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Unobservability Subspaces For Continuous-time Markovian Jump Systems with Application to Fault Diagnosis
Proceedings of the 7th IFAC Symposium on Fault Detection, Supervision and Safety of Technical Processes Barcelona, Spain, June 30 - July 3, 2009
Unobservability Subspaces For Continuous-time Markovian Jump Systems with Application to Fault Diagnosis N. Meskin ∗ , K. Khorasani ∗ ∗
Department of Electrical and Computer Engineering, Concordia University, 1455 de Maissouneuve Blvd. W., Montreal, Quebec H3G 1M8 Canada, Email: {n meskin,kash}@ece.concordia.ca Abstract: This paper introduces the notion of unobservability subspaces for continuous-time Markovian Jump System (MJS) with irreducible Markov process. First a geometric property related to the unobservable subspace of a Markovian jump system is presented. A new approach for determining the conditions for weak-observability of MJS systems is then developed. The concept of an unobservability subspace is presented and an algorithm for obtaining it is described. The necessary and sufficient conditions for solvability of the fundamental problem of residual generation (FPRG) for MJS systems are derived by utilizing our introduced unobservability subspaces. 1. INTRODUCTION A great deal of attention has recently been devoted to the Markovian jump systems (MJS) (Cao and Lam, 2000; Xing and Lam, 2006; Costa et al., 2005; Boukas, 2006) which comprise an important class of hybrid systems. This family of systems is generally modeled by a set of linear systems with transitions between models that are determined by a Markov chain taking values in a finite set. Markovian jump systems (MJS) are popular in modeling many practical systems where one may experience abrupt changes in system structure and parameters. These changes are quite common and do frequently occur in manufacturing systems, economic systems, communication systems, power systems (Mariton, 1990). Recently, Markovian jump systems have also gained interest for their capability in modeling behaviors and phenomenon in networks that are manifested among sensors, actuators and processors (Xiao et al., 2000; Gupta et al., 2003; Jin et al., 2006). In recent years, only a few work on fault detection and isolation (fault diagnosis) of MJS systems have appeared in the literature. In Zhang et al. (2004); Zhong et al. (2005); Mao et al. (2007), a robust fault detection (and not an isolation) filter for discrete-time Markovian jump systems is developed based on an H∞ filtering framework, in which the residual generator is also an MJS system. An LMI approach is developed for solving the problem. In Wang et al. (2006), a robust fault identification filter for a class of discrete-time Markovian jump systems with mode dependent time-delays and norm bounded uncertainty is developed based on an H∞ optimization technique. In the approach in Wang et al. (2006), the generated residual signal is an estimate of the fault signal. In Meskin and Khorasani (2008), a robust fault detection and isolation algorithm is developed for discrete-time Markovian jump system. This research is supported in part by a Strategic Projects Grant from Natural Sciences and Engineering Research Council of Canada (NSERC).
978-3-902661-46-3/09/$20.00 © 2009 IFAC
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However, the problem of fault isolation for continuous-time Markovian jump systems has not been completely solved and fully addressed in the above references. In this paper, we have adopted a geometric approach to the FDI problem of Markovian jump systems with irreducible Markov process. Towards this end, the first contribution of this paper is in the derivation of a geometric property for the unobservable subspace of MJS systems (Theorem 4). The notion of an unobservability subspace is then introduced for MJS systems (Definition 8). An algorithm is also proposed for obtaining this subspace. This is achieved by introducing an alternative definition of an unobservability subspace as presented in Theorem 13, which only depends on the matrices of the system. Based on this alternative definition, an algorithm for constructing the smallest unobservability subspace containing a given subspace is proposed (Algorithm 2). By utilizing the developed geometric framework, necessary and sufficient conditions for solvability of the fundamental problem of residual generation (FPRG) for MJS systems are derived (Theorem 17). To summarize, the main contribution of this work is to introduce and develop a theoretical foundation based on a geometric framework for solving the fault diagnosis problem for continuous-time Markovian jump systems. The remainder of this paper is organized as follows. In Section 2, the geometric characteristics of unobservable subspaces for MJS systems is developed and the notion of an unobservability subspace is introduced formally. In Section 3, the necessary and sufficient conditions for solving the fundamental problem of residual generation (FPRG) for MJS systems are obtained. Conclusions and future work are presented in Section 4. The following notation is used throughout this paper. Script letters X , U, Y, ..., denote real vector spaces. B = Im B denotes the image of B; Ker C denotes the kernel of C. We say a map C is monic if Ker C = 0 and is epic if Im C = Y. If a map C is epic, then C −r denotes a right
10.3182/20090630-4-ES-2003.0057
7th IFAC SAFEPROCESS (SAFEPROCESS’09) Barcelona, Spain, June 30 - July 3, 2009
inverse of C (i.e., CC −r = I)). For any positive integer k, k denotes the finite set {1, 2, · · · , k}. A subspace S ⊆ X is termed A-invariant if AS ⊆ S. For A-invariant subspace S ⊆ X , A : S denotes the restriction of A to S, and A : X /S denotes the map induced by A on the factor space X /S. For a linear system (C, A, B), < Ker C|A > denotes the unobservable subspace of (C, A). For a given subspace L, dim(L) denotes the dimension of L. The largest Ai invariant (i ∈ Ψ) subspace that is contained in K is denoted by << K|Ai >>i∈Ψ . 2. UNOBSERVABLE AND UNOBSERVABILITY SUBSPACES FOR MARKOVIAN JUMP SYSTEMS In this section, a geometric definition for the unobservable subspace of Markovian jump systems (MJS) is introduced. The notion of unobservability subspace is then formalized for the MJS systems that are governed by the dynamical system (1). To develop an algorithm for constructing such a subspace, an alternative definition of unobservability subspace is presented, which only depends on the matrices of the system. Based on this alternative definition, an algorithm for constructing the smallest unobservability subspace containing a given subspace is proposed. As shown in the next section, the unobservability subspace for MJS systems plays a central role in solving the fault diagnosis problem for Markovian jump systems. Consider the following Markovian jump system x(t) ˙ = Aλ(t) x(t), y(t) = Cλ(t) x(t)
(1)
with initial condition x(0) = x0 , λ(0) = i0 where x ∈ X is the continuous state of the system with dimension n; y ∈ Y is the output signal with dimensions q, and {λ(t), t ≥ 0} is a continuous-time irreducible Markov process taking values in the finite set Ψ = {1, ..., N }. The Markov process describes the switching between the different system modes and its evolution is governed by the following probability transitions: πij h + o(h) i = j P{λ(t + h) = j|λ(t) = i} = 1 + πii h + o(h), otherwise where πij is the transition rate from mode i to mode j with N πij ≥ 0 when i = j, πii = − j=1,j=i πij , and o(h) is a
function that satisfies limh→0 o(h) h = 0. The matrices Aλ(t) and Cλ(t) are known constant matrices for all λ(t) = i ∈ Ψ. For simplicity, we denote the matrices associated with λ(t) = i by Aλ(t) = Ai and Cλ(t) = Ci . We first start with the definition of weak observability for the Markovian jump system (1). Definition 1. (Costa and Val (2002)). The system (1) is said to be weakly (W-) observable when there exist td ≥ 0 and γ > 0 such that W td (x, i) ≥ γ|x|2 for each x ∈ X and i ∈ Ψ where t x (τ )Cλ(τ W t (x, i) = E ) Cλ(τ ) x(τ )dτ |x(0) = x, λ0 = i 0
In Costa and Val (2002), a collection of matrices O = {O1 , ..., ON } is introduced for testing the W-observability of a Markovian jump system according to the following procedure. Let Oi (0) = Ci Ci , i ∈ Ψ and define the sequence of matrices as Oi (k) = Ai Oi (k − 1) + Oi (k −
