FAULT DETECTION AND IDENTIFICATION BASED ON FULLY-DECOUPLED PARITY EQUATION

C. W. Chan1, Hua Song2, and Hong-Yue Zhang2

1 The University of Hong Kong, Hong Kong, China, Email: mechan@hku.hk, Fax: (852)28597906

2 Beijing University of Aeronautics and Astronautics, Beijing, China Keywords: Parity equation, fault detection and identification, recursive least squares method

Abstract A multiple fault detection and identification method based on the fully-decoupled parity equation for dynamic systems with known linear and unknown non-linear terms is presented. It is shown that the residuals generated from the fully-decoupled parity equations derived here are sensitive only to specific actuator or sensor faults, but are decoupled from other faults and the unknown nonlinear term. The conditions for the existence of these equations are also given. From the residuals generated from the fully-decoupled parity equation, faults are estimated using the recursive least squares method. The application of the proposed method to detect, isolate and identify faults in a simulated DC motor is also presented.

1 Introduction Fault detection, isolation and identification are now being integrated into practical control systems to improve the safety and reliability of these systems. Parity equation is a popular approach to detect faults. First, residuals are generated from the parity equations. The residuals are large, if a fault occurs; but small otherwise. Further, if the residuals are sensitive only to a specific fault, then it can be isolated and identified. Parity equations for systems with modeling errors arising from changes in the operating point are derived in [1], and optimal robust parity equations maximally sensitive to faults and minimally sensitive to modelling errors are given in [4], which are further refined in [9]. The optimal parity vector method for isolating faults is discussed in [3], but the identification of faults was not discussed.

A popular fault diagnosis technique for systems with unknown nonlinearity is the unknown input observer (UIO) method involving discounting first the unknown nonlinearity using its estimate [6]. Observers are proposed to decouple the effect of the unknown inputs [2]. However, fault identification has not been considered in these works. In this paper, a new approach not only to detect and isolate, but also to identify faults for systems with know linear and unknown nonlinear terms is presented. A fully-decoupled parity equation is derived for generating residuals sensitive only to specific sensor or actuator faults, but decoupled from the system state, the unknown nonlinearity and other faults. Faults can now be identified

from the residuals by the recursive least squares method. The performance of the proposed method is illustrated using a simulated DC motor with a shunt field circuit. From the simulation results, it is shown that the proposed method is able to identify, isolate and identify the faults.

2 Fully-decoupled equation Consider the following discrete nonlinear system,

+=++=+

),,()()(),,()()()1(

kuxGkCxkykuxFwkBukAxkx

ψ (1)

where, x(k)∈ Rn is the state, u(k)∈ Rp is the input, y(k)∈ Rq is the sensor output, w(x, u, k) ∈ Rr, ψ(x, u, k) ∈ Rr1 are the unknown nonlinear dynamic of the system, A, B, C, F and G are known matrices with appropriate dimensions. Define w(k) = w(x, u, k) and ψ(k) = ψ (x, u, k). For s > 0, the measurement equation of (1) is given by [3],

)()()()()( 0 kHkWHkUHskxHkY wc Ψ+++−= ψ (2) where Y(k) is the sensor output under normal conditions, s is the order of the parity space, Y(k)T = [yT(k–s) K yT(k)]T, U(k) = [uT(k–s), K uT(k)]T, W(k) = wT(k–s ) K wT(k)]T, Ψ(k) = [ψT(k–s) K ψT(k)]T, and H0, Hc, Hw and Hψ are:

,

=

−− 0

000

00

0

21 CBCABCA

CB

BCA

CABCB

H

ss

c L

L

L

L

M

=

−− 0

000

00

0

21 CFCAFFCA

CF

FCA

CAFCF

H

ss

w L

L

L

L

M

,

.;0

=

=

G

GG

H

CA

CAC

H

sMM

ψ

Definition 1：The parity space V is defined as [6, 8], }0|{ 0 == HvvV T (3)

where ν∈ Rp is the parity vector, H0 and s are given in (2).

Definition 2：The parity equation at time k is, [ )()()( kUHkZvkr cc

T −= ] (4) where r(k) is the residual, which may contain the fault. Uc(k) = [uc(k – s) K uc(k)]T，and Z(k) = [Z(k – s) K Z(k)]T. Uc(k) is the normal input to the actuator, and Z(k)

Control 2004, University of Bath, UK, September 2004 ID-135

is the sensor output that may contain faults. Under normal operating condition, Uc(k) = U(k) and Z(k) = Y(k). A fully decoupled parity equation is derived here for isolating and identifying faults, such that the residuals generated by (4) are sensitive only to specific faults.

