Fuzzy Sliding Mode Control For Turbocharged Diesel

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ables: The EGR rate and the Air Fuel Ratio (AFR) .... p1V1. (WHT (γTHT −T1)+WHEGR(γTHEGR −T1). +We(T1 −γTe)). ˙p2 = Rγ. V2. (WcTc −WHT THT ). ˙T2 = RT2. p2V2. (Wc(γTc −T2)+WHT (T2 −γTHT )). (1) where p1 is the intake manifold pressure and p2 is the pressure in the control volume before the Hp-Throttle.
Fuzzy Sliding Mode Control For Turbocharged Diesel Engine Samia Larguech

Sinda Aloui

MIS Laboratory University of Picardie Jules Verne 33 rue Saint Leu, 80039 Amiens, France. STA laboratory National School of Engineering of Sfax BP 1173, 3038 Sfax, Tunisia. Email: [email protected]

STA laboratory National School of Engineering of Sfax BP 1173, 3038 Sfax, Tunisia. Email: [email protected]

Olivier Pagès

Ahmed El Hajjaji

MIS Laboratory MIS Laboratory University of Picardie Jules Verne University of Picardie Jules Verne 33 rue Saint Leu, 80039 Amiens, France. 33 rue Saint Leu, 80039 Amiens, France. STA laboratory STA laboratory Email: [email protected] Email: [email protected]

Abdessattar Chaari STA laboratory National School of Engineering of Sfax BP 1173, 3038 Sfax, Tunisia. Email: [email protected]

In this work, Fuzzy Second Order and Adaptive Sliding Mode Control are developed for a Turbocharged Diesel Engine (TDE). In control design the TDE is represented by Multi-Output Multi-Input (MIMO) nonlinear model with partially unknown dynamics. To regulate the intake manifold pressure, the exhaust manifold pressure, the compressor flow and to estimate the unknown functions, a Sliding Mode Control (SMC) combined with Fuzzy Logic are firstly developed. Secondly to reduce the chattering phenomenon without deteriorating the tracking performance, two approaches are investigated. A special case of the Second Order Sliding Mode Controller (2-SMC): the super twisting sliding mode controller is developed. The results obtained using the Adaptive Sliding Mode Control (ASMC) are also presented to compare the performances of both methods. All parameter adaptive laws and robustifying control terms are derived based on Lyapunov stability analysis, so that convergence to zero of tracking errors and boudedness of all signals in the closed-loop system are guaranteed. Simulation re-

sults are given to show the efficiency of the proposed approaches.

Nomenclature HP − EGR(HEGR) HP − T (HT ) p, pa Ta Tx Wx e, eo ic, ec t f P Vx F γ ηx c,υ R cp cυ

High-pressure EGR. High-pressure Throttle. Pressure, ambient pressure. ambient Temperature. Temperature x ∈ {HT, HEGR, e, eo, t, 1, 3}. Flow rate x ∈ {HT, HEGR, c, t, e, eo, f }. Engine, engine-out. Inter-cooler, EGR-cooler. turbine. fuel. Power. Volume x ∈ {1, 2, 3}. Air Fraction. Specific heat ratio. Efficiency x ∈ {t, c, υ, ic, ec}. compressor, Volumetric. Gas constant. Heat at constant pressure. Engine at constant volume.

1

Introduction In control engineering, most physical systems are nonlinear. Despite this fact, control based on linear system may work fine in many cases, whereas in other situations nonlinear effects should be taken into account in order to get a stable control system. During the past several years, fuzzy systems and neural networks have received more and more attention and, in particular, fuzzy control has emerged as one of the most active areas for research in application of fuzzy set theory [1, 2, 3, 4]. Several adaptive fuzzy control schemes have been introduced for controlling both Single Input Single Output (SISO) [5, 6, 7] and Multiple Input Multiple Output (MIMO) nonlinear systems [8, 9, 10]. The turbocharged diesel engine is a Multi-Input Multi-Output (MIMO) system characterized by actuator constraints, strong coupling, and fast dynamics. In order to reduce exhaust emission, automotive industry has developed new efficient technologies. In fact, establishing an environmentally friendly diesel engine will become the most important issue in the future engine technology. The Selective Catalytic Reduction (SCR) is an advanced active emission control technology system that injects a liquid-reductant agent through a special catalyst into the exhaust stream of a diesel engine. The Diesel oxidation catalysts (DOCs) and the Diesel particulate filters (DPFs) are exhaust after-treatment devices that reduce emission from diesel engines. Actually, DOCs are widely used as a retrofit technology because they require little or no maintenance. Engine manufacturers have used DOCs in a variety of applications for many years. Normally, modern diesel engines are equipped with a Variable Geometry Turbocharger (VGT) and Exhaust Gas Recirculation (EGR) valves to control the NOx emissions coming from the oxidation of the nitrogen monoxide in the combustion chamber. The NOx emissions can be mainly identified with two feedback variables: The EGR rate and the Air Fuel Ratio (AFR) in the intake manifold which depend on the position of the EGR and VGT valve actuators [11]. To reduce the emissions, a coordinated control of the two actuators is needed. Considerable research efforts have been dedicated to the control of modern diesel engines. In literature, different control approaches of the combustion engine air path have been proposed [12], [13]. In [14], the authors proposed a Lyapunov control approach to handle a variable geometry turbocharger and exhaust gas recirculation valve for exhaust manifold pressure and fresh airflow rate regulation. The control law deals with uncertain parameters and nonlinear system properties. But, this control design requires the construction of Control Lyapunov Function which is not easy and may be quite restrictive. The authors, in [15], proposed a strategy to control the diesel engine air path. The control design is carried out under the sliding mode framework and has been tested on the simplified Jankovic turbocharged

