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Fault Diagnosis of a Class of Uncertain Nonlinear Systems with Lipschitz Nonlinearities
Proceedings of the 7th IFAC Symposium on Fault Detection, Supervision and Safety of Technical Processes Barcelona, Spain, June 30 - July 3, 2009
Fault Diagnosis of a Class of Uncertain Nonlinear Systems with Lipschitz Nonlinearities Xiaodong Zhang ∗ , Marios M. Polycarpou ∗∗ , and Thomas Parisini ∗∗∗ ∗
Dept. of Electrical Engineering, Wright State University, USA Dept. of Electrical and Computer Engineering, University of Cyprus, Cyprus ∗∗∗ Dept. of Electrical, Electronic and Computer Engineering, University of Trieste, Italy ∗∗
Abstract: This paper presents a fault detection and isolation scheme for a class of Lipschitz nonlinear systems with nonlinear and unstructured modeling uncertainty. This significantly extends previous results by considering a more general class of system nonlinearities which are modeled as functions of the system input and partially measurable state variables. A new fault diagnosis method is developed using adaptive estimation techniques. The fault isolability conditions are rigorously established via the use of the so-called fault mismatch functions, which are defined to characterize the mutual difference between pairs of faults. 1. INTRODUCTION In recent years, there has been significant research activity in the design and analysis of fault diagnosis and accommodation schemes. Considerable effort has been devoted to fault diagnosis of nonlinear systems in the framework of various kinds of assumptions and fault scenarios. The idea of using adaptive and learning techniques in fault diagnosis has been presented in (Farrell et al. [1993], Vemuri and Polycarpou [1997], Wang et al. [1997], Zhang et al. [2001], Jiang et al. [2004], Zhang et al. [2005]). In previous papers (Vemuri and Polycarpou [1997], Zhang et al. [2001, 2005]), the authors developed an adaptive approximation based fault diagnosis methodology for a class of nonlinear systems in which the known nonlinearity is represented as a function of measurable system signals (i.e., input and output variables). An assumption was made on the existence of a diffeomorphism that can transform more general nonlinear systems into the class of systems under consideration.
2. PROBLEM FORMULATION Consider a class of nonlinear multi-input-multi-output (MIMO) dynamic systems described by x˙ = Ax + ζ(x, u) + ϕ(x, u, t) + β(t − T0 )Dφ(y, u) (1) y = Cx
In this paper, we extend the previous results by considering a more general class of nonlinear systems without the need of diffeomorphism. More specifically, the known nonlinearity under consideration is a nonlinear function of the system input and state variables and satisfies a Lipschitz condition. With partially measurable states and the possible presence of modeling uncertainty, the design of fault diagnosis methods for such Lipschitz nonlinear systems is a challenging problem. Several researchers have investigated this problem and presented some interesting results under the framework of structured modeling uncertainty and faults. For instance, Vijayaraghavan et al. [2007] addressed the sensor fault diagnosis problem by assuming the absence of modeling uncertainty, and in (Yan and Edwards [2007], Chen and Saif [2007]) sliding mode observer based fault estimation methods were proposed under certain assumptions on the distribution matrices of the structured modeling uncertainty and faults. However, the modeling uncertainty is often unstructured, which makes it difficult to achieve robust fault diagnosis with respect to modeling uncertainty.
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In this research work, we develop an adaptive approximation based fault diagnosis scheme for a class of Lipschitz nonlinear systems with possibly nonlinear and unstructured modeling uncertainty. The objective is to detect the initiation of any faults as soon as possible and to determine whether a particular fault type in a partially known nonlinear fault set has occurred. The fault diagnosis method is presented with a rigorous analytical framework aimed at characterizing the behaviors of the proposed diagnostic scheme. The analysis of the fault diagnosis scheme focuses on: (i) determining adaptive thresholds for fault detection and fault isolation; (ii) deriving fault isolability conditions characterizing the class of faults that can be isolated by the proposed method, based on the so-called fault mismatch function, which provides a suitable measure of the mutual ”difference” between faults.
where x ∈ ℜn is the system state vector, u ∈ ℜm is the input vector, y ∈ ℜp is the output vector, ζ : ℜn × ℜm 7→ ℜn , ϕ : ℜn × ℜm × ℜ+ 7→ ℜn , φ : ℜp × ℜm 7→ ℜq are smooth vector fields. The constant matrices D ∈ ℜn×q and C ∈ ℜp×n with q ≤ p are of full rank, and (A, C) is an observable pair. The model given by x˙ N = AxN + ζ(xN , u) yN = CxN is the known nominal system model. The term ϕ in (1) represents the modeling uncertainty, and β(t−T0 )Dφ(y, u) denotes the changes in the system dynamics due to the occurrence of a fault. Specifically, β(t − T0 ) is the time profile of a fault which occurs at some unknown time T0 , φ(y, u) represents the nonlinear fault function, and D is the fault distribution matrix, which is possibly known. In this paper, we only consider the case of abrupt (sudden) faults, therefore, β(·) takes the form of a step function.
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7th IFAC SAFEPROCESS (SAFEPROCESS’09) Barcelona, Spain, June 30 - July 3, 2009
Moreover, the design and analysis in this paper is based on the assumption that only a single fault occurs.
Assumption 2. The functions η1 and η2 in (3), representing the unstructured modeling uncertainty, are possibly unknown nonlinear functions of z, u, and t, but bounded, i.e., ∀(z, y, u) ∈ Z × Y × U , ∀t ≥ 0,
Assumption 1. The fault distribution matrix D in (1) satisfies
|η1 (z, u, t)| ≤ η¯1 , |η2 (z, u, t)| ≤ η¯2 (y, u, t) ,
• rank(CD) = q • invariant zeros of (A,D,C) lie in the left half plane.
(6)
where the constant bound η¯1 and the bounding function η¯2 (y, u, t) are known, and η¯2 is uniformly bounded in Y × U × ℜ+ . Additionally, Z ⊂ ℜn , U ⊂ ℜm , and Y ⊂ ℜp are compact sets of admissible state variables, inputs, and outputs, respectively.
As described in Yan and Edwards [2007], under Assumption 1, there exists a linear change of coordinates z = T x = [z1⊤ z2⊤ ]⊤ with z1 ∈ ℜ(n−p) and z2 ∈ ℜp , such that A11 A12 −1 • T AT = , where the matrix A11 ∈ A21 A22 (n−p)×(n−p) ℜ is stable. 0 , where D2 ∈ ℜp×q . • TD = D2 • CT −1 = [0 Ip ], where Ip ∈ ℜp×p is an identity matrix.
Assumption 3. The system state vector z remains bounded before and after the occurrence of a fault, i.e., z(t) ∈ L∞ , ∀t ≥ 0. Assumption 4. The rates of change of each fault parameter vector θs (t) in (5) (s = 1, · · · , N ) are uniformly △ bounded, i.e., |θ˙s (t)| ≤ αs for all t ≥ 0, where θs (t) = i h ⊤ ⊤ ⊤ (θ1s (t)) , · · · , θps (t) , and αs is a known constant.
Therefore, in the new coordinate system, the system (1) is described by z˙1 = A11 z1 + A12 z2 + ρ1 (z, u) + η1 (z, u, t) z˙2 = A21 z1 + A22 z2 + ρ2 (z, u) + η2 (z, u, t) (2) +β(t − T0 )D2 φ(y, u) y = z2 , ρ1 η1 where = T ζ(T −1 z, u) and = T ϕ(T −1 z, u, t). ρ2 η2
Assumption 5. The known nonlinear terms ρ1 (z, u) and ρ2 (z, u) in (3) are uniformly Lipschitz in u ∈ U, i.e., ∀z, zˆ ∈ Z,
With a more general structure of the nonlinear fault model, system (2) can be extended to z˙1 = A11 z1 + A12 z2 + ρ1 (z, u) + η1 (z, u, t) z˙2 = A21 z1 + A22 z2 + ρ2 (z, u) + η2 (z, u, t) (3) +β(t − T0 )f (y, u) ¯ y = Cz2 ,
|ρ1 (z, u) − ρ1 (ˆ z , u)| ≤ γ1 |z − zˆ|
(7)
|ρ2 (z, u) − ρ2 (ˆ z , u)| ≤ γ2 |z − zˆ| ,
(8)
where γ1 and γ2 are the known Lipschitz constants for ρ1 (z, u) and ρ2 (z, u), respectively. Assumption 2 characterizes the class of modeling uncertainties under consideration. The bounds on the unstructured modeling uncertainties are needed in order to be able to distinguish between the effects of faults and modeling uncertainty (see Vemuri and Polycarpou [1997], Zhang et al. [2001]). It is worth noting that the unstructured modeling uncertainty considered in this paper is more general than the structured uncertainty considered for Lipschitz nonlinear systems in the fault diagnosis literature (e.g., Yan and Edwards [2007], Chen and Saif [2007], Vijayaraghavan et al. [2007]), which to achieve robustness additionally assumes that certain rank conditions are satisfied by the distribution matrix of the structured modeling uncertainty.
where f : ℜp × ℜm 7→ ℜp is a smooth vector field representing the nonlinear fault function under consideration, and C¯ ∈ ℜp×p is a nonsingular matrix. Clearly, (2) is a special case of (3) with f (y, u) = D2 φ(y, u) and C¯ = Ip . The main objective of this paper is to develop a robust fault detection and isolation (FDI) scheme using adaptive approximation techniques for nonlinear dynamic systems that can be transformed into (3). It is assumed that there are N types of possible fault types in the fault set F; specifically, the unknown fault function f (y, u) in (3) belongs to a finite set of fault types given by △ F = f 1 (y, u), . . . , f N (y, u) . (4)
Assumption 3 requires the boundedness of the state variables before and after the occurrence of a fault. Hence, it is assumed that the feedback control system is capable of retaining the boundedness of the state variables even in the presence of a fault. This is required for well posedness since the FDI design that we consider does not influence the closed-loop dynamics and stability.
