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5th IFAC Conference on 5th IFACand Conference onControl, Simulation and Modeling Engine Powertrain 5th IFACand Conference onControl,20-22, Available at www.sciencedirect.com Engine Powertrain Simulation Modeling Changchun, China, September 2018 and online 5th IFACand Conference onControl,20-22, Engine Powertrain Simulation Changchun, China, September 2018 and Modeling Engine and Powertrain Control,20-22, Simulation Changchun, China, September 2018 and Modeling Changchun, China, September 20-22, 2018
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IFAC PapersOnLine 51-31 (2018) 429–434
Chattering-Free Sliding Mode Control for Diesel Engine Air Path System with Chattering-Free Sliding Mode Control for Diesel Engine Air Path System with Actuatorfor Faults Chattering-Free Sliding Mode Control Diesel Engine Air Path System with Actuatorfor Faults Chattering-Free Sliding Mode Control Diesel Engine Air Path System with Actuator Faults Jian Zhang*, Long Liu*, Faults Xuemin Li*, Wenhui Li* Actuator Jian Zhang*, Long Liu*, Xuemin Li*, Wenhui Li*
Jian Zhang*, Long Liu*, Xuemin Li*, Wenhui Li* * College ofJian Power and Energy Harbin Engineering Zhang*, LongEngineering, Liu*, Li*, Wenhui Li* University, Xuemin * College of Power and Energy Harbin Engineering University, 150001, Harbin, China,Engineering, ( e-mail: [email protected], * College of Power and Energy Engineering, Harbin Engineering University, 150001, Harbin, China, ( e-mail: [email protected], [email protected], [email protected], [email protected] ) * College of Power and Energy Engineering, Harbin Engineering University, 150001, Harbin, China, ( e-mail: [email protected], [email protected], [email protected], [email protected] ) 150001, Harbin, China, ( e-mail: [email protected], [email protected], [email protected], [email protected] ) [email protected], [email protected], [email protected] ) Abstract: In this paper, a chattering-free sliding mode control law is constructed for the diesel engine air Abstract: this paper, a chattering-free control law is constructed for fault. the diesel engine air path systemInwith consideration of partial sliding loss of mode actuator effectiveness and additive Firstly, sliding Abstract: Inwith this paper, a chattering-free sliding mode control law is constructed for fault. the diesel engine air path system consideration of partial loss of actuator effectiveness and additive Firstly, sliding mode method andpaper, adaptive technique are employed to develop fault-tolerant With the Abstract: Inwith this a chattering-free sliding mode control law isa constructed for controller. the diesel engine air path system consideration of partial loss of actuator effectiveness and additive fault. Firstly, sliding mode method and law, adaptive are employed develop a fault-tolerant controller. With the adaptation update there technique isofno requirement of thetopriori knowledge of the upper bounds of the path system with consideration partial loss of actuator effectiveness and additive fault. Firstly, sliding mode method and law, adaptive are employed develop a fault-tolerant controller. With the adaptation update there technique is no of thetoand priori knowledge ofperformance, the upper bounds of the actuator faults. Then, to attenuate the requirement chattering behavior improve system the boundary mode method and adaptive technique are employed topriori develop a fault-tolerant controller. With adaptation update law, there is no requirement of the knowledge of the upper bounds of the actuator faults. Then, attenuate the chattering behavior and improve system performance, the boundary layer is introduced to to modify mode control method. Rigorous theoretical analysis is presented adaptation update law, there the is sliding no of the priori knowledge ofperformance, the upper bounds of the actuator faults. Then, to attenuate the requirement chattering behavior and improve system the boundary layer isonintroduced tostability modify the sliding mode control method. Rigorous theoretical analysis is to presented based Lyapunov theory, which demonstrates that the system trajectories converge a small actuator faults. Then, to attenuate the chattering behavior and improve system performance, the boundary layer is introduced to modify the sliding mode control method. Rigorous theoretical analysis is presented based on Lyapunov stability theory,Finally, which demonstrates that the system trajectories converge a small neighborhood around the origin. numerical simulation resultstheoretical are carried out tois to show the layer tostability modify the sliding mode control method. Rigorous analysis presented basedisonintroduced Lyapunov theory, which demonstrates that the system trajectories converge to a small neighborhood around the origin. Finally, numerical simulation results are carried out to show the effectiveness and validness of the proposed algorithms. based on Lyapunov stability theory,Finally, which demonstrates that the system a small neighborhood around the origin. numerical simulation resultstrajectories are carriedconverge out to to show the effectiveness and validness of the proposed algorithms. neighborhood around the origin. Finally, numerical simulation results are Ltd. carried out toreserved. show the effectiveness and validness of the proposed algorithms. © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier All rights Keywords: Control design, Diesel engine air path, Fault-tolerant, Adaptive control, Sliding mode effectiveness and validness of the proposed Keywords: Control design, Diesel engine airalgorithms. path, Fault-tolerant, Adaptive control, Sliding mode Keywords: Control design, Diesel engine air path, Fault-tolerant, Adaptive control, Sliding mode Keywords: Control design, Diesel engine air path, Fault-tolerant, Adaptivefault, control, mode uncertain additive etc. Sliding The unpredictable actuator faults 1. INTRODUCTION uncertain additive fault, etc. The unpredictable actuator have negative impact on system trajectories, and faults even 1. INTRODUCTION uncertain additive fault, etc. The unpredictable actuator have negative impact on system trajectories, and faults even 1. INTRODUCTION threaten the stability of the closed-loop system. In Nitsche et additive fault, etc. The unpredictable actuator During the past decades, diesel engines have become the uncertain have negative impact on closed-loop system trajectories, and faults even 1. INTRODUCTION threaten the stability of the system. In Nitsche et During the past decades, engines ofhave become the have al. (2010) the fault-tolerant control structure is established for negative impact on system trajectories, and even most common used power diesel plant because its high thermal threaten the stability of the control closed-loop system. In Nitschefor et During the past decades, diesel engines have become the al. (2010) the fault-tolerant structure is established most common used powerperformance. plant becauseInoforder its high the dieselthe engine air path with a system. jammedInEGR valve. efficiency reliable to thermal reduce stability of thesystem closed-loop Nitsche et During the and past decades, diesel engines ofhave become the threaten al. (2010) the fault-tolerant control structure is established for most common used power plant because its high thermal diesel engine path system with observer a jammedis EGR valve. efficiency and pollution, reliable performance. In order oftonitrogen reduce the In Hamouda etfault-tolerant al.air (2015), a control T-S fuzzy designed to environmental the emission problem al. (2010) the structure is established for most common powerperformance. plant becauseInoforder its high thermal diesel engine air path system with observer a jammedis EGR valve. efficiency and used reliable tonitrogen reduce the In Hamouda et al.states (2015), a T-S fuzzy designed to environmental pollution, the matter emission problem of andsystem faults, anda fault-tolerant control the dieselsystem engine air path with jammedis EGR valve. particulate of the engines has estimate oxides ( NOand efficiency reliable performance. In diesel order of tonitrogen reduce In Hamouda et al. (2015), a T-S fuzzy observer designed to x ) and environmental pollution, the emission problem estimate system by states and faults, and fault-tolerant control law is proposed using model predictive control approach. NO ) and particulate matter of the diesel engines has oxides ( In Hamouda et al.states (2015), a T-S fuzzy is designed to system and faults, andobserver fault-tolerant control environmental pollution, the matter emission problem of nitrogen gained ( increasing interest. Equipped with Exhaust Gas NOxxx ) and particulate of the diesel engines has estimate oxides law is proposed by using model predictive control approach. Sliding mode control isand amodel preferable approach toapproach. generate estimate system states faults, and fault-tolerant control gained increasing interest. Equipped with Exhaust Gas law is proposed by using predictive control particulate diesel engines has oxides ( NOx ) and Recirculation (EGR) valvematter andof the Variable Geometry mode control control algorithms is a preferable approach to generate fault-tolerant for the air path system, gained increasing interest. Equipped with Exhaust Gas Sliding law is proposed by using predictive control Recirculation (EGR) valve and to Variable Geometry Sliding mode control control is amodel preferable approach toapproach. generate Turbocharger (VGT), it is possible improve combustion fault-tolerant algorithms for the air path system, gained increasing interest. Equipped with Exhaust Gas however, chattering phenomenon always appears when Recirculation (EGR) valve and Variable Geometry Sliding mode control control algorithms is a preferable approach to generate Turbocharger (VGT), it is of possible to improve fault-tolerant foralways the airappears path system, and emission performance theand diesel engines.combustion Therefore, chattering phenomenon when Recirculation (EGR) it is valve Variable Geometry however, Turbocharger (VGT), possible to improve combustion system trajectories crossing the sliding manifold. Hence, fault-tolerant control algorithms for the air path system, and emission performance of the diesel engines. Therefore, however, chattering phenomenon always appears when efforts have been put it onisactively adjusting the combustion combustion Turbocharger (VGT), possible to improve system trajectories crossing the sliding manifold. Hence, and emission performance of the diesel engines. Therefore, attempts have been made to solve this drawback, boundary however, chattering phenomenon always appears when efforts have beenKebairi put on etactively adjusting the(2015); combustion system trajectories crossing the sliding manifold. Hence, process, see e.g., al.the (2015); Liengines. et al. Jung attempts have been made solve this drawback, and emission performance of diesel Therefore, efforts have been put on etactively adjusting the(2015); combustion layer technique (Hung et al.to(1993)), dynamical gainboundary schedule system trajectories crossing the sliding manifold. Hence, process, see e.g., Kebairi al. (2015); Li et al. Jung have been made solve this drawback, boundary et al. (2016) and therein. adjusting the combustion attempts layer technique (Hung et al.to(1993)), dynamical schedule efforts have beenreferences put on etactively method (Plestan et al.made (2010)), and higher order gain sliding mode process, see and e.g., Kebairi al. (2015); Li et al. (2015); Jung attempts have been to(1993)), solve this drawback, boundary et al. (2016) references therein. layer technique (Hung et al. dynamical gain schedule method (Plestan et (2007)) al. (2010)), and higher In order sliding mode process, see and e.g.,references Kebairi ettherein. al. (2015); Li et al. (2015); Jung (Laghrouche et al. (2016) et al. are employed. Ali et al. (2015), layer technique (Hung et al. (1993)), dynamical gain schedule Recently, control methods for the air path system have been method (Plestan et (2007)) al. (2010)), and higher order mode et al. (2016) and references therein. (Laghrouche et sliding al. are controller employed. In Ali sliding et al. (2015), Recently, control methods for the air path system have been super-twisting mode is designed for the method (Plestan et (2007)) al. (2010)), and higher In order sliding mode extensively studied. In Utkin et al. (2000), a sliding mode (Laghrouche et al. are employed. Ali et al. (2015), Recently, control methods for the air(2000), path system havemode been super-twisting sliding mode by controller is chattering designed effect for the extensively studied. In Utkin et al. a sliding diesel engine air path control which the is control is control designed to regulate theairVGT, the EGR et sliding al. (2007)) are controller employed.isIn designed Ali et al. (2015), Recently, methods for the path whereas system have been (Laghrouche super-twisting mode for the extensively studied. In regulate Utkin etthe al. VGT, (2000), a sliding mode diesel engineand air path control by which thehandled chatteringaseffect is control is designed to whereas the EGR eliminated actuator faults are well. super-twisting mode by controller designed effect for the valve is supposed controlled open-loop. Furthermore, extensively studied.totoInberegulate Utkin etthe al. VGT, (2000), a sliding mode diesel engineand airsliding path control which theishandled chattering is control is designed whereas the EGR eliminated actuator faults are as well. valve is supposed be controlled open-loop. Nevertheless, thepath upper bound of thethe addictive faults has engineand air control by which chattering effect is for tuning exhaust to manifold pressure and whereas freshFurthermore, airflow rate diesel control issupposed designed toberegulate the VGT, the EGR eliminated actuator faults are handled as well. valve is to controlled open-loop. Furthermore, Nevertheless, the upper bound of thecontrol addictive has for tuning exhaust manifold pressure andcontrol fresh airflow rate been involved to determine the proper gain,faults suchwell. that eliminated and actuator faults are handled as simultaneously, constructive Lyapunov approach is valve is supposed to be controlled open-loop. Nevertheless, upper bound of thecontrol addictive has for tuning exhaust manifold pressure andcontrol freshFurthermore, airflow rate involved the tocapacity determine the be proper gain,faults such that simultaneously, constructive Lyapunov approach is been its fault-tolerant adopted in Jankovic et al. (2000), and sliding mode algorithm Nevertheless, the upper could bound ofassured. thecontrol addictive faults has for tuning exhaust manifold pressure andcontrol fresh airflow rate been involved to determine the proper gain, such that simultaneously, constructive Lyapunov approach is fault-tolerant capacity could be assured. adopted in Jankovic et al. (2000), and sliding mode algorithm its been involved to determine the proper control gain, such that is developed for controlling VGT-EGR system in Upadhyay its fault-tolerant capacity could be assured. simultaneously, constructive Lyapunov control approach is Motivated by the issues mentioned above, the fault-tolerant adopted in Jankovic et al. (2000), and sliding mode algorithm is developed forAimed controlling VGT-EGR system in Upadhyay fault-tolerant could be assured. robust its et al. (2002). lowering adopted in Jankovic et al.at(2000), and NO sliding mode algorithm Motivated by thecapacity issues above, fault-tolerant x emission, control problem the mentioned diesel engine airthe path system is is developed for controlling VGT-EGR system in Upadhyay NOsystem robust Motivated et al. (2002).forAimed at lowering by the for issues mentioned above, the fault-tolerant x emission, control problem for the diesel engine air path systemlaw is is developed controlling VGT-EGR in Upadhyay sliding mode controllers designed NO in Wang (2008), which investigated in this paper. Adaptive sliding mode control emission, robust et al. (2002). Aimed atarelowering x Motivated by the for issues above, the fault-tolerant x control problem the mentioned diesel engine airmode path systemlaw is sliding mode controllers are designed in Wang (2008), which investigated in this paper. Adaptive sliding control NOx emission, robust is et al. (2002). at between loweringconventional facilitate smoothAimed switching combustion constructed tofor handle the engine partial airloss of system actuator control problem the diesel path is sliding mode controllers are designed in Wang (2008), which investigated in this paper. Adaptive sliding mode control law facilitate smooth switching between conventional combustion is constructedfaults to handle the partial actuator mode LTC mode by regulating intake charge and Adaptive additive faults.loss Theof auxiliary sliding and mode controllers arebetween designedconventional inengine Wang (2008), which effectiveness facilitate smooth switching combustion investigated in this paper. sliding mode control law is constructed to handle the partial loss of actuator mode and LTC mode regulating engine intake charge faults toand faults.factors The auxiliary amount and EGRswitching rate. by Xie et al. conventional (2016) proposes a new effectiveness parameters related the additive effectiveness and the facilitate smooth between combustion mode and by engine intake charge is constructed to handle the partial actuator effectiveness faults and additive faults.loss Theof auxiliary amount andLTC EGRmode rate. Xieregulating et al. (2016) proposes a new parameters related to the effectiveness factors and thus the decoupling method to regulate the VGT-EGR system by additive faults faults are estimated by adaptation update laws, mode and LTC mode by regulating engine intake charge amount and EGR rate. Xie et al. (2016) proposes a new effectiveness and additive faults. The auxiliary related to the by effectiveness factors laws, and thus the decoupling method disturbance to regulate rejection the VGT-EGR system by parameters additive faults are estimated adaptation update employing active control (ADRC), requirement of upper of the actuator faults is amount and method EGR rate. Xie et al. proposes a new parameters related to the bound effectiveness factors laws, and thus the decoupling to regulate the(2016) VGT-EGR system by the additive faults are estimated by adaptation update employing active disturbance rejection control (ADRC), the requirement of upper boundlayer of the actuator faults to is which is easymethod to implement and could tackle a wide range by of removed. In addition, boundary method is adopted decoupling to regulate the VGT-EGR system additive faults are estimated by adaptation update laws, thus employing active disturbance rejection control (ADRC), the requirement of upper bound of the actuator faults is which is easywithout to implement andthe could tackle a wide range of removed. In addition, boundary layer method is adopted to uncertainties returning control gains. solve the chattering phenomenon of the adaptive sliding employing active disturbance rejection control (ADRC), the requirement of upper bound of the actuator faults is which is easy to implement and could tackle a wide range of removed. In addition, boundary layer method is adopted to uncertainties without returning the control gains. solve control the chattering phenomenon of the adaptive sliding law. Rigorous theoretical analysis shows that which is easywithout to implement andthe could tackle a wide range of mode removed. In addition, boundary layer method is adopted to uncertainties returning control gains. solve the chattering phenomenon of the adaptive sliding In practical, the actuators of air path system would suffer mode control law. Rigorous theoretical analysis shows that system are driven into a small of the uncertainties without returning the control gains.would suffer the solve the states chattering phenomenon of neighbourhood the adaptive sliding In practical, actuators of including air path system mode control law. theoretical analysis shows from several the kinds of faults, loss-of-effectiveness, the system states areRigorous driven into a small neighbourhood ofthat the In practical, the actuators of air path system would suffer control law. Rigorous theoretical analysis showsofthat from several kinds of faults, including loss-of-effectiveness, mode the system states are driven into a small neighbourhood the In practical, actuators of including air path system would suffer from several the kinds of faults, loss-of-effectiveness, the system states are driven into a small neighbourhood of the Copyright 2018 IFAC 468Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2018, IFAC Federation of Automatic Control) from several kinds of(International faults, including loss-of-effectiveness, Copyright 2018 responsibility IFAC 468Control. Peer review©under of International Federation of Automatic Copyright © 2018 IFAC 468 10.1016/j.ifacol.2018.10.096 Copyright © 2018 IFAC 468
IFAC E-CoSM 2018 430 Changchun, China, September 20-22, 2018 Jian Zhang et al. / IFAC PapersOnLine 51-31 (2018) 429–434
equilibrium. Thus the dynamic performance and reliability of the air path system can be improved. The rest of this paper is organized as follows: the referred nonlinear diesel engine air path system with actuator faults and the control objective are described in Section 2. In Section 3, two fault-tolerant control laws are established by using adaptive technique and sliding mode control. Then, numerical simulation results and analysis are carried out in the followed section. Finally, conclusions and remarks are given in Section 5.