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N 1)Ai + j=1 πij Oj (k − 1). The matrix Oi is now defined according to Oi = [Oi (0) Oi (1) · · · Oi (n2 N − 1)] . Theorem 2. (Costa and Val (2002)). The MJS system (1) is W-observable if and only if Oi has a full rank for each i ∈ Ψ. By considering the above definition of W-observability, one can define the set of unobservable states as follows. Definition 3. A state x is said to be unobservable if W t (x, i) = 0 for all t ≥ 0 and i ∈ Ψ. Let Q denotes the unobservable set for the MJS system (1), i.e Q = {x|W t (x, i) = 0, ∀i ∈ Ψ, t ≥ 0}. It is shown in Costa and Val (2002) that for irreducible Markov processes, N {Oi } = N {Oj }, i, j ∈ Ψ, i = j and Q = N {Oi }. Therefore, Q is the subspace of X and is called the unobservable subspace of the Markovian jump system (1). The theorem introduced below characterizes a geometric property of Q. Theorem 4. An unobservable subspace Q for system (1) with irreducible Markov process is the largest Ai -invariant N (i ∈ Ψ) that is contained in K = i=1 KerCi . Proof: It follows from the above discussion that Q ⊆ KerCi , i ∈ Ψ, and hence Q ⊆ K. Let x ∈ Q. Our goal is to show that Ai x ∈ Q for all i ∈ Ψ (i.e. Q is Ai -invariant, i ∈ Ψ). Since x ∈ N {Oi (k − 1)} and x ∈ N {Oi (k)}, i ∈ Ψ, then Oi (k)x = Ai Oi (k − 1)x + N Oi (k − 1)Ai x + j=1 πij Oj (k − 1)x = Oi (k − 1)Ai x = 0. Hence, Ai x ∈ N {Oi (k − 1)} and Q is Ai -invariant for all i ∈ Ψ. Next we show that Q is the largest Ai -invariant (i ∈ Ψ) that is contained in K. Let V be an Ai -invariant (i ∈ Ψ) subspace that is contained in K. Clearly, V ⊆ N {Oi (0)}, i ∈ Ψ. Let V ⊆ N {Oi (k − 1)}, i ∈ Ψ and x ∈ V, then Oi (k)x = Ai Oi (k − 1)x + Oi (k − 1)Ai x + N j=1 πij Oj (k − 1)x = 0 since Oi (k − 1)x = 0, i ∈ Ψ and Ai x ∈ V (V is Ai -invariant). Hence, V ⊆ N {Oi (k)}, i ∈ Ψ and V ⊆ Q. This shows that Q contains all the subspaces that are Ai -invariant (i ∈ Ψ) and is contained in K. Let us now present definitions for stability and detectability of MJS systems. Definition 5. (Costa and Val (2002)). The system x(t) ˙ = Aλ(t) x(t) is mean square (MS) stable if for each x0 ∈ X and i0 ∈ Ψ, limt→∞ E{||x(t)||2 } = 0. Definition 6. (Costa and Val (2002)). We say that system (1) is MS-detectable when there exists maps Gi , i ∈ Ψ of appropriate dimension for which x(t) ˙ = (Aλ(t) + Gλ(t) Cλ(t) )x(t) is MS-stable. The following computational linear matrix inequalities can be used for testing the MS-detectability of a MJS system (Costa and Val, 2002). In other words, the MSdetectability of system (1) is equivalent to the feasibility of the set N Ai Xi + Xi Ai + Ci Li + Li Ci + πij Xj < 0, i ∈ Ψ (2) j=1
in the unknowns Xi > 0 and Li with appropriate dimensions. We are now in a position to introduce the notions
7th IFAC SAFEPROCESS (SAFEPROCESS’09) Barcelona, Spain, June 30 - July 3, 2009
of conditioned invariant and unobservability subspaces for Markovian jump systems. Definition 7. A subspace W is said to be conditioned invariant for system (1) if Ai (W ∩ KerCi ) ⊆ W, i ∈ Ψ. It is clear that if W is conditioned invariant for system (1), then W is (Ci , Ai )-invariant for all i ∈ Ψ. Therefore, there exist maps Di such that (Ai + Di Ci )W ⊆ W, i ∈ Ψ. We denote the class of conditioned invariant subspaces for system (1) by W. If W ∈ W, we write D(W) for the class of maps Di where (Ai + Di Ci )W ⊆ W, i ∈ Ψ. The notion of conditioned invariant subspace for system (1) is a dual to that of the robust maximal controlled invariant which is introduced in Basile and Marro (1987). By duality it can be shown that W is closed under the operation of subspace intersection, and hence for any given subspace L ⊆ X , the family of conditioned invariant subspaces that contains L (denoted by W(L)) has an infimal element which is denoted by W ∗ = inf W(L). The following algorithm can now be used for constructing W ∗ : Algorithm 1. The subspace W ∗ coincides with and is obtained from the last term of the following sequence W0 = L;
Wk = L +
N
Ai (Wk−1 ∩ Ker Ci )
Definition 8. A subspace S is an unobservability subspace for system (1) if there exist output injection maps Di : Y → X and measurement mixing maps Hi : Y → Y such that S is the unobservable subspace for system x(t) ˙ = (Aλ(t) + Dλ(t) Cλ(t) )x(t), y(t) = Hλ(t) Cλ(t) x(t) N
j=1 (KerHj Cj )|Ai
We are now in a position to state our next result. Theorem 10. A subspace S is an unobservability subspace for system (1) if and only if there exist maps Di , such that N S =<< j=1 (S + KerCj )|Ai + Di Ci >>i∈Ψ . Proof: The proof follows readily from Lemma 9 by taking Sˆj = KerHj Cj .
i=0
i.e. S =<<
N ˆ ˆ Proof: We have S ⊆ j=1 Sj ⊆ Sj and KerCj ⊆ N ˆ N Sˆj , j ∈ Ψ, so that j=1 (S + KerCj ) ⊆ j=1 Sj . N (S + KerC )|A >> Consequently, << j i i∈Ψ ⊆<< j=1 N ˆ j=1 Sj |Ai >>i∈Ψ = S. On the other hand, Ai S ⊆ S N N and j=1 (S + KerCj ) ⊇ S + j=1 KerCj ⊇ S. Hence N S ⊆<< j=1 (S + KerCj )|Ai >>i∈Ψ , and as a result N << j=1 (S +KerCj )|Ai >>i∈Ψ = S. To show the converse part, let {ci1 , ..., cir } be a basis for S + KerCi such that {cir−pi , ..., cir } is the basis for KerCi (dim(KerCi ) = pi ). Therefore, yij = Ci cij , j = 1, ..., r − pi − 1 are independent. Let {yi1 , ..., yiq } be a basis for Y, and define Hi yij = 0, j = 1, ..., r − pi − 1, Hi yij = yij , j = r − pi , ..., r. Consequently, KerCi + S = KerHi Ci , and thereN N fore i=1 KerCi + S = i=1 KerHi Ci .
+ Di Ci >>i∈Ψ .
Remark 1. It should be noted that the notion of unobservability subspace introduced in Definition 8 is the most general concept of u.o.s. and the ones that have been introduced in the literature for linear Massoumnia et al. (1989), LPV Bokor and Balas (2004), bilinear Hammouri et al. (2001) and time-delay Meskin and Khorasani (2009) systems can be considered as a special case of our proposed unobservability subspace. Indeed, for Ci = C, i ∈ Φα , (identical output measurements) the above definition coincides with the unobservability subspace for the LPV and the bilinear systems as presented in Bokor and Balas (2004) and Hammouri et al. (2001), respectively. We denote the class of all unobservability subspaces for system (1) by S. In the following our goal is to derive an alternative characterization for the unobservability subspace which is independent of the maps Di and Hi as used in Definition 8. As shown subsequently, this alternative definition provides us with means to obtain the unobservability subspaces more readily. The following lemma presents a result that is necessary for formulating our alternative definition. Lemma 9. Let Sˆj ⊆ X such that KerCj ⊆ Sˆj , j ∈ Ψ N ˆ N and << j=1 Sj |Ai >>i∈Ψ = S, then << j=1 (S + N KerCj )|Ai >>i∈Ψ = S. Conversely, if << j=1 (S + KerCj )|Ai >>i∈Ψ = S, there exist maps Hj : Y → Y, j ∈ N Ψ such that << j=1 KerHj Cj |Ai >>i∈Ψ = S
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The above theorem eliminates the need for maps Hi from the Definition 8. For a given unobservability subspace S, the measurement mixing maps Hi ’s can be computed from S by solving KerHi Ci = S + KerCi , i ∈ Ψ. Next, we try to the characterize the unobservability subspace by means of an algorithm that computes S without explicitly constructing (D1 , ..., DN ) ∈ D(S). For an arbitrary subspace N S ⊆ X , let us define a family G = {S : S = i=1 (S + (A−1 i S ∩KerCi ))}. Below we first show that G has a unique maximal member. Lemma 11. There exists a unique element S ∗ ∈ G such that S ⊆ S ∗ for every S ∈ G. Proof: Define a sequence S μ ⊆ X according to S 0 = X , N μ−1 ∩ KerCi )). The sequence and S μ = i=1 (S + (A−1 i S μ S is non-increasing since S 1 ⊆ S 0 and if S μ ⊆ S μ−1 , N N μ thenS μ+1 = i=1 (S + (A−1 i S ∩ KerCi )) ⊆ i=1 (S + −1 μ−1 μ ∩ KerCi )) = S . Therefore, there exist k ≤ n (Ai S such that S μ = S k and we set S ∗ = S k . Clearly S ∗ ∈ G. Next, we show that S ∗ is the maximal element. Let S ⊆ G, then S ⊆ S 0 and if S ⊆ S μ , we have N N −1 μ S = i=1 (S + (A−1 i S ∩ KerCi )) ⊆ i=1 (S + (Ai S ∩ μ+1 μ KerCi )) = S . Consequently, S ⊆ S for all μ, and hence S ⊆ S ∗ . The next lemma provides an important property of the maximal element S ∗ which will be used for introducing our suggested alternative characterization of the unobservability subspace of system (1). Lemma 12. Let S ∈ W and (D1 , ..., DN ) ∈ D, then S ∗ is the largest (Ai + Di Ci )-invariant (i ∈ Ψ) that is contained N N in j=1 (S + ker Cj ), i.e. S ∗ =<< j=1 (S + KerCj )|Ai + Di Ci >>i∈Ψ . Proof: First we show that any S ∈ G is (Ai + Di Ci )N invariant (i ∈ Ψ). We have S = i=1 (S + (A−1 i S ∩
7th IFAC SAFEPROCESS (SAFEPROCESS’09) Barcelona, Spain, June 30 - July 3, 2009
KerCi )) and (Aj + Dj Cj )S = (Aj + Dj Cj )
N
(S + (A−1 i S ∩ KerCi ))
i=1 −1 ⊆ (Aj + Dj Cj )(S + Aj S ∩ KerCj ) ⊆ (Aj + Dj Cj )S + Aj (A−1 j S ∩ KerCj ) ⊆ S + S ⊆ S N where we used the relationship S = i=1 (S + (A−1 i S N −1 S ∩KerC ) ⊇ S and A (A KerCi )) ⊇ S+ i=1 (A−1 i j i j S)
∩ ⊆ S . Therefore, (Aj + Dj Cj )S ⊆ S , j ∈ Ψ; and hence S ∈ W and (D1 , ..., DN ) ∈ D(S ).
