Control of diesel engine dual-loop EGR air-path systems by a singular perturbation method

Control Engineering Practice 21 (2013) 981–988

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Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac

Control of diesel engine dual-loop EGR air-path systems by a singular perturbation method$ Fengjun Yan a, Junmin Wang b,n a b

Department of Mechanical Engineering, McMaster University, Hamilton, ON, Canada L8S 4L7 Department of Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH 43210, USA

a r t i c l e i n f o

a b s t r a c t

Article history: Received 5 May 2012 Accepted 27 February 2013 Available online 22 April 2013

This paper presents a singular perturbation based method for controlling the dual-loop exhaust gas recirculation (DL-EGR) air-path systems on advanced diesel engines. A DL-EGR air-path system, consisting of a high-pressure loop EGR (HPL-EGR) and a low-pressure loop EGR (LPL-EGR), has significantly different time-scales (fast and slow) due to the inherent difference in the HPL-EGR’s and LPL-EGR’s corresponding control volumes. Such a feature of the DL-EGR systems makes the cooperative control of intake manifold gas conditions challenging. By considering the DL-EGR air-path system as a singularly perturbed system, a composite control law was devised to achieve systematic control of the air-path conditions including gas pressure, temperature, and oxygen fraction in the intake manifold. The effectiveness of the control method is experimentally evaluated on a medium-duty diesel engine. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Dual-loop EGR Diesel engines Air-path control

1. Introduction Advanced combustion modes, including homogenous charge compression ignition (HCCI) (Chiang & Stefanopoulou, 2007; Horibe, Harada, Ishiyama, & Shioji, 2009; Thring, 1989), low temperature diffusion combustion (LTDC) (Alriksson & Denbratt, 2006), and premixed charge compression ignition (PCCI) (Horibe et al., 2009), as well as the transitions between them (Wang & Chadwell, 2008), have imposed great challenges in the current field of diesel engine control. While such promising combustion modes may substantially contribute on engine fuel efficiency improvements and engine-out emission reductions, they are also sensitive to the intake gas conditions, including intake gas pressure, temperature, and oxygen fraction, during both steady state and transient operations. Moreover, the desired intake gas conditions may significantly differ for different combustion modes and vary with engine operating conditions as well (Wang, 2008a, 2008b). Thus, it is imperative to achieve precise and decoupled air-path gas condition control, which needs to be furnished by necessary air-path actuators and appropriate control methods as well. A variety of new diesel engine air-path actuators, such as variable geometry turbocharger (VGT), two-stage turbo-charging system, single- and dual-loop exhaust gas recirculation (EGR), and variable valve actuation (VVA), have been recently developed for providing n

Corresponding author. Tel.: þ 1 614 247 7275. E-mail address: [email protected] (J. Wang). ☆ This research is supported by National Science Foundation (NSF)—Control Systems Program Award CMMI-1029611. 0967-0661/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conengprac.2013.02.017

the authorities of controlling the intake manifold gas conditions in both steady-state and transient operations. With these engine actuators, various air-path system control law design methods have been investigated in the literature (Ammann, Fekete, Guzella, & Lattfelder, 2003; Utkin, Chang, Kolmanovsky, & Cook, 2000). In Ammann et al. (2003), the authors developed a coordinated EGR-VGT controller for passenger car diesel engines, through switching logic and a set of linear controllers. A sliding mode controller in Utkin et al. (2000) was employed for diesel engine control. Comparisons of different control algorithms were presented in Kolmanovsky, Moraal, Nieuwstadt, and Stefanopoulou (1998). In Wang (2008a), a hybrid robust air-path system control approach was developed for transient operations of diesel engines running LTDC and conventional combustion modes. In Plianos and Stobart (2008), a linear-quadratic-Gaussian control algorithm was designed for a two-stage turbo system with a highpressure loop EGR (HPL-EGR). As shown in Fig. 1, the exhaust gas from a low pressure loop EGR (LPL-EGR) is filtered by a diesel particulate filter (DPF) after passing through a diesel oxidation catalyst (DOC) and cooled by a cooler to ensure it is clean and cool. In contrast, the HPL-EGR gas is directly routed from the exhaust manifold and thus it is hot and unfiltered. To utilize the feature of high gas temperature in the HPL-EGR loop, the cooler in the HPL-EGR path was bypassed. Note that in general applications, the gas from HPL-EGR can be cooled by EGR cooler and thus will not necessarily have such a significant impact on intake temperature as in this case. The LPL-EGR loop provides cooled exhaust gas, which can be used to adjust the intake temperature and intake oxygen concentration. The turbocharger, HPL-EGR loop, and LPL-EGR valves together can enable the decoupling control of the pressure,

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Nomenclature 1 2 3 v a c t hegr legr hthr

intake manifold section before high-pressure throttle exhaust manifold valve ambient compressor turbo high pressure loop EGR valve low pressure loop EGR valve high pressure throttle

