɛ = ω ɛ = 1 x = ω, Ψ, Ψ q, i, i q T is the space state vector, u = u, u q T is the control vector an y = ω, i, i q T is the vector of measurable variables. The state variables are efine as follows: ω is the rotor spee, Ψ, Ψ q are rotor flux components, i, i q are stator current vector components. The variable T L is the unknown loa torque an V represents an exogenous input which is zero in case the IM works in unfaulty moe while is a boune unknown signal in presence of faults see the treatment in section 3. The variable ɛ represents the angular position of the rotating q reference frame with respect to the a-axis of the fixe stator reference frame a b; the relation between the original a b an transforme q variables is given by : e jɛ = x q = e jɛ x ab x ab = e jɛ x q [ ] cos ɛ sin ɛ sin ɛ cos ɛ x yz stans for two-imensional vectors in the y z reference frame. Vector function fx, ω an constant matrices B, an b are fx, ω = µ Ψ i q Ψ q i αψ + ω ω Ψ q + αl m i ω ω Ψ αψ q + αl m i q αβψ + βωψ q γi + ω i q βωψ + αβψ q ω i γi q

3 B = 1 σ 1 σ 1 J = b = 1 1 the positive constants in the moel are relate to electrical an mechanical parameters of IM as follow σ = L s 1 L2 m β = L m µ = 3 L s L r σl r 2 α = R r Rs γ = L r σ + αl mβ L m JL r with J the rotor inertia, R s, R r, L s, L r the stator/rotor resistances an inuctances respectively, L m the magnetizing inuctance. General specifications for spee controlle electric rives regar two IM outputs, namely spee an rotor flux moulus efine as [ ] ω y 1 = Ψ 2 + = Ψ2 q [ ] ω Ψ which are require to follow esire references using the stator voltage vector u as control inputs an the vector y as available measures, consiering the torque loa T L as an unknown isturbance. Furthermore steay state flux ecoupling, i.e. lim t Ψ q =, is also require. The tracking problem just formulate has been aresse an solve in several papers see Ortega et al., 1996, Peresaa an Tonielli, 2 an the reference therein. In the following the solution enote by inirect fiel oriente controller is briefly reviewe. Let ω an Ψ be enoting the reference signals for the angular velocity an the flux moulus respectively an efine the spee-flux moulus tracking errors as ω = ω ω Ψ = Ψ Ψ. Moreover let the current tracking errors be efine as ĩ = i i i = 1 αl m [ αψ + ĩ q = i q i q Ψ ] i q = 1 µψ [ k ω ω + ˆT + ω ] with k w a esign parameter an ˆT an estimate of the loa torque T L /J whose ynamics are specifie later. Moreover efine i 1 = 1 [ α αl Ψ + Ψ ] m i q1 = 1 [ µψ k ω µψ ĩ q k ω ω + ˆT ] + ω + 2 Ψ /Ψ i q. Then the following proposition presents the IFO controller yieling global stabilization of the tracking error see Peresaa an Tonielli, 2. Proposition 1. Let V t an let k ω, k ωi, k ζ be arbitrary positive constants. There exist positive k, kq an ki such that for all k k, k q kq an k i ki the state of the close loop system 1 with control law ˆT = k ωi ω ζ = k ζ Ψ ĩ q ω = ω + s ω u = u k := σ k ĩ + u I u q = u qk q := σ k q ĩ q + u I q i q ĩ s ω = αl m + βω Ψ Ψ u I = αβψ + γi ω i q + i 1 u I q = βωψ + γi q + ω i + i q1+ is boune for all t an 1 Ψ Ψ ĩ q + k i ζ lim sup ωt, Ψ t, Ψ q t =. t 3 4 The previous controller guarantees global asymptotic stability which inee can be prove to be exponential of the tracking error in case V is ientically zero. Note that, as stresse by the notation, the control laws u an u q are parameterize with the current gains k an k q which must be chosen sufficiently large. Finally it is worth stressing that, as verifie in Peresaa an Tonielli, 2 with the help of experimental activity, the IFO controller just erive secures a certain egree of robustness with respect to the rotor parameters which are usually characterize by high uncertainties. 3. THE IM MODEL IN PRESENCE OF FAULTS In this section we briefly review how the moel of the IM moifies in presence of faults which can be both of mechanical an electrical nature. Following the theory in Vas, 1994 it turns out that the presence of mechanical an electrical faults generates asymmetry of the IM yieling some slot harmonics

