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Fault Tolerant Control of Internal Combustion Engine subject to Intake Manifold Leakage*
8th IFAC Symposium on Fault Detection, Supervision and Safety of Technical Processes (SAFEPROCESS) August 29-31, 2012. Mexico City, Mexico
Fault Tolerant Control of Internal Combustion Engine subject to Intake Manifold Leakage ⋆ I. Djemili ∗,∗∗ A. Aitouche ∗,∗∗ V. Cocquempot ∗,∗∗∗ ∗ Automatic
Control Laboratory : LAGIS, Rue Paul Langevin, 59655, Villeneuve d’Ascq, France ∗∗ Hautes Etudes d’Ing´ enieur, 13 rue de Toul, 59046 Lille France {issam.djemili or abdel.aitouche}@hei.fr ∗∗∗ Lille 1 University, Rue Paul Langevin, 59655 Villeneuve d’Ascq France [email protected] Abstract: In this paper, a new FTC strategy of the diesel engine air path subject to intake manifold leakage described by Takagi-Sugeno model is presented. The proposed FTC design scheme integrates the state estimation, the leakage identification and the state feedback control law, to guaranty the stabilization of the faulty plant. State vector and leakage are estimated by an adaptive observer. The gains of the adaptive observer and feedback control law are obtained by solving a linear matrix inequality derived from the Lyapunov theory. The effectiveness and performances of the proposed approach are illustrated using the professional advanced diesel engine simulator AMEsim(LMS) platform in cosimulation with SIMULINK software. Keywords: Diesel engine, Air path, FTC, Adaptive Observer, Takagi-Sugeno models, LMI. 1. INTRODUCTION Diesel engines are recognized as the most common and preferred solutions for distributed power generating systems in numerous applications, such as automotive vehicles, ships, cranes, and electric power generators due to their low fuel consumption and durability. Until now, these advantages were counterbalanced by the high level of nitrogen oxides (NOx ) produced. One way to reduce the NOx formation during combustion is to use a post-treatment system. But for cost reasons, car makers prefer improving engine diagnosis and control to reduce such pollutant emission. Different Model-based diagnosis methods of automotive engines have been considered in earlier papers (see e.g. Gertler et al. (1995), Krishnaswami et al. (1995), Hsu et al. (1995), Kim et al. (1998)). However, the engines considered in these previous works were all gasoline-fuelled and did not include EGR (Exhaust Gas Recirculation) and VNT (Variable Nozzle Turbine). Both these components make the diagnosis problem significantly more difficult since the air flow through the EGRvalve, and also the exhaust side of the engine have to be taken into account. An interesting approach to model-based air-path faults detection for an engine which includes EGR and VNT can be found in Nyberg and Sutte (2004). In this work, 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 airpath of an automotive diesel engine has been developed in order to study the monotorability of the system in Djemili ⋆ This work was produced in the framework of SCODECE (Smart COntrol and Diagnosis for Economic and Clean Engine), a European territorial cooperation project part-funded by the European Regional Development Fund (ERDF) through the INTERREG IV A 2 Seas Programme, and the research department of the Region Nord Pas de Calais, France.
978-3-902823-09-0/12/$20.00 © 2012 IFAC
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et al. (2011a). Recently, an adaptive observer for intake leakage detection in diesel engine described by Takagi-Sugeno model was proposed in Djemili et al. (2011b). This approach allows to estimate simultaneously the system states and the leakage. Following this work, a Fault Tolerant control (FTC) strategy exploiting the estimation of the leakage, is proposed in this paper. Fault Tolerant Control aims at fulfilling the desired objectives in the presence of faults in components of such as sensors, actuators and/or system. Two FTC techniques are developed in the literature : the active and the passive techniques. The passive techniques can be viewed as a robust control laws where faults are considered as disturbances (see Chen et al. (1998)). On the other hand, the active fault tolerant control techniques consist on adapting the control law using the information provided by fault diagnosis algorithm (see Blanke et al. (2003)). This paper is organized as follows. The air path FTC problem is first formulated in section 2. Then, a mean value model of diesel engine is presented and transformed into a TS model in section 3. The FTC strategy of the diesel engine air path subject to an intake manifold leakage is presented in section 4. The proposed strategy is successfully evaluated using an advanced diesel engine professional simulator AMEsim(LMS) in section 5. Conclusion and future work are finally presented in the last section. 2. FTC PROBLEM FORMULATION The general problem of combustion control can be decomposed into three control issues : the air path control, the cylinder balancing control problem and the fuel path control issues. In this work, we are interested only in the air path control issues. 10.3182/20120829-3-MX-2028.00268
SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico
3. MODELING A model of the intake manifold subject to leakages based on a mean-value diesel engine model reported in Jankovic and Kolmanovsky (2000) and Kao and Moskwa (1995) is derived. This model will be used in the following to design an airpath observer that will provide the estimates of the system states and a variable that is directly related to the presence of the leakage. In this model, the heat exchangers (intercooler and the EGRcooler) are not included because the temperatures downstream the heat exchangers are measured by the sensors. For the sake of simplicity we do not take into account the turbocharger dynamics. Nomenclature used in the model is presented in Table 1. Figure 1. A schematic picture of an airpath system 3.1 Intake manifold subject to leakage modeling The diesel engine considered in this paper is a four-cylinder engine with a high-pressure Exhaust Gas Recirculation circuit (EGR) and a Variable Geometry Turbocharger (VGT). A schematic picture of an airpath system is shown in Fig.1. The air entering the engine is measured by an Air flow Meter. It then passes through the compressor, enters the intake manifold, and flows into the cylinders. The fuel is injected directly into the cylinders and burned, producing torque on the crank shaft. The hot exhaust gas is pumped out via the exhaust manifold. Part of the exhaust gas flows from the exhaust manifold through the turbine out of the engine and the other part is recirculated back into the intake manifold through the EGR valve. The turbine takes the energy from the exhaust gas to power the compressor. The scheme also shows the intercooler and the EGR-cooler that are used to reduce the intake manifold temperature. The engine is equipped with sensors measuring : the intake manifold temperature, the intake manifold pressure and the exhaust manifold pressure. It is also equipped with the AIR/Fuel Ratio sensor located downstream the turbine. It reflects the composition in the exhaust manifold. The control inputs to the engine are the injected fuel, the EGR-valve and the VGT. The