diesel engine model. In addition, many other techniques to control the diesel engine have been generated: the feedback linearization [16], [17]; the predictive control [18], [19]; fuzzy logic control [20] ... Known for its simple structure, its simple implementation and its robustness to external disturbances, the Sliding Mode Control (SMC) has been a topic of great interest in control theory and has represented a great potential for practical applications. However, it usually suffers from a main disadvantage: the chattering phenomenon. In the literature, several methods of chattering reduction have been reported. In [21], the authors suggested the boundary layers approach. This method consists in replacing the discontinuous switching action by a continuous saturation function. In fact, this approach is generally appropriate for low disturbances and it requires an approximation of the term of discontinuity. Furthermore, in [22] an asymptotic observer is presented to eliminate the chattering phenomenon. The application of such observer assumes that the unmodelled dynamics are completely unknown. Another way to solve this chattering problem is based on combining the SMC and intelligent controllers to approximate the switching control term such as in [23] and [24]. The free parameters of the adaptive fuzzy controller can be tuned on-line based on the Lyapunov approach [25] and [26]. In [27], the authors proposed a method to eliminate the chattering phenomenon by using an adaptive Proportional Integral controller (PI controller) for a SISO nonlinear system. The MIMO nonlinear systems are investigated in [28] and [29]. In this work, an air path fuzzy second order and adaptive sliding mode controller are designed for a turbocharged diesel engine. By considering that the structure of the controlled system is supposed partially unknown, our objective is to regulate the intake manifold pressure, the exhaust manifold pressure as well as the compressor flow. To reduce the chattering phenomenon, we proposed a comparison of different methods. First, a special case of the Second Order Sliding Mode Controller (2-SMC): the super twisting sliding mode controller is developed. Second, we decided to substitute the discontinuous term of the classical sliding mode law with an adaptive Proportional Derivative (PD) term. The sections of this paper are organized as follows. The description and the modeling of the turbocharged diesel engine are given in Section 2. In Section 3, we introduce the First Order Sliding Mode Control (1SMC).The fuzzy controller design is given in 4. The super twisting algorithm and the adaptive sliding mode control are developped in 5. Simulation results are presented in 5 to show the effectiveness of the proposed control strategies. Finally, Section 6 gives conclusions on the main works developed in this paper.

2 Diesel Engine Model 2.1 Description of the turbocharged diesel engine The system under consideration is a light-duty 4cylinder common rail diesel engine (Figure 1 [30]). The engine is provided with a Variable Geometry Turbocharger (VGT), High-Pressure loop EGR (HPEGR) and high-pressure throttle. The turbocharger consists of a turbine and a compressor mounted on the same shaft to compress the intake fresh air from Mass Air Flow (MAF). The turbine takes the energy from the exhaust gas to power the compressor which boosts the intake manifold pressure. The air mixture from the compressor and exhaust gas coming through the EGR valve is pumped from the intake manifold into the cylinders. The fuel is injected directly into the cylinders and burnt, producing the torque on the shaft. The hot exhaust gas is pumped out into the exhaust manifold. Part of the exhaust gas flows from the exhaust manifold through the turbocharger out of the engine and the other part is recirculated back into the intake manifold. The exhaust-treatment system consists in a Diesel Oxidation Catalyst (DOC), Diesel Particulate Filter (DPF) and Lean (NOx) Trap (LNT). An universal exhaust gas oxygen (UEGO) sensor with pressure compensation is installed at the exhaust manifold (close to the exhaust manifold runners). The intercooler and the EGR-cooler are used to reduce the intake manifold temperature. The highpressure throttle, the throttle valve equipped after the compressor/intercooler, is used to reduce the intake manifold pressure and therefore increases the EGR rate at light load for alternative combustion modes.

control systems, based on the mass and energy conservation as well as ideal gas laws, the engine intake dynamic model can be obtained as [31], [30]:

 Rγ   (WHT THT + WHEGR THEGR − We Te ) p˙1 =   V1     RT1   T˙1 = (WHT (γTHT − T1 ) + WHEGR (γTHEGR − T1 )    p1 V1  +We (T1 − γTe ))    Rγ   (Wc Tc − WHT THT ) p˙2 =    V2     RT2   T˙2 = (Wc (γTc − T2 ) + WHT (T2 − γTHT )) p2 V2 (1) where p1 is the intake manifold pressure and p2 is the pressure in the control volume before the Hp-Throttle. WHT , WHEGR , We and Wc represent respectively the High-pressure Throttle mass flow rate, the Highpressure EGR mass flow rate, the engine intake mass flow rate and the compressor mass flow rate. THT , THEGR , Te , T1 , Tc and T2 are respectively the temperature in the High-pressure Throttle, the High-pressure EGR, the engine, the intake manifold, the compressor and the control volume before the Hp-Throttle. c γ is the specific heat ratio = cvp and R is gas constant. where cp is heat at constant pressure and cυ is heat at constant volume. V1 and V2 are the volume of the intake manifold and the volume before the Hp Throttle. The engine intake gas flow rate We can be calculated by the speed density equation as follows:

We =

ηυ p1 Ne Vd 120RT1

(2)

where ηυ is the volume efficiency, Ne is the engine speed and Vd is the engine displacement. Similar modeling approach can give the dynamics of the pressure and temperature of the exhaust system as:

Figure 1: Schematic diagram of a TDE.