Consequently, we have S ∗ ∈ W. Next we show that for any subspace W such that it is (Ai + Di Ci )-invariant N (i ∈ Ψ) and is contained in j=1 (S + ker Cj ), we have W ⊆ S ∗ . If W ⊆ S then it follows that W ⊆ S ∗ , since S ⊆ S ∗ . Therefore, we consider the case where S ⊆ W. We have A−1 ∩ KerCi = (Ai + Di Ci )−1 W ∩ KerCi i W N N and as a result i=1 (S + A−1 i W ∩ KerCi ) = i=1 (S + (Ai + Di Ci )−1 W ∩ KerCi ). It is clear that W ⊆ S 0 . If W ⊆ S μ−1 , then N Sμ ⊇ (S + (Ai + Di Ci )−1 W ∩ KerCi )) i=1
⊇
N
(S + (W ∩ KerCi )) =
i=1
N
(W ∩ (S + KerCi )) = W
i=1
where we used the fact that W ⊆ (Ai +Di Ci )−1 W and the modular distributive rule (Wonham, 1985) (if S ⊆ W, then S + (W ∩ KerCi ) = W ∩ (S + KerCi )). As a consequence S μ ⊇ W; and hence S ∗ ⊇ W. This shows that S ∗ is the largest (Ai + Di Ci )-invariant (i ∈ Ψ) which is contained N in i=1 (S + KerCi ). We are in the position to introduce our proposed alternative characterization of an unobservability subspace for system (1). Theorem 13. Let S ⊆ X . Then S ∈ S if and only if a) S is conditioned invariant, and b) S = S ∗ where S ∗ is the maximal element of G. Proof: (If part) If a) and b) hold, then according to N Lemma 12, we have S =<< j=1 (S + KerCj )|Ai >>i∈Ψ , and hence using Theorem 10, we have S ∈ S. (Only if part) If S ∈ S, it follows that S ∈ W and according to Lemma 12, we have S = S ∗ . It can be shown that the class of unobservability subspaces of system (1) is close under the operation of subspace intersection and is nonempty. Hence the class of unobservability subspaces which contains a given subspace L ⊆ X (denoted by S(L)) has an infimal element S ∗ . This property is crucial for the application of unobservability subspace to the problem of fault diagnosis of MJS systems. The next algorithm provides a procedure for constructing S ∗ . Algorithm 2. Let W ∗ = inf W(L) and define the sequence Z μ according to N μ−1 Z0 = X ; Zμ = (W ∗ + (A−1 ∩ KerCi )) (3) i Z i=1
Then S ∗ = Z μ , whenever Z μ+1 = Z μ .
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To analyze the above algorithm note that the sequence Z μ is non-increasing and Z k+1 = Z k for k ≥ n−dim(W ∗ ). Let N Z ∗ = Z k . According to Lemma 12, Z ∗ =<< i=1 (W ∗ + KerCj )|Ai +Di Ci >>i∈Ψ for some (D1 , ..., DN ) ∈ D(W ∗ ). Using the same approach as in Lemma 9, one can obtain N the maps Hj ’s such that Z ∗ =<< i=1 (W ∗ +KerCj )|Ai + N Di Ci >>i∈Ψ =<< i=1 (KerHj Cj )|Ai + Di Ci >>i∈Ψ and therefore Z ∗ is an unobservability subspace according to Definition 8. Moreover, it follows that L ⊆ W ∗ ⊆ Z ∗ (Z ∗ ∈ W(L) and W = inf W(L)); hence Z ∗ ∈ S(L), and consequently S ∗ ⊆ Z ∗ . On the other hand, according to Theorem 13 we have N μ−1 S ∗ = S n where S 0 = X and S μ = i=1 (S ∗ + (A−1 ∩ i S KerCi )), μ ∈ n. Since W ∗ ⊆ S ∗ , it can be shown by induction that Z μ ⊆ S μ , μ ∈ n. Indeed, Z 0 = S 0 , and N μ−1 ∩ if Z μ−1 ⊆ S μ−1 , then Z μ = i=1 (W ∗ + (A−1 i Z N −1 μ−1 ∗ μ ∩ KerCi )) = S . ConKerCi )) ⊆ i=1 (S + (Ai S sequently, Z ∗ ⊆ S ∗ . It follows from the above algorithm and Lemma 12 that D(W ∗ ) ⊂ D(S ∗ ). Therefore, the maps Di ’s for S ∗ can be found from W ∗ and once S ∗ is found from Algorithm 2, the mapping Hi ’s can also be computed from S ∗ and equation KerHi Ci = S + KerCi . Finally, we introduce the notion of outer MS-detectability of an unobservability subspace. As shown in the next section, outer MS-detectability is a necessary condition for designing a stable residual generator for the fault diagnosis problem. Definition 14. A subspace S ∈ S is said to be outer MSdetectable if the factor system x(t) ˙ = A0λ(t) x(t), y(t) = Mλ(t) x(t) is MS-detectable where A0i = (Ai +Di Ci : X /S), Mi is the solution to Mi P = Hi Ci and P is the canonical projection of X on X /S. It is clear that the outer MS-detectability of a given unobservability subspace can be tested based on LMI (2). We are now in the position to formally introduce the fundamental problem in residual generation (FPRG) for the Markovian jump system (1). 3. FAULT DETECTION AND ISOLATION OF MARKOVIAN JUMP SYSTEMS In this section the Fundamental Problem in Residual Generation (FPRG) is investigated for the Markovian jump system (1). This problem was originally considered for linear systems in Massoumnia et al. (1989) and was extended to nonlinear systems in Persis and Isidori (2001). Similar problems are also considered for bilinear and LPV systems in Hammouri et al. (2001) and Bokor and Balas (2004), respectively. The objective in this section is to solve the FPRG for continuous-time Markovian jump systems (MJS) with irreducible Markov Process. Consider the following Markovian jump system x(t) ˙ = Aλ(t) x(t) + Bλ(t) u(t) + L1λ(t)) m1 (t) + L2λ(t) m2 (t) (4) y(t) = Cλ(t) x(t), x(0) = x0 , λ(0) = i0 where it is assumed that all the matrices are the same as in (1) and u is the input signal with dimension m. The
7th IFAC SAFEPROCESS (SAFEPROCESS’09) Barcelona, Spain, June 30 - July 3, 2009
matrices L1λ(t) , L2λ(t) represent the fault signatures and are monic and mi (t) ∈ Mi ⊂ X , i = 1, 2 denote the fault modes. For sake of simplicity in analysis and derivation of our results, we first consider the case with two faults. The more general case of multiple faults is considered at the end of this section. We denote the fault signatures associated with λ(t) = i by L1i and L2i . The fault modes together with the fault signatures can be used to model the effects of actuator faults, sensor faults and system faults on the dynamics of the system. For example, the effect of a fault in the i-th actuator may be represented by L1i as the i-th column of Bi and if an actuator fails, then m1 (t) = −ui (t). The FPRG problem is concerned with the design of a Markovian jump residual generator that is governed by the filter dynamics of the form w(t) ˙ = Fλ(t) w(t) − Eλ(t) y(t) + Kλ(t) u(t) r(t) = Mλ(t) w(t) − Hλ(t) y(t) (5) where w(t) ∈ F ⊂ X such that the response of r(t) is affected by the fault mode m1 (t) and is decoupled from m2 (t) and if m1 is identically zero then limt→∞ E||r(t)||2 = 0 for any input signal u(t). Define the extended space X e = X ⊕ F and U e = U ⊕ M2 , we can rewrite equations (4) and (5) as follows: e x˙ e (t) = Aeλ(t) xe (t) + Bλ(t) ue (t) + Le1 λ(t) m1 (t) e xe (t) r(t) = Hλ(t)
(6) 0 Fλ(t) ,
Aλ(t) where xe (t) ∈ X e and ue ∈ U e , Aeλ(t) = −Eλ(t) Cλ(t)
Bλ(t) L2λ(t)
L1 e e1 e λ(t) Bλ(t) = K , Lλ(t) = and Hλ(t) = 0 λ(t) 0
−Hλ(t) Cλ(t) Mλ(t) .
