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Fault Diagnosis of an Induction Drive Using Residual Directional Properties
Copyright @ IF AC Fault Detection, Supervision and Safety for Technical Processes, Budapest, Hungary, 2000
FAULT DIAGNOSIS OF AN INDUCTION DRIVE USING RESIDUAL DIRECTIONAL PROPERTIES
C. Combastel*, S. Gentil*,
J. P. Rognont
* Laboratoire d'Automatique de Grenoble, INPG,
UJF, CNRS UMR 5528,
t Laboratoire d 'Electrotechnique de Grenoble, INPG, UJF, CNRS UMR 5529, BP 46, F-38402 Saint Martin d'Heres Cedex, France Phone: 33 4 768264 13, Fax: 334 76826388 e-mail : Christophe. [email protected], Sylviane. [email protected], [email protected]
Abstract: Fault diagnosis is mainly a decoupling problem that can be solved through the elimination and/or estimation of unknown variables. In this paper, an elimination and estimation scheme based on residual directional properties is proposed. It is applied to the diagnosis of an induction drive. The interest in using the directions is that they contain the information about the relative orders of magnitude . Copyright © 2000 IFAC Keywords: Fault diagnosis, Induction machines, Fuzzy Logic, Gradient methods.
Contrary to signature tables, the relative order of magnitude of the residuals is taken into account in the decision procedure, improving thus the isolation ability. Residual directional properties can be also used in the residual generator itself: estimation methods involving a gradient calculation illustrates this aspect. After the problem formulation in section 2 and the description of the application in section 3, the diagnosis is carried out by using the residual directional properties. Two sets of residuals are derived from the motor electrical equations. A first set of residuals is used in section 4 to diagnose current sensor faults and open-circuit faults in the power switches. In section 5, a second set of residuals based on a gradient method allows to discriminate between the speed sensor fault and the other faults.
I . INTRODUCTION The applications requiring variable speed control are now often based on induction motors. Such motors are cheaper than DC motors but harder to control. Their maintenance is also easier as there is no brush. An induction drive consists of several components: a supply , a power electronics converter, a calculator for the control, sensors and the induction motor itself. The aim of on-line diagnosis is to detect, isolate and identify the faults that may occur in the different components of this system. Methods based on signal processing (mainly spectral analysis) have been widely used to diagnose grid supplied induction motors . Signal processing is a promising approach to isolate motor faults such as stator winding problems (insulation degradation or breakdown leading to shorted-turns), eccentricity or broken rotor bars (Thomson , 1999). In variable speed applications, the closed-loop and non stationary phenomena make signal processing-based approaches more difficult to apply . However, the availability of mathematical models such as those resulting from the Park transformation makes it possible to implement methods based on analytical redundancy (lsermann, 1997). Fixed direction residuals is a classical approach to fault diagnosis (Chen, et aI., 1995).
2. PROBLEM FORMULATION 2.1. Fault detection The availability of a mathematical model makes it possible to implement methods based on analytical redundancy. At first, a non linear system S in the state-space form is considered:
1101
2.2. Fault isolation .~
= g s (x, 11, I ,d) ,
x(O)
= Xo
(I)
The isolation problem can be formulated in the same way as the detection one, except that some variables modelling the faults have to be decoupled.
y=hs (x,u,l,d) where x is the a priori unknown state, u is the input, y is the (measured) output, I is the fault and d stands for the disturbances . The only requirement on faults and disturbances is the availability of a model like (I). Thus,fand d can be non-additive . Fault detection consists in deciding whether the system is faulty or not (frO or 1=0) independently from disturbances and operating point changes. A usual way to perform this task consists in designing a detection indicator )1 which is often boo lean or in the [0; IJ interval: ~i)
= gM (w, u, y),
w(O)
= Wo
Table 1: Fault Detection and Isolation (FDI) Purpose Detection Isolation of J;
/ )1(w,tI,y)=O~/=O
I J;
The isolation of the i1h fault consists in designing gM; and hM ; so that the fault indicator )1; satisfies (5) .
(2)
(5)
)1=h M (w,u , y)
gM , h M
Variable type: Known Unknown w,u x,d w, u fi,,;, x, d
Tested
)1, should be designed so as to be decoupled from all the unknown variables (jj,,;, x, d) . This is all the harder to satisfy as the fault number increases. However, under the single fault hypothesis, all the combinations involving more than one fault are assumed to be impossible, and thus do not require any decoupling. Therefore, )1; have to be decoupled only from {fj, x , d} for all j =I- i. In the following, a combination of elimination and estimation techniques is proposed to diagnose some faults in an induction motor. Special attention is paid on the residual directional properties.
(3)
Depending only on known variables, the on-line computation of)1 is implemented according to (2). In order to design gM and hM so that )1 fulfils the detection purposes (3), the influences of unknown variables can be assessed by substituting (1) for y in (2). This results in the following requirements for gM and hM :
(4) is difficult to fulfil due to the influence of the unknown variables x and d on )1. The aim of fault detection is thus to find gM and hM so that )1 is decoupled from x and d while being as sensitive as possible to f Two approaches can be used : elimination performs a direct decoupling whereas estimation performs an indirect decoupling (Zhang, et aI. , 1998; Krishnaswami and Rizzoni, 1994).
3. THE INDUCTION DRIVE Using the Concordia transform, the continuous model of a squirrel-cage induction motor can be expressed in the reference frame fixed to the stator (a/3): Xe = A(xm ).xe
(6)
+ B.(u + Iv)
Ye =C.X e + I,
Elimination (direct decoupling). Elimination consists in manipulating the original mathematical model so that unknown variables do not appear in the expression of )1. This case corresponds to a perfect decoupling. Approximate decoupling consists in minimising the influence of unknown variables on )1.