2.1 Mathematical model of diesel engine air path system
-
c t
Compressor isentropic efficiency
-
V1
Turbine isentropic efficiency Turbocharger mechanical efficiency Intake manifold volume
m3
V2
Exhaust manifold volume
m3
Vd
Engine volume cylinder
m3
Ta
Ambient temperature
T1
Intake manifold temperature
K K
T2
Exhaust manifold temperature
K
Specific gas constant
J Kg K
Ra
p1 k1 Wc Wegr ke p1 p2 k2 ke p1 Wegr Wt Wf
Pc
1
(1)
(2)
m Pt Pc
(3)
where the compressor mass flow Wc and turbine power Pt are given as k Wc Pc c (4) p1 1
Pt kt 1 p2 Wt
(5)
with the parameters calculated as kc c c pTa , kt c pt T2 ,
Variable Geometry Turbine
k1 RaT1 V1 , ke v NVd RaT1 and k2 RaT2 V2 .
Exhaust gas
To facilitate the control law design, the model (1)-(3) is rewritten into vector form
Exhaust manifold (m2 , p2 ,T2 )
x f x g1 x u1 g2 x u2
EGR cooler
where x p1
Cylinders
Intercooler
-
Considering the third-order model for the diesel engine air path system (Jankovic et al. (2000)):
A schematic representation of the diesel engine air path system is given in Fig. 1, and the corresponding variable are illustrated in Table 1. It can be observed that the turbocharger consists of a variable geometry turbine (VGT) and a compressor, which are connected by a shaft. As the VGT generating the energy of the exhaust gas in the exhaust manifold, the compressor is driven by the turbine via their common shaft. As a result, the mass of air provided to the engine cylinders are increased, so that higher power is achieved than non-turbocharged engine. A portion of the exhaust gas is delivered into the intake manifold through the exhaust gas recirculation valve (EGR). The recirculated exhaust gas lowers the combustion temperature and declines the formation of NOx .
Fresh air
Engine volumetric efficiency
m
2. PROBLEM FORMULATION
Compressor(Pc ,Wc )
v
p2
(6)
Pc represents the system states. The T
control variables are u1 Wegr , u2 Wt , respectively. And
EGR valve
the definition of f x , g1 x and g 2 x in (6) are given as
Intake manifold (m1 , p1 ,T1 )
Pc k1kc p 1 k1ke p1 1 f x k 2 ke p1 W f , Pc
Fig. 1. Schematic of a turbocharged diesel engine with EGR and VGT. Table 1. Nomenclature of the diesel engine variables Variables p1
Name Intake manifold pressure
Units Pa
p2
Exhaust manifold pressure
Pc
Compressor power
Pt
Turbine power
Pa W W
Wc
Compressor mass flow
Kg s
Wt
Turbine mass flow
Kg s
Wf
Fueling mass flow rate
Kg s
0 k1 g1 x k 2 , g2 x k2 0 K 1 p o 2
.
m kt . The specific values of the system parameters k1 , k 2 , k t , k e , k c , m , and mentioned where K o
above will be specified later. 469
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2.2 Air path system with actuator faults
431
Taking the time-derivative of s along (7) , thus it has
s f * x g * x E* t u F * t
The actuator faults of the air path system are taken into account, as a result the original model (6) is modified as x f x g1 x E1 t u1 F1 t g2 x E2 t u2 F2 t
with u u1 u2 being defined as
T
(7)
and f * x
(9)
, g * x , E* t , F * t
0 k1Wc k1ke p1 p1d k where E1 t and E2 t denote the effectiveness factor of the g* x 1 f * x , , k2 k2 k2 ke p1 k2W f p2 d actuators with 0 Ei t 1 , i 1, 2 . Note that Ei t 1 k1 F1 0 E indicates that the i th actuator works normally, and * E , F * t t 1 . 0 E k F F 0 Ei t 1 represents that the i th actuator partially loses 2 2 2 1
In the following context, E * t and F * t will be denoted
its effectiveness, i th actuator has failed completely if Ei t 0 . F1 t and F2 t denote the uncertain additive faults acting on each actuator, which are unknown but bounded.
as E * and F * for simplification. . Then, an adaptive sliding mode control law is proposed
uasmc g* x f * x F * hs sgn s
(10)
f * x F*
(12)
1
2.3 Control objectives To achieve the control requirements of the air path system, the compressor mass flow Wc and exhaust manifold pressure p2 should be adjusted to their desired set points. Nevertheless, the computation for the compressor mass flow is time consuming, because of the time derivative of Wc (Ali et al. (2012)). Hence, the pressure manifold set point p1d is
where h denotes the control gain, is a small positive constant. Denoting an auxiliary variable 1 1 , where
E
max *
with E I E * . Thus, F * and are used to
estimate F and , respectively. The adaptation update laws are given as
adopted instead of Wc to develop the control law. In addition, it is straightforward to obtain p1d :
F* s
1
k P (8) p1d c c 1 Wcd The tracking errors respect to the set points are defined as y1 p1 p1d and y2 p2 p2d , respectively. Thus, the control objective is to generate fault-tolerant control laws which steer the tracking errors to the equilibrium of the closed-loop system. As a result, the desired compressor mass flow Wc and exhaust manifold pressure p2 can be accurately tracked in the presence of actuator faults.
F * F * F * and are defined as the estimation errors of the adaptive parameters. Theorem 1. For the diesel engine air path system generated by (6). If the control law are designed as (10)-(12) with the adaptation law (13)-(14), the closed-loop system will achieve asymptotically stability.
Proof. Consider the following candidate Lyapunov function which is positive definite and proper:
1 1 *T * 1 2 (15) V1 sT s F F 2 2 2 Taking the time-derivative of (15) along system (9), it yields
In this section, two fault-tolerant control laws are designed for the air path system. Adaptive technique and sliding mode control are employed to solve the actuator faults without using any piror knowledge of the faults. However, chattering phenomenon always exits when the sliding mode control applied. Thus, the boundary layer method is introduced to modify the adaptive sliding mode control law and handle the chattering problem.