2.2

Modeling of the turbocharged diesel engine Similar as the derivation of mean-value engine models, which are widely used in production engine

 Rγ  (Weo Teo − T3 (WHEGR + Wt ))  p˙3 =   V3  RT3 T˙3 = (Weo (γTeo − T3 )   p3 V3    +Wt (T3 − γTt ) + WHEGR (T3 − γTHEGR ))

(3)

where p3 is the exhaust manifold pressure and T3 is the temperature of the exhaust system.

Weo = We + Wf is the gas flow rate coming out of the cylinders with Wf being the fueling rate. Teo the temperature coming out the cylinder and Wt is the turbine mass flow rate. In this work, the gas flow rates through gas handling devices (HP-throttle: WHT , EGR valve: WHEGR , and VGT: Wt ) are treated as control inputs. The gas mass flow rates through the HP-throttle valve and EGR valve can be modeled using the standard orifice flow equation as [14]:  pu Av (uv ) √RT   u " #    γ+1   γ+1    2(γ−1) 2(γ−1) 2 2   × γ 0.5 γ+1 , ppud ≤ γ+1    r  Wv = pu 2γ  √ A (u ) v v  γ−1 RTu   " #     γ2   γ+1   γ+1   γ 2(γ−1) pd pd pd 2   × , > −  pu pu pu γ+1 (4) where υ ∈ {HT, HEGR}, Av (uv ) is the effective area of the valves as a function of uv ∈ [0, 100] being the valve opening positions, pd is the downstream pressure and pu is the upstream pressure, and Tu is the up-stream temperature. The gas mass flow rate through turbine can be approximately modeled using a modified orifice equation [30]:  W t = At

pt , uV GT p3

 √

p3 ψ RT3



pt , uV GT p3

(5)

(6)

where Pc and Pt are respectively the compressor and the turbine power and τc is the time constant. pt 1− p3

k !

k

!



Pt = Wt Cp T3 ηt

1 Pc = Wc Cp T4 ηc



p2 p4

(7)

−1

The gas temperature reduction due to EGR cooler and intercooler are approximated using the linear heat exchanger effectiveness ηhe as well as the upstream temperature Tup and coolant temperature Tcoolant as follows: Tdown = ηhe Tcoolant + (1 − ηhe )Tup

(9)

So, for the EGR gas, when the path with EGR cooler is turned on by the switching valve, the high-pressure EGR temperature becomes: THEGR = ηec Tcoolant + (1 − ηec )T3

(10)

The temperature of the gas passing through the highpressure throttle is: THT = ηic Tcoolant + (1 − ηic )T2

(11)

where ηec and ηic are respectively the efficiency for the EGR cooler and for the inter-cooler.



The VGT position uV GT can be calculated inversely from (5) once the turbine flow rate is specified by the controller. For control design purpose, the turbocharger dynamics can be simplified as a first-order system: 1 P˙c = (Pt − Pc ) τc

p4 = pa with pa being the ambient pressure, which is assumed to be known and T4 = Ta being the inlet air temperature, which is usually available from an MAF sensor equipped on the engine.

(8)

where ηt is the turbine efficiency, ηc is the compressor cp efficiency and k = γ−1 γ and γ = cv .

Remark 1: Due to the difficulty of measuring the temperatures with sufficiently fast responses, we ignored their dynamics. However, observers based on the temperature dynamics can be developed to estimate the actual temperatures. The resultant dynamic model of the turbocharged diesel engine can be summarized as:  υ Ne Vd HT p1 + RγT WHT + RγTVEGR WHEGR p˙1 = − γη120V  V1  1 1  RγT RγT η P  c c c HT    W p ˙ = −  HT  V2 p2 k  2 V C T  −1 2 p 4 p4   Rγ Teo ηυ Ne Vd  p ˙ = p + T W − T W − T W  3 V3 eo f 3 HEGR 3 t 120RT1 1    k    C T η  p p t P 3  P˙c = − c + 1 − p3t Wt τtc τtc (12) The Air Fuel Ratio (AFR) is the mass ratio of air to fuel present in a combustion process such as in an internal combustion engine. If exactly enough air is provided to completely burn all of the fuel, the ratio is known as the stoichiometric mixture. For diesel engines, the combustion is usually lean, which means that air is in excess of stoichiometric amounts in the cylinder mixture. Thus, the exhaust gas contains unburned air and it could be recirculated back into the intake manifold through EGR valve. The fraction of the air (or

EGR gas) in cylinder is very important for combustion and emissions performance, especially for the alternative combustion modes, which are close to the stable edges. The dynamics of the intake manifold and exhaust manifold fresh air fractions can be described as:

kr1 =

2.3

RγTeo RγTEGR , kf = , K f = k f Wf V1 V3

Choice of system output for conventional diesel combustion mode

where Feo is the fresh air fraction coming out of the cylinder. Remark 2: In this work, we supposed that engine speed Ne , fueling rate Wf and exhaust manifold fresh air fractions F3 are known as external bounded disturbances. The system can be rewritten in a compact form as:

