- Email: [email protected]
Fault diagnosis viewed as a left invertibility problem
ISA Transactions 52 (2013) 652–661
Contents lists available at ScienceDirect
ISA Transactions journal homepage: www.elsevier.com/locate/isatrans
Fault diagnosis viewed as a left invertibility problem R. Martínez-Guerra a,n, J.L. Mata-Machuca a,c, J.J. Rincón-Pasaye b a
Departamento de Control Automático, CINVESTAV-IPN, Av. IPN 2508, 07360 DF, Mexico Facultad de Ingeniería Eléctrica, Universidad Michoacana de San Nicolás de Hidalgo, Ciudad Universitaria, Morelia, Mexico Instituto Politécnico Nacional, Unidad Profesional Interdisciplinaria en Ingeniería y Tecnologías Avanzadas, Academia de Mecatrónica, Av. IPN 2580, 07340 DF, Mexico b c
art ic l e i nf o
a b s t r a c t
Article history: Received 20 February 2011 Received in revised form 16 April 2013 Accepted 1 June 2013 Available online 6 July 2013 This paper was recommended for publication by Dr. Q.-G. Wang.
This work deals with the fault diagnosis problem, some new properties are found using the left invertibility condition through the concept of differential output rank. Two schemes of nonlinear observers are used to estimate the fault signals for comparison purposes, one of these is a proportional reduced order observer and the other is a sliding mode observer. The methodology is tested in a real time implementation of a three-tank system. & 2013 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Fault diagnosis Left invertibility condition Differential output rank Sliding mode observer Reduced order observer
1. Introduction A fault can be considered as a process degradation or degradation of the equipment performance caused by the change in the physical characteristic of the process, the input process or the external conditions. Industrial control systems have to deal with faults, therefore, fault diagnosis is a very important subject in control theory. System diagnosis helps us to detect and estimate faults in a process. In other words, the task of diagnosis is, from measurements of outputs and inputs, to reconstruct the fault vector. The fault detection and isolation problem have been studied for more than three decades, many papers dealing with this problem can be found, see for instance the surveys [1–4] and the books [5–7]. For the case of nonlinear systems a variety of approaches have been proposed [8–13], such as those based upon differential geometric methods [14,15], and on the other hand, alternative approaches based on an algebraic and differential framework can be found in [16–20]. Currently, the diagnosis problem is playing an important role in modern industrial processes. This has led control theory into a wide variety of model-based approaches which rely on descriptions via differential and/or difference equations, contrary to other standpoints developed mainly among computer scientist (see [18,19] and references therein). The primary objectives of fault n
Corresponding author. Tel.: +52 5557473800; fax: +52 5557473982. E-mail addresses: [email protected] (R. Martínez-Guerra), [email protected] (J.L. Mata-Machuca), [email protected] (J.J. Rincón-Pasaye).
diagnosis are fault detectability and isolability, i.e., the possible location and determination of the faults present in a system and the time of their occurrences. The tasks of fault detection and isolation are to be accomplished by measuring only the input and the output variables. This paper focuses on the diagnosis of nonlinear systems and the goal is to determine malfunctions in the dynamics. In this communication, the outputs are mainly signals obtained from the sensors. Their number is important to know whether a system is diagnosable or not. In this paper, the diagnosis problem is tackled as a left invertibility problem throughout the concept of differential output rank ρ. Two schemes of observers are proposed in order to estimate the fault signals, one of them is a reduced-order observer based on a freemodel approach and another is a sliding-mode observer based on a Generalized Observability Canonical Form (GOCF) [18]. Both schemes are proved to possess asymptotic convergence properties. The class of systems for which this methodology can be applied contains systems that depend on the inputs and their time derivatives in a polynomial form. The type of faults considered in this work is additive and bounded, however, the algebraic approach can also be used to deal with multiplicative faults. These proposals are applied in this paper to a three-tank system [21]. The Amira DTS200 three-tank system [22] has been widely considered for the experimental fault diagnosis study, see for instance [15,17,23], even recently, one work based on the geometric approach has been reported [15]. We can also mention one previous work with the three-tank system using the differential algebraic
0019-0578/$ - see front matter & 2013 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2013.06.001
R. Martínez-Guerra et al. / ISA Transactions 52 (2013) 652–661
approach [17]; in that work the authors only report a numerical simulation study and not a real-time experiment, also they only solve the simplest case in which three measured outputs are available to estimate two faults, that is to say, they do not analyze the minimal number of measurements to attack the diagnosis problem as we do in the present work. The intention of choosing the three-tank system example is to clarify the proposed methodology and to highlight the simplicity and flexibility of the present approach. The three-tank system is known as a system with parameter uncertainties, so this work also deals with these uncertainties by means of algebraic parameter estimation, considering that there is no simultaneous presence of uncertainty and faults, in the same way as it is considered in [17]. This paper is organized as follows. In Section 2, some definitions of differential algebra are given. In Section 3, we discuss the left invertibility condition and we present some examples. In Sections 4 and 5 we give a brief description of the proposed observers. In Section 6 the three-tank system is analyzed. Finally, in Section 7 we illustrate this methodology with some experimental results.
2. Some definitions Some basic definitions are introduced. Further details can be found in [17,18] and references therein. Definition 1. Let L and K be differential fields. A differential field extension L=K is given by K and L such that: (1) K is a subfield of L and (2) the derivation of K is the restriction to K of the derivation of L. Example 1. R〈et 〉=R is a differential field extension, where R D _ ¼ 0. R〈et 〉. et being a solution of x−x Definition 2. Let ξ ¼ ðξ1 ; ξ2 ; …; ξn Þ satisfies an algebraic differential coefficients in K it is called dependent, otherwise ξ is called independent.
be a set of elements of L. If it _ ξ…Þ € equation Pðξ; ξ; ¼ 0 with differentially Kalgebraically differentially Kalgebraically
Definition 3. Any set of elements of L which is differentially Kalgebraically independent and maximal with respect to inclusion forms is a differential transcendence basis of L=K. Two such bases have the same cardinality. This is called the differential ○ transcendence degree of L=K and denoted by diff tr d L=K. Definition 4. Let G, K〈u〉 be differential fields. A nominal dynamic consists of a finitely generated differential algebraic extension G=K〈u〉, ðG ¼ K〈u; ξ〉; ξ∈GÞ. Any element of G satisfies an algebraic differential equation with coefficients over K in the components of u and their time derivatives. Example 2. Consider the following differential equation: u_ 2 y þ 4u€ ¼ 0 In this case, y is algebraic over K〈u〉, therefore, it can be seen as a dynamic of the form K〈u; y〉=K〈u〉 where K ¼ R and y∈K〈u; y〉. Definition 5. Any unknown variable x in a dynamic is said to be algebraically observable with respect to K〈u; y〉 if x satisfies a differential algebraic equation with coefficients over K in the components of u, y and a finite number of their derivatives. Any dynamic with output y is said to be algebraically observable if, and only if, any state variable has this property.
Example 3. Let us consider the following system: 8 2 _ > < x 1 ¼ 3x1 x2 þ u1 x_ 2 ¼ x1 þ x32 u2 > :y¼x ;
653
ð1Þ
2
since x1 and x2 satisfy the polynomials x1 þ y3 u2 −y_ ¼ 0 and x2 −y ¼ 0, respectively, then x1 ; x2 are algebraically observable over R〈u; y〉 and by applying Definition 5, system (1) is algebraically observable. Definition 6. A fault is not a permitted deviation of at least one characteristic property or parameter of any process in relation to the development of the same parameter under normal conditions. Faults are defined as transcendent elements over K〈u〉, therefore, a system with the presence of faults is a differential transcendental extension, denoted as K〈u; f ; y〉=K〈u〉, where f is a vector that includes the faults and their time derivatives. Definition 7. Let G, K〈u〉 be differential fields. A fault dynamic consists of a finitely generated differential algebraic extension G=K〈u; f 〉, G ¼ K〈u; f ; ξ〉; ξ∈G. Any element of G satisfies an algebraic differential equation with coefficients over K in the components of u; f and their time derivatives. Definition 8 (Algebraic observability condition). A fault f ∈G is said to be diagnosable if it is algebraically observable over R〈u; y〉, i.e., if it is possible to estimate the fault from the available measurements of the system. Let us consider the class of nonlinear systems with faults described by the following equation: ( _ ¼ Aðx; uÞ xðtÞ ð2Þ yðtÞ ¼ hðx; uÞ; where x ¼ ðx1 ; …; xn ÞT ∈Rn is a state vector, u ¼ ðu1 ; …; um Þ∈Rm is a known input vector, f ¼ ðf 1 ; …; f μ Þ∈Rμ is an unknown input vector, u ¼ ðu; f Þ∈Rmþμ , yðtÞ∈Rp is the output vector. A and h are assumed to be analytical vector functions. Example 4. Let us consider the nonlinear system with one fault ðf 1 Þ on the actuator and one fault ðf 2 Þ on the sensor of output y1 : 8 x_ 1 ¼ x1 x2 þ f 1 þ u > > > > < x_ 2 ¼ x1 ð3Þ y1 ¼ x1 þ f 2 > > > > : y ¼ x2 : 2
Since f 1 , f2 satisfy the differential algebraic equations f 1 −y€ 2 þ y2 y_ 2 þ u ¼ 0 f 2 −y1 þ y_ 2 ¼ 0
ð4Þ
the system (3) is diagnosable and the faults can be reconstructed from the knowledge of u, y and their time derivatives. Remark 1. The diagnosability condition is independent of the observability of a system. Example 5. Let us consider the system 8 x_ ¼ x1 x2 þ f þ u > > > 1 > < x_ 2 ¼ x1 x_ 3 ¼ x3 f þ u > > > > : y ¼ x2 :
ð5Þ
In this case f is diagnosable. However, x3 is not algebraically observable.
654
R. Martínez-Guerra et al. / ISA Transactions 52 (2013) 652–661
differential polynomials of the form
3. On the left invertibility condition We have some definitions concerning the differential output rank of a system. Definition 9. The differential output rank ρ of a system is equal to the differential transcendence degree of the differential extension K〈y〉 over the differential field K, i.e., o
ρ ¼ diff tr d K〈y〉=K: Property 1 (Kolchin [24]). Let K, L, M be the differential fields such that K⊂L⊂M. Then o
o
o
diff tr d ðM=KÞ ¼ diff tr d ðM=LÞ þ diff tr d ðL=KÞ
□
ð6Þ
Property 2. The differential output rank ρ of a system is smaller than or equal to minðm; pÞ: o
ρ ¼ diff tr d K〈y〉=K≤minðm; pÞ;
hr ðy1 ; …; yp Þ ¼ 0
ð14Þ
and if it is possible to find r independent relations of the form (14), then the differential output rank is given by ρ ¼ p−r, that is to say, there exist only p−r independent outputs. Example 6. Consider the following system with one input and two outputs: 9 x_ 1 ¼ u > = with y1 ¼ x1 u x_ 2 ¼ x1 ð15Þ > y ¼x ; x_ ¼ x u 3
1
2
3
Replacing y1 and y2, 9 y_ 1 u_ > − 2 y1 ¼ u > > > u u = y1 _x 2 ¼ > u> > > y_ 2 ¼ y1 ;
ð16Þ
where m and p are the total number of inputs and outputs, respectively. A proof of Property 2 can be given in the following manner: an input–output system, with input u ¼ ðu1 ; …; um Þ and output y ¼ ðy1 ; :; yp Þ, is defined by the next conditions:
Only one equation of (16) can be written in the form of (14). Then we have the relation
ðu1 ; …; um Þ are differentially Kalgebraically independent, i.e.,
where only one output is differentially independent and then ρ ¼ 1.