V1 s T g * x I E uasmc f * x F *
1
F *T F *
1
(16)
Substituting control law (11)-(14) into (16), it further has 3.1 Adaptive sliding mode fault-tolerant controller
V1 hs T s s 1 s s f * x F *
To facilitate the control law design, a sliding manifold is defined as s p pd , with s s1 and p p1d
s2 , p p1
(13)
(14) s where and are positive constants to be designed. Then,
3. MAIN RESULTS
T
(11)
p2
T
h s s 1 s 2
p2d . T
s 470
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controller (20) can be refer to the proof of Theorem 1, thus it is omitted here.
h s s 1 s 2
1 s
Case 2:
(17)
If s , the controller is rewritten as
2
h s s
s 1 (21) ucfasmc g * x f * x F * hs 2 Given the definition of h , , and the adaptation update
Clearly, V1 0 , and given the definition of V1 , it implies that
s L , F * L , L . Integrating (17), it shows that s L2
L . Also, it is easy to verify that s L , which
law for F , , and taking the time derivative of V 2 , it has
further leads to lim s 0 . Hence, p1 and p2 will track their t
desired set points asymptotically.
2
V2 h s
Remark 1. In the above analysis, is considered
2
2
s
1 s
s s f * x F * 2 as a positive constant, which requires that 1 always holds. In order to guarantee such condition, the initial value 2 s 2 of t is set as t0 1 . Thus, given the adaptation law h s 2 1 s
(22)
s and the definition of , it is reasonable to conclude that 1 1 1 .
s s 2
3.2 Chattering-free fault-tolerant controller
h s s
2
2
It is obvious that V2 0 if any of the following inequality holds
As it is known that chattering phenomenon would occur when system trajectories cross the sliding manifold, which deteriorates dynamic performance of the closed-loop system. Aimed at handling this problem, the boundary layer method is employed to redesign the control scheme.
1
s
Considering the modified fault-tolerant control law:
s
g * x 1 f * x F * hs sgn s , s ucfasmc 1 * * * 2 s , s g x f x F hs (18) with denoting the boundary layer, and the definition of ,
4h 1 4
s
(23) (24) (25)
It implies that the state trajectories converge to a small set containing the equilibrium of the closed-loop system defined as 1 1 s t lim s t s t t 4h 4 (26) s t Given the two cases above, the conclusion can be drawn that the trajectories of the closed-loop system are uniformly ultimately bounded (UUB). This completes the proof.
, F , are as same as (11)-(14), respectively. Theorem 2. For the diesel engine air path system generated by (6). If the control law (18) with the corresponding variable definition and adaptation update laws (13)-(14). Then, the trajectories of the closed-loop system are finally converge to a small neighbourhood around the equilibrium. Proof. Construct the Lyapunov candidate function:
1 1 *T * 1 2 V2 sT s F F 2 2 2
1 s 1 2 4
(19)
Remark 2. From inequality (26), it can be observed that the value of the boundary layer effects on the accuracy of the tracking error, the smaller value of leads to the smaller set of s . Moreover, by choosing larger control gain h , the tracking performance would be more precise.
Case 1: If s , the control law yields
ucfasmc g* x f * x F * hs sgn s (20) which possesses the same form as controller (11). Therefore, the stability analysis of the closed-loop system under 1
4. SIMULATION AND ANALYSIS To illustrate the effectiveness of the proposed control laws, numerical simulation are carried out in this section. According to (Larsen et al. (2000)), the parameters of system 471
IFAC E-CoSM 2018 Changchun, China, September 20-22, 2018 Jian Zhang et al. / IFAC PapersOnLine 51-31 (2018) 429–434
(1)-(5) are selected as k1 143.91 , k1 1715.5 , kt 391.365 , ke 0.028 , kc 0.0025 , m 0.95 , 0.285 , 0.15 . The initial value are set as p1 1.32 , p2 1.35 , Pc 5.605 The control requirements for the system states are addressed in Table 2.
2-4, the chattering problem has been solved by CFAFMC. Moreover, define the tracking errors as e1 Wcd Wc and e2 p2 d p2 . At around 30 seconds, the tracking errors converge to zero with the accuracies of e1 0.002 and
e2 0.05 under CFAFMC, while the accuracies turn out to be e1 0.005 and
Table 2. Control requirements for the air path system Variables
Set point 1
Set point 2
Wc Kg s
0.01
0.07
p2 Bar
1.15
1.55
Wf Kg h
3
7
433
e2 0.2 when AFMC adopted. Hence,
it can be concluded that tracking performance is further improved as chattering behaviour is attenuated.
In the numerical simulation, the adaptive sliding mode faulttolerant controller (10)-(14) is denoted as ASMC, and the modified fault-tolerant controller (18) with the corresponding adaptation update laws is denoted as CFASMC. The control parameters are selected as h 5 , 0.005 ， 0.005 ， 0.0001 , 0.01 . Besides, the additive fault is set as
F t 103 4sin 0.1t 2cos 0.1t and the effectiveness T
factor is chosen to be Fig. 3. Response curves of p2 under ASMC.
if t 25 I 22 E t 0.85 0.05sin 0.2 t I 22 if t 25 Firstly, the ASMC is employed. The time histories of Wc and p2 are given in Figs. 2 and 3, respectively. It can be seen that the errors between the set points and the corresponding actual states are considerable small. Hence, the proposed control law is capable of handling the actuator faults while guarantee accurate tracking of Wcd and p2d . The response
curves of Wegr and Wt are presented in Fig. 4. Obviously, high frequency chattering behaviour impacts on both the system states and the control input due to the discontinuous term in the control law. Fig. 4. Response curves of Wegr and Wt under ASMC.
Fig. 2. Response curves of Wc under ASMC. Then, the response curves of Wc , p2 , Wegr and Wt under
Fig. 5. Response curves of Wc under CFASMC.
CFAFMC are carried out in Figs. 5-7. Compared with Figs. 472
IFAC E-CoSM 2018 434 Changchun, China, September 20-22, 2018 Jian Zhang et al. / IFAC PapersOnLine 51-31 (2018) 429–434
Fig. 6. Response curves of p2 under CFASMC.
Fig. 7. Response curves of Wegr and Wt under CFASMC. 5. CONCLUSIONS In this paper, two fault-tolerant control schemes for the air path system of the tubocharged diesel engine are designed. Initially, an adaptive sliding mode control law is constructed, by which the requirement of the piror knowledge of the actuator faults is removed. Then, the boundary layer method is adopted to handling the chattering phenomenon by modifying the discontinuous control term. Lyapunov stability theory is used to analyze the state trajectories of the closedloop system, and numerical results show the effectiveness and robustness of the proposed control laws. Our future work will focus on developing fault estimation method for the actuators of the diesel engine air path system to further upgrade the performance of the fault-tolerant control law. REFERENCES Ali, S.A. and Langlois, N. (2012). A robust sliding mode control strategy for turbocharged diesel engine air path, The Proceeding of the 13th IFAC Symposium on Control in Transportation Systems, Sofia, Bulgaria, 262-267. Ali, S.A., Guermouche, M., and Langlois, N. (2015). Faulttolerant control based super-twisting algorithm for the diesel engine air path subject to loss-of-effectiveness and additive actuator faults, Applied Mathematical Modelling, 39(15), 4309-4329. 473
Hamouda, L.B., Ayadi, M., and Langlois, N. (2015). Fuzzy fault tolerant predictive control for a diesel engine air path, International Journal of Control, Automation and Systems, 14(2), 443-451. Hung, J.Y., Gao, W., and Hung, J.C. (1993). Variable structure control: a survey. IEEE Transactions on Industrial Electronics, 40(1), 2-22. Jankovic, M., Jankovic, M., and Kolmanovsky, I. (2000). Constructive Lyapunov control design for turbocharged diesel engines, IEEE Transaction on Control Systems Technology, 8(2), 288-299. Jung, H. and Choi, S. (2016). Model based burnt gas fraction controller designed of diesel engine with VGT/dual loop EGR system, International Journal of Automotive Technology, 17(4), 555-566. Kebairi, A., Becherif, M., and Bagdouri, M.E. (2015). Modelling, simulation and identification of an engine air path electromechanical actuator, Control Engineering Practice, 34, 88-97. Laghrouche, S., Plestan, F., and Glumineau, A. (2007). Higher order sliding mode control based on integral sliding mode, Automatica, 43(3), 531-537. Larsen, M., Jankovic, M., and Kokotovic, P.V. (2000). Indirect passivation design for a diesel engine model, The Proceedings of the 2000 IEEE International Conference on Control Applications, Anchorage, USA, 309-314. Li, Y., Zhou, X., Hu, Y.F., and Chen, H. (2015). Air path system control of turbocharged gasoline engine based on fuzzy PID, The Proceeding of the 27th Chinese Control and Decision Conference, 941-946. Nitsche, R., Bitzer, M., Khaldi, M.E., and Bloch, G. (2010). Fault tolerant oxygen control of a diesel engine air system, The Proceeding of the 6th IFAC Symposium Advances in Automotive Control, 578-583. Plestan, F., Shtessel, Y., Bregeault, V., and Poznyak, A. (2010). New methodologies for adaptive sliding mode control. International Journal of Control, 83(9), 19071919. Upadhyay, D., Utkin, V.I., and Rizzoni, G. (2002). Multivariable control design for intake flow regulation of a diesel engine using sliding mode, IFAC Proceedings Volumes, 35(1), 277-282. Utkin, V.I., Chang, H.C., Kolmanovsky, I., and Cook, J.A. (2000). Sliding mode control for variable geometry turbocharged diesel engines, The Proceeding of the American Control Conference, Chicago, USA, 584-588. Wang, J.M. (2008). Hybrid robust air-path control for diesel engines operating conventional and low temperature, IEEE Transaction on Control Systems Technology, 16(6), 1138-1151. Xie, H., Song, K., Tatsumi, J., Zheng, Q.L., Zhang, H., and Gao, Z.Q. (2016). On decoupling control of the VGTEGR system in diesel engine: a new framework, IEEE Transactions on Control Systems Technology, 24(5), 1788-1796.
