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STRUCTURAL ANALYSIS FOR THE SENSOR LOCATION PROBLEM IN FAULT DETECTION AND ISOLATION
STRUCTURAL ANALYSIS FOR THE SENSOR LOCATION PROBLEM IN FAULT DETECTION AND ISOLATION Christian Commault ∗ Jean-Michel Dion ∗ Sameh Yacoub Agha ∗
∗
Laboratoire d’Automatique de Grenoble. LAG-CNRS, ENSIEG-INPG BP 46 38402 Saint Martin d’H`eres, France. Email: {Christian.Commault,JeanMichel.Dion,Sameh.Yacoub-Agha}@inpg.fr
Abstract: In this paper we consider linear systems with faults. We present a structural analysis for sensor location in the Fault Detection and Isolation problem. We deal with this problem when the system under consideration is structured, that is, the entries of the system matrices are either fixed zeros or free parameters. With such structured systems one can associate a graph. We define a new set of separators (Irreducible Input Separators) which allows us to get sets of system variables in which additional sensors must be implemented to solve the considered c 2006 IFAC. FDI Problem.Copyright ° Keywords: Linear systems, Structured systems, Fault detection, Sensor location.
1. INTRODUCTION
The paper is structured as follows. The problem is formulated in Section 2. Structured systems are presented in Section 3. In Section 4, we define Irreducible Input Separators and give their properties. Application to the sensor location problem is made in Section 5. Illustrative examples are given in Section 6.
The FDI problem has received considerable attention in the past ten years (Chen and Patton, 1999; Frank, 1996). When this FDI problem is not solvable with the existing sensors we look for a solution using additional sensors, see for example (Raghuraj et al., 1999)
2. PROBLEM FORMULATION We consider the FDI problem in the framework of structured systems (Lin, 1974; Dion et al., 2003; Commault et al., 2002). In this paper we define a new set of separators (Irreducible Input Separators) which allows us to get sets of system variables in which additional sensors must be implemented to solve the considered FDI Problem. This paper is an extension of previous results (Commault et al., 2005) in which only particular sets of separators were considered.
2.1 Observer-based FDI problem Let us consider the following linear time-invariant system : ½ x(t) ˙ = Ax(t) + Lf (t) Σ (1) y(t) = Cx(t) + M f (t) where x(t) ∈ Rn is the state vector, f (t) ∈ Rr the fault vector and y(t) ∈ Rp the measured output
896
z(t) ∈ Rq , where zi (t) is the measurement obtained from the i-th additional sensor. Define now the composite system denoted by Σc . ˙ = Ax(t) + Lf (t) x(t) Σc y(t) = Cx(t) + M f (t) (6) z(t) = Hx(t) + P f (t)
vector. A, C, L and M are matrices of appropriate dimensions. Note that the control input effects are not considered here since, for any observer-based FDI problem, it is well known that these can be taken into account in the observer structure without loss of generality. A dedicated residual set is designed using a bank of r observers for system (1), according to the dedicated observer scheme (Chen and Patton, 1999). Each residual will be designed to be sensitive to a single fault while remaining insensitive to the other faults. The ith observer of this bank of r observers is designed for a system of type (1) as follows: i x ˆ˙ (t) = Aˆ xi (t) + K i (y(t) − C x ˆi (t))
In the next sections we will consider the following sensor location problem for FDI: in which parts of the system should we implement additional sensors in such a way that the FDI problem is solvable on the composite system? If these additional sensors are necessary we look for an implementation which minimizes their number, and give results concerning the additional sensors location.
(2)
Our study will be achieved in the framework of structured systems that we introduce now.
where x ˆi (t) ∈ Rn is the state of the ith observer, i K is the observer gain to be designed such that x ˆi (t) asymptotically converges to x(t), when no fault is considered. The residuals are defined as : ri (t) = Qi (y(t) − C x ˆi (t)), for i = 1, . . . , r
3. LINEAR STRUCTURED SYSTEMS In this part we recall some definitions and results on linear structured systems. More details can be found in (Dion et al., 2003). We consider linear systems as described in (1), but with parameterized entries and denoted by ΣΛ ½ x(t) ˙ = Ax(t) + Lf (t) ΣΛ (7) y(t) = Cx(t) + M f (t)
(3)
where Qi is a 1 × p matrix. Definition 1. The bank of observer-based FDI problem consists in finding, if possible, matrices K i and Qi , such that, for i = 1, 2, . . . , r, A − K i C is stable, and the fault to residual transfer matrix is non zero, proper and diagonal, i.e. the transfer form the faults to the residuals has the form t11 (s) 0 · · · 0 0 t22 (s) · · · 0 r(s) = . (4) .. .. f (s) .. .. . . . 0
0
This system is called a linear structured systems if · ¸ A L the entries of the composite matrix J = C M are either fixed zeros or independent parameters (not related by algebraic equations). Λ = {λ1 , λ2 , . . . , λk } denotes the set of independent parameters of the composite matrix J. For the sake of simplicity the dependence of the system matrices on Λ will not be made explicit in the notation. A structured system represents a large class of parameter dependent linear systems. The structure is given by the location of the fixed zero entries of J. For such systems one can study generic properties i.e. properties which are true for almost all values of the parameters collected in Λ (Murota, 1987; Reinschke, 1988). More precisely a property is said to be generic (or structural) if it is true for all values of the parameters (i.e. any Λ ∈ Rk ) outside a proper algebraic variety of the parameter space. A directed graph G(ΣΛ ) = (Z, W ) can be easily associated with the structured · ¸ system ΣΛ of type A L (7) where the matrix is structured: C M • the vertex set is Z = F ∪ X ∪ Y where F , X and Y are the fault, state and output sets given by {f1 , f2 , . . . , fr }, {x1 , x2 , . . . , xn } and {y1 , y2 , . . . , yp } respectively,
· · · trr (s)
where tii (s) 6= 0 for i = 1, 2, . . . , r. The solvability conditions for this problem will be detailed further. These conditions express in particular that there must exist a sufficient number of measured outputs to allow the detection and isolation of the faults.