temperature, and oxygen concentration (Grondin, Moulin, & Chauvin, 2009; Wang, 2008a; Yan & Wang, 2009). Without using LPL-EGR, the typical single HPL-EGR system cannot fully decouple the intake manifold gas temperature and oxygen fraction, both of which are influenced by EGR loop. A mean-value model and air-fraction estimation methods for diesel engines with a DL-EGR and a singlestage turbocharger were proposed in Wang (2008b). In Chauvin, Corde, Petit, and Rouchon (2008), the authors developed a DL-EGR control method, which deals with the HPL-EGR and LPL-EGR separately by a motion planning method. Cooperative control of dual-loop EGR systems with explicit considerations on the DL-EGR system timescale difference is still absent. Through dual-loop EGR cooperative control in conjunction with VGT control, the three intake gas conditions (i.e. pressure, temperature, and oxygen fraction) can be controlled to a large extent. Such a flexibility of the intake gas condition control is essential in further advanced combustion mode control. The DL-EGR air-path system considered in this paper is illustrated in Fig. 1. The objective here is to control the gas conditions, i.e., gas pressure, temperature, and oxygen fraction in the intake manifold. As shown in Fig. 1, there are three actuators (HPL-EGR valve, HP-throttle, and LPL-EGR valve) that can influence the three intake manifold conditions (gas pressure, temperature, and oxygen fraction). An exhaust valve with fixed opening is equipped at the end of the exhaust pipe to provide the necessary back pressure for the LPL-EGR. While the HPL-EGR valve and HP-throttle valve can directly adjust the intake manifold conditions, the effects of LPL-EGR valve need to be transferred through the volume upstream HP-throttle. Thus, cooperative control of the gas conditions in the

gas flow rate through x (kg/s) engine intake gas flow rate (kg/s) pressure in x (Pa) volume of x (m3) oxygen fraction in x temperature at x (K) ideal gas constant EGR transport delay (s) engine speed (rpm) specific heat ratio engine displacement volume (m3) engine volumetric efficiency

Wx W in px Vx Fx Tx R d Ne γ Vd ηv

two volumes (intake manifold and the volume upstream HP-throttle) is necessary in the DL-EGR air-path system. To be noted, the LPL-EGR loop (including the volume before HP-throttle) has a substantially larger volume and consequently a much slower dynamics than those of the HPL-EGR. In the authors’ previous conference paper (Yan & Wang, 2011), a singular perturbation control method for addressing the DL-EGR systems’ different time-scale issues was preliminarily studied with GT-Power engine simulation results. This paper significantly extends the preliminary study by including comprehensive analysis and extensive engine experimental investigation results of the method. To achieve better transient control of the intake manifold gas conditions, the drawback of slow LPL-EGR dynamics needs to be compensated by the HPL-EGR and VGT control, which have a wider bandwidth. Since the geometries of air-path systems are different according to the lay-outs of the engines, the transient control objective here is to control the transient intake gas dynamic responses at the same level as the HPL-EGR alone, which can be seen at the 81 s in Fig. 2 for this engine configuration. In the experimental test bench for this research, a medium-duty diesel engine, the volume of intake manifold is around 0.3 L and the volume for the LPL-EGR is about 4 L. Figs. 3 and 2 show the step changes of LPL-EGR and HPL-EGR valves and the corresponding dynamics of the intake gas conditions. In Fig. 3, the step drops of the LPL-EGR and HPL-EGR valves are indicated in solid line and dashed line circles, respectively. The exemplifying dynamics of the measured corresponding intake gas conditions are indicated in the circles in Fig. 2 at 31 s and 81 s, respectively. As can be seen, the LPL-EGR control shows obviously slower dynamics than that of

F1

0.22 0.2 0.18 0

10

20

30

40

50

60

70

80

90

100

110

70

80

90

100

110

70

80

90

100

110

Time [s]

p1(Pa)

1.2

x 10

5

1.15 1.1 0

10

20

30

40

50

60

Time [s] T1 (K)

315 310 305 300 0

10

20

30

40

50

60

Time [s] Fig. 1. A schematic diagram for a diesel engine with dual-loop EGR.

Fig. 2. Experimentally measured dynamic responses under HPL-EGR and LPL-EGR step changes.

F. Yan, J. Wang / Control Engineering Practice 21 (2013) 981–988

2.1. Intake manifold model

0.8 0.6

uhegr

983

Based on the mass and energy conservations as well as the ideal gas law, the intake manifold pressure, temperature, and oxygen fraction models can be derived as (Chauvin et al., 2008; Killingsworth, Aceves, Flowers, & Krstic, 2006; Wang, 2008a)

0.4 0.2 0 0

10

20

30

40

50

60

70

80

90

100

110



RT 1 ðW hegr þW hthr −W in Þ, V1

ð1Þ



 RT 1
W hegr ðγT hegr −T 1 Þ þW hthr ðγT 2 −T 1 Þ þ W in ðT 1 −γT e Þ , p1 V 1

ð2Þ



 RT 1
W hegr ðF 3 −F 1 Þ þ W hthr ðF 2 −F 1 Þ : p1 V 1

ð3Þ

p1 ¼

Time [s] 0.8

T1 ¼

ulegr

0.6 0.4

F1 ¼

0.2 0 0

10

20

30

40

50

60

70

80

90

100

110

Time [s] Fig. 3. HPL-EGR and LPL-EGR step changes.