4 in the stator wining. In the two-phase moel, it is possible to moel this effect thinking of a sinusoial component which corrupts the stator currents, i.e. i A i A + A sinɛ e t + φ i B i B + A cosɛ e t + φ, t ɛ e t = 2π 5 f e ττ, 6 an f e is a function which epens on the specific fault. For example faults cause by rotor asymmetry ue to broken bars or ynamic eccentricity yiels harmonic component at the frequency f e = f rbb = 1 ± 2ks w f 7 s w = ω ω is the slip, f is the supply frequency an k is a positive integer. On the other han faults generate by stator asymmetry ue, for instance, to short circuit or static eccentricity generates harmonic components at the frequency f e = f ssc = f. 8 As far as the amplitue A an the phase φ in 5 are concerne, they epen on the entity of the rotor or stator asymmetry an then can not be consiere known since epen on the specific fault severity. Similarly, once the variables are expresse in the q reference frame, it turns out that the stator currents in presence of stator or rotor asymmetries moify as i i + A sinɛ e t + ɛ t + φ i q i q + A cosɛ e t + ɛ t + φ. 9 ɛ, introuce in the previous section, enotes the angular position of the q reference frame. Few assumptions are one in the following to simplify relation 9. First of all, for sake of simplicity, we shall concentrate on the case in which the possible frequencies which characterize the sinusoial aitive terms in 9 are constant. This assumption is automatically fulfille in case the reference angular velocity ω is constant an, moreover, if a possible fault is allowe to arise just when the steay state has been reache. As a matter of fact, uner these assumptions, it is easy to realize that in steay state bearing in min 6, 7, 8 an the efinition of the slip s ω in 4 ɛ e t + ɛ t = 2πft + ω + s ωt + ɛ := Ω 1 t + ɛ 1 as far as faults concerning stator asymmetries are concerne, an ɛ e t + ɛ t = 2π1 ± 2ks ωft+ + ω + s ωt + ɛ := Ω 2,±k t + ɛ 11 as far as faults concerning rotor asymmetries are concerne. In the previous expressions s ω enotes the constant steay state reache by s ω which turns out to be s ω := αl ˆT m. 12 µψ 2 while ɛ enotes the unknown position of the reference frame once the fault occurs. As a further hypothesis we assume that the frequencies Ω 1 an Ω 2,±k efine in 1 an 11 are perfectly known. This, as clear from the previous computations, amounts to asking perfect knowlege of the IM parameters. Finally, as far as the aitive term arising in presence of rotors asymmetry is concerne, we assume that all the terms with frequency characterize by k > 1 see 11 are negligible with respect to the ominant first components with frequencies Ω 2,±1. This allows to moel the presence of rotor faults as two harmonics at frequencies respectively Ω 2,1 an Ω 2, 1. However it is worth noting that, by increasing the complexity of the regulator presente in the next section, this assumption can be softene ealing with an arbitrarily large, but finite, value of k. As a result of this brief iscussion, we have that in the setup above presente the presence of faults in the IM shows with harmonic components on the stator currents with known frequency, the latter epenent on the kin of fault which belongs to the two possible classes rotor or stator faults, an unknown amplitue an phase, the latter epening on the fault severity. In particular it is easy to realize that, efining the exosystem with an S = ẇ = Sw w IR 6 13 Ss S r S s = Ω1 Ω 1 Ω 2,1 S r = Ω 2,1 Ω 2,2 Ω 2,2 the aitive perturbing terms in 9 can be thought as a suitable combination of the exosystem state, namely with i i + Q w i q i q + Q q w. Q := Q q := In this way the uncertainty on the amplitue an phase of the aitive sinusoial terms in the faulty conition reflects in that on the initial state of the exosystem.