In order to investigate the criteria for determining whether a nonzero m1 (t) affects the residual signal r(t), the notion of an input observability for the Markovian jump system (6) is defined and formalized below. Definition 15. The input signal m1 (t) in system (4) is said to be input observable if Le1 i , i ∈ Ψ is monic and the image of Le1 ’s does not intersect with the unobservable subspace i of system (6). Based on the above definition, the FPRG problem can now be formally stated as the problem of designing the dynamical filter (5) such that (a) r is decoupled from ue , (b) m1 is input observable in the augmented system (6) and (c) limt→0 E{||r(t)||2 } = 0, for m1 (t) = 0, ∀ i0 ∈ Ψ and ∀xe0 ∈ X e . We need to first derive a preliminary result for obtaining the solvability condition for the FPRG problem. The following embedding map Q : X → X e is x defined according to Massoumnia et al. (1989): Qx = 0 x e −1 where if V ⊆ X , we have Q V = {x|x ∈ X , 0 ∈ V}. Our first result is the generalization of the Proposition 1 that was obtained in Massoumnia et al. (1989) to the Markovian jump system (4). Lemma 16. Let S e be the unobservable subspace of system (6). The unobservability subspace for system (4) is given by Q−1 S e . Proof: First we show that S = Q−1 S e is conditioned invariant. Let x ∈ S ∩ KerCi , we to show that Ai x ∈ S.
need This follows by noting that A0i x = −EAiiCi F0i x0 since
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S e is Aei -invariant i ∈ Ψ. Therefore, Ai x ∈ S and S is conditioned Next if x ∈ S, then Qx ∈ S e and there x invariant. N fore 0 ∈ i=1 KerHie . This shows that Hi Ci x = 0; and N hence x ∈ ∩N i=1 KerHi Ci and S ⊆ ∩i=1 KerHi Ci . Finally, according to the definition of the unobservable subspace S e (the largest Ae -invariant subspace in ∩KerHie ), S is the largest conditioned invariant contained in ∩N i=1 KerHi Ci , and therefore S ∈ S. We are now in the position to derive the solvability condition for the FPRG problem corresponding to the Markovian jump system (4). Theorem 17. The FPRG problem has a solution for the augmented MJS system (6) only if S ∗ L1j = 0, j ∈ Ψ (7) N where S ∗ = inf S( i=1 L2i ). On the other hand, if the above S ∗ exists such that it is also outer MS-detectable, then the FPRG problem is then guaranteed to have a solution. Proof: (Only if part) Let S e be an unobservable subspace of system (6). To satisfy the condition (a) of FPRG, we should have Bie ⊂ S e , i ∈ Ψ. Hence, L2i ⊂ Q−1 Bie ⊂ Q−1 S e = S and by invoking Lemma 16, we obtain N L2i ) S ∈ S(
(8)
i=1
For condition (b) to hold, according to the Definition 15, Le1 i should be monic (which is already assumed to hold) e and Le1 i ∩ S = 0, i ∈ Ψ. Thus e 1 Q−1 (Le1 i∈Ψ (9) i ∩ S ) = Li ∩ S = 0, Therefore, equations (8) and (9) hold only if equation (7) is true. (if part): Given the unobservability subspace S ∗ which is outer MS-detectable, there exist output injection maps Di ’s and measurement mixing maps Hi ’s such that N S ∗ =<< j=1 Ker Hj Cj |Ai + Di Ci >>i∈Ψ where Hj is the solution to equation KerHj Cj = S ∗ + KerCj . Let P be the canonical projection of X on X /S ∗ and Mi , i ∈ Ψ be a unique solution to Mi P = Hi Ci and A0i = (Ai + Di Ci : X /S ∗ ) where P (Ai + Di Ci ) = A0i P, i ∈ Ψ. Due to the fact that S ∗ is assumed to be outer MS-detectable, there exist Gi , i ∈ Ψ such that x(t) ˙ = (A0λ(t) + Gλ(t) Mλ(t) )x(t) is MS-stable. Let us define Fi = A0i + Gi Mi , Ei = P (Di + P −r Gi Hi ), Ki = P Bi , i ∈ Ψ, and e(t) = w(t) − P x(t). By using equation (5) we obtain e(t) ˙ = Fi w(t) − Ei y(t) + Ki u(t) − P (Ai x(k) + Bi u(k) + L1i m1 (t)) + L2i m2 (t)) = Fi w(t) − P L1i m1 (t) − P (Ai + Di Ci )x(t) − Gi Hi Ci x(t) = Fi w(t) − P L1i m1 (t) − A0i P x(t) − Gi Mi P x(t) = Fi e(t) − P L1i m1 (t)
Note that P L2i = 0, i ∈ Ψ, since L2i ∈ S ∗ , i ∈ Ψ. Also r(t) = Mi w(t) − Hi y(t) = Mi w(t) − Hi Ci x(t) = Mi e(t) Consequently, the error dynamics can be written according to e(t) ˙ = Fλ(t) e(t) − P L1λ(t) m1 (t) r(t) = Mλ(t) e(t)
(10)
7th IFAC SAFEPROCESS (SAFEPROCESS’09) Barcelona, Spain, June 30 - July 3, 2009
It is clear that the fault mode m2 (t) does not affect the residual signal r(t) and since the dynamics (10) is observable, condition (b) also holds. Moreover, for m1 (t) = 0 the system is MS-stable. To conclude this section, we now consider a Markovian jump system that has multiple faults and that is governed by the following dynamical system k x(t) ˙ = Aλ(t) x(t) + Bλ(t) u(t) + Ljλ(t) mj (t) j=1
x(0) = x0 , λ(0) = i0 (11) y(t) = Cλ(t) x(t) where all the matrices are the same as in system (4), Ljλ(k) , j ∈ k are the fault signatures, and mj (t) ∈ Mj , j ∈ k are the fault modes. We denote the fault signatures associated with λ(t) = i by Lji , i ∈ Ψ, j ∈ k. The Extended Fundamental Problem in Residual Generation (EFPRG) (Massoumnia et al., 1989) for the Markovian jump system (11) is now defined as the problem of generating k residual signals rl (t), l ∈ k from the Markovian jump detection filters (5) such that a fault in the j-th component mj can only affect the residual signal rj (t) and no other residual signals rl (t), l = j. It should be pointed out that a solution to the EFRPG problem amounts to indeed both detecting and isolating multiple and simultaneous faults in all the system components. The solvability condition for the EFPRG problem is obtained by invoking the solvability condition that was developed earlier for the FPRG problem as follows: Theorem 18. The EFPRG problem has a solution for system (11) only if Sj∗ Lji = 0, i ∈ Ψ, j ∈ k where N k Sj∗ = inf S( v=1 l=1,l=j Llv ). On the other hand, if the above Sj∗ , j ∈ k exist such that they are outer MSdetectable, the EFPRG problem is then guaranteed to have a solution. Proof: The proof is immediate by following along the lines that we developed for the proof of Theorem 17. . 4. CONCLUSIONS A geometric approach to the problem of fault diagnosis of continuous-time linear Markovian jump systems is developed in this paper. Starting with a new geometric characterization of the unobservable subspace of a Markovian jump system, the concept of unobservability subspaces is formalized and an algorithm for constructing these subspaces is presented. By invoking the notion of an unobservability subspace, the necessary and sufficient conditions for solving the fundamental problem of residual generation (FPRG) for Markovian jump systems is formally investigated. REFERENCES G. Basile and G. Marro. On the robust controlled invariant. Systems & Control Letters, 9(3):191–195, 1987. J. Bokor and G. Balas. Detection filter design for LPV systems: A geometric approach. Automatica, 40(3):511– 518, 2004. El-Kebir Boukas. Stochastic switching systems: analysis and design. Birkhauser, Boston, 2006.
