Application of non-linear observers to on-line estimation of indicated

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Application of non-linear observers to on-line estimation of indicated torque in automotive engines Yunsong Wang* and Fulei Chu Department of Precision Instruments and Mechanology, Tsinghua University, Beijing 100084, People’s Republic of China The manuscript was received on 8 December 2003 and was accepted after revision for publication on 30 June 2004. DOI: 10.1243/095440705X6434

Abstract: Estimation of the indicated torque for automotive engines is profitable for evaluation of engine combustion conditions, optimization of engine performance, diagnosis of engine faults and control of automotive transmission. On the basis of the crankshaft–wheel system dynamics of the automotive multicylinder engine without including load torque, this paper describes mainly the application of three non-linear observers to the on-line estimation of indicated torque and analyses the similarities and differences between them. Of all the proposed observers the second-order sliding mode observers are most robust and exact. The classical sliding mode observers are slightly more accurate than the high gain observers, but have the undesired chattering effect. These observers require crank angle and crankshaft speed measurements. Experimental results strongly demonstrate that the on-line estimation techniques of the indicated torque based on non-linear observers are valid, not only for steady state operation but also under transients. Keywords: high gain observer, classical sliding mode observer, second-order sliding mode observer, engine, vehicle, indicated torque, estimation

1 INTRODUCTION Because of the complexity of today’s automotive systems, an improved and more extensive diagnosis for service companies as well as in the vehicle has become necessary [1]. More severe legislated emission regulations require on-board diagnosis of all emission relevant components for vehicles, which also saves time and reduces cost in engine maintenance and failure repair. Nowadays, modern on-board diagnosis systems are mainly based on simple threshold supervision or plausibility checks of measured signals as well as on signal processing methods such as the frequency analysis of the engine speed signal. Apparently, these approaches will no longer be sufficient for the increasing demands. To keep up with this, model-based fault diagnosis methods investigated and developed in recent years will come into operation [2], one of which is the torque estimation in automotive engines. It is not only profitable for the evaluation of engine combustion conditions but is also made accessible for * Corresponding author: Department of Precision Instruments and Mechanology, Tsinghua University, Beijing, 100084, People’s Republic of China. email: [email protected]

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optimization of engine performance—subject to emission and fuel consumption constraints—and for transmission control. The simple and immediate method used to obtain the indicated torque of engines is by calculation from the measurements of in-cylinder pressure [3]. Recently many types of pressure transducers have been developed for this purpose, which can provide a very high degree of precision. However, because of the high cost of the transducer hardware, the poor durability of current cylinder pressure transducers and little room in an automotive engine to install them and associated equipment, direct cylinder pressure measurements are not extensively found in the automotive industry. For these reasons, the alternative of using direct methods for torque estimation becomes attractive. They can be classified into two types: one is based on engine speed fluctuations and the other is based on linear or non-linear observers. The former usually uses the techniques of mathematical analysis and/or signal processing to obtain the indicated torque through modelling of the crankshaft dynamics [4–9], but considerable errors exist in these estimation methods at high speed and low load. The latter has a better robust performance and accuracy [10–13]. The idea of using a dynamical Proc. IMechE. Vol. 219 Part D: J. Automobile Engineering

