In this paper, a method for formulating and solving the optimal sensor placement problem for an electro-pneumatic actuator is presented. The approach minimizes the number of additional sensors while maintaining maximum possible diagnosability and isolability. The proposed strategy is based on a Binary Diagnostic Matrix. Proposed isolability measure distinguish weak and strong isolability. It uses the branch-and-cut algorithm to find a solution.
The quality of diagnosis is often characterized using the fault isolability. Different methods of Fault Detection and Isolation (FDI) can be compared with it.
Available measurements strongly affect the performance of an FDI system for a given industrial process. Additional sensors providing additional information about a process can improve the performance of an FDI system. From a practical point of view, it is vital to achieving the best possible FDI system performance with minimal additional costs. The problem of optimal sensor selection can be understood as a combinatorial problem of selecting the optimal set of measurements.
In recent years, numerous papers discussed different
problems of the optimal sensor placement. The required
minimum fault isolability of the diagnostic system is
usually considered [1; 2] . Some of the proposed methods also
maximize designed fault isolability using heuristic methods,
e.g., genetic algorithms [
The model-based FDI considers faults as deviations from
nominal values of process parameters or as unknown
process inputs. If system model and measurements behave
differently, then faults are detected. In [
This paper presents the method of an optimal sensor placement for diagnostic purposes using linear constraints and a linear objective function. The method uses a new measure of isolability proposed by the author. The main contribution of this metric is that both weak and unidirectionally strong isolability properties are considered and distinguished. Various additional optimization constraints are analyzed. The model of an electro–pneumatic actuator was used as an example illustrating the procedure.
The paper is organized as follows. In Section 2, the preliminary definitions used in this work are given. Section 3 defines the measure of fault isolability. Section 4 presents the proposed optimization procedure. Section 5 describes the example of an electro–pneumatic actuator. Conclusions and final remarks section finalizes this paper. 2
A signal sensitive to faults is considered as a diagnostic
signal in FDI. A symptom is a value of a diagnostic signal
which indicates fault or faults. In the case of multi-valued
diagnostic signals, one fault type may generate different
values of each diagnostic signal. A fault signature is a vector
of diagnostic signal values associated with a particular fault
[
There are different definitions of fault isolability.
Generally, faults are considered isolable when at least some of
their signatures are different [
Binary Diagnostic Matrix (BDM) or Incidence Matrix is a form of notation of a relationship specified by the Cartesian product of diagnostic signals sets S = fsj : j = 1; 2; :::; J g f3 1 1 f4 and faults F = ffi : i = 1; 2; :::; ng. Each row displays sensitivity of a given diagnostic signal to each fault. Each column Vi = [v1;i; v2;i; ; vJ;i]T of binary diagnostic matrix V can be associated with a fault fi. Often column Vi is called signature of fault fi. An example of binary diagnostic matrix is shown in Table 1.
The basic definition of isolability can be formulated in the
context of BDM in the following way [
Faults fk; fm 2 F are weakly isolable if their signatures are different.
In the example from Table 1 all faults with exception of
a pair (f2; f3) are weakly isolable. A weak isolability in
some applications is not sufficient. It is possible that due to
a different sensitivity of diagnostic tests or process
dynamics some signals appear earlier and match a signature of a
different, weakly isolated fault. In the above example, the
appearance of only the signal s1 may be insufficient to
indicate the fault f1 reliably. Later signals s2 or s3 may appear
indicating faults f2, f3 or f5. Therefore a stronger
isolability property is required [
A structure is unidirectionally strongly isolating if it is weakly isolating and if no column in the structure matrix can be obtained from any other column by turning an arbitrary number of “1”s into “0”s or by turning an arbitrary number of “0”s into “1”s.
In a unidirectionally strongly isolating structure, each pair of faults differs in at least two entries. Firstly, where “1” is in the first column and “0” in the other one and secondly, where “0” is in the first column and “1” in the other one. A weak isolability is a necessary condition for a strong isolability.
In Table 1 faults f5 and f6 are unidirectionally strongly isolable.
From Definition 2 following statement can be extrapolated:
The signature Vi is excluding a fault fk if Vi is different than Vk and Vi cannot be obtained from Vk by turning “1”s into “0”s.