o
diff tr d K〈u〉=K ¼ m
ð7Þ
ðy1 ; …; yp Þ are differentially algebraic over K〈u〉, i.e., K〈u; y〉=K〈u〉 is differentially algebraic or o
diff tr d K〈u; y〉=K〈u〉 ¼ 0
ð8Þ
y1 −y_ 2 ¼ 0
Example 7. Consider the following system with three inputs and two outputs: 9 x_ 1 ¼ u1 > = x_ 2 ¼ u2 with y1 ¼ x1 ð17Þ > y2 ¼ x2 x3 ; x_ 3 ¼ u3 Substituting y1 and y2,
Consider the field tower K⊂K〈u〉⊂K〈u; y〉
ð9Þ
By Property 1, o
o
o
diff tr d K〈u; y〉=K ¼ diff tr d K〈u; y〉=K〈u〉 þ diff tr d K〈u〉=K
ð10Þ
Replacing (7) and (8) into (10), we obtain o
diff tr d K〈u; y〉=K ¼ m K⊂K〈y〉⊂K〈u; y〉
ð12Þ
By using Property 1, o
o
diff tr d K〈u; y〉=K ¼ diff tr d K〈u; y〉=K〈y〉 þ diff tr d K〈y〉=K
ð13Þ
Substituting (11) into (13), o
o
m ¼ diff tr d K〈u; y〉=K〈y〉 þ diff tr d K〈y〉=K Since the differential transcendence degree is not negative, we have that o
ρ ¼ diff tr d K〈y〉=K≤m o
In a similar manner, y ¼ ðy1 ; :; yp Þ and ρ ¼ diff tr d K〈y〉=K≤p. Finally, o
y_ 2 u3 − y x3 x23 2 y_ 2 u2 − y x2 x22 2
¼ ¼ ¼
9 u1 > > > > > = u2 > > > > > u3 > > ;
ð18Þ
ð11Þ
Now, let us consider the field tower
o
y_ 1
ρ ¼ diff tr d K〈y〉=K≤minðm; pÞ:
□
The differential output rank ρ is also the maximum number of outputs that are related by a differential polynomial equation with coefficients over K (independent of x and u). A practical way, for certain simple cases, to determine the differential output rank is by taking into account all possible
By (18), we have that in this case there exists no any differential equation in which the outputs appear (independent of x and u). We conclude that the two outputs are differentially dependent and therefore ρ ¼ 2. Definition 10. A system is left-invertible if, and only if, the differential output rank is equal to the total number of inputs, i.e., ρ ¼ m. Example 8. Let us consider once again the system (15). In this case we have that m ¼ 1, p¼ 2, and ρ ¼ 1, then the system is left invertible (ρ ¼ m), this implies that it will be possible to recover the input by means of the available outputs. In fact, the input can be found since the following differential equation is satisfied: uy_ 1 ¼ u3 þ y_ 2 u_ which is a Bernoulli equation and by means of a change of variable z ¼ u−2 it is transformed in the linear differential equation z_ þ 2
2 y_ 1 z¼ ; y_ 2 y_ 2
where y_ 1 and y_ 2 are known functions of t.
R. Martínez-Guerra et al. / ISA Transactions 52 (2013) 652–661
Example 9. Now, let us consider the system with two inputs and one output: ) x_ 1 ¼ u1 ð19Þ x_ 2 ¼ u2 with y1 ¼ x1 x2 we have that ρ ¼ p ¼ 1, m ¼2, then the system (19) is not left invertible. Proposition 1 (Fliess [25]). Let us consider a class of systems given by (2). A system is said to be left invertible if and only if o
o
ρ ¼ diff tr d K〈y〉=K ¼ diff tr d K〈u; f 〉=K:
□
Property 1 is the main tool used to prove the following theorem that looks quite natural. The theorem shows the relationship between the diagnosability and the left invertibility condition.
655
Example 10. Let us consider the system x_ 1 ¼ x2 þ f 1 þ f 2 x_ 2 ¼ x1 þ f 1
with
y1 ¼ x1
)
y2 ¼ x2
ð26Þ
The differential output rank of (26) is 2 since there exists no relation hr such that hr ðy1 ; y2 Þ ¼ 0: In fact, because ρ is equal to the number of faults, we conclude that the system (26) is left invertible, in other words, f1 and f2 are diagnosable. To verify the result, we substitute y1 and y2 in (26), then y_ 1 ¼ y2 þ f 1 þ f 2 y_ 2 ¼ y1 þ f 1
ð27Þ
From (27), Theorem 1. If system (2) is left invertible, then the fault vector f can be obtained by means of the output vector.
f 1 ¼ y_ 2 −y1 f 2 ¼ y_ 1 −y_ 2 þ y1 −y2
Proof. Let us consider the following field towers: K⊂K〈u〉⊂K〈u; f 〉⊂K〈u; y; f 〉;
ð20Þ
K⊂K〈y〉⊂K〈u; y〉⊂K〈u; y; f 〉;
ð21Þ
From (20) and Property 1, we have o
o
diff tr d K〈u; y; f 〉=K ¼ diff tr d K〈u; y; f 〉=K〈u; f 〉 o
o
þdiff tr d K〈u; f 〉=K〈u〉 þ diff tr d K〈u〉=K
Example 11. Consider the system that describes the growth of methanol in a bioreactor, where f represents the presence of an unexpected catalyst in the substrate concentration that produces wrong measures of methanol: ) x_ 1 ¼ f μðx2 Þx1 þ ux1 with y ¼ x2 ð28Þ x_ 2 ¼ −sðx2 Þx1 þ u½B−x2
¼0þμþm ¼mþμ
ð22Þ where o
From Proposition 1, diff tr d K〈y〉=K ¼ m þ μ. By using this fact in (21) we obtain o
o
diff tr d K〈u; y; f 〉=K ¼ diff tr d K〈u; y; f 〉=K〈u; y〉 o
o
þdiff tr d K〈u; y〉=K〈y〉 þ diff tr d K〈y〉=K o
¼ diff tr d K〈u; y; f 〉=K〈u; y〉 o
þdiff tr d K〈u; y〉=K〈y〉 þ m þ μ
ð23Þ
represents the density of the methylomonas x1 x2 represents the methanol concentration u constant of dilution rate μð Þ specific growth rate of substrate, μð Þ 4 0 sð Þ consumption rate of substrate, sð Þ 4 0 In this case the differential output rank is equal to 1 since there does not exist any relation hr (independent of x1, x2, u and f) such that hr ðyÞ ¼ 0:
From (22) and (23) we have o
o
diff tr d K〈u; y; f 〉=K〈u; y〉 þ diff tr d K〈u; y〉=K〈y〉 þ m þ μ ¼ m þ μ: This implies that o
o
diff tr d K〈u; y; f 〉=K〈u; y〉 ¼ −diff tr d K〈u; y〉=K〈y〉
ð24Þ
Since the transcendence degree is always positive, we have the following: o
diff tr d K〈u; y; f 〉=K〈u; y〉 ¼ 0
ð25Þ
This means that f is differentially algebraic over K〈u; y〉. Thus, the diagnosability condition is satisfied and the theorem is proven. □
The system (28) is left invertible because the differential output rank is equal to the number of faults and inputs. This implies that system (28) is diagnosable, where the fault can be expressed as follows: _ _ ysðyÞ− € _ _ ½uðB−yÞ−u y− sðyÞ½uðB−yÞ− y uðB−yÞ−y_ −u sðyÞ s2 ðyÞ f¼ uðB−yÞ−y_ μðyÞ sðyÞ
ð29Þ
_ From (29), we have that the fault is diagnosable if ½uðB−yÞ−y≠0. Moreover, the unknown state x1 is algebraically observable since it satisfies an equation in terms of y, i.e., x1 ¼
uðB−yÞ−y_ : sðyÞ
4. Reduced-order observer 3.1. Illustrative examples In this section we present some academic examples in which the left invertibility condition is applied.
Let us consider system (2). 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
656
R. Martínez-Guerra et al. / ISA Transactions 52 (2013) 652–661
Lemma 2. If a fault signal fi, i∈f1; …; μg, of system (2) is algebraically observable and can be written in the following form:
unknown fault vector. The new extended system is given by _ ¼ Aðx; uÞ xðtÞ _f ¼ Ωðx; uÞ
f i ¼ ai y_ þ bi ðu; yÞ
yðtÞ ¼ hðx; uÞ
ð30Þ T
nþmþμ
μ
where Ωðx; uÞ ¼ ðΩ1 ðx; uÞ; …; Ωμ ðx; uÞÞ : R -R is an uncertain function. Note that a classic Luenberger observer cannot be constructed because the term Ωðx; uÞ is unknown. This problem is overcome by using a reduced order uncertainty observer in order to estimate the failure variable f. Next lemma describes the construction of a proportional reduced order observer for (30).
f^ i ¼ ki ðf i −f^ i Þ;
Proof. Let us define the estimation error εi ðtÞ as
2
γ_ i ¼ −ki γ i þ ki bi ðu; yÞ−ki ai y;
ð40Þ
□
ðrÞ
ð41Þ
where r is the maximum order of the output time derivatives. Introducing the following change of coordinates: ð34Þ
exp½−ki ðt−τÞ dτ
by solving the integral we have N 0≤jεi ðtÞj≤expð−ki tÞjεi0 j þ ½1−expð−ki tÞ ki When t-∞, 0≤lim supjεi ðtÞj≤lim supt-∞ expð−ki tÞjεi0 j t-∞ N þlim sup ½1−expð−ki tÞ k t-∞ i simplifying, N ki
and the proof is completed.
From (37) and (39) we have
_ ψ ðf ; y; y; y…; y ; u; u…Þ ¼0
From H1,
t-∞
ð39Þ
ð33Þ
0
0≤lim supjεi ðtÞj≤
̇ _ γ_ i ≜f^ i −ki ai y:
Consider the nonlinear system with faults given by (2), assuming that the fault vector f is algebraically observable over R〈u; y〉 and therefore it satisfies a differential algebraic polynomial
then, by applying the triangle and Schwarz inequalities, the following is obtained: Z t jεi ðtÞj≤expð−ki tÞjεi0 þ expð−ki tÞj expðki τÞΩi ðτÞ dτ
0
ð38Þ
we get
ð32Þ
0
t
ð37Þ
5. Sliding-mode observer
0
Z 0≤jεi ðtÞj≤expð−ki tÞjεi0 j þ N
ð36Þ
Proof. From (31) and (35) we obtain
where γ i ∈C .
The dynamics of the error εi ðtÞ can be expressed as
where εi0 is the initial condition. Then Eq. (33) yields Z t jεi ðtÞj ¼ expð−ki tÞ εi0 þ expðki τÞΩi ðτÞ dτ
γ i ð0Þ ¼ γ i0 ∈R
i i
i
1
εi ðtÞ ¼ f i −f^ i
The solution of Eq. (32) is given by Z t εi ðtÞ ¼ expð−ki tÞ εi0 þ expðki τÞΩi ðτÞ dτ
i
γ i ≜f^ i −ki ai y;
is a reduced order observer for system (30), where f^ i denotes the estimate of fault fi and ki ∈Rþ ∀i ¼ 1; …; μ are positive real coefficients that determine the desired convergence rate of the observer.
ε_ i ðtÞ þ ki εðtÞ ¼ Ωi ðx; uÞ
2
γ_ i ¼ −ki γ i þ ki bi ðu; yÞ−ki ai y; f^ ¼ γ þ k a y
Let us define
ð31Þ
1≤i≤μ
where ai ¼ ½ai1 …; aim ∈R is a constant vector and bi ðu; yÞ is a bounded function, then there exists a function γ i ∈C 1 , such that the reduced order observer (31) can be written as the following asymptotically stable system:
̇ f^ i ¼ ki ai y_ þ ki bi ðu; yÞ−ki f^ i :
Lemma 1. If the following hypotheses are satisfied: H1: Ωðx; uÞ is bounded, i.e., jΩi ðx; uÞj≤N∈Rþ ∀1≤i≤μ. H2: f(t) is algebraically observable over R〈u; y〉. Then the system ̇
ð35Þ m
□
Remark 2. Sometimes the output time derivatives (which are unknown) appear in the algebraic equation of the fault, then it is necessary to use an auxiliary variable to avoid using them as is described in the next lemma.
