Fault diagnosis via a polynomial observer

2011 8th International Conference on Electrical Engineering, Computing Science and Automatic Control.

Fault diagnosis via a polynomial observer H. Aguilar Sierra, R. Martínez-Guerra, J. L. Mata-Machuca Department of Automatic Control, CINVESTAV-IPN, Mexico D.F., Mexico E-mail: {haguilar, rguerra, jamata}@ctrl.cinvestav.mx

Abstract— The fault diagnosis problem of a class of nonlinear systems based on a differential approach is used to determine fault diagnosability with the minimum number of measurements from the system. In order to reconstruct the faults on the system, a polynomial observer is proposed, which includes in its structure corrections terms of high order. Another two schemes of nonlinear observers are used for reconstructing the faults for comparison purposes, one of them being a reduced order observer and the other a sliding mode observer.The approach was tested in a real-time experimental setting Amira DTS-200. Index Terms— Fault diagnosis, polynomial observer, reduced order observer, sliding mode observer, differential algebra.

I. I NTRODUCTION Fault diagnosis is an important problem in process engineering. The fault diagnosis problem has been studied for more than three decades, there are many papers on process of fault diagnosis ranging from approaches based on quantitative models, qualitative models and approaches based on process history data [1]. For the approaches based on quantitative models, can be found approaches based upon differential geometric methods [2], [3]. On the other hand, alternative approaches has been proposed based on an algebraic and differential framework [4] and [5]. This paper deals with nonlinear systems diagnosis and the goal is to find malfunctions in the system, based on input/output measurements. The outputs are mainly measured signals obtained from sensors, their number is important in order to know if the system is diagnosable or not. The fault diagnosis problem is considered as the problem of observing the fault signals. So a system is called diagnosable if the faults satisfy the so-called algebraic observability condition [10]. The main contribution of this article consists of the solution of the fault diagnosis problem by means of a polynomial observer for the case of multiples available measurements. In addition, another two schemes of observer are proposed in order to estimate the fault signals for comparison purposes, one of them is a reduced order observer and the other is a sliding mode observer based on partial change of coordinates. This paper is organized as follows: In section II A definition related on Observability and Diagnosability is given. In section III the relation between the number of faults and the number of available measurements in terms of the differential transcendence degree concept is given. An asymptotic polynomial for the fault signals is presented in section IV. In section V the Amira DTS200 three-tank system model is described, and the diagnosability condition is evaluated. In section VI the experimental results are shown for the fault and state

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estimation with the three different observers. Finally, in section VII the paper is closed with some concluding remarks. II. O BSERVABILITY AND D IAGNOSABILITY C ONDITION The observability and diagnosability notion of a system, linear or nonlinear in the differential algebra approach need a basic definition. Further details can be found in [4] and [10]. Definition 1: Algebraic observability and diagnosability condition. A fault f 2 G is said to be diagnosable if it is possible to estimate the fault from the available measurements of the system, i.e., f is diagnosable if it is algebraically observable over the differential field R hu; yi[4]. Let us consider the class of nonlinear systems with faults described by the following equation x_ (t) = g (x; u; f ) y (t) = h (x; u)

(1)

Where x = (x1 ; x2 ; :::; xn ) 2 Rn is a state vector, u = (u1 ; u2 ; :::; um ) 2 Rm is a know input vector, f = (f1 ; f2 ; :::; f ) 2 R is a unknown input vector, y (t) 2 Rp is the output vector (available measurements), g and h are assumed to be analytical vector functions. Remark 1: The diagnosability condition is independent of the observability of a system. Example : Consider the following nonlinear system 8 x_ 1 = x1 x2 + f > > < x_ 2 = x22 + x1 + u (2) x_ 3 = x3 x2 > > : y = x2 Since f satisfies the differential algebraic equation f y• y y_ + y 3 uy + u_ = 0, then the system (2) is diagnosable and the fault can be reconstruct knowing u, y and their time derivatives. However the state x3 is not algebraically observable. III. R ELATION BETWEEN NUMBER OF FAULTS AND THE NUMBER OF AVAILABLE MEASUREMENTS