Each fault function type f s , s = 1, · · · , N , is described by ⊤ △ f s (y, u) = (θ1s (t))⊤ g1s (y, u), · · · , (θps (t))⊤ gps (y, u) , (5)
where each θis (t), i = 1, · · · , p , is an unknown parameter vector assumed to belong to as known compact and convexs set Θsi (i.e., θis ∈ Θsi ⊂ ℜli ) and gis : ℜp × ℜm 7→ ℜli is a known smooth vector field. As discussed in Zhang et al. [2001], the fault model described by (4) and (5) characterizes a general class of nonlinear faults where the vector field gis represents the functional structure of the s-th fault affecting the i-th state equation, while the unknown parameter vector θis characterizes the timevarying “magnitude” of the fault. Throughout the paper, the following assumptions are introduced:
In Assumption 4, known bounds on the rates of change of θs (t) are assumed. In practice, the rate bounds αs can be set by exploiting some a priori knowledge on the fault developing dynamics. In the case of a constant fault size, we simply set αs = 0. Remark 1. The nominal system dynamics in (3) is similar to the model used in Yan and Edwards [2007]. The objective of this paper is to develop a robust FDI scheme using adaptive approximation techniques, while Yan and Edwards [2007] developed a fault estimation
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method using a sliding mode observer, which requires additional assumptions on the distribution matrices of the modeling uncertainty terms η1 and η2 as well as the fault function f in (3).
Lemma 2. Consider the system described by (3) and the fault detection estimator described by (9). Let k0 and λ0 be positive constants chosen such that ||eA11 t || ≤ k0 e−λ0 t . Assume that λ0 > k0 γ1 , where γ1 is the Lipschitz constant given in (7). Then, for 0 ≤ t < T0 , the state estimation error z˜1 (t) satisfies: k0 η¯1 k0 η¯1 |˜ z1 (t)| ≤ e−(λ0 −k0 γ1 )t , + k0 ω1 − λ0 − k0 γ1 λ0 − k0 γ1 (13) where ω1 is a constant bound for |z1 (0)|, which will be defined later on.
3. FAULT DETECTION AND ISOLATION ARCHITECTURE The fault detection and isolation architecture is based on a bank of N + 1 nonlinear adaptive estimators, where N is the number of different nonlinear fault types in the fault set F. One of the nonlinear adaptive estimators is the fault detection estimator (FDE) used for detecting the occurrence of any faults, while the remaining N nonlinear adaptive estimators are fault isolation estimators (FIEs), which are activated after fault detection for the purpose of determining the particular type of fault that has occurred.
Proof: From (10), we have
z˜1 (t) =
3.1 Fault Detection Scheme
Zt
eA11 (t−τ ) [ρ1 (z, u) − ρ1 (ˆ z , u) + η1 (z, u, τ )] dτ
0
Based on the system model given by (3), the FDE is chosen as follows: z , u) , zˆ1 (0) = 0 zˆ˙ 1 = A11 zˆ1 + A12 C¯ −1 y + ρ1 (ˆ zˆ˙ 2 = A21 zˆ1 + A22 zˆ2 + ρ2 (ˆ z , u) + L(y − yˆ) , zˆ2 (0) = 0 (9) yˆ = C¯ zˆ2 ,
+ eA11 t z˜1 (0) . By using (6), (7), and the triangle inequality, we obtain
|˜ z1 (t)| ≤ γ1
Zt
k0 e−λ0 (t−τ ) |z(τ ) − zˆ(τ )| dτ + k0 e−λ0 t |˜ z1 (0)|
0
where zˆ1 , zˆ2 , and yˆ denote the estimated state and output variables, respectively, L ∈ ℜp×p is a design gain matrix, △ △ and zˆ = [(ˆ z1 )⊤ (C¯ −1 y)⊤ ]⊤ . Let z˜1 = z1 − zˆ1 and
k0 η¯1 + 1 − e−λ0 t λ0 Zt k0 η¯1 ≤ γ1 k0 e−λ0 (t−τ ) |z(τ ) − zˆ(τ )| dτ + λ0 0 η¯1 + k0 ω1 − e−λ0 t , (14) λ0 where k0 and λ0 are positive constants chosen such that ||eA11 t || ≤ k0 e−λ0 t (since A11 is stable, such constants k0 and λ0 always exist (see Ioannou and Sun [1995])), and ω1 is a (possibly conservative) constant bound for |z1 (0)|, such that |˜ z1 (0)| = |z1 (0)| ≤ ω1 . Based on Assumption 3, such a constant bound ω1 can always be chosen.
△
z˜2 = z2 − zˆ2 denote the state estimation errors, and △ y˜ = y− yˆ denote the output estimation error. Then, before fault occurrence (i.e., for t < T0 ), from (3) and (9), we have z˜˙ 1 = A11 z˜1 + ρ1 (z, u) − ρ1 (ˆ z , u) + η1 (10) z˜˙ 2 = A21 z˜1 + A22 z˜2 + ρ2 (z, u) − ρ2 (ˆ z , u) − L(y − yˆ) + η2 = A¯22 z˜2 + A21 z˜1 + ρ2 (z, u) − ρ2 (ˆ z , u) + η2 ¯ 2 − zˆ2 ) = C¯ z˜2 , y˜ = C(z
(11) (12)
△ ¯ Note that, since C¯ is nonsingular, where A¯22 = A22 −LC. we can always choose L to make A¯22 stable.
Note that
In the analysis of the state estimation error z˜1 (t), we will need the following Lemmas.
z(τ ) − zˆ(τ ) =
z1 (τ ) − zˆ1 (τ ) z2 (τ ) − C¯ −1 y(τ )
=
z˜1 (τ ) 0
.
(15)
Lemma 1 . Let t0 be a given time-instant and c0 , c1 , Therefore, we have c2 , λ be nonnegative constants, and κ(t) a nonnegative |z(τ ) − zˆ(τ )| = |˜ z1 (τ )| . (16) piecewise continuous function of time. If h(t) satisfies the inequality By substituting (16) into (14), we obtain Zt h(t) ≤ c0 e−λ(t−t0 ) +c1 +c2 e−λ(t−τ ) κ(τ )h(τ )dτ , ∀t ≥ t0 , Zt η¯1 −λ0 t z1 (τ )| dτ |˜ z1 (t)| ≤ k0 ω1 − e + k0 γ1 e−λ0 (t−τ ) |˜ t0 λ0 then 0 Rt k0 η¯1 κ(s)ds c2 + . (17) h(t) ≤ (c0 + c1 )e−λ(t−t0 ) e t0 λ0 Zt Rt κ(s)ds c Now, applying Lemma 1 to (17) with c0 = k0 ω1 − λη¯10 dτ, ∀t ≥ t0 . + c1 λ e−λ(t−τ ) e 2 τ (we can always choose ω1 to ensure that c0 ≥ 0 ), t0 c1 = kλ0 η0¯1 , c2 = k0 γ1 , and κ = 1, the proof of (13) can Proof: The proof of the above lemma can be found on be immediately concluded. page 103 of the book by Ioannou and Sun [1995].
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3.2 Fault Isolation Estimators and Decision Scheme
Next, we analyze the state estimation error z˜2 (t). The solution of (11) is given by z˜2 (t) =
Zt
eA22 (t−τ ) [A21 z˜1 + ρ2 (z, u) − ρ2 (ˆ z , u)] dτ
+
Zt
Now, assume that a fault is detected at some time Td ; accordingly, at t = Td the fault isolation estimators (FIEs) are activated. Each FIE corresponds to one potential fault type. Specifically, the following N nonlinear adaptive estimators are used as isolation estimators: for s = 1, · · · , N , s z s , u) , zˆ1s (Td ) = 0 zˆ˙ 1 = A11 zˆ1s + A12 C¯ −1 y + ρ1 (ˆ s zˆ˙ 2 = A21 zˆ1s + A22 zˆ2s + ρ2 (ˆ z s , u) + L(y − yˆs ) ˙ s s s s (23) +fˆ (y, u, θˆ ) + Ω θˆ , zˆ2s (Td ) = 0 ˙Ωs = A¯22 Ωs + Gs (y, u), Ωs (Td ) = 0 yˆs = C¯ zˆ2s ,
¯
0
¯
¯
eA22 (t−τ ) η2 (z, u, τ )dτ + eA22 t z˜2 (0) .