=

−− 0

0

0

0000

00

***2

*

*1

*

*

M

M

L

L

M

CBCABBCA

CB

BCA

CABCB

H

ss

ci

(9)

where B* is B with the ith column corresponding to the ith actuator being removed. Since the parity vector sensitive to the ith actuator must satisfy:

Lemma 1：The residuals r(k) generated by the fully-decoupled parity equation (4) can be used to detect, isolate and identify fault, if there exists a fully-decoupled parity vector v, such that r(k) is: 0][ 0 =ciw

Ti HHHHv ψ (10)

hence the parity equation constructed by the parity vector that satisfies (10) is referred as the fully-decoupled parity equation. The condition for the existence of nonzero solutions of (10) is given in Theorem 1.

(1) decoupled from the system state, (2) decoupled from the unknown nonlinear part W(k) and

Ψ(k), which may contain the unknown inputs and uncertainties,

Theorem 1 Let [H0 Hw Hψ Hci] consist of nx independent columns, the necessary and sufficient condition for (10) to have nonzero solutions is:

(3) sensitive only to specific actuator faults or sensor faults, but decoupled from each other.

Proof: Condition (1) can be obtained from (1). For a system operating normally, r(k) obtained from (2) is, 1−>

qns x (11)

)]()()([

)()([)(

0 kHkWHskxHv

kUHkYvkr

wT

ccT

Ψ++−=

−=

ψ (5) and the sufficient condition is:

1)(,1)(1

1 −+>−+−+

+> prqprq

rns (12) From (3), r(k) is decoupled from the system state, when the system is normal. If the unknown nonlinear terms W(k) and Ψ(k) are identically zero, r(k) is zero only if the system is operating normally. If vT[HwW(k) + HψΨ(k)] is identically zero, then condition (2) follows, and hence r(k) is decoupled from W(k) and Ψ(k). If conditions (1) and (2) are satisfied, r(k) is non-zero only if the system is faulty. Therefore, faults can be detected from r(k), as follows,

where n is the dimension of the system state, q and p are the number of sensors and actuators, r and r1 are the dimension of the unknown nonlinear terms in (1).

<≥

normalfaulty

rr

krh

h)( (6)

Proof: The left null space and hence the nonzero solution of (10) exists, if and only if the number of independent rows of [H0 Hw Hψ Hci] is greater than nx. From (1), and the dimensions of H0, Hw, Hψ and Hci, which are (s + 1)q × n, (s + 1)q × (s + 1) r, (s + 1)q × r1 and (s + 1)q × (s + 1) (p - 1), it follows that the dimension of [H0 Hw Hψ Hci] is (s + 1)q × [n + r1 + (s + 1)(r + p – 1)]. Therefore, the necessary and sufficient condition for the existence of nonzero solutions is: (s + 1)q > nx, or s > nx /q – 1.

where rh is the threshold obtained from a statistical table. The residual r(k) generated from (4) can be used to detect, but not isolate fault, since r(k) is a function of actuator or sensor faults. However, if conditions (3) is satisfied, r(k) is sensitive only to specific faults, and insensitive to other faults. Consequently, faults can be isolated and identified from r(k), using a fault model to be described later. Conditions (2) and (3) are the key results in deriving the fully decoupled parity equation.

Further, if the number of independent rows of [H0 Hw Hψ Hci] is greater than the number of independent columns, then nonzero solutions of (11) exist, giving the sufficient condition: (s + 1)qn + r1 + (s + 1) (r + p – 1). From (12), an appropriate window with size s can be selected to obtain the fully-decoupled parity vector to construct H0, Hw, Hψ and Hci. If q > (r + p) – 1. If q > n + r1 + (r + p), then s = 0, implying the hardware redundancy of the system is enough to isolate faults and the spatial redundancy is unnecessary.

