# Actuator Fault Diagnosis with Robustness to Sensor Distortion

Authors: Qinghua Zhang
Journal: Journal of Control Science and Engineering
 (2008)

---

## Abstract

Actuator fault diagnosis is often studied under strong assumptions on available sensors.
Typically, it is assumed that the sensors are either fault free or sufficiently redundant. The
purpose of this paper is to present a new method foractuator fault diagnosis which is robust
to sensor distortion. It does not require sensor redundancy to compensate sensor distortion.
The essential assumption is that sensor distortions are strictly monotonous. Despite the
nonlinear and unknown nature of distortions, such sensors still provide useful information
for fault diagnosis. The robustness of the presented diagnosis method is analyzed, as well
as its ability to detect actuator faults. A numerical example is provided to illustrate its efficiency.


---

## Body

## 1. Introduction

In the design of modern control systems, fault
diagnosis is often considered for component faults, sensor faults, and actuator
faults. Various methods for fault diagnosis are generally based on the
processing of sensor signals [1–3]. Many methods for actuator fault diagnosis
assume reliable fault-free sensors, see, for instance, [4, 5]. For methods simultaneously dealing with actuator and
sensor faults, it is typically assumed that there is sufficient redundancy
among the sensors such that at any moment, the valid sensors can provide the
information required for faults diagnosis. For mass production, it is important
to use as less sensors as possible to reduce production cost. In such
situations, fault diagnosis cannot uniquely rely on redundant sensors.
Moreover, still for the purpose of cost reduction, sensors used in mass
production may be highly nonlinear. If the sensor nonlinearity is well known,
it can often be electronically compensated. Unfortunately, sometimes sensor
nonlinearity varies within the production, and for each piece in use, the
nonlinearity varies during its normal life duration. For example, most oxygen
sensors used in cars equipped with catalytic converters are highly nonlinear. They
roughly indicate if the oxygen concentration is over or below a reference
value. Nevertheless, such sensors are sufficient for the purpose of engine
control. It is then important to develop fault diagnosis methods relying on the
same sensors. In this paper, unknown nonlinear behaviors of sensors are
generally called sensor distortion.The purpose of this paper is to present a method for
actuator fault diagnosis which is robust to sensor distortion. It is assumed
that each sensor can be affected by anunknown and arbitrary, but strictly monotonous, nonlinearity. The monotonousness is a weak
assumption since any non monotonous distortion would make the sensor
information useless. Remark that saturation is not a strictly monotonous
nonlinearity, and thus is excluded in the proposed method. Nevertheless, a
correctly working sensor should not be saturated when it is in its normal
working range.Robust methods for fault diagnosis has been studied in
different contexts by many authors. In [6], a method based on adaptive wavelet analysis is
proposed. The method studied in [7] uses generalized frequency response functions.
Kullback discrimination information is used as a fault detection index in [8]. In [9], some faults of induction motors are detected by
Fourier analysis of frequency signatures. (See the references cited in these
publications for more information.) Compared to these existing results, the
novelty of the method presented in this paper is its ability to deal with
unknown nonlinear sensor distortions without requiring sensor redundancy.The preliminary results presented in [10] are further developed in this paper, in particular,
the analysis of the proposed robust detection method is completed with the
characterization of the faults which can be detected.The paper is organized as follows. The problem
considered in this paper is formulated in Section2. Fault detection and
isolation are, respectively, studied in Sections 3 and 4. A numerical example is presented in Section 5. Some concluding remarks are given in Section 6.


