# Actuator Fault Diagnosis with Application to a Diesel Engine Testbed

Authors: Boulaïd Boulkroune; Abdel Aitouche; Vincent Cocquempot; Li Cheng; Zhijun Peng
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

This work addresses the issues of actuator fault detection and isolation for diesel engines. We are particularly interested in faults affecting the exhaust gas recirculation (EGR) and the variable geometry turbocharger (VGT) actuator valves. A bank of observer-based residuals is designed using a nonlinear mean value model of diesel engines. Each residual on the proposed scheme is based on a nonlinear unknown input observer and designed to be insensitive to only one fault. By using this scheme, each actuator fault can be easily isolated since only one residual goes to zero while the others do not. A decision algorithm based on multi-CUSUM is used. The performances of the proposed approach are shown through a real application to
a Caterpillar 3126b engine.


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## Body

## 1. Introduction

On-board diagnosis of automotive engines has become increasingly important because of environmentally based legislative regulations such as OBDII (On-Board Diagnostics-II) [1]. On-board diagnosis is also needed to guarantee high-performance engine behavior. Today, due to the legislation, the majority of the code in modern engine management systems is dedicated to diagnosis.Model-based diagnosis of automotive engines has been considered in earlier papers (see, e.g., [2, 3]), to name only a few. However, the engines investigated in these previous works were all gasoline-fuelled and did not include exhaust gas recirculation (EGR) and variable geometry turbocharger (VGT). Both of these components make the diagnosis problem significantly more difficult since the air flows through the EGR valve, and also the exhaust side of the engine has to be taken into account. An interesting approach to model-based air-path faults detection for an engine which includes EGR and VGT can be found in [4, 5]. By using several models in parallel, where each one is sensitive to one kind of fault, predicted outputs are compared and a diagnosis is provided. The hypothesis test methodology proposed in [4] deals with the multifault detection in air-path system. In [5] the authors propose an extended adaptive Kalman filter to find which faulty model best matches with measured data; then a structured hypothesis allows going back to the faults. A structural analysis for air path of an automotive diesel engine was developed in order to study the monitorability of the system [6–8]. Other approaches to detect intake leakages in diesel engines based on adaptive observers are proposed in [9, 10] and recently in [11]. Note that, in all these approaches, the leakage size is assumed to be constant. To overcome this limitation, an approach based on a nonlinear unknown input observer (NIUO), for intake leakage detection, is proposed in [12]. Noa priori assumption about the leakage size is made. In [13, 14], an interesting method for bias compensation in model-based estimation using model augmentation is proposed. The extended Kalman filter (EKF) is used for estimating the states of the augmented model. Also, the observability of the augmented model is well discussed. Recently, an automated model-based and data-driven design methodology for automotive engine fault detection and isolation (FDI) is proposed in [15]. This methodology, which combines model-based sequential residual generation and data-driven statistical residual evaluation, is used to create a complete FDI system for an automotive diesel engine.The problem of designing unknown input observers (UIO) has received great attention in literature. This problem is motivated by certain applications such as fault diagnosis and control system design. If this issue in linear case is well solved, it remains an open problem in the nonlinear case. The first unknown input observers dedicated to linear systems were proposed in [16, 17]. Necessary and sufficient conditions of the existence of the UIO have been well established. New sufficient conditions formulated in terms of linear matrix inequality (LMI) were given by [18]. This result is extended in [12] to cover a wide class of nonlinear systems that cannot be treated by the previous approach as well as nonlinear systems with a large Lipschitz constant.Our aim is to address the issue of actuator fault detection and isolation for diesel engines. Indeed, actuators (EGR and VGT) fault diagnosis is necessary and crucial to guarantee its healthy operation. In this work, a fault detection and isolation (FDI) system is developed. The proposed FDI system is composed of two parts: residual generation and decision system. A multiobserver strategy is used for residual generation. A mean model of diesel engine is exploited for the design of a set of nonlinear unknown input observers. These observers are designed in order to estimate the states behavior without any knowledge of the unknown inputs. The sufficient conditions of the existence of the NUIO are given in terms of linear matrix inequalities (LMIs). The advantage of this method is that no a priori assumption on the unknown input is required and also can be employed for a wider class of nonlinear systems. To achieve fault detection and isolation, a decision system based on a statistical approach, multi-CUSUM (cumulative sum), is used to process the resulting set of residuals.This paper is organized as follows. The experimental setup is described in Section2. The diesel engine model and its validation are described in Section 3. A nonlinear unknown input observer is presented in Section 4. Section 5 describes the residual generation system while the decision system is presented in Section 6. The experimental results and discussion are presented in Section 7. Finally, conclusion and future works are given in the last section.The notations used in this paper are quite standard. LetR denote the set of real numbers. The set of p by q real matrices is denoted as Rp×q. AT and A-1 represent the transpose of A and its left inverse (assuming A has full column rank), respectively. Ir represents the identity matrix of dimension r. (*) is used for the blocks induced by symmetry. · represents the usual Euclidean norm. L2r denotes the Lebesgue space. fa,i denotes the ith component of the vector fa.