3 Isolation of faults 3.1 Isolation of Actuator Faults To isolate actuator faults assuming that the sensors are operating normally, the parity vector must satisfy (3) to ensure the residuals are decoupled from the system state. The residuals are decoupled from the unknown nonlinear terms W(k) and Ψ(k), if [3],

3.2 Isolation of sensor faults

0=wT Hv , v (7) 0=ψHT

The parity vector must satisfy the following equation for the residuals generated from the fully-decoupled parity equation to be sensitive only to a specific actuator fault:

piHv ciT

i ,,2,10 L== (8)

In isolating sensor faults, it is assumed the actuators are operating normally. The fully-decoupled parity equation is constructed such that residuals generated from it are sensitive only to specific sensor faults, but insensitive to other sensor faults. The fully-decoupled parity equation (4) for each sensor is obtained from H0i, Hci, Hwi and Hψi and with C and G replaced by Ci and Gi obtained from C and G by setting the ith row corresponding to the ith sensor to 0. If the sensors are operating normally, then

where p is the number of actuators in the system, vi is the parity vector sensitive to the ith actuator, Hci is given by, [ ])()()()( 0 kHkWHskxHvkr iwii

Tii Ψ++−= ψ (13)

Control 2004, University of Bath, UK, September 2004 ID-135

For ri(k) to be decoupled from the system state, vi must satisfy:

00 =iT

i Hv (14) 0=Hv ψ

From (10), and

, then

,0)(,0)(0 ==− kWHvskxHv wT

iT

i

)(kWTi

[ )()()( kUHkUHvkr cccT

ii −= ] (22) The parity vector vi should also satisfy the following condition if it is decoupled from W(k) and Ψ(k):

0=wiT

i Hv , v (15) 0=iT

i Hψ

Also, , where H0=ciT

i Hv ci is obtained by removing the ith column of B in Hc. Since ri(k) obtained from (22) is sensitive only to the ith actuator, and insensitive to other actuators, then (20) becomes,

It follows that if the parity vector is insensitive to the ith sensor, the following condition holds,

EkkUkkU ici )()()()( λη += (23) 0][ 0 =iwiiT

i HHHv ψ (16) where E = [1 1 L 1]T is a (s + 1)p dimension vector. Rewriting (22) gives, For vi to be insensitive to the ith sensor, the elements in

H0i, Hci, Hwi and Hψi corresponding to the ith sensor must be 0. The parity equation constructed by vi satisfying (16) is referred to as the fully-decoupled parity equation. The condition for the existence of vi is given in Theorem 2. )()(

])()()1)([(

)]())()()(([)(

kkEkHkUHkv

kUHEkkUkHvkr

ii

iccciTi

ccicicTii

θφλη

λη

=+−=

−+=

(24)

Theorem 2 Let ni be the number of independent rows in The necessary and sufficient condition

for the existence of nonzero solutions of (16) is: ].[ ***

0 iwii HHH ψ

After taking into account modeling and measurement noise n(k), (24) becomes,

)()()()( knkkkr iii += θφ (25)

1−>qns x (17) [ Hi =φ

and the sufficient condition is: 1,1

11 +>−

−−+> rq

rqrns (18)

where , θ]),()( EHvkUvk cTicc

Ti i(k) = [(ηi(k) –1),

λi(k)]T. Assuming n(k) is a zero mean white noise with a covariance matrix of R(k), then from (25), can be obtained by the recursive least squares method,

)(ˆ kθ

1)]()1()()[()1()( −−+−= kkPkIkkPkK TT φφφ (26) where, , and are obtained from H*0iH *

wiH *iHψ 0i, Hwi and

Hψi by removing the ith row (a ‘0’ vector) of Ci, Di and Gi. The proof of this theorem is similar to that of theorem 1. From (17) and (18), H0i, Hci, Hwi and HψI are constructed, once s is selected.

)]1(ˆ)()()[()1(ˆ)(ˆ −−+−= kkkrkKkk θφθθ (27) )1()()()1()( −−−= kPkkKkPkP φ (28)

where K(k) and P(k) are the gain matrix and the error covariance matrix respectively, and I is the unit matrix. From (25), the fully-decoupled parity vector must satisfy the following condition to ensure the fault in the ith actuator appears in the residuals:

4 Identification of faults

0≠cTi Hv (29) 4.1 Identification of Actuator Faults

This result is summarized in Theorem 3. Actuator faults arising from a change in the scaling factor and/or the constant bias can be described by the following eqiation [6]:

λη += yz (19)

Theorem 3 The fully-decoupled parity equation (24) can detect, isolate and identify actuator faults, if the fully-decoupled parity vector satisfies (10) and (29).

where z is the fault, y is the output without fault, η the scaling factor, and λ the bias. If an actuator is operating normally, it is clear from (19) that η = 1 and λ = 0. If there are a constant output fault, then η = 0, and λ is a non-zero constant. For scaling factor faults, η ≠ 1 and λ = 0, and for constant bias faults, η = 1 and λ ≠ 1. If the ith actuator is faulty, then, (19) becomes,