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Y. Cao and J. Lam. Robust h1 control of uncertain Markovian jump systems with time-delay. IEEE Trans. Autom. Control, 45(1):77–83, 2000. E. F. Costa and J. B. R. Do Val. On the observability and detectability of continuous-time Markov jump linear systems. Siam J. Control Optim., 41(4):1295–1314, 2002. O. L. V. Costa, M. D. Fragoso, and R. P. Marques. Discrete-Time Markov Jump systems. Springer-Verlag, London, 2005. Vijay Gupta, Richard M. Murray, and Babak Hassibi. On the control of jump linear Markov systems with Markov state estimation. Proceeding of American Control Conference, pages 2893–2898, 2003. H. Hammouri, P. Kabore, and M. Kinnaert. A geometric approach to fault detection and isolation for bilinear systems. IEEE Trans. Autom. Control, 6(9):1451–1455, 2001. Zhipu Jin, Vijay Gupta, and Richard M. Murray. State estimation over packet dropping networks using multiple description coding. Automatica, 42(9):1441–1452, 2006. Z. Mao, B. Jiang, and P. Shi. H∞ fault detection filter design for networked control systems modeled by discrete Markovian jump systems. IET Control Theory and Application, 1(5):1336–1343, 2007. M. Mariton. Jump linear systems in automatic control. Marcel Dekker Inc., New York, 1990. M. A. Massoumnia, G. C. Verghese, and A. S. Willsky. Failure detection and identification. IEEE Trans. Autom. Control, 34(3):316–321, 1989. N. Meskin and K. Khorasani. Robust fault detection and isolation of time-delay systems using a geometric approach. Automatica, 2009. doi:10.1016/j.automatica.2009.02.019. N. Meskin and K. Khorasani. Geometric approach to robust fault detection and isolation of Markovian jump systems. Proceeding of American Control Conference, 2008. C. D. Persis and A. Isidori. A geometric approach to nonlinear fault detection and isolation. IEEE Trans. Autom. Control, 46(6):853–865, 2001. H. Wang, C. Wang, H. Gao, and L. Wu. An LMI approach to fault detection and isolation filter design for Markovian jump system with mode-dependent timedelays. Proceedings of the American Control Conference, pages 5686–5691, 2006. W. M. Wonham. Linear multivariable control - a geometric approach. New York:Springer, third edition, 1985. L. Xiao, A. Hassibi, and J. How. Control with random communication delays via a discrete-time jump system approach. Proceeding of American Control Conference, pages 2199–2204, 2000. J. Xing and J. Lam. Fixed-order robust H∞ filter design for Markovian jump systemswith uncertain switching probabilities. IEEE Trans. Signal Process., 54(4):1421– 1430, 2006. P. Zhang, S. X Ding, P. M. Frank, and M. Sader. Fault detection of networked control systems with missing measurements. Proceeding of the 5th Asian Control Conference, pages 1258–1263, 2004. M. Zhong, H. Ye, P. Shi, and G. Wang. Fault detection for Markovian jump systems. IEE Proc. Control Theory Application, 152(4):397–402, 2005.
Unobservability Subspaces For Continuous-time Markovian Jump Systems with Application to Fault Diagnosis N. Meskin ∗ , K. Khorasani ∗ ∗
Department of Electrical and Computer Engineering, Concordia University, 1455 de Maissouneuve Blvd. W., Montreal, Quebec H3G 1M8 Canada, Email: {n meskin,kash}@ece.concordia.ca Abstract: This paper introduces the notion of unobservability subspaces for continuous-time Markovian Jump System (MJS) with irreducible Markov process. First a geometric property related to the unobservable subspace of a Markovian jump system is presented. A new approach for determining the conditions for weak-observability of MJS systems is then developed. The concept of an unobservability subspace is presented and an algorithm for obtaining it is described. The necessary and sufficient conditions for solvability of the fundamental problem of residual generation (FPRG) for MJS systems are derived by utilizing our introduced unobservability subspaces. 1. INTRODUCTION A great deal of attention has recently been devoted to the Markovian jump systems (MJS) (Cao and Lam, 2000; Xing and Lam, 2006; Costa et al., 2005; Boukas, 2006) which comprise an important class of hybrid systems. This family of systems is generally modeled by a set of linear systems with transitions between models that are determined by a Markov chain taking values in a finite set. Markovian jump systems (MJS) are popular in modeling many practical systems where one may experience abrupt changes in system structure and parameters. These changes are quite common and do frequently occur in manufacturing systems, economic systems, communication systems, power systems (Mariton, 1990). Recently, Markovian jump systems have also gained interest for their capability in modeling behaviors and phenomenon in networks that are manifested among sensors, actuators and processors (Xiao et al., 2000; Gupta et al., 2003; Jin et al., 2006). In recent years, only a few work on fault detection and isolation (fault diagnosis) of MJS systems have appeared in the literature. In Zhang et al. (2004); Zhong et al. (2005); Mao et al. (2007), a robust fault detection (and not an isolation) filter for discrete-time Markovian jump systems is developed based on an H∞ filtering framework, in which the residual generator is also an MJS system. An LMI approach is developed for solving the problem. In Wang et al. (2006), a robust fault identification filter for a class of discrete-time Markovian jump systems with mode dependent time-delays and norm bounded uncertainty is developed based on an H∞ optimization technique. In the approach in Wang et al. (2006), the generated residual signal is an estimate of the fault signal. In Meskin and Khorasani (2008), a robust fault detection and isolation algorithm is developed for discrete-time Markovian jump system. This research is supported in part by a Strategic Projects Grant from Natural Sciences and Engineering Research Council of Canada (NSERC).
978-3-902661-46-3/09/$20.00 © 2009 IFAC
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However, the problem of fault isolation for continuous-time Markovian jump systems has not been completely solved and fully addressed in the above references. In this paper, we have adopted a geometric approach to the FDI problem of Markovian jump systems with irreducible Markov process. Towards this end, the first contribution of this paper is in the derivation of a geometric property for the unobservable subspace of MJS systems (Theorem 4). The notion of an unobservability subspace is then introduced for MJS systems (Definition 8). An algorithm is also proposed for obtaining this subspace. This is achieved by introducing an alternative definition of an unobservability subspace as presented in Theorem 13, which only depends on the matrices of the system. Based on this alternative definition, an algorithm for constructing the smallest unobservability subspace containing a given subspace is proposed (Algorithm 2). By utilizing the developed geometric framework, necessary and sufficient conditions for solvability of the fundamental problem of residual generation (FPRG) for MJS systems are derived (Theorem 17). To summarize, the main contribution of this work is to introduce and develop a theoretical foundation based on a geometric framework for solving the fault diagnosis problem for continuous-time Markovian jump systems. The remainder of this paper is organized as follows. In Section 2, the geometric characteristics of unobservable subspaces for MJS systems is developed and the notion of an unobservability subspace is introduced formally. In Section 3, the necessary and sufficient conditions for solving the fundamental problem of residual generation (FPRG) for MJS systems are obtained. Conclusions and future work are presented in Section 4. The following notation is used throughout this paper. Script letters X , U, Y, ..., denote real vector spaces. B = Im B denotes the image of B; Ker C denotes the kernel of C. We say a map C is monic if Ker C = 0 and is epic if Im C = Y. If a map C is epic, then C −r denotes a right
10.3182/20090630-4-ES-2003.0057
7th IFAC SAFEPROCESS (SAFEPROCESS’09) Barcelona, Spain, June 30 - July 3, 2009