speed engine is also considered to be an input. In the air path control, it is desirable to control the masses aspirated by the cylinder (Masp,air and Masp,bg ) with two air path actuators (EGR valve and the VGT ). In practical situation, the considered masses can not be measured. Equivalent variables can be considered. Controlling those two masses is equivalent to controlling the intake manifold pressure Pint (being a reflection of Masp,air + Masp,bg ) and the burned gas M rate Fint (representing to ratio Masp,airasp,bg +Masp,bg ). Set points are often chosen to to reduce the NOx emissions. In the engine life service, leakages can occur in the intake manifold, that will result in mismatches between the mixture in the cylinder and the reference mixture. These mismatches can have dramatic effects. It generates extra noises and pollution. In this context, the aim of this work is to design a new FTC strategy to control the EGR valve and the VGT in presence of such leakages. The proposed FTC approach should maintain the intake manifold pressure and the burned gas rate closed to the set points given by the torque controller and preserve stability conditions in the presence of the leakage. 601
Applying the first law of thermodynamics (energy conservation principle), by considering that the heat transfer to the surroundings is negligible, leads to the expression of the variation of the intake manifold pressure as a function of the aspirated flow Dasp , the manifold air flow Dair , the EGR flow Degr and the leakage mass flow Dleak . The manifold dynamics pressure Pint is described by the following equation:
γ RTint P˙int = (Dair + Degr − Dasp − Dleak ) (1) Vint where Vint is the manifold volume, R is the perfect gas constant relative to the air and Tint is the temperature in the intake manifold . The aspirated flow is given by ( ) Pint Pint Ne Dasp = ηvol Ne , Vcyl (2) Tint RTint 120 where Vcyl is the cylinders’ volume. ηvol is the volumetric efficiency which is experimentally ( derived ) and eventually defined int . Ne is the engine speed. through a look-up table ηvol Ne , PTint The leakage mass flow rate from the intake manifold is given by ( ) Pint Patm Dleak = θ √ σ (3) Pint RTint ( ) where θ is the leakage size. The function σ PPatm is given by int v u )2 ( ) γ +1 u ( P γ γ 2 γ P u down down t − γ − 1 P P up up ( ) γ ( ) ( ) Pdown σ = i f Pdown ≥ Pdown γ −1 Pup Pup v Pup u( γ +1 ) u γ −1 2 t otherwise, γ +1 (4) where the subscripts ”up” and ”down” stand for upstream and downstream values across a section θ . The dynamic of burned gas fraction in the intake manifold Fint is derived as
SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico
RTint F˙int = (Degr (Fexh − Fint ) − Dair Fint ) (5) PintVint The Air/Fuel Ratio sensor located downstream the turbine allows us to obtain a reflection of the composition in the exhaust manifold Fexh . The composition of the EGR Fegr is that of the exhaust manifold Fexh , i.e. Fegr = Fexh . 3.2 State space representation Let us set x1 = Pint , x2 = Fint , u1 = Dair , u2 = Degr and note : Ne αint = γ VRTintint and βint = RT1int Vcyl 120 . The state space nonlinear model of the intake, with constant leakage, is obtained from Eq.(1) and Eq.(5). x˙1 = − f1 (x1 ) x1 + αint u1 + αint u2 − f4 (x1 ) θ x˙2 = − f2 (x1 , u1 , u2 ) x2 + f3 (x1 ) u2 (6) θ˙ = 0 The considered output is y = x1 . The nonlinear functions are given by f1 (x1 ) = αint βint ηvol (x1 , Ne ) αint (u1 + u2 ) f2 (x1 , u1 , u2 ) = x1 αint Fexh f3 (x1 ) = x1 ( ) x1 Patm σ f4 (x1 ) = αint √ Pint RTint
∀i ∈ {1, 2, 3} , fi ≤ fi ≤ fi (7) The representation (6) is one of many possible reformulations of the engine model. These will be subsequently used to derive a TS model using the sector nonlinearity approach Tanaka and Wang (2001). Quantity Total mass in the i.m Air mass in the i.m Atmospheric pressure Pressure in the i.m Fraction of burned gas in the i.m Fraction of burned gas in the e.m. Temperature in the i.m Manifold air flow EGR flow Aspirated flow into the cylinders Engine speed Volume of the i.m Volume of the cylinders Ideal Gas constant Ratio of specific heats Volumetric efficiency Air/Fuel equivalence ratio
A TS model is a nonlinear model composed of linear models blended together with some nonlinear functions. The purpose is to obtain an exact representation of a nonlinear model in a compact set of the state variables. The way to derive one TS model from a nonlinear model is to use two nonlinear functions (called membership functions in the TS model) to describe each nonlinearity. This systematic method, called the sector nonlinearity approach (Tanaka and Wang (2001)), results in a TS model with r = 2nl linear models (rules), where nl is the number of nonlinearities of the model. The TS model may be expressed under the following state space form r x˙ (t) = ∑ µi (ξ (t)) (Ai x (t) + Bi u (t) + Ψi θ (t))
i=1
θ˙ (t) = 0 y (t) = Cx (t)
(8)
where x (t) ∈ ℜ2 is the state vector, u (t) ∈ ℜ2 is the control input vector, θ (t) is the the leakage size, y (t) ∈ ℜ is the output vector and ξ (t) is called the premise vector. Ai , Bi , ψi and C are matrices with appropriate dimensions. The nonlinear scalar functions µi (ξ (t)) are assumed to be positive and to verify the convex sum property. To obtain these functions, each bounded function f can be easily split using two nonlinear functions M1 (.) ≥ 0, M2 (.) ≥ 0 such that M1 (.) + M2 (.) = 1 (see, Tanaka and Wang (2001)) as follows
Model (6) is nonlinear, thus a nonlinear observer has to be designed. One way to take into account the nonlinearities of the model is to use a polytopic approach such as the Takagi and Sugeno’s modeling (Takagi and Sugeno (1985)). Since x1 ̸= 0, the functions f1 , f2 and f3 can be assumed to be bounded such that
Symb. Mint Mint,air Patm Pint Fint Fexh Tint Dair Degr Dasp Ne Vint Vcyl R γ ηvol λ
3.3 TS fuzzy modeling
Unit Kg Kg Pa Pa − − K Kg.s−1 Kg.s−1 Kg.s−1 rpm m3 m3( ) J. KgK −1 − − −
f (.) − f f − f (.) +f = f M1 (.) + f M2 (.) (9) f−f f−f This so-called sector nonlinearity approach can be repeated for each nonlinear function of the model. Note that the arguments of the nonlinear functions µi (.) are known and that solving the LMI problem is harder without this condition. f (.) = f
By considering the nonlinear model (6), the constant matrices of the considered model (8) are given below [ ] − f1 0 Ai={1,2,3,4} = 0 − f2 [ ] − f1 0 Ai={5,6,7,8} = 0 − f2 ] [ − f1 0 Ai={9,10,11,12} = 0 −f2 ] [ − f1 0 Ai={13,14,15,16} = 0 − f2 [ Bi={1,2,5,6,9,10,13,14} = [ Bi={3,4,7,8,11,12,15,16} =
αint αint 0 f3 αint αint 0 f3 [
Ψi={1,3,5,7,9,11,13,15} =
Table 1. Nomenclature. (i.m. refers to the intake manifold)
[
− f4 0
− f4 Ψi={2,4,6,8,10,12,14,16} = 0 602
] ]
] ]
SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico
r ( ) x˙ f (t) = ∑ µi (ξ (t)) Ai x f (t) + Bi u f (t) + Ψi θ (t)
C=[1 0] 4. PROPOSED FTC STRATEGY BASED ON AN ADAPTIVE OBSERVER
i=1
4.1 The structure of the proposed FTC scheme The proposed scheme is illustrated in Figure 2. The method uses an adaptive observer to detect and estimate the leakage size in the intake manifold. The proposed FTC approach should maintain the intake manifold pressure and the burned gas rate closed to the set points given by the torque controller and preserve stability conditions in the presence of the leakage.