4.2 Sensor Faults Identification

The fault model of the ith sensor can be described by the following model [7]:

)()()( kfkykz iii += (30)

)()()()( kkukku iicii λη += (20)

where, zi(k) is the actual output of the ith sensor and yi(k) is the normal output. In matrix form,

)()()( kfkykz += (31) where, ui(k) is the input of the ith actuator, uic(k) is the normal input, ηi(k) is the scaling factor, and λi(k) is the bias. After the residuals are generated from the fully-decoupled parity equations, the parameters of the fault model (19) can be estimated from the residuals using the recursive least squares method. From (2) and (4), the residuals sensitive to the ith actuator are given by:

where z(k) = [z1(k) z2(k) … zq(k)]T, y(k) = [y1(k) y2(k) … yq(k)]T, f(k) = [f1(k) f2(k) … fq(k)]T. Then

)()()( * kfIkYkZ ⋅+= (32)

)]())()()()([()( 0

kUHkHkWHkUHskxHvkr

cc

wcT

ii−Ψ+

++−=

ψ (21)

where Z(k) = [zT(k – s) zT(k – s + 1) … zT(k)]T is the actual sensor output, I* = [I0 I0 L I0]T is a matrix with dimension (s + 1)q x q and I0 is a q x q identity matrix. The residual insensitive to the ith sensor is,

Control 2004, University of Bath, UK, September 2004 ID-135

)]()()([

)]()([)(* kUHkfIkYv

kUHkZvkr

cii

iTi

ciiTii

−+=

−= (33)

where yi and fi are replaced by 0 in Yi(k) and f i(k). When the actuators are operating normally, the residuals insen-sitive to the ith sensor can be obtained from (13) and (33),

)]()()()()()([)(

*0

kUHkfIkHkWHkUHskxHvkr

cii

i

wiciiT

ii−+Ψ+

++−=ψ

(34)

From (16), v and

, hence

,0)(,0)(0 ==− kWHvskxH wiT

iiT

i

0)( =kWHv iT

i ψ

)()( * kfIvkr iTii = (35)

)()()()( * kkkfvIkr θφ== (36) where r(k) = [r1(k) r2(k) … rq(k)]T, ν = [ν1 ν2 … νq]T. For systems with measurement noise, n(k), (36) becomes

)()()()( knkkkr += θφ (37) where θ(k) = f(k), φ(k) = νI*, and n(k) are defined in (25). From (31), sensor faults can be estimated by the re-cursive least squares algorithms as discussed previously.

5 Example The proposed method is applied to detect and identify faults of the DC motor with a shunt field circuit described by the nonlinear ordinary differential equations [9]

=−=

=+−−=

=+−=

0

0

0

)0(,

)0(,1

)0(,1

rrrafr

aaa

rfa

aa

aa

fff

ff

ff

JDii

JM

iiVL

iLMi

LRi

iiVL

iLR

i

ωωωω

ω

&

&

&

(38)

where, if is the field current, ia the armature current and ωr the speed of revolution, V the input voltage, Rf and Lf the field resistance and inductance, Ra and La the armature resistance and inductance, M the mutual inductance between Lf and La, D and J the viscous resistance and the moment of inertia of the load. Let x = [if ia ωr]T be the state, and y = [if ia if + ia ωr]T be the measurement output, and u = V, (38) can be rewritten in discrete time as [10],

)(100

010

)(0

0832.0002235.0

)(9948.000

06839.00

00

8825.0)1(

kwku

kxkx

+

+

=+

(39)

)(

1000

0110

0101

)( kxky

= (40)

where w is the unknown nonlinear term in (1) in discrete form. From (39) and (40), there are one actuator and four sensors in this system. Assuming the covariance matrix

R(k) is a 2 x 2 diagonal matrix with 0.02 along its diagonal. The parity vectors can be obtained from (10) and (16), and residuals sensitive to specific faults can be computed by (4). The fault models can be estimated from (25) and (37) by the recursive least squares method.

(1) Fault identification of actuator The residuals sensitive before any fault occurs are shown in Fig. 1. When a fault given by the fault model (19) with η = 2 and λ = 10 occurs at 10 seconds, the residuals are computed and are shown in Fig. 2. The identification of the fault under normal operating condition is shown in Fig. 3, and under faulty condition in Fig. 4. Clearly, the residuals generated from the fully-decoupled parity equation is sensitive only to the specific actuator fault, and is independent of the system state and the unknown nonlinear term, indicating that the actuator fault can be detected, isolated and identified.