2.2 Sensor location for FDI Consider again the system (1). In general the above defined FDI problem has no solution using only the existing sensors on the system. In this case we consider new sensors which could be implemented on the system. We assume that these new sensors are fault free. Define the new output vector z which collects the new measurements: z(t) = Hx(t) + P f (t),
(5)
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Theorem 4. Consider the structurally observable system with r faults ΣΛ as defined in (7) and the associated graph G(ΣΛ ). The bank of observerbased diagonal FDI problem of Definition 1, is generically solvable if and only if:
• the arc set is W = {(fi , xj )|Lji 6= 0} ∪ {(xi , xj )|Aji 6= 0} ∪ {(xi , yj )|Cji 6= 0} ∪ {(fi , yj )| Mji 6= 0}, where Aji (resp. Cji ,Lji ,Mji ) denotes the entry (j, i) of the matrix A (resp. C,L,M ). Moreover, recall that a directed path in G(ΣΛ ) from a vertex iµ0 to a vertex iµq is a sequence of arcs (iµ0 , iµ1 ), (iµ1 , iµ2 ), . . . , (iµq−1 , iµq ) such that iµt ∈ Z for t = 0, 1, . . . , q and (iµt−1 , iµt ) ∈ W for t = 1, 2, . . . , q. Moreover, if iµ0 ∈ F and, iµq ∈ Y , P is called a fault-output path. A path which is such that iµ0 = iµq is called a circuit. If i0 ∈ V1 and, il ∈ V2 , where V1 and V2 are two subsets of Z, P is called a V1 -V2 path. Moreover, if the only vertices of P which belong to V1 ∪ V2 are i0 and il , P is called a direct V1 -V2 path. A set of paths with no common vertex is said to be vertex disjoint. A V1 -V2 linking of size k is a set of k vertex disjoint V1 -V2 paths. A linking is maximal when k is maximal. All the previous definitions can be extended to a composite structured system ΣcΛ with associated graph G(ΣcΛ ) where ΣcΛ is defined as ˙ = Ax(t) + Lf (t) x(t) ΣcΛ y(t) = Cx(t) + M f (t) (8) z(t) = Hx(t) + P f (t)
k=r
(9)
where k is the size of a maximal linking in G(ΣΛ ) Example 1 Consider the following structured system ΣΛ which is of type (7) with two faults and one output: · ¸ · ¸ 0 0 λ2 0 A= ,L = , λ1 0 0 λ3
£ ¤ C = 0 λ4 , M =
·
0 0 0 0
¸
The non zero entries of these matrices are the free parameters Λ = (λ1 , . . . , λ4 ). The associated graph G(ΣΛ ) is given in Figure 1. It is clear that
As a first example of these results, recall the graph characterization of the structural observability, which will be useful later (Lin, 1974; Murota, 1987). Proposition 2. Let ΣΛ be the linear structured system defined by (7) with its associated graph G(ΣΛ ). The system (in fact the pair (C, A)) is structurally observable if and only if:
f1
x1
f2
x2
y1
Fig. 1. Graph G(ΣΛ ) of Example 1 no solution for the F DI problem exists in our example because the condition (9) is not satisfied, k = 1 < r = 2. By measuring the state vertex x1 for example with a new additional sensor, we will have k = 2 = r and the F DI problem will become solvable.
• there exists a state-output path starting from any state vertex in X, • there exists a set of vertex disjoint circuits and state-output paths which cover all state vertices.
Consider now the system ΣΛ defined in (7) whose transfer matrix is TΛ (s) = C(sI − A)−1 L + M . We can calculate the generic rank of TΛ (s) by using the following result (van der Woude, 1991).
4. STRUCTURAL ANALYSIS VIA IRREDUCIBLE INPUT SEPARATORS Consider the graph G(ΣΛ ) = (Z, W ) of a structured system of type (7) with vertex set Z and edge set W .
Theorem 3. Let ΣΛ be the linear structured system defined by (7) with its associated graph G(ΣΛ ). The generic rank of TΛ (s) is equal to the size of a maximal fault-output linking in G(ΣΛ ).
Definition 5. (van der Woude, 2000) A separator S is a set of vertices such that any fault-output path has at least one vertex in S. The dimension of a separator is the number of elements in S.
Give now the result concerning the diagonal FDI problem by using a bank of observers, which was stated first in (Commault et al., 2002).
Definition 6. A separator of dimension d is said to be irreducible if it does not contain a separator of dimension d0 < d.