## 2. Experimental Installation

The testbed is built with a Caterpillar 3126b truck engine coupled to a SCHORCH dynamometer controlled by CP Cadet Software. The Caterpillar engine is presented in Figure1. The front and back view of the testbed are shown by the pictures numbered (1) and (2), respectively. Engine’s components are inlet manifold (3), encoder for measuring of the engine speed (4), intake air flow meter (5), exhaust manifold (6), GT3782VA variable geometry turbocharger, (7) and exhaust gas valve (8). In order to enable a transient control of the EGR and VGT, a dSPACE MicroAutoBox 1401/1501 real-time controller is connected (see Figure 2). Apart from the standard OEM electronic sensors built in for ECU and dynamometer control, additional sensors and actuators have been wired in specifically for the MicroAutoBox. These sensors and actuators are listed in Table 1.Table 1
Sensors and actuators list.
Sensors
Actuators
Inlet temperature sensor
EGR valve drive actuator
Inlet air flow meter
VGT vanes drive actuator
Inlet pressure sensor
Pre-turbo exhaust pressure sensor
Acceleration pedal position sensor
Engine speed sensor
Inlet manifold oxygen sensor
Exhaust manifold oxygen sensor
Exhaust opacity sensor (AVL Opacimeter 439)
Exhaust emission sensor (Testo 350 Engine test kit)
EGR position feedback sensor
VGT position feedback sensorFigure 1
Caterpillar testbed 3126, Sussex University.Figure 2
Experimental installation schematic, Sussex University.The data and control signal flow are illustrated in Figure2. The engine is connected with two control platforms, which are CP Cadet and dSPACE Control Desk. The engine tests are conducted and monitored by CP Cadet Platform while the EGR and VGT valve positions can be adjusted through dSPACE Control Desk in real time. Testing data can be collected from both platforms and used for data analysis purpose.The specification of the Caterpillar 3126b midrange truck engine is given in Table3.