## 2. Problem Statement

The considered linear state-space system with sensor distortion is formulated as(1a)x˙(t)=Ax(t)+Bdiag(u(t))θ+w(t),(1b)z(t)=Cx(t)+v(t),(1c)y(t)=h(z(t)),
where x(t)∈ℝn is the state, u(t)∈ℝl the input, z(t)∈ℝm the output before sensor distortion,
y(t)∈ℝm the output after sensor distortion, and w(t)∈ℝn and v(t)∈ℝm represent bounded
modeling uncertainties. The notation diag(u(t)) denotes the diagonal matrix formed by the
components of the input vector u(t), and the vector θ∈ℝl is a coefficient vector introduced to
describe the efficiency loss of actuators (multiplicative actuator faults). For fault-free actuators,
θ takes the nominal value θ0.For notation simplicity, the parenthesis(t) of the time-dependent variables will not be
written unless necessary.Sensor distortion is modeled by the component-wisely defined nonlinear functionh:
let zi
and yi be, respectively, the
ith component of z and y,
i=1,…,m, then
(2)yi=hi(zi).Assumption 1.
Each sensor distortionhi:ℝ→ℝ is an unknown, but a strictly monotonously
increasing, function. In other words, for any ξ1, ξ2∈ℝ,
(3)ξ1<ξ2⟹hi(ξ1)<hi(ξ2).Assumption 2.
The system dynamics matrixA is asymptotically stable, that is, the eigenvalues of
A have negative real parts.Remark 1.
No additive uncertainty is assumed in the sensor distortion equation (1c) since the arbitrary unknown monotonous nonlinear function can take into account some sensor distortion
uncertainty. On the other hand, if it is assumed that
(4)y=h(z)+ε,
where ε is some additive uncertainty, define(5)v=h−1[h(Cx+v)+ε]−Cx,z=Cx+v,then (4) can be replaced by
(6)y=h(z)which is in the form of (1c). Of course, in order to limit v, some regularity of h should be assumed.With the above formulation, the problem considered in this paper is the detection and
isolation of multiplicative actuator faults, modeled as changes in the coefficient vectorθ, despite
the unknown sensor distortions.