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IFAC PapersOnLine 51-31 (2018) 429–434
Chattering-Free Sliding Mode Control for Diesel Engine Air Path System with Chattering-Free Sliding Mode Control for Diesel Engine Air Path System with Actuatorfor Faults Chattering-Free Sliding Mode Control Diesel Engine Air Path System with Actuatorfor Faults Chattering-Free Sliding Mode Control Diesel Engine Air Path System with Actuator Faults Jian Zhang*, Long Liu*, Faults Xuemin Li*, Wenhui Li* Actuator Jian Zhang*, Long Liu*, Xuemin Li*, Wenhui Li*
Jian Zhang*, Long Liu*, Xuemin Li*, Wenhui Li* * College ofJian Power and Energy Harbin Engineering Zhang*, LongEngineering, Liu*, Li*, Wenhui Li* University, Xuemin * College of Power and Energy Harbin Engineering University, 150001, Harbin, China,Engineering, ( e-mail: [email protected], * College of Power and Energy Engineering, Harbin Engineering University, 150001, Harbin, China, ( e-mail: [email protected], [email protected], [email protected], [email protected] ) * College of Power and Energy Engineering, Harbin Engineering University, 150001, Harbin, China, ( e-mail: [email protected], [email protected], [email protected], [email protected] ) 150001, Harbin, China, ( e-mail: [email protected], [email protected], [email protected], [email protected] ) [email protected], [email protected], [email protected] ) Abstract: In this paper, a chattering-free sliding mode control law is constructed for the diesel engine air Abstract: this paper, a chattering-free control law is constructed for fault. the diesel engine air path systemInwith consideration of partial sliding loss of mode actuator effectiveness and additive Firstly, sliding Abstract: Inwith this paper, a chattering-free sliding mode control law is constructed for fault. the diesel engine air path system consideration of partial loss of actuator effectiveness and additive Firstly, sliding mode method andpaper, adaptive technique are employed to develop fault-tolerant With the Abstract: Inwith this a chattering-free sliding mode control law isa constructed for controller. the diesel engine air path system consideration of partial loss of actuator effectiveness and additive fault. Firstly, sliding mode method and law, adaptive are employed develop a fault-tolerant controller. With the adaptation update there technique isofno requirement of thetopriori knowledge of the upper bounds of the path system with consideration partial loss of actuator effectiveness and additive fault. Firstly, sliding mode method and law, adaptive are employed develop a fault-tolerant controller. With the adaptation update there technique is no of thetoand priori knowledge ofperformance, the upper bounds of the actuator faults. Then, to attenuate the requirement chattering behavior improve system the boundary mode method and adaptive technique are employed topriori develop a fault-tolerant controller. With adaptation update law, there is no requirement of the knowledge of the upper bounds of the actuator faults. Then, attenuate the chattering behavior and improve system performance, the boundary layer is introduced to to modify mode control method. Rigorous theoretical analysis is presented adaptation update law, there the is sliding no of the priori knowledge ofperformance, the upper bounds of the actuator faults. Then, to attenuate the requirement chattering behavior and improve system the boundary layer isonintroduced tostability modify the sliding mode control method. Rigorous theoretical analysis is to presented based Lyapunov theory, which demonstrates that the system trajectories converge a small actuator faults. Then, to attenuate the chattering behavior and improve system performance, the boundary layer is introduced to modify the sliding mode control method. Rigorous theoretical analysis is presented based on Lyapunov stability theory,Finally, which demonstrates that the system trajectories converge a small neighborhood around the origin. numerical simulation resultstheoretical are carried out tois to show the layer tostability modify the sliding mode control method. Rigorous analysis presented basedisonintroduced Lyapunov theory, which demonstrates that the system trajectories converge to a small neighborhood around the origin. Finally, numerical simulation results are carried out to show the effectiveness and validness of the proposed algorithms. based on Lyapunov stability theory,Finally, which demonstrates that the system a small neighborhood around the origin. numerical simulation resultstrajectories are carriedconverge out to to show the effectiveness and validness of the proposed algorithms. neighborhood around the origin. Finally, numerical simulation results are Ltd. carried out toreserved. show the effectiveness and validness of the proposed algorithms. © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier All rights Keywords: Control design, Diesel engine air path, Fault-tolerant, Adaptive control, Sliding mode effectiveness and validness of the proposed Keywords: Control design, Diesel engine airalgorithms. path, Fault-tolerant, Adaptive control, Sliding mode Keywords: Control design, Diesel engine air path, Fault-tolerant, Adaptive control, Sliding mode Keywords: Control design, Diesel engine air path, Fault-tolerant, Adaptivefault, control, mode uncertain additive etc. Sliding The unpredictable actuator faults 1. INTRODUCTION uncertain additive fault, etc. The unpredictable actuator have negative impact on system trajectories, and faults even 1. INTRODUCTION uncertain additive fault, etc. The unpredictable actuator have negative impact on system trajectories, and faults even 1. INTRODUCTION threaten the stability of the closed-loop system. In Nitsche et additive fault, etc. The unpredictable actuator During the past decades, diesel engines have become the uncertain have negative impact on closed-loop system trajectories, and faults even 1. INTRODUCTION threaten the stability of the system. In Nitsche et During the past decades, engines ofhave become the have al. (2010) the fault-tolerant control structure is established for negative impact on system trajectories, and even most common used power diesel plant because its high thermal threaten the stability of the control closed-loop system. In Nitschefor et During the past decades, diesel engines have become the al. (2010) the fault-tolerant structure is established most common used powerperformance. plant becauseInoforder its high the dieselthe engine air path with a system. jammedInEGR valve. efficiency reliable to thermal reduce stability of thesystem closed-loop Nitsche et During the and past decades, diesel engines ofhave become the threaten al. (2010) the fault-tolerant control structure is established for most common used power plant because its high thermal diesel engine path system with observer a jammedis EGR valve. efficiency and pollution, reliable performance. In order oftonitrogen reduce the In Hamouda etfault-tolerant al.air (2015), a control T-S fuzzy designed to environmental the emission problem al. (2010) the structure is established for most common powerperformance. plant becauseInoforder its high thermal diesel engine air path system with observer a jammedis EGR valve. efficiency and used reliable tonitrogen reduce the In Hamouda et al.states (2015), a T-S fuzzy designed to environmental pollution, the matter emission problem of andsystem faults, anda fault-tolerant control the dieselsystem engine air path with jammedis EGR valve. particulate of the engines has estimate oxides ( NOand efficiency reliable performance. In diesel order of tonitrogen reduce In Hamouda et al. (2015), a T-S fuzzy observer designed to x ) and environmental pollution, the emission problem estimate system by states and faults, and fault-tolerant control law is proposed using model predictive control approach. NO ) and particulate matter of the diesel engines has oxides ( In Hamouda et al.states (2015), a T-S fuzzy is designed to system and faults, andobserver fault-tolerant control environmental pollution, the matter emission problem of nitrogen gained ( increasing interest. Equipped with Exhaust Gas NOxxx ) and particulate of the diesel engines has estimate oxides law is proposed by using model predictive control approach. Sliding mode control isand amodel preferable approach toapproach. generate estimate system states faults, and fault-tolerant control gained increasing interest. Equipped with Exhaust Gas law is proposed by using predictive control particulate diesel engines has oxides ( NOx ) and Recirculation (EGR) valvematter andof the Variable Geometry mode control control algorithms is a preferable approach