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Definition 7. An irreducible separator S is an irreducible input separator (IIS) of dimension d if for any irreducible separator S 0 of dimension d0 , such that any direct path from inputs to S contains a vertex in S 0 , we have d0 > d. This means that a separator is an IIS if there is no irreducible separator of lower or equal dimension between inputs and this separator.
f1
y1
xf 15
f2
x7
f3
x1
x3
x2
x4
y2 y3
x6
f4
Notice that these IIS include the set of separators defined in (Commault et al., 2005). This will be illustrated on the example. Among all the irreducible input separators, the separator which has the minimal dimension can be proved to be unique (van der Woude, 2000). It is called the minimal input separator and denoted S∗. S ∗ can be obtained using standard maximum flow algorithms as the Ford and Fulkerson algorithm (T.C.Hu, 1982). The dimension of S ∗ is equal to the maximal size of a fault-output linking,. S ∗ is indeed the first bottleneck between faults and outputs. S ∗ may contain fault, state and output vertices. We will give now an important property of the IIS.
Fig. 2. Graph G(ΣΛ ) of Example 2 - S31 ={f1 , f2 , x2 }, S32 ={f3 , f4 , x1 } are two irreducible input separators of dimension 3. In (Commault et al., 2005) we considered only some specific disjoint separators i.e {x1 , x2 }, {f1 , f2 , f3 , f4 }. We will show now that the set of IIS can be endowed with a lattice structure, which is a partially ordered set in which all nonempty finite subsets have both a supremum and an infimum (Blyth and Janowitz., 1972). Definition 9. Consider the structured system ΣΛ with its associated graph G(ΣΛ ). Consider an IIS S of G(ΣΛ ). Define Ts as the set of all vertices in any direct path from F to S in G(ΣΛ ) except for the vertices of S.
Theorem 8. Consider the structured system ΣΛ and its associated graph G(ΣΛ ). A separator S of dimension d is an IIS if and only if:
Consider the following partial order relation: S Â S 0 if Ts ⊂ Ts0 . We have the following:
• There exists a F -S linking of size d in G(ΣΛ ). • For any separator S 0 such that any direct path from F to S contains a vertex in S 0 , the maximal size of a F − S 0 linking in G(ΣΛ ) is of dimension d0 > d.
Proposition 10. Consider the structured system ΣΛ and its associated graph G(ΣΛ ). The set of IIS with the above defined partial order has a lattice structure. Furthermore
PROOF. We give only a sketchy proof. Assume that the F -S 0 linking is of size d0 ≤ d, then S 0 will be a separator of a lower dimension than S and located between F and S, contradicting the fact that S is an IIS.
• The infimum of this lattice is the minimal input separator S ∗ . • The supremum of this lattice is the set of fault vertices F . Remark 11. On any S ∗ -F path in the lattice oriented graph, the order is total.
Example 2 Consider the structured system ΣΛ whose associated graph is depicted in Figure 2
F From this graph we can remark that: - The set {x3 , y1 , y2 , y3 } is a separator of dimension 4. but it is not an irreducible separator because it contains a separator {y1 , y2 , y3 } of dimension 3. - {x3 , x4 },{x3 , x2 },{x1 , x4 },{x1 , x2 }, are all irreducible separators of dimension 2, and among them only {x1 , x2 } is an irreducible input separator of minimal dimension and it is the minimal input separator S ∗ of the system.
S32
S 13
S* Fig. 3. The lattice structure corresponding to the set of IIS of Example 2
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Corollary 13. The minimal number of additional sensors we have to add to solve our F DI problem is equal to r − k. where k is the dimension of S ∗ . These sensors must measure variables in Ts ∗ .
In Example 2 the set of four IIS has a lattice structure as illustrated in Figure 3 5. APPLICATION TO THE SENSOR LOCATION PROBLEM We will now use the properties of the irreducible input separators to tackle the sensor location problem.
6. ILLUSTRATIVE EXAMPLES Consider the structured system of Example 2 with its associated graph depicted in Figure 2. If we apply Theorem 12 to the IIS obtained in Example 2, we get:
Theorem 12. Consider the linear structurally observable system ΣΛ defined by (7) with its associated graph G(ΣΛ ). Consider an irreducible input separator S of dimension d of G(ΣΛ ) and its associated set Ts defined previously. Consider the composite system ΣcΛ with additional sensors z(t) defined in (8) and its associated graph G(ΣcΛ ). The F DI problem on ΣcΛ has a solution only if there are at least r − d additional sensors which measure vertices of Ts .
• {x1 , x2 } = S ∗ is an IIS of dimension 2 and Ts∗ is {f1 , f2 , f3 , f4 , x5 , x6 , x7 }. In any solution of the sensor location problem we must measure 2 variables in this Ts∗ . • {x1 , f3 , f4 } is an IIS of dimension 3 and Ts is {f1 , f2 , x5 , x7 }. In any solution of the sensor location problem we must measure one variable in this Ts . • {x2 , f1 , f2 } is an IIS of dimension 3 and Ts is {f3 , f4 , x6 }. In any solution of the sensor location problem we must measure one variable in this Ts . • {f1 , f2 , f3 , f4 } is an IIS of dimension 4 and the corresponding Ts is ∅.
PROOF. Assume that the problem has a solution on ΣcΛ and therefore that we have a size r linking between F and Y ∪Z. This linking is composed of: • F − Y paths. • F −Z paths whose last edge has initial vertex in Ts . • F − Z paths whose last edge has not initial vertex in Ts .