η1 ¼ y;
ðr−1Þ
η2 ¼ y; …; ηr ¼ y
ð42Þ
we obtain the following representation of (41) which is the socalled Generalized Observability Canonical Form [18]:
η1 ¼ η2
η2 ¼ η3 ⋮
ðr−1Þ
ηr ¼ Φðf ; η1 ; η2 …ηr ; u; u; ‥ u Þ y ¼ η1 ;
ð43Þ
where Φð Þ is considered as an unmodeled dynamic. The observer structure: The following system is a sliding-mode observer for the system (43): ^ η^ ̇ i ¼ η^ 2 þ m1 signðy−yÞ ⋮ ^ η^ ̇ r−1 ¼ η^ r þ mr−1 signðy−yÞ ^ η^ ̇ r ¼ mr signðy−yÞ with y^ ¼ η^ 1
ð44Þ
where mj 4 0, ∀ 1≤j≤r, and 8 ^ 40 if ðy−yÞ > <1 ^ o0 ^ −1 if ðy− yÞ signðy−yÞ ¼ > : undefined if ðy−yÞ ^ ¼ 0: Then returning to the original coordinates and taking into account (41), the fault can be estimated from the following relationship: ðrÞ
_ ψ ðf^ ; η^ ; η^ ̇ ; η^€ …; η^ ; u; u…Þ ¼0
ð45Þ
R. Martínez-Guerra et al. / ISA Transactions 52 (2013) 652–661
Observer convergence analysis: We will analyze the convergence properties of the proposed observer considering the presence of a noise signal δ contaminating the output measurements, such that y ¼ η1 þ δ:
ð46Þ
ei ¼ ðηi −^η i Þ=m; i ¼ 2…; r;
ð47Þ
where m 40, it follows that the estimation error vector e ¼ ½e1 … er T verifies the relationship
e ¼ Aμ e−K signðCe þ δÞ þ Δs
ð48Þ
where μ 4 0 is a regularizing parameter, 2 3 −μ m 0 … 0 2 3 m1 6 0 −μ m 0 7 6 7 6 m2 7 6 7 6 7 ⋮ 7 0 −μ Aμ ¼ 6 7; 6 0 7; K ¼ 6 4 ⋮ 5 6 7 ⋱ m5 4 mr 0 0 0 … −μ 2 C ¼ ½1 0 … 0
μe1
ð53Þ
VðeÞ≜eT Pe ¼ ∥e∥2P
ð54Þ
where 0 o P ¼ P T ∈Rrr is the solution of the Riccati equation (51). By taking the time derivative of (54) and taking into account (48) it yields V_ ðeÞ ¼ 2eT P e_ ¼ 2eT P½Aμ e−K signðCe þ δÞ þ Δs; −1
according to assumption A3, K ¼ kP C T , then the previous equation can be written as T V_ ðeÞ ¼ 2eT PAμ e−2ke C T signðCe þ δÞ þ 2eT PΔs:
By using the following matrix inequality: X T Y þ Y T X≤X T Λs X þ Y T Λ−1 s Y which is valid for any X, Y∈Rrm , 0 o Λs ¼ ΛTs ∈Rrr , then it follows that
3
6 7 ⋮ 6 7 7 Δs ¼ 6 6 μer−1 7 4 5 Φ þ μer
and
and the function ½þ is defined as follows: ( x if x≥0 ½xþ ¼ 0 if x o0:
Proof. Let V(e) be the following Lyapunov candidate function:
Let us define the state estimation errors as e1 ¼ η1 −^η 1 ;
657
T T V_ ðeÞ≤eT ðPAμ þ ATμ PÞe−2ke C T signðCe þ δÞ þ eT PΛ−1 s Pe þ ðΔsÞ Λs Δs ;
from assumption A1 the following is obtained: ðΔsÞT Λs Δs ≤∥Δs∥2 ∥Λs ∥≤½L0s þ ðL1s þ ∥Aμ ∥Þ∥e∥2 ∥Λs ∥;
are uncertainty terms.
then Assumption 1. There exist nonnegative constants L0s , L1s , such that the following generalized quasi-Lipschitz condition holds:
T V_ ðeÞ≤eT ðPAμ þ ATμ P þ PΛ−1 s P þ Q Þe−e Qe T
∥Δs∥≤L0s þ ðL1s þ ∥Aμ ∥Þ∥e∥:
Assumption 2. The additive output noise δ is bounded, namely jδj≤δþ o ∞;
ð50Þ
PAμ þ
þ PRP þ Q ¼ 0
ð51Þ
with R≔Λ−1 s þ 2∥Λs ∥L1s I;
Λs ¼ ΛTs 40;
has a positive definite solution P ¼ P T 4 0.
ð52Þ
where
2∥Λs ∥L20s þ 4kδþ λmin ðP
Q QP
−1=2
ð55Þ
In order to eliminate the discontinuity contained in the function signð Þ in (55) the following inequality, valid for any x, y∈R, is considered:
x signðx þ yÞ≥jxj−2jyj:
ð56Þ
Now using (56) in (55) the following inequality is obtained: V_ ðeÞ≤−eT Qe−2kjCej þ 2L20s ∥Λs ∥ þ 4kδþ ;
ð57Þ
which can be rewritten as V_ ðeÞ≤−∥e∥2Q þ 2L20s ∥Λs ∥ þ 4kδþ ; that is to say, ð58Þ
where α≜λmin ðP −1=2 Q T QP −1=2 Þ 4 0; β ¼ 2L20s ∥Λs ∥ þ 4kδþ :
V ¼ VðeÞ ¼ ∥e∥2P ≔eT Pe; T
T V_ ðeÞ≤−eT Qe−2ke C T signðCe þ δÞ þ 2L20s ∥Λs ∥:
V_ ðeÞ≤−αQ VðeÞ þ β;
Theorem 2. If assumptions from A1 to A3 are satisfied, then ½V−V n þ -0
and taking (51) into account, it follows that
and furthermore jx þ yj≥jxj−jyj, then
Remark 3. A1 only limits the maximum slope present in the uncertainty term Δs which depends on the Lipschitz properties of ηr . A2 is a standard assumption that allows us to avoid involving the statistic behavior of the noise signal. The expression (51) from A3 has a positive definite solution if the matrix Aμ is stable, which is true for any μ 4 0. Since P 4 0, there exists k 4 0 such that −1 K ¼ kP C T , then A3 provides an additional degree of freedom to choose the gain k which can be used to establish the size of the region defined by μ. ~
−1=2
T V_ ðeÞ≤eT ðPAμ þ ATμ P þ PRP þ Q Þe−eT Qe−2ke C T signðCe þ δÞ
x signðx þ yÞ ¼ ðx þ yÞsignðx þ yÞ−y signðx þ yÞ ≥jx þ yj−jyj
Q ¼ Q 0 þ 2ðL1s þ ∥Aμ ∥2 ÞI
V n≔
from the definition of matrix R in assumption A3, the previous expression can be rewritten as
þ2L20s ∥Λs ∥;
Assumption 3. There exists a positive definite matrix Q 0 ¼ Q T0 4 0, such that the following matrix Riccati equation: ATμ P
−2ke C T signðCe þ δÞ þ 2½L20s þ ðL1s þ ∥Aμ ∥Þ2 ∥e∥2 ∥Λs ∥;
ð49Þ
Þ
;
Now, considering the following differential equation related to (58): V_ ðeÞ ¼ −αV þ β;
ð59Þ
658
R. Martínez-Guerra et al. / ISA Transactions 52 (2013) 652–661
which is linear and stable and such that V-V n as t-∞, where V n is the single equilibrium point of Eq. (59): Vn ¼
β ≥0; α
it follows that the function Gt ≜½V−V n 2þ where ½þ is defined as in (53), according to (58) satisfies (for any V≠V n Þ G_ t ≤−2½V −V n þ ½−αV þ β≤0 subtracting −αV n þ β ¼ 0, it yields G_ t ≤−2αðV−V n Þ½V −V n þ ≤0 that is to say, G_ t ≤−2αGt ≤0: Integrating the last inequality it follows that Z t Gτ dτ; Gt −G0 ≤−2α 0
in other words Z t Gτ dτ≤G0 −Gt ≤G0 2α
6.2. Diagnosability analysis
0
According to Theorem 1 we need two or more measured outputs, this can only happen in the following cases:
then Z
t
lim 2α
t-∞
where u1 ¼ q1 and u2 ¼ q2 are the manipulable input flows, xi ¼ hi is the level in the tank i. A is the transversal constant section of any of the identical tanks, and qij represents the water flow from tank i to tank j (1≤i; j≤3) which according to the generalized Torricelli's rule, is valid for laminar flow qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qij ¼ ai S signðhi −hj Þ 2gjhi −hj j ð61Þ pffiffiffiffiffiffiffiffiffiffiffi with q20 ¼ a2 S 2gh2 , where S is the transversal area of the pipe that interconnects the tanks (see Fig. 1) and ai are the output flow coefficients, which are not exactly known, so they are considered as uncertain parameters. We assume the existence of actuator faults denoted by f1 and f2 (μ ¼ 2), each one of these faults represents a variation in the respective pump driver gain, which can be originated by an electronic component malfunction, or even by a leakage or an obstruction in the pump pipes. The system (60) has four state regions in which the corresponding model is differentiable [17], any of these regions can be chosen to do the analysis, just avoiding the loss of differentiability by crossing from one to another. In this work x1 4 x3 4 x2 40 is the only considered region of operation, which experimentally is easy to operate.
0
Gτ dτ≤G0 :
From Barbalat Lemma [26], it follows Gt -0, which is equivalent to say ½V−V n þ -0. □ Remark 4. Theorem 2 states that the weighted estimation error norm V(e) asymptotically converges to the zone bounded by V n . In other words, it is ultimately bounded. 6. Application to the three-tank system 6.1. Description of the three-tank system The Amira DTS200 is described in Fig. 1. The corresponding model with faults is given by the following equations [22]: 1 x_ 1 ¼ ðu1 −q13 þ f 1 Þ A 1 x_ 2 ¼ ðu2 þ q32 −q20 þ f 2 Þ A 1 x_ 3 ¼ ðq13 −q32 Þ A
Case Case Case Case
0: p ¼ 3 (h1 , h2 , and h3 measurable). 1: p ¼ 2 (h1 not measurable, h2 and h3 measurable). 2: p ¼ 2 (h2 not measurable, h1 and h3 measurable). 3. p¼ 2 (h3 not measurable, h1 and h2 measurable).
6.2.1. Case 0 The simplest case (and the only one reported in previous works [17], with numerical results) takes place when we can measure the full state vector, that is to say, we have three outputs: y1 ¼ x1 , y2 ¼ x2 , y3 ¼ x3 ; in this case, from (60) we have pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð62Þ f 1 ¼ Ay_ 1 þ a1 S 2gðy1 −y3 Þ−u1 f 2 ¼ Ay_ 2 −a3 S
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 2gðy3 −y2 Þ þ a2 S 2gy2 −u2
ð63Þ
System (60) is left invertible because the differential output rank is equal to 2. This means that faults f1 and f2 are diagnosable.