to generate fault-tolerant for the air path system, gained increasing interest. Equipped with Exhaust Gas Sliding law is proposed by using predictive control Recirculation (EGR) valve and to Variable Geometry Sliding mode control control is amodel preferable approach toapproach. generate Turbocharger (VGT), it is possible improve combustion fault-tolerant algorithms for the air path system, gained increasing interest. Equipped with Exhaust Gas however, chattering phenomenon always appears when Recirculation (EGR) valve and Variable Geometry Sliding mode control control algorithms is a preferable approach to generate Turbocharger (VGT), it is of possible to improve fault-tolerant foralways the airappears path system, and emission performance theand diesel engines.combustion Therefore, chattering phenomenon when Recirculation (EGR) it is valve Variable Geometry however, Turbocharger (VGT), possible to improve combustion system trajectories crossing the sliding manifold. Hence, fault-tolerant control algorithms for the air path system, and emission performance of the diesel engines. Therefore, however, chattering phenomenon always appears when efforts have been put it onisactively adjusting the combustion combustion Turbocharger (VGT), possible to improve system trajectories crossing the sliding manifold. Hence, and emission performance of the diesel engines. Therefore, attempts have been made to solve this drawback, boundary however, chattering phenomenon always appears when efforts have beenKebairi put on etactively adjusting the(2015); combustion system trajectories crossing the sliding manifold. Hence, process, see e.g., al.the (2015); Liengines. et al. Jung attempts have been made solve this drawback, and emission performance of diesel Therefore, efforts have been put on etactively adjusting the(2015); combustion layer technique (Hung et al.to(1993)), dynamical gainboundary schedule system trajectories crossing the sliding manifold. Hence, process, see e.g., Kebairi al. (2015); Li et al. Jung have been made solve this drawback, boundary et al. (2016) and therein. adjusting the combustion attempts layer technique (Hung et al.to(1993)), dynamical schedule efforts have beenreferences put on etactively method (Plestan et al.made (2010)), and higher order gain sliding mode process, see and e.g., Kebairi al. (2015); Li et al. (2015); Jung attempts have been to(1993)), solve this drawback, boundary et al. (2016) references therein. layer technique (Hung et al. dynamical gain schedule method (Plestan et (2007)) al. (2010)), and higher In order sliding mode process, see and e.g.,references Kebairi ettherein. al. (2015); Li et al. (2015); Jung (Laghrouche et al. (2016) et al. are employed. Ali et al. (2015), layer technique (Hung et al. (1993)), dynamical gain schedule Recently, control methods for the air path system have been method (Plestan et (2007)) al. (2010)), and higher order mode et al. (2016) and references therein. (Laghrouche et sliding al. are controller employed. In Ali sliding et al. (2015), Recently, control methods for the air path system have been super-twisting mode is designed for the method (Plestan et (2007)) al. (2010)), and higher In order sliding mode extensively studied. In Utkin et al. (2000), a sliding mode (Laghrouche et al. are employed. Ali et al. (2015), Recently, control methods for the air(2000), path system havemode been super-twisting sliding mode by controller is chattering designed effect for the extensively studied. In Utkin et al. a sliding diesel engine air path control which the is control is control designed to regulate theairVGT, the EGR et sliding al. (2007)) are controller employed.isIn designed Ali et al. (2015), Recently, methods for the path whereas system have been (Laghrouche super-twisting mode for the extensively studied. In regulate Utkin etthe al. VGT, (2000), a sliding mode diesel engineand air path control by which thehandled chatteringaseffect is control is designed to whereas the EGR eliminated actuator faults are well. super-twisting mode by controller designed effect for the valve is supposed controlled open-loop. Furthermore, extensively studied.totoInberegulate Utkin etthe al. VGT, (2000), a sliding mode diesel engineand airsliding path control which theishandled chattering is control is designed whereas the EGR eliminated actuator faults are as well. valve is supposed be controlled open-loop. Nevertheless, thepath upper bound of thethe addictive faults has engineand air control by which chattering effect is for tuning exhaust to manifold pressure and whereas freshFurthermore, airflow rate diesel control issupposed designed toberegulate the VGT, the EGR eliminated actuator faults are handled as well. valve is to controlled open-loop. Furthermore, Nevertheless, the upper bound of thecontrol addictive has for tuning exhaust manifold pressure andcontrol fresh airflow rate been involved to determine the proper gain,faults suchwell. that eliminated and actuator faults are handled as simultaneously, constructive Lyapunov approach is valve is supposed to be controlled open-loop. Nevertheless, upper bound of thecontrol addictive has for tuning exhaust manifold pressure andcontrol freshFurthermore, airflow rate involved the tocapacity determine the be proper gain,faults such that simultaneously, constructive Lyapunov approach is been its fault-tolerant adopted in Jankovic et al. (2000), and sliding mode algorithm Nevertheless, the upper could bound ofassured. thecontrol addictive faults has for tuning exhaust manifold pressure andcontrol fresh airflow rate been involved to determine the proper gain, such that simultaneously, constructive Lyapunov approach is fault-tolerant capacity could be assured. adopted in Jankovic et al. (2000), and sliding mode algorithm its been involved to determine the proper control gain, such that is developed for controlling VGT-EGR system in Upadhyay its fault-tolerant capacity could be assured. simultaneously, constructive Lyapunov control approach is Motivated by the issues mentioned above, the fault-tolerant adopted in Jankovic et al. (2000), and sliding mode algorithm is developed forAimed controlling VGT-EGR system in Upadhyay fault-tolerant could be assured. robust its et al. (2002). lowering adopted in Jankovic et al.at(2000), and NO sliding mode algorithm Motivated by thecapacity issues above, fault-tolerant x emission, control problem the mentioned diesel engine airthe path system is is developed for controlling VGT-EGR system in Upadhyay NOsystem robust Motivated et al. (2002).forAimed at lowering by the for issues mentioned above, the fault-tolerant x emission, control problem for the diesel engine air path systemlaw is is developed controlling VGT-EGR in Upadhyay sliding mode controllers designed NO in Wang (2008), which investigated in this paper. Adaptive sliding mode control emission, robust et al. (2002). Aimed atarelowering x Motivated by the for issues above, the fault-tolerant x control problem the mentioned diesel engine airmode path systemlaw is sliding mode controllers are designed in Wang (2008), which investigated in this paper. Adaptive sliding control NOx emission, robust is et al. (2002). at between loweringconventional facilitate smoothAimed switching combustion constructed tofor handle the engine partial airloss of system actuator control problem the diesel path is sliding mode controllers are designed in Wang (2008), which investigated in this paper. Adaptive sliding mode control law facilitate smooth switching between conventional combustion is constructedfaults to handle the partial actuator mode LTC mode by regulating intake charge and Adaptive additive faults.loss Theof auxiliary sliding and mode controllers arebetween designedconventional inengine Wang (2008), which effectiveness facilitate smooth switching combustion investigated in this paper. sliding mode control law is constructed to handle the partial loss of actuator mode and LTC mode regulating engine intake charge faults toand faults.factors The auxiliary amount and EGRswitching rate. by Xie et al. conventional (2016) proposes a new effectiveness parameters related the additive effectiveness and the facilitate smooth between combustion mode and by engine intake charge is constructed to handle the partial actuator effectiveness faults and additive faults.loss Theof auxiliary amount andLTC EGRmode rate. Xieregulating et al. (2016) proposes a new parameters related to the effectiveness factors and thus the decoupling method to regulate the VGT-EGR system by additive faults faults are estimated by adaptation update laws, mode and LTC mode by regulating engine intake charge amount and EGR rate. Xie