To solve this problem we need two additional sensors and all the solutions are: {f1 , f3 }, {f1 , f4 }, {f1 , x6 }, {f2 , f3 }, {f2 , f4 }, {f2 , x6 }, {x5 , f3 }, {x5 , f4 }, {x5 , x6 }, {x7 , f3 }, {x7 , f4 }, {x7 , x6 }. It is easy to check that these solutions satisfy the above given constraints.
The three types of paths are illustrated in Figure 4. Consider a F − Z path whose last edge has
Z
Example 3 Consider the structured system with 9 faults and 3 outputs whose graph is depicted in Figure 5. The same graph was studied in another context in (Murota, 1987).
xi
F
S
Y
Ts
The IIS are: • S ∗ = {y1 , y2 , y3 } has dimension 3 and Ts∗ correspondent is {f1 , f2 , f3 , f4 , f5 , f6 , f7 , f8 , f9 , x1 , x2 , x3 , x4 , x5 , x6 , x7 } and in any solution for the sensor location problem we must measure 6 variables in this set. • {f1 , f2 , x3 , x5 } is an IIS of dimension 4 and Ts correspondent is {f3 , f4 , f5 , f6 , f7 , f8 , f9 , x2 , x7 } and in any solution for the sensor location problem we must measure 5 variables in this set. • {f1 , f2 , f3 , f4 , f5 , x5 } is an IIS of dimension 6 with Ts correspondent {f6 , f7 , f8 , f9 , x7 } and in any solution for the sensor location problem we must measure 3 variables in this set.
Fig. 4. Possible types of paths in a F − Y ∪Z linking initial vertex xi out of Ts . There is a path from xi to Y since the system is observable. By definition of S and Ts any F − xi path has a vertex in S. Then in this linking the F −Y paths and the F −Z paths whose last edge has not initial vertex in Ts , have a vertex in S. Then there are at most d such paths. Therefore there are at least r − d paths F − Z whose last edge has initial vertex in Ts . So for any separator S and for any solution of the F DI problem, there are at least r − d new sensors measuring vertices in Ts .
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REFERENCES
• {f1 , f2 , x3 , f5 , f6 , x7 } is another IIS of dimension 6 with Ts correspondent {f3 , f4 , x2 , f7 , f8 , f9 } and in any solution for the sensor location problem we must measure 3 variables in this set. • {f1 , f2 , f3 , f4 , f5 , f6 , x7 } is an IIS of dimension 7 with Ts correspondent {f7 , f8 , f9 } and in any solution for the sensor location problem we must measure 2 variables in this set. • {f1 , f2 , f3 , f4 , f5 , f6 , f7 , f8 , f9 } is an IIS of dimension 9 and Ts correspondent is ∅.
Blyth, T.S. and M.F. Janowitz. (1972). Residuation Theory.. Pergamon Press, Oxford,. Chen, J. and R.J. Patton (1999). Robust modelbased fault diagnosis for dynamic systems. Kluwer Academic Publishers. Commault, C., J.M. Dion and S.Yacoub Agha (2005). A system decomposition for sensor location in fault detection and isolation. IFAC world congress. Prague. Commault, C., J.M. Dion, O. Sename and R. Motyeian (2002). Observer-based fault detection and isolation for structured systems. IEEE Trans. Automat. Control 47(12), 2074–2079. Dion, J.M., C. Commault and J. van der Woude (2003). Generic properties and control of linear structured systems: a survey. Automatica 39(7), 1125–1144. Frank, P.M. (1996). Analytical and qualitative model-based fault diagnosis - a survey and some new results. European Journal of Control 2, 6–28. Lin, C.T. (1974). Structural controllability. IEEE Trans. Automat. Control 19, 201–208. Murota, K. (1987). Systems Analysis by Graphs and Matroids. Vol. 3 of Algorithms and Combinatorics. Springer-Verlag New-York, Inc. Raghuraj, R., M. Bhushan and R. Rengaswamy (1999). Locating sensors in complex chemical plants based on fault diagnosis observability criteria. AIChE Journal 45(2), 310–322. Reinschke, K.J. (1988). Multivariable control : A graph-theoretic approach. Vol. 108 of Lect. Notes in Control and Information Sciences. Springer-Verlag. T.C.Hu (1982). Combinatorial Algorithms. Addison-Wesley publishing company. van der Woude, J.W. (1991). A graph theoretic characterization for the rank of the transfer matrix of a structured system. Math. Control Signals Systems 4, 33–40. van der Woude, J.W. (2000). The generic number of invariant zeros of a structured linear system. SIAM Journal. of Control 38, 1–21.
These IIS are illustrated in Figure 6, the solutions of the sensor location problem in F DI will satisfy Theorem 12 for all these IIS. f1
x1
f2 f3
yf1
x2
f4
x3
x4
y2
x5
x6
y3
f5 f6
f7 f8
x7
f9
Fig. 5. Graph G(ΣΛ ) of Example 4 f1
x1
f2 f3
yf1
x2
f4
x3
x4
y2
x5
x6
y3
f5 f6 f7
f8
x7
f9
Fig. 6. The IIS separators of G(ΣΛ )
7. CONCLUDING REMARKS In this paper we have presented a structural analysis which is well suited for the sensor location in the Fault Detection and Isolation problem (FDI). We have dealt with this problem when the system under consideration is structured. The analysis proposed is based on new separators called irreducible input separators and extends and refines previous works.