y f (t) = Cx f (t) θ˙ (t) = 0
(12)
The main objective of this paper is to propose a new control strategy which can be used to determine the control inputs u f (t) such that : • the closed-loop system (12) is stable, • x f (t) converges asymptotically to xre f (t) even in the T presence of the fault. xre f (t) = [ Pint,re f Fint,re f ] is the set points vector given by (10). The following control strategy is used : r ( ( ) ) u f (t) = ∑ µ j (ξ (t)) −S j θˆ (t) + G j x (t) − xˆ f (t) + u (t) j=1
(13) where θˆ is the fault estimate and u (t) is the nominal control input (no fault condition). The FTC design needs
Figure 2. The FTC strategy scheme for the airpath dynamic system By taking into account the gear-box configuration, the driver’s request (accelerator position) is turned into a torque control objective under the form of an Indicated Mean Effective Pressure (IMEP) set point. Then, the set points for the intake manifold pressure and the Burned Gas Ratio (BGR) are given by experimentally calibrated static maps on the (IMEPre f ,Ne ) operating range. The engine speed Ne is considered as an external input. The set points vector is defined as ( ) Pint,re f = f pressure ( IMEPre f ,)Ne (10) Fint,re f = fBGR IMEPre f , Ne
• to determine xˆ f (t) and θˆ , • to design G j in such a way that the closed-loop system, including the state and fault estimations, is stable. The nominal control input u (t) is an state feedback control law computed using a Parallel Distributed Compensation (PDC) scheme as described in Wang et al. (1996). To estimate simultaneously the system states xˆ f (t) and a variable that is directly related to the presence of leakage, namely θˆ , an adaptive observer for system (12) is used : r . ( ) xˆ f = ∑ µi (ξ ) Ai xˆ f (t) + Bi u f (t) + Ψi θˆ (t) + Ki (y f (t) −Cxˆ f (t)) .
i=1
(14)
r
θˆ (t) = ∑ µi (ξ ) (Li (y f (t) −Cxˆ f (t))) i=1
A multivariate control (using the EGR valve and the VGT as actuators ) of the masses has to be designed to reach the set points given by the torque controller. This FTC strategy as described below exploits the states and the leakage size given by the adaptive observer. The outputs of the FTC block : Degr,re f and Dair,re f , are used to determine the opening area of the EGR valve (Vegr ) and the VGT position (Vtur ). PID controllers are added to the structure in order to provide further accuracy and robustness. The goal of the second feedback is to control Degr and Dair toward the feasible references set points Degr,re f and Dair,re f obtained by the FTC controller. The main purpose of the first feedback is to give a feasible and continuous set points for the second feedback action (see Figure 2). 4.2 FTC strategy Let us consider the following reference model : r x˙ (t) = ∑ µi (ξ (t)) (Ai x (t) + Bi u (t)) i=1 y (t) = Cx (t)
(11)
The extended error system, containing the two error dynamics: x f (t) − xˆ f (t) and θ (t) − θˆ (t), can be expressed as (
x˙ f (t) − x˙ˆ f (t) θ˙ (t) − θ˙ˆ (t)
)
i=1
603
(
Ai − KiC −Ψi −LiC 0
)(
x f (t) − xˆ f (t) θ (t) − θˆ (t)
) (15)
The tracking error dynamic e (t) = x (t) − x f (t), is given by
A e (t) + Bi u (t) i −S j θˆ (t) r r e˙ (t) = ∑ ∑ µi (ξ (t)) µ j (ξ (t)) −Bi +G j (x (t) − xˆ f (t)) i=1 j=1 +u (t)
(16)
−Ψi θ (t)
Let us assume that S j satisfies the following equality Bi S j = Ψi . Thus ( ) r r (Ai −(Bi G j ) e (t) ) e˙ (t) = ∑ ∑ µi (ξ (t)) µ j (ξ (t)) (17) −Ψi θ (t) − θˆ (t) i=1 j=1 An extended error system e˜ (t), containing the tracking error e (t), the state estimation error x f (t) − xˆ f (t) and the fault estimation error f (t) − fˆ (t), can be expressed as r
Let us also consider a faulty system (intake manifold subject to leakage) described by the following equation :
r
= ∑ µi ξ (t)
r
e˙˜ (t) = ∑ ∑ µi (ξ (t)) µ j (ξ (t)) A˜ i j e˜ (t) i=1 j=1
(18)
SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico
e (t) e˜ (t) = x f (t) − xˆ f (t) θ (t) − θˆ (t) A˜ i j =
Ai − Bi G j 0 −Ψi 0 Ai − KiC −Ψi 0 −LiC 0
) 1.9
The main result of this paper is given in the following Theorem Theorem 1. The tracking error e (t), the state estimation error x f (t) − xˆ f (t) and the fault estimation error f (t) − fˆ (t) converge asymptotically to zero, if there exists symmetric positive definite matrices X and P2 , gain matrices H1i and H2i such that the following LMI are verified : A X + XAT i
i
0 −ΨTi ∗ ∗
0 −Ψi −Bi G j ATi P2 −CT H1iT + P2 Ai − H1iC −CT H2iT − P2 Ψi 0 −ΨTi P2 − H2iC 0 0 ∗ ∗ −I ∗ ∗ ∗
i.m Pressure obtained by the nominal controller i.m Pressure obtained by the FTC strategy
1.85
X 0 0 0 −I
Intake Manifold Pressure [bar]
(
strategy increases the air mass flow Dair and decreases the EGR mass flow Degr using the EGR valve and the VGT as actuators, as shown respectively in Figure 5 and Figure 6.
1.8
1.75
1.7
1.65
1.6 0
1
2
3
4
5
6
7
8
9
Time [s]
Figure 4. AMEsim results : Intake manifold Pressure histories
60
90 88 86
The gain of the controller are G j and the gains of the observer are given by Li = H2i and Ki = P2−1 H1i .
Air mass flow [g/s]
with :
84 82 80 78 76 74
Proof : the proof is given in the appendix
72 1
2
3
4
5. MODEL IN THE LOOP : RESULTS The developed FTC strategy has been tested on a four-cylinder diesel engine model running on AMEsimr platform in cosimulation with SIMULINK.