et al. (2016) proposes a new effectiveness and additive faults. The auxiliary related to the by effectiveness factors laws, and thus the decoupling method disturbance to regulate rejection the VGT-EGR system by parameters additive faults are estimated adaptation update employing active control (ADRC), requirement of upper of the actuator faults is amount and method EGR rate. Xie et al. proposes a new parameters related to the bound effectiveness factors laws, and thus the decoupling to regulate the(2016) VGT-EGR system by the additive faults are estimated by adaptation update employing active disturbance rejection control (ADRC), the requirement of upper boundlayer of the actuator faults to is which is easymethod to implement and could tackle a wide range by of removed. In addition, boundary method is adopted decoupling to regulate the VGT-EGR system additive faults are estimated by adaptation update laws, thus employing active disturbance rejection control (ADRC), the requirement of upper bound of the actuator faults is which is easywithout to implement andthe could tackle a wide range of removed. In addition, boundary layer method is adopted to uncertainties returning control gains. solve the chattering phenomenon of the adaptive sliding employing active disturbance rejection control (ADRC), the requirement of upper bound of the actuator faults is which is easy to implement and could tackle a wide range of removed. In addition, boundary layer method is adopted to uncertainties without returning the control gains. solve control the chattering phenomenon of the adaptive sliding law. Rigorous theoretical analysis shows that which is easywithout to implement andthe could tackle a wide range of mode removed. In addition, boundary layer method is adopted to uncertainties returning control gains. solve the chattering phenomenon of the adaptive sliding In practical, the actuators of air path system would suffer mode control law. Rigorous theoretical analysis shows that system are driven into a small of the uncertainties without returning the control gains.would suffer the solve the states chattering phenomenon of neighbourhood the adaptive sliding In practical, actuators of including air path system mode control law. theoretical analysis shows from several the kinds of faults, loss-of-effectiveness, the system states areRigorous driven into a small neighbourhood ofthat the In practical, the actuators of air path system would suffer control law. Rigorous theoretical analysis showsofthat from several kinds of faults, including loss-of-effectiveness, mode the system states are driven into a small neighbourhood the In practical, actuators of including air path system would suffer from several the kinds of faults, loss-of-effectiveness, the system states are driven into a small neighbourhood of the Copyright 2018 IFAC 468Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2018, IFAC Federation of Automatic Control) from several kinds of(International faults, including loss-of-effectiveness, Copyright 2018 responsibility IFAC 468Control. Peer review©under of International Federation of Automatic Copyright © 2018 IFAC 468 10.1016/j.ifacol.2018.10.096 Copyright © 2018 IFAC 468
IFAC E-CoSM 2018 430 Changchun, China, September 20-22, 2018 Jian Zhang et al. / IFAC PapersOnLine 51-31 (2018) 429–434
equilibrium. Thus the dynamic performance and reliability of the air path system can be improved. The rest of this paper is organized as follows: the referred nonlinear diesel engine air path system with actuator faults and the control objective are described in Section 2. In Section 3, two fault-tolerant control laws are established by using adaptive technique and sliding mode control. Then, numerical simulation results and analysis are carried out in the followed section. Finally, conclusions and remarks are given in Section 5.
2.1 Mathematical model of diesel engine air path system
-
c t
Compressor isentropic efficiency
-
V1
Turbine isentropic efficiency Turbocharger mechanical efficiency Intake manifold volume
m3
V2
Exhaust manifold volume
m3
Vd
Engine volume cylinder
m3
Ta
Ambient temperature
T1
Intake manifold temperature
K K
T2
Exhaust manifold temperature
K
Specific gas constant
J Kg K
Ra
p1 k1 Wc Wegr ke p1 p2 k2 ke p1 Wegr Wt Wf
Pc
1
(1)
(2)
m Pt Pc
(3)
where the compressor mass flow Wc and turbine power Pt are given as k Wc Pc c (4) p1 1
Pt kt 1 p2 Wt
(5)
with the parameters calculated as kc c c pTa , kt c pt T2 ,
Variable Geometry Turbine
k1 RaT1 V1 , ke v NVd RaT1 and k2 RaT2 V2 .
Exhaust gas
To facilitate the control law design, the model (1)-(3) is rewritten into vector form
Exhaust manifold (m2 , p2 ,T2 )
x f x g1 x u1 g2 x u2
EGR cooler
where x p1
Cylinders
Intercooler
-
Considering the third-order model for the diesel engine air path system (Jankovic et al. (2000)):
A schematic representation of the diesel engine air path system is given in Fig. 1, and the corresponding variable are illustrated in Table 1. It can be observed that the turbocharger consists of a variable geometry turbine (VGT) and a compressor, which are connected by a shaft. As the VGT generating the energy of the exhaust gas in the exhaust manifold, the compressor is driven by the turbine via their common shaft. As a result, the mass of air provided to the engine cylinders are increased, so that higher power is achieved than non-turbocharged engine. A portion of the exhaust gas is delivered into the intake manifold through the exhaust gas recirculation valve (EGR). The recirculated exhaust gas lowers the combustion temperature and declines the formation of NOx .
Fresh air
Engine volumetric efficiency
m
2. PROBLEM FORMULATION
Compressor(Pc ,Wc )
v
p2
(6)
Pc represents the system states. The T
control variables are u1 Wegr , u2 Wt , respectively. And
EGR valve
the definition of f x , g1 x and g 2 x in (6) are given as
Intake manifold (m1 , p1 ,T1 )
Pc k1kc p 1 k1ke p1 1 f x k 2 ke p1 W f , Pc
Fig. 1. Schematic of a turbocharged diesel engine with EGR and VGT. Table 1. Nomenclature of the diesel engine variables Variables p1
Name Intake manifold pressure
Units Pa
p2
Exhaust manifold pressure
Pc
Compressor power
Pt
Turbine power
Pa W W
Wc
Compressor mass flow
Kg s
Wt
Turbine mass flow
Kg s
Wf
Fueling mass flow rate
Kg s
0 k1 g1 x k 2 , g2 x k2 0 K 1 p o 2
.
m kt . The specific values of the system parameters k1 , k 2 , k t , k e , k c , m , and mentioned where K o
above will be specified later. 469
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2.2 Air path system with actuator faults
431
Taking the time-derivative of s along (7) , thus it has
s f * x g * x E* t u F * t
The actuator faults of the air path system are taken into account, as a result the original model (6) is modified as x f x g1 x E1 t u1 F1 t g2 x E2 t u2 F2 t
with u u1 u2 being defined as
T
(7)
and f * x
(9)
, g * x , E* t , F * t
0 k1Wc k1ke p1 p1d k where E1 t and E2 t denote the effectiveness factor of the g* x 1 f * x , , k2 k2 k2 ke p1 k2W f p2 d actuators with 0 Ei t 1 , i 1, 2 . Note that Ei t 1 k1 F1 0 E indicates that the i th actuator works normally, and * E , F * t t 1 . 0 E k F F 0 Ei t 1 represents that the i th actuator partially loses 2 2 2 1
In the following context, E * t and F * t will be denoted
its effectiveness, i th actuator has failed completely if Ei t 0 . F1 t and F2 t denote the uncertain additive faults acting on each actuator, which are unknown but bounded.
as E * and F * for simplification. . Then, an adaptive sliding mode control law is proposed
uasmc g* x f * x F * hs sgn s
(10)
f * x F*
(12)
1
2.3 Control objectives To achieve the control requirements of the air path system, the compressor mass flow Wc and exhaust manifold pressure p2 should be adjusted to their desired set points. Nevertheless, the computation for the compressor mass flow is time consuming, because of the time derivative of Wc (Ali et al. (2012)). Hence, the pressure manifold set point p1d is
where h denotes the control gain, is a small positive constant. Denoting an auxiliary variable 1 1 , where
E
max *
with E I E * . Thus, F * and are used to
estimate F and , respectively. The adaptation update laws are given as
adopted instead of Wc to develop the control law. In addition, it is straightforward to obtain p1d :
F* s
1
k P (8) p1d c c 1 Wcd The tracking errors respect to the set points are defined as y1 p1 p1d and y2 p2 p2d , respectively. Thus, the control objective is to generate fault-tolerant control laws which steer the tracking errors to the equilibrium of the closed-loop system. As a result, the desired compressor mass flow Wc and exhaust manifold pressure p2 can be accurately tracked in the presence of actuator faults.