A conventional diesel combustion mode is characterized, as detailed in [30], by a high fresh air flow rate, a relatively low EGR rate, a high AFR (Air Fuel Ratio) (20∼28), a high intake manifold pressure (above ambient pressure), a lower exhaust gas temperature at light-load. The general state model of the turbocharged diesel engine includes five states and only three available control inputs (corresponding to HP-throttle, EGR, and VGT). At this combustion mode and in order to fulfill the square property of designing the MIMO tracking controller, we need to select the controlled outputs. Knowing that intake manifold pressure (p1 ), fresh air charge (Wc ) and EGR rate (p3 ) play an important role for this combustion mode beside fueling parameter, the authors of [30] have chosen the following controlled outputs:

x˙ = a(x) + B(x)u

 T y = p1 Wc p3

RT1 F˙1 = ((1 − F1 )WHT + (F3 − F1 )WHEGR ) p1 V1

RT3 F˙3 = Weo (Feo − F3 ) p3 V3

(13)

(14)

(15)

(16)

We note T

T

x˙ = [x˙ 1 x˙ 2 x˙ 3 x˙ 4 x˙ 5 ] , u = [u1 u2 u3 ]

 T y = y1 y2 y3 where x1 = p1 , x2 = p2 , x3 = p3 , x4 = 1, Pc , x5 = F −k1 x1   k2 x4   xk2 −pka   u1 = WHT ,u2 = WHEGR , u3 = Wt . a =  k3 x1 + Kf    x4  −τ tc   0 kht1 kr1 0  −kht2  0 0    0 −kr3 −kt3  B=    0 0 kt1 − kxt2 k   3 k4 (1 − x5 ) k4 (F3 − x5 ) 0 where: γηυ Ne Vd RγTc ηc pka γTeo ηυ Ne Vd k1 = , k2 = , k3 = 120V1 V2 Cp Ta 120V3 T1

k4 =

kt1 =

RγTHT RγTHT RT1 , kht1 = , kht2 = p1 V1 V1 V2

Cp T3 ηt Cp T3 ηt pka RγT3 , kt2 = , kr3 = kt3 = τtc τtc V3

where y1 = x1 = p1 , y2 = Wc and y3 = x3 = p3 . Fresh airflow rate through the compressor can be calculated as: Wc = Cp T4

P η  c c k p2 p4

 −1

(17)

The time derivative of fresh airflow rate (Wc ) is given by: Wc W˙ c = − + c(x)u3 − d(x)k2c Wc2 + d(x)Wc kht2 u1 τtc (18) where:   k  ηc T3 ηt 1 − xp3t kpk−1 RγTc    , d(x) = k 2 c(x) = , k2c = k k V2 p2 − pa x2 τtc Ta −1 pa Remark 3: We notice that the system has a singularity at x2 = p2 = pa . As solution, we will choose p2 as constant such that p2 6= pa .

Finally, the dynamics of the output variables are described as:

the following expression: S˙ = f (x) + G(x)u − y˙d + ke

y˙ = f (x) + G(x)u

(23)

(19) The equivalent control term is defined as:

 −k1 x1  T 2 c u = u1 u2 u3 , f (x) =  − W τtc − d(x)k2c Wc k3 x1 + kf W f 

ueq = G−1 (x)[−f (x) + y˙d − ke]

The switching control term is defined by the following expression:



 kht1 kr1 0 G(x) = d(x)Wc kht2 0 c(x)  0 −kr3 −kt3

usw = G−1 (x)(−ηsign(S)), η > 0

3

First order Sliding Mode Control (1-SMC) To develop the sliding mode approach for the MIMO system, two steps are required. First, the choice of sliding surface and second the calculation of the control law.

3.1

Integral Sliding Surface The system tracking error is defined as: e = y − yd

(20)

where yd is the vector of desired trajectories.  T yd = yd1 yd2 yd3 The integral vector sliding surface is defined by the following expression: Zt S = e+k

 T e(τ )dτ = s1 s2 s3

(21)

0

where

(24)

(25)

 T where sign(S) = sign(s1 ) sign(s2 ) sign(s3 ) and (sign is the signum function.) Thus, the classical sliding mode control law is given by: u = G−1 (x)(−f (x) + y˙d − ke − ηsign(S))

(26)

In this work, our objective is to develop an adaptive sliding mode control law for a class of nonlinear system: The Turbocharged Diesel Engine and to force vector "y" to follow given reference signals "yd ". First, we have started with known nonlinear functions f (x) and G(x) in order to develop the classical sliding mode control. Secondly, to test our approaches and our developed control laws, we assume that the nonlinear functions f and G are partially unknown so the control law (26) is no longer applied. In order to approximate those functions, we will introduce adaptive fuzzy logic systems.

4

Design of the fuzzy controller Step 1 : Approximation of f (x) and G(x) In this paper, fuzzy logic systems are used to approximate nonlinear functions f (x) and G(x). A fuzzy logic system consists of four parts: the knowledge base, the fuzzifier, the fuzzy inference engine working on fuzzy rules and the defuzzifier as shown in Figure.2.

  k = diag( k1 k2 k3 ); ki > 0, ∀i = 1, 2, 3

3.2

Control law The control law includes two terms: continuous term ueq , known as the equivalent control and switching term usw , known as the discontinuous control. Control law is designed as follows: u = ueq + usw

(22)

Equivalent control law ueq is determined by S˙ = 0. The time derivative of the sliding surface is defined by

Figure 2: Block diagram of a fuzzy logic system [32].