## 3. Engine Model and Validation

The considered diesel engine is a six-cylinder engine with a high-pressure EGR and VGT. A principle illustration scheme of the air-path system is shown in Figure3. It consists of two parts: the turbocharger and exhaust gas recirculation. The turbocharger is a turbine driven by the exhaust gas and connected via a common shaft to the compressor, which compresses the air in the intake. The exhaust gas recirculation (EGR) allows for recirculating gas from the exhaust manifold to the intake manifold. First, the mixture of air coming from the compressor and exhaust gas coming through the EGR valve enters the intake manifold before injecting it into the cylinders. Then, the fuel is injected directly in the cylinders and burned, producing the torque on the crank shaft. Exhaust gases are expelled into the exhaust manifold. As shown in Figure 3, part of exhaust gas comes out from the exhaust manifold through the turbine and the other part is recirculated through EGR valve. We noted that the temperature of gases (compressed air and EGR gas) entering the intake manifold is reduced using the intercooler and the EGR cooler.Figure 3
Turbocharged air-intake system schematic.The mean value engine modeling approach is one of the most considered approaches in the literature [19]. It uses temporal and spatial averages of relevant temperatures, pressures, and mass flow rates. The engine model is derived based on the laws of conservation of mass and energy and also on the ideal gas law (see, for instance, [5, 20]). As an example, the pressure dynamics in the inlet manifold is obtained by differentiating the ideal gas law equation PV=mRT. The complete considered mean value model is expressed as follows [5]:(1)P˙Inlet=1VInletRAircp,Aircv,AirWHFMTCAC+RExhcp,Exhcv,ExhWEGRTEGR-RInletcp,Inletcv,InletWInletTInlet,m˙Air=WHFM-mAirmAir+mEGRWInlet,m˙EGR=WEGR-mEGRmAir+mEGRWInlet,m˙Exh=WExh-WTurb-WEGRwith(2)Ψp1p0=2κκ-1p1p02/κ-p1p0(κ+1)/κifp1p0≥2κ+1κ/(κ-1)κ2κ+1(κ+1)/(κ-1)otherwise,WEGR=AEGRPExhRExhTExhΨκExhPInletPExh,WInlet=fvolNEng,PInletTInletRInletNEngPInletTInletRInletVEng120,TInlet=PInletVInletmAir+mEGRRInlet,TEGR=PInletPExh(κExh-1)/κExhTExh,WExh=WInlet+WFuel,TExh=TInlet+QLHVhWFuel,NEngcp,ExhWInlet+WFuel,PExh=mExhRExhTExhVExh,WTurb=PExhTExhτPExhPAtm,uXVNT,RInlet=RAirmAir+RExhmEGRmAir+mEGR,cv,Inlet=cv,AirmAir+cv,ExhmEGRmAir+mEGR,cp,Inlet=cv,Inlet+RInlet,AEGR=AEGRmaxfEGRuEGR,where PInlet is the pressure in intake manifold. mAir and mEGR are, respectively, the mass of air and EGR-gas in intake manifold. mExh represents the mass of exhaust gas in exhaust manifold. The other model variables and constant parameters with their value are listed in Table 4. The temperatures TEGR and TExh are assumed to be constant and equal to 329.436 K and 837 K, respectively. AEGRmax is the maximum opening area for the EGR. The EGR valve (VGT) is closed when uEGR=0% (uXVGT=0%) and open when uEGR=100% (uXVGT=100%). The static functions, fvol, fEGR, h, and τ, are represented as interpolation in lookup tables. All these parameters are estimated using weighted least squares optimization approach. It is worth noting that the compressor and the CAC are not considered in the model as in [3] since the mass flow and the temperature after the charge-air cooler (CAC) are known variables because they are measured by the production sensors. The considered control inputs are the VGT vane position uXVGT and the EGR valve position uEGR. Notice that the following variables: TCAC, WHFM, NEng, WFuel, and TExh are considered as measurable signals. The measured outputs are the inlet and exhaust pressures.We draw the reader’s attention that another more general mean value model composed of eight states is developed, parameterized, and validated in [21]. The obtained model describes the gas flow dynamics including the dynamics in the manifold pressures, EGR, turbocharger, and actuators.The mean value model described in the previous section was simulated and compared with real measurements obtained from the Caterpillar testbed. The results are depicted in Figures4–7. The input variables used for the validation purpose are illustrated in Figure 4. The measured and modeled outputs (PInlet and PExh) are presented in Figure 5. The absolute value of the difference between the real and estimated variables is shown in Figure 6. One can conclude that the used model can reproduce engine dynamic behavior with good accuracy since the obtained average error is below 3%. Besides, the output measurements are very noisy as shown in Figure 5. Furthermore, the air and exhaust gas mass flows (mAir and mExh) are also presented in Figure 7.Figure 4
Input variables: (a) EGR position [%], (b) VGT position [%], and (c) engine speed [min-1].
(a)
(b)
(c)Figure 5
Modeled (blue color) and measured (red color) inlet and exhaust pressures.
(a)
(b)Figure 6
Absolute value of the difference between the measured and modeled pressures: (a) inlet pressure error and (b) exhaust pressure error.
(a)
(b)Figure 7
Air and EGR mass flows (mAir and mEGR).
(a)
(b)