ð60Þ
6.2.2. Case 1 We consider only the outputs y2 ¼ x2 and y3 ¼ x3 . By taking into account (60) we have pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ay_ 3 ¼ a1 S 2gðx1 −y3 Þ−a3 S 2gðy3 −y2 Þ; ð64Þ we get x1 ¼ y3 þ
1 2ga21 S2
Ay_ 3 þ a3 S
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 2gðy3 −y2 Þ
ð65Þ
Then, by replacing x1 in (62) we obtain a set of two differential equations with coefficients in R〈u; y〉 with two unknowns f1 and f2, this means that system (60) is left invertible (i.e., faults f1 and f2 are diagnosable) with the two considered outputs.
Fig. 1. Schematic diagram of the three-tank system.
6.2.3. Case 2 We consider only the outputs: y1 ¼ x1 and y3 ¼ x3 . By taking into account (64) we obtain pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 x2 ¼ y3 − −Ay_ 3 þ a1 S 2gðy1 −y3 Þ : ð66Þ 2 2 2ga3 S
R. Martínez-Guerra et al. / ISA Transactions 52 (2013) 652–661
659
Fig. 2. (a) Evolution of the parameter identification. Coefficients a1, a2, a3. (b) Validation of the estimated model. Levels h1, h2, h3.
From (63) in a similar way we can obtain system (60) that is left invertible (i.e., faults f1 and f2 are diagnosable) with the two considered outputs.
6.2.4. Case 3 We consider only the outputs y1 ¼ x1 and y2 ¼ x2 . By taking into account (62) we get x3 ¼ y1 −
1 2ga21 S2
ð−A y_ 1 þ f 1 þ u1 Þ2 :
ð67Þ
From (63) we can only obtain one differential equation involving the two faults, therefore, system (60) is not left invertible, i.e., faults f1 and f2 are not diagnosable with the two considered outputs.
6.3. Fault reconstruction We present two novel observers to obtain effective fault estimations, as well as they can be used to estimate time derivatives as follows. Reduced order observer: Let us consider the following time derivative to be estimated: _ η ¼ y:
ð69Þ
introducing the change of variable η^ ¼ γ þ k3 y
2
γ_ ¼ −k3 γ−k3 y
ð71Þ
then (71) together with (70) constitutes an asymptotic estimator for η. Sliding-mode observer: We introduce the following change of variables: η1 ¼ y, η2 ¼ η1 , then we obtain the following observer: ) η^̇ 1 ¼ η^ 2 þ m1 signðy−^η 1 Þ ð72Þ η^̇ 2 ¼ m2 signðy−^η 1 Þ which can be used to estimate η2 from the knowledge of y. 7. Experimental results We verified the real time performance of the proposed estimators in a laboratory setting of the Amira DTS200 system. The known parameter values for the utilized system are A ¼0.0149 m2, S ¼ 5 10−5 m2 and the unknown parameters a1 , a2 , and a3 . The sample time in all the experiments was 0.001 s, this was chosen to be so small in order to get the best performance from the slidingmode observer. The experimental results are described as follows. 7.1. Identification results
ð68Þ
According to (31), we propose the observer structure ^ η^ ̇ ¼ k3 ðη−ηÞ
and from (69) and (70) we can get γ_ ¼ −k3 η^ , then again from (70)
ð70Þ
With no presence of faults, the unknown parameters a1 , a2 , and a3 are algebraically observable, in other words, they can be expressed in terms of inputs, outputs and time derivatives of these variables, respectively, as follows: q1 −Ay_ 1 ffi a1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S 2gðy1 −y3 Þ
ð73Þ
660
a2 ¼
R. Martínez-Guerra et al. / ISA Transactions 52 (2013) 652–661
q1 þ q2 −Aðy_ 1 þ y_ 2 þ y_ 3 Þ pffiffiffiffiffiffiffiffiffiffi S 2gy2
q −Aðy_ 1 þ y_ 3 Þ ffi a3 ¼ 1pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S 2gðy3 −y2 Þ
ð74Þ
ð75Þ
Estimations of these parameters were obtained by estimating the time derivatives involved in the expressions (73)–(75); meanwhile the values for the input flows were q1 ¼ 0:000025 m3 =s and q2 ¼ 0:000020 m3 =s, along 1000 s in these conditions the evolution of the estimated values for the unknown coefficients is shown in Fig. 2a. At the end of the identification process the estimated values for the flow parameters were obtained. These are the values that were used in the fault estimation schemes (62) and (63) as they were implemented in the conditions described in the next section. These values were the following: a1 ¼ 0:418;
a2 ¼ 0:789;
a3 ¼ 0:435
ð76Þ
In Fig. 2b the simulated and the measured actual levels are shown in order to give a visual comparison between the actual and the estimated model, the actual level measurements are drawn in a gray color, while the levels obtained by simulating the model using the estimated values given by (76) for the flow coefficients are shown in black color.
7.2. Fault estimation results In all the experiments described in this subsection the input flows were maintained constant as q1 ¼ 0:00002 m3 =s and q2 ¼ 0:000015 m3 =s, also two faults were artificially generated through the following expressions: f 1 ¼ 0:00005½1þ sin ð0:2te−0:01t ÞUðt−220Þ, f 2 ¼ 0:00005½1 þ sin ð0:05te−0:001t ÞUðt−300Þ, where UðtÞ is the unit step function.
As we do not know the dynamics Φ, we can take as a reference the Lipschitz constants of the fault signals, which are 10:6 10−7 and 11:25 10−7 , then we choose L1s bigger enough, for example L1s ¼ 0:001, in a similar way, we choose m ¼0.1, μ ¼ 1, Λs ¼ 20, Q 0 ¼ I, then R ¼ 0:09I, Q ¼ 3:2122I, with these parameters we obtain 20:4009 −1:2107 P¼ 40 −1:2107 20:5446 The two proposed schemes for fault estimation were evaluated in case 1 (x1 not measurable), the results are described as follows. Only the two outputs y2 ¼ x2 and y3 ¼ x3 were taken into account; an estimation for the unknown state x1 was necessary to be obtained. In Fig. 3 we show the resulting estimations achieved with the reduced-order observer. A low-pass filter was necessary in order to reduce the effect of the measurement noise, we chose a second-order Butterwort filter whose transfer function is given by Gf ðsÞ ¼ 1=ð32s2 þ 8s þ 1Þ. The gain values chosen for both the fault observers were k1 ¼ k2 ¼ 2, and for the state observer x1 , kx1 ¼ 0:3. As we can observe the estimation results with this scheme are good (Fig. 3). A sliding-mode observer was also tested in this case. In Fig. 4 the corresponding results achieved with the sliding-mode observer are shown. It is worth to mention that with this observer it was not necessary to include the reducing noise filter providing the inherent robustness of this observer. The gain values chosen for the fault and state observers were m1 ¼ 0:1 and m2 ¼ 0:01. As we can observe from Fig. 4, this scheme also provides good estimation results.
8. Concluding remarks We have tackled the fault diagnosis problem in nonlinear systems using the condition of left invertibility through the
Fig. 3. Fault diagnosis for unknown h1 using the reduced order observer: (a) Levels. (b) Actual and estimated f1. (c) Actual and estimated f2.
R. Martínez-Guerra et al. / ISA Transactions 52 (2013) 652–661
661
Fig. 4. Fault diagnosis for unknown h1 using the sliding mode observer: (a) Levels. (b) Actual and estimated f1. (c) Actual and estimated f2.
concept of differential output rank. The usefulness of Theorems 1, 2 and Lemma 2 was shown; this allowed the estimation of two simultaneous faults with less measurements. The theoretical and simulation results were tested in a real-time implementation (three-tank system). The experimental results for the two observers showed similar performance, however the proposed slidingmode observer is more robust against measurement noise, as it was expected. References [1] Alcorta García E, Frank P. Deterministic nonlinear observer-based approaches to fault diagnosis: a survey. Control Engineering Practice 1997;5:663–70. [2] Frank P, Ding X. Survey of robust residual generation and evaluation methods in observer-based fault detection systems. Journal of Process Control 1977;7: 403–24. [3] Willsky A. A survey of design methods in observer-based fault detection systems. Automatica 1976;1(2):601–11. [4] Massoumnia Verghese G, Willsky A. Failure detection and identification. IEEE Transactions on Automatic Control 1989;34:316–21. [5] Chen J, Patton R. Robust model-based fault diagnosis for dynamic systems. Kluwer Academic Publishers; 1999. [6] Blanke M, Kinnaert M, Lunze J, Staroswiecki M. Diagnosis and fault-tolerant control. Berlin: Springer; 2003. [7] Noura H, Theilliol D, Ponsart JC, Chamseddine A. Fault-tolerant control systems: design and practical applications. London: Springer; 2009. [8] Zhang Dan, Yu Li, Wang Qing-Guo. Fault detection for a class of network-based nonlinear systems with communication constraints and random packet dropouts. ISA Transactions 2011;25(10):876–98. [9] Hu Di, Sarosh Ali, Dong Yun-Feng. A novel KFCM based fault diagnosis method for unknown faults in satellite reaction wheels. ISA Transactions 2012;51 (2):309–16. [10] Wang Tao, Xie Wenfang, Zhang Youmin. Sliding mode fault tolerant control dealing with modeling uncertainties and actuator faults. ISA Transactions 2012;51(3):386–92. [11] Tong Shaocheng, Li Han-Xiong. Fuzzy adaptive sliding-mode control for MIMO nonlinear systems. IEEE Transactions on Fuzzy Systems 2003;11(3):354–60.
[12] Tong Shaocheng, Li Yongming. Observer-based fuzzy adaptive control for strict-feedback nonlinear systems. Fuzzy Sets and Systems 2009;160(12): 1749–64. [13] Wang Qing-Guo, Zhang Yu. A fast algorithm for reduced-order modeling. ISA Transactions 1999;38(3):225–30. [14] De Persis C, Isidori A. A geometric approach to nonlinear fault detection and isolation. IEEE Transactions on Automatic Control 2001;46(6):853–65. [15] Join C, Ponsart J-C, Sauter D, Theilliol D. Nonlinear filter design for fault diagnosis: application to the three-tank system. IEE Proceedings—Control Theory and Applications 2005;152(1):55–64. [16] Fliess M, Join C, Mounier H. An introduction to nonlinear fault diagnosis with an application to a congested internet router. In: Abdallah CT, Chiasson J, editors. Advances in communication control networks. Lecture notes in control and information sciences, vol. 308. Berlin: Springer; 2005. p. 327–43. [17] Join C, Sira-Ramírez H, Fliess M. Control of an uncertain three tank system via on-line parameter identification and fault detection. In: Proceedings of 16th triennial world IFAC conference (IFAC’05), Prague, Czech Republic, July 2005. [18] Fliess M. Nonlinear control theory and differential algebra. In: Byrnes C, Kurzhanski A, editors. Modelling and adaptive control. Lecture notes in control and information sciences, vol. 105. Berlin: Springer; 1988. p. 134–45. [19] Fliess M, Join C, Sira-Ramírez H. Robust residual generation for nonlinear fault diagnosis: an algebraic setting with examples. International Journal of Control 2004;14(77). [20] Fliess M, Join C, Sira-Ramirez H. Non-linear estimation is easy. International Journal of Modelling Identification and Control 2008;4(1):12–27. [21] Nagy AM, Marx B, Mourot G, Schutz G, Ragot J. State estimation of the threetank system using a multiple model. In: 48th IEEE conference on decision and control, Shanghai, PR China, December 2009. p. 7795–800. [22] Amira DTS200: Laboratory setup three tank system. Duisburgh, Germany: Amira Gmbh; 1996. [23] Theilliol D, Noura H, Ponsart J-C. Fault diagnosis and accommodation of a three-tank system based on analytical redundancy. ISA Transactions 2002;41: 365–82. [24] Kolchin E. Differential algebra and algebraic groups. New York: Academic Press; 1973. [25] Fliess M. A note on invertibility of nonlinear input–output differential systems. System & Control Letters 1986;8:147–51. [26] Khalil HK. Nonlinear systems. Prentice Hall; 2002.