The fuzzy system consists of a collection of fuzzy IF-THEN rules: Rl : IF x1 is Al1 and x2 is Al2 and...and xn is Aln T HEN z is B l  i where Ali ∈ A1i , ..., An , i = 1, ..., n, B l are fuzzy i sets defined respectively for xi , i = 1, ..., n and z, and n Q ni is the total number of rules. N= i=1

Using the singleton fuzzifier, product inference engine, and center average defuzzifier, the final output of the fuzzy logic system can be expressed as:

 θgT11 . . . θgT1p   where θgT =  ...  ; Φ(x) = diag [ξ(x) . . . ξ(x)] T T θgp1 . . . θgpp Step 2 : Control law design Using the same reasoning as the first order sliding mode control in order to develop the control law and by replacing functions f (x) and G(x) with fˆ(x θf ) and ˆ |θg ) respectively, the fuzzy control law is described G(x by the following expression: 

h i ˆ −1 (x |θg ) −fˆ(x θf ) + y˙ d − k(y − yd ) − ηsign(S) u=G (32)

T

z = θ ξ(x)

(27)

where θT = [θ1 , θ2 , ..., θN ] is a vector grouping all consequent parameters and ξ T = [ξ1 , ξ2 , ..., ξN ] is a set fuzzy basis functions defined as: n Q

ξk (x) =

µAk (xi ) i i=1 N n P Q (

k=1 i=1

(28)

µAk (xi ))

5

Reduction of Chattering phenomenon The presence of the signum in the expression of the control law (32) leads to the chattering phenomenon which can excite the high frequency dynamics. In order to reduce this phenomenon and to achieve the control objective, two approaches are presented: the super twisting algorithm and the adaptive sliding mode control design.

i

5.1 where µAk (xi ) is the membership function and reprei

sents the fuzzy meaning of the symbol Aki . θk is the value of the singleton associated with B k . To approximate functions fi (x) and gij (x), i, j = ˆ 1, ..., 3, we used outputs fi (x θfi ) and gˆij (x θgij ) obtained from 27:

fˆi (x θfi ) = θfTi ξ(x), gˆij (x θgij ) = θgTij ξ(x)

(29)

T  fˆ(x θf ) = fˆ1 (x θf1 ) fˆ2 (x θf2 ) fˆ3 (x θf3 ) = θfT ξ(x) (30)   θf = θf 1 θf 2 θf 3 where θfi and θgij are the vectors of adjustable pah iT rameters defined as: θfi = θf1i ...θfNi and θgij = iT h θg1ij ...θgMij  gˆ11 (x |θg11 ) . . . gˆ1p (x θg1p )  .. T ˆ |θg ) =  G(x   = θg Φ(x) . gˆp1 (x θgp1 ) . . . gˆpp (x θgpp ) (31) p=3 

Super-twisting algorithm (2-SMC) The main feature of this strategy is that the discontinuous part appears on the derivative of the control law. So, by calculating the system control law, it becomes continuous and limits the chattering phenomenon. The super twisting algorithm has become the prototype of second-order sliding mode algorithm. In conventional SMC design, the control target is to move the system state into sliding surfaces S = 0. Super-Twisting Algorithm Second-Order Sliding Mode Controller aims to S = S˙ = 0. That is, the system states converge to zero at the intersection of S and S˙ in the state space. Actually, this algorithm is only applied for the systems whose relative degree is equal to one. It achieves finite time convergence by means of a continuous action, without using information about derivatives of the sliding constraint. Thus, chattering associated to traditional sliding-mode controllers is reduced. The algorithm convergence is governed by the rotation around the origin of the phase diagram (also called Twisting). The switching control term of the super twisting algorithm is defined by two terms: u1 and u2 . usw = −u1 − u2

(33)

p where u˙1 = αsign(S) and u2 = β |S|sign(S) |.|: denotes the norm of a vector. The equivalent control term is defined as: h i ˆ −1 (x |θg ) −fˆ(x θf ) + y˙ d − ke ueq = G

(34)

The switching control term is defined by the following expression:

1 1 ˜T ˜ 1 ˜T ˜ V = SS T + θf θf + θ θg 2 2γf 2γg g

Zt usw = −

Let us consider the following Lyapunov function:

αsign(S)dτ − β

p

|S|sign(S)

(39)

(35) where γf and γg are positive adaptive gains. The time derivative of V is given by:

0

where α and β are positive constant gains. Thus, the control law is given by:

˙ + 1 tr(θ˜T θ˜˙f ) + 1 tr(θ˜gT θ˜˙g ) V˙ = tr(S T S) f γf γg

(40)

ˆ −1 (x |θg )[−fˆ(x θf ) + y˙ d − k(y − yd ) u=G Zt −

αsign(S)dτ − β

p

|S|sign(S)]

(36)

0

Parameter vectors θf and θg are adjusted on line by the following adaptive laws:

The time derivative of the sliding surface is given by equation (23). By replacing the control law with new ˆ |θg ) × expression (36) and adding and subtracting (G(x u), (23) is rewritten as follows: ˆ |θg ) ]u S˙ =[f (x) − fˆ(x θf ) ] + [G(x) − G(x Zt



θ˙f = γf ξf (x)S T θ˙g = γg Φ(x)U S T

− (37)

Theorem 1: Consider the class of MIMO nonlinear systems (19), if the control law (36) is applied, where ˆ |θg ) are given by (30) and functions fˆ(x θf ) and G(x (31), respectively. Parameters vectors θf and θg are adjusted on-line by applying adaptive laws (37). Then, the proposed control scheme guarantees the following properties: -The semi-global asymptotic stability is guaranteed; -The tracking errors converge to zero. Proof: First, let us define the following variables: Optimal parameter vectors θf∗ and θg∗ of the previous fuzzy systems:



θf∗ = arg min( sup f (x) − fˆ(x θf ) ) θf ∈Ωθ

f

θg∗

αsign(S)dτ − β

p

|S|sign(S)