## 4. Nonlinear Unknown Input Observer Design

In this section, we will present briefly a nonlinear unknown input observer developed in [12]. It was proposed for a class of Lipschitz nonlinear systems with large Lipschitz constant. The sufficient existence conditions for this observer are formulated in terms of LMIs.Obviously, engine model (1) can be put into the form (3a) and (3b). One can see that this form contains three parts: one linear parameter varying part, a nonlinear term with known variables, and a nonlinear state-dependent part. By considering the form (3a) and (3b), this will allow us to tackle our synthesis problem based on Lyapunov theory for LPV systems. Thus, let us consider the general class of nonlinear systems described by the following equations:(3a)x˙=∑j=1nρ‍ρjAjx+Bggυ,y,u+fx,u+Bffa+Bw¯w¯,(3b)y=Cx+Dw¯w¯,where x∈Rnx is the state vector, u∈Rnu is the control input vector, fa∈Rnfa represents the actuator faults assimilated as unknown inputs, y∈Rny is the output vector, w¯∈Rnw¯ is the vector of disturbances/noises, and υ∈Rnυ is the vector of measurable signals. Notice that υ contains any other measured signals which have no link with the system output (y) (e.g., temperature measurement, air mass flow (WHFM), etc.). Aj, for j=1,…,nρ, Bg, Bf, Bw¯, C, and Dw¯ are constant matrices with appropriate dimensions. Without loss of generality, the matrix Bf is assumed to be of full column rank. The functions gυ,y,u and fx,u are nonlinear. No assumption is made about the function gυ,y,u. However, the function fx,u is assumed to be once differentiable with large Lipschitz constant. The weighting functions ρj are assumed to be known and depend on measurable variables. They verify the convex sum property:(4)∑j=1nρ‍ρj=1,ρj≥0,∀j∈1,…,nρ.Let us first rewrite Bffa as follows:(5)Bffa=∑i=1nfa‍Bf,ifa,i,where Bf,i is the distribution matrix of the fault fa,i.The aim is to construct an NUIO-based residual generator which is insensitive to only one actuator fault (e.g.,Bf,ifa,i). By choosing the unknown input (scalar) d=fa,i, the system (3a) and (3b) can be rewritten as(6a)x˙=∑j=1nρ‍ρjAjx+Bggυ,y,u+fx,u+Bdd+Bww,(6b)y=Cx+Dww,where Bd=Bf,i, Bw=[Bw¯B¯f], Dw=[Dw¯0], and w=w¯f¯a with f¯a the actuator fault vector fa without the ith component. B¯f is the matrix Bf without the ith column.It is worth noting that the necessary condition for the existence of a solution to the unknown input observer design is the following ([18, 22] for a more explanation): Rank(CBd)=Rank(Bd), where Bd is a matrix of full column rank as is a column of the matrix Bf which is assumed before to be full column rank.The considered residual generator for the system (6a) and (6b) is given by(7a)z˙=Nρz+Ggυ,y,u+Mfx^,u+Lρy,(7b)x^=z-Ey,(7c)r=Πr(y-Cx^)with N(ρ)=∑j=1nρ‍ρjNj and L(ρ)=∑j=1nρ‍ρjLj. Πr is a known matrix. x^ represents the state estimation vector. Matrices N, G, M, L, and E are the observer gains and matrices which must be determined such that x^ converges asymptotically to x. Notice that the index ρ is omitted where it is not necessary to simplify the notations.By defining the state estimation error ase(t)=x^(t)-x(t), the error dynamics can be expressed as(8)e˙=Ne+(G-MBg)g(υ,y,u)+NM+LC-MAx-MBdd+Mf~+KDw-MBww-EDww˙with f~=f(x^,u)-f(x,u) and K=L+NE. Now, if the following matrix equations are satisfied:(9a)Nρ=MAρ-KρC,witheachNjstable,(9b)L(ρ)=K(ρ)×(I+CE)-MA(ρ)E,(9c)G=MBg,(9d)M=I+EC,(9e)MBd=0,e(t) goes to zero asymptotically if w=0 and is invariant with respect to the unknown input d(t). The notation I stands for the identity matrix.Conditions (9d) and (9e) are equivalent to ECBd=-Bd. One necessary condition to have for ECBd=-Bd is that CBd is of full column rank since Bd is of full column rank. If CBd is of full column rank, then all possible solutions of