5 Time [s]
6
7
8
9
Figure 5. AMEsim results : Air flow histories (Dair ) obtained by the proposed FTC strategy 5.2 5
EGR mass flow [g/s]
4.8 4.6 4.4 4.2 4 3.8 3.6 0
1
2
3
4
5 Time [s]
6
7
8
9
Figure 6. AMEsim results : EGR flow histories (Degr ) obtained by the proposed FTC strategy Figure 3. AMESim/SIMULINK co-simulation platform 5 4.5 4 Leakage size [mm2]
The simulation is performed considering engine running at 3000 rpm with the following fault scenario : { 0, t < 5s θ (t) = 5 mm2 , t ≥ 5s
3.5 3 2.5 2 1.5 1 0.5
Figures 4-6 show the results obtained by applying the proposed FTC strategy. As a result Figure 6 clearly shows that the faults can be estimated with a very high accuracy. Moreover, from Figure 4 it can be observed that the intake manifold pressure obtained using the FTC strategy is equal to the pressure obtained using the nominal low control before the occurrence of the leakage. On the case of nominal controller, we can see the pressure drop of 200 mBar due to the gas mass flow rate through the leakage. In the other hand, the proposed approach maintains the intake manifold pressure closed to the set points given by the torque controller in the presence of the leakage. The burned gas rate is not affected by the leakage. To maintain the pressure closed to the set points given by the torque controller in the presence of the leakage, our FTC 604
0
0
1
2
3
4
5 Time [s]
6
7
8
9
10
Figure 7. Leakage size estimate 6. CONCLUSION In this paper, we proposed an FTC strategy of the diesel engine airpath subject to intake manifold leakage described by TakagiSugeno’s model. The proposed approach is an integrated FTC design procedure of the leakage identification and fault tolerant control scheme. Leakage identification is based on the use of an adaptive observer, while the FTC controller is implemented as an adaptive state feedback controller. The effectiveness and performances of the proposed approach have been illustrated using a professional advanced diesel engine simulator
SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico
AMEsim(LMS) platform in co-simulation with SIMULINK software. Due to its implementation simplicity, this strategy has the potential to be used on real-time system control which would be the subject of our future work. REFERENCES Blanke, M., Kinnaert, M., Lunze, J., and Staroswiecki, M. (2003). Diagnosis and Fault-Tolerant. Springer-Verlag Berlin Heidelberg. Chen, J., Patton, R., and Chen, Z. (1998). An lmi approach to approach to fault-tolerant control of uncertain systems. In In Proceeding of the IEEE Conference on Decision and Control, volume 1, 175–180. Djemili, I., Aitouche, A., and Cocquempot, V. (2011a). Structural analysis for air path of an automotive diesel engine. In IEEE-International Conference on Communications, Computing and Control Applications (CCCA’11). Hammamet, Tunisia. Djemili, I., Aitouche, A., and Cocquempot, V. (2011b). Adaptive observer for intake leakage detection in diesel engines described by takagi-sugeno model. In 19th Mediterranean Conference on Control & Automation (MED). Gertler, J., Costin, M., Fang, X., and Hira, R. (1995). Mode based diagnosis for automotive engines - algorithm development and testing on a production vehicle. IEEE Transactions on Control Systems Technology, 3(1), 61–69. Hsu, P., Lin, K., and Shen, L. (1995). Diagnosis of multiple sensors and actuartors failures in automotive engines. IEEE Transactions on Vehicular Technology, 44(4), 779–789. Jankovic, M. and Kolmanovsky, I. (2000). Constructive lyapounov control design for turbocharged diesel engines. IEEE Transactions on Control Systems Technology, 8, 288–299. Kao, K. and Moskwa, J. (1995). Turbocharged diesel engine modeling for nonlinear engine control and state estimation. ASME Journal of Dynamic Systems, Measurement and Control, 117(1), 20–30. Kim, Y., Rizzoni, G., and Utkin, V. (1998). Automotive engine diagnosis and control via nonlineare estimation. IEEE Control Systems, 18(5), 84–99. Krishnaswami, V., Luh, G., and Rizzoni, G. (1995). Nonlinear parity equation based residual generation for diagnosis of automotive engine faults. Control Engineering Practice, 10(10), 1385–1392. Nyberg, M. and Sutte, T. (2004). Model based diagnosis of the air path of an automotive diesel engine. Control Engineering Practice 12., 513–525. Takagi, T. and Sugeno, M. (1985). Fuzzy identification of systems and its applications to modeling and control. IEEE Transactions on Systems, Man and Cybernetics,, 15(1), 116– 132. Tanaka, K. and Wang, H.O. (2001). Fuzzy control systems design and analysis: A linear matrix inequality approach. In New York: Wiley, Wiley- Interscience, ISBN 0-471-32324190000. Wang, H., Tanaka, K., and Griffin, M. (1996). An approach to fuzzy control of nonlinear system : stability and design issues. IEEE Transaction on Fuzzy Systems, 4(1), 14–23. Appendix A. PROOF Lemma 2. Let us consider two matrices X and Y of appropriate dimensions. The following inequality is verified for each matrix Q : X T Y + XY T ≤ X T Q−1 X +Y QY T 605
The proof of the Theorem 1 is established using the following Lyapunov’s function : V (e˜ (t)) = e(t) ˜ T Pe˜ (t) , P = PT > 0
(A.1)
where the matrix P is defined as follows : ( ) P1 0 0 P = 0 P2 0 0 0 P3 The derivative of V (e˜ (t)) is written as
( ) r r ˜ T Mi j e˜ (t) V˙ (e˜ (t)) = ∑ ∑ µi (ξ (t)) µ j (ξ (t)) e(t)
(A.2)
i=1 j=1
where Mi j = Z
((
P1 Ai − P1 Bi G j 0 −P1 Ψi 0 P2 Ai − P2 KiC −P2 Ψi 0 −P3 LiC 0
))
Z (X) denotes the Hermitian of the matrix X, i.e., Z (X) = X T + X The derivative of the Lyapunov function is semi negative definite if the following inequality is satisfied Mi j 6 0 i, j = 1, . . . , r (A.3) using the lemma of congruence, P1−1 0 0 P1−1 0 0 Mi j < 0 ⇔ 0 I 0 Mi j 0 I 0 6 0 (A.4) 0 0 I 0 0 I we obtain the following inequality 1 Ωi j 0 −Ψi 0 Ω2i j −CT LiT P3 − P2 Ψi 6 0 T T −Ψi −Ψi P2 − P3 LiC 0
(A.5)
where Ω1i j = XATi − XGTj BTi + Ai X − Bi G j X and Ω2i j = ATi P2 − CT KiT P2 + P2 Ai − P2 KiC with X = P1−1 . In order to use the lemma 2 , the inequality A.5 can be written as
Ai X + XATi 0 −Ψi 0 ATi P2 −CT KiT P2 + P2 Ai − P2 KiC −CT LiT P3 − P2 Ψi + −ΨTi −ΨTi P2 − P3 LiC 0 T T −Bi G j X X −Bi G j 0 0 + 0 0 6 0 0 0 0 0
(A.6)
Using the lemma 2, we obtain
Ai X + XATi 0 −Ψi 0 ATi P2 −CT KiT P2 + P2 Ai − P2 KiC −CT LiT P3 − P2 Ψi + −ΨTi −ΨTi P2 − P3 LiC 0 T T −Bi G j −Bi G j X X 0 Q−1 0 + 0 Q 0 6 0 0 0 0 0
(A.7)
where Q is a symmetric definite positive matrix. Using the Schur complement and changes of variable H1i = P2 Ki and H2i = P3 Li , we obtain the LMI of theorem 1 with Q = I. The matrix P3 does not appear in the LMI, so it is chosen as identity matrix. Therefore, Li = H2i .