5 Recalling the current ynamics in the un-faulty operative conition reporte in the previous section, a simple computation shows that, once the perturbing terms Q w an Q q w are ae, the i, i q ynamics moify as i = αβψ + βωψ q γi + ω i q + 1 σ u + γq w + Q Sw + ω Q q w i q = βωψ + αβψ q ω i γi q σ u + ω Q w γq q w + Q q Sw. In view of this it is reaily seen that the moel of the IM in presence of faults is given by 1 with the exogenous input V equal to γq w + Q V = Sw + ω Q q w. 15 ω Q w γq q w + Q q Sw or, in a more compact form, with Γ := V = Γ Γ q w = Γw γ l1 γ l 2 γ l 3 l 1 γ l 2 γ l 3 γ l 1 = ω + s ω + Ω 1 l 2 = ω + s ω + Ω 2,1 l 3 = ω + s ω + Ω 2, 1. Note that with the above formalism both electrical or mechanical or simultaneous faults are allowe with the first two components of the exosystem state which take into account for stator faults, while the last four for rotor faults. 4. A FAULT TOLERANT GLOBAL IFO CONTROLLER aitional terms to the IFO controller 3, able to asymptotically tackle the exogenous isturbance V t achieving in this way fault tolerance an, so oing, also to asymptotically estimate the faults an their gravity. To this en we make use of a well-establishe theoretical machinery evelope in the context of the nonlinear output regulation see Byrnes et al., 1997 by esigning an internal moel of the exosystem in orer to asymptotically reproucing the unknown term V t. We show how to esign two aitional control laws, enote by u im control inputs as an uim u = u k + u im u q = u qk q, k i + u im q q, such that choosing the 16 u k an u qk q represent the stanar global IFO controller 3, then the tracking objective is achieve also in presence of the exogenous fault-generate term Γw. Clearly in 16 the role of u im, uim q is to compensate for the effects of the faults while u k, u qk q, k i take care of the tracking objective as in the un-faulty operative moe. In orer to recover the global result of proposition 1, we esign the terms u im q as saturation functions, uim of the state of the internal moel see the expression 19 below. To this en we shall assume that the exogenous term Γw is lower an upper boune by known values an we assume the existence of a λ max > such that 2 Γw λ max. 17 Then the following main new result can be prove. Proposition 2. Let Ω IR 6 be an arbitrary compact set such that if w Ω then 17 hols for some λ max. Moreover let F an G be arbitrary IR 6 IR 6 an IR 6 IR 2 matrices such that F is Hurwitz an F, G is a controllable pair an let M be the unique nonsingular matrix solution of the following Sylvester equation 3 MS F M = GΓ. 18 In the un-faulty moe wt an hence V t is ientically zero an the IFO controller u, u q presente in proposition 1 meets the esign objective of steering the tracking error to zero. On the other han the presence of some kin of asymmetry, which as above explaine is a symptom of incipient fault, reflects in an initial state of the exosystem ifferent from zero which yiels a boune perturbing term V t. If the fault an hence wt was perfectly known then the control law [ ] u t = u q t [ ] u t u σγwt qt woul achieve fault tolerance by counteracting the exogenous term V t. Since the fault is not a priori known we propose a controller, esigne aing Then fix any λ λ max an consier the control law 16 with u im u im q ξ = Sξ + M 1 F GĨ [ GψGĨ+ ] + Ghξ, Ĩ GKĨ G ki ζ = sat λ ψ Mξ ψ GĨ = sat λ ψ q Mξ ψ q GĨ 19 2 Note that this assumption amounts to asking that the initial state w of the exosystem, which is relate to the entity of the fault, belongs to an arbitrary large compact set. 3 Existence an uniqueness of the matrix M follow from the fact that S an F have isjoint spectrum. The fact that M is nonsingular can be easily prove using observability of the pairs S, Γ an controllability of the pair F, G.