(18)
0
Therefore, for each component of the output estimation △ error, i.e., y˜j (t) = C¯j z˜2 (t), j = 1, · · · , p, where C¯j is the ¯ by using (18), (8), (16), (6), j-th row vector of matric C, and the triangle inequality, it can be shown that |˜ yj (t)| ≤ kj
Zt
e
−λj (t−τ )
where zˆ1s , zˆ2s , and yˆs denote the estimated state and output variables, respectively, L ∈ ℜp×p is a design △ △ gain matrix, zˆs = [(ˆ z1s )⊤ (C¯ −1 y)⊤ ]⊤ , fˆs (y, u, θˆs ) = s [(θˆ1s )⊤ g1s (y, u), · · · , (θˆps )⊤ gps (y, u)], and θˆis ∈ ℜli , for i = 1, · · · , p, is the estimate of the fault parameter vector provided by the s-th isolation estimator. It is noted that, according to (5), the fault approximation model fˆs is linear in the adjustable weights θˆs . Consequently, the gradient △ matrix Gs = ∂ fˆs (y, u, θˆs ) / ∂ θˆs = diag [(g1s )⊤ , · · · , (gps )⊤ ] does not depend on θˆs .
(||A21 || + γ2 ) |˜ z1 (τ )| + η¯2 dτ
0
+ kj ω2 e−λj t , (19) where kj and λj are positive constants chosen such that ¯ |C¯j eA22 t | ≤ kj e−λj t (since A¯22 is stable, constants kj and λj satisfying the above inequality always exist (see Ioannou and Sun [1995])), and ω2 is a (possibly conservative) bound for |z2 (0)|, such that |˜ z2 (0)| = |z2 (0)| ≤ ω2 . Based on Assumption 3, such a constant bound ω2 can always be chosen. By using (13) and (19), we obtain |˜ yj (t)| < kj
Zt 0
e−λj (t−τ ) (||A21 || + γ2 ) χ(τ ) + η¯2 (y, u, τ ) dτ
+ kj ω2 e−λj t ,
(20)
where
The adaptation in the isolation estimators arises due to the i⊤ h △ ⊤ ⊤ unknown parameter vector θs = (θ1s ) , · · · , (θns ) . s ˆ The adaptive law for adjusting θ is derived using the Lyapunov synthesis approach (see for example Ioannou and Sun [1995]). Specifically, the learning algorithm is chosen as follows n o ˙ θˆs = PΘs ΓΩs⊤ C¯ ⊤ y˜s , (24) △
where y˜s (t) = y(t) − yˆs (t) denotes the output estimation error of the s-th estimator, Γ > 0 is a symmetric, positivedefinite learning rate matrix, and PΘs is the projection operator restricting θˆs to the corresponding known set Θs (in order to guarantee stability of the learning algorithm in the presence of modeling uncertainty, as described in Ioannou and Sun [1995], Farrell and Polycarpou [2006])). The stability of the FIE can be proved following a similar procedure as reported in (Vemuri and Polycarpou [1997], Zhang et al. [2001]). Due to page limitations, the proof is omitted in this paper.
k0 η¯1 k0 η¯1 χ(t) = e−(λ0 −k0 γ1 )t . + k0 ω1 − λ0 − k0 γ1 λ0 − k0 γ1 (21) △
According to (20), we have the following decision algorithm: Fault Detection Decision Scheme: The decision on the occurrence of a fault (detection) is made when the modulus of at least one component of the output estimation error (i.e., | y˜j (t) |, for some j) exceeds its corresponding threshold given by
The fault isolation decision scheme is based on the following intuitive principle: if fault s occurs at time T0 and is detected at time Td , then a set of adaptive threshold △ νj (t) = kj functions {µsj (t), j = 1, · · · , p} can be designed for the s0 th isolation estimator, such that the j-th component of yjs (t)| ≤ µsj (t), for all + kj ω2 e−λj t . (22) its output estimation error satisfies |˜ t > Td . Consequently, for each s = 1, · · · , N , such a set of adaptive thresholds {µsj (t), j = 1, · · · , p} can be associated Remark 2. The adaptive thresholds νj (t) can be easily with the output estimation of the s-th isolation estimator. implemented using linear filtering techniques (Zhang et In the fault isolation procedure, if, for a particular isolation al. [2001]). Note that, ω1 and ω2 in (21) and (22) are estimator s, there exists some j ∈ {1, · · · , p}, such that (possibly conservative) bounds for the unknown initial the j-th component of its output estimation error satisfies yjs (t)| > µsj (t) for some finite time t > Td , then the conditions. However, since their effects decreases exponen- |˜ −(λ0 −k0 γ1 )t −λj t tially (i.e., they are multiplied by e and e , possibility of the occurrence of fault s can be excluded. respectively), they will not affect significantly the FDI Based on this intuitive idea, the following fault isolation decision scheme is devised: performance. Zt
−λj (t−τ ) (||A21 || + γ2 ) χ(τ ) + η¯2 (y, u, τ ) dτ e
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7th IFAC SAFEPROCESS (SAFEPROCESS’09) Barcelona, Spain, June 30 - July 3, 2009
Fault Isolation Decision Scheme: If, for each r ∈ {1, · · · , N }\{s} , there exist some finite time tr > Td and some j ∈ {1, · · · , p}, such that |˜ yjr (tr )| > µrj (tr ), then the occurrence of fault s is concluded.
Now, based on (30) and the solution of (29), as well as Assumption 2, 4 and 5, after some algebraic manipulations, it can be shown that
4. ADAPTIVE THRESHOLDS FOR FAULT ISOLATION
Td
+kj e−λj (t−Td ) |¯ z2s (Td )| + |(C¯j Ωs )⊤ | |θ˜s | . Note that (27) is in the same form as (10). Hence, from (13), (16) and (26), we have
e−λj (t−τ ) (||A21 || + γ2 )χs (τ ) + η¯2 + αs ||Ωs ||
+
k0 η¯1 λ0 − k0 γ1
k0 η¯1 , λ0 − k0 γ1
(26)
s
+ ρ1 (z, u) − ρ1 (ˆ z , u) + η1 (z, u, t) +
A21 z˜1s
µsj (t) = kj
Zt
e
−λj (t−τ )
(||A21 || + γ2 )χs (τ ) + η¯2 (y, u, τ )
Td
+α ||Ω || dτ + kj e−λj (t−Td ) ω2 + |C¯j Ωs |κs , (31)
(27)
s
s
+ ρ2 (z, u) − ρ2 (ˆ z , u) + η2 (z, u, t) ˙ (28) +f s (y, u) − fˆ(y, u, θˆs ) − Ωs θˆs . s s s s ˆs ˆ By substituting f (y, u) = G θ and f = G θ into (28) and by using Gs = Ω˙ s − A¯22 Ωs (see (23)), we have
s
which again can be easily implemented using linear filtering techniques (Zhang et al. [2001, 2005]). 5. FAULT ISOLABILITY ANALYSIS For our purpose, a fault is said to be isolable if the fault isolation scheme is able to reach a correct decision in finite time. Intuitively, faults are isolable if they are mutually different according to a certain measure quantifying the difference in the effects that different faults have on measurable outputs and on the estimated quantities in the isolation scheme. In this respect, we introduce the fault mismatch function between the s–th fault and the r–th fault: △ hsr (t) = C¯j Ωs θs − Ωr θˆr , (32)
s z s , u) z˜˙ 2 = A¯22 z˜2s + Ωs θ˜s + A21 z˜1s + ρ2 (z, u) − ρ2 (ˆ
d s ˜s (Ω θ ) − Ωs θ˙s (t) . dt △ = z˜2s + Ωs θ˜s , the above equation can be
+η2 (z, u, t) − By letting z¯2s rewritten as
× ||Ωs || dτ + kj e−λj (t−Td ) |¯ z2s (Td )| + |(C¯j Ωs )⊤ | |θ˜s | .
