Mixed Approach to Fault Diagnosis in Linear Systems

Copyright <0 IFAC Fault Detection, Supervision and Safety for Technical Processes, Kingston Upon Hull, UK, 1997

MIXED APPROACH TO FAULT DIAGNOSIS IN LINEAR SYSTEMS E. Alcorta Garcia,

P. M. Frank,

Department of Measurement and Control, Gerhard- M ercator- University- G H- Duisburg Bismarckstr. 81 BB, D-47048 Duisburg, Germany lax +49(203) 3792928 e-mail: ealcorta@uni-duisburg. de

Abstract: In this paper an approach to the detection and isolation of faults in linear time invariant systems is presented. Based on state and parameter estimation two residual generators are defined. The first is an observer-based residual which is used to the detection of faults and it is given in a parametrised form . The second one is based on parameter estimation and it is used to solve the fault isolation problem. The estimator take advantage of the structure of the parametrised residual. As a result, advantages of both approaches could be added. With the joint utilization of parameter estimation and observers to the generation of residuals, the existing fault diagnosis approaches can be improved. Copyright © 1998 IFAC Keywords: Linear systems, fault diagnosis , parameter estimation, parametrised residual .

the observer based are rather similar as pointed out in (Gertler 1991 , Magni and Mouyon 1994, Patton and Chen 1994, Wiinnenberg 1990) .

1. INTRODUCTION

Automatic control systems are susceptible to faults (due to their physical nature) that cause an unacceptable deterioration of the performance or even lead to dangerous situations. The need to avoid these dangerous situations and to more reliable systems motive the developing of fault diagnosis methods . A large variety of techniques to fault detection and isolation of faults have been proposed in the last decades as can be seen, for example, in the survey papers (Frank 1990, Gertler 1991, Isermann 1984, Patton 1994, Willsky 1976) . The most of the approaches can be classified into one of the three groups: i) Parity space, ii) Observer based and iii) Parameter identification based.

The utilization of observer-based residuals allows the detection of faults even if they are time variant and/or the system is not persistently excited (evidently if the input-output measurements are zero no fault can be detected) . The fault isolation is a very difficult task because the coupling between the parameters . The parameter estimation-based approach allows the isolation of all the possible faults (changes in the mathematical parameters) in a system even if their occurrence is simultaneous. In order to achieve the fault isolation via parameter estimation approach two conditions have to be satisfied: the system must be persistently excited and the faults must be constant or slow variable.

The approaches in each group have been developed independently, however , the parity space and

133

o ...

A desired characteristic of in a residual generator is that it contains the detectability of the observerbased residual generator approaches and the isolability of the parameter estimation-based methods.

o

In a pioneer work (Delmaire et al. 1994) establish a relationship between a parity space (as in (Chow and Willsky 1984)) and parameter estimation for SISO systems. Using a generalised parity space (as defined in (Gertler and Singer 1990)) in (Gertler 1995) the MIMO case is considered as well as some ideas about the joint use of both approaches. Different to the above approaches in (Alcorta-Garda and Frank 1996) a relationship between parameter estimation and observer-based approaches to residual generation in linear timeinvariant MIMO systems was established. The relationship is given in a form of parametrised residual , i.e. an observer-based residual which is formed as the multiplication of a parameter estimationbased residual and a regressor .

The elements gi and bi with i = 1, ... , n of 9 and b represent the actual mathematical parameters of the system (1) . It is assumed that the fault will change the nominal value (gi and bi) of the actual mathematical parameters (gi and bi) . In order to detect the faults consider the following fault detection observer

i(t) = (Aa

r(t) = cw(t)p(t)

where gT = [gl . .. gn ], bT [0 ... 0 1] and

pT(t) =

[gi -g1(t)

... g~ -gn(t)

bi - b1(t) ... b~ - bn(t) J wet) = [6 ... ~n (1 ... (n J

(4)

+ gaC)~i(t) + eiy(t) (i(t) = (Aa + gac)(i(t) + eiu(t)

(6)

(5)

and

(i(t) = (Aa

(7)

where i = 1" " , n . From the particular form of the matrix Aa + goc follows that