Let consider the system (1). The fault vector f is unknown and it can be assimilated as a state with uncertain dynamics. Then, in order to estimate it, the state vector is extended to deal with the unknown fault vector. The new extended system is given by 8 < x_ (t) = g (x; u; f ) 1 j (3) f_j = Aj (x; u; f ) : y (t) = h (x; u)

The following results from the theory of differential algebraic field extensions are an useful tool to determine whether a

2011 8th International Conference on Electrical Engineering, Computing Science and Automatic Control.

fault can be reconstructed from the know inputs and available measurements. Definition 2: A maximal family of elements of the field K that are K-differentially independent is denominated differential transcendence basis of extension L=K and the cardinality of this base is called the differential transcendence degree denoted by: dif f tr do L=K. Property 1: Let K,L,M be differential fields such that K L M. Then o

o

o

dif f tr d (M=K) = dif f tr d (M=L) + dif f tr d (L=K)

[Kij ] 1

H3

T

8 > > > ex = > > > > > > > > <

IV. P OLYNOMIAL O BSERVER The system (3) can be expressed in the following form 8 < x_ (t) = Ax + (x; u; f ) 1 k (6) f_k = Ak (x; u; f ) , : y (t) = Cx

where A = (A1 ; A2 ; :::; A ) 2 R is a unknown bounded function i.e., kAj k N < 1, N 2 R+ and (x; u) is a nonlinear function that satisfies the Lipschitz condition with u = (u; f ), i.e., k (x; u)

(^ x; u)k

L kx

x ^k

and u uniformly bounded. The following system is an observer for the system (6) 8 > > x ^ (t) = A^ x + (x; u) + > > > p X m > X > 2j 1 > > + Kij (yi Ci x ^) < i=1 j=1 m X

> > > > f^k (t) = Kkl fk > > > > l=1 > : y (t) = Ci x

where, x ^ 2 Rn , [Kij ] 1

i p 1 j m

f^k

A

X i=1

Ki1 Ci

P +P

A

X i=1

2j 1

Kij (Ci ex )

(9)

+ [ (x; u) Ak

Kk1 ek

(^ x; u)] m X 2l Kkj (ek )

1

j=2

(10)

where P satisfies the assumption H2. The proof of the theorem is given in two parts, as follows: (i) The time derivate of V1 along the trajectories of (9) is = =

T e_ T x P ex + ex P e_ x 0 !T p X @ eT A K C P +P i1 i x i=1

2l 1

+

i=1 j=2

V = V 1 + V2 V1 = eTx P ex V2 = 12 e2k

V_ 1

(8)

ex

+2eT x P [ (x; u) p X m X

(^ x; u)]

Kij (Ci ex )2j

2

2 T ex

A

p X i=1

!1 Ki1 Ci A ex +

(P Ki;j Ci )T + (P Kij Ci ) ex

i=1 j=2

; Kkl

p

Ki1 Ci

The following theorem proves the observer convergence. Theorem 3: For the nonlinear system (6), suppose that x (t) exists for all t 0, the nonlinear function (x; u; f ) satisfies de Lipschitz condition (7), x (t) and f (t) are algebraically observable. If there exists a matrix P positive definite and observer gains Kij and Kkl > 0 such that the system (8) is an observer of the system (6), then the estimator error converges asymptotically to zero. Proof: Consider the following Lyapunov function candidate

1 k 1 l m

; Kij ; Kkl > 0,

The next inequality [7], which is based on the Lipschitz condition (7) 2eT xP [

!

Ki1 Ci +L2 P P +I+"I = 0

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(x; u)

(^ x; u)]

T L2 eT x P P ex + ex ex

(11)

Now, applying the Rayleigh’s inequality [8] and consider the assumption H3, we obtain

eT x P Kij Ci ex p

i=1 p X m X

!