ECBd=-Bd can be expressed as follows [18]:(10)E=U+YVwith U=-Bd(CBd)† and V=(I-CBd(CBd)†) where Y can be any compatible matrix and X†=(XTX)-1XT.Then, the error dynamics becomes(11)e˙=MA-KCe+Mf~+KDw-MBww-EDww˙.In order to minimize the effect of disturbances on the observer error, theH∞ performance criterion can be used. However, the presence of the term w˙ makes the task difficult because it should be discarded from the derivative of the Lyapunov function as we will see later. Another solution is to add a negative term depending on w˙Tw˙ as proposed in [23]. This solution needs to modify the classical H∞ criterion. In this work, the modified H∞ criterion presented in [23] is used.The modifiedH∞ estimation problem consists in computing the matrices N and L such that(12a)limt→∞et=0forw(t)=0,(12b)eL2nx≤γw1,2rforw(t)≠0;e(0)=0,where ·k,pr represents the Sobolev norm (see [23]).Then to satisfy (12a)-(12b), it is sufficient to find a Lyapunov function Υ such that(13)Γ=Υ˙+eTe-γ2wTw-γ2w˙Tw˙<0,where Υ=eTPe with P a positive definite symmetric matrix. From (11), Γ is given by(14)Γ=eTMA-KCTP+PMA-KCe+eTPMf~+Mf~TPe+eTP(KDw-MBw)w+wTKDw-MBwTPe-eTPEDww˙-w˙TEDwTPe+eTe-γ2wTw-γ2w˙Tw˙.Before introducing our main result, let us present the modified mean value theorem [24] for a vector function.Theorem 1 (see [24]).
Let the canonical basis of the vectorial spaceRs for all s≥1 be defined by(15)Es=esi∣esi=0,…,0,1︷ith,0,…,0︸scomponentsT,i=1⋯s.Let f(x):Rn→Rn be a vector function continuous on a,b∈Rn and differentiable on convex hull of the set (a,b). For s1,s2∈a,b, there exist δijmax⁡ and δijmin⁡ for i=1⋯n and j=1⋯n such that(16)fs2-fs1=∑i,j=1n,n‍Hijmax⁡δijmax⁡+∑i,j=1n,n‍Hijmin⁡δijmin⁡s2-s1δijmax⁡,δijmin⁡≥0,δijmax⁡+δijmin⁡=1,where (i)
hijmax⁡≥max⁡(∂fi/∂xj) and hijmin⁡≤min⁡(∂fi/∂xj),(ii)
Hijmax⁡=en(i)enT(j)hijmax⁡ and Hijmin⁡=en(i)enT(j)hijmin⁡.The proof of this theorem is given in [24].In our case, the nonlinear functionf depends on the state vector x and also on the known input u. Although the previous theorem is applicable in our case, u is bounded.Now, we can give a sufficient condition under which the observer given by (7a), (7b), and (7c) is a NUIO. Thus, the negativity of Γ is ensured by the following theorem.Theorem 2 (see [25]).
The observer errore(t) converges asymptotically towards zero if there exist matrices K¯k, Y¯, a positive definite symmetric matrix P, and a positive scalar μ such that the following LMIs are satisfied:(17a)P>0,(17b)Ξijkmax⁡Φk-PYDw-Y¯VDw(*)-μI0(*)(*)-μI<0,(17c)Ξijkmin⁡Φk-PYDw-Y¯VDw(*)-μI0(*)(*)-μI<0∀i=1,…,n, j=1,…,n and k=1,…,nρ, where(18)Φk=-K¯kDw+PI+UCBw+Y¯VCBw,μ=γ2,Ξijkmax⁡=I+UCA+H¯ijmax⁡TP+PI+UCA+H¯ijmax⁡-CTK¯kT-K¯kC+A+H¯ijmax⁡TCTVTY¯T+Y¯VCA+H¯ijmax⁡+I,Ξijkmin⁡=I+UCA+H¯ijmin⁡TP+PI+UCA+H¯ijmin⁡-CTK¯kT-K¯kC+A+H¯ijmin⁡TCTVTY¯T+Y¯VCA+H¯ijmin⁡+I,H¯ijmax⁡=ZHHijmax⁡,H¯ijmin⁡=ZHHijmin⁡with ZH=n×n. Solving LMIs (17a)–(17c) leads to determine matrices P, Y¯, and K¯k. The matrices Kk and Y can be obtained from Kk=P-1K¯k and Y=P-1Y¯. The other matrices N and L can then be deduced easily from (9a) and (9b), respectively.Proof.
The proof is omitted. A sketch of this proof is presented in [25].Notice that if there exist terms such that∂fi/∂xj=0, then the scaling factor ∑i,j=1n,n(δijmax⁡+δijmin⁡) is less than 1. Consequently, the scaling factor Z¯H must be redefined as follows:(19)Z¯H=∑i,j=1n,n‍δijmax⁡+δijmin⁡=n×n-n0,∑i,j=1n,nδijmax⁡+δijmin⁡Z¯H=1,where n0 is the number of terms in ∂fi/∂xj that equal zero.The procedure to design the NUIO parameters is summarized by the following algorithm.Algorithm 3.
NUIO design is as follows:(1)
computeU and V from  (10),(2)
determine matricesY and Kk, with k=1,…,nρ from the LMI sets (17a)–(17c),(3)
compute the observer matricesE, M, G, N, and L from (10), (9d), (9c), (9b), and (9a), respectively.