Fault Tolerant Control of Internal Combustion Engine subject to Intake Manifold Leakage ⋆ I. Djemili ∗,∗∗ A. Aitouche ∗,∗∗ V. Cocquempot ∗,∗∗∗ ∗ Automatic
Control Laboratory : LAGIS, Rue Paul Langevin, 59655, Villeneuve d’Ascq, France ∗∗ Hautes Etudes d’Ing´ enieur, 13 rue de Toul, 59046 Lille France {issam.djemili or abdel.aitouche}@hei.fr ∗∗∗ Lille 1 University, Rue Paul Langevin, 59655 Villeneuve d’Ascq France [email protected] Abstract: In this paper, a new FTC strategy of the diesel engine air path subject to intake manifold leakage described by Takagi-Sugeno model is presented. The proposed FTC design scheme integrates the state estimation, the leakage identification and the state feedback control law, to guaranty the stabilization of the faulty plant. State vector and leakage are estimated by an adaptive observer. The gains of the adaptive observer and feedback control law are obtained by solving a linear matrix inequality derived from the Lyapunov theory. The effectiveness and performances of the proposed approach are illustrated using the professional advanced diesel engine simulator AMEsim(LMS) platform in cosimulation with SIMULINK software. Keywords: Diesel engine, Air path, FTC, Adaptive Observer, Takagi-Sugeno models, LMI. 1. INTRODUCTION Diesel engines are recognized as the most common and preferred solutions for distributed power generating systems in numerous applications, such as automotive vehicles, ships, cranes, and electric power generators due to their low fuel consumption and durability. Until now, these advantages were counterbalanced by the high level of nitrogen oxides (NOx ) produced. One way to reduce the NOx formation during combustion is to use a post-treatment system. But for cost reasons, car makers prefer improving engine diagnosis and control to reduce such pollutant emission. Different Model-based diagnosis methods of automotive engines have been considered in earlier papers (see e.g. Gertler et al. (1995), Krishnaswami et al. (1995), Hsu et al. (1995), Kim et al. (1998)). However, the engines considered in these previous works were all gasoline-fuelled and did not include EGR (Exhaust Gas Recirculation) and VNT (Variable Nozzle Turbine). Both these components make the diagnosis problem significantly more difficult since the air flow through the EGRvalve, and also the exhaust side of the engine have to be taken into account. An interesting approach to model-based air-path faults detection for an engine which includes EGR and VNT can be found in Nyberg and Sutte (2004). In this work, 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 airpath of an automotive diesel engine has been developed in order to study the monotorability of the system in Djemili ⋆ This work was produced in the framework of SCODECE (Smart COntrol and Diagnosis for Economic and Clean Engine), a European territorial cooperation project part-funded by the European Regional Development Fund (ERDF) through the INTERREG IV A 2 Seas Programme, and the research department of the Region Nord Pas de Calais, France.
978-3-902823-09-0/12/$20.00 © 2012 IFAC
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et al. (2011a). Recently, an adaptive observer for intake leakage detection in diesel engine described by Takagi-Sugeno model was proposed in Djemili et al. (2011b). This approach allows to estimate simultaneously the system states and the leakage. Following this work, a Fault Tolerant control (FTC) strategy exploiting the estimation of the leakage, is proposed in this paper. Fault Tolerant Control aims at fulfilling the desired objectives in the presence of faults in components of such as sensors, actuators and/or system. Two FTC techniques are developed in the literature : the active and the passive techniques. The passive techniques can be viewed as a robust control laws where faults are considered as disturbances (see Chen et al. (1998)). On the other hand, the active fault tolerant control techniques consist on adapting the control law using the information provided by fault diagnosis algorithm (see Blanke et al. (2003)). This paper is organized as follows. The air path FTC problem is first formulated in section 2. Then, a mean value model of diesel engine is presented and transformed into a TS model in section 3. The FTC strategy of the diesel engine air path subject to an intake manifold leakage is presented in section 4. The proposed strategy is successfully evaluated using an advanced diesel engine professional simulator AMEsim(LMS) in section 5. Conclusion and future work are finally presented in the last section. 2. FTC PROBLEM FORMULATION The general problem of combustion control can be decomposed into three control issues : the air path control, the cylinder balancing control problem and the fuel path control issues. In this work, we are interested only in the air path control issues. 10.3182/20120829-3-MX-2028.00268
SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico
3. MODELING A model of the intake manifold subject to leakages based on a mean-value diesel engine model reported in Jankovic and Kolmanovsky (2000) and Kao and Moskwa (1995) is derived. This model will be used in the following to design an airpath observer that will provide the estimates of the system states and a variable that is directly related to the presence of the leakage. In this model, the heat exchangers (intercooler and the EGRcooler) are not included because the temperatures downstream the heat exchangers are measured by the sensors. For the sake of simplicity we do not take into account the turbocharger dynamics. Nomenclature used in the model is presented in Table 1. Figure 1. A schematic picture of an airpath system 3.1 Intake manifold subject to leakage modeling The diesel engine considered in this paper is a four-cylinder engine with a high-pressure Exhaust Gas Recirculation circuit (EGR) and a Variable Geometry Turbocharger (VGT). A schematic picture of an airpath system is shown in Fig.1. The air entering the engine is measured by an Air flow Meter. It then passes through the compressor, enters the intake manifold, and flows into the cylinders. The fuel is injected directly into the cylinders and burned, producing torque on the crank shaft. The hot exhaust gas is pumped out via the exhaust manifold. Part of the exhaust gas flows from the exhaust manifold through the turbine out of the engine and the other part is recirculated back into the intake manifold through the EGR valve. The turbine takes the energy from the exhaust gas to power the compressor. The scheme also shows the intercooler and the EGR-cooler that are used to reduce the intake manifold temperature. The engine is equipped with sensors measuring : the intake manifold temperature, the intake manifold pressure and the exhaust manifold pressure. It is also equipped with the AIR/Fuel Ratio sensor located downstream the turbine. It reflects the composition in the exhaust manifold. The control inputs to the engine are the injected fuel, the EGR-valve and the VGT. The speed engine is also considered to be an input. In the air path control, it is desirable to control the masses aspirated by the cylinder (Masp,air and Masp,bg ) with two air path actuators (EGR valve and the VGT ). In practical situation, the considered masses can not be measured. Equivalent variables can be considered. Controlling those two masses is equivalent to controlling the intake manifold pressure Pint (being a reflection of Masp,air + Masp,bg ) and the burned gas M rate Fint (representing to ratio Masp,airasp,bg +Masp,bg ). Set points are often chosen to to reduce the NOx emissions. In the engine life service, leakages can occur in the intake manifold, that will result in mismatches between the mixture in the cylinder and the reference mixture. These mismatches can have dramatic effects. It generates extra noises and pollution. In this context, the aim of this work is to design a new FTC strategy to control the EGR valve and the VGT in presence of such leakages. The proposed FTC approach should maintain the intake manifold pressure and the burned gas rate closed to the set points given by the torque controller and preserve stability conditions in the presence of the leakage. 601