Here, T e is the temperature of the gas into the engine cylinders, which can be approximated as intake manifold temperature T 1 . The engine intake gas flow rate W in can be estimated by the speed–density model as W in ¼

ηv p1 Ne V d : 120RT 1

ð4Þ

HPL-EGR. Regarding the intake oxygen fraction, the approximated time constant is about 7 s for LPL-EGR and 0.5 s for HPL-EGR. Regarding the intake pressure, the time constant is approximately 10 s for LPL-EGR and 1 s for HPL-EGR. Regarding the intake temperature, the time constant is about 20 s for the LPL-EGR and 5 s for the HPL-EGR. To be noted, since the system is complex and nonlinear, and cannot be directly viewed as a first-order system, the time constants here are only roughly approximated by the experimental results and are provided as a indicator for the DL-EGR transient performances. Such a significant difference in time scale of the dynamics makes the coordination of a dual-loop EGR system challenging. For such systems with different time scale dynamics, the singular perturbation control methods, described in Saberi and Khali (1984) and Castro-Linares, Alvarez-Gallegos, and Vasquez-Lopez (2001), offer some effective means. In the singular perturbation control methods, the original system can be decomposed into two separated subsystems with lower dimensions, which describe the different time scale dynamics correspondingly. Respective control laws can be designed for the different dynamic scale subsystems, and the final control law can be therefore derived by the so-called composite feedback control method, which combines the control signals from subsystems together (Castro-Linares et al., 2001; Saberi & Khali, 1984). The stability analysis of the singular perturbation method was addressed in Saberi and Khali (1984). In this paper, a systematic air-path control method that cooperates DL-EGR valves and HP-throttle for simultaneously controlling the intake manifold gas pressure, temperature, and oxygen fraction in a seamless way is devised. Such a new air-path control method essentially expands the conventional diesel engine control scope by adding the temperature dimension, which is crucial for control of advanced combustion modes. The arrangement of the rest of this paper is as follows. In Section 2, the control-oriented model of the air-path system is introduced. Section 3 describes the decomposition control method for DL-EGR air-path control (viewed as a singularly perturbed system). In Section 4, such a control methodology is experimentally investigated on a medium-duty diesel engine. Conclusive remarks are presented in Section 5.

where ηv is the volumetric efficiency, N e is the engine speed, V d is the displacement of the engine cylinder and R is the ideal gas constant. To be noted, the accuracy of Eq. (4) highly relies on the volumetric efficiency, which needs to be carefully calibrated. Note that Eq. (1) is based on the assumption (Chauvin, Corde, & Petit, 2006; Chauvin et al., 2008) that the intake temperature variations are small. Such an assumption is reasonable in this model because the maximal temperature variation in the test results in this research is within 5 K per second (around 1.6% in percentage). The temperature variation is trivial comparing to the mass flow rate changes. Since the temperature measurement by standard thermocouples has a transient first-order delay, a pressure-based temperature reconstruction technique in Yan and Wang (2012) was used for the transient temperature measurement correction.

2. Air-path modeling

2.3. Resultant control-oriented model of dual-loop EGR system

In this section, a control-oriented model for a diesel engine DL-EGR air-path system is introduced.

In a typical DL-EGR air-path system, LPL-EGR has a large volume (from LPL-EGR valve to HP throttle), i.e. V 2 4 4V 1 .

2.2. Control volume before HP-throttle The gas dynamics in the control volume before the HP-throttle can be described as 

 RT 2
W legr ðγT legr −T 2 Þ þ W a ðγT a −T 2 Þ þW hthr ðT 2 −γT 1 Þ , p2 V 2

ð5Þ




 RT 2  W legr F 3 ðt−dÞ−F 2 þ W a ðF a −F 2 Þ : p2 V 2

ð6Þ

T2 ¼ F2 ¼

Here t is the time and d is the EGR gas transportation delay from the exhaust oxygen sensor location to the LPL-EGR valve. It can be estimated as a function of mass flow rate through the EGR loop (Zhang, Grigoriadis, Franchek, & Makki, 2007), i.e., d¼

δ W legr

ð7Þ

where δ is the coefficient determined by experimental data. Eq. (5) holds with the assumption that the inter-cooler after compressor can compensate for the temperature increasing effect caused by the compressor. Since the temperatures of LPL-EGR, ambience, and inter-cooler are at the same level, which is significantly lower than the ones of the HPL-EGR gas, the assumption here will not influence the nature of the dynamics described in (5).

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Denote α1 ¼ RT 1 =p1 V 2 , α2 ¼ RT 2 =p2 V 2 , ε ¼ V 1 =V 2 . x1 ¼ T 2 , x2 ¼ F 2 , z1 ¼ p1 , z2 ¼ T 1 , z3 ¼ F 1 , u1 ¼ W hegr , u2 ¼ W hthr , u3 ¼ W legr . In the engine experimental setup considered in this paper, the value of ε is 0.075. Then according to (1)–(6), the resultant models turn out to be

states in the control volume before HP-throttle, need to be utilized, which will be detailed in the rest of this section.



The control law design was based on the assumptions that the sensors have properly measured signals and the actuators are mass flow rates through the EGR valves and the high-pressure throttle. The control law can be designed with respect to the fast and slow parts separately by decomposing the above system into reduced and boundary-layer subsystems. The stability of overall system will be analyzed in the next sub-section. A reduced system can be derived by letting ε ¼ 0, i.e., neglecting the fast dynamics of ½z1 ,z2 ,z3 T . In this reduced system z1,s , z2,s , z3,s , u1,s , u2,s , and u3,s are used instead of z1 , z2 , z3 , u1 , u2 , and u3 in the original system. Thus, the following holds that z1 ¼ z1,s þ z1,f , z2 ¼ z2,s þ z2,f , z3 ¼ z3,s þz3,f , u1 ¼ u1,s þ u1,f , u2 ¼ u2,s þu2,f , u3 ¼ u3,s þ u3,f . z1,f , z2,f , z3,f , u1,f , u2,f , and u3,f refer to the counterparts of the fast dynamics. When ε ¼ 0, by (10)–(12), the following equations holds:

x1 ¼ α2 ðγT legr −x1 Þu3 þα2 ðγT a −x1 ÞW a þα2 ðx1 −γz2 Þu2 ¼ f 1 ðx,z,uÞ,

ð8Þ



x2 ¼ α2 ½F 3 ðt−dÞ−x2 u3 þ α2 ðF a −x2 ÞW a ¼ f 2 ðx,z,uÞ,

ð9Þ



εz1 ¼ α1 z1 u1 þα1 z1 u2 −α1 z1 W in ¼ g 1 ðx,z,uÞ,

ð10Þ



εz2 ¼ α1 ðγT hegr −z2 Þu1 þ α1 ðγx1 −z2 Þu2 þ α1 ðz2 −γT e ÞW in ¼ g 2 ðx,z,uÞ, ð11Þ 