## 3. Fault Detection

The main difficulty of the problem formulated in the previous section is caused by theunknown
sensor distortions. The key question is how to use the information provided by such sensors.
Because of the unknown nature of hi, for each measured value of yi=hi(zi), the corresponding
value of zi is completely unknown, and even the sign of zi is unknown. The strict monotonousness
of hi assumed in Assumption 1 is not helpful in this aspect. However, it is important to make
the following observation. For any two time instants t and τ, Assumption 1 implies that
(7)sign[zi(t)−zi(τ)]=sign[yi(t)−yi(τ)].In other words, the relative sign of zi at different time instants is known from the sensor output
yi measured at these time instants. Remark that t and τ are two arbitrary and independent time
instants, and either one can be earlier than the other one. Since the absolute value |zi(t)−zi(τ)| is completely unknown, the relative sign is thus the only information about zi(t)−zi(τ) provided
by the sensor output. This information will be the basis for the design of fault detection and
isolation algorithms in this paper. Such information can also be used for the identification of
Wiener systems [11].LetZ(s), U(s), W(s), and V(s) be, respectively, the Laplace transforms of z(t), u(t), w(t),
and v(t). It is then derived from (1a), (1b), and (1c) (by assuming zero initial state) that
(8)Z(s)=C(sI−A)−1Bdiag(U(s))θ+C(sI−A)−1W(s)+V(s).
Define
(9)Φ(t)=L−1[C(sI−A)−1Bdiag(U(s))],(10)ζ(t)=L−1[C(sI−A)−1W(s)+V(s)],
where 𝔏−1 is the inverse Laplace transform operator. Then
(11a)z(t)=Φ(t)θ+ζ(t),(11b)y(t)=h(z(t)).Notice thatΦ(t)∈ℝm×l depends on u(t) and can be computed through (9), whereas ζ(t)∈ℝm depends on modeling uncertainties w(t), v(t) and is unknown.Remark 2.
Zero initial condition of the statex has been assumed when (8) was derived. The
asymptotic stability condition (Assumption 2) is not explicitly used in the above reasoning, but
it ensures well-behaved computation of  Φ through (9). To some extent, this stability condition
allows also to tolerate nonzero initial states which are asymptotically forgotten. If the system
was not asymptotically stable, then, in principle, an observer should be used in the computation
of Φ. However, it is difficult to design observers with sensors distorted by unknown nonlinear
functions.Letϕi be the ith row of Φ and ζi be the ith component of ζ, then the ith row of
(11a) writes(12)zi(t)=ϕi(t)θ+ζi(t).
For any two time instants t and τ, it is obvious that
(13)sign[zi(t)−zi(τ)][zi(t)−zi(τ)]≥0.
Substitute zi(t) and zi(τ) with the last equality, then, for constant θ,
(14)sign[zi(t)−zi(τ)][(ϕi(t)−ϕi(τ))θ+ζi(t)−ζi(τ)]≥0.
Remind the relative sign equality (7), then
(15)sign[yi(t)−yi(τ)][(ϕi(t)−ϕi(τ))θ+ζi(t)−ζi(τ)]≥0,
or equivalently(16)−sign[yi(t)−yi(τ)][ϕi(t)−ϕi(τ)]θ≤sign[yi(t)−yi(τ)][ζi(t)−ζi(τ)].Letθ0 be the nominal value of θ (corresponding to fault-free actuators), a residual ri(t,τ)
for actuator fault detection can be generated as
(17)ri(t,τ)=−sign[yi(t)−yi(τ)][ϕi(t)−ϕi(τ)]θ0.
It then follows from (16) that, in the fault-free case,
(18)ri(t,τ)≤sign[yi(t)−yi(τ)][ζi(t)−ζi(τ)].
This result leads to the following proposition.Proposition 1.
If|ζi(t)|≤λ for some constant λ and any t, then the residual ri(t,τ) defined
in (17) satisfies, in the fault-free case, that is, θ=θ0, the inequality
(19)ri(t,τ)≤2λ.Proof.
Let us first consider the caseyi(t)≠yi(τ), then it follows from the inequality (18) that
(20)ri(t,τ)≤|ri(t,τ)|≤|ζi(t)−ζi(τ)|≤|ζi(t)|+|ζi(τ)|≤2λ.
Now for the case yi(t)=yi(τ), it follows trivially from the inequality (18) that
(21)ri(t,τ)≤0≤2λ.Remind thatζi(t) is a component of the variable ζ(t) depending on the bounded modeling
uncertainties w(t) and v(t), as defined in (10). The boundedness of w(t) and v(t), together