(7)
Xm = A(xm,xe,f')
w=xm+lw
where:
A(xm ) =
Estimation (indirect decoupling). The indirect decoupling of an unknown variable is based on its estimation . The estimate is used in a forward model (or in an observer) of the system to compensate for the influence of the unknown variable. Unless the estimation is perfect, only approximate decoupling can be achieved. The design of gM and hM is often performed in two steps: residual generation and decision . No matter how the diagnosis is decomposed, it is before anything a decoupling problem. Moreover, it can be noticed that the decoupling of unknown variables can be carried out by the residual generation on the one hand and by the decision on the other hand. This is often used to achieve fault isolation.
l
0
-RJ
0
0
R, a
p.xmL,/a
-~/a
- p.xmL,/a
R,/a
P'Xm
0
0/
[- r J{3
'
I ~~/~ 0
(8)
-R
Ye = [l,a]
(9)
l'h
(6) represents the electrical (subscript e) equations and (7) is the mechanical one (subscript m). R" R" L" L" L m , p refer to the usual electrical parameters of an induction motor. I,
1102
currents. fluxes and voltages, the measured motor speed and, respectively, the load torque. Assuming that the machine is balanced, the Concordia transform matrix K23 (resp. Kn) allows to change the reference frame from (abc) (resp. (ab» to (af3) (or inversely). The measurements are those usually used for control purposes: stator currents (i.,,,, l,b) and OJ. Thus, y = [Ye w( Sensor faults (/J = [Ji\{/ flxbf,fw) are modelled by additive inputs. A classical vectorial control is used to control the torque and an outer loop controls the speed . A very simple model of the faultfree Pulse Width Modulated (PWM) inverter (Fig. I) is a unit gain : this means that the stator voltages calculated by the control, V,,,, V,b, V,,, are those actually applied to the induction motor, V ,lIm",,,,, V,h"""'''' V.c"IO'''''. In order to deal with open-circuit faults in the power switches, the model of the PWM inverter is that of Fig. 2. fv = [fvstlfV.,bfv",f represents the actuator faults in (6).
input of the electrical equations (6), an input-output sub-model eliminating r can be built. After discretisation, it can be expressed with a discrete Linear Time Varying (LTV) transfer G such that: G(z,W- fw).(u+ fv )-(Ye - f,) =0 ~
r = Ye -G(z,W).u ~
.\(1
v..1/,
--J: -J::
Faulty switch (open-circuit) S(1+
vs " V II/fl10r '\"h
-J;: v.. VH
V.I.e
"'Ofor
.H
5,,' 5,: 5,; 5c +
S,'
r = -h(z,u,y,O)
Fig. 3 shows the residual generation scheme. A usual approach to discretize the electrical equations (6) consists in developing the corresponding matrix exponential until order two. The on-line calculation of r is thus implemented as follows:
Fig. I. Structure of the PWM inverter
V.WI II/%r
(12)
Fig. 3 : Residual generation
E : DC .HIpp/.\' "a/Wile (output of a three-phase rectifier)
v.'"
h(z,u,y,j)=O
A residual can be calculated in order to check whether this subsystem (consisting of the electrical equations) is faulty or not. Such a residual IS computed from the available information (u. y):
c
V
(11)
Fault model
Xe .k + 1 = Ad k .xe.k + Bd k .u k
iv.", < 0 iv.", > 0 iv.,-!J 0 iv« < 0 Iv,,· >0
(13)
r = Ye -C,Xe.k Ad k
= I + A(wk ).T, +~(A(Wk ).T, f Bd.
Fig. 2. Model of open-circuit faults in the power switches
=T,{/+~A(W.).T,}B
r = [rl m rlshf represents the residuals consistent with the a and b stator currents. (13) can be interpreted as a particular reduced order unknown input observer. It performs an elimination of Xm and r and an estimation of x,. As r satisfies the detection requirements, a detection indicator can be directly derived from r. Its directional properties are now studied in order to isolate the faults.
In the following, 1;, refers to variations of the electrical parameters modelling motor faults (broken bars, etc ... ). The link between the application and the problem general formulation (I) is given by (10).
4.2. Residual directional properties
4. DIAGNOSIS OF CURRENT SENSOR FAULTS AND PWM INVERTER FAULTS
The influence of faults on the residuals can be assessed from the model using a first order development. This approximation is only valid for small fault amplitudes unless the residual is a linear function of the faults .
4.1. Residual Generation
The residual generation has been studied in order to fulfil the following two requirements: the method does not depend on the type of control and, above all, the residuals have to be perfectly decoupled from the load torque variations (as the load torque is a priori unknown). A systematic approach for the application of model partitioning to fault diagnosis can be found in (Combastel, et aI., 1999). Here, a simple physical reasoning gives the same result: by using (J) as an
(14)
~
r ==
dh(z, u, y, j) df
1103
·f = ·~)j(z.u,Y).fj {=O
J"
is not mentioned in section 4 to simplify the notations. The wmments about fw in this section are also applicable to J" . The incidence matrix derived from (14) has the following form :
r ~ [O *
(15)
*1** ** *1*]·[/'] * * Iv
0
Iw
It can be noticed that a decision based on a Boolean signature table only allows to isolate the current sensor faults (15). It is thus necessary to take more information into account about the residual behaviours for fault isolation to become possible. This is why attention is paid to the residual directional properties: the relative order of magnitude of the residual vector elements is then taken into account by the decision procedure. In other words, different residual sensitivities in response to different faults become sufficient to perform fault isolation. The residual directional properties can be deduced from ( 14). The column vectors related to fr (D, = [I O]T and D2 = [0 I]T) are constant and are distinct. So, when a single current sensor fault occurs, r has a fixed direction which is characteristic of the occurrence of this fault. The column vector related to f,y, Df>, can be calculated at each sample time k from the filtering of u with the discrete transfer JG(z,w-fw)/Jfw- Do,k can be interpreted as a time varying direction for r which is characteristic of the occurrence offw- The column vectors related to fv, Dj to D" cannot be directly calculated in time domain: the direction of r actually depends on the temporal evolution of f". As faults are a priori unknown, the corresponding directions of r cannot be computed without a supplementary assumption: the dynamic response of the residual to fv is thus assumed to be close to the steady state response:
G(z , w)·fv "" G(l,w)·fv
sinusoids, r becomes a non zero rotating vector when fw occurs. In this situation, the direction of r is periodically the same as the direction of a current sensor fault (for instance). Consequently, the decision at time k has to take past information into account. The example of the speed sensor fault shows that the isolation ability depends on the system excitation when directions are not fixed: a lack of excitation can lead to the same directions for two faults, whereas enough excitation may result in different residual directions allowing to distinguish between the same two faults. 4.3. Fault Diagnosis (Detection, Isolation, Identification)
A fault is detected if r becomes different from zero i.e. if F(q) .lrkl exceeds a (fixed or fuzzy) threshold. F(q) stands for a discrete time filter that can be used to reduce the influence of noise. (18) J1z(.) stands for a function modelling the membership
to Z (zero). D(j) is a detection indicator represented by a fuzzy set. With the same notations, D(f;) refers to a fuzzy set the membership to which represents the decision about a possible occurrence of fi. Under the single fault hypothesis, the membership of the residuals to D(f;) is obtained from a three step reasoning: a) When fi occurs, a fault is detected by the detection indicator (18). b) The angle (denoted by L. ) between the direction related to fi, Dj, and the residual vector r is close to zero at time k. c) As the information at time k is not sufficient to obtain a relevant decision, a filtering based on a forgetting factor A is introduced; the purpose of such a filtering is to reduce the influence of noise and to discriminate between two faults that may have the same direction at time k but not over a larger period of time. The three step reasoning results in the following expression for the decision related to fi :
(16)
In practice, if the variations of fv are slow compared to the dynamic of G(z,w), (16) remains true. Moreover, if Iv is a step fault, the response to fv is constant after a transient depending on the dynamic of G(z,w). In the case of the induction motor, this is not restrictive as G(z,w) models the fast electrical dynamics. The steady-state gain of G leads to fixed residual directions in response to fv.
'" [: :1XR.I', -,' -:]1 JG('a;~- J.' 'l[~l ~ D =LD r k ""
k ·fk
mj,k+1 =A.m j.k +(l-A) ·J1z(Di,kLrk) J10(f,)(rk
)
(19)
= J10(J)(rk ) /\ mi,k
A definition of the angle (L.) either taking vector orientation into account or not allows either to isolate faults that have different signs or not. This is used to isolate PWM inverter faults (Fig. 2). The identification of fault amplitudes under the assumption of single faults results from a least square criterion :
(17)
J.k .f;.k
8 i .k T
.T k
(20)
8 i.k T .8 i.k
The directions of the current sensor faults and the directions of the PWM inverter faults are fixed and are distinct (17). Isolation of these faults is thus possible under the single fault hypothesis . Moreover, as the stator voltages are balanced three-phase
To improve the identification of fi,b the estimate can be filtered or/and the criterion can be defined over a temporal window. It is focused on the fault indicator related to an open-circuit fault (OCF) of S,,' to
1104
illustrate the application to the induction drive (Fig.
rejection specifications. The adaptation of f w IS hased on an on-line gradient algorithm minimising the criterion J=pTp:
4).
····· ·-·--···----·1
•
axAr
( il/w
j_ aA(w-jw)
A'
e
+
A(
_fA
W
ap
Iw
= ajw = -
)(axe A
W·
ilfw
o. Ca) Residual responses to an OCF of Sa'
A X
a/w
j
(22)
C aXe
K 22 '
.
ajw
The residual generator resulting from (21) and (22) is discretised by developing the corresponding matrix exponential until order 2. (23) shows the adaptation of j w : it is based on the residual directional
( -
properties of p i.e. on the gradient of p. A
T
A
fw.k+1 = fw,k -TJ·8 i""k ·Pk (b)
11 DU v", >0) (rk
(c) I1D(fvm>O)(rk )
)
when an OCF of SIl' occurs.
Let us interpret the adaptation of j w in order to
whenfw occurs.
precise the diagnosis scheme. p=o means that the adaptive parallel simulation fits the system real behaviour (known through the available measurements). Assuming the electrical model is correct, if fw occurs, a value of j w exists such that
Fig. 4. Fault indicator for an open-circuit fault of Sa . As illustrated by f.1 DU"", >0) (rk
)
(23)
in Fig. 4, f.1 DU,) (rk )
is decoupled from x and d due to the properties of r. By focusing on a particular direction for each fault (i.e. by eliminating the other directions), an approximate decoupling of jj"i is achieved by f.1 DU, ) (fk ) .
When the adaptation converges, the minimisation of J=pTp leads to J=O, and j w is correctly estimated. If some faults do not act on the motor as fw does, the algorithm minimises J through the adaptation of j w ' but does not manage to null J
p=0.
5. DIAGNOSIS OF SPEED SENSOR FAULTS
(or p). Therefore, p is decoupled from fw apart from the noise, the model errors and the transients before the estimation convergence. The proposed method is thus more suited to diagnose slow developing faults. However, when jj;:;=O, the two elements of vector p become very noisy sinusoids: a low-pass filtering of p or IIpll is inefficient to detect slight changes.
The direct application to the induction drive of the method proposed in section 4 shows that the diagnosis (isolation, identification) of the speed sensor fault becomes less accurate as the fault amplitude increases. This is due to the first order development involved in (14). Therefore, results of section 4 are used to diagnose fi and fv. In order to discriminate fw from the other faults (including 1;,), a second set of residuals, p, has been derived from the motor electrical equations. As explained in section 4. I, P is decoupled from x and d:
(24)
p/, and p/ are two residuals resulting from scalar
jw)'x, +B.u
;£0
= A(w-
p
= Kn .(y,
(21 )
products. The orthogonal direction of the gradient is unique as p E ~2. p/, represents the component of p that can be locally interpreted by a variation of fur The adaptation of f w ma k es Pk" cI ose to zero. Pk ~
-C.x,)
•
The closer
jw
is to f wo the more decoupled is p from
A
represents the component of p that cannot be locally interpreted by a variation of fw . p/, which is not a sinusoid when 1;,:;=0, is thus suitable to discriminate betweenfwand1;,.