on the geometric properties of set Θs (see Zhang et al. [2001]). Hence, the following threshold function is chosen:
Proof: Denote the state estimation errors of the s-th △ △ isolation estimator by z˜1s (t) = z1 (t) − zˆ1s (t) and z˜2s (t) = z2 (t) − zˆ2s (t). By using (23) and (3), in the presence of fault s, the state estimation error of the s-th estimator, for t > Td , satisfies = A11 z˜1s = A¯22 z˜2s
Td
e−λj (t−τ ) (||A21 || + γ2 )χs (τ ) + η¯2 + αs
Although Lemma 3 provides an upper bound on the output estimation error of the s-th estimator, the right-hand side of (25) cannot be directly used as a threshold function for fault isolation because θ˜s (t) is not available. However, since the estimate θˆs belongs to the known compact set Θs , we s ˆs have θ − θ (t) ≤ κs (t) for a suitable κs (t) depending
e−(λ0 −k0 γ1 )(t−Td )
△ and θ˜s (t) = θˆs (t) − θs (t) represents the fault parameter vector estimation error.
s z˜˙ 1 s z˜˙ 2
Zt
(25) Now, inequality (25) follows directly from the initial conditiion zˆ2s (Td ) = 0, Ωs (Td ) = 0, and |z2s (Td )| ≤ ω2 .
where △ χs (t) = k0 ω1 −
e−λj (t−τ ) [ η¯2 (y, u, τ ) + αs ||Ωs || ] dτ
Td
|˜ yjs (t)| ≤ kj
×dτ + kj e−λj (t−Td ) ω2 + |(C¯j Ωs )⊤ | |θ˜s | ,
Zt
+kj
Lemma 3. If fault s occurs, where s ∈ {1, · · · , N }, then for all t > Td , the j-th component of the output estimation error of the s-th isolation estimator satisfies: |˜ yjs (t)| ≤ kj
e−λj (t−τ ) [ ||A21 ||˜ z1s (τ )| + γ2 |z(τ ) − zˆs (τ )| ] dτ
Td
The threshold functions µsj (t) clearly play a significant role in the proposed fault decision scheme. The following lemma provides a bounding function for the output estimation error of the s-th isolation estimator in the case that fault s occurs.
Zt
Zt
|˜ yjs (t)| ≤ kj
s z¯˙ 2 = A¯22 z¯2s + A21 z˜1s + ρ2 (z, u) − ρ2 (ˆ z s , u) + η2 − Ωs θ˙s (t) .
(29)
j
By defining each component of the output estimation error △ as y˜js (t) = yj (t) − yˆjs (t), j = 1, · · · p, and by using (23) and (3), we have: (30) y˜js (t) = C¯j z˜2s (t) = C¯j z¯2s (t) − Ωs θ˜s .
where r, s = 1, · · · , N, r 6= s. From a qualitative point of view, hsr j (t) can be considered as a filtered version of the difference between the actual fault function Gs θs and its estimate Gr θˆr generated by isolation estimator r whose structure does not match the actual fault s. Recalling
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that each FIE corresponds to one of the particular type of nonlinear faults in the fault class. Consequently, if fault s occurs, its estimate Gr θˆr associated with FIE r is determined by the structure of FIE r, which in turn is determined by fault r. Therefore, the fault mismatch function hsr (t), defined as the ability of FIE r to learn fault s, provides a measure of the difference between fault s and r.
Note that (36) is in the same form as (29). Therefore, by using (36) and (37) and by following the reasoning reported in the proof of Lemma 3, we have |˜ yjr (t)| ≥ |hsr j (t)| −
+|η2 (z, u, τ ) − Ωs θ˙s | dτ − kj e−λj (t−Td ) |z2r (Td )| .
Theorem 1. Consider the fault isolation scheme described by (23) and (31). Suppose that a fault s, s = 1, · · · , N , occurring at time t = T0 is detected at time t = Td . Then fault s is isolable if, for each r ∈ {1, · · · , N }\{s} , there exist some time tr > Td and some j ∈ {1, · · · , p} , r such that the fault mismatch function hsr j (t ) satisfies the following inequality: |hsr j (t)| ≥ 2kj
e
Td
+ kj
Zt Td
−λj (t−τ )
¯ |C¯j eA22 (t−τ ) | (||A21 || + γ2 )χr (τ )
Td
The following theorem characterizes (in a non-closed form) the class of nonlinear faults that are isolable by the proposed method:
Zt
Zt
Now taking into account the corresponding adaptive threshold µrj (t) given by (31), we can conclude that, if condition (33) is satisfied at time t = tr , we obtain |˜ yjr (tr )| > µrj (tr ), which implies that the possibility of the occurrence of fault r can be excluded at time t = tr . 6. CONCLUSION In this paper, a robust fault diagnosis scheme for a class of Lipschitz uncertain nonlinear system is presented. Robustness and fault sensitivity are enhanced via the use of adaptive thresholds in the diagnostic decision-making stage. The fault isolability conditions characterizing the class of isolable faults are rigorously investigated.
(||A21 || + γ2 )χr (τ ) + η¯2 dτ
e−λj (t−τ ) αs ||Ωs || + αr ||Ωr || dτ
REFERENCES
+ |C¯j Ωr |κr (t) + 2ω2 kj e−λj (t−Td ) . (33) Proof: Denote the state estimation errors of the r-th △ △ isolation estimator by z˜1r (t) = z1 (t) − zˆ1r (t) and z˜2r (t) = r z2 (t) − zˆ2 (t). By using (23) and (3), in the presence of fault s, the state estimation error of the r-th estimator for t > Td satisfies
W. Chen and M. Saif, ”A sliding mode observer-based strategy for fault detection, isolation, and estimation in a class of Lipschitz nonliner systems,” International Journal of System Science, vol. 38, pp. 943–955, 2007. J. A. Farrell, T. Berger, and B. D. Appleby, ”Using learning techniques to accommodate unanticipated faults,” IEEE Control Sys. Mag., vol. 13, pp. 40–49, 1993. r z˜˙ 1 = A11 z˜1r + ρ1 (z, u) − ρ1 (ˆ z r , u) + η1 (z, u, t) (34) J. Farrell and M. M. Polycarpou. Adaptive Approximation Based Control. Hoboken, NJ: J. Wiley, 2006. r z˜˙ 2 = A¯22 z˜2r + A21 z˜1r + ρ2 (z, u) − ρ2 (ˆ z r , u) + η2 (z, u, t) P. A. Ioannou and J. Sun. Robust Adaptive Control. ˙ Prentice Hall, Englewood Cliffs, NJ, 1996. (35) + Gs θs − Gr θˆr − Ωr θˆr . B. Jiang, M. Staroswiecki, and V. Cocquempot, ”Fault ˙ s − A¯22 Ωs and Gr = Ω ˙ r − A¯22 Ωr By substituting Gs = Ω diagnosis based on adaptive observer for a class of into (35), we obtain nonlinear systems with unknown parameters,” Int. J. Control, vol 77, no. 4, pp. 415-426, 2004. r z r , u) K. Vijayaraghavan, R. Rajamani and J. Bokor, ”Quantiz˜˙ 2 = A¯22 z˜2r + Ωr θˆr − Ωs θs + A21 z˜1r + ρ2 (z, u) − ρ2 (ˆ tative fault estimation for a class of nonlinear systems,” d r ˆr Int. Journal of Control, vol. 80, pp. 64–74, no. 1, 2007. + η2 (z, u, t) − Ω θ − Ωs θs − Ωs θ˙s . dt A. T. Vemuri and M. M. Polycarpou, ”Robust nonlinear △ fault diagnosis in input-output systems,” Int. Journal By defining z¯2r (t) = z˜2r (t) + Ωr θˆr − Ωs θs , the above of Control, vol. 68, pp. 343-360, 1997. differential equation can be rewritten as H. Wang, Z. J. Huang, and S. Daley, “On the use of r adaptive updating rules for actuator and sensor fault z¯˙ 2 = A¯22 z¯2r + A21 z˜1r + ρ2 (z, u) − ρ2 (ˆ z r , u) + η2 − Ωs θ˙s . diagnosis,” Automatica, vol. 33, pp. 217–225, 1997. (36) X. Yan and C. Edwards, ”Nonlinear robust fault resconstruction and estimation using a sliding mode observer,” △ The j-th component of the output estimation error y˜jr (t) = Automatica, vol. 43, pp. 1605-1614, 2007. yj (t) − yjr (t), j = 1, · · · p, is given by: X. Zhang, M. M. Polycarpou, and T. Parisini, ”Robust fault isolation of a class of nonlinear input-output sysy˜jr (t) = C¯j z˜2 (t) = C¯j (¯ z2r (t) − Ωr θˆr + Ωs θs ) . tems,” Int. Journal of Control, vol. 74, pp. 1295-1310, By using (32), i.e., the definition of the fault mismatch 2001. function, the above equation can be rewritten as follows X. Zhang, T. Parisini, and M. M. Polycarpou, ”Sensor Bias y˜jr (t) = C¯j z¯2r (t) + hsr Fault Isolation in a Class of Noninear Systems,” IEEE j (t) . Trans. on Automatic Control, vol. 50, pp. 370-376, 2005. By applying the triangle inequality, we have |˜ yjr (t)| ≥ |hsr ¯2r (t)| . j (t)| − |Cj z
(37)
17
Fault Diagnosis of a Class of Uncertain Nonlinear Systems with Lipschitz Nonlinearities Xiaodong Zhang ∗ , Marios M. Polycarpou ∗∗ , and Thomas Parisini ∗∗∗ ∗
Dept. of Electrical Engineering, Wright State University, USA Dept. of Electrical and Computer Engineering, University of Cyprus, Cyprus ∗∗∗ Dept. of Electrical, Electronic and Computer Engineering, University of Trieste, Italy ∗∗
Abstract: This paper presents a fault detection and isolation scheme for a class of Lipschitz nonlinear systems with nonlinear and unstructured modeling uncertainty. This significantly extends previous results by considering a more general class of system nonlinearities which are modeled as functions of the system input and partially measurable state variables. A new fault diagnosis method is developed using adaptive estimation techniques. The fault isolability conditions are rigorously established via the use of the so-called fault mismatch functions, which are defined to characterize the mutual difference between pairs of faults. 1. INTRODUCTION In recent years, there has been significant research activity in the design and analysis of fault diagnosis and accommodation schemes. Considerable effort has been devoted to fault diagnosis of nonlinear systems in the framework of various kinds of assumptions and fault scenarios. The idea of using adaptive and learning techniques in fault diagnosis has been presented in (Farrell et al. [1993], Vemuri and Polycarpou [1997], Wang et al. [1997], Zhang et al. [2001], Jiang et al. [2004], Zhang et al. [2005]). In previous papers (Vemuri and Polycarpou [1997], Zhang et al. [2001, 2005]), the authors developed an adaptive approximation based fault diagnosis methodology for a class of nonlinear systems in which the known nonlinearity is represented as a function of measurable system signals (i.e., input and output variables). An assumption was made on the existence of a diffeomorphism that can transform more general nonlinear systems into the class of systems under consideration.