Contents lists available at ScienceDirect
ISA Transactions journal homepage: www.elsevier.com/locate/isatrans
Fault diagnosis viewed as a left invertibility problem R. Martínez-Guerra a,n, J.L. Mata-Machuca a,c, J.J. Rincón-Pasaye b a
Departamento de Control Automático, CINVESTAV-IPN, Av. IPN 2508, 07360 DF, Mexico Facultad de Ingeniería Eléctrica, Universidad Michoacana de San Nicolás de Hidalgo, Ciudad Universitaria, Morelia, Mexico Instituto Politécnico Nacional, Unidad Profesional Interdisciplinaria en Ingeniería y Tecnologías Avanzadas, Academia de Mecatrónica, Av. IPN 2580, 07340 DF, Mexico b c
art ic l e i nf o
a b s t r a c t
Article history: Received 20 February 2011 Received in revised form 16 April 2013 Accepted 1 June 2013 Available online 6 July 2013 This paper was recommended for publication by Dr. Q.-G. Wang.
This work deals with the fault diagnosis problem, some new properties are found using the left invertibility condition through the concept of differential output rank. Two schemes of nonlinear observers are used to estimate the fault signals for comparison purposes, one of these is a proportional reduced order observer and the other is a sliding mode observer. The methodology is tested in a real time implementation of a three-tank system. & 2013 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Fault diagnosis Left invertibility condition Differential output rank Sliding mode observer Reduced order observer
1. Introduction A fault can be considered as a process degradation or degradation of the equipment performance caused by the change in the physical characteristic of the process, the input process or the external conditions. Industrial control systems have to deal with faults, therefore, fault diagnosis is a very important subject in control theory. System diagnosis helps us to detect and estimate faults in a process. In other words, the task of diagnosis is, from measurements of outputs and inputs, to reconstruct the fault vector. The fault detection and isolation problem have been studied for more than three decades, many papers dealing with this problem can be found, see for instance the surveys [1–4] and the books [5–7]. For the case of nonlinear systems a variety of approaches have been proposed [8–13], such as those based upon differential geometric methods [14,15], and on the other hand, alternative approaches based on an algebraic and differential framework can be found in [16–20]. Currently, the diagnosis problem is playing an important role in modern industrial processes. This has led control theory into a wide variety of model-based approaches which rely on descriptions via differential and/or difference equations, contrary to other standpoints developed mainly among computer scientist (see [18,19] and references therein). The primary objectives of fault n
Corresponding author. Tel.: +52 5557473800; fax: +52 5557473982. E-mail addresses: [email protected] (R. Martínez-Guerra), [email protected] (J.L. Mata-Machuca), [email protected] (J.J. Rincón-Pasaye).
diagnosis are fault detectability and isolability, i.e., the possible location and determination of the faults present in a system and the time of their occurrences. The tasks of fault detection and isolation are to be accomplished by measuring only the input and the output variables. This paper focuses on the diagnosis of nonlinear systems and the goal is to determine malfunctions in the dynamics. In this communication, the outputs are mainly signals obtained from the sensors. Their number is important to know whether a system is diagnosable or not. In this paper, the diagnosis problem is tackled as a left invertibility problem throughout the concept of differential output rank ρ. Two schemes of observers are proposed in order to estimate the fault signals, one of them is a reduced-order observer based on a freemodel approach and another is a sliding-mode observer based on a Generalized Observability Canonical Form (GOCF) [18]. Both schemes are proved to possess asymptotic convergence properties. The class of systems for which this methodology can be applied contains systems that depend on the inputs and their time derivatives in a polynomial form. The type of faults considered in this work is additive and bounded, however, the algebraic approach can also be used to deal with multiplicative faults. These proposals are applied in this paper to a three-tank system [21]. The Amira DTS200 three-tank system [22] has been widely considered for the experimental fault diagnosis study, see for instance [15,17,23], even recently, one work based on the geometric approach has been reported [15]. We can also mention one previous work with the three-tank system using the differential algebraic
0019-0578/$ - see front matter & 2013 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2013.06.001
R. Martínez-Guerra et al. / ISA Transactions 52 (2013) 652–661
approach [17]; in that work the authors only report a numerical simulation study and not a real-time experiment, also they only solve the simplest case in which three measured outputs are available to estimate two faults, that is to say, they do not analyze the minimal number of measurements to attack the diagnosis problem as we do in the present work. The intention of choosing the three-tank system example is to clarify the proposed methodology and to highlight the simplicity and flexibility of the present approach. The three-tank system is known as a system with parameter uncertainties, so this work also deals with these uncertainties by means of algebraic parameter estimation, considering that there is no simultaneous presence of uncertainty and faults, in the same way as it is considered in [17]. This paper is organized as follows. In Section 2, some definitions of differential algebra are given. In Section 3, we discuss the left invertibility condition and we present some examples. In Sections 4 and 5 we give a brief description of the proposed observers. In Section 6 the three-tank system is analyzed. Finally, in Section 7 we illustrate this methodology with some experimental results.
2. Some definitions Some basic definitions are introduced. Further details can be found in [17,18] and references therein. Definition 1. Let L and K be differential fields. A differential field extension L=K is given by K and L such that: (1) K is a subfield of L and (2) the derivation of K is the restriction to K of the derivation of L. Example 1. R〈et 〉=R is a differential field extension, where R D _ ¼ 0. R〈et 〉. et being a solution of x−x Definition 2. Let ξ ¼ ðξ1 ; ξ2 ; …; ξn Þ satisfies an algebraic differential coefficients in K it is called dependent, otherwise ξ is called independent.
be a set of elements of L. If it _ ξ…Þ € equation Pðξ; ξ; ¼ 0 with differentially Kalgebraically differentially Kalgebraically
Definition 3. Any set of elements of L which is differentially Kalgebraically independent and maximal with respect to inclusion forms is a differential transcendence basis of L=K. Two such bases have the same cardinality. This is called the differential ○ transcendence degree of L=K and denoted by diff tr d L=K. Definition 4. Let G, K〈u〉 be differential fields. A nominal dynamic consists of a finitely generated differential algebraic extension G=K〈u〉, ðG ¼ K〈u; ξ〉; ξ∈GÞ. Any element of G satisfies an algebraic differential equation with coefficients over K in the components of u and their time derivatives. Example 2. Consider the following differential equation: u_ 2 y þ 4u€ ¼ 0 In this case, y is algebraic over K〈u〉, therefore, it can be seen as a dynamic of the form K〈u; y〉=K〈u〉 where K ¼ R and y∈K〈u; y〉. Definition 5. Any unknown variable x in a dynamic is said to be algebraically observable with respect to K〈u; y〉 if x satisfies a differential algebraic equation with coefficients over K in the components of u, y and a finite number of their derivatives. Any dynamic with output y is said to be algebraically observable if, and only if, any state variable has this property.
Example 3. Let us consider the following system: 8 2 _ > < x 1 ¼ 3x1 x2 þ u1 x_ 2 ¼ x1 þ x32 u2 > :y¼x ;
653
ð1Þ
2
since x1 and x2 satisfy the polynomials x1 þ y3 u2 −y_ ¼ 0 and x2 −y ¼ 0, respectively, then x1 ; x2 are algebraically observable over R〈u; y〉 and by applying Definition 5, system (1) is algebraically observable. Definition 6. A fault is not a permitted deviation of at least one characteristic property or parameter of any process in relation to the development of the same parameter under normal conditions. Faults are defined as transcendent elements over K〈u〉, therefore, a system with the presence of faults is a differential transcendental extension, denoted as K〈u; f ; y〉=K〈u〉, where f is a vector that includes the faults and their time derivatives. Definition 7. Let G, K〈u〉 be differential fields. A fault dynamic consists of a finitely generated differential algebraic extension G=K〈u; f 〉, G ¼ K〈u; f ; ξ〉; ξ∈G. Any element of G satisfies an algebraic differential equation with coefficients over K in the components of u; f and their time derivatives. Definition 8 (Algebraic observability condition). A fault f ∈G is said to be diagnosable if it is algebraically observable over R〈u; y〉, i.e., if it is possible to estimate the fault from the available measurements of the system. Let us consider the class of nonlinear systems with faults described by the following equation: ( _ ¼ Aðx; uÞ xðtÞ ð2Þ yðtÞ ¼ hðx; uÞ; where x ¼ ðx1 ; …; xn ÞT ∈Rn is a state vector, u ¼ ðu1 ; …; um Þ∈Rm is a known input vector, f ¼ ðf 1 ; …; f μ Þ∈Rμ is an unknown input vector, u ¼ ðu; f Þ∈Rmþμ , yðtÞ∈Rp is the output vector. A and h are assumed to be analytical vector functions. Example 4. Let us consider the nonlinear system with one fault ðf 1 Þ on the actuator and one fault ðf 2 Þ on the sensor of output y1 : 8 x_ 1 ¼ x1 x2 þ f 1 þ u > > > > < x_ 2 ¼ x1 ð3Þ y1 ¼ x1 þ f 2 > > > > : y ¼ x2 : 2
Since f 1 , f2 satisfy the differential algebraic equations f 1 −y€ 2 þ y2 y_ 2 þ u ¼ 0 f 2 −y1 þ y_ 2 ¼ 0
ð4Þ
the system (3) is diagnosable and the faults can be reconstructed from the knowledge of u, y and their time derivatives. Remark 1. The diagnosability condition is independent of the observability of a system. Example 5. Let us consider the system 8 x_ ¼ x1 x2 þ f þ u > > > 1 > < x_ 2 ¼ x1 x_ 3 ¼ x3 f þ u > > > > : y ¼ x2 :
ð5Þ
In this case f is diagnosable. However, x3 is not algebraically observable.
654
R. Martínez-Guerra et al. / ISA Transactions 52 (2013) 652–661
differential polynomials of the form
3. On the left invertibility condition We have some definitions concerning the differential output rank of a system. Definition 9. The differential output rank ρ of a system is equal to the differential transcendence degree of the differential extension K〈y〉 over the differential field K, i.e., o
ρ ¼ diff tr d K〈y〉=K: Property 1 (Kolchin [24]). Let K, L, M be the differential fields such that K⊂L⊂M. Then o
o
o
diff tr d ðM=KÞ ¼ diff tr d ðM=LÞ þ diff tr d ðL=KÞ
□
ð6Þ
Property 2. The differential output rank ρ of a system is smaller than or equal to minðm; pÞ: o
ρ ¼ diff tr d K〈y〉=K≤minðm; pÞ;
hr ðy1 ; …; yp Þ ¼ 0
ð14Þ
and if it is possible to find r independent relations of the form (14), then the differential output rank is given by ρ ¼ p−r, that is to say, there exist only p−r independent outputs. Example 6. Consider the following system with one input and two outputs: 9 x_ 1 ¼ u > = with y1 ¼ x1 u x_ 2 ¼ x1 ð15Þ > y ¼x ; x_ ¼ x u 3
1
2
3
Replacing y1 and y2, 9 y_ 1 u_ > − 2 y1 ¼ u > > > u u = y1 _x 2 ¼ > u> > > y_ 2 ¼ y1 ;
ð16Þ
where m and p are the total number of inputs and outputs, respectively. A proof of Property 2 can be given in the following manner: an input–output system, with input u ¼ ðu1 ; …; um Þ and output y ¼ ðy1 ; :; yp Þ, is defined by the next conditions:
Only one equation of (16) can be written in the form of (14). Then we have the relation
ðu1 ; …; um Þ are differentially Kalgebraically independent, i.e.,
where only one output is differentially independent and then ρ ¼ 1.