Applying the first law of thermodynamics (energy conservation principle), by considering that the heat transfer to the surroundings is negligible, leads to the expression of the variation of the intake manifold pressure as a function of the aspirated flow Dasp , the manifold air flow Dair , the EGR flow Degr and the leakage mass flow Dleak . The manifold dynamics pressure Pint is described by the following equation:
γ RTint P˙int = (Dair + Degr − Dasp − Dleak ) (1) Vint where Vint is the manifold volume, R is the perfect gas constant relative to the air and Tint is the temperature in the intake manifold . The aspirated flow is given by ( ) Pint Pint Ne Dasp = ηvol Ne , Vcyl (2) Tint RTint 120 where Vcyl is the cylinders’ volume. ηvol is the volumetric efficiency which is experimentally ( derived ) and eventually defined int . Ne is the engine speed. through a look-up table ηvol Ne , PTint The leakage mass flow rate from the intake manifold is given by ( ) Pint Patm Dleak = θ √ σ (3) Pint RTint ( ) where θ is the leakage size. The function σ PPatm is given by int v u )2 ( ) γ +1 u ( P γ γ 2 γ P u down down t − γ − 1 P P up up ( ) γ ( ) ( ) Pdown σ = i f Pdown ≥ Pdown γ −1 Pup Pup v Pup u( γ +1 ) u γ −1 2 t otherwise, γ +1 (4) where the subscripts ”up” and ”down” stand for upstream and downstream values across a section θ . The dynamic of burned gas fraction in the intake manifold Fint is derived as
SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico
RTint F˙int = (Degr (Fexh − Fint ) − Dair Fint ) (5) PintVint The Air/Fuel Ratio sensor located downstream the turbine allows us to obtain a reflection of the composition in the exhaust manifold Fexh . The composition of the EGR Fegr is that of the exhaust manifold Fexh , i.e. Fegr = Fexh . 3.2 State space representation Let us set x1 = Pint , x2 = Fint , u1 = Dair , u2 = Degr and note : Ne αint = γ VRTintint and βint = RT1int Vcyl 120 . The state space nonlinear model of the intake, with constant leakage, is obtained from Eq.(1) and Eq.(5). x˙1 = − f1 (x1 ) x1 + αint u1 + αint u2 − f4 (x1 ) θ x˙2 = − f2 (x1 , u1 , u2 ) x2 + f3 (x1 ) u2 (6) θ˙ = 0 The considered output is y = x1 . The nonlinear functions are given by f1 (x1 ) = αint βint ηvol (x1 , Ne ) αint (u1 + u2 ) f2 (x1 , u1 , u2 ) = x1 αint Fexh f3 (x1 ) = x1 ( ) x1 Patm σ f4 (x1 ) = αint √ Pint RTint
∀i ∈ {1, 2, 3} , fi ≤ fi ≤ fi (7) The representation (6) is one of many possible reformulations of the engine model. These will be subsequently used to derive a TS model using the sector nonlinearity approach Tanaka and Wang (2001). Quantity Total mass in the i.m Air mass in the i.m Atmospheric pressure Pressure in the i.m Fraction of burned gas in the i.m Fraction of burned gas in the e.m. Temperature in the i.m Manifold air flow EGR flow Aspirated flow into the cylinders Engine speed Volume of the i.m Volume of the cylinders Ideal Gas constant Ratio of specific heats Volumetric efficiency Air/Fuel equivalence ratio
A TS model is a nonlinear model composed of linear models blended together with some nonlinear functions. The purpose is to obtain an exact representation of a nonlinear model in a compact set of the state variables. The way to derive one TS model from a nonlinear model is to use two nonlinear functions (called membership functions in the TS model) to describe each nonlinearity. This systematic method, called the sector nonlinearity approach (Tanaka and Wang (2001)), results in a TS model with r = 2nl linear models (rules), where nl is the number of nonlinearities of the model. The TS model may be expressed under the following state space form r x˙ (t) = ∑ µi (ξ (t)) (Ai x (t) + Bi u (t) + Ψi θ (t))
i=1
θ˙ (t) = 0 y (t) = Cx (t)
(8)
where x (t) ∈ ℜ2 is the state vector, u (t) ∈ ℜ2 is the control input vector, θ (t) is the the leakage size, y (t) ∈ ℜ is the output vector and ξ (t) is called the premise vector. Ai , Bi , ψi and C are matrices with appropriate dimensions. The nonlinear scalar functions µi (ξ (t)) are assumed to be positive and to verify the convex sum property. To obtain these functions, each bounded function f can be easily split using two nonlinear functions M1 (.) ≥ 0, M2 (.) ≥ 0 such that M1 (.) + M2 (.) = 1 (see, Tanaka and Wang (2001)) as follows
Model (6) is nonlinear, thus a nonlinear observer has to be designed. One way to take into account the nonlinearities of the model is to use a polytopic approach such as the Takagi and Sugeno’s modeling (Takagi and Sugeno (1985)). Since x1 ̸= 0, the functions f1 , f2 and f3 can be assumed to be bounded such that
Symb. Mint Mint,air Patm Pint Fint Fexh Tint Dair Degr Dasp Ne Vint Vcyl R γ ηvol λ
3.3 TS fuzzy modeling
Unit Kg Kg Pa Pa − − K Kg.s−1 Kg.s−1 Kg.s−1 rpm m3 m3( ) J. KgK −1 − − −
f (.) − f f − f (.) +f = f M1 (.) + f M2 (.) (9) f−f f−f This so-called sector nonlinearity approach can be repeated for each nonlinear function of the model. Note that the arguments of the nonlinear functions µi (.) are known and that solving the LMI problem is harder without this condition. f (.) = f
By considering the nonlinear model (6), the constant matrices of the considered model (8) are given below [ ] − f1 0 Ai={1,2,3,4} = 0 − f2 [ ] − f1 0 Ai={5,6,7,8} = 0 − f2 ] [ − f1 0 Ai={9,10,11,12} = 0 −f2 ] [ − f1 0 Ai={13,14,15,16} = 0 − f2 [ Bi={1,2,5,6,9,10,13,14} = [ Bi={3,4,7,8,11,12,15,16} =
αint αint 0 f3 αint αint 0 f3 [
Ψi={1,3,5,7,9,11,13,15} =
Table 1. Nomenclature. (i.m. refers to the intake manifold)
[
− f4 0
− f4 Ψi={2,4,6,8,10,12,14,16} = 0 602
] ]
] ]
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r ( ) x˙ f (t) = ∑ µi (ξ (t)) Ai x f (t) + Bi u f (t) + Ψi θ (t)
C=[1 0] 4. PROPOSED FTC STRATEGY BASED ON AN ADAPTIVE OBSERVER
i=1
4.1 The structure of the proposed FTC scheme The proposed scheme is illustrated in Figure 2. The method uses an adaptive observer to detect and estimate the leakage size in the intake manifold. The proposed FTC approach should maintain the intake manifold pressure and the burned gas rate closed to the set points given by the torque controller and preserve stability conditions in the presence of the leakage.