2. PROBLEM FORMULATION Consider a class of nonlinear multi-input-multi-output (MIMO) dynamic systems described by x˙ = Ax + ζ(x, u) + ϕ(x, u, t) + β(t − T0 )Dφ(y, u) (1) y = Cx
In this paper, we extend the previous results by considering a more general class of nonlinear systems without the need of diffeomorphism. More specifically, the known nonlinearity under consideration is a nonlinear function of the system input and state variables and satisfies a Lipschitz condition. With partially measurable states and the possible presence of modeling uncertainty, the design of fault diagnosis methods for such Lipschitz nonlinear systems is a challenging problem. Several researchers have investigated this problem and presented some interesting results under the framework of structured modeling uncertainty and faults. For instance, Vijayaraghavan et al. [2007] addressed the sensor fault diagnosis problem by assuming the absence of modeling uncertainty, and in (Yan and Edwards [2007], Chen and Saif [2007]) sliding mode observer based fault estimation methods were proposed under certain assumptions on the distribution matrices of the structured modeling uncertainty and faults. However, the modeling uncertainty is often unstructured, which makes it difficult to achieve robust fault diagnosis with respect to modeling uncertainty.
978-3-902661-46-3/09/$20.00 © 2009 IFAC
In this research work, we develop an adaptive approximation based fault diagnosis scheme for a class of Lipschitz nonlinear systems with possibly nonlinear and unstructured modeling uncertainty. The objective is to detect the initiation of any faults as soon as possible and to determine whether a particular fault type in a partially known nonlinear fault set has occurred. The fault diagnosis method is presented with a rigorous analytical framework aimed at characterizing the behaviors of the proposed diagnostic scheme. The analysis of the fault diagnosis scheme focuses on: (i) determining adaptive thresholds for fault detection and fault isolation; (ii) deriving fault isolability conditions characterizing the class of faults that can be isolated by the proposed method, based on the so-called fault mismatch function, which provides a suitable measure of the mutual ”difference” between faults.
where x ∈ ℜn is the system state vector, u ∈ ℜm is the input vector, y ∈ ℜp is the output vector, ζ : ℜn × ℜm 7→ ℜn , ϕ : ℜn × ℜm × ℜ+ 7→ ℜn , φ : ℜp × ℜm 7→ ℜq are smooth vector fields. The constant matrices D ∈ ℜn×q and C ∈ ℜp×n with q ≤ p are of full rank, and (A, C) is an observable pair. The model given by x˙ N = AxN + ζ(xN , u) yN = CxN is the known nominal system model. The term ϕ in (1) represents the modeling uncertainty, and β(t−T0 )Dφ(y, u) denotes the changes in the system dynamics due to the occurrence of a fault. Specifically, β(t − T0 ) is the time profile of a fault which occurs at some unknown time T0 , φ(y, u) represents the nonlinear fault function, and D is the fault distribution matrix, which is possibly known. In this paper, we only consider the case of abrupt (sudden) faults, therefore, β(·) takes the form of a step function.
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Moreover, the design and analysis in this paper is based on the assumption that only a single fault occurs.
Assumption 2. The functions η1 and η2 in (3), representing the unstructured modeling uncertainty, are possibly unknown nonlinear functions of z, u, and t, but bounded, i.e., ∀(z, y, u) ∈ Z × Y × U , ∀t ≥ 0,
Assumption 1. The fault distribution matrix D in (1) satisfies
|η1 (z, u, t)| ≤ η¯1 , |η2 (z, u, t)| ≤ η¯2 (y, u, t) ,
• rank(CD) = q • invariant zeros of (A,D,C) lie in the left half plane.
(6)
where the constant bound η¯1 and the bounding function η¯2 (y, u, t) are known, and η¯2 is uniformly bounded in Y × U × ℜ+ . Additionally, Z ⊂ ℜn , U ⊂ ℜm , and Y ⊂ ℜp are compact sets of admissible state variables, inputs, and outputs, respectively.
As described in Yan and Edwards [2007], under Assumption 1, there exists a linear change of coordinates z = T x = [z1⊤ z2⊤ ]⊤ with z1 ∈ ℜ(n−p) and z2 ∈ ℜp , such that A11 A12 −1 • T AT = , where the matrix A11 ∈ A21 A22 (n−p)×(n−p) ℜ is stable. 0 , where D2 ∈ ℜp×q . • TD = D2 • CT −1 = [0 Ip ], where Ip ∈ ℜp×p is an identity matrix.
Assumption 3. The system state vector z remains bounded before and after the occurrence of a fault, i.e., z(t) ∈ L∞ , ∀t ≥ 0. Assumption 4. The rates of change of each fault parameter vector θs (t) in (5) (s = 1, · · · , N ) are uniformly △ bounded, i.e., |θ˙s (t)| ≤ αs for all t ≥ 0, where θs (t) = i h ⊤ ⊤ ⊤ (θ1s (t)) , · · · , θps (t) , and αs is a known constant.
Therefore, in the new coordinate system, the system (1) is described by z˙1 = A11 z1 + A12 z2 + ρ1 (z, u) + η1 (z, u, t) z˙2 = A21 z1 + A22 z2 + ρ2 (z, u) + η2 (z, u, t) (2) +β(t − T0 )D2 φ(y, u) y = z2 , ρ1 η1 where = T ζ(T −1 z, u) and = T ϕ(T −1 z, u, t). ρ2 η2
Assumption 5. The known nonlinear terms ρ1 (z, u) and ρ2 (z, u) in (3) are uniformly Lipschitz in u ∈ U, i.e., ∀z, zˆ ∈ Z,
With a more general structure of the nonlinear fault model, system (2) can be extended to z˙1 = A11 z1 + A12 z2 + ρ1 (z, u) + η1 (z, u, t) z˙2 = A21 z1 + A22 z2 + ρ2 (z, u) + η2 (z, u, t) (3) +β(t − T0 )f (y, u) ¯ y = Cz2 ,
|ρ1 (z, u) − ρ1 (ˆ z , u)| ≤ γ1 |z − zˆ|
(7)
|ρ2 (z, u) − ρ2 (ˆ z , u)| ≤ γ2 |z − zˆ| ,
(8)
where γ1 and γ2 are the known Lipschitz constants for ρ1 (z, u) and ρ2 (z, u), respectively. Assumption 2 characterizes the class of modeling uncertainties under consideration. The bounds on the unstructured modeling uncertainties are needed in order to be able to distinguish between the effects of faults and modeling uncertainty (see Vemuri and Polycarpou [1997], Zhang et al. [2001]). It is worth noting that the unstructured modeling uncertainty considered in this paper is more general than the structured uncertainty considered for Lipschitz nonlinear systems in the fault diagnosis literature (e.g., Yan and Edwards [2007], Chen and Saif [2007], Vijayaraghavan et al. [2007]), which to achieve robustness additionally assumes that certain rank conditions are satisfied by the distribution matrix of the structured modeling uncertainty.
where f : ℜp × ℜm 7→ ℜp is a smooth vector field representing the nonlinear fault function under consideration, and C¯ ∈ ℜp×p is a nonsingular matrix. Clearly, (2) is a special case of (3) with f (y, u) = D2 φ(y, u) and C¯ = Ip . The main objective of this paper is to develop a robust fault detection and isolation (FDI) scheme using adaptive approximation techniques for nonlinear dynamic systems that can be transformed into (3). It is assumed that there are N types of possible fault types in the fault set F; specifically, the unknown fault function f (y, u) in (3) belongs to a finite set of fault types given by △ F = f 1 (y, u), . . . , f N (y, u) . (4)
Assumption 3 requires the boundedness of the state variables before and after the occurrence of a fault. Hence, it is assumed that the feedback control system is capable of retaining the boundedness of the state variables even in the presence of a fault. This is required for well posedness since the FDI design that we consider does not influence the closed-loop dynamics and stability.