A. Observer convergence analysis

(7)

x ^0 = x ^ (t0 ) and f^0 = f^ (t0 ) are arbitrarily initial conditions. For this observer the following hypotheses are considered: H1: f (t) is algebraically observable on R hu; yi H2 The gains [Ki1 ]1 i p can be chosen such that algebraic Riccati equation has a symmetric and positive definite solution P for some F > 0. !T

p X

A

> > > > > > > > > > > : ek =

p

(5)

0

For analyzing the estimation error, we define e = [ex ; ek ] , where ex = x x ^ and ek = fk f^k . From systems (6) and (8), the dynamics of estimation error is given by

This property is an important tool to proof the theorems 1 and 2, see [4]. Theorem 1:.Assume that the system (1) is diagnosable, then the number of faults is less or equal to the number of available measurements (outputs), i.e.

dif f tr do K hu; yi =K hui =

T

(P Kij Ci ) + (P Kij Ci )

min

(4)

Theorem 2: The system (1) is diagnosable if and only if

is chosen such that

i p 2 j m

min

P Kij Ci + (P Kij Ci )T kex k2

for 2 i m. Hence, combining the inequalities (11) and (12)

(12)

2011 8th International Conference on Electrical Engineering, Computing Science and Automatic Control.

2

V_ 1

p X

4 A eT x

Ki1 Ci

i=1

!T

P +P

A

p X m X

Ki1 Ci

i=1

+ L2 P P + I ex 2

p X

Kij (Ci ex )2j

!

+

P Kij Ci + (P Kij Ci )T kex k2

2 min

i=1 j=2

eT x

2

p X

4 A

Ki1 Ci

i=1

!T

P +P

A

p X

Ki1 Ci

i=1

+ L2 P P + I ex

!

+

F kex k2

=

(ii) On the other hand, considering the second term of the Lyapunov function candidate, we have V_ 2

= ek e_ k = ek

Ak

= ek Ak

m X

Kk1 ek Kk1 e2k

2l 1

Kkl (ek )

l=2 m X

V. A PPLICATION OF THE THREE - TANK SYSTEM

!

A. Description of the three-tank system

2l

Kkl (ek )

l=2

Kk1 e2k

jek j jAk j =

2

jek j N Kk1 jek j Kk1 jek j N jek j

V_ 2 is negative inside the set jek j > N=Kk1 , i.e., exists F > 0 such that Kk1 jek j N = F > 0: Now we proves that jek j is upper bounded. Let B, C upper 2 bounds of V2 (ek ). With C > N 2 =2Kk1 , the solution in the set fV2 (ek ) Cg will remain there for all t 0, due to that V_ 2 is negative in V2 = C. Hence, the solution of e_ k are 2 uniformly bounded [9]. Furthermore, if N 2 =2Kl1 < B < C, then V_ 2 will be negative in the setfB V2 Cg. In this set V2 will decrease monotonously until the solution of the set fV2 Bg. According pto [9] the solution is ultimately bounded 2B. For example, if we defined B and with bound jek j 2 2 C as B = N 2 =2Kk1 and C = 2N 2 =2Kk1 . Then, the ultimate bound is jek j

N Kk1

V_ 2

F jek j

Fig 1. Diagram of the Three-Tank System

The Amira DTS200 [6] system consists of three cylinders (T1 , T2 y T3 ) with transversal constant section A, which are connected in series one to another one by means cylindrical tubes with transversal section S (Figure 1). The corresponding nominal model is given by the following system 8 1 dh1 > > = (q1 q13 + u1 ) > > dt A < 1 dh2 (13) = (q2 + q32 q20 + u2 ) > dt A > > > : dh3 = 1 (q13 q32 ) dt A According to the generalized rule of Torricelli, this is valid for laminar flow p q13 = a1 Ssign (h1 h3 ) p2g jh1 h3 j (14) q32 = a3 Ssign p (h3 h2 ) 2g jh3 h2 j q20 = a2 S 2gh2 A completed description of the system (13) is given in the following table. TABLE 1 VARIABLES AND PARAMETERS OF THE AMIRA DTS 200

Hence

Finally, from (i) and (ii), we conclude that V_

2

F kex k

In what follows, it is considered that the state vector is T T x = [x1 x2 x3 ] = [h1 h2 h3 ] and the input vector is u = T T [u1 u2 ] = [q1 q2 ] .

F jek j < 0

and the theorem is proven.

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2011 8th International Conference on Electrical Engineering, Computing Science and Automatic Control.

The system (13) has four regions of operations in which the corresponding model is differentiable, in this paper we only consider the operation region x1 x3 x2 .