y f (t) = Cx f (t) θ˙ (t) = 0
(12)
The main objective of this paper is to propose a new control strategy which can be used to determine the control inputs u f (t) such that : • the closed-loop system (12) is stable, • x f (t) converges asymptotically to xre f (t) even in the T presence of the fault. xre f (t) = [ Pint,re f Fint,re f ] is the set points vector given by (10). The following control strategy is used : r ( ( ) ) u f (t) = ∑ µ j (ξ (t)) −S j θˆ (t) + G j x (t) − xˆ f (t) + u (t) j=1
(13) where θˆ is the fault estimate and u (t) is the nominal control input (no fault condition). The FTC design needs
Figure 2. The FTC strategy scheme for the airpath dynamic system By taking into account the gear-box configuration, the driver’s request (accelerator position) is turned into a torque control objective under the form of an Indicated Mean Effective Pressure (IMEP) set point. Then, the set points for the intake manifold pressure and the Burned Gas Ratio (BGR) are given by experimentally calibrated static maps on the (IMEPre f ,Ne ) operating range. The engine speed Ne is considered as an external input. The set points vector is defined as ( ) Pint,re f = f pressure ( IMEPre f ,)Ne (10) Fint,re f = fBGR IMEPre f , Ne
• to determine xˆ f (t) and θˆ , • to design G j in such a way that the closed-loop system, including the state and fault estimations, is stable. The nominal control input u (t) is an state feedback control law computed using a Parallel Distributed Compensation (PDC) scheme as described in Wang et al. (1996). To estimate simultaneously the system states xˆ f (t) and a variable that is directly related to the presence of leakage, namely θˆ , an adaptive observer for system (12) is used : r . ( ) xˆ f = ∑ µi (ξ ) Ai xˆ f (t) + Bi u f (t) + Ψi θˆ (t) + Ki (y f (t) −Cxˆ f (t)) .
i=1
(14)
r
θˆ (t) = ∑ µi (ξ ) (Li (y f (t) −Cxˆ f (t))) i=1
A multivariate control (using the EGR valve and the VGT as actuators ) of the masses has to be designed to reach the set points given by the torque controller. This FTC strategy as described below exploits the states and the leakage size given by the adaptive observer. The outputs of the FTC block : Degr,re f and Dair,re f , are used to determine the opening area of the EGR valve (Vegr ) and the VGT position (Vtur ). PID controllers are added to the structure in order to provide further accuracy and robustness. The goal of the second feedback is to control Degr and Dair toward the feasible references set points Degr,re f and Dair,re f obtained by the FTC controller. The main purpose of the first feedback is to give a feasible and continuous set points for the second feedback action (see Figure 2). 4.2 FTC strategy Let us consider the following reference model : r x˙ (t) = ∑ µi (ξ (t)) (Ai x (t) + Bi u (t)) i=1 y (t) = Cx (t)
(11)
The extended error system, containing the two error dynamics: x f (t) − xˆ f (t) and θ (t) − θˆ (t), can be expressed as (
x˙ f (t) − x˙ˆ f (t) θ˙ (t) − θ˙ˆ (t)
)
i=1
603
(
Ai − KiC −Ψi −LiC 0
)(
x f (t) − xˆ f (t) θ (t) − θˆ (t)
) (15)
The tracking error dynamic e (t) = x (t) − x f (t), is given by
A e (t) + Bi u (t) i −S j θˆ (t) r r e˙ (t) = ∑ ∑ µi (ξ (t)) µ j (ξ (t)) −Bi +G j (x (t) − xˆ f (t)) i=1 j=1 +u (t)
(16)
−Ψi θ (t)
Let us assume that S j satisfies the following equality Bi S j = Ψi . Thus ( ) r r (Ai −(Bi G j ) e (t) ) e˙ (t) = ∑ ∑ µi (ξ (t)) µ j (ξ (t)) (17) −Ψi θ (t) − θˆ (t) i=1 j=1 An extended error system e˜ (t), containing the tracking error e (t), the state estimation error x f (t) − xˆ f (t) and the fault estimation error f (t) − fˆ (t), can be expressed as r
Let us also consider a faulty system (intake manifold subject to leakage) described by the following equation :
r
= ∑ µi ξ (t)
r
e˙˜ (t) = ∑ ∑ µi (ξ (t)) µ j (ξ (t)) A˜ i j e˜ (t) i=1 j=1
(18)
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e (t) e˜ (t) = x f (t) − xˆ f (t) θ (t) − θˆ (t) A˜ i j =
Ai − Bi G j 0 −Ψi 0 Ai − KiC −Ψi 0 −LiC 0
) 1.9
The main result of this paper is given in the following Theorem Theorem 1. The tracking error e (t), the state estimation error x f (t) − xˆ f (t) and the fault estimation error f (t) − fˆ (t) converge asymptotically to zero, if there exists symmetric positive definite matrices X and P2 , gain matrices H1i and H2i such that the following LMI are verified : A X + XAT i
i
0 −ΨTi ∗ ∗
0 −Ψi −Bi G j ATi P2 −CT H1iT + P2 Ai − H1iC −CT H2iT − P2 Ψi 0 −ΨTi P2 − H2iC 0 0 ∗ ∗ −I ∗ ∗ ∗
i.m Pressure obtained by the nominal controller i.m Pressure obtained by the FTC strategy
1.85
X 0 0 0 −I
Intake Manifold Pressure [bar]
(
strategy increases the air mass flow Dair and decreases the EGR mass flow Degr using the EGR valve and the VGT as actuators, as shown respectively in Figure 5 and Figure 6.
1.8
1.75
1.7
1.65
1.6 0
1
2
3
4
5
6
7
8
9
Time [s]
Figure 4. AMEsim results : Intake manifold Pressure histories
60
90 88 86
The gain of the controller are G j and the gains of the observer are given by Li = H2i and Ki = P2−1 H1i .
Air mass flow [g/s]
with :
84 82 80 78 76 74
Proof : the proof is given in the appendix
72 1
2
3
4
5. MODEL IN THE LOOP : RESULTS The developed FTC strategy has been tested on a four-cylinder diesel engine model running on AMEsimr platform in cosimulation with SIMULINK.