Each fault function type f s , s = 1, · · · , N , is described by ⊤ △ f s (y, u) = (θ1s (t))⊤ g1s (y, u), · · · , (θps (t))⊤ gps (y, u) , (5)
where each θis (t), i = 1, · · · , p , is an unknown parameter vector assumed to belong to as known compact and convexs set Θsi (i.e., θis ∈ Θsi ⊂ ℜli ) and gis : ℜp × ℜm 7→ ℜli is a known smooth vector field. As discussed in Zhang et al. [2001], the fault model described by (4) and (5) characterizes a general class of nonlinear faults where the vector field gis represents the functional structure of the s-th fault affecting the i-th state equation, while the unknown parameter vector θis characterizes the timevarying “magnitude” of the fault. Throughout the paper, the following assumptions are introduced:
In Assumption 4, known bounds on the rates of change of θs (t) are assumed. In practice, the rate bounds αs can be set by exploiting some a priori knowledge on the fault developing dynamics. In the case of a constant fault size, we simply set αs = 0. Remark 1. The nominal system dynamics in (3) is similar to the model used in Yan and Edwards [2007]. The objective of this paper is to develop a robust FDI scheme using adaptive approximation techniques, while Yan and Edwards [2007] developed a fault estimation
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method using a sliding mode observer, which requires additional assumptions on the distribution matrices of the modeling uncertainty terms η1 and η2 as well as the fault function f in (3).
Lemma 2. Consider the system described by (3) and the fault detection estimator described by (9). Let k0 and λ0 be positive constants chosen such that ||eA11 t || ≤ k0 e−λ0 t . Assume that λ0 > k0 γ1 , where γ1 is the Lipschitz constant given in (7). Then, for 0 ≤ t < T0 , the state estimation error z˜1 (t) satisfies: k0 η¯1 k0 η¯1 |˜ z1 (t)| ≤ e−(λ0 −k0 γ1 )t , + k0 ω1 − λ0 − k0 γ1 λ0 − k0 γ1 (13) where ω1 is a constant bound for |z1 (0)|, which will be defined later on.
3. FAULT DETECTION AND ISOLATION ARCHITECTURE The fault detection and isolation architecture is based on a bank of N + 1 nonlinear adaptive estimators, where N is the number of different nonlinear fault types in the fault set F. One of the nonlinear adaptive estimators is the fault detection estimator (FDE) used for detecting the occurrence of any faults, while the remaining N nonlinear adaptive estimators are fault isolation estimators (FIEs), which are activated after fault detection for the purpose of determining the particular type of fault that has occurred.
Proof: From (10), we have
z˜1 (t) =
3.1 Fault Detection Scheme
Zt
eA11 (t−τ ) [ρ1 (z, u) − ρ1 (ˆ z , u) + η1 (z, u, τ )] dτ
0
Based on the system model given by (3), the FDE is chosen as follows: z , u) , zˆ1 (0) = 0 zˆ˙ 1 = A11 zˆ1 + A12 C¯ −1 y + ρ1 (ˆ zˆ˙ 2 = A21 zˆ1 + A22 zˆ2 + ρ2 (ˆ z , u) + L(y − yˆ) , zˆ2 (0) = 0 (9) yˆ = C¯ zˆ2 ,
+ eA11 t z˜1 (0) . By using (6), (7), and the triangle inequality, we obtain
|˜ z1 (t)| ≤ γ1
Zt
k0 e−λ0 (t−τ ) |z(τ ) − zˆ(τ )| dτ + k0 e−λ0 t |˜ z1 (0)|
0
where zˆ1 , zˆ2 , and yˆ denote the estimated state and output variables, respectively, L ∈ ℜp×p is a design gain matrix, △ △ and zˆ = [(ˆ z1 )⊤ (C¯ −1 y)⊤ ]⊤ . Let z˜1 = z1 − zˆ1 and
k0 η¯1 + 1 − e−λ0 t λ0 Zt k0 η¯1 ≤ γ1 k0 e−λ0 (t−τ ) |z(τ ) − zˆ(τ )| dτ + λ0 0 η¯1 + k0 ω1 − e−λ0 t , (14) λ0 where k0 and λ0 are positive constants chosen such that ||eA11 t || ≤ k0 e−λ0 t (since A11 is stable, such constants k0 and λ0 always exist (see Ioannou and Sun [1995])), and ω1 is a (possibly conservative) constant bound for |z1 (0)|, such that |˜ z1 (0)| = |z1 (0)| ≤ ω1 . Based on Assumption 3, such a constant bound ω1 can always be chosen.
△
z˜2 = z2 − zˆ2 denote the state estimation errors, and △ y˜ = y− yˆ denote the output estimation error. Then, before fault occurrence (i.e., for t < T0 ), from (3) and (9), we have z˜˙ 1 = A11 z˜1 + ρ1 (z, u) − ρ1 (ˆ z , u) + η1 (10) z˜˙ 2 = A21 z˜1 + A22 z˜2 + ρ2 (z, u) − ρ2 (ˆ z , u) − L(y − yˆ) + η2 = A¯22 z˜2 + A21 z˜1 + ρ2 (z, u) − ρ2 (ˆ z , u) + η2 ¯ 2 − zˆ2 ) = C¯ z˜2 , y˜ = C(z
(11) (12)
△ ¯ Note that, since C¯ is nonsingular, where A¯22 = A22 −LC. we can always choose L to make A¯22 stable.
Note that
In the analysis of the state estimation error z˜1 (t), we will need the following Lemmas.
z(τ ) − zˆ(τ ) =
z1 (τ ) − zˆ1 (τ ) z2 (τ ) − C¯ −1 y(τ )
=
z˜1 (τ ) 0
.
(15)
Lemma 1 . Let t0 be a given time-instant and c0 , c1 , Therefore, we have c2 , λ be nonnegative constants, and κ(t) a nonnegative |z(τ ) − zˆ(τ )| = |˜ z1 (τ )| . (16) piecewise continuous function of time. If h(t) satisfies the inequality By substituting (16) into (14), we obtain Zt h(t) ≤ c0 e−λ(t−t0 ) +c1 +c2 e−λ(t−τ ) κ(τ )h(τ )dτ , ∀t ≥ t0 , Zt η¯1 −λ0 t z1 (τ )| dτ |˜ z1 (t)| ≤ k0 ω1 − e + k0 γ1 e−λ0 (t−τ ) |˜ t0 λ0 then 0 Rt k0 η¯1 κ(s)ds c2 + . (17) h(t) ≤ (c0 + c1 )e−λ(t−t0 ) e t0 λ0 Zt Rt κ(s)ds c Now, applying Lemma 1 to (17) with c0 = k0 ω1 − λη¯10 dτ, ∀t ≥ t0 . + c1 λ e−λ(t−τ ) e 2 τ (we can always choose ω1 to ensure that c0 ≥ 0 ), t0 c1 = kλ0 η0¯1 , c2 = k0 γ1 , and κ = 1, the proof of (13) can Proof: The proof of the above lemma can be found on be immediately concluded. page 103 of the book by Ioannou and Sun [1995].
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3.2 Fault Isolation Estimators and Decision Scheme
Next, we analyze the state estimation error z˜2 (t). The solution of (11) is given by z˜2 (t) =
Zt
eA22 (t−τ ) [A21 z˜1 + ρ2 (z, u) − ρ2 (ˆ z , u)] dτ
+
Zt
Now, assume that a fault is detected at some time Td ; accordingly, at t = Td the fault isolation estimators (FIEs) are activated. Each FIE corresponds to one potential fault type. Specifically, the following N nonlinear adaptive estimators are used as isolation estimators: for s = 1, · · · , N , s z s , u) , zˆ1s (Td ) = 0 zˆ˙ 1 = A11 zˆ1s + A12 C¯ −1 y + ρ1 (ˆ s zˆ˙ 2 = A21 zˆ1s + A22 zˆ2s + ρ2 (ˆ z s , u) + L(y − yˆs ) ˙ s s s s (23) +fˆ (y, u, θˆ ) + Ω θˆ , zˆ2s (Td ) = 0 ˙Ωs = A¯22 Ωs + Gs (y, u), Ωs (Td ) = 0 yˆs = C¯ zˆ2s ,
¯
0
¯
¯
eA22 (t−τ ) η2 (z, u, τ )dτ + eA22 t z˜2 (0) .