(sI - A*)-1ei = Ti(sI -

Ar + cT gD-1 e1 (8)

where A* = Aa + gac and the Ti's are constant matrices whose elements are the coefficients of the numerator polynomials of (sI -Aa +gac)-1ei and i = 1, ... , n. Therefore equations (6)-(7) can be expressed as

+ gacf vet) + e1y(t) J(t) = (Aa + gacf t9(t) + e1 u(t) vet) = (Aa ~i

= T;v

(i

= Tit9

(9) (10)

i = 1, ... , n

A block diagram of the parametrised residual can be seen in figure 1. Note that the parametrised residual is in a well known linear regression form (Johansson 1993) . As can be observed from equation (3) and the definition of the vector p (equation (4)) , the observerbased residual is a time-variant map of the difference between actual and the nominal parameters.

(1) c

(3)

where

Consider the linear time invariant SISO system in observable canonical form (Narendra and Annaswamy 1989) :

=cx(t)

(2)

The parameterised residual in continuous time is given by

2. PARAMETRISED RESIDUAL

yet)

- ga)y(t)

where g*, b* represent the nominal parameters. If no faults are present the nominal and the actual parameters are same.

The paper is organized as follows : In section 2 the parametrised residual is reviewed. The proposed approach is considered in section 3. Section 4 shows some simulations . In section 5 some conclusions are given.

= (Aa + gc)x(t) + bu(t)

+ gac)x(t) + b*u(t) + (g*

r(t) = cx(t) - yet)

The purpose of this paper is to consider the joint use of parameter estimation and observers to residual generation. For sake of simplicity only the SIsb case is considered in this work. The proposed approach is based on the parametrised residual introduced in (Alcorta-Garda and Frank 1996) and some well known results from the adaptive control theory (Narendra and Annaswamy 1989) . An observer-based residual is utilized for the detection task . A second residual based on the parametrised version of the first one is defined. It has the form of a estimator and it will be utilized to solve the fault isolation task. The proposed schema is based on the cooperative application of parameter estimation and observerbased approaches. The approach is exemplified using simulations.

x(t)

1

Aa =

=

134

r

b

9

b*

Fig. 1. Structure of the parametrised residual

(A" = Ao

+ goe)

The regression form of the parametrised residual suggest the further processing of it by a parameter estimation-like method. This is considered in the next section .

Fig. 2. Structure of the Mixed approach to residual generation 3.1 Convergence Analysis

First, we present a well-known result from the adaptive control literature which will be invoked to made conclusions about boundedness of related functions . See e.g. (Narendra and Annaswamy 1989) for details.

3. MIXED APPROACH TO RESIDUAL GENERATION Following the structure of the parametrized residual (3) and (9)-(11), an algorithm for the estimation of p in (3) is considered. Actually, the estimation algorithm is a modification of an estimator used in the adaptive control of robots manipulators (Tang and Arteaga 1994). The modification of the algorithm allows the application to linear systems. A characteristic of this identifier is that the vector p can be even if the excitation is not persistent (but sufficient (Tang and Arteaga 1994)) .

lR+ _ lRn " continuous and differentiable functions. Define V : lRn ., x lRny _ lR as:

Lemma 1. Let x : lR+

get) Z(t)

vex,

(11)

= -AZ(t) + 12
Z(O)

(12)

=0

y)

~ -( x T

yT)

(~ ~)

(;)

(15)

where Q is a symmetric positive definite n" x nr matrix, then

= -[.AI + 6Z(t)]g(t) + 12
y :

where P is a symmetric positive definite (n" + ny) x (n" + ny) matrix. If the time derivative of Vex , y) is given by:

The propose adaptive parametrised residual is given by :

pet) = -6g(t) - 11
_ lRn ., and

(1) x E Lr;,; (2) y E L':;f (3) x E L~"

(13)

= eW(t)p, c(t) = r - r , 0 and A > O. The evolution of the vector p gives us information about the fault

where r(t)

where L m is the space of bounded m-vector functions and L';' is the space of m-vector square integrable functions .

11 , 12,6

occurrence .

If in addition , :i: E L~", then limt_oo x(t)

The above identifier (11 )-( 13) is a modified version of one proposed in (Tang and Arteaga 1994) for the adaptive control of robots manipulators . This algorithm gives a smoother parameter error (c(t)) and parameter convergence under a weaker excitation condition . These characteristics are also desired in fault diagnosis. As pointed out in (Tang and Arteaga 1994) the weaker excitation condition can be satisfied in a transitory time (or when a fault occur) . A schematic of the mixed approach can be found in figure 2.