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system to generate estimates of the system states— an observer—can be traced to Lunberger, who proposed a method for linear systems that now bears his name. In this approach, the observer system is driven by the control input and by the difference between the output of the observer and the output of the system. The latter should ideally become zero. An observer provides an estimation of variables for a dynamic system by using the known system’s inputs and outputs measured by transducers or sensors, and the mathematical model of the system. In standard control theory, the aim of an observer is to estimate precisely the running value of the state of the system. In most controlled systems, the dimension of the output vector is less than that of the state vector for several reasons (e.g. technical, cost, etc.). On the other hand, many control laws derived from the state space concept often require a knowledge of the state. Therefore, state observers are often necessary for actual implementation of the controller. Linear observer theory has been well established since the work of Lunberger in 1964 and Kalman and Bucy in 1961. This naturally suggests the exploration of ideas to generate a sliding mode on the subspace for which the output error is zero. In particular, it is of interest to explore the possibilities of using sliding mode observers for robust state construction. In fact, the sliding mode observers have been developed for automotive engineering with a variety of objectives, from reconstruction of cylinder pressure waveforms for off-line analysis of the combustion process to attempts at an on-line estimation of combustion pressure or indicated torque for monitoring, diagnostics and control purposes. However, all the investigations on the torque estimation problem are with regard to rig-testing engines, other than the ones on production vehicles, and the relative experiments are carried out on engine-testing stands in a laboratory cell indoors, instead of on roads outdoors. The differences in the experimental conditions between indoors and outdoors may introduce many estimation errors into torque estimations [14]. For example, in the case of rig-testing investigations in the laboratory, the load torque essential for an indicated torque estimation must be simulated and generated by a dynamometer, but the simulated load torque is very different from actual load-bearing of the engines on vehicles operating on roads outdoors, and is not suited to on-line torque estimation. Moreover, most of the torque estimation observers are based on the dynamics describing the crankshaft rotation of single-cylinder engines rather than multicylinder Proc. IMechE. Vol. 219 Part D: J. Automobile Engineering

ones. It is known that the crankshaft dynamic properties of multicylinder engines are different from single-cylinder ones in many ways. The objective of the current paper is to build a crankshaft–wheel system dynamic model for automotive multicylinder engines without including load torque, to describe how non-linear observers can estimate the indicated torque of automotive engines in real time and to report supporting vehicle validation results.

2 STIFF CRANKSHAFT MODEL OF THE AUTOMOTIVE ENGINE According to the crankshaft–wheel system dynamics of the multicylinder internal combustion (IC) engine, the dynamic equation describing the rotation of the crankshaft is J(Q)Q¨ =T −T −T −T (1) i r fp l where J(Q) is the equivalent lumped inertia of the kinetic parts of the dynamic system including crankshaft, wheel, piston and connecting rod, etc., T the i indicated torque, T the reciprocating torque, T the r fp friction and pumping loss torque and T the load l torque. The crank–connecting rod mechanism is a timevarying system, whose inertia is the function of the crank angle, Q J(Q)=m R2 [ f (Q)]2+m R2 (2) 1 c 2 c where R is the crank radius, m the equivalent c 1 reciprocating lumped mass and m the equivalent 2 rotating lumped mass. According to the crank angle geometry, the two geometric functions are defined as f (Q)=sin Q+

l sin (2Q) 2 √1−l2 sin2 Q

g(Q)=cos Q+

l sin (2Q)

√1−l2 sin2 Q

+

l3 sin2 (2Q) 4 √(1−l2 sin2 Q)3

The indicated torque T is generated by the i cylinder indicated pressure P, through the crank angle geometric function N N T = ∑ T (k) =A R ∑ [P(k) f (Q−w )] (3) i i p c k k=1 k=1 where A is the area of piston top, N the number p of cylinders and w the firing phase of the kth k cylinder relative to the first cylinder for the four-stroke multicylinder engine 4p w = (k−1) k N D21803 © IMechE 2005

On-line estimation of indicated torque in automotive engines

The reciprocating torque T is produced by the r reciprocating movement of the piston and part of the connecting rod N T = ∑ T (k) =m R2 r r 1 c k=1 N × ∑ {[Q¨ f (Q−w )+Q˙ 2g(Q−w )] f (Q−w )} k k k k=1 (4) The friction and pumping loss torque T is created fp by the sliding friction between the piston and ring and the pumping action of the engine. These losses have traditionally been lumped together due to the difficulty in separating the effects of one from the others N T = ∑ T (k) =DQ˙ (5) fp fp k=1 where D is the damping coefficient. The indicated torque (3), reciprocating torque (4) and loss torque (5) are substituted into the dynamic equation (1) respectively, and then the non-linear multicylinder IC engine dynamic model is rewritten as 1 Q¨ = N J(Q)+m R2 ∑ [ f (Q−w )]2 1 c k k=1 N × A R ∑ P(k) f (Q−w ) p c k k=1