Opposite does not have to be true.
Definition 4. Faults fk; fm 2 F are weakly isolated iff each alternative fault signature (fk) excludes the fault fm, or each alternative fault signature (fm) excludes the fault fk.
If faults are mutually excluding each other, then they are strongly isolable.
Definition 5. Faults fk; fm 2 F are unidirectional strongly isolable iff each alternative fault signature (fk) excludes the fault fm, and each alternative fault signature (fm) excludes the fault fk.
In Table 1 signature V2 is excluding f1. Opposite is not true so they are not strongly isolable.
Commonly, in FDI an exoneration assumption is accepted. It states that a lack of symptoms exonerates a fault. It means that all symptoms must appear for a fault isolation. This assumption is not always valid. Due to dynamics of symptoms and different sensitivity to faults, they may not appear simultaneously or may even not appear at all. 3
An implementation of the measure of isolability for a Binary
Diagnostic Matrix was proposed in [
D: F
F ! f0; 1g; where: F is the set of faults and fk 2 F; k = 1 : : : K are particular faults. It is assumed that the value D (fk; fm) = 1 when the appearance of all symptoms of the fault fk excludes the fault fm. If this is not true, then D (fk; fm) = 0. 2. Calculate the value of the measure as: =
1 (K 1) K
K K X X D (fk; fm):
In the case of multi-valued diagnostic signals, the
conditional isolability metric was proposed in [
The formula (3) generalizes the first step of calculation of the proposed measure. It can be understood as a fraction of all alternative signatures of fk that excludes fm. In the case of binary diagnostic signals, there is always only one alternative signature (fk). The value of D (fk; fm) is then equal to 0 or 1. Consequently, in the case of binary diagnostic signals, the formula (3) is equivalent to formulation below the formula (1).
The proposed metric of isolability makes it possible to distinguish unidirectional strong isolability from weak isolability. If D (fk; fm) = 1 _ D (fm; fk) = 1, then the signature of the fault fk excludes the fault fm or the signature of the fault fm excludes the fault fk. Therefore, according to Definition 4, the faults are weakly isolable. Moreover, if D (fk; fm) = 1 ^ D (fm; fk) = 1, then the signature of the fault fk excludes the fault fm and the signature of the fault fm excludes the fault fk. Thus, faults fk and fm are unidirectionally strongly isolable as defined in Definition 5.
The value of the presented measure of isolability can be interpreted as a mean fraction of all diagnoses that can be (1) (2) excluded, after the occurrence of a single fault. The measure of isolability takes the maximal value when all pairs of faults are unidirectionally strongly isolable. In such a case, each single fault signature excludes (K 1) other faults (the fault does not exclude itself). Then PkK=1PKmm6==k1D (fk; fm) = (K 1) K and the value of the measure of isolability is equal to ((KK 11))KK
= 1. 4
In this section, only binary diagnostic signals are analyzed. If a fault fk is unisolable from fm, then D(fk; fm) is equal to 0. Otherwise, it is equal to 1. The value of D(fk; fm) can be calculated in the following way: xDk;m = D(fk; fm) = max xsj : vj;k 6= 0 ^ vj;m = 0 ; xsj (4) where: xsj is the decision variable, which indicates that jth diagnostic signal is available. This formula states that D(fk; fm) is equal to 1 if at least one diagnostic signal sj is sensitive to the fault fk and not sensitive to the fault fm. The shorthand notation xDk;m will be used instead of D(fk; fm) as a variable in the description of an optimal sensor placement problem.
Similarly, the variable xsj can be expressed as: 0 xsj min fxi : xi is necessary to calculate sj g ; xi (5) where: xi is the decision variable, which indicates that ith sensor is available. If even one of the sensors necessary for the diagnostic signal sj is unavailable, then this signal cannot be used. The inequality relation is used, because, even if all required sensors are available, the diagnostic signal may be not of interest, e.g., due to a too high cost of development of necessary models.
In Tab. 2 an example of a simple BDM is presented. There are three faults and three diagnostic signals. Each diagnostic signal requires two sensors to be available.
The objective function maxximize 16 PkK=1 PKmm6==k1 xDk;m with constraints (6) is a difficult, constrained, non-linear optimisation problem. 4.1
Generally, it is not possible to determine which faults will be detectable before solving the basic optimal sensor placement problem. In practice, detectability of the most important faults is often required. In a special case, this requirement can refer to all faults.