fw. This is an example of indirect decoupling. (21) is an adaptive residual generator with a single parameter tracking. This alternative allows to deal with slow parameter changes and to by-pass the problem of persistent process excitation (Hofling and Isermann , 1996). As voltages are sinusoids (apart from the influence of the Pulse Width Modulation in high frequencies), the natural excitation of u is a priori sufficient to adapt the single parameter j w . This is important to warrant that the on-line FDI scheme does not disturb the control or harmonic
JID(fw) =)lz(F(q).p;)/\(l-JIZ(jW.k» JI DU p) = (I - )l z (F (q). p ;
»/\ (I - )l D([ I1 Iv )) )
(25)
(26)
The fault indicators related to fw and jj, are given by (25) and (26) where F(q) stands for a low-pass filter
1105
rcducing thc influence of nOise and
J1 D([!!
sensor faults are identified. The identification of an open-circuit inverter fault has no real sense: such a fault is entirely characterised by thc faulty switch that is determined through the orientation of the directions used for fault isolation. The case of shortcircuit inverter faults has not been considered here: in practice, hardware protection makes the global system inactivc when such a fault occur. As the direction deduced from the residual derivatives at forO (see (14» is not sufficient to diagnose fw> the adaptation of j ill allows to fit more precisely the system behaviour. The directional properties of the second residual vector are then given by its gradient calculated at j ill (and not at forO as before). The gradient is then used to adjust a unique parameter: this allows to relax the excitation requirements. In spite of the computation burden, the isolation and the identification of the speed sensor fault have been achieved. Moreover, the motor faults (modelled by parameter variations) are detected and isolated from all the sensor and the actuator faults.
f,,])
indicates the occurrence of a current scnsor or an actuator fault. The application of the adaptive residual generator to the induction drive is now dealt with. The induction drive and the FDI system are simulated during I Os. The sampling time is 0.2ms. A 1200rpm speed reference step is applied at t=0.2s. The load torque is a ramp starting at t=0.2s and finishing at t=2s with an amplitude of 20Nm. In Fig. 5, RIO corresponds to 10% ramp variations of both R, and R, from their nominal value. L5 corresponds to 5% ramp variations of both L, and Lr • All parameter ramp variations start at t=2s and finish at t=7s. SJO corresponds to a speed sensor fault modelled by a ramp starting at t=5s and finishing at t=6s. The amplitude of the fault is 10% of the nominal speed (2000rpm): 12.6 rad/s.
j (rad/s)
F(q).p/
ill
510
lOa --------.,-.---.--'5~ . 5 - - - - - - - - - -, - .- - - -- -.
o
.---------.
-\
~
10
'5~
10 - - - - -- --- - :-- ----- --
510, RIO
5 - -- - - - - -- -, ---- - -- - -
o. -5
L5
•
-5
o
+
.
-
-
-
-
-
~-
-
-
-
-
.
-
-
-
-
10
Chen, I ., R. I . Patton and H.Y. Zhang (1995). Design of Robust Structured and Directional Residuals for Fault Isolation via Unknown Input Observers. Proceedings of 3rd European Control Conference, pp348-353 , Rome, Italy. Combastel, c., S. Petropol, S. Gentil and S. Lesecq (2000). Model-based and Wavelet Approaches to On-line Fault Detection: Application to an Induction Drive, IFACIIFlPIIEEE MCPL '2000, 2nd Conference on Management and Control of Production and Logistics, Grenoble, France. Combastel, c., S. Gentil and I.P. Rognon (1999). Fault Detection and Isolation using Local Models, Comparison with Unknown Input Observers. ECC'99, European Control Conference, Karlsruhe, Germany. Hofting T. and R. Isermann (1996). Fault Detection based on Adaptive Parity Equations and Single Parameter Tracking. Control Eng. Practice, Vol. 4, No . 10, pp.1361-1369. Isermann, R. (1997). Supervision, fault-detection and fault-diagnosis methods - An introduction, Control Eng. Practice, 5(5), pp. 639-652. Krishnaswami V. and G. Rizzoni (1994). Nonlinear Parity Equation Residual Generation for Fault Detection and Isolation. IFAC 5afeprocess '94, pp.317-322, Espoo, Finland. Thomson, W.T. (1999). A review of on-line condition monitoring techniques for three-phase squirrel-cage induction motors - Past, present and future . Proceedings of IEEE SDEMPED '99, pp318, Gij6n, Spain. Zhang, Q., M. Basseville and A. Benveniste (1998). Fault Detection and Isolation in Nonlinear Dynamic Systems: A Combined Input-Output and Local Approach . Automatica, Vol. 34, No. 11 , pp.1359-1373.
•
5
10
:Ee --- ------~- --- --- - ~------ - ---
·1
5
REFERENCES
- --- ---- -
o
o
- ------- -
10
~§----- ---i---·--- ---
10
'5~ o
;
~
-\
·1
5
10 - - -. - - -- - - -; - - - - - - - - --
5 . --
1 ----.- . . -- , .-- - -----:
o
---- ------
o
: 2§
0
.