5 Time [s]
6
7
8
9
Figure 5. AMEsim results : Air flow histories (Dair ) obtained by the proposed FTC strategy 5.2 5
EGR mass flow [g/s]
4.8 4.6 4.4 4.2 4 3.8 3.6 0
1
2
3
4
5 Time [s]
6
7
8
9
Figure 6. AMEsim results : EGR flow histories (Degr ) obtained by the proposed FTC strategy Figure 3. AMESim/SIMULINK co-simulation platform 5 4.5 4 Leakage size [mm2]
The simulation is performed considering engine running at 3000 rpm with the following fault scenario : { 0, t < 5s θ (t) = 5 mm2 , t ≥ 5s
3.5 3 2.5 2 1.5 1 0.5
Figures 4-6 show the results obtained by applying the proposed FTC strategy. As a result Figure 6 clearly shows that the faults can be estimated with a very high accuracy. Moreover, from Figure 4 it can be observed that the intake manifold pressure obtained using the FTC strategy is equal to the pressure obtained using the nominal low control before the occurrence of the leakage. On the case of nominal controller, we can see the pressure drop of 200 mBar due to the gas mass flow rate through the leakage. In the other hand, the proposed approach maintains the intake manifold pressure closed to the set points given by the torque controller in the presence of the leakage. The burned gas rate is not affected by the leakage. To maintain the pressure closed to the set points given by the torque controller in the presence of the leakage, our FTC 604
0
0
1
2
3
4
5 Time [s]
6
7
8
9
10
Figure 7. Leakage size estimate 6. CONCLUSION In this paper, we proposed an FTC strategy of the diesel engine airpath subject to intake manifold leakage described by TakagiSugeno’s model. The proposed approach is an integrated FTC design procedure of the leakage identification and fault tolerant control scheme. Leakage identification is based on the use of an adaptive observer, while the FTC controller is implemented as an adaptive state feedback controller. The effectiveness and performances of the proposed approach have been illustrated using a professional advanced diesel engine simulator
SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico
AMEsim(LMS) platform in co-simulation with SIMULINK software. Due to its implementation simplicity, this strategy has the potential to be used on real-time system control which would be the subject of our future work. REFERENCES Blanke, M., Kinnaert, M., Lunze, J., and Staroswiecki, M. (2003). Diagnosis and Fault-Tolerant. Springer-Verlag Berlin Heidelberg. Chen, J., Patton, R., and Chen, Z. (1998). An lmi approach to approach to fault-tolerant control of uncertain systems. In In Proceeding of the IEEE Conference on Decision and Control, volume 1, 175–180. Djemili, I., Aitouche, A., and Cocquempot, V. (2011a). Structural analysis for air path of an automotive diesel engine. In IEEE-International Conference on Communications, Computing and Control Applications (CCCA’11). Hammamet, Tunisia. Djemili, I., Aitouche, A., and Cocquempot, V. (2011b). Adaptive observer for intake leakage detection in diesel engines described by takagi-sugeno model. In 19th Mediterranean Conference on Control & Automation (MED). Gertler, J., Costin, M., Fang, X., and Hira, R. (1995). Mode based diagnosis for automotive engines - algorithm development and testing on a production vehicle. IEEE Transactions on Control Systems Technology, 3(1), 61–69. Hsu, P., Lin, K., and Shen, L. (1995). Diagnosis of multiple sensors and actuartors failures in automotive engines. IEEE Transactions on Vehicular Technology, 44(4), 779–789. Jankovic, M. and Kolmanovsky, I. (2000). Constructive lyapounov control design for turbocharged diesel engines. IEEE Transactions on Control Systems Technology, 8, 288–299. Kao, K. and Moskwa, J. (1995). Turbocharged diesel engine modeling for nonlinear engine control and state estimation. ASME Journal of Dynamic Systems, Measurement and Control, 117(1), 20–30. Kim, Y., Rizzoni, G., and Utkin, V. (1998). Automotive engine diagnosis and control via nonlineare estimation. IEEE Control Systems, 18(5), 84–99. Krishnaswami, V., Luh, G., and Rizzoni, G. (1995). Nonlinear parity equation based residual generation for diagnosis of automotive engine faults. Control Engineering Practice, 10(10), 1385–1392. Nyberg, M. and Sutte, T. (2004). Model based diagnosis of the air path of an automotive diesel engine. Control Engineering Practice 12., 513–525. Takagi, T. and Sugeno, M. (1985). Fuzzy identification of systems and its applications to modeling and control. IEEE Transactions on Systems, Man and Cybernetics,, 15(1), 116– 132. Tanaka, K. and Wang, H.O. (2001). Fuzzy control systems design and analysis: A linear matrix inequality approach. In New York: Wiley, Wiley- Interscience, ISBN 0-471-32324190000. Wang, H., Tanaka, K., and Griffin, M. (1996). An approach to fuzzy control of nonlinear system : stability and design issues. IEEE Transaction on Fuzzy Systems, 4(1), 14–23. Appendix A. PROOF Lemma 2. Let us consider two matrices X and Y of appropriate dimensions. The following inequality is verified for each matrix Q : X T Y + XY T ≤ X T Q−1 X +Y QY T 605
The proof of the Theorem 1 is established using the following Lyapunov’s function : V (e˜ (t)) = e(t) ˜ T Pe˜ (t) , P = PT > 0
(A.1)
where the matrix P is defined as follows : ( ) P1 0 0 P = 0 P2 0 0 0 P3 The derivative of V (e˜ (t)) is written as
( ) r r ˜ T Mi j e˜ (t) V˙ (e˜ (t)) = ∑ ∑ µi (ξ (t)) µ j (ξ (t)) e(t)
(A.2)
i=1 j=1
where Mi j = Z
((
P1 Ai − P1 Bi G j 0 −P1 Ψi 0 P2 Ai − P2 KiC −P2 Ψi 0 −P3 LiC 0
))
Z (X) denotes the Hermitian of the matrix X, i.e., Z (X) = X T + X The derivative of the Lyapunov function is semi negative definite if the following inequality is satisfied Mi j 6 0 i, j = 1, . . . , r (A.3) using the lemma of congruence, P1−1 0 0 P1−1 0 0 Mi j < 0 ⇔ 0 I 0 Mi j 0 I 0 6 0 (A.4) 0 0 I 0 0 I we obtain the following inequality 1 Ωi j 0 −Ψi 0 Ω2i j −CT LiT P3 − P2 Ψi 6 0 T T −Ψi −Ψi P2 − P3 LiC 0
(A.5)
where Ω1i j = XATi − XGTj BTi + Ai X − Bi G j X and Ω2i j = ATi P2 − CT KiT P2 + P2 Ai − P2 KiC with X = P1−1 . In order to use the lemma 2 , the inequality A.5 can be written as
Ai X + XATi 0 −Ψi 0 ATi P2 −CT KiT P2 + P2 Ai − P2 KiC −CT LiT P3 − P2 Ψi + −ΨTi −ΨTi P2 − P3 LiC 0 T T −Bi G j X X −Bi G j 0 0 + 0 0 6 0 0 0 0 0
(A.6)
Using the lemma 2, we obtain
Ai X + XATi 0 −Ψi 0 ATi P2 −CT KiT P2 + P2 Ai − P2 KiC −CT LiT P3 − P2 Ψi + −ΨTi −ΨTi P2 − P3 LiC 0 T T −Bi G j −Bi G j X X 0 Q−1 0 + 0 Q 0 6 0 0 0 0 0
(A.7)
where Q is a symmetric definite positive matrix. Using the Schur complement and changes of variable H1i = P2 Ki and H2i = P3 Li , we obtain the LMI of theorem 1 with Q = I. The matrix P3 does not appear in the LMI, so it is chosen as identity matrix. Therefore, Li = H2i .