=0 000

Lemma 2 . Given

Proof. From equation (13) Z(t) can be obtained as :

J t

Z(t)

= 12

e-A(t-T)cpT(r)
o applying the norm 11

135

* 1100

to Z (t) result

(16)

J t

With the condition (19) and the fact that the term gT(t)G(t) > 0 results

e- A(t-T)II
IIZ(t)lloo :5

o

V(t) :5 - 60' pT (t)p(t)

L;::

Z(t) EL;::

With the help of Lemma 1, we conclude the points 1. and 2. of the theorem. Now equation (23) can be rewritten as

(18)

V(t) :5 -KV(t)

Q.E .D. ator (11)-(13) satisfy the following: (1) p E L~n n L;:: (2) g(t) E L~n n L~ (3) If there exists a tl, 0' > 0 such that

Remark 1. The condition (19) is called in (Tang and Arteaga 1994) Sufficient Excitation (SE) and it is a weaker condition than the well known Persistently Excitation (PE) condition.

J t

~ 12

(24)

with K = 20'6 . It follows from (24) that under condition (19) the estimation error p( t) will converge exponentially to zero with a rate at least as fast as K. Q.E.D.

Theorem 1. The estimator-based residual gener-

Z(t)

(23)

/1

because
e-A(t-T)
o

2:: 0'/ Vt 2:: tl

(19)

3.2 Advantages of the mixed approach

than p -> 0 exponentially for t 2:: tl Proof First, note that since g(t) from (11), (12) and (16) that eAtg(t)

=
The novelty of the proposed schema result from the following characteristics: (1) Explicit use of the structure of the parametrized residual. This allows the obtention of an observer-based residual as an intermediate step in the estimation of p. The combined use of parameter estimation and observerbased methods is realised in an interesting way because the structure of both approaches is utilized. (2) The considered estimation algorithm will estimate the elements of the vector p under weak excitation conditions , however if no excitation is present the parameters are not updated. In the presence of faults, the changes in the signals will produce enough information for the update of p. No divergence of the estimation (under reasonable perturbations) is guaranteed. This is not necessary the case with others estimators (Anderson 1985).

+ ).eAtg(t) =/2e At
J t

12

eAT
o and this can be rewritten as

Since the initial conditions of Z(t) and g(t) are zero, from this last equation it follows that

J t

g(t) = 12

e-A(t-r)
o

= Z(t)p(t)

(20)

Now define a Lyapunov-like function

1 V(t) = -2 pT (t)p(t)

4. SIMULATION RESULTS

(21)

In this section, a numerical example is used to shown the applicability of the proposed approach .

/1

Its time derivative along (11) is given by

Consider the following model of a DC motor coupled through a gear system to a symmetric disc as a load . Assuming a separately excited DC motor with viscous friction and neglecting the armature inductance, a classic description of the system is given by the following state equation (Kuo 1987) in observable canonical form

where (20) has been used to obtain the last equation .

136

d [X1(t)] _ dt X2(t) -

[01 -~0 1[X1(t)] [f{ 1 X2(t) + 0 vet)

y(t)=[O

1]