The detectability of a given fault can be interpreted as the possibility to distinguish this fault from the state without faults. To satisfy detectability requirements, an additional constraint can be added in the following way:
D(fk; faultless state) = max xsj : vj;k 6= 0 xsj = 1: (7) This ensures that there is at least one signal sensitive to fault fk.
If the problem with this additional constraint becomes infeasible, then it is impossible to meet the detectability requirements.
The detectability requirements for the problem introduced in Example 1 can be formulated in the following way:
For some critical subset of faults, it may be beneficial to require the solution of the optimal sensor placement problem to isolate these faults. These requirements can be fulfilled by introducing additional equality constraints. For example, if it is important that a fault fk is isolable from a fault fm, then the following constraint should be added:
If unidirectional strong isolability is desired, then two constraints should be added: xDk;m = 1: xDk;m = 1; xDm;k = 1: xD3;1 = 1; xD3;2 = 1: (8) (9) (10) (11)
If the isolability requirements cannot be satisfied, then the constrained problem will be infeasible.
For the diagnostic system introduced in Example 1, if it is required that the fault f3 is isolable from both f1 and f2, then the following constraints should be added: 5
To demonstrate an example of the optimal sensor placement
formulation, an electro–pneumatic valve actuator will be
discussed. Fig. 1 illustrates the actuator [
0; xi; ssj 2 f0; 1g; i; j = 1 : : : 3: The pairs of faults for which D(fk; fm) = 0 were omitted.
SP
CV
Air pressure sensor
PVP E/P
Ps X f4 f1 f2
CVI Electronic controller
f3 PVX
Pneumatic Positioner feedback servo-motor
Control valve Air supply system
Pz F f5 f6 of an electronic controller, an electro–pneumatic converter, a servo–motor, a control valve, and an electro–mechanical stem position feedback. The list of available measurements includes the control value CV, the control value of the electro–pneumatic transducer CVI, the stem displacement measurement X, and the pressure in the chamber of the servo–motor. Tab. 3 lists the considered faults. Tab. 4 Using Tab. 4, the maximum value of the metric of isolability can be calculated as: =
1 (K 1) K
XK k=1
XK 9 mm6==k1 D (fk; fm) = 30 = 0:3: (12)
To find the diagnostic structure with = 0:3 and the minimum number of required sensors the optimal sensor placement problem should be formulated as (13). minimize x s.t.
xCV + xCV I + xP s + xX 1 XK XK xDk;m = 0:3; 30 xCV ; xX ; xCV I ; xX ; xP s; xP s; xCV ; xP s; xCV I ; xs3 ; xs3 ; xs3 ; xs3 ;
(13) xs1 + xs2 ; xs1 + xs2 + xs3 ; xs1 + xs2 + xs3 ; xs1 + xs2 + xs3 ; xs1 + xs2 + xs3 ; k; m = 1 : : : 6; j = 1 : : : 5: xCV ; xCV I ; xX ; xP s; xsj ;xDk;m 2 f0; 1g;
To ensure that all faults are detectable the following constraints should be added:
xs1 + xs2 + xs4 + xs5
The problem (13) with (14) was solved using a Coin– or branch–and–cut (Cbc) solver and a PuLP modeler. The following solution was returned by the solver: xCV = 1:0; xCV I = 0:0; xP s = 1:0; xX = 1:0. Consequently, the optimal sensor set for given constraints is fCV; P s; Xg and the resulting BDM is presented in Tab. 5. All of the considered faults are detectable and the value of the isolability measure is = 0:3.
In this paper, the sensor placement problem was addressed for an electro–pneumatic actuator. A key contribution of this work is the introduction of a new measure of fault isolability as an objective function or constraint to Linear Programming problem. It distinguishes weak and unidirectionally strong isolability. A strategy of introducing new variables which allow obtaining BILP problem was presented. This strategy makes it possible to use efficient tools to find optimal sensors sets.
In this paper, the method was applied to a Binary Diagnostic Matrix, but the proposed measure of fault isolability can describe multi-valued systems such as Fault Information Systems (FIS).