1) Reduced Order Observer: We propose a reduced order observer [10]. Let us consider the following time derivative to be estimated = y_ (20) we propose the observer structure

B. Considered faults and measurements of the system The nominal model (13) is transformed into the following, where the additive faults f1 and f2 ( = 2) are considered in the actuators that control the input flows u1 = q1 , u2 = q2 : 8 > > x_ 1 = > > < x_ 2 = > > > > : x_ 3 =

^=K(

^)

(21)

now, applying the next change of coordinates ^ = D + Ky

1 (u1 q13 + u1 + f1 ) A 1 (u2 + q32 q20 + u2 + f2 ) A 1 (q13 q32 ) A

(15)

(22)

and from (20) and (21) we can get D_ = from (22) D_ = KD K 2 y

K^, then again (23)

Hence (23) and (22) constitute an asymptotic estimator of .

C. Diagnosability analysis For the theorem 1, for the diagnosability is required that the number of faults be less or equal that the number or available measurements. The system (15) consists of two faults. In order to determine observability and the diagnosability of the system (15), we have to evaluate the algebraic observability condition from definition 5 for f1 and f2 . and the unknown state x1 . We only consider the region of operation x1 x3 x2 . For this case we only consider two outputs y2 = x2 , and y3 = x3 . In this case from (15) we have p f1 = Ax_ 1 + a1 S 2g (x1 y3 ) u1 (16) f2 = Ay_ 2

x1 = y3 +

a3 S

p 2g (y3

y2 ) + a2 S

p

2gy2

p 1 Ay_ 3 + a3 S 2g (y3 2 2 2ga1 S

y2 )

u2

(17)

2

(18)

The algebraic observability condition for x1 is deduced directly from (18), which is an equation with coefficients in R hu; yi. Replacing (18) into (16) and (17), we have f1

=

Ax_ 1 + r +a1 S

(2g

1) y3 +

1 2 a2 1S

Ay_ 3 + a3 S

u1

f2

=

Ay_ 2

a3 S

p 2g (y3

y2 ) + a 2 S

p 2gy2

p 2g (y3 u2

2) Sliding Mode Observer: We propose a sliding mode observer [10], then we introduce the following change of variables: 1 = y, 2 = _ 1 , then we obtain the following observer ^1 = ^2 + m1 sign (y y^) (24) ^2 = m2 sign (y y^) which can be used to estimate

2

from the output y.

3) Polynomial Observer: We introduce the following change of variables 1

= y,

2

= y, _ ... ,

r

= y (r

1)

(25)

Then the system (1) can be expressed as follows _ = A + ( ; u) y= 1

(26)

So, by the theorem 3, the observer has the following structure p X m X 2j 1 ^ = A^ + (^; u) + Kij (y Ci x ^) (27) i=1 j=1

Then, the polynomial observer (27) is used to estimate the variables 1 = y, 2 = y, _ ... , r = y (r 1) by means of the available output y.

2

y2 )

VI. FAULT ESTIMATION RESULTS For the uncertain parameters ai from the system (13), we

(19) consider the following identification results

It is easy to check from (19) that dif f tr d K hu; yi =K hui = 2 and for the theorem 2 that the system is diagnosable with the two considered outputs.

a1 = 0:4663, a2 = 0:7654, a3 = 0:4616

o

D. Fault reconstruction We consider the outputs y2 = x2 and y3 = x3 , then for this case we need to estimate the time derivatives from outputs y2 and y3 . In this section a methodology appears to reconstruct first r 1 time derivatives from the available output y.

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For all the experiments reported in this section the input 3 flows were maintained constant as u1 = q1 = 0:00002 ms and 3 u2 = q2 = 0:000015 ms , for 3000 seconds. The two faults were artificially generated through the following equations: f1 f2

= =

5 5

10

6

10

6

1 + sin 0:2te 1 + sin 0:05te

0:01t 0:001t

where U (t) is the unit step function.

U (t

U (t

220) 330)

2011 8th International Conference on Electrical Engineering, Computing Science and Automatic Control.