(18)
0
Therefore, for each component of the output estimation △ error, i.e., y˜j (t) = C¯j z˜2 (t), j = 1, · · · , p, where C¯j is the ¯ by using (18), (8), (16), (6), j-th row vector of matric C, and the triangle inequality, it can be shown that |˜ yj (t)| ≤ kj
Zt
e
−λj (t−τ )
where zˆ1s , zˆ2s , and yˆs denote the estimated state and output variables, respectively, L ∈ ℜp×p is a design △ △ gain matrix, zˆs = [(ˆ z1s )⊤ (C¯ −1 y)⊤ ]⊤ , fˆs (y, u, θˆs ) = s [(θˆ1s )⊤ g1s (y, u), · · · , (θˆps )⊤ gps (y, u)], and θˆis ∈ ℜli , for i = 1, · · · , p, is the estimate of the fault parameter vector provided by the s-th isolation estimator. It is noted that, according to (5), the fault approximation model fˆs is linear in the adjustable weights θˆs . Consequently, the gradient △ matrix Gs = ∂ fˆs (y, u, θˆs ) / ∂ θˆs = diag [(g1s )⊤ , · · · , (gps )⊤ ] does not depend on θˆs .
(||A21 || + γ2 ) |˜ z1 (τ )| + η¯2 dτ
0
+ kj ω2 e−λj t , (19) where kj and λj are positive constants chosen such that ¯ |C¯j eA22 t | ≤ kj e−λj t (since A¯22 is stable, constants kj and λj satisfying the above inequality always exist (see Ioannou and Sun [1995])), and ω2 is a (possibly conservative) bound for |z2 (0)|, such that |˜ z2 (0)| = |z2 (0)| ≤ ω2 . Based on Assumption 3, such a constant bound ω2 can always be chosen. By using (13) and (19), we obtain |˜ yj (t)| < kj
Zt 0
e−λj (t−τ ) (||A21 || + γ2 ) χ(τ ) + η¯2 (y, u, τ ) dτ
+ kj ω2 e−λj t ,
(20)
where
The adaptation in the isolation estimators arises due to the i⊤ h △ ⊤ ⊤ unknown parameter vector θs = (θ1s ) , · · · , (θns ) . s ˆ The adaptive law for adjusting θ is derived using the Lyapunov synthesis approach (see for example Ioannou and Sun [1995]). Specifically, the learning algorithm is chosen as follows n o ˙ θˆs = PΘs ΓΩs⊤ C¯ ⊤ y˜s , (24) △
where y˜s (t) = y(t) − yˆs (t) denotes the output estimation error of the s-th estimator, Γ > 0 is a symmetric, positivedefinite learning rate matrix, and PΘs is the projection operator restricting θˆs to the corresponding known set Θs (in order to guarantee stability of the learning algorithm in the presence of modeling uncertainty, as described in Ioannou and Sun [1995], Farrell and Polycarpou [2006])). The stability of the FIE can be proved following a similar procedure as reported in (Vemuri and Polycarpou [1997], Zhang et al. [2001]). Due to page limitations, the proof is omitted in this paper.
k0 η¯1 k0 η¯1 χ(t) = e−(λ0 −k0 γ1 )t . + k0 ω1 − λ0 − k0 γ1 λ0 − k0 γ1 (21) △
According to (20), we have the following decision algorithm: Fault Detection Decision Scheme: The decision on the occurrence of a fault (detection) is made when the modulus of at least one component of the output estimation error (i.e., | y˜j (t) |, for some j) exceeds its corresponding threshold given by
The fault isolation decision scheme is based on the following intuitive principle: if fault s occurs at time T0 and is detected at time Td , then a set of adaptive threshold △ νj (t) = kj functions {µsj (t), j = 1, · · · , p} can be designed for the s0 th isolation estimator, such that the j-th component of yjs (t)| ≤ µsj (t), for all + kj ω2 e−λj t . (22) its output estimation error satisfies |˜ t > Td . Consequently, for each s = 1, · · · , N , such a set of adaptive thresholds {µsj (t), j = 1, · · · , p} can be associated Remark 2. The adaptive thresholds νj (t) can be easily with the output estimation of the s-th isolation estimator. implemented using linear filtering techniques (Zhang et In the fault isolation procedure, if, for a particular isolation al. [2001]). Note that, ω1 and ω2 in (21) and (22) are estimator s, there exists some j ∈ {1, · · · , p}, such that (possibly conservative) bounds for the unknown initial the j-th component of its output estimation error satisfies yjs (t)| > µsj (t) for some finite time t > Td , then the conditions. However, since their effects decreases exponen- |˜ −(λ0 −k0 γ1 )t −λj t tially (i.e., they are multiplied by e and e , possibility of the occurrence of fault s can be excluded. respectively), they will not affect significantly the FDI Based on this intuitive idea, the following fault isolation decision scheme is devised: performance. Zt
−λj (t−τ ) (||A21 || + γ2 ) χ(τ ) + η¯2 (y, u, τ ) dτ e
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7th IFAC SAFEPROCESS (SAFEPROCESS’09) Barcelona, Spain, June 30 - July 3, 2009
Fault Isolation Decision Scheme: If, for each r ∈ {1, · · · , N }\{s} , there exist some finite time tr > Td and some j ∈ {1, · · · , p}, such that |˜ yjr (tr )| > µrj (tr ), then the occurrence of fault s is concluded.
Now, based on (30) and the solution of (29), as well as Assumption 2, 4 and 5, after some algebraic manipulations, it can be shown that
4. ADAPTIVE THRESHOLDS FOR FAULT ISOLATION
Td
+kj e−λj (t−Td ) |¯ z2s (Td )| + |(C¯j Ωs )⊤ | |θ˜s | . Note that (27) is in the same form as (10). Hence, from (13), (16) and (26), we have
e−λj (t−τ ) (||A21 || + γ2 )χs (τ ) + η¯2 + αs ||Ωs ||
+
k0 η¯1 λ0 − k0 γ1
k0 η¯1 , λ0 − k0 γ1
(26)
s
+ ρ1 (z, u) − ρ1 (ˆ z , u) + η1 (z, u, t) +
A21 z˜1s
µsj (t) = kj
Zt
e
−λj (t−τ )
(||A21 || + γ2 )χs (τ ) + η¯2 (y, u, τ )
Td
+α ||Ω || dτ + kj e−λj (t−Td ) ω2 + |C¯j Ωs |κs , (31)
(27)
s
s
+ ρ2 (z, u) − ρ2 (ˆ z , u) + η2 (z, u, t) ˙ (28) +f s (y, u) − fˆ(y, u, θˆs ) − Ωs θˆs . s s s s ˆs ˆ By substituting f (y, u) = G θ and f = G θ into (28) and by using Gs = Ω˙ s − A¯22 Ωs (see (23)), we have
s
which again can be easily implemented using linear filtering techniques (Zhang et al. [2001, 2005]). 5. FAULT ISOLABILITY ANALYSIS For our purpose, a fault is said to be isolable if the fault isolation scheme is able to reach a correct decision in finite time. Intuitively, faults are isolable if they are mutually different according to a certain measure quantifying the difference in the effects that different faults have on measurable outputs and on the estimated quantities in the isolation scheme. In this respect, we introduce the fault mismatch function between the s–th fault and the r–th fault: △ hsr (t) = C¯j Ωs θs − Ωr θˆr , (32)
s z s , u) z˜˙ 2 = A¯22 z˜2s + Ωs θ˜s + A21 z˜1s + ρ2 (z, u) − ρ2 (ˆ
d s ˜s (Ω θ ) − Ωs θ˙s (t) . dt △ = z˜2s + Ωs θ˜s , the above equation can be
+η2 (z, u, t) − By letting z¯2s rewritten as
× ||Ωs || dτ + kj e−λj (t−Td ) |¯ z2s (Td )| + |(C¯j Ωs )⊤ | |θ˜s | .
on the geometric properties of set Θs (see Zhang et al. [2001]). Hence, the following threshold function is chosen:
Proof: Denote the state estimation errors of the s-th △ △ isolation estimator by z˜1s (t) = z1 (t) − zˆ1s (t) and z˜2s (t) = z2 (t) − zˆ2s (t). By using (23) and (3), in the presence of fault s, the state estimation error of the s-th estimator, for t > Td , satisfies = A11 z˜1s = A¯22 z˜2s
Td
e−λj (t−τ ) (||A21 || + γ2 )χs (τ ) + η¯2 + αs
Although Lemma 3 provides an upper bound on the output estimation error of the s-th estimator, the right-hand side of (25) cannot be directly used as a threshold function for fault isolation because θ˜s (t) is not available. However, since the estimate θˆs belongs to the known compact set Θs , we s ˆs have θ − θ (t) ≤ κs (t) for a suitable κs (t) depending
e−(λ0 −k0 γ1 )(t−Td )
△ and θ˜s (t) = θˆs (t) − θs (t) represents the fault parameter vector estimation error.