εz3 ¼ α1 ðF 3 −z3 Þu1 þ α1 ðx2 −z3 Þu2 ¼ g 3 ðx,z,uÞ:

ð12Þ T

For the sake of simplicity, vector fields ½f 1 ð⋅Þ,f 2 ð⋅Þ and ½g 1 ð⋅Þ,g 2 ð⋅Þ,g 3 ð⋅ÞT are denoted as f ð⋅Þ and gð⋅Þ, respectively. To be noted, α1 and α2 are in the similar scale level, according to their definitions (functions of pressure, temperature, and V 2 ). As discussed in the previous sections, the control objective is to control the intake manifold gas pressure p1 , temperature T 1 , and oxygen fraction F 1 simultaneously. The references of x1 , x2 , z1 , z2 , and z3 are denoted as x1,d , x2,d , z1,d , z2,d , and z3,d , respectively. z1,d , z2,d , and z3,d are based on the control objective. x1,d and x2,d will be designed later to facilitate the control algorithm. 2.4. Control actuators To realize the aforementioned air-path control objective, three actuators can be considered, including HPL-EGR valve, LPL-EGR valve, and HP-throttle. To reduce the complexity of the model, in this control algorithm design, the nonlinear effects of the actuators were bypassed by assuming the gas flow rates through these valves can be well controlled by the orifice equation. Thus the gas flow rate through these valves can be estimated as "   #)1=2   ( C D Av p pd 1=γ 2γ p ðγ−1Þ=γ W v ¼ pffiffiffiffiffiffiffiffiu 1− d , ð13Þ γ−1 pu RT d pu
γ=ðγ−1Þ }, and for subsonic flow {pd =pu 4 2=ðγ þ 1Þ

ðγ þ 1Þ=2ðγ−1Þ C D Av p pffiffiffi 2 W v ¼ pffiffiffiffiffiffiffiffiu γ , ð14Þ γ þ1 RT d for choked flow pd =pu ≤½2=ðγ þ1Þγ=ðγ−1Þ , where Av is the effective opening area for the valve, pu is the upstream stagnation pressure, T d is the downstream stagnation temperature, and pd is the downstream stagnation pressure. Adjusting the valve openings can control the effective areas in these equations. In this sense, if the gas flow rates can be determined as inputs by the control algorithm, then each required effective area can be calculated by the pressure difference across the valve and the downstream gas temperature.

3.1. Control algorithm design for the decomposed subsystems

α1 z1,s u1,s þα1 z1,s u2,s −α1 z1,s W in ¼ 0,

ð15Þ

α1 ðγT hegr −z2,s Þu1,s þ α1 ðγx1 −z2,s Þu2,s þ α1 ðz2,s −γT e ÞW in ¼ 0,

ð16Þ

α1 ðF 3 −z3,s Þu1,s þα1 ðx2 −z3,s Þu2,s ¼ 0:

ð17Þ

i.e., u1,s þ u2,s −W in ¼ 0,

ð18Þ

T hegr u1,s þx1 u2,s −T e W in ¼ 0,

ð19Þ

F 3 u1,s þ x2 u2,s : u1,s þ u2,s

ð20Þ

z3,s ¼

Without influencing the above conditions, here z2,s is selected to satisfy following equation: α2 ðx1 −γz2,s Þu2,s ¼ α2 ðx1 −γz2,d Þu2 :

ð21Þ

i.e., z2,s ¼

ðγz2,d −x1 Þu2 þ x1 u2,s : γu2,s

ð22Þ

Through (21) and (22), the slowly-varying part of z2 , i.e., z2,s , can be derived, which can be taken care of by the LPL-EGR and HP-throttle. To be noted, such a selection of z2,s will be used in the stability analysis in the next section. By (18) and (19) u1,s ¼

T e −x1 W , T hegr −x1 in

ð23Þ

u2,s ¼

T hegr −T e W : T hegr −x1 in

ð24Þ

The reduced dynamic system, therefore, can be derived as 

3. Singular perturbation control methodology In this section, a composite control law is devised based on the singular perturbation control method (Saberi & Khali, 1984). Unlike typical singular perturbation problems, there are two direct actuators (HP-throttle and HPL-EGR valve openings) and three control variables (pressure, temperature, and oxygen fraction) in intake manifold, as can be seen from Fig. 1. Note that LPL-EGR valve can be used to tune the gas conditions passing through the HP-throttle instead of directly controlling the intake manifold conditions. Therefore, additional authorities, including the desired

x1 ¼ α2 ðγT legr −x1 Þu3,s þ α2 ðγT a −x1 ÞW a þ α2 ðx1 −γz2,s Þu2,s ¼ f 1 ðx,zs ,us Þ, ð25Þ 

x2 ¼ α2 ½F 3 ðt−dÞ−x2 u3,s þ α2 W a ðF a −x2 Þ ¼ f 2 ðx,zs ,us Þ:

ð26Þ

With respect to (25) and (26), choose a Lyapunov function candidate (LFC) as V ¼ 12 θ1 ðx1 −x1,d Þ2 þ 12θ1 ðx2 −x2,d Þ2 : Then, 

ð∇ðx−xd Þ VÞ⋅½f ðx,zs ,us Þ−x1,d 

ð27Þ

F. Yan, J. Wang / Control Engineering Practice 21 (2013) 981–988 



¼ θ1 ðx1 −x1,d Þðf 1 ðx,zs ,us Þ−x1,d Þ þ θ1 ðx2 −x2,d Þðf 2 ðx,zs ,us Þ−x2,d Þ: ð28Þ Here ∇ðx−xd Þ V indicates the derivative of V with respect to x−xd . To assure that (28) is negative definite, let 

f 1 ðx,zs ,us Þ−x1,d ¼ −k1 ðx1 −x1,d Þ,

ð29Þ



f 2 ðx,zs ,us Þ−x2,d ¼ −k1 ðx2 −x2,d Þ,

ð30Þ

where k1 is a positive constant. Then, the followings hold: 

u3,s ¼

−k1 ðx2 −x2,d Þ þ x2,d −α2 W a ðF a −x2 Þ , α2 ½F 3 ðt−dÞ−x2 

ð31Þ



x1,d ¼ k1 ðx1 −x1,d Þ þ α2 ðγT legr −x1 Þu3,s þ α2 ðγT a −x1 ÞW a þ α2 ðx1 −γz2,s Þu2,s :

ð32Þ Through (32), the desired trajectories of x1 can be derived. Let τ ¼ t=ε, then a boundary-layer subsystem can be defined as dz1 ¼ α1 z1 u1,f þ α1 z1 ðu2,s ðx1 Þ þu2,f Þ−α1 z1 W in ¼ g 1 ðx,z,uf Þ, dτ

ð33Þ

dz2 ¼ α1 ðγT hegr −z2 Þu1,f þ α1 ðγx1 −z2 Þðu2,s ðx1 Þ þ u2,f Þ þ α1 ðz2 −γT e ÞW in dτ ¼ g 2 ðx,z,uf Þ, dz3 ¼ α1 ðF 3 −z3 Þu1,f þ α1 ðγx2 −z2 Þðu2,s ðx1 Þ þ u2,f Þ ¼ g 3 ðx,z,uf Þ: dτ

1 θ2 ‖z−zd ‖2 : 2





þ θ2 ðz2 −z2,d Þðg 2 ðx,z,uf Þ−εz2,d Þ þ θ2 ðz3 −z3,d Þðg 3 ðx,z,uf Þ−εz3,d Þ: ð37Þ To guarantee that (37) is negative definite, let 

ð38Þ



g 2 ðx,z,uf Þ−εz2,d ¼ −k2 ðz2 −z2,d Þ, 

ð40Þ

where k2 is a positive constant. Then, the following control law can be derived by solving (38)–(40) as 

−k2 ðz2 −z2,d Þ þ εz2,d −α1 ðz1 −γT e ÞW in −α1 ðz2 −γT e ÞA , α1 γT hegr þ α1 γx1

u2,f ¼ A−u1,f −

To be noted, the underlying essence of (32) and (44) is that the low-pressure loop EGR cannot fully achieve the fast transient performance (due to its large control volume). If the desired values of x1 and x2 have properly designed dynamics or values (as in Eqs. (32) and (44)), the transient performance will be improved. Therefore, the fast transient requirements can be transmitted to the HPL-EGR valve and HP-throttle through x1,d and x2,d . By this modification, according to (35), Eq. (40) becomes 

g 3 ðx,z,uf Þ−εz3,d ¼ −k2 ðz3 −z3,d Þ þα1 γðx2 −x2,d Þu2,f :

ð45Þ

To be noted, Eqs. (40) and (43) are only for the description of the control law design. By replacing x2 by x2,d , Eq. (40) does not hold and instead Eq. (45) holds. All the stability proofs with a Lyapunov function in appendix are based on Eq. (45). Thus, (28) and (37) can be rewritten as 

ð46Þ



≤−θ2 k2 ‖z−zd ‖2 þ β‖z−zd ‖‖x−xd ‖,

ð47Þ

where β ¼ α1 maxðu2,f Þ, and ‖d‖ represents the 2-norm. The composite control law, therefore, can be generated as u1 ¼ u1,s þ u1,f ,

ð48Þ

u2 ¼ u2,s þ u2,f ,

ð49Þ

u3 ¼ u3,s ,

ð50Þ

where u1,s refers to Eq. (23), u1,f refers to Eq. (41), u2,s refers to Eq. (24), u2,f refers to Eq. (42), and u3,s refers to Eq. (31). 3.2. Stability analysis for overall system

ð39Þ

g 3 ðx,z,uf Þ−εz3,d ¼ −k2 ðz3 −z3,d Þ,

u1,f ¼

ð44Þ

ð∇ðz−zd Þ ΓÞ½gðx,z,uf Þ−εzd  ¼ −θ2 k2 ‖z−zd ‖2 þ θ2 ðz3 −z3,d Þα1 γðx2 −xd Þu2,f

ð∇ðz−zd Þ ΓÞ⋅½gðx,z,uf Þ−εzd  ¼ θ2 ðz1 −z1,d Þðg 1 ðx,z,uf Þ−εz1,d Þ

g 1 ðx,z,uf Þ−εz1,d ¼ −k2 ðz1 −z1,d Þ,

α1 ðF 3 −z3 Þu1,f þα1 z3 ðA−u1,f Þ−εz3,d þ k2 ðz3 −z3,d Þ α1 ðu1,f −AÞ

ð35Þ

Then, the partial derivative of Γ with respect to ðz−zd Þ along gðx,z,uf Þ−ε̇z d can be written as 



x2,d ¼

ð∇ðx−xd Þ VÞ⋅½f ðx,zs ,us Þ−x1,d  ¼ −θ1 k1 ‖x−xd ‖2 :