901
∗
Laboratoire d’Automatique de Grenoble. LAG-CNRS, ENSIEG-INPG BP 46 38402 Saint Martin d’H`eres, France. Email: {Christian.Commault,JeanMichel.Dion,Sameh.Yacoub-Agha}@inpg.fr
Abstract: In this paper we consider linear systems with faults. We present a structural analysis for sensor location in the Fault Detection and Isolation problem. We deal with this problem when the system under consideration is structured, that is, the entries of the system matrices are either fixed zeros or free parameters. With such structured systems one can associate a graph. We define a new set of separators (Irreducible Input Separators) which allows us to get sets of system variables in which additional sensors must be implemented to solve the considered c 2006 IFAC. FDI Problem.Copyright ° Keywords: Linear systems, Structured systems, Fault detection, Sensor location.
1. INTRODUCTION
The paper is structured as follows. The problem is formulated in Section 2. Structured systems are presented in Section 3. In Section 4, we define Irreducible Input Separators and give their properties. Application to the sensor location problem is made in Section 5. Illustrative examples are given in Section 6.
The FDI problem has received considerable attention in the past ten years (Chen and Patton, 1999; Frank, 1996). When this FDI problem is not solvable with the existing sensors we look for a solution using additional sensors, see for example (Raghuraj et al., 1999)
2. PROBLEM FORMULATION We consider the FDI problem in the framework of structured systems (Lin, 1974; Dion et al., 2003; Commault et al., 2002). In this paper we define a new set of separators (Irreducible Input Separators) which allows us to get sets of system variables in which additional sensors must be implemented to solve the considered FDI Problem. This paper is an extension of previous results (Commault et al., 2005) in which only particular sets of separators were considered.
2.1 Observer-based FDI problem Let us consider the following linear time-invariant system : ½ x(t) ˙ = Ax(t) + Lf (t) Σ (1) y(t) = Cx(t) + M f (t) where x(t) ∈ Rn is the state vector, f (t) ∈ Rr the fault vector and y(t) ∈ Rp the measured output
896
z(t) ∈ Rq , where zi (t) is the measurement obtained from the i-th additional sensor. Define now the composite system denoted by Σc . ˙ = Ax(t) + Lf (t) x(t) Σc y(t) = Cx(t) + M f (t) (6) z(t) = Hx(t) + P f (t)
vector. A, C, L and M are matrices of appropriate dimensions. Note that the control input effects are not considered here since, for any observer-based FDI problem, it is well known that these can be taken into account in the observer structure without loss of generality. A dedicated residual set is designed using a bank of r observers for system (1), according to the dedicated observer scheme (Chen and Patton, 1999). Each residual will be designed to be sensitive to a single fault while remaining insensitive to the other faults. The ith observer of this bank of r observers is designed for a system of type (1) as follows: i x ˆ˙ (t) = Aˆ xi (t) + K i (y(t) − C x ˆi (t))
In the next sections we will consider the following sensor location problem for FDI: in which parts of the system should we implement additional sensors in such a way that the FDI problem is solvable on the composite system? If these additional sensors are necessary we look for an implementation which minimizes their number, and give results concerning the additional sensors location.
(2)
Our study will be achieved in the framework of structured systems that we introduce now.
where x ˆi (t) ∈ Rn is the state of the ith observer, i K is the observer gain to be designed such that x ˆi (t) asymptotically converges to x(t), when no fault is considered. The residuals are defined as : ri (t) = Qi (y(t) − C x ˆi (t)), for i = 1, . . . , r
3. LINEAR STRUCTURED SYSTEMS In this part we recall some definitions and results on linear structured systems. More details can be found in (Dion et al., 2003). We consider linear systems as described in (1), but with parameterized entries and denoted by ΣΛ ½ x(t) ˙ = Ax(t) + Lf (t) ΣΛ (7) y(t) = Cx(t) + M f (t)
(3)
where Qi is a 1 × p matrix. Definition 1. The bank of observer-based FDI problem consists in finding, if possible, matrices K i and Qi , such that, for i = 1, 2, . . . , r, A − K i C is stable, and the fault to residual transfer matrix is non zero, proper and diagonal, i.e. the transfer form the faults to the residuals has the form t11 (s) 0 · · · 0 0 t22 (s) · · · 0 r(s) = . (4) .. .. f (s) .. .. . . . 0
0
This system is called a linear structured systems if · ¸ A L the entries of the composite matrix J = C M are either fixed zeros or independent parameters (not related by algebraic equations). Λ = {λ1 , λ2 , . . . , λk } denotes the set of independent parameters of the composite matrix J. For the sake of simplicity the dependence of the system matrices on Λ will not be made explicit in the notation. A structured system represents a large class of parameter dependent linear systems. The structure is given by the location of the fixed zero entries of J. For such systems one can study generic properties i.e. properties which are true for almost all values of the parameters collected in Λ (Murota, 1987; Reinschke, 1988). More precisely a property is said to be generic (or structural) if it is true for all values of the parameters (i.e. any Λ ∈ Rk ) outside a proper algebraic variety of the parameter space. A directed graph G(ΣΛ ) = (Z, W ) can be easily associated with the structured · ¸ system ΣΛ of type A L (7) where the matrix is structured: C M • the vertex set is Z = F ∪ X ∪ Y where F , X and Y are the fault, state and output sets given by {f1 , f2 , . . . , fr }, {x1 , x2 , . . . , xn } and {y1 , y2 , . . . , yp } respectively,
· · · trr (s)
where tii (s) 6= 0 for i = 1, 2, . . . , r. The solvability conditions for this problem will be detailed further. These conditions express in particular that there must exist a sufficient number of measured outputs to allow the detection and isolation of the faults.