0510
Fig. 5. Application of the adaptive residual generation to the induction drive Fig. 5 shows that j ill is correctly estimated when fw occurs (case SIO). p/ becomes non zero only when parameter deviations occur (RIO or L5) . Small inductance variations can be easily detected with p/ (case L5) . It can be also noticed that the diagnosis of fw is not very sensitive to resistance variations (case S 10, RIO). Thi s is important to ensure that the diagnosis is robust to resistance variations due to thermal constraints : such constraints exist even in the case of a fault-free behaviour and can lead to important resistance variations (more than 50% for R,). A fault detection which is robust to the important (but very slow) resistance variations is out of the scope of this article but it is dealt with in (Combastel , et aI., 2000) . It is also based on a single parameter tracking. Assuming that resistances are rather wellknown at the detection time, fault diagnosis can be carried out as mentioned in this paper. 6. CONCLUSION Due to the availability of a mathematical model, analytical redundancy can be applied to the induction drive. Current sensor faults and open-circuit PWM inverter faults are detected and isolated using the (distinct) directions of the residual vector. Current
[106
FAULT DIAGNOSIS OF AN INDUCTION DRIVE USING RESIDUAL DIRECTIONAL PROPERTIES
C. Combastel*, S. Gentil*,
J. P. Rognont
* Laboratoire d'Automatique de Grenoble, INPG,
UJF, CNRS UMR 5528,
t Laboratoire d 'Electrotechnique de Grenoble, INPG, UJF, CNRS UMR 5529, BP 46, F-38402 Saint Martin d'Heres Cedex, France Phone: 33 4 768264 13, Fax: 334 76826388 e-mail : Christophe. [email protected], Sylviane. [email protected], [email protected]
Abstract: Fault diagnosis is mainly a decoupling problem that can be solved through the elimination and/or estimation of unknown variables. In this paper, an elimination and estimation scheme based on residual directional properties is proposed. It is applied to the diagnosis of an induction drive. The interest in using the directions is that they contain the information about the relative orders of magnitude . Copyright © 2000 IFAC Keywords: Fault diagnosis, Induction machines, Fuzzy Logic, Gradient methods.
Contrary to signature tables, the relative order of magnitude of the residuals is taken into account in the decision procedure, improving thus the isolation ability. Residual directional properties can be also used in the residual generator itself: estimation methods involving a gradient calculation illustrates this aspect. After the problem formulation in section 2 and the description of the application in section 3, the diagnosis is carried out by using the residual directional properties. Two sets of residuals are derived from the motor electrical equations. A first set of residuals is used in section 4 to diagnose current sensor faults and open-circuit faults in the power switches. In section 5, a second set of residuals based on a gradient method allows to discriminate between the speed sensor fault and the other faults.
I . INTRODUCTION The applications requiring variable speed control are now often based on induction motors. Such motors are cheaper than DC motors but harder to control. Their maintenance is also easier as there is no brush. An induction drive consists of several components: a supply , a power electronics converter, a calculator for the control, sensors and the induction motor itself. The aim of on-line diagnosis is to detect, isolate and identify the faults that may occur in the different components of this system. Methods based on signal processing (mainly spectral analysis) have been widely used to diagnose grid supplied induction motors . Signal processing is a promising approach to isolate motor faults such as stator winding problems (insulation degradation or breakdown leading to shorted-turns), eccentricity or broken rotor bars (Thomson , 1999). In variable speed applications, the closed-loop and non stationary phenomena make signal processing-based approaches more difficult to apply . However, the availability of mathematical models such as those resulting from the Park transformation makes it possible to implement methods based on analytical redundancy (lsermann, 1997). Fixed direction residuals is a classical approach to fault diagnosis (Chen, et aI., 1995).
2. PROBLEM FORMULATION 2.1. Fault detection The availability of a mathematical model makes it possible to implement methods based on analytical redundancy. At first, a non linear system S in the state-space form is considered:
1101
2.2. Fault isolation .~
= g s (x, 11, I ,d) ,
x(O)
= Xo
(I)
The isolation problem can be formulated in the same way as the detection one, except that some variables modelling the faults have to be decoupled.
y=hs (x,u,l,d) where x is the a priori unknown state, u is the input, y is the (measured) output, I is the fault and d stands for the disturbances . The only requirement on faults and disturbances is the availability of a model like (I). Thus,fand d can be non-additive . Fault detection consists in deciding whether the system is faulty or not (frO or 1=0) independently from disturbances and operating point changes. A usual way to perform this task consists in designing a detection indicator )1 which is often boo lean or in the [0; IJ interval: ~i)
= gM (w, u, y),
w(O)
= Wo
Table 1: Fault Detection and Isolation (FDI) Purpose Detection Isolation of J;
/ )1(w,tI,y)=O~/=O
I J;
The isolation of the i1h fault consists in designing gM; and hM ; so that the fault indicator )1; satisfies (5) .
(2)
(5)
)1=h M (w,u , y)
gM , h M
Variable type: Known Unknown w,u x,d w, u fi,,;, x, d
Tested
)1, should be designed so as to be decoupled from all the unknown variables (jj,,;, x, d) . This is all the harder to satisfy as the fault number increases. However, under the single fault hypothesis, all the combinations involving more than one fault are assumed to be impossible, and thus do not require any decoupling. Therefore, )1; have to be decoupled only from {fj, x , d} for all j =I- i. In the following, a combination of elimination and estimation techniques is proposed to diagnose some faults in an induction motor. Special attention is paid on the residual directional properties.
(3)
Depending only on known variables, the on-line computation of)1 is implemented according to (2). In order to design gM and hM so that )1 fulfils the detection purposes (3), the influences of unknown variables can be assessed by substituting (1) for y in (2). This results in the following requirements for gM and hM :
(4) is difficult to fulfil due to the influence of the unknown variables x and d on )1. The aim of fault detection is thus to find gM and hM so that )1 is decoupled from x and d while being as sensitive as possible to f Two approaches can be used : elimination performs a direct decoupling whereas estimation performs an indirect decoupling (Zhang, et aI. , 1998; Krishnaswami and Rizzoni, 1994).
3. THE INDUCTION DRIVE Using the Concordia transform, the continuous model of a squirrel-cage induction motor can be expressed in the reference frame fixed to the stator (a/3): Xe = A(xm ).xe
(6)
+ B.(u + Iv)
Ye =C.X e + I,
Elimination (direct decoupling). Elimination consists in manipulating the original mathematical model so that unknown variables do not appear in the expression of )1. This case corresponds to a perfect decoupling. Approximate decoupling consists in minimising the influence of unknown variables on )1.