ð36Þ



singularity of (43) can be avoided by properly selecting the control parameters. Since x2 is a state in the system, it cannot be predesigned to satisfy (43). Therefore, in the control law, x2,d is used instead of x2 . Then (43) becomes

ð34Þ

Regarding to the dynamics of z1 , z2 , z3 in the boundary-layer subsystem, choose a LFC as Γ¼

985

T e −x1 W , T hegr −x1 in

Theorem 1. The vector field f ðx,z,uÞ fulfills the following condition, i.e., a positive constant m exists, such that ‖f ðx,z,uÞ−f ðx,zs ,us Þ‖≤m‖z−zd ‖:

ð51Þ

Proof. See Appendix A.1. ð41Þ ð42Þ

Theorem 2. The control law designed in subsection III-B can guarantee that z1 , z2 , z3 in the system (8)–(12) converge to their references, z1,d , z2,d , z3,d , asymptotically, if the following condition holds



x2 ¼

α1 ðF 3 −z3 Þu1,f þ α1 z3 ðA−u1,f Þ−εz3,d þ k2 ðz3 −z3,d Þ , α1 ðu1,f −AÞ

ð43Þ

where

k1 k2 4 βm:

ð52Þ

Proof. See Appendix A.2.



−k2 ðz1 −z1,d Þ þεz1,d þ α1 z1 W in A¼ : α1 z1 

As can be seen, when z1,d is not extremely large and k2 is selected properly small, the value of A will be around W in . Thus the

To be noted, the proposed method can achieve certain robustness to model uncertainties by tuning the converging parameters k1 and k2 . A detail mathematic robustness analysis is not in the scope of this research.

F. Yan, J. Wang / Control Engineering Practice 21 (2013) 981–988

4. Experimental validations 4.1. Experimental setup The foregoing control methodology was experimentally investigated on a medium-duty diesel engine. The test bench set-up is shown in Fig. 4. The engine is equipped by a dual-loop EGR system, HP-throttle, and VGT. The control input signals utilized here are mass flow rates through the HPL-EGR valve, LPL-EGR valve, and the HP-throttle. The controlled outputs here are the intake manifold pressure p1 , temperature T 1 , and oxygen fraction F 1 . The temperatures and pressures in the intake and exhaust manifolds of the diesel engine were measured by thermocouples and pressure sensors. The oxygen fractions in intake and exhaust manifolds were measured by wide-band λ sensors. The control algorithm was implemented on a dSPACE MicroAutoBox real-time controller. In the experiments, the engine ran at a constant speed, i.e. 1500 rpm, and the fuel injection amount was 40 mg per cylinder per cycle. To provide the backpressure for the LPL-EGR, an exhaust valve with a constant opening area was equipped at the end of tailpipe.

changing the oxygen fraction by a single loop EGR will cause the intake temperature change simultaneously (HPL-EGR will increase the intake temperature and LPL-EGR will reduce the intake temperature). The capability of controlling the three intake manifold gas conditions in a decoupled fashion by appropriately coordinating the dual-loop EGR air-path system actuators was exhibited. As can be seen in Fig. 6, the intake gas pressure and oxygen fraction were required to change, while the intake gas temperature needs to remain constant. If only HPL-EGR was used, the intake temperature would be largely affected when the oxygen fraction changes. By adding the LPL-EGR, the decoupling of intake temperature and oxygen fraction control can be achieved. To be noted, during the transient, the intake pressure and oxygen 0.16

F1

986

Actual 0.14

Desired 0

20

40

60

80

100

120

140

Time [s] x 105

p1 (Pa)

1.15 1.1

Actual Desired

1.05 0

20

40

60

80

100

120

140

Time [s] T1 (K)

315

Actual Desired

310 305 0

20

40

60

80

100

120

140

u_hegr

Time [s]

0.4 0.2 0

20

40

60

80

100

120

140

80

100

120

140

80

100

120

140

Cycle 1

u_htr

To show the effectiveness of the proposed control algorithm, particularly the capability of controlling the three intake manifold gas conditions in a decoupled fashion by appropriately coordinating the DL-EGR air-path system actuators, three different engine operating cases are considered. As HPL-EGR is mostly used for low-load conditions and LPL-EGR is mostly used for high-load conditions, a region where engine would experience combined usage of both HPL-EGR and LPL-EGR was selected for the experimental investigations. In CASE 1, the control objective is to control the gas conditions in the intake manifold to track the desired trajectories. The control performances are shown in Fig. 5. Comparing with the performances in Fig. 5 and dynamics of HPL-EGR and LPL-EGR in Fig. 2, CASE 1 shows good performance of the singularly perturbed control law in the intake gas condition tracking control, i.e. the slow dynamics of LPL-EGR control were well compensated. The actuator traces are shown in Fig. 5 and the openings of HPL-EGR valve, HP-throttle, and the LPL-EGR valves are denoted as u_hegr, u_legr, and u_htr, with 0 indicating fully closed and 1 indicating fully open. Similar denotations apply for Fig. 7 as well. CASE 2 illustrates a decoupling aspect of the proposed control method. In CASE 2, the intake pressures and oxygen fractions vary according to the desired time-varying references and the intake temperatures are maintained close to a constant level. Such an effect requires coordinating control of dual-loop EGR, since

0.5 0 0

20

40

60

Cycle 0.25

u_legr

4.2. Experimental results

0.2 0.15 0

20

40

60

Cycle Fig. 5. Tracking control performance for intake gas conditions.