2.2 Sensor location for FDI Consider again the system (1). In general the above defined FDI problem has no solution using only the existing sensors on the system. In this case we consider new sensors which could be implemented on the system. We assume that these new sensors are fault free. Define the new output vector z which collects the new measurements: z(t) = Hx(t) + P f (t),
(5)
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Theorem 4. Consider the structurally observable system with r faults ΣΛ as defined in (7) and the associated graph G(ΣΛ ). The bank of observerbased diagonal FDI problem of Definition 1, is generically solvable if and only if:
• the arc set is W = {(fi , xj )|Lji 6= 0} ∪ {(xi , xj )|Aji 6= 0} ∪ {(xi , yj )|Cji 6= 0} ∪ {(fi , yj )| Mji 6= 0}, where Aji (resp. Cji ,Lji ,Mji ) denotes the entry (j, i) of the matrix A (resp. C,L,M ). Moreover, recall that a directed path in G(ΣΛ ) from a vertex iµ0 to a vertex iµq is a sequence of arcs (iµ0 , iµ1 ), (iµ1 , iµ2 ), . . . , (iµq−1 , iµq ) such that iµt ∈ Z for t = 0, 1, . . . , q and (iµt−1 , iµt ) ∈ W for t = 1, 2, . . . , q. Moreover, if iµ0 ∈ F and, iµq ∈ Y , P is called a fault-output path. A path which is such that iµ0 = iµq is called a circuit. If i0 ∈ V1 and, il ∈ V2 , where V1 and V2 are two subsets of Z, P is called a V1 -V2 path. Moreover, if the only vertices of P which belong to V1 ∪ V2 are i0 and il , P is called a direct V1 -V2 path. A set of paths with no common vertex is said to be vertex disjoint. A V1 -V2 linking of size k is a set of k vertex disjoint V1 -V2 paths. A linking is maximal when k is maximal. All the previous definitions can be extended to a composite structured system ΣcΛ with associated graph G(ΣcΛ ) where ΣcΛ is defined as ˙ = Ax(t) + Lf (t) x(t) ΣcΛ y(t) = Cx(t) + M f (t) (8) z(t) = Hx(t) + P f (t)
k=r
(9)
where k is the size of a maximal linking in G(ΣΛ ) Example 1 Consider the following structured system ΣΛ which is of type (7) with two faults and one output: · ¸ · ¸ 0 0 λ2 0 A= ,L = , λ1 0 0 λ3
£ ¤ C = 0 λ4 , M =
·
0 0 0 0
¸
The non zero entries of these matrices are the free parameters Λ = (λ1 , . . . , λ4 ). The associated graph G(ΣΛ ) is given in Figure 1. It is clear that
As a first example of these results, recall the graph characterization of the structural observability, which will be useful later (Lin, 1974; Murota, 1987). Proposition 2. Let ΣΛ be the linear structured system defined by (7) with its associated graph G(ΣΛ ). The system (in fact the pair (C, A)) is structurally observable if and only if:
f1
x1
f2
x2
y1
Fig. 1. Graph G(ΣΛ ) of Example 1 no solution for the F DI problem exists in our example because the condition (9) is not satisfied, k = 1 < r = 2. By measuring the state vertex x1 for example with a new additional sensor, we will have k = 2 = r and the F DI problem will become solvable.
• there exists a state-output path starting from any state vertex in X, • there exists a set of vertex disjoint circuits and state-output paths which cover all state vertices.
Consider now the system ΣΛ defined in (7) whose transfer matrix is TΛ (s) = C(sI − A)−1 L + M . We can calculate the generic rank of TΛ (s) by using the following result (van der Woude, 1991).
4. STRUCTURAL ANALYSIS VIA IRREDUCIBLE INPUT SEPARATORS Consider the graph G(ΣΛ ) = (Z, W ) of a structured system of type (7) with vertex set Z and edge set W .
Theorem 3. Let ΣΛ be the linear structured system defined by (7) with its associated graph G(ΣΛ ). The generic rank of TΛ (s) is equal to the size of a maximal fault-output linking in G(ΣΛ ).
Definition 5. (van der Woude, 2000) A separator S is a set of vertices such that any fault-output path has at least one vertex in S. The dimension of a separator is the number of elements in S.
Give now the result concerning the diagonal FDI problem by using a bank of observers, which was stated first in (Commault et al., 2002).
Definition 6. A separator of dimension d is said to be irreducible if it does not contain a separator of dimension d0 < d.