(7)
Xm = A(xm,xe,f')
w=xm+lw
where:
A(xm ) =
Estimation (indirect decoupling). The indirect decoupling of an unknown variable is based on its estimation . The estimate is used in a forward model (or in an observer) of the system to compensate for the influence of the unknown variable. Unless the estimation is perfect, only approximate decoupling can be achieved. The design of gM and hM is often performed in two steps: residual generation and decision . No matter how the diagnosis is decomposed, it is before anything a decoupling problem. Moreover, it can be noticed that the decoupling of unknown variables can be carried out by the residual generation on the one hand and by the decision on the other hand. This is often used to achieve fault isolation.
l
0
-RJ
0
0
R, a
p.xmL,/a
-~/a
- p.xmL,/a
R,/a
P'Xm
0
0/
[- r J{3
'
I ~~/~ 0
(8)
-R
Ye = [l,a]
(9)
l'h
(6) represents the electrical (subscript e) equations and (7) is the mechanical one (subscript m). R" R" L" L" L m , p refer to the usual electrical parameters of an induction motor. I,
1102
currents. fluxes and voltages, the measured motor speed and, respectively, the load torque. Assuming that the machine is balanced, the Concordia transform matrix K23 (resp. Kn) allows to change the reference frame from (abc) (resp. (ab» to (af3) (or inversely). The measurements are those usually used for control purposes: stator currents (i.,,,, l,b) and OJ. Thus, y = [Ye w( Sensor faults (/J = [Ji\{/ flxbf,fw) are modelled by additive inputs. A classical vectorial control is used to control the torque and an outer loop controls the speed . A very simple model of the faultfree Pulse Width Modulated (PWM) inverter (Fig. I) is a unit gain : this means that the stator voltages calculated by the control, V,,,, V,b, V,,, are those actually applied to the induction motor, V ,lIm",,,,, V,h"""'''' V.c"IO'''''. In order to deal with open-circuit faults in the power switches, the model of the PWM inverter is that of Fig. 2. fv = [fvstlfV.,bfv",f represents the actuator faults in (6).
input of the electrical equations (6), an input-output sub-model eliminating r can be built. After discretisation, it can be expressed with a discrete Linear Time Varying (LTV) transfer G such that: G(z,W- fw).(u+ fv )-(Ye - f,) =0 ~
r = Ye -G(z,W).u ~
.\(1
v..1/,
--J: -J::
Faulty switch (open-circuit) S(1+
vs " V II/fl10r '\"h
-J;: v.. VH
V.I.e
"'Ofor
.H
5,,' 5,: 5,; 5c +
S,'
r = -h(z,u,y,O)
Fig. 3 shows the residual generation scheme. A usual approach to discretize the electrical equations (6) consists in developing the corresponding matrix exponential until order two. The on-line calculation of r is thus implemented as follows:
Fig. I. Structure of the PWM inverter
V.WI II/%r
(12)
Fig. 3 : Residual generation
E : DC .HIpp/.\' "a/Wile (output of a three-phase rectifier)
v.'"
h(z,u,y,j)=O
A residual can be calculated in order to check whether this subsystem (consisting of the electrical equations) is faulty or not. Such a residual IS computed from the available information (u. y):
c
V
(11)
Fault model
Xe .k + 1 = Ad k .xe.k + Bd k .u k
iv.", < 0 iv.", > 0 iv.,-!J
(13)
r = Ye -C,Xe.k Ad k
= I + A(wk ).T, +~(A(Wk ).T, f Bd.
Fig. 2. Model of open-circuit faults in the power switches
=T,{/+~A(W.).T,}B
r = [rl m rlshf represents the residuals consistent with the a and b stator currents. (13) can be interpreted as a particular reduced order unknown input observer. It performs an elimination of Xm and r and an estimation of x,. As r satisfies the detection requirements, a detection indicator can be directly derived from r. Its directional properties are now studied in order to isolate the faults.
In the following, 1;, refers to variations of the electrical parameters modelling motor faults (broken bars, etc ... ). The link between the application and the problem general formulation (I) is given by (10).
4.2. Residual directional properties
4. DIAGNOSIS OF CURRENT SENSOR FAULTS AND PWM INVERTER FAULTS
The influence of faults on the residuals can be assessed from the model using a first order development. This approximation is only valid for small fault amplitudes unless the residual is a linear function of the faults .
4.1. Residual Generation
The residual generation has been studied in order to fulfil the following two requirements: the method does not depend on the type of control and, above all, the residuals have to be perfectly decoupled from the load torque variations (as the load torque is a priori unknown). A systematic approach for the application of model partitioning to fault diagnosis can be found in (Combastel, et aI., 1999). Here, a simple physical reasoning gives the same result: by using (J) as an
(14)
~
r ==
dh(z, u, y, j) df
1103
·f = ·~)j(z.u,Y).fj {=O
J"
is not mentioned in section 4 to simplify the notations. The wmments about fw in this section are also applicable to J" . The incidence matrix derived from (14) has the following form :
r ~ [O *
(15)
*1** ** *1*]·[/'] * * Iv
0
Iw
It can be noticed that a decision based on a Boolean signature table only allows to isolate the current sensor faults (15). It is thus necessary to take more information into account about the residual behaviours for fault isolation to become possible. This is why attention is paid to the residual directional properties: the relative order of magnitude of the residual vector elements is then taken into account by the decision procedure. In other words, different residual sensitivities in response to different faults become sufficient to perform fault isolation. The residual directional properties can be deduced from ( 14). The column vectors related to fr (D, = [I O]T and D2 = [0 I]T) are constant and are distinct. So, when a single current sensor fault occurs, r has a fixed direction which is characteristic of the occurrence of this fault. The column vector related to f,y, Df>, can be calculated at each sample time k from the filtering of u with the discrete transfer JG(z,w-fw)/Jfw- Do,k can be interpreted as a time varying direction for r which is characteristic of the occurrence offw- The column vectors related to fv, Dj to D" cannot be directly calculated in time domain: the direction of r actually depends on the temporal evolution of f". As faults are a priori unknown, the corresponding directions of r cannot be computed without a supplementary assumption: the dynamic response of the residual to fv is thus assumed to be close to the steady state response:
G(z , w)·fv "" G(l,w)·fv
sinusoids, r becomes a non zero rotating vector when fw occurs. In this situation, the direction of r is periodically the same as the direction of a current sensor fault (for instance). Consequently, the decision at time k has to take past information into account. The example of the speed sensor fault shows that the isolation ability depends on the system excitation when directions are not fixed: a lack of excitation can lead to the same directions for two faults, whereas enough excitation may result in different residual directions allowing to distinguish between the same two faults. 4.3. Fault Diagnosis (Detection, Isolation, Identification)
A fault is detected if r becomes different from zero i.e. if F(q) .lrkl exceeds a (fixed or fuzzy) threshold. F(q) stands for a discrete time filter that can be used to reduce the influence of noise. (18) J1z(.) stands for a function modelling the membership
to Z (zero). D(j) is a detection indicator represented by a fuzzy set. With the same notations, D(f;) refers to a fuzzy set the membership to which represents the decision about a possible occurrence of fi. Under the single fault hypothesis, the membership of the residuals to D(f;) is obtained from a three step reasoning: a) When fi occurs, a fault is detected by the detection indicator (18). b) The angle (denoted by L. ) between the direction related to fi, Dj, and the residual vector r is close to zero at time k. c) As the information at time k is not sufficient to obtain a relevant decision, a filtering based on a forgetting factor A is introduced; the purpose of such a filtering is to reduce the influence of noise and to discriminate between two faults that may have the same direction at time k but not over a larger period of time. The three step reasoning results in the following expression for the decision related to fi :
(16)
In practice, if the variations of fv are slow compared to the dynamic of G(z,w), (16) remains true. Moreover, if Iv is a step fault, the response to fv is constant after a transient depending on the dynamic of G(z,w). In the case of the induction motor, this is not restrictive as G(z,w) models the fast electrical dynamics. The steady-state gain of G leads to fixed residual directions in response to fv.