F * F * F * and are defined as the estimation errors of the adaptive parameters. Theorem 1. For the diesel engine air path system generated by (6). If the control law are designed as (10)-(12) with the adaptation law (13)-(14), the closed-loop system will achieve asymptotically stability.
Proof. Consider the following candidate Lyapunov function which is positive definite and proper:
1 1 *T * 1 2 (15) V1 sT s F F 2 2 2 Taking the time-derivative of (15) along system (9), it yields
In this section, two fault-tolerant control laws are designed for the air path system. Adaptive technique and sliding mode control are employed to solve the actuator faults without using any piror knowledge of the faults. However, chattering phenomenon always exits when the sliding mode control applied. Thus, the boundary layer method is introduced to modify the adaptive sliding mode control law and handle the chattering problem.
V1 s T g * x I E uasmc f * x F *
1
F *T F *
1
(16)
Substituting control law (11)-(14) into (16), it further has 3.1 Adaptive sliding mode fault-tolerant controller
V1 hs T s s 1 s s f * x F *
To facilitate the control law design, a sliding manifold is defined as s p pd , with s s1 and p p1d
s2 , p p1
(13)
(14) s where and are positive constants to be designed. Then,
3. MAIN RESULTS
T
(11)
p2
T
h s s 1 s 2
p2d . T
s 470
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controller (20) can be refer to the proof of Theorem 1, thus it is omitted here.
h s s 1 s 2
1 s
Case 2:
(17)
If s , the controller is rewritten as
2
h s s
s 1 (21) ucfasmc g * x f * x F * hs 2 Given the definition of h , , and the adaptation update
Clearly, V1 0 , and given the definition of V1 , it implies that
s L , F * L , L . Integrating (17), it shows that s L2
L . Also, it is easy to verify that s L , which
law for F , , and taking the time derivative of V 2 , it has
further leads to lim s 0 . Hence, p1 and p2 will track their t
desired set points asymptotically.
2
V2 h s
Remark 1. In the above analysis, is considered
2
2
s
1 s
s s f * x F * 2 as a positive constant, which requires that 1 always holds. In order to guarantee such condition, the initial value 2 s 2 of t is set as t0 1 . Thus, given the adaptation law h s 2 1 s
(22)
s and the definition of , it is reasonable to conclude that 1 1 1 .
s s 2
3.2 Chattering-free fault-tolerant controller
h s s
2
2
It is obvious that V2 0 if any of the following inequality holds
As it is known that chattering phenomenon would occur when system trajectories cross the sliding manifold, which deteriorates dynamic performance of the closed-loop system. Aimed at handling this problem, the boundary layer method is employed to redesign the control scheme.
1
s
Considering the modified fault-tolerant control law:
s
g * x 1 f * x F * hs sgn s , s ucfasmc 1 * * * 2 s , s g x f x F hs (18) with denoting the boundary layer, and the definition of ,
4h 1 4
s
(23) (24) (25)
It implies that the state trajectories converge to a small set containing the equilibrium of the closed-loop system defined as 1 1 s t lim s t s t t 4h 4 (26) s t Given the two cases above, the conclusion can be drawn that the trajectories of the closed-loop system are uniformly ultimately bounded (UUB). This completes the proof.
, F , are as same as (11)-(14), respectively. Theorem 2. For the diesel engine air path system generated by (6). If the control law (18) with the corresponding variable definition and adaptation update laws (13)-(14). Then, the trajectories of the closed-loop system are finally converge to a small neighbourhood around the equilibrium. Proof. Construct the Lyapunov candidate function:
1 1 *T * 1 2 V2 sT s F F 2 2 2
1 s 1 2 4
(19)
Remark 2. From inequality (26), it can be observed that the value of the boundary layer effects on the accuracy of the tracking error, the smaller value of leads to the smaller set of s . Moreover, by choosing larger control gain h , the tracking performance would be more precise.
Case 1: If s , the control law yields
ucfasmc g* x f * x F * hs sgn s (20) which possesses the same form as controller (11). Therefore, the stability analysis of the closed-loop system under 1
4. SIMULATION AND ANALYSIS To illustrate the effectiveness of the proposed control laws, numerical simulation are carried out in this section. According to (Larsen et al. (2000)), the parameters of system 471
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(1)-(5) are selected as k1 143.91 , k1 1715.5 , kt 391.365 , ke 0.028 , kc 0.0025 , m 0.95 , 0.285 , 0.15 . The initial value are set as p1 1.32 , p2 1.35 , Pc 5.605 The control requirements for the system states are addressed in Table 2.
2-4, the chattering problem has been solved by CFAFMC. Moreover, define the tracking errors as e1 Wcd Wc and e2 p2 d p2 . At around 30 seconds, the tracking errors converge to zero with the accuracies of e1 0.002 and
e2 0.05 under CFAFMC, while the accuracies turn out to be e1 0.005 and
Table 2. Control requirements for the air path system Variables
Set point 1
Set point 2
Wc Kg s
0.01
0.07
p2 Bar
1.15
1.55
Wf Kg h
3
7
433
e2 0.2 when AFMC adopted. Hence,
it can be concluded that tracking performance is further improved as chattering behaviour is attenuated.
In the numerical simulation, the adaptive sliding mode faulttolerant controller (10)-(14) is denoted as ASMC, and the modified fault-tolerant controller (18) with the corresponding adaptation update laws is denoted as CFASMC. The control parameters are selected as h 5 , 0.005 ， 0.005 ， 0.0001 , 0.01 . Besides, the additive fault is set as
F t 103 4sin 0.1t 2cos 0.1t and the effectiveness T
factor is chosen to be Fig. 3. Response curves of p2 under ASMC.
if t 25 I 22 E t 0.85 0.05sin 0.2 t I 22 if t 25 Firstly, the ASMC is employed. The time histories of Wc and p2 are given in Figs. 2 and 3, respectively. It can be seen that the errors between the set points and the corresponding actual states are considerable small. Hence, the proposed control law is capable of handling the actuator faults while guarantee accurate tracking of Wcd and p2d . The response
curves of Wegr and Wt are presented in Fig. 4. Obviously, high frequency chattering behaviour impacts on both the system states and the control input due to the discontinuous term in the control law. Fig. 4. Response curves of Wegr and Wt under ASMC.
Fig. 2. Response curves of Wc under ASMC. Then, the response curves of Wc , p2 , Wegr and Wt under
Fig. 5. Response curves of Wc under CFASMC.
CFAFMC are carried out in Figs. 5-7. Compared with Figs. 472
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Fig. 6. Response curves of p2 under CFASMC.
Fig. 7. Response curves of Wegr and Wt under CFASMC. 5. CONCLUSIONS In this paper, two fault-tolerant control schemes for the air path system of the tubocharged diesel engine are designed. Initially, an adaptive sliding mode control law is constructed, by which the requirement of the piror knowledge of the actuator faults is removed. Then, the boundary layer method is adopted to handling the chattering phenomenon by modifying the discontinuous control term. Lyapunov stability theory is used to analyze the state trajectories of the closedloop system, and numerical results show the effectiveness and robustness of the proposed control laws. Our future work will focus on developing fault estimation method for the actuators of the diesel engine air path system to further upgrade the performance of the fault-tolerant control law. REFERENCES Ali, S.A. and Langlois, N. (2012). A robust sliding mode control strategy for turbocharged diesel engine air path, The Proceeding of the 13th IFAC Symposium on Control in Transportation Systems, Sofia, Bulgaria, 262-267. Ali, S.A., Guermouche, M., and Langlois, N. (2015). Faulttolerant control based super-twisting algorithm for the diesel engine air path subject to loss-of-effectiveness and additive actuator faults, Applied Mathematical Modelling, 39(15), 4309-4329. 473
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