with Assumption 2, implies the boundedness of ζ(t). For practical convenience, the residual threshold should be directly derived from the assumed bounds of w(t) and v(t). It would require
nontrivial error bound propagation through (10). Techniques of interval analysis or
set-membership computation [12] can be applied for this purpose.
This topic is not further discussed in this paper.Proposition1 guarantees the absence of false detection if fault detection is made by comparing the residual ri(t,τ) with the threshold 2λ. However, it does not tell what are the faults which
can be detected with such a decision rule. In general, robust fault detection methods are based
on conservative decision rules, preventing the detection of some faults. In publications about
robust fault detection, typically robustness results are provided, but the analysis about the set
of faults which can be detected is usually absent, because it is often difficult to characterize the
detectable faults in a robust detection framework. In contrast, for the method proposed in this
paper, the faults which can be detected are clearly characterized as follows.Proposition 2.
Assume that the system matrix pair(A,B) is controllable, θ≠0, and θ0≠0.
For the faulty actuator parameter vector θ≠θ0, if there does not exist any positive real number
α such that θ=αθ0, then there exist an input signal u(t) and time instants t and τ such that
(22)ri(t,τ)>2λ
for each i=1,…,m, where λ is a positive constant such that |ζi(t)|≤λ.Proof.
This proof applies to the residualri(t,τ) for each i=1,…,m.
Define the vector(23)ν=β(‖θ‖−1θ−‖θ0‖−1θ0),
where the norm ∥θ∥=θTθ and β is a positive number to be specified later. Then
(24)νTθ=β(‖θ‖−‖θ0‖−1θ0Tθ)=β‖θ‖(1−‖θ0‖−1‖θ‖−1θ0Tθ)≥0,
where the last inequality follows from the fact that |θ0Tθ|≤∥θ0∥∥θ∥.
Notice that the equality(25)|θ0Tθ|=‖θ0‖‖θ‖
would imply θ=αθ0 for some α>0, that is, the case excluded in Proposition 2. Therefore,
(26)νTθ=β(‖θ‖−‖θ0‖−1θ0Tθ)>0.
Similarly, it is also shown that(27)νTθ0=β(‖θ‖−‖θ0‖−1θ0Tθ)<0.
It then follows that
(28)sign[νTθ]νTθ0<0.
For given values of θ and θ0, the value of β in (23) is chosen large enough such that
(29)|νTθ|>2λ,|νTθ0|>2λ.
Now, let us considerϕi, the ith row of the matrix Φ defined in (9). If there is an input signal u(t) such that ϕi(t)−ϕi(τ)=νT for two time instants t and τ, then the inequality (28) leads to
(30)sign[(ϕi(t)−ϕi(τ))θ](ϕi(t)−ϕi(τ))θ0<0.
Notice that the existence of such input signals is ensured by the controllability of the matrix
pair(A,B).
Because|(ϕi(t)−ϕi(τ))θ|=|νTθ|>2λ≥|ζi(t)−ζi(τ)|, the sign of (ϕi(t)−ϕi(τ))θ+(ζi(t)−ζi(τ)) is determined by the sign of (ϕi(t)−ϕi(τ))θ, thus
(31)sign[(ϕi(t)−ϕi(τ))θ+(ζi(t)−ζi(τ))]=sign[(ϕi(t)−ϕi(τ))θ].
This last equality, together with the inequality (30), leads to
(32)sign[(ϕi(t)−ϕi(τ))θ+(ζi(t)−ζi(τ))](ϕi(t)−ϕi(τ))θ0<0,
or equivalently
(33)sign[zi(t)−zi(τ)](ϕi(t)−ϕi(τ))θ0<0,sign[yi(t)−yi(τ)](ϕi(t)−ϕi(τ))θ0<0.
Remind that |[ϕi(t)−ϕi(τ)]θ0|=|νTθ0|>2λ, then
(34)sign[yi(t)−yi(τ)][ϕi(t)−ϕi(τ)]θ0<−2λ.
Therefore, the residual as defined in (17) satisfies
(35)ri(t,τ)>2λ.Ift is the current time instant, then ri(t,τ) can be computed for different past time instant τ. In order to reduce the effect of modeling uncertainties, ri(t,τ) can be averaged over different
values of τ in a sliding window. The computation of the residul from sampled signals is
summarized as follows.Residual Generation Algorithm Summary
Assume thatΦ(t) and y(t) are sampledFor notation simplicity, the sampling period is assumed to be 1 here. at discrete time instants 1,2,…,N. Choose the
sliding window length 0<L<N for residual averaging. For each sensor number i=1,2,…,m,
a residual is computed, for k≥L+1, through the formulas
(36)ri(k,s)=−sign[yi(k)−yi(s)][ϕi(k)−ϕi(s)]θ0,r¯i(k)=1L∑s=k−Lk−1max⁡(ri(k,s),0).
Notice that the max function is used to exclude negative values of ri(k,s) in the computation
of the average value.