'" [: :1XR.I', -,' -:]1 JG('a;~- J.' 'l[~l ~ D =LD r k ""
k ·fk
mj,k+1 =A.m j.k +(l-A) ·J1z(Di,kLrk) J10(f,)(rk
)
(19)
= J10(J)(rk ) /\ mi,k
A definition of the angle (L.) either taking vector orientation into account or not allows either to isolate faults that have different signs or not. This is used to isolate PWM inverter faults (Fig. 2). The identification of fault amplitudes under the assumption of single faults results from a least square criterion :
(17)
J.k .f;.k
8 i .k T
.T k
(20)
8 i.k T .8 i.k
The directions of the current sensor faults and the directions of the PWM inverter faults are fixed and are distinct (17). Isolation of these faults is thus possible under the single fault hypothesis . Moreover, as the stator voltages are balanced three-phase
To improve the identification of fi,b the estimate can be filtered or/and the criterion can be defined over a temporal window. It is focused on the fault indicator related to an open-circuit fault (OCF) of S,,' to
1104
illustrate the application to the induction drive (Fig.
rejection specifications. The adaptation of f w IS hased on an on-line gradient algorithm minimising the criterion J=pTp:
4).
····· ·-·--···----·1
•
axAr
( il/w
j_ aA(w-jw)
A'
e
+
A(
_fA
W
ap
Iw
= ajw = -
)(axe A
W·
ilfw
o. Ca) Residual responses to an OCF of Sa'
A X
a/w
j
(22)
C aXe
K 22 '
.
ajw
The residual generator resulting from (21) and (22) is discretised by developing the corresponding matrix exponential until order 2. (23) shows the adaptation of j w : it is based on the residual directional
( -
properties of p i.e. on the gradient of p. A
T
A
fw.k+1 = fw,k -TJ·8 i""k ·Pk (b)
11 DU v", >0) (rk
(c) I1D(fvm>O)(rk )
)
when an OCF of SIl' occurs.
Let us interpret the adaptation of j w in order to
whenfw occurs.
precise the diagnosis scheme. p=o means that the adaptive parallel simulation fits the system real behaviour (known through the available measurements). Assuming the electrical model is correct, if fw occurs, a value of j w exists such that
Fig. 4. Fault indicator for an open-circuit fault of Sa . As illustrated by f.1 DU"", >0) (rk
)
(23)
in Fig. 4, f.1 DU,) (rk )
is decoupled from x and d due to the properties of r. By focusing on a particular direction for each fault (i.e. by eliminating the other directions), an approximate decoupling of jj"i is achieved by f.1 DU, ) (fk ) .
When the adaptation converges, the minimisation of J=pTp leads to J=O, and j w is correctly estimated. If some faults do not act on the motor as fw does, the algorithm minimises J through the adaptation of j w ' but does not manage to null J
p=0.
5. DIAGNOSIS OF SPEED SENSOR FAULTS
(or p). Therefore, p is decoupled from fw apart from the noise, the model errors and the transients before the estimation convergence. The proposed method is thus more suited to diagnose slow developing faults. However, when jj;:;=O, the two elements of vector p become very noisy sinusoids: a low-pass filtering of p or IIpll is inefficient to detect slight changes.
The direct application to the induction drive of the method proposed in section 4 shows that the diagnosis (isolation, identification) of the speed sensor fault becomes less accurate as the fault amplitude increases. This is due to the first order development involved in (14). Therefore, results of section 4 are used to diagnose fi and fv. In order to discriminate fw from the other faults (including 1;,), a second set of residuals, p, has been derived from the motor electrical equations. As explained in section 4. I, P is decoupled from x and d:
(24)
p/, and p/ are two residuals resulting from scalar
jw)'x, +B.u
;£0
= A(w-
p
= Kn .(y,
(21 )
products. The orthogonal direction of the gradient is unique as p E ~2. p/, represents the component of p that can be locally interpreted by a variation of fur The adaptation of f w ma k es Pk" cI ose to zero. Pk ~
-C.x,)
•
The closer
jw
is to f wo the more decoupled is p from
A
represents the component of p that cannot be locally interpreted by a variation of fw . p/, which is not a sinusoid when 1;,:;=0, is thus suitable to discriminate betweenfwand1;,.