o
diff tr d K〈u〉=K ¼ m
ð7Þ
ðy1 ; …; yp Þ are differentially algebraic over K〈u〉, i.e., K〈u; y〉=K〈u〉 is differentially algebraic or o
diff tr d K〈u; y〉=K〈u〉 ¼ 0
ð8Þ
y1 −y_ 2 ¼ 0
Example 7. Consider the following system with three inputs and two outputs: 9 x_ 1 ¼ u1 > = x_ 2 ¼ u2 with y1 ¼ x1 ð17Þ > y2 ¼ x2 x3 ; x_ 3 ¼ u3 Substituting y1 and y2,
Consider the field tower K⊂K〈u〉⊂K〈u; y〉
ð9Þ
By Property 1, o
o
o
diff tr d K〈u; y〉=K ¼ diff tr d K〈u; y〉=K〈u〉 þ diff tr d K〈u〉=K
ð10Þ
Replacing (7) and (8) into (10), we obtain o
diff tr d K〈u; y〉=K ¼ m K⊂K〈y〉⊂K〈u; y〉
ð12Þ
By using Property 1, o
o
diff tr d K〈u; y〉=K ¼ diff tr d K〈u; y〉=K〈y〉 þ diff tr d K〈y〉=K
ð13Þ
Substituting (11) into (13), o
o
m ¼ diff tr d K〈u; y〉=K〈y〉 þ diff tr d K〈y〉=K Since the differential transcendence degree is not negative, we have that o
ρ ¼ diff tr d K〈y〉=K≤m o
In a similar manner, y ¼ ðy1 ; :; yp Þ and ρ ¼ diff tr d K〈y〉=K≤p. Finally, o
y_ 2 u3 − y x3 x23 2 y_ 2 u2 − y x2 x22 2
¼ ¼ ¼
9 u1 > > > > > = u2 > > > > > u3 > > ;
ð18Þ
ð11Þ
Now, let us consider the field tower
o
y_ 1
ρ ¼ diff tr d K〈y〉=K≤minðm; pÞ:
□
The differential output rank ρ is also the maximum number of outputs that are related by a differential polynomial equation with coefficients over K (independent of x and u). A practical way, for certain simple cases, to determine the differential output rank is by taking into account all possible
By (18), we have that in this case there exists no any differential equation in which the outputs appear (independent of x and u). We conclude that the two outputs are differentially dependent and therefore ρ ¼ 2. Definition 10. A system is left-invertible if, and only if, the differential output rank is equal to the total number of inputs, i.e., ρ ¼ m. Example 8. Let us consider once again the system (15). In this case we have that m ¼ 1, p¼ 2, and ρ ¼ 1, then the system is left invertible (ρ ¼ m), this implies that it will be possible to recover the input by means of the available outputs. In fact, the input can be found since the following differential equation is satisfied: uy_ 1 ¼ u3 þ y_ 2 u_ which is a Bernoulli equation and by means of a change of variable z ¼ u−2 it is transformed in the linear differential equation z_ þ 2
2 y_ 1 z¼ ; y_ 2 y_ 2
where y_ 1 and y_ 2 are known functions of t.
R. Martínez-Guerra et al. / ISA Transactions 52 (2013) 652–661
Example 9. Now, let us consider the system with two inputs and one output: ) x_ 1 ¼ u1 ð19Þ x_ 2 ¼ u2 with y1 ¼ x1 x2 we have that ρ ¼ p ¼ 1, m ¼2, then the system (19) is not left invertible. Proposition 1 (Fliess [25]). Let us consider a class of systems given by (2). A system is said to be left invertible if and only if o
o
ρ ¼ diff tr d K〈y〉=K ¼ diff tr d K〈u; f 〉=K:
□
Property 1 is the main tool used to prove the following theorem that looks quite natural. The theorem shows the relationship between the diagnosability and the left invertibility condition.
655
Example 10. Let us consider the system x_ 1 ¼ x2 þ f 1 þ f 2 x_ 2 ¼ x1 þ f 1
with
y1 ¼ x1
)
y2 ¼ x2
ð26Þ
The differential output rank of (26) is 2 since there exists no relation hr such that hr ðy1 ; y2 Þ ¼ 0: In fact, because ρ is equal to the number of faults, we conclude that the system (26) is left invertible, in other words, f1 and f2 are diagnosable. To verify the result, we substitute y1 and y2 in (26), then y_ 1 ¼ y2 þ f 1 þ f 2 y_ 2 ¼ y1 þ f 1
ð27Þ
From (27), Theorem 1. If system (2) is left invertible, then the fault vector f can be obtained by means of the output vector.
f 1 ¼ y_ 2 −y1 f 2 ¼ y_ 1 −y_ 2 þ y1 −y2
Proof. Let us consider the following field towers: K⊂K〈u〉⊂K〈u; f 〉⊂K〈u; y; f 〉;
ð20Þ
K⊂K〈y〉⊂K〈u; y〉⊂K〈u; y; f 〉;
ð21Þ
From (20) and Property 1, we have o
o
diff tr d K〈u; y; f 〉=K ¼ diff tr d K〈u; y; f 〉=K〈u; f 〉 o
o
þdiff tr d K〈u; f 〉=K〈u〉 þ diff tr d K〈u〉=K
Example 11. Consider the system that describes the growth of methanol in a bioreactor, where f represents the presence of an unexpected catalyst in the substrate concentration that produces wrong measures of methanol: ) x_ 1 ¼ f μðx2 Þx1 þ ux1 with y ¼ x2 ð28Þ x_ 2 ¼ −sðx2 Þx1 þ u½B−x2
¼0þμþm ¼mþμ
ð22Þ where o
From Proposition 1, diff tr d K〈y〉=K ¼ m þ μ. By using this fact in (21) we obtain o
o
diff tr d K〈u; y; f 〉=K ¼ diff tr d K〈u; y; f 〉=K〈u; y〉 o
o
þdiff tr d K〈u; y〉=K〈y〉 þ diff tr d K〈y〉=K o
¼ diff tr d K〈u; y; f 〉=K〈u; y〉 o
þdiff tr d K〈u; y〉=K〈y〉 þ m þ μ
ð23Þ
represents the density of the methylomonas x1 x2 represents the methanol concentration u constant of dilution rate μð Þ specific growth rate of substrate, μð Þ 4 0 sð Þ consumption rate of substrate, sð Þ 4 0 In this case the differential output rank is equal to 1 since there does not exist any relation hr (independent of x1, x2, u and f) such that hr ðyÞ ¼ 0:
From (22) and (23) we have o
o
diff tr d K〈u; y; f 〉=K〈u; y〉 þ diff tr d K〈u; y〉=K〈y〉 þ m þ μ ¼ m þ μ: This implies that o
o
diff tr d K〈u; y; f 〉=K〈u; y〉 ¼ −diff tr d K〈u; y〉=K〈y〉
ð24Þ
Since the transcendence degree is always positive, we have the following: o
diff tr d K〈u; y; f 〉=K〈u; y〉 ¼ 0
ð25Þ
This means that f is differentially algebraic over K〈u; y〉. Thus, the diagnosability condition is satisfied and the theorem is proven. □
The system (28) is left invertible because the differential output rank is equal to the number of faults and inputs. This implies that system (28) is diagnosable, where the fault can be expressed as follows: _ _ ysðyÞ− € _ _ ½uðB−yÞ−u y− sðyÞ½uðB−yÞ− y uðB−yÞ−y_ −u sðyÞ s2 ðyÞ f¼ uðB−yÞ−y_ μðyÞ sðyÞ
ð29Þ
_ From (29), we have that the fault is diagnosable if ½uðB−yÞ−y≠0. Moreover, the unknown state x1 is algebraically observable since it satisfies an equation in terms of y, i.e., x1 ¼
uðB−yÞ−y_ : sðyÞ
4. Reduced-order observer 3.1. Illustrative examples In this section we present some academic examples in which the left invertibility condition is applied.
Let us consider system (2). 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
656
R. Martínez-Guerra et al. / ISA Transactions 52 (2013) 652–661
Lemma 2. If a fault signal fi, i∈f1; …; μg, of system (2) is algebraically observable and can be written in the following form:
unknown fault vector. The new extended system is given by _ ¼ Aðx; uÞ xðtÞ _f ¼ Ωðx; uÞ
f i ¼ ai y_ þ bi ðu; yÞ
yðtÞ ¼ hðx; uÞ
ð30Þ T
nþmþμ
μ
where Ωðx; uÞ ¼ ðΩ1 ðx; uÞ; …; Ωμ ðx; uÞÞ : R -R is an uncertain function. Note that a classic Luenberger observer cannot be constructed because the term Ωðx; uÞ is unknown. This problem is overcome by using a reduced order uncertainty observer in order to estimate the failure variable f. Next lemma describes the construction of a proportional reduced order observer for (30).
f^ i ¼ ki ðf i −f^ i Þ;
Proof. Let us define the estimation error εi ðtÞ as
2
γ_ i ¼ −ki γ i þ ki bi ðu; yÞ−ki ai y;
ð40Þ
□
ðrÞ
ð41Þ
where r is the maximum order of the output time derivatives. Introducing the following change of coordinates: ð34Þ
exp½−ki ðt−τÞ dτ
by solving the integral we have N 0≤jεi ðtÞj≤expð−ki tÞjεi0 j þ ½1−expð−ki tÞ ki When t-∞, 0≤lim supjεi ðtÞj≤lim supt-∞ expð−ki tÞjεi0 j t-∞ N þlim sup ½1−expð−ki tÞ k t-∞ i simplifying, N ki
and the proof is completed.
From (37) and (39) we have
_ ψ ðf ; y; y; y…; y ; u; u…Þ ¼0
From H1,
t-∞
ð39Þ
ð33Þ
0
0≤lim supjεi ðtÞj≤
̇ _ γ_ i ≜f^ i −ki ai y:
Consider the nonlinear system with faults given by (2), assuming that the fault vector f is algebraically observable over R〈u; y〉 and therefore it satisfies a differential algebraic polynomial
then, by applying the triangle and Schwarz inequalities, the following is obtained: Z t jεi ðtÞj≤expð−ki tÞjεi0 þ expð−ki tÞj expðki τÞΩi ðτÞ dτ
0
ð38Þ
we get
ð32Þ
0
t
ð37Þ
5. Sliding-mode observer
0
Z 0≤jεi ðtÞj≤expð−ki tÞjεi0 j þ N
ð36Þ
Proof. From (31) and (35) we obtain
where γ i ∈C .
The dynamics of the error εi ðtÞ can be expressed as
where εi0 is the initial condition. Then Eq. (33) yields Z t jεi ðtÞj ¼ expð−ki tÞ εi0 þ expðki τÞΩi ðτÞ dτ
γ i ð0Þ ¼ γ i0 ∈R
i i
i
1
εi ðtÞ ¼ f i −f^ i
The solution of Eq. (32) is given by Z t εi ðtÞ ¼ expð−ki tÞ εi0 þ expðki τÞΩi ðτÞ dτ
i
γ i ≜f^ i −ki ai y;
is a reduced order observer for system (30), where f^ i denotes the estimate of fault fi and ki ∈Rþ ∀i ¼ 1; …; μ are positive real coefficients that determine the desired convergence rate of the observer.
ε_ i ðtÞ þ ki εðtÞ ¼ Ωi ðx; uÞ
2
γ_ i ¼ −ki γ i þ ki bi ðu; yÞ−ki ai y; f^ ¼ γ þ k a y
Let us define
ð31Þ
1≤i≤μ
where ai ¼ ½ai1 …; aim ∈R is a constant vector and bi ðu; yÞ is a bounded function, then there exists a function γ i ∈C 1 , such that the reduced order observer (31) can be written as the following asymptotically stable system:
̇ f^ i ¼ ki ai y_ þ ki bi ðu; yÞ−ki f^ i :
Lemma 1. If the following hypotheses are satisfied: H1: Ωðx; uÞ is bounded, i.e., jΩi ðx; uÞj≤N∈Rþ ∀1≤i≤μ. H2: f(t) is algebraically observable over R〈u; y〉. Then the system ̇
ð35Þ m
□
Remark 2. Sometimes the output time derivatives (which are unknown) appear in the algebraic equation of the fault, then it is necessary to use an auxiliary variable to avoid using them as is described in the next lemma.