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Definition 7. An irreducible separator S is an irreducible input separator (IIS) of dimension d if for any irreducible separator S 0 of dimension d0 , such that any direct path from inputs to S contains a vertex in S 0 , we have d0 > d. This means that a separator is an IIS if there is no irreducible separator of lower or equal dimension between inputs and this separator.
f1
y1
xf 15
f2
x7
f3
x1
x3
x2
x4
y2 y3
x6
f4
Notice that these IIS include the set of separators defined in (Commault et al., 2005). This will be illustrated on the example. Among all the irreducible input separators, the separator which has the minimal dimension can be proved to be unique (van der Woude, 2000). It is called the minimal input separator and denoted S∗. S ∗ can be obtained using standard maximum flow algorithms as the Ford and Fulkerson algorithm (T.C.Hu, 1982). The dimension of S ∗ is equal to the maximal size of a fault-output linking,. S ∗ is indeed the first bottleneck between faults and outputs. S ∗ may contain fault, state and output vertices. We will give now an important property of the IIS.
Fig. 2. Graph G(ΣΛ ) of Example 2 - S31 ={f1 , f2 , x2 }, S32 ={f3 , f4 , x1 } are two irreducible input separators of dimension 3. In (Commault et al., 2005) we considered only some specific disjoint separators i.e {x1 , x2 }, {f1 , f2 , f3 , f4 }. We will show now that the set of IIS can be endowed with a lattice structure, which is a partially ordered set in which all nonempty finite subsets have both a supremum and an infimum (Blyth and Janowitz., 1972). Definition 9. Consider the structured system ΣΛ with its associated graph G(ΣΛ ). Consider an IIS S of G(ΣΛ ). Define Ts as the set of all vertices in any direct path from F to S in G(ΣΛ ) except for the vertices of S.
Theorem 8. Consider the structured system ΣΛ and its associated graph G(ΣΛ ). A separator S of dimension d is an IIS if and only if:
Consider the following partial order relation: S Â S 0 if Ts ⊂ Ts0 . We have the following:
• There exists a F -S linking of size d in G(ΣΛ ). • For any separator S 0 such that any direct path from F to S contains a vertex in S 0 , the maximal size of a F − S 0 linking in G(ΣΛ ) is of dimension d0 > d.
Proposition 10. Consider the structured system ΣΛ and its associated graph G(ΣΛ ). The set of IIS with the above defined partial order has a lattice structure. Furthermore
PROOF. We give only a sketchy proof. Assume that the F -S 0 linking is of size d0 ≤ d, then S 0 will be a separator of a lower dimension than S and located between F and S, contradicting the fact that S is an IIS.
• The infimum of this lattice is the minimal input separator S ∗ . • The supremum of this lattice is the set of fault vertices F . Remark 11. On any S ∗ -F path in the lattice oriented graph, the order is total.
Example 2 Consider the structured system ΣΛ whose associated graph is depicted in Figure 2
F From this graph we can remark that: - The set {x3 , y1 , y2 , y3 } is a separator of dimension 4. but it is not an irreducible separator because it contains a separator {y1 , y2 , y3 } of dimension 3. - {x3 , x4 },{x3 , x2 },{x1 , x4 },{x1 , x2 }, are all irreducible separators of dimension 2, and among them only {x1 , x2 } is an irreducible input separator of minimal dimension and it is the minimal input separator S ∗ of the system.
S32
S 13
S* Fig. 3. The lattice structure corresponding to the set of IIS of Example 2
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Corollary 13. The minimal number of additional sensors we have to add to solve our F DI problem is equal to r − k. where k is the dimension of S ∗ . These sensors must measure variables in Ts ∗ .
In Example 2 the set of four IIS has a lattice structure as illustrated in Figure 3 5. APPLICATION TO THE SENSOR LOCATION PROBLEM We will now use the properties of the irreducible input separators to tackle the sensor location problem.
6. ILLUSTRATIVE EXAMPLES Consider the structured system of Example 2 with its associated graph depicted in Figure 2. If we apply Theorem 12 to the IIS obtained in Example 2, we get:
Theorem 12. Consider the linear structurally observable system ΣΛ defined by (7) with its associated graph G(ΣΛ ). Consider an irreducible input separator S of dimension d of G(ΣΛ ) and its associated set Ts defined previously. Consider the composite system ΣcΛ with additional sensors z(t) defined in (8) and its associated graph G(ΣcΛ ). The F DI problem on ΣcΛ has a solution only if there are at least r − d additional sensors which measure vertices of Ts .
• {x1 , x2 } = S ∗ is an IIS of dimension 2 and Ts∗ is {f1 , f2 , f3 , f4 , x5 , x6 , x7 }. In any solution of the sensor location problem we must measure 2 variables in this Ts∗ . • {x1 , f3 , f4 } is an IIS of dimension 3 and Ts is {f1 , f2 , x5 , x7 }. In any solution of the sensor location problem we must measure one variable in this Ts . • {x2 , f1 , f2 } is an IIS of dimension 3 and Ts is {f3 , f4 , x6 }. In any solution of the sensor location problem we must measure one variable in this Ts . • {f1 , f2 , f3 , f4 } is an IIS of dimension 4 and the corresponding Ts is ∅.
PROOF. Assume that the problem has a solution on ΣcΛ and therefore that we have a size r linking between F and Y ∪Z. This linking is composed of: • F − Y paths. • F −Z paths whose last edge has initial vertex in Ts . • F − Z paths whose last edge has not initial vertex in Ts .