s z˜˙ 1 s z˜˙ 2
Zt
(25) Now, inequality (25) follows directly from the initial conditiion zˆ2s (Td ) = 0, Ωs (Td ) = 0, and |z2s (Td )| ≤ ω2 .
where △ χs (t) = k0 ω1 −
e−λj (t−τ ) [ η¯2 (y, u, τ ) + αs ||Ωs || ] dτ
Td
|˜ yjs (t)| ≤ kj
×dτ + kj e−λj (t−Td ) ω2 + |(C¯j Ωs )⊤ | |θ˜s | ,
Zt
+kj
Lemma 3. If fault s occurs, where s ∈ {1, · · · , N }, then for all t > Td , the j-th component of the output estimation error of the s-th isolation estimator satisfies: |˜ yjs (t)| ≤ kj
e−λj (t−τ ) [ ||A21 ||˜ z1s (τ )| + γ2 |z(τ ) − zˆs (τ )| ] dτ
Td
The threshold functions µsj (t) clearly play a significant role in the proposed fault decision scheme. The following lemma provides a bounding function for the output estimation error of the s-th isolation estimator in the case that fault s occurs.
Zt
Zt
|˜ yjs (t)| ≤ kj
s z¯˙ 2 = A¯22 z¯2s + A21 z˜1s + ρ2 (z, u) − ρ2 (ˆ z s , u) + η2 − Ωs θ˙s (t) .
(29)
j
By defining each component of the output estimation error △ as y˜js (t) = yj (t) − yˆjs (t), j = 1, · · · p, and by using (23) and (3), we have: (30) y˜js (t) = C¯j z˜2s (t) = C¯j z¯2s (t) − Ωs θ˜s .
where r, s = 1, · · · , N, r 6= s. From a qualitative point of view, hsr j (t) can be considered as a filtered version of the difference between the actual fault function Gs θs and its estimate Gr θˆr generated by isolation estimator r whose structure does not match the actual fault s. Recalling
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7th IFAC SAFEPROCESS (SAFEPROCESS’09) Barcelona, Spain, June 30 - July 3, 2009
that each FIE corresponds to one of the particular type of nonlinear faults in the fault class. Consequently, if fault s occurs, its estimate Gr θˆr associated with FIE r is determined by the structure of FIE r, which in turn is determined by fault r. Therefore, the fault mismatch function hsr (t), defined as the ability of FIE r to learn fault s, provides a measure of the difference between fault s and r.
Note that (36) is in the same form as (29). Therefore, by using (36) and (37) and by following the reasoning reported in the proof of Lemma 3, we have |˜ yjr (t)| ≥ |hsr j (t)| −
+|η2 (z, u, τ ) − Ωs θ˙s | dτ − kj e−λj (t−Td ) |z2r (Td )| .
Theorem 1. Consider the fault isolation scheme described by (23) and (31). Suppose that a fault s, s = 1, · · · , N , occurring at time t = T0 is detected at time t = Td . Then fault s is isolable if, for each r ∈ {1, · · · , N }\{s} , there exist some time tr > Td and some j ∈ {1, · · · , p} , r such that the fault mismatch function hsr j (t ) satisfies the following inequality: |hsr j (t)| ≥ 2kj
e
Td
+ kj
Zt Td
−λj (t−τ )
¯ |C¯j eA22 (t−τ ) | (||A21 || + γ2 )χr (τ )
Td
The following theorem characterizes (in a non-closed form) the class of nonlinear faults that are isolable by the proposed method:
Zt
Zt
Now taking into account the corresponding adaptive threshold µrj (t) given by (31), we can conclude that, if condition (33) is satisfied at time t = tr , we obtain |˜ yjr (tr )| > µrj (tr ), which implies that the possibility of the occurrence of fault r can be excluded at time t = tr . 6. CONCLUSION In this paper, a robust fault diagnosis scheme for a class of Lipschitz uncertain nonlinear system is presented. Robustness and fault sensitivity are enhanced via the use of adaptive thresholds in the diagnostic decision-making stage. The fault isolability conditions characterizing the class of isolable faults are rigorously investigated.
(||A21 || + γ2 )χr (τ ) + η¯2 dτ
e−λj (t−τ ) αs ||Ωs || + αr ||Ωr || dτ
REFERENCES
+ |C¯j Ωr |κr (t) + 2ω2 kj e−λj (t−Td ) . (33) Proof: Denote the state estimation errors of the r-th △ △ isolation estimator by z˜1r (t) = z1 (t) − zˆ1r (t) and z˜2r (t) = r z2 (t) − zˆ2 (t). By using (23) and (3), in the presence of fault s, the state estimation error of the r-th estimator for t > Td satisfies
W. Chen and M. Saif, ”A sliding mode observer-based strategy for fault detection, isolation, and estimation in a class of Lipschitz nonliner systems,” International Journal of System Science, vol. 38, pp. 943–955, 2007. J. A. Farrell, T. Berger, and B. D. Appleby, ”Using learning techniques to accommodate unanticipated faults,” IEEE Control Sys. Mag., vol. 13, pp. 40–49, 1993. r z˜˙ 1 = A11 z˜1r + ρ1 (z, u) − ρ1 (ˆ z r , u) + η1 (z, u, t) (34) J. Farrell and M. M. Polycarpou. Adaptive Approximation Based Control. Hoboken, NJ: J. Wiley, 2006. r z˜˙ 2 = A¯22 z˜2r + A21 z˜1r + ρ2 (z, u) − ρ2 (ˆ z r , u) + η2 (z, u, t) P. A. Ioannou and J. Sun. Robust Adaptive Control. ˙ Prentice Hall, Englewood Cliffs, NJ, 1996. (35) + Gs θs − Gr θˆr − Ωr θˆr . B. Jiang, M. Staroswiecki, and V. Cocquempot, ”Fault ˙ s − A¯22 Ωs and Gr = Ω ˙ r − A¯22 Ωr By substituting Gs = Ω diagnosis based on adaptive observer for a class of into (35), we obtain nonlinear systems with unknown parameters,” Int. J. Control, vol 77, no. 4, pp. 415-426, 2004. r z r , u) K. Vijayaraghavan, R. Rajamani and J. Bokor, ”Quantiz˜˙ 2 = A¯22 z˜2r + Ωr θˆr − Ωs θs + A21 z˜1r + ρ2 (z, u) − ρ2 (ˆ tative fault estimation for a class of nonlinear systems,” d r ˆr Int. Journal of Control, vol. 80, pp. 64–74, no. 1, 2007. + η2 (z, u, t) − Ω θ − Ωs θs − Ωs θ˙s . dt A. T. Vemuri and M. M. Polycarpou, ”Robust nonlinear △ fault diagnosis in input-output systems,” Int. Journal By defining z¯2r (t) = z˜2r (t) + Ωr θˆr − Ωs θs , the above of Control, vol. 68, pp. 343-360, 1997. differential equation can be rewritten as H. Wang, Z. J. Huang, and S. Daley, “On the use of r adaptive updating rules for actuator and sensor fault z¯˙ 2 = A¯22 z¯2r + A21 z˜1r + ρ2 (z, u) − ρ2 (ˆ z r , u) + η2 − Ωs θ˙s . diagnosis,” Automatica, vol. 33, pp. 217–225, 1997. (36) X. Yan and C. Edwards, ”Nonlinear robust fault resconstruction and estimation using a sliding mode observer,” △ The j-th component of the output estimation error y˜jr (t) = Automatica, vol. 43, pp. 1605-1614, 2007. yj (t) − yjr (t), j = 1, · · · p, is given by: X. Zhang, M. M. Polycarpou, and T. Parisini, ”Robust fault isolation of a class of nonlinear input-output sysy˜jr (t) = C¯j z˜2 (t) = C¯j (¯ z2r (t) − Ωr θˆr + Ωs θs ) . tems,” Int. Journal of Control, vol. 74, pp. 1295-1310, By using (32), i.e., the definition of the fault mismatch 2001. function, the above equation can be rewritten as follows X. Zhang, T. Parisini, and M. M. Polycarpou, ”Sensor Bias y˜jr (t) = C¯j z¯2r (t) + hsr Fault Isolation in a Class of Noninear Systems,” IEEE j (t) . Trans. on Automatic Control, vol. 50, pp. 370-376, 2005. By applying the triangle inequality, we have |˜ yjr (t)| ≥ |hsr ¯2r (t)| . j (t)| − |Cj z
(37)
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