η1 ¼ y;
ðr−1Þ
η2 ¼ y; …; ηr ¼ y
ð42Þ
we obtain the following representation of (41) which is the socalled Generalized Observability Canonical Form [18]:
η1 ¼ η2
η2 ¼ η3 ⋮
ðr−1Þ
ηr ¼ Φðf ; η1 ; η2 …ηr ; u; u; ‥ u Þ y ¼ η1 ;
ð43Þ
where Φð Þ is considered as an unmodeled dynamic. The observer structure: The following system is a sliding-mode observer for the system (43): ^ η^ ̇ i ¼ η^ 2 þ m1 signðy−yÞ ⋮ ^ η^ ̇ r−1 ¼ η^ r þ mr−1 signðy−yÞ ^ η^ ̇ r ¼ mr signðy−yÞ with y^ ¼ η^ 1
ð44Þ
where mj 4 0, ∀ 1≤j≤r, and 8 ^ 40 if ðy−yÞ > <1 ^ o0 ^ −1 if ðy− yÞ signðy−yÞ ¼ > : undefined if ðy−yÞ ^ ¼ 0: Then returning to the original coordinates and taking into account (41), the fault can be estimated from the following relationship: ðrÞ
_ ψ ðf^ ; η^ ; η^ ̇ ; η^€ …; η^ ; u; u…Þ ¼0
ð45Þ
R. Martínez-Guerra et al. / ISA Transactions 52 (2013) 652–661
Observer convergence analysis: We will analyze the convergence properties of the proposed observer considering the presence of a noise signal δ contaminating the output measurements, such that y ¼ η1 þ δ:
ð46Þ
ei ¼ ðηi −^η i Þ=m; i ¼ 2…; r;
ð47Þ
where m 40, it follows that the estimation error vector e ¼ ½e1 … er T verifies the relationship
e ¼ Aμ e−K signðCe þ δÞ þ Δs
ð48Þ
where μ 4 0 is a regularizing parameter, 2 3 −μ m 0 … 0 2 3 m1 6 0 −μ m 0 7 6 7 6 m2 7 6 7 6 7 ⋮ 7 0 −μ Aμ ¼ 6 7; 6 0 7; K ¼ 6 4 ⋮ 5 6 7 ⋱ m5 4 mr 0 0 0 … −μ 2 C ¼ ½1 0 … 0
μe1
ð53Þ
VðeÞ≜eT Pe ¼ ∥e∥2P
ð54Þ
where 0 o P ¼ P T ∈Rrr is the solution of the Riccati equation (51). By taking the time derivative of (54) and taking into account (48) it yields V_ ðeÞ ¼ 2eT P e_ ¼ 2eT P½Aμ e−K signðCe þ δÞ þ Δs; −1
according to assumption A3, K ¼ kP C T , then the previous equation can be written as T V_ ðeÞ ¼ 2eT PAμ e−2ke C T signðCe þ δÞ þ 2eT PΔs:
By using the following matrix inequality: X T Y þ Y T X≤X T Λs X þ Y T Λ−1 s Y which is valid for any X, Y∈Rrm , 0 o Λs ¼ ΛTs ∈Rrr , then it follows that
3
6 7 ⋮ 6 7 7 Δs ¼ 6 6 μer−1 7 4 5 Φ þ μer
and
and the function ½þ is defined as follows: ( x if x≥0 ½xþ ¼ 0 if x o0:
Proof. Let V(e) be the following Lyapunov candidate function:
Let us define the state estimation errors as e1 ¼ η1 −^η 1 ;
657
T T V_ ðeÞ≤eT ðPAμ þ ATμ PÞe−2ke C T signðCe þ δÞ þ eT PΛ−1 s Pe þ ðΔsÞ Λs Δs ;
from assumption A1 the following is obtained: ðΔsÞT Λs Δs ≤∥Δs∥2 ∥Λs ∥≤½L0s þ ðL1s þ ∥Aμ ∥Þ∥e∥2 ∥Λs ∥;
are uncertainty terms.
then Assumption 1. There exist nonnegative constants L0s , L1s , such that the following generalized quasi-Lipschitz condition holds:
T V_ ðeÞ≤eT ðPAμ þ ATμ P þ PΛ−1 s P þ Q Þe−e Qe T
∥Δs∥≤L0s þ ðL1s þ ∥Aμ ∥Þ∥e∥:
Assumption 2. The additive output noise δ is bounded, namely jδj≤δþ o ∞;
ð50Þ
PAμ þ
þ PRP þ Q ¼ 0
ð51Þ
with R≔Λ−1 s þ 2∥Λs ∥L1s I;
Λs ¼ ΛTs 40;
has a positive definite solution P ¼ P T 4 0.
ð52Þ
where
2∥Λs ∥L20s þ 4kδþ λmin ðP
Q QP
−1=2
ð55Þ
In order to eliminate the discontinuity contained in the function signð Þ in (55) the following inequality, valid for any x, y∈R, is considered:
x signðx þ yÞ≥jxj−2jyj:
ð56Þ
Now using (56) in (55) the following inequality is obtained: V_ ðeÞ≤−eT Qe−2kjCej þ 2L20s ∥Λs ∥ þ 4kδþ ;
ð57Þ
which can be rewritten as V_ ðeÞ≤−∥e∥2Q þ 2L20s ∥Λs ∥ þ 4kδþ ; that is to say, ð58Þ
where α≜λmin ðP −1=2 Q T QP −1=2 Þ 4 0; β ¼ 2L20s ∥Λs ∥ þ 4kδþ :
V ¼ VðeÞ ¼ ∥e∥2P ≔eT Pe; T
T V_ ðeÞ≤−eT Qe−2ke C T signðCe þ δÞ þ 2L20s ∥Λs ∥:
V_ ðeÞ≤−αQ VðeÞ þ β;
Theorem 2. If assumptions from A1 to A3 are satisfied, then ½V−V n þ -0
and taking (51) into account, it follows that
and furthermore jx þ yj≥jxj−jyj, then
Remark 3. A1 only limits the maximum slope present in the uncertainty term Δs which depends on the Lipschitz properties of ηr . A2 is a standard assumption that allows us to avoid involving the statistic behavior of the noise signal. The expression (51) from A3 has a positive definite solution if the matrix Aμ is stable, which is true for any μ 4 0. Since P 4 0, there exists k 4 0 such that −1 K ¼ kP C T , then A3 provides an additional degree of freedom to choose the gain k which can be used to establish the size of the region defined by μ. ~
−1=2
T V_ ðeÞ≤eT ðPAμ þ ATμ P þ PRP þ Q Þe−eT Qe−2ke C T signðCe þ δÞ
x signðx þ yÞ ¼ ðx þ yÞsignðx þ yÞ−y signðx þ yÞ ≥jx þ yj−jyj
Q ¼ Q 0 þ 2ðL1s þ ∥Aμ ∥2 ÞI
V n≔
from the definition of matrix R in assumption A3, the previous expression can be rewritten as
þ2L20s ∥Λs ∥;
Assumption 3. There exists a positive definite matrix Q 0 ¼ Q T0 4 0, such that the following matrix Riccati equation: ATμ P
−2ke C T signðCe þ δÞ þ 2½L20s þ ðL1s þ ∥Aμ ∥Þ2 ∥e∥2 ∥Λs ∥;
ð49Þ
Þ
;
Now, considering the following differential equation related to (58): V_ ðeÞ ¼ −αV þ β;
ð59Þ
658
R. Martínez-Guerra et al. / ISA Transactions 52 (2013) 652–661
which is linear and stable and such that V-V n as t-∞, where V n is the single equilibrium point of Eq. (59): Vn ¼
β ≥0; α
it follows that the function Gt ≜½V−V n 2þ where ½þ is defined as in (53), according to (58) satisfies (for any V≠V n Þ G_ t ≤−2½V −V n þ ½−αV þ β≤0 subtracting −αV n þ β ¼ 0, it yields G_ t ≤−2αðV−V n Þ½V −V n þ ≤0 that is to say, G_ t ≤−2αGt ≤0: Integrating the last inequality it follows that Z t Gτ dτ; Gt −G0 ≤−2α 0
in other words Z t Gτ dτ≤G0 −Gt ≤G0 2α
6.2. Diagnosability analysis
0
According to Theorem 1 we need two or more measured outputs, this can only happen in the following cases:
then Z
t
lim 2α
t-∞
where u1 ¼ q1 and u2 ¼ q2 are the manipulable input flows, xi ¼ hi is the level in the tank i. A is the transversal constant section of any of the identical tanks, and qij represents the water flow from tank i to tank j (1≤i; j≤3) which according to the generalized Torricelli's rule, is valid for laminar flow qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qij ¼ ai S signðhi −hj Þ 2gjhi −hj j ð61Þ pffiffiffiffiffiffiffiffiffiffiffi with q20 ¼ a2 S 2gh2 , where S is the transversal area of the pipe that interconnects the tanks (see Fig. 1) and ai are the output flow coefficients, which are not exactly known, so they are considered as uncertain parameters. We assume the existence of actuator faults denoted by f1 and f2 (μ ¼ 2), each one of these faults represents a variation in the respective pump driver gain, which can be originated by an electronic component malfunction, or even by a leakage or an obstruction in the pump pipes. The system (60) has four state regions in which the corresponding model is differentiable [17], any of these regions can be chosen to do the analysis, just avoiding the loss of differentiability by crossing from one to another. In this work x1 4 x3 4 x2 40 is the only considered region of operation, which experimentally is easy to operate.
0
Gτ dτ≤G0 :
From Barbalat Lemma [26], it follows Gt -0, which is equivalent to say ½V−V n þ -0. □ Remark 4. Theorem 2 states that the weighted estimation error norm V(e) asymptotically converges to the zone bounded by V n . In other words, it is ultimately bounded. 6. Application to the three-tank system 6.1. Description of the three-tank system The Amira DTS200 is described in Fig. 1. The corresponding model with faults is given by the following equations [22]: 1 x_ 1 ¼ ðu1 −q13 þ f 1 Þ A 1 x_ 2 ¼ ðu2 þ q32 −q20 þ f 2 Þ A 1 x_ 3 ¼ ðq13 −q32 Þ A
Case Case Case Case
0: p ¼ 3 (h1 , h2 , and h3 measurable). 1: p ¼ 2 (h1 not measurable, h2 and h3 measurable). 2: p ¼ 2 (h2 not measurable, h1 and h3 measurable). 3. p¼ 2 (h3 not measurable, h1 and h2 measurable).
6.2.1. Case 0 The simplest case (and the only one reported in previous works [17], with numerical results) takes place when we can measure the full state vector, that is to say, we have three outputs: y1 ¼ x1 , y2 ¼ x2 , y3 ¼ x3 ; in this case, from (60) we have pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð62Þ f 1 ¼ Ay_ 1 þ a1 S 2gðy1 −y3 Þ−u1 f 2 ¼ Ay_ 2 −a3 S
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 2gðy3 −y2 Þ þ a2 S 2gy2 −u2
ð63Þ
System (60) is left invertible because the differential output rank is equal to 2. This means that faults f1 and f2 are diagnosable.