## 5. Residual Generation

This section addresses the problem of actuator faults isolation based on a bank of residual generators (see Figure8). Each residual on the proposed scheme is an NUIO based on model (6a) and (6b). Notice that the system (1) can be easily rewritten in a state space form as (3a) and (3b) (see Appendix C). Furthermore, each residual is designed to be insensitive to only one fault. Thus, the actuator faults can be easily isolated since only one residual goes to zero while the others do not. Therefore, for our application, the combination of two observers is sufficient to detect and isolate any faulty actuator. It is also assumed that only a single actuator fault can occur at one time.Figure 8
FDI system.To construct the bank of residual generators we have to(i)
design an NUIO which generates the residualr1 insensitive to fa,1 by taking (20)Bd=RExhTEGRAEGRmaxcp,ExhVInletcv,Exh0AEGRmax-AEGRmaxT,(ii)
design an NUIO which generates the residualr2 insensitive to fa,2 by taking (21)Bd=0001T.To obtainBd as in (20), it suffices to replace in the original system the input vector u by u+fa, where fa represents the actuator fault vector. Thus, Bd can be obtained easily by gathering the terms multiplying the vector fa in one matrix.Each residual,ri(t) (i=1,2), computed from (7a), (7b), and (7c), is a 1-dimensional vector associated with one of the actuators uEGR and uXVGT. By stacking the two vectors, one defines r¯(t)=r1′t,r2′t′. In the absence of faults, this vector of residuals has zero mean, while, upon occurrence of a single step-like fault, the effect on each of its components is indicated in Table 2. A “×” is placed when the fault located in the corresponding column affects the mean of the residual component on the corresponding row, and a “0” when the residual component presents very low sensitivity to the fault. The sampled vector r¯(k) can be rewritten as(22)r¯(k)=r¯0(k)+∑l=12‍νlΓl(k)1{k≥k0}δl,where the sample number k corresponds to the time instant t=kTs with Ts denoting the sampling period, r¯0(k) is the fault-free residual, Γl(k) is the dynamic profile of the change on r¯(k) due to a unit step-like fault fl, and νl is the magnitude of fault l. Model (22) will be used as the basis for the decision system design in the next section.Table 2
Effect of the faults on the residuals.
r
-
f
1
f
2
r
1
0
×
r
2
×
0Table 3
Caterpillar 3126B engine characteristics.
Description
Model
Caterpillar 3126B
Type of engine
Inline, 4-stroke
Number of cylinders
6
Number of inlet valves
2
Number of exhaust valves
1
Firing order
1-5-3-6-2-4
Type of combustion
Direct injection
Maximum torque
1166 Nm @ 1440 rpm
Maximum power
224 kw @ 2200 rpm
Idle speed
700 rpm
Maximum speed
2640 rpm
Geometrical characteristics
Bore
110 mm
Stroke
127 mm
Compression ratio
16
Total displacement
7.25 liter
Connecting rod length
199.9 mm
Crank throw radius
63.5 mm
Injection system
Type
HEUI
Injection pressure
200–145 bar
Injection orifices number
6
Type of combustion
Direct injection
Geometrical characteristics of manifolds and pipes
Intake manifold
5 L
Exhaust manifold
0.945 LTable 4
The variables used in the engine model.
Symb.
Quantity
Value/unit
P
Inlet
Pressure in intake manifold
Pa
p
Atm
Atmospheric pressure
Pa
W
Turb
Exhaust mass flow past the turbine
kg·s−1
u
XVNT
Position of VNT vanes
%
N
Eng
Engine speed
min−1
W
Fuel
Mass flow of injected fuel
kg·s−1
Q
LHV
Lower heating value
J·kg−1
A
EGR
Effective area of EGR valve
m2
R
Inlet
Gas constant in intake manifold
J·(kg·K)
W
Inlet
Mass flow into engine inlet ports
kg·s−1
T
Inlet
Temperature in the intake manifold
K
m
Air
Mass of air in intake manifold
kg
m
EGR
Mass of EGR gas in intake manifold
kg
W
Exh
Exhaust mass flow into the exhaust manifold
kg·s−1
m
Exh
Mass of exhaust gas in exhaust manifold
kg
P
Exh
Pressure in exhaust manifold
Pa
c
p
,
Inlet
Specific heat at const. pres. in intake manifold
J·(kg·K)
c
v
,
Inlet
Specific heat at const. vol. in intake manifold
J·(kg·K)
κ
Ratio of specific heats
c
p
/
c
v
W
HFM
Air mass flow past the air mass flow sensor
kg·s−1
W
EGR
EGR mass flow into intake manifold
kg·s−1
V
Inlet
Volume of intake manifold
1.8
×
1
0
-
3 m3
R
Air
Gas constant of air
288.2979 J·(kg·K)
c
p
,
Air
Specific heat at const. pres. of air
1067.4 J·(kg·K)
c
v
,
Air
Specific heat at const. vol. of air
779.1021 J·(kg·K)
R
Exh
Gas constant of exhaust gas of exhaust gas
290.155 J·(kg·K)
c
p
,
Exh
Specific heat at const. pres. of exhaust gas
1288.08 J·(kg·K)
c
v
,
Exh
Specific heat at const. vol. of exhaust gas
997.9250 J·(kg·K)
T
CAC
Temperature of the air after the charge-air cooler
322.5 K
T
EGR
Temperature of EGR gas flow into the i.m.
322 K
T
Exh
Temperature in exhaust manifold
837 K
V
Eng
Engine displacement
7.239
×
1
0
-
3 m3
V
Exh
Volume of exhaust manifold
9.4552
×
1
0
-
3 m3One can see that residuals are designed in a deterministic framework. However, due to measurement noise and system disturbances, parameters uncertainties, and variations, the computed residuals are stochastic signals. As a consequence, a multi-CUSUM method will be used for the decision step. Another way would have been to consider the noise and stochastic disturbances characteristics to design robust residuals, analyze the stochastic characteristics of the residuals, and design the decision algorithm with respect to these characteristics. However, this would be too complex to do for our application and would lie anyway on given assumptions on the stochastic disturbances.