0

From (38), we obtain: ˆ θg∗ )u + w f (x) + G(x)u = fˆ(x θf∗ ) + G(x

(42)

According to (42), (41) is written as follows: ˆ θ∗ ) − G(x ˆ |θg ) ]u + w S˙ =[fˆ(x θf∗ ) − fˆ(x θf ) ] + [G(x g Zt −

αsign(S)dτ − β

p |S|sign(S)

0

(43) From θ˜f = θf∗ − θf and θ˜g = θg∗ − θg , (43) is rewritten as follows:

x∈Ωx



ˆ |θg ) = arg min( sup G(x) − G(x

)

S˙ =θ˜fT ξ(x) + θ˜gT Φ(x)u + w −

θg ∈Ωθg x∈Ωx

Zt αsign(S)dτ 0

where Ωθf and Ωθg denote the sets of suitable bounds on θf and θg , respectively. Ωx denotes the compact set of the suitable bounds on x. Assume that the constraint sets Ωθf and Ωθg are specified as:

 Ωθf = θf : θf ≤ Mf , Ωθg = {θg : kθg k ≤ Mg } Mf and Mg are predefined parameters. This assumption is essential for the universal approximation theorem. The fuzzy approximation error is written: ˆ θ∗ )]u w = f (x) − fˆ(x θf∗ ) + [G(x) − G(x g

(41)

(38)

(44)

p −β |S|sign(S) By substituting (44) in (40) and with (30) and (31) we obtain:  i 1 h V˙ =tr θ˜fT ξ(x) + θ˜gT Φ(x)u S T + tr(θ˜fT θ˜˙f ) γf t Z 1 + tr(θ˜gT θ˜˙g ) − αtr(S T sign(S)dτ ) γg 0 h i p T −βtr(S |S|sign(S)) + tr wS T

i 1 h  V˙ = tr θ˜fT γf ξ(x)S T + θ˜˙f γf i i h 1 h  + tr θ˜gT γg Φ(x)uS T + θ˜˙g + tr wS T γg (45) t Z p −αtr(S T sign(S)dτ ) − βtr(S T |S|sign(S))

where kpj and kdj , j = 1 . . . 3 are the control gains adjusted online from an adaptive law. The adaptive PD term derived from (47) can be rewritten as:  T uP D = ρ (S |θρ ) = ρ1 (s1 |θρ1 ) . . . ρp sp θρp ,p = 3

0

Using the fact that θ˜˙f = −θ˙f and θ˜˙g = −θ˙g , and by substituting (37) in (45), we obtain:

iT h uP D = Θ(S)θρ = θT ρj Θ(s1 ) . . . θT ρp Θ(sp ) , p = 3

(48) where θρj is the adjustable parameter vector given by p h iT   V˙ = − αtr(S T sign(S)dτ ) − βtr(S T |S|sign(S)) d sj (t) are the θρj = kpj ,kdj and ΘT (sj ) = sj (t), dτ h iT 0 i h regressive vectors, j=1...3, θρ = θpT1 . . . θpTp , Θ(S) = T +tr wS   diag ΘT (s1 )...ΘT (sp ) . By replacing the discontinuous term of the classical sliding mode law with the adaptive Proportional Derivative (PD) term and adding a term of robustness u1 in Zt p order to cancel the effect of the error of approximation, V˙ ≤ (S T w)−αtr(S T sign(S)dτ )−βtr(S T |S|sign(S)) the control law is written as follows: Zt

0

h i ˆ −1 (x |θg ) −fˆ(x θf ) + y˙ d − ke − uP D + u1 (49) u=G p V˙ ≤ |S|w − α|S|t − β|S| |S|

where 

V˙ ≤ |S|(w − αt − β

p

|S|))

uP D = Θ(S)θρ u1 = w ˆP D

(50)

The parameter vectors θf , θg and θρ are adjusted online by the following adaptive laws: V˙ ≤ 0, ∀(αt + β

p |S|) > w

(46)   θ˙f = γf ξf (x)S T θ˙ = γg Φ(x)U S T  ˙g θρ = γρ Θ(S)T S

√ w−β |S| Which mean that ∀t > | |, V˙ ≤ 0; so we guarα antee the semi-global asymptotic stability and all the tracking errors asymptotically converge to zero.

5.2

Fuzzy Adaptive Sliding Mode Control In this section, we propose to compare the performances of the proposed approach (the super twisting sliding mode control) with the Adaptive SMC method inspired from the method proposed [33]. In fact, in [33] the authors has substituted the discontinuous term of the classical sliding mode law with an adaptive Proportional Derivative (PD) term. The expression of the Proportional Derivative term is written as follows:  d s1 (t) kp1 s1 (t) + kd1 dτ   uP D =  ... ,p = 3 

d kpp sp (t) + kdp dτ sp (t)

(47)

(51)

where γf > 0 , γg > 0 and γρ > 0 are the adaptation gains. More details can be found in [33].

6

Simulation Results The numerical illustration considers the nonlinear diesel engine model defined in section 2 and the computations are performed with MATLAB. Intake manifold pressure (p1 ), exhaust manifold pressure (p3 )and compressor c ) ranges of variation  flow (W 5 , 1.15 × 105 (P a), p ∈ are respectively: p ∈ 1.1 × 10 1 3   1.3 × 105 , 1.4 × 105 (P a) and Wc ∈ [0.02, 0.022] (kg/s). Desired trajectories p1d , Wcd and p3d are chosen based on the simulation results shown in [30]. In this work, we started with known nonlinear functions f and G based on the chosen Turbocharged

Table 1: The parameter values of TDE

5

x 10

p1 Desired

Intake manifold pressure p1 (Pa)

1.15

p1

1.14

1.13

1.12

1.11

1.1 5

10

15

20

25 Time (sec)

30

35

40

45

50

Figure 3: Intake manifold pressure response p1 (Pa) (1SMC).