inverse of C (i.e., CC −r = I)). For any positive integer k, k denotes the finite set {1, 2, · · · , k}. A subspace S ⊆ X is termed A-invariant if AS ⊆ S. For A-invariant subspace S ⊆ X , A : S denotes the restriction of A to S, and A : X /S denotes the map induced by A on the factor space X /S. For a linear system (C, A, B), < Ker C|A > denotes the unobservable subspace of (C, A). For a given subspace L, dim(L) denotes the dimension of L. The largest Ai invariant (i ∈ Ψ) subspace that is contained in K is denoted by << K|Ai >>i∈Ψ . 2. UNOBSERVABLE AND UNOBSERVABILITY SUBSPACES FOR MARKOVIAN JUMP SYSTEMS In this section, a geometric definition for the unobservable subspace of Markovian jump systems (MJS) is introduced. The notion of unobservability subspace is then formalized for the MJS systems that are governed by the dynamical system (1). To develop an algorithm for constructing such a subspace, an alternative definition of unobservability subspace is presented, which only depends on the matrices of the system. Based on this alternative definition, an algorithm for constructing the smallest unobservability subspace containing a given subspace is proposed. As shown in the next section, the unobservability subspace for MJS systems plays a central role in solving the fault diagnosis problem for Markovian jump systems. Consider the following Markovian jump system x(t) ˙ = Aλ(t) x(t), y(t) = Cλ(t) x(t)
(1)
with initial condition x(0) = x0 , λ(0) = i0 where x ∈ X is the continuous state of the system with dimension n; y ∈ Y is the output signal with dimensions q, and {λ(t), t ≥ 0} is a continuous-time irreducible Markov process taking values in the finite set Ψ = {1, ..., N }. The Markov process describes the switching between the different system modes and its evolution is governed by the following probability transitions: πij h + o(h) i = j P{λ(t + h) = j|λ(t) = i} = 1 + πii h + o(h), otherwise where πij is the transition rate from mode i to mode j with N πij ≥ 0 when i = j, πii = − j=1,j=i πij , and o(h) is a
function that satisfies limh→0 o(h) h = 0. The matrices Aλ(t) and Cλ(t) are known constant matrices for all λ(t) = i ∈ Ψ. For simplicity, we denote the matrices associated with λ(t) = i by Aλ(t) = Ai and Cλ(t) = Ci . We first start with the definition of weak observability for the Markovian jump system (1). Definition 1. (Costa and Val (2002)). The system (1) is said to be weakly (W-) observable when there exist td ≥ 0 and γ > 0 such that W td (x, i) ≥ γ|x|2 for each x ∈ X and i ∈ Ψ where t x (τ )Cλ(τ W t (x, i) = E ) Cλ(τ ) x(τ )dτ |x(0) = x, λ0 = i 0
In Costa and Val (2002), a collection of matrices O = {O1 , ..., ON } is introduced for testing the W-observability of a Markovian jump system according to the following procedure. Let Oi (0) = Ci Ci , i ∈ Ψ and define the sequence of matrices as Oi (k) = Ai Oi (k − 1) + Oi (k −
954
N 1)Ai + j=1 πij Oj (k − 1). The matrix Oi is now defined according to Oi = [Oi (0) Oi (1) · · · Oi (n2 N − 1)] . Theorem 2. (Costa and Val (2002)). The MJS system (1) is W-observable if and only if Oi has a full rank for each i ∈ Ψ. By considering the above definition of W-observability, one can define the set of unobservable states as follows. Definition 3. A state x is said to be unobservable if W t (x, i) = 0 for all t ≥ 0 and i ∈ Ψ. Let Q denotes the unobservable set for the MJS system (1), i.e Q = {x|W t (x, i) = 0, ∀i ∈ Ψ, t ≥ 0}. It is shown in Costa and Val (2002) that for irreducible Markov processes, N {Oi } = N {Oj }, i, j ∈ Ψ, i = j and Q = N {Oi }. Therefore, Q is the subspace of X and is called the unobservable subspace of the Markovian jump system (1). The theorem introduced below characterizes a geometric property of Q. Theorem 4. An unobservable subspace Q for system (1) with irreducible Markov process is the largest Ai -invariant N (i ∈ Ψ) that is contained in K = i=1 KerCi . Proof: It follows from the above discussion that Q ⊆ KerCi , i ∈ Ψ, and hence Q ⊆ K. Let x ∈ Q. Our goal is to show that Ai x ∈ Q for all i ∈ Ψ (i.e. Q is Ai -invariant, i ∈ Ψ). Since x ∈ N {Oi (k − 1)} and x ∈ N {Oi (k)}, i ∈ Ψ, then Oi (k)x = Ai Oi (k − 1)x + N Oi (k − 1)Ai x + j=1 πij Oj (k − 1)x = Oi (k − 1)Ai x = 0. Hence, Ai x ∈ N {Oi (k − 1)} and Q is Ai -invariant for all i ∈ Ψ. Next we show that Q is the largest Ai -invariant (i ∈ Ψ) that is contained in K. Let V be an Ai -invariant (i ∈ Ψ) subspace that is contained in K. Clearly, V ⊆ N {Oi (0)}, i ∈ Ψ. Let V ⊆ N {Oi (k − 1)}, i ∈ Ψ and x ∈ V, then Oi (k)x = Ai Oi (k − 1)x + Oi (k − 1)Ai x + N j=1 πij Oj (k − 1)x = 0 since Oi (k − 1)x = 0, i ∈ Ψ and Ai x ∈ V (V is Ai -invariant). Hence, V ⊆ N {Oi (k)}, i ∈ Ψ and V ⊆ Q. This shows that Q contains all the subspaces that are Ai -invariant (i ∈ Ψ) and is contained in K. Let us now present definitions for stability and detectability of MJS systems. Definition 5. (Costa and Val (2002)). The system x(t) ˙ = Aλ(t) x(t) is mean square (MS) stable if for each x0 ∈ X and i0 ∈ Ψ, limt→∞ E{||x(t)||2 } = 0. Definition 6. (Costa and Val (2002)). We say that system (1) is MS-detectable when there exists maps Gi , i ∈ Ψ of appropriate dimension for which x(t) ˙ = (Aλ(t) + Gλ(t) Cλ(t) )x(t) is MS-stable. The following computational linear matrix inequalities can be used for testing the MS-detectability of a MJS system (Costa and Val, 2002). In other words, the MSdetectability of system (1) is equivalent to the feasibility of the set N Ai Xi + Xi Ai + Ci Li + Li Ci + πij Xj < 0, i ∈ Ψ (2) j=1
in the unknowns Xi > 0 and Li with appropriate dimensions. We are now in a position to introduce the notions
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of conditioned invariant and unobservability subspaces for Markovian jump systems. Definition 7. A subspace W is said to be conditioned invariant for system (1) if Ai (W ∩ KerCi ) ⊆ W, i ∈ Ψ. It is clear that if W is conditioned invariant for system (1), then W is (Ci , Ai )-invariant for all i ∈ Ψ. Therefore, there exist maps Di such that (Ai + Di Ci )W ⊆ W, i ∈ Ψ. We denote the class of conditioned invariant subspaces for system (1) by W. If W ∈ W, we write D(W) for the class of maps Di where (Ai + Di Ci )W ⊆ W, i ∈ Ψ. The notion of conditioned invariant subspace for system (1) is a dual to that of the robust maximal controlled invariant which is introduced in Basile and Marro (1987). By duality it can be shown that W is closed under the operation of subspace intersection, and hence for any given subspace L ⊆ X , the family of conditioned invariant subspaces that contains L (denoted by W(L)) has an infimal element which is denoted by W ∗ = inf W(L). The following algorithm can now be used for constructing W ∗ : Algorithm 1. The subspace W ∗ coincides with and is obtained from the last term of the following sequence W0 = L;
Wk = L +
N
Ai (Wk−1 ∩ Ker Ci )
Definition 8. A subspace S is an unobservability subspace for system (1) if there exist output injection maps Di : Y → X and measurement mixing maps Hi : Y → Y such that S is the unobservable subspace for system x(t) ˙ = (Aλ(t) + Dλ(t) Cλ(t) )x(t), y(t) = Hλ(t) Cλ(t) x(t) N
j=1 (KerHj Cj )|Ai
We are now in a position to state our next result. Theorem 10. A subspace S is an unobservability subspace for system (1) if and only if there exist maps Di , such that N S =<< j=1 (S + KerCj )|Ai + Di Ci >>i∈Ψ . Proof: The proof follows readily from Lemma 9 by taking Sˆj = KerHj Cj .
i=0
i.e. S =<<
N ˆ ˆ Proof: We have S ⊆ j=1 Sj ⊆ Sj and KerCj ⊆ N ˆ N Sˆj , j ∈ Ψ, so that j=1 (S + KerCj ) ⊆ j=1 Sj . N (S + KerC )|A >> Consequently, << j i i∈Ψ ⊆<< j=1 N ˆ j=1 Sj |Ai >>i∈Ψ = S. On the other hand, Ai S ⊆ S N N and j=1 (S + KerCj ) ⊇ S + j=1 KerCj ⊇ S. Hence N S ⊆<< j=1 (S + KerCj )|Ai >>i∈Ψ , and as a result N << j=1 (S +KerCj )|Ai >>i∈Ψ = S. To show the converse part, let {ci1 , ..., cir } be a basis for S + KerCi such that {cir−pi , ..., cir } is the basis for KerCi (dim(KerCi ) = pi ). Therefore, yij = Ci cij , j = 1, ..., r − pi − 1 are independent. Let {yi1 , ..., yiq } be a basis for Y, and define Hi yij = 0, j = 1, ..., r − pi − 1, Hi yij = yij , j = r − pi , ..., r. Consequently, KerCi + S = KerHi Ci , and thereN N fore i=1 KerCi + S = i=1 KerHi Ci .
+ Di Ci >>i∈Ψ .