## 6. Decision System

In this section, we will present a decision system based on a statistical approach proposed in [26]. This approach, named multi-CUSUM, was developed in [27] and refined in [28]. Indeed, in model (22), it is seen that an actuator fault induces a change in the mean of the residual vector r¯(k). Thus, the problem amounts to detecting and isolating a step-like signal within a white Gaussian noise sequence. Therefore, the change detection/isolation problem can be stated as the following hypothesis testing:(23)H0:L(r¯(i))=N(μ0,Σ),i=1,…,k,Hl:L(r¯(i))=N(μ0,Σ),i=1,…,k0-1,Hl:L(r¯(i))=N(μl,Σ),i=k0,…,k,where k0∈[1,k], and μl=μ0+νlΓl is the mean value of the residual sequence when the lth fault has occurred, for l∈{1,2}. The decision system is based on a combination of CUSUM decision functions [27], each of them involving the log-likelihood ratio between hypotheses Hl and H0, namely,(24)skl,0=ln⁡plr¯kp0r¯k,where pl is the probability density function of r¯(k) under the hypothesis Hl. Under the Gaussian hypothesis, the log-likelihood ratio can be rewritten as(25)sk(l,0)=μl-μ0TΣ-1r¯(k)-12(μl+μ0).The CUSUM decision function is defined recursively as(26)gk(l,0)=max⁡(0,gk-1(l,0)+sk(l,0)).In order to estimate the fault occurrence time, the number of successive observations for which the decision function remains strictly positive is computed as Nl(k)=Nl(k-1)1{gk-1(l,0)>0}+1. To decide whether fault l has occurred, one has to check that, on average, all the likelihood ratios sk(l,j), for 1≤j≠l≤2, are significantly larger than zero. Noticing that sk(l,j)=sk(l,0)-sk(j,0), one can build a CUSUM algorithm to decide between hypotheses Hl and Hj by taking into account the difference between gk(l,0) and gk(j,0). Hence a decision that Hl holds can be issued when the following decision function becomes positive:(27)g¯l=min⁡0≤j≠l≤2(gk(l,0)-gk(j,0)-hl,j)for l=1,2 and gk(0,0)=0.Like in [26, 27], it is advisable to consider only two values for the thresholds hl,j, for each l=1,…,nf, namely,(28)hl,j=hld,forj=0,hli,for1≤j≠l≤nf,where hld is the detection threshold and hli is the isolation threshold. Notice that the mean detection delay, the mean time before false alarm, and the probability of a false isolation depend on the choice of these thresholds. Indeed, the thresholds hld and hli can be linked to the mean detection delay (τ¯) for the fault l thanks to the following expression [27] assuming that hld=γ¯hli where γ¯≥1 is a constant:(29)τ¯=max⁡hldκ¯l,0,hlimin⁡j≠0,l⁡(κ¯l,j)ashli⟶∞,where κ¯l,j is the Kullback-Leibler information defined as (30)κ¯l,j=12μl-μjTΣ-1(μl-μj).When consideringhl,d=hl,i=hl, (31) is used instead of (27):(31)gl*=min⁡0≤j≠l≤2(gk(l,0)-gk(j,0))and an alarm is generated when a gl*≥hl.The reader can refer to [26, 27] for more detailed information.


## 7. Experimental Results and Discussion

The proposed fault detection and isolation approach has been tested on the caterpillar 3126 engine located at Sussex University, UK. The aim is to detect and isolate any additive actuator fault. The detection delayτ¯ should be lower than 0.01 s in average. This delay is defined as the difference between the alarm time and the actual fault occurrence time.


### 7.1. Design Parameters

For the design of the residual generators, the gain matrices given by (9a)–(9e) should be determined. Unfortunately, we cannot present their values due to pages limitation. Notice that the LMIs (17a)–(17c) are solved by using the software YALMIP toolbox, which is a toolbox for modeling and optimization in Matlab. Matrix Πr for the two residual generators is chosen as Πr=0,1. The sampling period Ts is set as 10-3 s.Remark 4.
It is worth noting that no feasible solution is obtained with the approach presented in [18] due to the Lipschitz constant value of the nonlinear function f (see (C.7)). In fact, the Lipschitz constant value is bigger than 1010 in our case. Other methods can be used, or extended to our case, as that proposed in [23] for estimating the state and unknown input vectors. Unfortunately, these methods are computationally demanding since the number of LMIs, in each residual generator, that should be solved is equal to NLMI=2n×nf, where nf is the number of nonlinearities in the system. So, in our case NLMI=24×4=65536. In addition, it is only proposed for the standard systems with a linear time-invariant part (Ax). We know that the number of LMIs will be strongly increased if this method is extended to our LPV case.The covariance matrixΣ and the mean μ0 used for designing the multi-CUSUM algorithm are estimates of the variance and the mean of r¯ obtained from a set of simulation data generated in healthy operating conditions. Assuming that all considered faults should be detected and isolated with a mean detection delay τ¯≤0.1 s, the threshold is selected as hl=1000.