5

x 10

p3 Desired

1.42

Exhaust manifold pressure p3(Pa)

Diesel Engine Model and we have developed the classical sliding mode control. Then, we have assumed that those functions are partially unknown and we have introduced fuzzy logic systems to approximate them. For that purpose, three memberships functions    1 yi +c 2 are chosen: µP ositive = exp − 2 σ , µN egative =       2 2 , µZero = exp − 12 yσi exp − 12 yiσ−c where i = 1, 2, 3; c and σ are chosen differently for each yi . In fact, to cover the controllability region, 32 possible combinations should be obtained. This consequently leads not only to better performances but also to a relative complexity and a long computation time. The different parameter values are summarized into Table 1.

p3

1.4 1.38 1.36 1.34 1.32 1.3 1.28 1.26

Variable

Value.Unit

Ne

1500 rpm

Wf

6 kg/h

Vd

0.002 m3

V1

0.006 m3

V2

0.002 m3

V3

m3

0

5

10

15

20

25 Time (sec)

30

35

40

45

Figure 4: Exhaust manifold pressure response p3 (Pa) (1-SMC).

0.0225 W

c

Ta

300 K

T1

313 K

T3

762.7 K

0.0215

0.021

0.0205

0.02

0.0195

THT

343.2 K

TEGR

773.6 K

Tc

300 K

R

287 J/KgK

cp

1014.4 J/KgK



727.4 J/KgK

ηc

0.61

ηt

0.76

ηυ

0.87

τt c

0.11

Desired

Wc

0.022

Fresh airflow rate Wc (Kg/s)

0.001

50

0

5

10

15

20

25 Time (sec)

30

35

40

45

50

Figure 5: Fresh airflow rate response Wc (kg/s) (1SMC).

e1 (p1)

10

0

−10

0 −4 x 10

5

10

15

20

25

30

35

40

45

50

0

5

10

15

20

25

30

35

40

45

50

0

5

10

15

20

25 Time (sec)

30

35

40

45

50

e2 (Wc)

2 0

−2

e3 (p3)

10 0 −10

Figure 6: Evolution of tracking errors (e1 , e2 , e3 )(1SMC).

0.018 0.0178 0.0176

Control law u1 (Kgs−1)

The simulation results of the turbocharged diesel engine are shown in Figures 2-22. In order to show the efficiency of the proposed methods, we have presented the simulation results of the first order Sliding Mode Control (1-SMC) (Figures 2-8), the second order Sliding Mode Control (2-SMC) (Figures 9-15) and the Fuzzy Adaptive Sliding Mode Control (ASMC) (Figures 1622) .

0.0174 0.0172 0.017 0.0168 0.0166 0.0164

• Simulation trajectories of the first order sliding mode control

0

5

10

15

20

25 Time (sec)

30

35

40

45

Figure 7: Evolution of control law u1 (1-SMC).

50

−3

3.55

x 10

0.025 Wc Desired 0.024

Fresh airflow rate Wc (kg/s)

3.5

Control law u2 (Kgs−1)

3.45 3.4 3.35 3.3 3.25

0.022 0.021 0.02 0.019 0.018

3.2 3.15

Wc

0.023

0

5

10

15

20

25 Time (sec)

30

35

40

45

0.017

50

Figure 8: Evolution of control law u2 (1-SMC).

0

5

10

15

20

25 Time (sec)

30

35

40

45

50

Figure 12: Fresh airflow rate response Wc (kg/s) (2SMC).

10 −3

e1 (p1)

x 10

0 −10 2

e2 (Wc)

8

7.5

0

5

10

15

20

25 Time (sec)

30

35

40

45

Figure 9: Evolution of control law u3 (1-SMC).

15

20

25

30

35

40

45

50

5

10

15

20

25

30

35

40

45

50

5

10

15

20

25 Time (sec)

30

35

40

45

50

0

0

0.022

0.02

0.018

0.016

0.014 5

1.16

10

Figure 13: Evolution of tracking errors (e1 , e2 , e3 )(2SMC).

Control law u1 (Kgs−1)

• Simulation trajectories of the fuzzy second order sliding mode control

5

0

−10

The figures 3-9 represents the simulation results of the first order Sliding Mode Control (1-SMC). Those trajectories show clearly the presence of the chattering phenomenon.

0 −4 x 10

−2 0 10

50

e3 (p3)

Control law u3 (Kgs−1)

8.5

x 10

0

5

10

15

20

25 Time (sec)

30

35

40

45

50

1.15

p1

Figure 14: Evolution of control law u1 (2-SMC).

1.14 1.13 1.12

−3

1.11

5

1.1 1.09

x 10

4.5

0

5

10

15

20

25 Time (sec)

30

35

40

45

50

Figure 10: Intake manifold pressure response p1 (Pa) (2SMC).