H

N −m R2 Q˙ 2 ∑ [ f (Q−w )g(Q−w )]−DQ˙ −T 1 c k k l k=1 (6)

It is well known that the dynamic equation of the running vehicle is

A

B

(7)

where v is the longitudinal vehicle speed, A the v frontal area of the vehicle, C the aerodynamic drag D coefficient, f the road rolling resistance coefficient, r g the gravity acceleration, i the transmission rate, i g 0 the rate of main reducing gear, m the vehicle mass, r the wheel radius, d the conversion coefficient of rotating mass, g the transmission efficiency and r T the air density. If the vehicle wheels do not slip on the ground, and the viscous damping and friction losses resulting from the coupling between the engine flywheel and the vehicle clutch are neglected, the following relation may exist between the running speed of the vehicle and the crankshaft speed of the engine on D21803 © IMechE 2005

board r r v = v= Q˙ v i i i i g0 g0

(8)

r Q¨ v˙ = v i i g0

(9)

When car wheels roll on rigid road surfaces, the wheel tyres are subjected to metamorphosis, which is dependent on the rigidity of the tyres relative to the road surfaces, due to the radial load. Strictly speaking, the radius of a car tyre rolling on the road is smaller than the nominal one of the tyre without bearing any weight, but neglecting this tyre radius difference resulting from tyre rigidity is acceptable in engineering applications. Therefore, the tyre radius in equations (8) and (9) indicates the nominal size described in the tyre specifications. Substitute equations (8) and (9) into equation (7) and then the load torque can be obtained as

C

A B

D

r r 2 dmr 1 T= Q˙ 2+ mg f +mgi+ rAC Q¨ l i i g r D 2 i i i i g0 T g0 g0 (10) Therefore, the combination of equations (6) and (10) yields the dynamic stiff crankshaft model of the automotive engine in the time domain

G

1 i i g 1 g 0 T T −mg f −mgi− rAC v2 v˙ = v dm l r D v r 2

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Q¨ =

C

1 mgr T −DQ˙ − ( f +i )−M(Q)Q˙ 2 I(Q) i i i g r g0 T

D

(11)

where

A B

dm r 2 N I(Q)=J(Q)+m R2 ∑ [ f (Q−w )]2+ 1 c k g i i T g0 k=1 (12)

A B

r 3 rAC D 2g i i T g0 N +m R2 ∑ [ f (Q−w )g(Q−w )] 1 c k k k=1

M(Q)=

(13)

Consider the following coordinate transformations Q˙ =

dQ =v dt

(14)

Q¨ =

dv dv dv dQ = =v dt dQ dt dQ

(15)

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Equation (11) can then be changed in the crankangular domain into the form

C

dv 1 mgr = ( f +i )−M(Q)v2 T −Dv− i dQ vI(Q) i i g r g0 T

D (16)

There are several popular algorithms based on nonlinear observers for estimations such as sliding mode observers, high gain observers, Luenberger observers and ‘dirty derivative’ filters [15, 16]. Their similarities are that they all rely on an intuitive observation if a controller, which can also be interpreted as an observer for the dynamic system of interest, is designed [17]. In this paper, high gain observers, classical sliding mode observers and second-order sliding mode observers are chosen and designed for estimation of the indicated torque of the automotive engine. 3.1 High gain observer In order to design the high gain observer for indicated torque, equation (16) is rewritten as dv =y(Q)+u dQ

(17)

C

Apparently, the following inequality holds true 1 du 2 du +l e2 −2e (24) 1 u u dQ l dQ 1 If there is a positive constant a such that |du/dQ|∏a, combining equation (24) with equation (23), the upper bound of |e | is obtained u