fw. This is an example of indirect decoupling. (21) is an adaptive residual generator with a single parameter tracking. This alternative allows to deal with slow parameter changes and to by-pass the problem of persistent process excitation (Hofling and Isermann , 1996). As voltages are sinusoids (apart from the influence of the Pulse Width Modulation in high frequencies), the natural excitation of u is a priori sufficient to adapt the single parameter j w . This is important to warrant that the on-line FDI scheme does not disturb the control or harmonic
JID(fw) =)lz(F(q).p;)/\(l-JIZ(jW.k» JI DU p) = (I - )l z (F (q). p ;
»/\ (I - )l D([ I1 Iv )) )
(25)
(26)
The fault indicators related to fw and jj, are given by (25) and (26) where F(q) stands for a low-pass filter
1105
rcducing thc influence of nOise and
J1 D([!!
sensor faults are identified. The identification of an open-circuit inverter fault has no real sense: such a fault is entirely characterised by thc faulty switch that is determined through the orientation of the directions used for fault isolation. The case of shortcircuit inverter faults has not been considered here: in practice, hardware protection makes the global system inactivc when such a fault occur. As the direction deduced from the residual derivatives at forO (see (14» is not sufficient to diagnose fw> the adaptation of j ill allows to fit more precisely the system behaviour. The directional properties of the second residual vector are then given by its gradient calculated at j ill (and not at forO as before). The gradient is then used to adjust a unique parameter: this allows to relax the excitation requirements. In spite of the computation burden, the isolation and the identification of the speed sensor fault have been achieved. Moreover, the motor faults (modelled by parameter variations) are detected and isolated from all the sensor and the actuator faults.
f,,])
indicates the occurrence of a current scnsor or an actuator fault. The application of the adaptive residual generator to the induction drive is now dealt with. The induction drive and the FDI system are simulated during I Os. The sampling time is 0.2ms. A 1200rpm speed reference step is applied at t=0.2s. The load torque is a ramp starting at t=0.2s and finishing at t=2s with an amplitude of 20Nm. In Fig. 5, RIO corresponds to 10% ramp variations of both R, and R, from their nominal value. L5 corresponds to 5% ramp variations of both L, and Lr • All parameter ramp variations start at t=2s and finish at t=7s. SJO corresponds to a speed sensor fault modelled by a ramp starting at t=5s and finishing at t=6s. The amplitude of the fault is 10% of the nominal speed (2000rpm): 12.6 rad/s.
j (rad/s)
F(q).p/
ill
510
lOa --------.,-.---.--'5~ . 5 - - - - - - - - - -, - .- - - -- -.
o
.---------.
-\
~
10
'5~
10 - - - - -- --- - :-- ----- --
510, RIO
5 - -- - - - - -- -, ---- - -- - -
o. -5
L5
•
-5
o
+
.
-
-
-
-
-
~-
-
-
-
-
.
-
-
-
-
10
Chen, I ., R. I . Patton and H.Y. Zhang (1995). Design of Robust Structured and Directional Residuals for Fault Isolation via Unknown Input Observers. Proceedings of 3rd European Control Conference, pp348-353 , Rome, Italy. Combastel, c., S. Petropol, S. Gentil and S. Lesecq (2000). Model-based and Wavelet Approaches to On-line Fault Detection: Application to an Induction Drive, IFACIIFlPIIEEE MCPL '2000, 2nd Conference on Management and Control of Production and Logistics, Grenoble, France. Combastel, c., S. Gentil and I.P. Rognon (1999). Fault Detection and Isolation using Local Models, Comparison with Unknown Input Observers. ECC'99, European Control Conference, Karlsruhe, Germany. Hofting T. and R. Isermann (1996). Fault Detection based on Adaptive Parity Equations and Single Parameter Tracking. Control Eng. Practice, Vol. 4, No . 10, pp.1361-1369. Isermann, R. (1997). Supervision, fault-detection and fault-diagnosis methods - An introduction, Control Eng. Practice, 5(5), pp. 639-652. Krishnaswami V. and G. Rizzoni (1994). Nonlinear Parity Equation Residual Generation for Fault Detection and Isolation. IFAC 5afeprocess '94, pp.317-322, Espoo, Finland. Thomson, W.T. (1999). A review of on-line condition monitoring techniques for three-phase squirrel-cage induction motors - Past, present and future . Proceedings of IEEE SDEMPED '99, pp318, Gij6n, Spain. Zhang, Q., M. Basseville and A. Benveniste (1998). Fault Detection and Isolation in Nonlinear Dynamic Systems: A Combined Input-Output and Local Approach . Automatica, Vol. 34, No. 11 , pp.1359-1373.
•
5
10
:Ee --- ------~- --- --- - ~------ - ---
·1
5
REFERENCES
- --- ---- -
o
o
- ------- -
10
~§----- ---i---·--- ---
10
'5~ o
;
~
-\
·1
5
10 - - -. - - -- - - -; - - - - - - - - --
5 . --
1 ----.- . . -- , .-- - -----:
o
---- ------
o
: 2§
0
.
0510
Fig. 5. Application of the adaptive residual generation to the induction drive Fig. 5 shows that j ill is correctly estimated when fw occurs (case SIO). p/ becomes non zero only when parameter deviations occur (RIO or L5) . Small inductance variations can be easily detected with p/ (case L5) . It can be also noticed that the diagnosis of fw is not very sensitive to resistance variations (case S 10, RIO). Thi s is important to ensure that the diagnosis is robust to resistance variations due to thermal constraints : such constraints exist even in the case of a fault-free behaviour and can lead to important resistance variations (more than 50% for R,). A fault detection which is robust to the important (but very slow) resistance variations is out of the scope of this article but it is dealt with in (Combastel , et aI., 2000) . It is also based on a single parameter tracking. Assuming that resistances are rather wellknown at the detection time, fault diagnosis can be carried out as mentioned in this paper. 6. CONCLUSION Due to the availability of a mathematical model, analytical redundancy can be applied to the induction drive. Current sensor faults and open-circuit PWM inverter faults are detected and isolated using the (distinct) directions of the residual vector. Current
[106