The three proposed schemes for fault diagnosis were evaluated for the case when x2 = y2 and x3 = y3 are available, for this reason an estimation for the unknown state x1 was necessary to be obtained. In the Figure 2 we show the estimation of unknown state x1 and the fault estimation results and as we can observe, the reconstruction results of signals are good. The gain values for the fault observers were K1 = 0:045 and K2 = 1:25 In the same way, in the Figure 3 we show the state and fault estimation results and also the reconstruction results of signals are good. The gain values for the fault observers were m11 = 0:0195, m12 = 0:00055 and m21 = 0:0165, m22 = 0:0005. Finally in the Figure 4 the corresponding results achieved wiht the polynomial observer are shown. The gain values for the fault observers were K11 = 0:045, K12 = 23, K13 = 19 and K21 = 0:196, K22 = 25, K23 = 15. It is worth to mention that with this observer we estimated the full state vector and the both fault signals. As we can observe, this scheme also provides good results.

Fig 3. Sliding mode observer a) Level estimation b) Fault reconstruction f1 c) Fault reconstruction f2

Fig 2. Reduce order observer a) Level estimation b) Fault reconstruction f1 c) Fault reconstruction f2

The performance of the proposed observer is evaluated using the following cost function, with "0 = 0:0001, in this case is defined as Jt =

1 t + "0

Zt 0

2

kek ( )k d

(28)

Figure 5 shows the performance of three observers proposed to estimate the fault f1 . We observe that the polynomial observer converges much faster than the other two. In the same way, in Figure 6 we observe that the performance of the three observers for the estimation of the fault f2 is similar.

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Fig 4. Polynomial observer a) Level estimation b) Fault reconstruction f1 c) Fault reconstruction f2

2011 8th International Conference on Electrical Engineering, Computing Science and Automatic Control.

was tested in a real-time experimental setting Amira DTS-200. From the fault reconstruction result we can see that for the three different observers a similar performance and estimation results. Finally the proposed polynomial observer presents a good performance. R EFERENCES

Fig 5. Performance evaluation of observers for the estimation error of fault f1

Fig 6. Performance evaluation of observers for the estimation error of fault f2

VII. C ONCLUDING R EMARKS In this work the fault diagnosis problem for a class of nonlinear systems using the differential algebraic approach was used. The method consists of considering to the fault as an augmented state of the system. To achieve the reconstruction of the unknown states of the extended system, we proposed a polynomial observer and another two schemes of nonlinear observers for comparison purposes. The approach

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[1] Venkat. V., Raghunathan R., Kewen Y., Suraya N. K. "A review of process fault detection and diagnosis, Part I: Quantitative model-based methods". Computers & Chemical Engineering, vol 27, Issue 3, pp 293311, 2003. [2] Join C., Ponsart J.-C., Sauter D. and Theilliol D. "Nonlinear filter design for fault diagnosis: application to the three-tank system", IEE Proc. Control Theory Appl., vol. 152, No. 1, pp 55-64, 2005. [3] M. Massoumnia, "A geometric approach to the synthesis of failure detection filters", IEEE Transactions on Automatic Control, vol. 31, pp. 839–846, 1986. [4] J. Cruz-Victoria, R. Martinez-Guerra and J. Rincon-Pasaye "On nonlinear systems diagnosis using differential and algebraic methods" Jorunal of the Franklin Institute, Vol.345, pp.102-118, 2008. [5] Fliess M., Join C. and Sira-Ramírez H., "Robust residual generation for nonlinear fault diagnosis: an algebraic setting with examples", International Journal of Control, vol. 14, no. 77, 2004. [6] Amira DTS200: "Laboratory setup three tank system", Amira Gmbh, Duisburgh, Germany, 1996 [7] S. Raghavan and J. Hedrick (1994), "Observer design for a class of nonlinear systems", Int. J. Control, vol. 59, pp. 515–528, 1994. [8] R. Horn, and C. Johnson, Matrix analysis, Cambridge University Press, 1985. [9] H. Khalil, Nonlinear systems, Third edition, Prentice Hall, 2002. [10] J. Rincón-Pasaye, R. Martínez-Guerra, A. Soria-Lopez "Fault diagnosis in nonlinear systems: An application to a three-tank system" American Control Conference, 2008.