ð60Þ
6.2.2. Case 1 We consider only the outputs y2 ¼ x2 and y3 ¼ x3 . By taking into account (60) we have pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ay_ 3 ¼ a1 S 2gðx1 −y3 Þ−a3 S 2gðy3 −y2 Þ; ð64Þ we get x1 ¼ y3 þ
1 2ga21 S2
Ay_ 3 þ a3 S
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 2gðy3 −y2 Þ
ð65Þ
Then, by replacing x1 in (62) we obtain a set of two differential equations with coefficients in R〈u; y〉 with two unknowns f1 and f2, this means that system (60) is left invertible (i.e., faults f1 and f2 are diagnosable) with the two considered outputs.
Fig. 1. Schematic diagram of the three-tank system.
6.2.3. Case 2 We consider only the outputs: y1 ¼ x1 and y3 ¼ x3 . By taking into account (64) we obtain pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 x2 ¼ y3 − −Ay_ 3 þ a1 S 2gðy1 −y3 Þ : ð66Þ 2 2 2ga3 S
R. Martínez-Guerra et al. / ISA Transactions 52 (2013) 652–661
659
Fig. 2. (a) Evolution of the parameter identification. Coefficients a1, a2, a3. (b) Validation of the estimated model. Levels h1, h2, h3.
From (63) in a similar way we can obtain system (60) that is left invertible (i.e., faults f1 and f2 are diagnosable) with the two considered outputs.
6.2.4. Case 3 We consider only the outputs y1 ¼ x1 and y2 ¼ x2 . By taking into account (62) we get x3 ¼ y1 −
1 2ga21 S2
ð−A y_ 1 þ f 1 þ u1 Þ2 :
ð67Þ
From (63) we can only obtain one differential equation involving the two faults, therefore, system (60) is not left invertible, i.e., faults f1 and f2 are not diagnosable with the two considered outputs.
6.3. Fault reconstruction We present two novel observers to obtain effective fault estimations, as well as they can be used to estimate time derivatives as follows. Reduced order observer: Let us consider the following time derivative to be estimated: _ η ¼ y:
ð69Þ
introducing the change of variable η^ ¼ γ þ k3 y
2
γ_ ¼ −k3 γ−k3 y
ð71Þ
then (71) together with (70) constitutes an asymptotic estimator for η. Sliding-mode observer: We introduce the following change of variables: η1 ¼ y, η2 ¼ η1 , then we obtain the following observer: ) η^̇ 1 ¼ η^ 2 þ m1 signðy−^η 1 Þ ð72Þ η^̇ 2 ¼ m2 signðy−^η 1 Þ which can be used to estimate η2 from the knowledge of y. 7. Experimental results We verified the real time performance of the proposed estimators in a laboratory setting of the Amira DTS200 system. The known parameter values for the utilized system are A ¼0.0149 m2, S ¼ 5 10−5 m2 and the unknown parameters a1 , a2 , and a3 . The sample time in all the experiments was 0.001 s, this was chosen to be so small in order to get the best performance from the slidingmode observer. The experimental results are described as follows. 7.1. Identification results
ð68Þ
According to (31), we propose the observer structure ^ η^ ̇ ¼ k3 ðη−ηÞ
and from (69) and (70) we can get γ_ ¼ −k3 η^ , then again from (70)
ð70Þ
With no presence of faults, the unknown parameters a1 , a2 , and a3 are algebraically observable, in other words, they can be expressed in terms of inputs, outputs and time derivatives of these variables, respectively, as follows: q1 −Ay_ 1 ffi a1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S 2gðy1 −y3 Þ
ð73Þ
660
a2 ¼
R. Martínez-Guerra et al. / ISA Transactions 52 (2013) 652–661
q1 þ q2 −Aðy_ 1 þ y_ 2 þ y_ 3 Þ pffiffiffiffiffiffiffiffiffiffi S 2gy2
q −Aðy_ 1 þ y_ 3 Þ ffi a3 ¼ 1pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S 2gðy3 −y2 Þ
ð74Þ
ð75Þ
Estimations of these parameters were obtained by estimating the time derivatives involved in the expressions (73)–(75); meanwhile the values for the input flows were q1 ¼ 0:000025 m3 =s and q2 ¼ 0:000020 m3 =s, along 1000 s in these conditions the evolution of the estimated values for the unknown coefficients is shown in Fig. 2a. At the end of the identification process the estimated values for the flow parameters were obtained. These are the values that were used in the fault estimation schemes (62) and (63) as they were implemented in the conditions described in the next section. These values were the following: a1 ¼ 0:418;
a2 ¼ 0:789;
a3 ¼ 0:435
ð76Þ
In Fig. 2b the simulated and the measured actual levels are shown in order to give a visual comparison between the actual and the estimated model, the actual level measurements are drawn in a gray color, while the levels obtained by simulating the model using the estimated values given by (76) for the flow coefficients are shown in black color.
7.2. Fault estimation results In all the experiments described in this subsection the input flows were maintained constant as q1 ¼ 0:00002 m3 =s and q2 ¼ 0:000015 m3 =s, also two faults were artificially generated through the following expressions: f 1 ¼ 0:00005½1þ sin ð0:2te−0:01t ÞUðt−220Þ, f 2 ¼ 0:00005½1 þ sin ð0:05te−0:001t ÞUðt−300Þ, where UðtÞ is the unit step function.
As we do not know the dynamics Φ, we can take as a reference the Lipschitz constants of the fault signals, which are 10:6 10−7 and 11:25 10−7 , then we choose L1s bigger enough, for example L1s ¼ 0:001, in a similar way, we choose m ¼0.1, μ ¼ 1, Λs ¼ 20, Q 0 ¼ I, then R ¼ 0:09I, Q ¼ 3:2122I, with these parameters we obtain 20:4009 −1:2107 P¼ 40 −1:2107 20:5446 The two proposed schemes for fault estimation were evaluated in case 1 (x1 not measurable), the results are described as follows. Only the two outputs y2 ¼ x2 and y3 ¼ x3 were taken into account; an estimation for the unknown state x1 was necessary to be obtained. In Fig. 3 we show the resulting estimations achieved with the reduced-order observer. A low-pass filter was necessary in order to reduce the effect of the measurement noise, we chose a second-order Butterwort filter whose transfer function is given by Gf ðsÞ ¼ 1=ð32s2 þ 8s þ 1Þ. The gain values chosen for both the fault observers were k1 ¼ k2 ¼ 2, and for the state observer x1 , kx1 ¼ 0:3. As we can observe the estimation results with this scheme are good (Fig. 3). A sliding-mode observer was also tested in this case. In Fig. 4 the corresponding results achieved with the sliding-mode observer are shown. It is worth to mention that with this observer it was not necessary to include the reducing noise filter providing the inherent robustness of this observer. The gain values chosen for the fault and state observers were m1 ¼ 0:1 and m2 ¼ 0:01. As we can observe from Fig. 4, this scheme also provides good estimation results.
8. Concluding remarks We have tackled the fault diagnosis problem in nonlinear systems using the condition of left invertibility through the
Fig. 3. Fault diagnosis for unknown h1 using the reduced order observer: (a) Levels. (b) Actual and estimated f1. (c) Actual and estimated f2.
R. Martínez-Guerra et al. / ISA Transactions 52 (2013) 652–661
661
Fig. 4. Fault diagnosis for unknown h1 using the sliding mode observer: (a) Levels. (b) Actual and estimated f1. (c) Actual and estimated f2.
concept of differential output rank. The usefulness of Theorems 1, 2 and Lemma 2 was shown; this allowed the estimation of two simultaneous faults with less measurements. The theoretical and simulation results were tested in a real-time implementation (three-tank system). The experimental results for the two observers showed similar performance, however the proposed slidingmode observer is more robust against measurement noise, as it was expected. References [1] Alcorta García E, Frank P. Deterministic nonlinear observer-based approaches to fault diagnosis: a survey. Control Engineering Practice 1997;5:663–70. [2] Frank P, Ding X. Survey of robust residual generation and evaluation methods in observer-based fault detection systems. Journal of Process Control 1977;7: 403–24. [3] Willsky A. A survey of design methods in observer-based fault detection systems. Automatica 1976;1(2):601–11. [4] Massoumnia Verghese G, Willsky A. Failure detection and identification. IEEE Transactions on Automatic Control 1989;34:316–21. [5] Chen J, Patton R. Robust model-based fault diagnosis for dynamic systems. Kluwer Academic Publishers; 1999. [6] Blanke M, Kinnaert M, Lunze J, Staroswiecki M. Diagnosis and fault-tolerant control. Berlin: Springer; 2003. [7] Noura H, Theilliol D, Ponsart JC, Chamseddine A. Fault-tolerant control systems: design and practical applications. London: Springer; 2009. [8] Zhang Dan, Yu Li, Wang Qing-Guo. Fault detection for a class of network-based nonlinear systems with communication constraints and random packet dropouts. ISA Transactions 2011;25(10):876–98. [9] Hu Di, Sarosh Ali, Dong Yun-Feng. A novel KFCM based fault diagnosis method for unknown faults in satellite reaction wheels. ISA Transactions 2012;51 (2):309–16. [10] Wang Tao, Xie Wenfang, Zhang Youmin. Sliding mode fault tolerant control dealing with modeling uncertainties and actuator faults. ISA Transactions 2012;51(3):386–92. [11] Tong Shaocheng, Li Han-Xiong. Fuzzy adaptive sliding-mode control for MIMO nonlinear systems. IEEE Transactions on Fuzzy Systems 2003;11(3):354–60.
[12] Tong Shaocheng, Li Yongming. Observer-based fuzzy adaptive control for strict-feedback nonlinear systems. Fuzzy Sets and Systems 2009;160(12): 1749–64. [13] Wang Qing-Guo, Zhang Yu. A fast algorithm for reduced-order modeling. ISA Transactions 1999;38(3):225–30. [14] De Persis C, Isidori A. A geometric approach to nonlinear fault detection and isolation. IEEE Transactions on Automatic Control 2001;46(6):853–65. [15] Join C, Ponsart J-C, Sauter D, Theilliol D. Nonlinear filter design for fault diagnosis: application to the three-tank system. IEE Proceedings—Control Theory and Applications 2005;152(1):55–64. [16] Fliess M, Join C, Mounier H. An introduction to nonlinear fault diagnosis with an application to a congested internet router. In: Abdallah CT, Chiasson J, editors. Advances in communication control networks. Lecture notes in control and information sciences, vol. 308. Berlin: Springer; 2005. p. 327–43. [17] Join C, Sira-Ramírez H, Fliess M. Control of an uncertain three tank system via on-line parameter identification and fault detection. In: Proceedings of 16th triennial world IFAC conference (IFAC’05), Prague, Czech Republic, July 2005. [18] Fliess M. Nonlinear control theory and differential algebra. In: Byrnes C, Kurzhanski A, editors. Modelling and adaptive control. Lecture notes in control and information sciences, vol. 105. Berlin: Springer; 1988. p. 134–45. [19] Fliess M, Join C, Sira-Ramírez H. Robust residual generation for nonlinear fault diagnosis: an algebraic setting with examples. International Journal of Control 2004;14(77). [20] Fliess M, Join C, Sira-Ramirez H. Non-linear estimation is easy. International Journal of Modelling Identification and Control 2008;4(1):12–27. [21] Nagy AM, Marx B, Mourot G, Schutz G, Ragot J. State estimation of the threetank system using a multiple model. In: 48th IEEE conference on decision and control, Shanghai, PR China, December 2009. p. 7795–800. [22] Amira DTS200: Laboratory setup three tank system. Duisburgh, Germany: Amira Gmbh; 1996. [23] Theilliol D, Noura H, Ponsart J-C. Fault diagnosis and accommodation of a three-tank system based on analytical redundancy. ISA Transactions 2002;41: 365–82. [24] Kolchin E. Differential algebra and algebraic groups. New York: Academic Press; 1973. [25] Fliess M. A note on invertibility of nonlinear input–output differential systems. System & Control Letters 1986;8:147–51. [26] Khalil HK. Nonlinear systems. Prentice Hall; 2002.