6 hξ, Ĩ = h ξ, Ĩ h q ξ, Ĩ := satλ ψ := Mξ ψ GĨ ψ, Mξ + ψ GĨ sat λ ψ q Mξ ψ q GĨ ψ qmξ + ψ q GĨ ψ Γ ψ = := M 1, ψ q Γ q K := k, Ĩ := k q ĩ Ψ. ĩ q an sat λ s := sgns min{ s, λ} is the classical saturation function. Then there exist k, k q, ki, M x, M ξ, α x, α ξ all positive such that for all k k, k q kq, k i ki an for any x, ξ, w IR 5 IR 6 Ω the state of the close loop system 1, 13, 15, 16, 19, is boune an ωt, Ψt, Ψ q t ω, Ψ, Ψ q M x e α xt 2 ξt wt ξ w M ξ e α ξt. 21 Note that the controller guarantees fault tolerance, since the tracking specifications are asymptotically exponentially fulfille as claime in 2, an also fault etection an isolation since the state ξ of the internal moel asymptotically reprouces that of the exosystem see 21. The fault etection an isolation phase is carrie out testing the state of the internal moel unit which automatically compensates for the effect of the fault. In other wors the fault is etecte an isolate just after its effect has been compensate. This means that the controller just esigne is implicitly fault tolerant with respect to the sie effects of any fault belonging to the aforementione classes an no explicit reconfiguration is require. Moreover it is worth stressing that, as the stanar IFO controller in proposition 1, also this fault tolerant moification guarantees global stabilization as far as the state of the IM is concerne. This is achieve recovering the same structure of the IFO controller u, u q in proposition 1 with only the current gains k, k q which eventually nee to be re-tune as a consequence of the fault-tolerant moification. This is an important fact since it allows to recover the same performances an robustness of the IFO controller in the unfaulty operative moe an moreover to have the esign of the global stabilizer which is somewhat inepenent from that of the FDI an FTC unit. angular velocity of the IM in orer to enforce esire flux an spee profiles, can be enriche with an internal moel of the fault in orer to achieve fault tolerance an also fault etection an isolation. The esign of the internal moel unit can be consiere inepenent from that of the stabilizing IFO unit as only the current gains are eventually require to be re-tune. We have shown how the internal moel unit can be esigne in orer to have global tracking of the esire references an semi-global tolerance to possible faults. 6. REFERENCES Bellini, A., F. Filippetti, G. Franceschini an C. Tassoni 2. Close-loop control impact on the iagnosis of inuction motor faults. IEEE Transactions on Inustry Applications 365, Byrnes, C.I., F. Delli Priscoli, A. Isiori an W. Kang Structurally stable output regulation of nonlinear systems. Automatica 332, Frank, P.M Fault iagnosis in ynamic systems using analitical an knowlege base reunancy: a survey an some new results. Automatica 263, Marino, R., S. Peresaa an P. Valigi Aaptive input-output linearizing control of inuction motors. IEEE Transactions on Automatic Control 382, Ortega, R., P.J. Nicklassonan an G. Espinosa On spee control of inuction motors. Automatica 323, Paoli, A., L. Marconi an C. Bonivento 21. A fault-tolerant strategy for inuction motors. Proceeings of the 4th IEEE Conference on Decision an Control pp Peresaa, S. an A. Tonielli 2. High-performance robust spee-flux tracking controller for inuction motor. International Journal on Aaptive Control an Signal Processing 14, Vas, P Parameter estimation, conition monitoring an iagnosis of electrical machines. Oxfor science publications. Oxfor. 5. CONCLUSIONS In this paper we have presente a new iea for ealing with fault tolerant control systems esign presenting the esign of a fault tolerant control unit for an Inuction Motors. We have shown how an Inirect Fiel Oriente controller processing the currents an the