Remark 1. It should be noted that the notion of unobservability subspace introduced in Definition 8 is the most general concept of u.o.s. and the ones that have been introduced in the literature for linear Massoumnia et al. (1989), LPV Bokor and Balas (2004), bilinear Hammouri et al. (2001) and time-delay Meskin and Khorasani (2009) systems can be considered as a special case of our proposed unobservability subspace. Indeed, for Ci = C, i ∈ Φα , (identical output measurements) the above definition coincides with the unobservability subspace for the LPV and the bilinear systems as presented in Bokor and Balas (2004) and Hammouri et al. (2001), respectively. We denote the class of all unobservability subspaces for system (1) by S. In the following our goal is to derive an alternative characterization for the unobservability subspace which is independent of the maps Di and Hi as used in Definition 8. As shown subsequently, this alternative definition provides us with means to obtain the unobservability subspaces more readily. The following lemma presents a result that is necessary for formulating our alternative definition. Lemma 9. Let Sˆj ⊆ X such that KerCj ⊆ Sˆj , j ∈ Ψ N ˆ N and << j=1 Sj |Ai >>i∈Ψ = S, then << j=1 (S + N KerCj )|Ai >>i∈Ψ = S. Conversely, if << j=1 (S + KerCj )|Ai >>i∈Ψ = S, there exist maps Hj : Y → Y, j ∈ N Ψ such that << j=1 KerHj Cj |Ai >>i∈Ψ = S
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The above theorem eliminates the need for maps Hi from the Definition 8. For a given unobservability subspace S, the measurement mixing maps Hi ’s can be computed from S by solving KerHi Ci = S + KerCi , i ∈ Ψ. Next, we try to the characterize the unobservability subspace by means of an algorithm that computes S without explicitly constructing (D1 , ..., DN ) ∈ D(S). For an arbitrary subspace N S ⊆ X , let us define a family G = {S : S = i=1 (S + (A−1 i S ∩KerCi ))}. Below we first show that G has a unique maximal member. Lemma 11. There exists a unique element S ∗ ∈ G such that S ⊆ S ∗ for every S ∈ G. Proof: Define a sequence S μ ⊆ X according to S 0 = X , N μ−1 ∩ KerCi )). The sequence and S μ = i=1 (S + (A−1 i S μ S is non-increasing since S 1 ⊆ S 0 and if S μ ⊆ S μ−1 , N N μ thenS μ+1 = i=1 (S + (A−1 i S ∩ KerCi )) ⊆ i=1 (S + −1 μ−1 μ ∩ KerCi )) = S . Therefore, there exist k ≤ n (Ai S such that S μ = S k and we set S ∗ = S k . Clearly S ∗ ∈ G. Next, we show that S ∗ is the maximal element. Let S ⊆ G, then S ⊆ S 0 and if S ⊆ S μ , we have N N −1 μ S = i=1 (S + (A−1 i S ∩ KerCi )) ⊆ i=1 (S + (Ai S ∩ μ+1 μ KerCi )) = S . Consequently, S ⊆ S for all μ, and hence S ⊆ S ∗ . The next lemma provides an important property of the maximal element S ∗ which will be used for introducing our suggested alternative characterization of the unobservability subspace of system (1). Lemma 12. Let S ∈ W and (D1 , ..., DN ) ∈ D, then S ∗ is the largest (Ai + Di Ci )-invariant (i ∈ Ψ) that is contained N N in j=1 (S + ker Cj ), i.e. S ∗ =<< j=1 (S + KerCj )|Ai + Di Ci >>i∈Ψ . Proof: First we show that any S ∈ G is (Ai + Di Ci )N invariant (i ∈ Ψ). We have S = i=1 (S + (A−1 i S ∩
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KerCi )) and (Aj + Dj Cj )S = (Aj + Dj Cj )
N
(S + (A−1 i S ∩ KerCi ))
i=1 −1 ⊆ (Aj + Dj Cj )(S + Aj S ∩ KerCj ) ⊆ (Aj + Dj Cj )S + Aj (A−1 j S ∩ KerCj ) ⊆ S + S ⊆ S N where we used the relationship S = i=1 (S + (A−1 i S N −1 S ∩KerC ) ⊇ S and A (A KerCi )) ⊇ S+ i=1 (A−1 i j i j S)
∩ ⊆ S . Therefore, (Aj + Dj Cj )S ⊆ S , j ∈ Ψ; and hence S ∈ W and (D1 , ..., DN ) ∈ D(S ).
Consequently, we have S ∗ ∈ W. Next we show that for any subspace W such that it is (Ai + Di Ci )-invariant N (i ∈ Ψ) and is contained in j=1 (S + ker Cj ), we have W ⊆ S ∗ . If W ⊆ S then it follows that W ⊆ S ∗ , since S ⊆ S ∗ . Therefore, we consider the case where S ⊆ W. We have A−1 ∩ KerCi = (Ai + Di Ci )−1 W ∩ KerCi i W N N and as a result i=1 (S + A−1 i W ∩ KerCi ) = i=1 (S + (Ai + Di Ci )−1 W ∩ KerCi ). It is clear that W ⊆ S 0 . If W ⊆ S μ−1 , then N Sμ ⊇ (S + (Ai + Di Ci )−1 W ∩ KerCi )) i=1
⊇
N
(S + (W ∩ KerCi )) =
i=1
N
(W ∩ (S + KerCi )) = W
i=1
where we used the fact that W ⊆ (Ai +Di Ci )−1 W and the modular distributive rule (Wonham, 1985) (if S ⊆ W, then S + (W ∩ KerCi ) = W ∩ (S + KerCi )). As a consequence S μ ⊇ W; and hence S ∗ ⊇ W. This shows that S ∗ is the largest (Ai + Di Ci )-invariant (i ∈ Ψ) which is contained N in i=1 (S + KerCi ). We are in the position to introduce our proposed alternative characterization of an unobservability subspace for system (1). Theorem 13. Let S ⊆ X . Then S ∈ S if and only if a) S is conditioned invariant, and b) S = S ∗ where S ∗ is the maximal element of G. Proof: (If part) If a) and b) hold, then according to N Lemma 12, we have S =<< j=1 (S + KerCj )|Ai >>i∈Ψ , and hence using Theorem 10, we have S ∈ S. (Only if part) If S ∈ S, it follows that S ∈ W and according to Lemma 12, we have S = S ∗ . It can be shown that the class of unobservability subspaces of system (1) is close under the operation of subspace intersection and is nonempty. Hence the class of unobservability subspaces which contains a given subspace L ⊆ X (denoted by S(L)) has an infimal element S ∗ . This property is crucial for the application of unobservability subspace to the problem of fault diagnosis of MJS systems. The next algorithm provides a procedure for constructing S ∗ . Algorithm 2. Let W ∗ = inf W(L) and define the sequence Z μ according to N μ−1 Z0 = X ; Zμ = (W ∗ + (A−1 ∩ KerCi )) (3) i Z i=1
Then S ∗ = Z μ , whenever Z μ+1 = Z μ .
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To analyze the above algorithm note that the sequence Z μ is non-increasing and Z k+1 = Z k for k ≥ n−dim(W ∗ ). Let N Z ∗ = Z k . According to Lemma 12, Z ∗ =<< i=1 (W ∗ + KerCj )|Ai +Di Ci >>i∈Ψ for some (D1 , ..., DN ) ∈ D(W ∗ ). Using the same approach as in Lemma 9, one can obtain N the maps Hj ’s such that Z ∗ =<< i=1 (W ∗ +KerCj )|Ai + N Di Ci >>i∈Ψ =<< i=1 (KerHj Cj )|Ai + Di Ci >>i∈Ψ and therefore Z ∗ is an unobservability subspace according to Definition 8. Moreover, it follows that L ⊆ W ∗ ⊆ Z ∗ (Z ∗ ∈ W(L) and W = inf W(L)); hence Z ∗ ∈ S(L), and consequently S ∗ ⊆ Z ∗ . On the other hand, according to Theorem 13 we have N μ−1 S ∗ = S n where S 0 = X and S μ = i=1 (S ∗ + (A−1 ∩ i S KerCi )), μ ∈ n. Since W ∗ ⊆ S ∗ , it can be shown by induction that Z μ ⊆ S μ , μ ∈ n. Indeed, Z 0 = S 0 , and N μ−1 ∩ if Z μ−1 ⊆ S μ−1 , then Z μ = i=1 (W ∗ + (A−1 i Z N −1 μ−1 ∗ μ ∩ KerCi )) = S . ConKerCi )) ⊆ i=1 (S + (Ai S sequently, Z ∗ ⊆ S ∗ . It follows from the above algorithm and Lemma 12 that D(W ∗ ) ⊂ D(S ∗ ). Therefore, the maps Di ’s for S ∗ can be found from W ∗ and once S ∗ is found from Algorithm 2, the mapping Hi ’s can also be computed from S ∗ and equation KerHi Ci = S + KerCi . Finally, we introduce the notion of outer MS-detectability of an unobservability subspace. As shown in the next section, outer MS-detectability is a necessary condition for designing a stable residual generator for the fault diagnosis problem. Definition 14. A subspace S ∈ S is said to be outer MSdetectable if the factor system x(t) ˙ = A0λ(t) x(t), y(t) = Mλ(t) x(t) is MS-detectable where A0i = (Ai +Di Ci : X /S), Mi is the solution to Mi P = Hi Ci and P is the canonical projection of X on X /S. It is clear that the outer MS-detectability of a given unobservability subspace can be tested based on LMI (2). We are now in the position to formally introduce the fundamental problem in residual generation (FPRG) for the Markovian jump system (1). 3. FAULT DETECTION AND ISOLATION OF MARKOVIAN JUMP SYSTEMS In this section the Fundamental Problem in Residual Generation (FPRG) is investigated for the Markovian jump system (1). This problem was originally considered for linear systems in Massoumnia et al. (1989) and was extended to nonlinear systems in Persis and Isidori (2001). Similar problems are also considered for bilinear and LPV systems in Hammouri et al. (2001) and Bokor and Balas (2004), respectively. The objective in this section is to solve the FPRG for continuous-time Markovian jump systems (MJS) with irreducible Markov Process. Consider the following Markovian jump system x(t) ˙ = Aλ(t) x(t) + Bλ(t) u(t) + L1λ(t)) m1 (t) + L2λ(t) m2 (t) (4) y(t) = Cλ(t) x(t), x(0) = x0 , λ(0) = i0 where it is assumed that all the matrices are the same as in (1) and u is the input signal with dimension m. The
7th IFAC SAFEPROCESS (SAFEPROCESS’09) Barcelona, Spain, June 30 - July 3, 2009
matrices L1λ(t) , L2λ(t) represent the fault signatures and are monic and mi (t) ∈ Mi ⊂ X , i = 1, 2 denote the fault modes. For sake of simplicity in analysis and derivation of our results, we first consider the case with two faults. The more general case of multiple faults is considered at the end of this section. We denote the fault signatures associated with λ(t) = i by L1i and L2i . The fault modes together with the fault signatures can be used to model the effects of actuator faults, sensor faults and system faults on the dynamics of the system. For example, the effect of a fault in the i-th actuator may be represented by L1i as the i-th column of Bi and if an actuator fails, then m1 (t) = −ui (t). The FPRG problem is concerned with the design of a Markovian jump residual generator that is governed by the filter dynamics of the form w(t) ˙ = Fλ(t) w(t) − Eλ(t) y(t) + Kλ(t) u(t) r(t) = Mλ(t) w(t) − Hλ(t) y(t) (5) where w(t) ∈ F ⊂ X such that the response of r(t) is affected by the fault mode m1 (t) and is decoupled from m2 (t) and if m1 is identically zero then limt→∞ E||r(t)||2 = 0 for any input signal u(t). Define the extended space X e = X ⊕ F and U e = U ⊕ M2 , we can rewrite equations (4) and (5) as follows: e x˙ e (t) = Aeλ(t) xe (t) + Bλ(t) ue (t) + Le1 λ(t) m1 (t) e xe (t) r(t) = Hλ(t)
(6) 0 Fλ(t) ,
Aλ(t) where xe (t) ∈ X e and ue ∈ U e , Aeλ(t) = −Eλ(t) Cλ(t)
Bλ(t) L2λ(t)
L1 e e1 e λ(t) Bλ(t) = K , Lλ(t) = and Hλ(t) = 0 λ(t) 0
−Hλ(t) Cλ(t) Mλ(t) .