### 7.2. Validation Step

For illustrating the performance of this approach, the following scenario is considered. First, a positive step-like change in the EGR actuator appears whent∈6.5,8 s. Next, a single negative step-like fault in the VGT actuator is introduced in the time interval 8.9,11.5 s.The experiment is performed with engine average speedNEng=1800 rpm, where the minimum and maximum value of NEng (N_Eng and N¯Eng) are 1700 rpm and 1900 rpm, respectively. The initial conditions for the two observers are randomly chosen as follows:(32)z10=1202×10-710-10T,z20=1202×10-710-10T.The experimental results are shown in Figure9. First, the actuators (uEGR and uXVGT) behavior is illustrated in Figure 9(a). Then, the normalized residuals r1 and r2 are shown in Figure 9(b). Finally, the decision functions resulting from the multi-CUSUM algorithm are presented in Figure 9(c). It is clear from the last figures (Figure 9(c)) that only the exact decision function g2*, corresponding to an occurred fault in uXVGT, crosses the threshold. It means that this fault is correctly detected and isolated. The obtained mean detection delay τ¯, given by (29), is equal to 0.0890 s. As expected, this fault is detected and isolated with a mean detection delay τ¯≤0.1 s.Figure 9
Experimental results: (a) EGR and VGT actuators, (b) residuals (ri), and (c) multi-CUSUM decision functions.
(a)
(b)
(c)


## 7.1. Design Parameters

For the design of the residual generators, the gain matrices given by (9a)–(9e) should be determined. Unfortunately, we cannot present their values due to pages limitation. Notice that the LMIs (17a)–(17c) are solved by using the software YALMIP toolbox, which is a toolbox for modeling and optimization in Matlab. Matrix Πr for the two residual generators is chosen as Πr=0,1. The sampling period Ts is set as 10-3 s.Remark 4.
It is worth noting that no feasible solution is obtained with the approach presented in [18] due to the Lipschitz constant value of the nonlinear function f (see (C.7)). In fact, the Lipschitz constant value is bigger than 1010 in our case. Other methods can be used, or extended to our case, as that proposed in [23] for estimating the state and unknown input vectors. Unfortunately, these methods are computationally demanding since the number of LMIs, in each residual generator, that should be solved is equal to NLMI=2n×nf, where nf is the number of nonlinearities in the system. So, in our case NLMI=24×4=65536. In addition, it is only proposed for the standard systems with a linear time-invariant part (Ax). We know that the number of LMIs will be strongly increased if this method is extended to our LPV case.The covariance matrixΣ and the mean μ0 used for designing the multi-CUSUM algorithm are estimates of the variance and the mean of r¯ obtained from a set of simulation data generated in healthy operating conditions. Assuming that all considered faults should be detected and isolated with a mean detection delay τ¯≤0.1 s, the threshold is selected as hl=1000.


## 7.2. Validation Step

For illustrating the performance of this approach, the following scenario is considered. First, a positive step-like change in the EGR actuator appears whent∈6.5,8 s. Next, a single negative step-like fault in the VGT actuator is introduced in the time interval 8.9,11.5 s.The experiment is performed with engine average speedNEng=1800 rpm, where the minimum and maximum value of NEng (N_Eng and N¯Eng) are 1700 rpm and 1900 rpm, respectively. The initial conditions for the two observers are randomly chosen as follows:(32)z10=1202×10-710-10T,z20=1202×10-710-10T.The experimental results are shown in Figure9. First, the actuators (uEGR and uXVGT) behavior is illustrated in Figure 9(a). Then, the normalized residuals r1 and r2 are shown in Figure 9(b). Finally, the decision functions resulting from the multi-CUSUM algorithm are presented in Figure 9(c). It is clear from the last figures (Figure 9(c)) that only the exact decision function g2*, corresponding to an occurred fault in uXVGT, crosses the threshold. It means that this fault is correctly detected and isolated. The obtained mean detection delay τ¯, given by (29), is equal to 0.0890 s. As expected, this fault is detected and isolated with a mean detection delay τ¯≤0.1 s.Figure 9
Experimental results: (a) EGR and VGT actuators, (b) residuals (ri), and (c) multi-CUSUM decision functions.
(a)
(b)
(c)


## 8. Conclusion

The problem of actuator fault detection and isolation for diesel engines is treated in this paper. The faults affecting the EGR system and VGT actuator valves are considered. A bank of NUIO has been used. Each residual is designed to be insensitive to only one fault. By using this kind of scheme and by assuming that only a single fault can occur at one time, each actuator fault can be easily isolated since only one residual goes to zero while the others do not. A multi-CUSUM algorithm for statistical change detection and isolation is used as a decision system. Fault detection/isolation is achieved within the imposed timeslot. Experimental results are presented to demonstrate the effectiveness of the proposed approach.



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*Source: 189860-2015-05-19.xml*