Fig. 4. A V8 medium-duty diesel engine test bench.

F. Yan, J. Wang / Control Engineering Practice 21 (2013) 981–988 Actual Desired

F1

0.2

0.15 0

5

10

15

20

25

30

35

40

Time [s] Actual

x 105

Desired

p1 (Pa)

1.3

PID control method, in terms of overshooting and transient speed. To be noted, by carefully tuning the PID parameters and adding feed-forward controls (through extensive calibrations), better transient performance of the PID control can be achieved. However, the cost is the extensive and costly calibration efforts on the feed-forward mapping and parameter selections/tuning, which however are not involved or necessary for the proposed singular perturbation control method.

1.25 1.2

5. Conclusions

1.15 0

5

10

15

20

25

30

35

40

Time [s] Actual

320

T1 (K)

987

Desired 310 300 0

5

10

15

20

25

30

35

40

Time [s] Fig. 6. Tracking control for decoupling intake gas conditions.

0.2

F1

Actual Desired

By decomposing the original system into two lower dimension sub-systems and designing the control laws separately, a composite feedback control law was derived to address the different time scales in DL-EGR systems. The effectiveness of such a control method for controlling the intake manifold gas conditions in diesel engine DL-EGR air-path systems within the specific operating region where the HPL-EGR and LPL-EGR overlap was demonstrated by experimental results on a medium-duty diesel engine. It was observed that the proposed method can appropriately coordinate the DL-EGR air-path actuators and the different timescale dynamics for achieving decoupled control of the intake manifold gas conditions, which is desirable for advanced combustion mode engines.

0.15 0

0.5

1

1.5

2

2.5

3

3.5

P1 (Pa)

Time (s) x 10

5

Actual Desired

1.1 1.05 1 0

0.5

1

Appendix A 1.5

2

2.5

3

3.5

A.1 Proof of Theorem 1

T1 (K)

Time (s)

By (45), (22), (8) and (25), the following holds:

325 320 0

0.5

1

1.5

2

2.5

3

3.5

 

ð53Þ

Time (s) u_hegr

 

f ðx,zÞ−f ðx,zs Þ ¼ α2 ðx1 −γz2 Þu2 −α2 ðx1 −γz2,s Þ ¼ α2 u2 γðz2 −z2,d Þ≤mz2 −z2,d ,

315

where m ¼ maxðα2 u2 γÞ. Considering (9) and (29)

1 0.5 0 0

0.5

1

1.5

2

2.5

3

3.5

f 2 ðx,zÞ−f 2 ðx,zs Þ ¼ 0:

ð54Þ

Time (s)

With (53) and (54), conclusion (51) can be easily achieved.

u_htr

1 0.5 0 0

0.5

1

1.5

2

2.5

3

3.5

u_legr

Time (s)

A.2 Proof of Theorem 2 Choose an overall LFC as: M ¼ ð1−λÞVðxÞ þλΓðzÞ, where λ is a constant such that 0 oλ o 1. According to (51), (46) and (47)

0.25 0.2 0

0.5

1

1.5

2

2.5

3

3.5

Time (s) Fig. 7. Tracking control for decoupling intake gas conditions (small time scale).

fraction had some mismatches, and then went back to the set values. This is because of the stronger coupling effect comparing to that in CASE 1. Another small time-scale transient test with actuator traces are shown in Fig. 7. In CASE 3, the control law was validated by comparing it to a conventional proportional-integral-derivative (PID) control algorithm, in which the intake temperature was controlled by HPL-EGR valve; the intake pressure was controlled by the mass flow rate through HP-throttle; and the oxygen fraction was controlled by LPL-EGR, separately. The performances of the two control methods are compared in Fig. 8. As can be observed, the singular perturbation method shows better transient performances than that of the

h i     M ¼ ð1−λÞV þλΓ ¼ ð1−λÞ⋅ð∇ðx−xd Þ VÞ⋅ f ðx,z,uÞ−x1,d   λ dz  þ ∇ðz−zd Þ Γ ε −εzd ε dτ h i  ¼ ð1−λÞ⋅ð∇ðx−xd Þ VÞ⋅ f ðx,zs ,us Þ−x1,d
 þ ð1−λÞ⋅ð∇ðx−xd Þ VÞ⋅ f ðx,z,uÞ−f ðx,zs ,us Þ  λ þ ⋅ð∇ðz−zd Þ ΓÞ⋅ðgðx,z,uf Þ−εzd Þ ε

¼ −θ1 ð1−λÞk1 ‖x−xd ‖2 þ θ1 ð1−λÞm‖x−xd ‖‖z−zd ‖ λ β −θ2 k2 ‖z−zd ‖2 þ θ2 ‖z−zd ‖‖x−xd ‖ ε ε !T ! ‖x−xd ‖ ‖x−xd ‖ Ω ¼− : ‖z−zd ‖ ‖z−zd ‖

ð55Þ

988

F. Yan, J. Wang / Control Engineering Practice 21 (2013) 981–988

x 105

340

0.2

1.2 1.15

300

0

1.1 1.05

PID 280

0.18

F1

p1(Par)

T1 (K)

320

0.16

PID 1

50

0

Time [s]

20

40

0

Time [s]

SP

SP

0.18

F1

p1(Pa)

T1 (K)

1.15 1.1 1.05

280

0

50

40

0.2

1.2

300

20

Time [s]

x 105 340

320

PID

1

0.16 0

Time [s]

20

Time [s]

40

SP 0

20

40

Time [s]

Fig. 8. Comparisons between the singular perturbation and the PID control methods.