In order to investigate the criteria for determining whether a nonzero m1 (t) affects the residual signal r(t), the notion of an input observability for the Markovian jump system (6) is defined and formalized below. Definition 15. The input signal m1 (t) in system (4) is said to be input observable if Le1 i , i ∈ Ψ is monic and the image of Le1 ’s does not intersect with the unobservable subspace i of system (6). Based on the above definition, the FPRG problem can now be formally stated as the problem of designing the dynamical filter (5) such that (a) r is decoupled from ue , (b) m1 is input observable in the augmented system (6) and (c) limt→0 E{||r(t)||2 } = 0, for m1 (t) = 0, ∀ i0 ∈ Ψ and ∀xe0 ∈ X e . We need to first derive a preliminary result for obtaining the solvability condition for the FPRG problem. The following embedding map Q : X → X e is x defined according to Massoumnia et al. (1989): Qx = 0 x e −1 where if V ⊆ X , we have Q V = {x|x ∈ X , 0 ∈ V}. Our first result is the generalization of the Proposition 1 that was obtained in Massoumnia et al. (1989) to the Markovian jump system (4). Lemma 16. Let S e be the unobservable subspace of system (6). The unobservability subspace for system (4) is given by Q−1 S e . Proof: First we show that S = Q−1 S e is conditioned invariant. Let x ∈ S ∩ KerCi , we to show that Ai x ∈ S.
need This follows by noting that A0i x = −EAiiCi F0i x0 since
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S e is Aei -invariant i ∈ Ψ. Therefore, Ai x ∈ S and S is conditioned Next if x ∈ S, then Qx ∈ S e and there x invariant. N fore 0 ∈ i=1 KerHie . This shows that Hi Ci x = 0; and N hence x ∈ ∩N i=1 KerHi Ci and S ⊆ ∩i=1 KerHi Ci . Finally, according to the definition of the unobservable subspace S e (the largest Ae -invariant subspace in ∩KerHie ), S is the largest conditioned invariant contained in ∩N i=1 KerHi Ci , and therefore S ∈ S. We are now in the position to derive the solvability condition for the FPRG problem corresponding to the Markovian jump system (4). Theorem 17. The FPRG problem has a solution for the augmented MJS system (6) only if S ∗ L1j = 0, j ∈ Ψ (7) N where S ∗ = inf S( i=1 L2i ). On the other hand, if the above S ∗ exists such that it is also outer MS-detectable, then the FPRG problem is then guaranteed to have a solution. Proof: (Only if part) Let S e be an unobservable subspace of system (6). To satisfy the condition (a) of FPRG, we should have Bie ⊂ S e , i ∈ Ψ. Hence, L2i ⊂ Q−1 Bie ⊂ Q−1 S e = S and by invoking Lemma 16, we obtain N L2i ) S ∈ S(
(8)
i=1
For condition (b) to hold, according to the Definition 15, Le1 i should be monic (which is already assumed to hold) e and Le1 i ∩ S = 0, i ∈ Ψ. Thus e 1 Q−1 (Le1 i∈Ψ (9) i ∩ S ) = Li ∩ S = 0, Therefore, equations (8) and (9) hold only if equation (7) is true. (if part): Given the unobservability subspace S ∗ which is outer MS-detectable, there exist output injection maps Di ’s and measurement mixing maps Hi ’s such that N S ∗ =<< j=1 Ker Hj Cj |Ai + Di Ci >>i∈Ψ where Hj is the solution to equation KerHj Cj = S ∗ + KerCj . Let P be the canonical projection of X on X /S ∗ and Mi , i ∈ Ψ be a unique solution to Mi P = Hi Ci and A0i = (Ai + Di Ci : X /S ∗ ) where P (Ai + Di Ci ) = A0i P, i ∈ Ψ. Due to the fact that S ∗ is assumed to be outer MS-detectable, there exist Gi , i ∈ Ψ such that x(t) ˙ = (A0λ(t) + Gλ(t) Mλ(t) )x(t) is MS-stable. Let us define Fi = A0i + Gi Mi , Ei = P (Di + P −r Gi Hi ), Ki = P Bi , i ∈ Ψ, and e(t) = w(t) − P x(t). By using equation (5) we obtain e(t) ˙ = Fi w(t) − Ei y(t) + Ki u(t) − P (Ai x(k) + Bi u(k) + L1i m1 (t)) + L2i m2 (t)) = Fi w(t) − P L1i m1 (t) − P (Ai + Di Ci )x(t) − Gi Hi Ci x(t) = Fi w(t) − P L1i m1 (t) − A0i P x(t) − Gi Mi P x(t) = Fi e(t) − P L1i m1 (t)
Note that P L2i = 0, i ∈ Ψ, since L2i ∈ S ∗ , i ∈ Ψ. Also r(t) = Mi w(t) − Hi y(t) = Mi w(t) − Hi Ci x(t) = Mi e(t) Consequently, the error dynamics can be written according to e(t) ˙ = Fλ(t) e(t) − P L1λ(t) m1 (t) r(t) = Mλ(t) e(t)
(10)
7th IFAC SAFEPROCESS (SAFEPROCESS’09) Barcelona, Spain, June 30 - July 3, 2009
It is clear that the fault mode m2 (t) does not affect the residual signal r(t) and since the dynamics (10) is observable, condition (b) also holds. Moreover, for m1 (t) = 0 the system is MS-stable. To conclude this section, we now consider a Markovian jump system that has multiple faults and that is governed by the following dynamical system k x(t) ˙ = Aλ(t) x(t) + Bλ(t) u(t) + Ljλ(t) mj (t) j=1
x(0) = x0 , λ(0) = i0 (11) y(t) = Cλ(t) x(t) where all the matrices are the same as in system (4), Ljλ(k) , j ∈ k are the fault signatures, and mj (t) ∈ Mj , j ∈ k are the fault modes. We denote the fault signatures associated with λ(t) = i by Lji , i ∈ Ψ, j ∈ k. The Extended Fundamental Problem in Residual Generation (EFPRG) (Massoumnia et al., 1989) for the Markovian jump system (11) is now defined as the problem of generating k residual signals rl (t), l ∈ k from the Markovian jump detection filters (5) such that a fault in the j-th component mj can only affect the residual signal rj (t) and no other residual signals rl (t), l = j. It should be pointed out that a solution to the EFRPG problem amounts to indeed both detecting and isolating multiple and simultaneous faults in all the system components. The solvability condition for the EFPRG problem is obtained by invoking the solvability condition that was developed earlier for the FPRG problem as follows: Theorem 18. The EFPRG problem has a solution for system (11) only if Sj∗ Lji = 0, i ∈ Ψ, j ∈ k where N k Sj∗ = inf S( v=1 l=1,l=j Llv ). On the other hand, if the above Sj∗ , j ∈ k exist such that they are outer MSdetectable, the EFPRG problem is then guaranteed to have a solution. Proof: The proof is immediate by following along the lines that we developed for the proof of Theorem 17. . 4. CONCLUSIONS A geometric approach to the problem of fault diagnosis of continuous-time linear Markovian jump systems is developed in this paper. Starting with a new geometric characterization of the unobservable subspace of a Markovian jump system, the concept of unobservability subspaces is formalized and an algorithm for constructing these subspaces is presented. By invoking the notion of an unobservability subspace, the necessary and sufficient conditions for solving the fundamental problem of residual generation (FPRG) for Markovian jump systems is formally investigated. REFERENCES G. Basile and G. Marro. On the robust controlled invariant. Systems & Control Letters, 9(3):191–195, 1987. J. Bokor and G. Balas. Detection filter design for LPV systems: A geometric approach. Automatica, 40(3):511– 518, 2004. El-Kebir Boukas. Stochastic switching systems: analysis and design. Birkhauser, Boston, 2006.
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