## 4. Fault Isolation

After the detection of an actuator fault, the purpose of fault isolation is to figure out which
actuators are faulty. In terms of (11a) and (11b), it amounts to deciding which components of θ
have deviated from the nominal value θ0.It should be first remarked that, because of the arbitrary unknown functionh, it is not
possible to detect or isolate any proportional changes in all the components of θ. In other
words, for any value α∈ℝ, the parameter vector θ1=αθ0 cannot be distinguished from θ0
based on the known signals Φ(t) and y(t). For the same reason, if all the components of θ,
except one, have changed, it is not possible to determine which one has not changed.After having clarified the limitation related to unknown sensor distortion, let us look for an
algorithm for fault isolation. The basic idea is to design residuals similar to (17), but capable
of rejecting some actuator faults. Each designed residual should be insensitive to some of the
possible actuator faults, whereas sensitive to the others. The actually occurred fault can then
be isolated by comparing such residuals.LetP be a permutation matrix, that is, a matrix obtained by permuting the rows of the l×l
identity matrix Il (remind that l is the number of actuators). Divide P into two sub-matrices
Pf and Ps, respectively, composed of lf and ls=l−lf rows of P. Then it can be easily verified
that (37)PfTPf+PsTPs=Il.The notationsPfθ and Psθ will be used to select, respectively, the assumed faulty components and sound (or fault-free) components of θ. It is derived from (11a) that
(38)z(t)=Φ(t)PfTPfθ+Φ(t)PsTPsθ+ζ(t).
If Ps is assumed to select the components of θ corresponding to sound actuators, then, even
after the occurrence of actuator faults, the equality
(39)Psθ=Psθ0
still holds. Define
(40)θf≜Pfθ, then
(41)z(t)=Φ(t)PfTθf+Φ(t)PsTPsθ0+ζ(t).For the purpose of fault isolation, different partitions ofP into (Pf,Ps) should be considered.
For each particular partition, a residual will be designed to be insensitive to changes in the
corresponding subvector θf=Pfθ. Such a residual is said to be rejecting changes in θf. The rejection method used in the following is the estimation of θf from measure signals.For two time instantst, τ, Assumption 1 implies (by rewriting inequality (15)) that(42)sign[yi(t)−yi(τ)][(ϕi(t)−ϕi(τ))PfTθf+(ϕi(t)−ϕi(τ))×PsTPsθ0+ζi(t)−ζi(τ)]≥0.
For a given set of signals Φ(t), y(t) sampled at discrete time instants 1,2,…,N, the value of
θf can be estimated by minimizing the error term ζi(t)−ζi(τ) in the inequality (42) where the
time instants t, τ are replaced by all pairs among the sampling instants 1,2,…,N.Becauseyi(t)<yi(s) and yi(s)<yi(τ) imply yi(t)<yi(τ), there would be too much
redundancy if all the possible pairs among 1,2,…,N were considered in (42). In order to
reduce redundancy, for each sensor yi, the data samples are sorted according to the values
of yi(t), and only the neighboring pairs are considered. The algorithm is summarized as
follows.Residual Generation Algorithm Summary
For each chosen partition ofP into (Pf,Ps),
the residuals rejecting changes in θf=Pfθ are computed as follows. For each sensor number
i=1,2,…,m, sort the data such that
(43)yi(k1i)≤yi(k2i)≤⋯≤yi(kNi).
Solve the constrained optimization problem
(44)minθf⁡max1≤i≤m,1≤j≤N−1⁡ζi(j),
subject to the constraints, for i=1,2,…,m and j=1,2,…,N−1,
(45)[ϕi(kji)−ϕi(kj+1i)]PfTθf+[ϕi(kji)−ϕi(kj+1i)]PsTPsθ0+ζi(j)≤0.
The corresponding sequences
(46)ζi(1),ζi(2),…,ζi(N−1),i=1,2,…,m,
are the residuals rejecting changes in θf.Remark that the inequality (45) corresponds to the inequality (42) where the omitted sign[yi(kji)−yi(kj+1i)] is always negative due to the sorted sequence, and accordingly, the inequality changes from “≥” to “≤.”The constrained minimization (44)-(45) can be easily reformulated in the form of a standard
linear programing problem. There are efficient numerical algorithms for its solution
[13].Proposition 3.
If the true parameter vectorθ governing (11a) satisfies Psθ=Psθ0, and if
|ζi(t)|≤λ for some constant λ and any t, then the residual ζi(j), solution of the constrained
optimization problem (44)-(45), satisfies
(47)ζi(j)≤2λ.Proof.
The proof of this result is quite straightforward. Let us first derive from (11a), (11b), and (37)
that (48)[ϕi(kji)−ϕi(kj+1i)]PfTPfθ+[ϕi(kji)−ϕi(kj+1i)]PsTPsθ+ζi(kji)−ζi(kj+1i)=zi(kji)−zi(kj+1i)≤0,
where the indices kji, kj+1i come from the sequence sorting yi(k) as in (43).
It is assumed thatPsθ=Psθ0. Then the above inequality shows that there exist a value of
θf (equal to Pfθ) and a value of ζi(j) (equal to ζi(kji)−ζi(kj+1i)) such that the inequality (45)
is satisfied. These values of θf and ζi(j) do not necessarily correspond to the solution of the
constrained optimization problem (44)-(45), however, the optimal solution certainly has a value
of ζi(j) not larger than ζi(kji)−ζi(kj+1i). Therefore,
(49)ζi(j)≤ζi(kji)−ζi(kj+1i)≤2λ.Property (47) has been proved under the assumption Psθ=Psθ0. It can be shown that (47) holds also if Psθ=αPsθ0 for any α∈ℝ because of the unknown sensor distortion function.For fault isolation, various matricesPs are assumed and the corresponding residuals are
computed. Then property (47) can be used to decide if each assumed Ps is correct or not.
Alternatively, for different matrices Ps of the same size, the values of the residuals can also be
compared for fault isolation.Keep in mind that fault isolation cannot distinguish the cases such thatPsθ=αPsθ0. For
reliable fault isolation, the following assumption is required.Assumption 3.
There exists a permutation matrixP such that, for Ps composed of some rows of P,
(50)Psθ=Psθ0,
and there does not exist any Ps≠Ps (except for Ps composed of a subset of permutated rows of
Ps) such that
(51)Psθ=αPsθ0,
for some scalar real value α.