Here, " θ1 ð1−λÞk1 Ω¼ −θ2 βε

−θ1 ð1−λÞm θ2 λε k2

# :

If the control law can guarantee that Ω is positive definite, then the asymptotic stability can be therefore ensured. As θ1 ð1−λÞ 4 0, this condition is equivalent to that the determinant of Ω is positive, i.e.,     ð56Þ Ω ¼ θ1 θ2 =ε½λk1 k2 ð1−λÞ−βð1−λÞm 4 0: Thus, (52) holds. As can be realized, for given λ, m, ε, and β, by choosing k1 and k2 properly, the condition (56) can be satisfied. Then, the derivative of M ¼ ð1−λÞVðxÞ þλΓðzÞ are negative definite with respect to the tracking errors. Thus, the conclusion of Theorem 2 holds. References Alriksson, M, & Denbratt, I. (2006). Low temperature combustion in a heavy duty diesel engine using high levels of EGR. In Proceedings of the SAE conference, 2006-01-0075. Ammann, M., Fekete, N. P., Guzella, L. , &Lattfelder, A. H. G. (2003). Model-based control of the VGT and EGR in a turbocharged common-rail diesel engine: Theory and passenger car implementation. SAE Paper 2003-01-0357. Castro-Linares, R, Alvarez-Gallegos, J A, & Vasquez- Lopez, V (2001). Sliding mode control and state estimation for a class of nonlinear singularly perturbed systems. Dynamics and Control, 11, 25–46. Chauvin, J., Corde, G., & Petit, N. (2006). Constrained motion planning for the airpath of a diesel HCCI engine. In Proceedings of the 45th IEEE conference on decision & control (pp. 3589–3596). Chauvin, J, Corde, G, Petit, N, & Rouchon, P (2008). Motion planning for experimental airpath control of a diesel homogeneous charge-compression ignition engine. Control Engineering Practice, 16(9), 1081–1091. Chiang, C, & Stefanopoulou, A G (2007). Stability analysis in homogeneous charge compression ignition (HCCI) engines with high dilution. IEEE Transactions on Control Systems Technology, 15(2), 209–219.

Grondin, O., Moulin, P., & Chauvin, J. (2009). Control of a turbocharged diesel engine fitted with high pressure and low pressure exhaust gas recirculation systems. In Proceedings of 48th IEEE conference on decision and control. Horibe, N, Harada, S, Ishiyama, T, & Shioji, M (2009). Improvement of premixed charge compression ignition-based combustion by two-stage injection. International Journal of Engine Research, 10(71), 71–80. Killingsworth, N. J., Aceves, S. M., Flowers, D. L., &Krstic, M. (2006). A simple HCCI engine model for control. In Proceedings of the 2006 IEEE international conference on control applications (pp. 2424–2430). Kolmanovsky, I., Moraal, M., Nieuwstadt, M., &Stefanopoulou, A. (1998). Issues in modeling and control of intake flow in variable geometry turbocharged engines. In Proceedings 18th IFIP conference on system modelling and optimization (pp. 436–445). Plianos, A., & Stobart, R. (2008). Modeling and control of diesel engines equipped with a two-stage turbo-system. SAE Paper, 2008-01-1018. Saberi, A, & Khali, H (1984). Quadratic-type Lyapunov functions for singularly perturbed systems. IEEE Transactions on Automatic Control, AC, 29(6), 542–550. Thring, R. H. (1989). Homogeneous-charge compression ignition engine. SAE Paper 892068. Utkin, V. I., Chang, H., Kolmanovsky, I., &Cook, J. (2000). Sliding mode control for variable geometry turbocharged diesel engines. In Proceedings of the American control conference, Chicago, IL, June. Wang, J (2008a). Hybrid robust air-path control for diesel engines operating conventional and low temperature combustion modes. IEEE Transactions on Control Systems Technology, 16(6), 1138–1151. Wang, J (2008b). Air fraction estimation for multiple combustion mode diesel engines with dual-loop EGR systems. Control Engineering Practice, 16(12), 1479–1486. Wang, J., & Chadwell, C. (2008). On the advanced air-path control for multiple and alternative combustion mode engines. SAE Paper 2008-01-1730. Yan, F., & Wang, J. (2009). Enabling air-path systems for homogeneous charge compression ignition (HCCI) engine transient control. In Proceedings of the ASME dynamic systems and control conference. Yan, F., & Wang, J. (2011). Control of dual loop EGR air-path systems for advanced combustion diesel engines by a singular perturbation methodology. In Proceedings of the 2011 American control conference (invited paper) (pp. 1561–1566). Yan, F, & Wang, J (2012). Pressure-based transient intake manifold temperature reconstruction in diesel engines. Control Engineering Practice, 20(5), 531–538. Zhang, F, Grigoriadis, K M, Franchek, M A, & Makki, I H (2007). Linear parametervarying lean burn air-fuel ratio control for a spark ignition engine. Journal of Dynamic Systems, Measurement, and Control, 129(4), 404–414.