To solve this problem we need two additional sensors and all the solutions are: {f1 , f3 }, {f1 , f4 }, {f1 , x6 }, {f2 , f3 }, {f2 , f4 }, {f2 , x6 }, {x5 , f3 }, {x5 , f4 }, {x5 , x6 }, {x7 , f3 }, {x7 , f4 }, {x7 , x6 }. It is easy to check that these solutions satisfy the above given constraints.
The three types of paths are illustrated in Figure 4. Consider a F − Z path whose last edge has
Z
Example 3 Consider the structured system with 9 faults and 3 outputs whose graph is depicted in Figure 5. The same graph was studied in another context in (Murota, 1987).
xi
F
S
Y
Ts
The IIS are: • S ∗ = {y1 , y2 , y3 } has dimension 3 and Ts∗ correspondent is {f1 , f2 , f3 , f4 , f5 , f6 , f7 , f8 , f9 , x1 , x2 , x3 , x4 , x5 , x6 , x7 } and in any solution for the sensor location problem we must measure 6 variables in this set. • {f1 , f2 , x3 , x5 } is an IIS of dimension 4 and Ts correspondent is {f3 , f4 , f5 , f6 , f7 , f8 , f9 , x2 , x7 } and in any solution for the sensor location problem we must measure 5 variables in this set. • {f1 , f2 , f3 , f4 , f5 , x5 } is an IIS of dimension 6 with Ts correspondent {f6 , f7 , f8 , f9 , x7 } and in any solution for the sensor location problem we must measure 3 variables in this set.
Fig. 4. Possible types of paths in a F − Y ∪Z linking initial vertex xi out of Ts . There is a path from xi to Y since the system is observable. By definition of S and Ts any F − xi path has a vertex in S. Then in this linking the F −Y paths and the F −Z paths whose last edge has not initial vertex in Ts , have a vertex in S. Then there are at most d such paths. Therefore there are at least r − d paths F − Z whose last edge has initial vertex in Ts . So for any separator S and for any solution of the F DI problem, there are at least r − d new sensors measuring vertices in Ts .
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REFERENCES
• {f1 , f2 , x3 , f5 , f6 , x7 } is another IIS of dimension 6 with Ts correspondent {f3 , f4 , x2 , f7 , f8 , f9 } and in any solution for the sensor location problem we must measure 3 variables in this set. • {f1 , f2 , f3 , f4 , f5 , f6 , x7 } is an IIS of dimension 7 with Ts correspondent {f7 , f8 , f9 } and in any solution for the sensor location problem we must measure 2 variables in this set. • {f1 , f2 , f3 , f4 , f5 , f6 , f7 , f8 , f9 } is an IIS of dimension 9 and Ts correspondent is ∅.
Blyth, T.S. and M.F. Janowitz. (1972). Residuation Theory.. Pergamon Press, Oxford,. Chen, J. and R.J. Patton (1999). Robust modelbased fault diagnosis for dynamic systems. Kluwer Academic Publishers. Commault, C., J.M. Dion and S.Yacoub Agha (2005). A system decomposition for sensor location in fault detection and isolation. IFAC world congress. Prague. Commault, C., J.M. Dion, O. Sename and R. Motyeian (2002). Observer-based fault detection and isolation for structured systems. IEEE Trans. Automat. Control 47(12), 2074–2079. Dion, J.M., C. Commault and J. van der Woude (2003). Generic properties and control of linear structured systems: a survey. Automatica 39(7), 1125–1144. Frank, P.M. (1996). Analytical and qualitative model-based fault diagnosis - a survey and some new results. European Journal of Control 2, 6–28. Lin, C.T. (1974). Structural controllability. IEEE Trans. Automat. Control 19, 201–208. Murota, K. (1987). Systems Analysis by Graphs and Matroids. Vol. 3 of Algorithms and Combinatorics. Springer-Verlag New-York, Inc. Raghuraj, R., M. Bhushan and R. Rengaswamy (1999). Locating sensors in complex chemical plants based on fault diagnosis observability criteria. AIChE Journal 45(2), 310–322. Reinschke, K.J. (1988). Multivariable control : A graph-theoretic approach. Vol. 108 of Lect. Notes in Control and Information Sciences. Springer-Verlag. T.C.Hu (1982). Combinatorial Algorithms. Addison-Wesley publishing company. van der Woude, J.W. (1991). A graph theoretic characterization for the rank of the transfer matrix of a structured system. Math. Control Signals Systems 4, 33–40. van der Woude, J.W. (2000). The generic number of invariant zeros of a structured linear system. SIAM Journal. of Control 38, 1–21.
These IIS are illustrated in Figure 6, the solutions of the sensor location problem in F DI will satisfy Theorem 12 for all these IIS. f1
x1
f2 f3
yf1
x2
f4
x3
x4
y2
x5
x6
y3
f5 f6
f7 f8
x7
f9
Fig. 5. Graph G(ΣΛ ) of Example 4 f1
x1
f2 f3
yf1
x2
f4
x3
x4
y2
x5
x6
y3
f5 f6 f7
f8
x7
f9
Fig. 6. The IIS separators of G(ΣΛ )
7. CONCLUDING REMARKS In this paper we have presented a structural analysis which is well suited for the sensor location in the Fault Detection and Isolation problem (FDI). We have dealt with this problem when the system under consideration is structured. The analysis proposed is based on new separators called irreducible input separators and extends and refines previous works.
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