Control law u2 (Kgs−1)

Intake manifold pressure p1 (Pa)

p1 Desired

4 3.5 3 2.5 2 1.5

0

5

10

15

20

25 Time (sec)

30

35

40

45

50

Figure 15: Evolution of control law u2 (2-SMC). 5

1.42

x 10

−3

p3

10

1.38 1.36 1.34 1.32 1.3 1.28

x 10

9.5

Control law u3 (Kgs−1)

Exhaust manifold pressure p3 (Pa)

p3 Desired 1.4

0

5

10

15

20

25 Time (sec)

30

35

40

45

50

9 8.5 8 7.5 7 6.5

Figure 11: Exhaust manifold pressure response p3 (Pa) (2-SMC).

0

5

10

15

20

25 Time (sec)

30

35

40

45

Figure 16: Evolution of control law u3 (2-SMC).

50

0.018 0.0178 0.0176

Control law u1 (Kgs−1)

By comparing the simulation results of both methods the first and the second order fuzzy sliding mode control, we notice that the 2-SMC have better tracking performances. The evolution of the control laws also show the reduction of the chattering phenomenon.

0.0174 0.0172 0.017 0.0168 0.0166

• Simulation trajectories of the Adaptive Sliding Mode Control (ASMC) 5

1.16

0

5

10

15

20

25 Time (sec)

30

35

40

45

50

Figure 21: Evolution of control law u1 (ASMC).

x 10

p1 Desired p1

1.15

Intake manifold pressure p1 (Pa)

0.0164

1.14 1.13 1.12 1.11 1.1 1.09

0

5

10

15

20

25 Time (sec)

30

35

40

45

50

−3

3.55

x 10

3.5 3.45

Control law u2 (Kgs−1)

Figure 17: Intake manifold pressure response p1 (Pa) (ASMC).

3.4 3.35 3.3 3.25

5

1.42

3.2

x 10

Desired p3

Exhaust manifold pressure p3 (Pa)

3.15

p3

1.4

0

5

10

15

20

25 Time (sec)

30

35

40

45

50

1.38

Figure 22: Evolution of control law u2 (ASMC).

1.36 1.34 1.32 1.3 1.28

0

5

10

15

20

25 Time (sec)

30

35

40

45

50

Figure 18: Exhaust manifold pressure response p3 (Pa) (ASMC). Control law u3 (Kgs−1)

0.022 Wc Desired Wc

Fresh airflow rate Wc (Kg/s)

−3

8.5

0.0215

x 10

8

0.021 7.5 0.0205

0

5

10

15

20

25 Time (sec)

30

35

40

45

50

0.02

Figure 23: Evolution of control law u3 (ASMC). 0

5

10

15

20

25 Time (sec)

30

35

40

45

50

Fresh airflow rate response Wc (kg/s)

Figure 19: (ASMC).

e1 (p1)

10 0 −10

e2 (Wc)

2

5

10

15

20

25

30

35

40

45

50

0

5

10

15

20

25

30

35

40

45

50

0

5

10

15

20

25 Time (sec)

30

35

40

45

50

0 −2 10

e3 (p3)

0 −4 x 10

0

For evaluating the tracking performance of the First Order Sliding Mode Control system (1-SMC), the Second Order Sliding Mode Control system (2-SMC) and the Adaptive Sliding Mode Control (ASMC) system, we adopt a quantitative comparison of tracking errors by using the integral of the absolute error (IAE). This (IAE) criteria integrates the absolute error over time: Z∞ (IAE) = |e(t)| dt 0

−10

Figure 20: Evolution of tracking errors (e1 , e2 , e3 )(ASMC).

In Table 2, we have reported the (IAE) of different algorithms cited before.

Table 2: Integral Absolute Error (IAE) values for various methods

IAE

Methods 1-SMC

2-SMC

ASMC

e1

29.12

e2

0.0005595

0.0003729

9.67310−7

e3

146.2

99.67

5.137

21.59

2.469

Remark 4: In this work, the sensors in the simulations are supposed to be ideal with excellent precision. The errors from nonlinearity, the repeatability and the hysteresis of the sensors are supposed to be close to zero. We are aware that these operating conditions are not realistic, but the objective, here, is to evaluate the performance of three control laws. Note that, if we decide to take into account the measurement accuracy of the sensors (10−2 ), we can always conclude that ASMC has the best tracking performance. The evolution of the tracking errors given in Figures 6, 13 ,20, shows that both of the 2-SMC and the ASMC have better performances rather than 1-SMC. In fact, we notice better tracking performance for both methods (2-SMC and ASMC). Besides, we notice that the ASMC has better transient response. By comparing the evolution of the control laws, we notice the elimination of the chattering phenomenon for both approaches (2-SMC and ASMC). In this work, the 2-SMC and the ASMC not only guarantee the stability of the system but also the reduction of the chattering phenomenon as well. Based on the results in Table 2, we remark that the ASMC has better results. Although, this approach (ASMC) requires the addition of the Proportional Derivative term (PD) in order to replace the discontinuous one which can complicate the calculation of the control law and make the simulation more difficult compared to the super twisting approach, this method present better tracking performances comparing to the 2-SMC.

7

The purpose is to compare different methods. The simulation results of the turbocharged diesel engine for different approaches are presented to show the effectiveness of the proposed control methods.

Conclusions In this paper, a fuzzy sliding mode controller is developed to control the air path of a turbocharged diesel engine. The structure of the control system is assumed to be unknown so inference fuzzy system are used to estimate those unknown functions. In the second part, we proposed solutions in order to reduce the chattering phenomenon. In fact, a special case of the Second Order Sliding Mode Controller: the super twisting sliding mode controller is developed. Furthermore, an additional method is also presented in which we combined an adaptive PD controller into classical sliding mode.

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