(2) Fault identification of sensors A constant off-set of 10 units occurs in sensors 1 and 2, and 0.1 units in sensors 3 and 4 after 10 seconds. The estimated parameters of the fault model are shown in Fig. 5, which are close to the actual fault. Next, consider the case when sensors 1 and 4 are operating normally, whilst sensors 2 and 3 have developed a fault with a constant off-set of 3.5 and 2 units respectively after 10 seconds. The estimated faults are shown in Fig. 6, showing that the estimated faults are close to the actual ones, illustrating that the proposed method is able to detect, isolate and identify successfully sensor faults.

6 Conclusions This paper presents an approach for fault diagnosis in the system with unknown nonlinearity based on the fully-decoupled parity equation. The residuals generated from the fully-decoupled parity equation is sensitive only to specific faults and is decoupled from the system state, the unknown nonlinearity and other faults. The parameters of the fault models can be estimated from the residuals by the recursive least squares method. The performance of the proposed method is illustrated by using it to detect and identify successfully multiple actuator or sensor faults in a simulated DC motor.

Acknowledgements This project is supported by the HKSAR RGC Grant (HKU 7050/02E) and the National Natural Science Foundation of China (60234010)

References [1] E. Y. Chow and A. S. Willsky. “Analytical

redundancy and the design of robust failure detection systems”, IEEE Trans. Automatic Control, AC-29(7), pp. 603-614, (1984).

[2] M. Darouach, M. Zasadzinski and S. J. Xu. “Full-order observer for linear systems with unknown

Control 2004, University of Bath, UK, September 2004 ID-135

inputs”, IEEE Trans. Automatic Control, 39, pp. 606-609, (1994).

0 10 20 30 40 50

0

2

4

tima/s

scale

0 10 20 30 40 500

5

10

time/s

bias/V

[3] H. Jin and H. Y. Zhang. “FDI of slow-grown faults of dynamic systems [J].”, IEEE Transactions on Aerospace and Electronic Systems, 35(4), pp. 1122-1128, (1999).

[4] X. C. Lou, A. S. Willsky and G. C. Verghese. “Optimally robust redundancy relations for failure detection in uncertain systems”, Automatica, 22(3), pp. 333-344, (1986).

Fig. 4 Estimated parameters of actuator fault model (faulty)

[5] H. Song and, H. Y. Zhang. “An approach to sensor fault diagnosis based on fully-decoupled parity equation and parameter estimate”, Proceedings of the 4th World Congress on Intelligent Control and Automation, pp. 2750-2754, (2002).

0 100 200 300 400 5000510

f10 100 200 300 400 5000510

f2

0 100 200 300 400 500-0.2

0

0.2f3

0 100 200 300 400 500-0.05

0

0.05

time/s

f4

[6] Jay F. Tu and J. L., Stein. “Model error compensation for observer design”, International Journal of Control, 69, pp. 329-345, (1998).

[7] Y. Wang. On-line fault diagnosis of nonlinear dynamical systems using recurrent neural networks, Ph.D. Thesis, The University of Hong Kong, (2000).

[8] X. Wen, H. Y. Zhang and I. Zhou. Fault diagnosis and fault tolerant control for control system, Machine Industry Press, (1998).

[9] H. Y. Zhang and R. J. Patton. “Optimal design of robust analytical redundancy for uncertain systems”, IEEE Region 10 Conference on “Computer, Communication, Control and Power Engineering”, (1993).

0 10 20 30 40 50-0.02

0

0.02

time/s

Fig. 5 Estimation parameters of Sensor faults occurring at 10s, [ ]Tf 01.01010=

0 100 200 300 400 500-0.05

0

0.05

f1

0 100 200 300 400 500024

f2

0 100 200 300 400 500-1012

f3

0 100 200 300 400 500-0.05

0

0.05

time/s

f4

Fig. 1 Residuals for actuator (normal)

0 10 20 30 40 50-0.05

0

0.05

time/s Fig. 2 Residuals for actuator (Faulty)

0 10 20 30 40 50-1

0

1

2

time/s

scale

0 10 20 30 40 50-1

0

1

time/s

bias/V

Fig. 6 Estimation of sensor faults occurring at10s, Fig. 3 Estimated parameters of actuator fault model

(normal) f = [0 3.5 2 0]T

Control 2004, University of Bath, UK, September 2004 ID-135