## 5. Numerical Example

In this section, the presented fault diagnosis method will be illustrated with a simulated distillation
column.In [14, page 223], a distillation column is modeled with a transfer function
matrix with three control inputs: top draw flow rate, side draw flow rate, and bottom temperature
control, and three outputs: top draw composition, side draw composition, and bottom reflux temperature.
In order to illustrate the possibility of fault diagnosis with only one output sensor, let us
consider the model relating the three inputs and one of the outputs, the bottom reflux temperature.
The transfer function model is
(52)Z(s)=4.38e−20s33s+1U1(s)+4.42e−22s44s+1U2(s)+7.2019s+1U3(s).
Note that the known time delays at the inputs do not cause any serious difficulty for fault
diagnosis, though the on-line computation has to be delayed accordingly.Numerical simulation is first made in continuous time. The three input variables are randomly
drawn with uniform distributions ranged within the intervals [20, 40] (mol/min), [10, 30] (mol/min) and [10, 20] (°C). The simulated noise-free output is disturbed by an additive bandlimited
noise with noise power 0.01 and unitary sample time. Then the sensor distortion function
(53)h(z)=1001+e−0.2(z−320)+10
is applied. In this monotonously increasing function, the additive constant 10 has been added
to illustrate the robustness of the proposed method to sensor bias (shifting error). Finally, the
distorted sensor output y(t) sampled at the period of one minute is corrupted by a random noise
uniformly distributed within the interval [−0.05, 0.05] (°C). The simulated data is recorded
during 2000 minutes after an initial simulation of 1000 minutes to avoid the initial transient
period. At the beginning, all the actuators are fault free. At the 1000th minute (of the recorded
duration), a factor of 0.8 is applied to the first actuator, simulating an actuator fault.The residual for fault detection, computed with the averaging window lengthL=100, is
illustrated in Figure 1. In Figure 1(a) showing the residual in full scale, there is a strong
transient behavior of the residual after occurrence of the actuator fault (at the 1000th minute).
This transient behavior is in favor of a fast detection of the actuator fault. In order to view
better the residual outside the transient period, it is plotted in Figure 1(b) in a finer scale.
The detected fault is clearly confirmed by the residual after the transient period.Fault detection residual. The same residual is plotted twice at different scales. The time
unit is the minute.
(a)(b)For fault isolation, three residuals are computed with the signals from the 1801th minute to
the 2000th minute. Each of the three residuals is designed to reject a fault affecting one of the three
actuators. The residuals rejecting the faults of actuator 1, 2, and 3 are, respectively, plotted in
the top, middle, and bottom pictures of Figure2. The first residual is clearly smaller than the
two others, indicating that the hypothesis of a fault affecting the first actuator is the most likely
one, which corresponds to the actually simulated fault.Fault isolation residuals. The time unit is the minute. Top: residual rejecting the fault of
actuator 1. Middle: residual rejecting the fault of actuator 2. Bottom: residual rejecting the fault
of actuator 3.
(a)(b)(c)


## 6. Conclusion

Despite unknown nonlinear distortions of sensors, the information provided by such sensors
is still useful for fault diagnosis, even when there is no redundant sensors, if the distortions
are strictly monotonous. The monotonousness is a weak assumption since nonmonotonous
distortion would make the sensor information useless. The main idea of the method presented
in this paper is about how to use the information provided by such sensors. Because of the
unknown nature of nonlinear distortion, neither the absolute value of the measured physical
variable nor its sign can be determined from the sensor signal. The strict monotonousness of the nonlinear distortion is not helpful in this aspect. However, for any two different time
instants, the relative sign of the measured variable is preserved by the monotonous nonlinear
distortion. By using the information residing in the relative sign of sensor signals, the method
for actuator fault diagnosis presented in this paper is conceptually robust to sensor distortions,
as illustrated by the numerical example presented in this paper.



---

*Source: 723292-2008-01-01.xml*