Chow, E. Y. and A. S. Willsky (1984) . Analytical redundancy and the design of robust failure detection systems. IEEE Trans. on Autom. Control AC-29(7), 603-614 . Delmaire, G ., J. P. Cassar and M. Staroswiecki (1994) . Identification and parity space techniques for failure detection in siso systems including modellins errors .. In : Proc. of the 33rd. Conf. on Decision and Control. Vol. Florida, USA . pp . 2279-2285 . Frank , P. M. (1990) . Fault diagnosis in dynamic systems using analytical and knowledgebased redundancy - a survey and some new results. Automatica 26, 459-474. Gertler, J . (1991) . Analytical redundancy methods in fault detection and diagnosis. In : IFAC/IMACS Symp . SA FEPRO CESS, Baden-Baden, Germany. pp . 9-21. Gertler, J . (1995) . Diagnosing parametric faults : from parameter estimation to parity relations. In : American Control Conference. Seatie, Washington . pp . 1615-1620. Gertler, J . and D. Singer (1990) . A new structural framework for parity equation-based failure detection and isolation . Automatica 26(2) , 381-388 . Isermann, R . (1984) . Process fault detection based on modeling and estimation methods-a survey. Automatica 20, 387-404 . Johansson , R. (1993) . System Modelling Identification. Englewood Cliffs, NJ : Prentice Hall. Kuo , B. C . (1987). Automatic Control Systems. Englewood Cliffs, NJ : Prentice Hall. Magni, J-F . and Ph . Mouyon (1994) . On the residual generation by observer and parity space approaches. IEEE Trans . on Automa. Control 39(2) , 441-447 . Narendra, K. S. and A . Annaswamy (1989) . Stable Adaptive Systems . Englewood Cliffs, NJ : Prentice Hall. Patton, R. J . (1994) . Robust model-based fault diagnosis: the state of the art . In : Proc. IFAC Symp . SAFEPROCESS '94, Espoo Finland. pp . 1-24. Patton , R. J. and J. Chen (1994). A review of parity space approaches to fault diagnosis for aerospace systems. founal of Guidance, Control 8 Dynamics 17(2) , 278-285 . Tang, Y. and M. A. Arteaga (1994) . Adaptive control of robot manipulators based on passivity. IEEE Trans . Automatic Control 39(9) , 18711875 . Willsky, A. S. (1976) . A survey of design methods for failure detection in dynamic systems. Automatica 12, 601-611 . Wiinnenberg , J . (1990) . Observer-Based Fault Dynamic Systems. VDIDetection in Fortschrittsber ., VDI-Verlag, Reihe 8, Nr. 222 . Diisseldorf, Germany.

[X1(t)] X2(t)

where vet) is the armature voltage, f{ ~ 7.50470 and T ~ 0.0483 are the nominal parameters of the system . For sake of simplicity we assume that the changes in the system parameters occur independently, even if this does not represent the reality, in order to test the proposed algorithm. A state feedback is used to have closed loop poles in -4 and -3 . The reference signal is assigned to be a pulse generator. The following parameters were used for the estimator 6 400, 50 , ,2 15 and A = 0.15.

=

,1 =

=

Two fault has been simulated . The first is 30% changes in T. The estimator residual as well as the observer based residual can be found in figures 3 and 4 respectively. The second fault is a 10% changes on K. the results can be found in figures 5 and 6. As can be seen from the figures 4 and 6, the observer based residual detect first the faults . Some seconds later, the fault can be isolated (figures 3 and 5) .

5. CONCLUSIONS

A mixed approach to residual generation has been presented. Fault detection is realised via an observer-based residual in parametrised form . The approach take advantage of the structure of the parametrised residual to use a parameter estimation-like approach in order to achieve fault isolation . The estimator considered is a modified in such a way that only a weak excitation condition is required . However, the convergence rate depends on the excitation. The combined used of parameter and state estimation based approaches could be useful to improve the detection and isolation of faults Further work include the test of the proposed algorithm in a physical system.

6. REFERENCES Alcorta-Garcia, E . and P. M. Frank (1996) . On the relationship between observer and parameter identification-based approaches to fault detection . In : IFAC- World Congres 1996. Vol. N. pp . 25- 29 . Anderson , B. O . (1985). Adaptive systems, lack of persistency of excitation and bursting phenomena. Automatica 21(3) , 247-258.

137

10

o

~

-10 -20 -30

40 30

~Pz ~

o

5

10

y

20

~

-40

-50

50

-

PI

15

10

/'

o

'"

-10

20

o

5

time

15

10

20

time

Figure 3. Estimator residual, fault I

0.4

Pz

Figure 5. Estimator residual, fault 2

,..-----r----r----..-----,

0.5 ,...------..---"""T""----r------, r

r

0.2 t-----I-II--f\---I\I-of\---.".-+ofr---4\r---4Ii

-0.5 '--_ _--'::-_ _ _~-----'----:-'. o 5 10 15 20

-0.40.:----5-=----1-:':0:----~15;:-----:::'20· time

time

Figure 4. Observer based residual, fault I

Figure 6. Observer based residual, fault 2

138