S

A B

a 2 [e (0)]2 e−l1Q+ (25) u l 1 It can be seen from equation (25) that by choosing the observer gain l properly, |e | can be made 1 u arbitrarily small for any Q>0. Then the estimates of u and T can be derived from i uˆ =l v−z (26) 1 Tˆ =uˆvI(Q)=v(l v−z) i 1 dm r 2 N × f (Q)+m R2 ∑ [ f (Q−w )]2+ 1 c k g i i T g0 k=1 (27) |e |∏ u

G

A BH

3.2 Sliding mode observer

D

ˆ dv =y(Q)+g sign (e ) v dQ (18)

u=

(23)

The sliding mode observer for the indicated torque is given by

where 1 mgr −Dv− ( f+i )−M(Q)v2 vI(Q) i i g g0 T

de2 du u =−2l e2 −2e 1 u u dQ dQ

A B

3 DESIGN OF NON-LINEAR OBSERVERS

y(Q)=

By multiplying both sides by 2e , equation (22) is u transformed into

T i vI(Q)

(19)

The variable y(Q) is indirectly measured, but u is an unknown input that is to be estimated. Now introduce two auxiliary variables e and z, which u satisfy the following equations: e =l v−z−u u 1 dz =−l z+l y(Q)+l2 v 1 1 1 dQ

(20) (21)

where l is a positive observer gain. By combining 1 equations (17), (20) and (21), the derivative of e is u obtained: de du u =−l e − 1 u dQ dQ Proc. IMechE. Vol. 219 Part D: J. Automobile Engineering

(22)

(28)

which basically repeats what is known about the system (17) with an additional discontinuous injection. The error dynamics follows from subtraction of equation (28) from equation (17), where the gain g>0, e =v−v ˆ is the switching function v e˙ =u−g sign (e ) (29) v v Ideally, in the sliding surface, i.e. e =0 and Q>Q , v 0 where 2 Q = , 0 g |e (0)| 0 v v(Q )=v ˆ (Q ) 0 0

g g>b+ 0 , 2

|u|∏b

so that u=[g sign (e )] v eq (30)

The operator [.] outputs the value of its diseq continuous argument that would satisfy the invariance conditions of the sliding motion (e =0, e˙ =0) that v v this discontinuous input induces. The equivalent D21803 © IMechE 2005

On-line estimation of indicated torque in automotive engines

value operator, [.] , can be approximately realized eq by a high bandwidth low-pass filter according to the equivalent control methodology, i.e. tu˙ˆ +uˆ =g sign (e ) (t>0) (31) v Therefore, an estimate of the indicated torque can be obtained in the same form as equation (27). The estimation error is |u−uˆ|∏ √[u(0)−uˆ(0)]2e−gQ+(ta)2

(32)

Assuming that t=1/g, from equations (32) and (25) it can be seen that the estimation error for the sliding mode observer has the same form as the high gain observer. In other words, the two observers above have almost the same estimation accuracy. Nevertheless, implementation of sliding mode observers is troublesome because of the discontinuous injection needed to ensure robust properties and the possibility that the so-called chattering phenomenon can arise [18]. In particular, the latter may lead to large undesired oscillation that will damage the observer. On the contrary, the high gain observer does not have this defect. 3.3 Second-order sliding mode observer The chattering effect is intrinsic in the classical sliding mode techniques mentioned above. In order to cancel it, many measures have been taken such as the quasi-sliding mode, high gain control and continuous switching function, etc. The recently invented high-order sliding modes generalize the classical sliding mode idea. They are characterized by a discontinuous control acting on the higher-order time derivatives of the sliding variables instead of influencing its first-order time derivative, as happens in classical sliding modes [19]. Preserving the main advantages of the original approach with respect to robustness and ease of implementation, at the same time they totally remove the chattering effect and guarantee an even higher accuracy in the presence of plant and/or control device imperfections [20]. In all high-order sliding modes, the second-order sliding modes (SOSM) are most widely used in practical engineering applications. Many second-order mode algorithms exist in the sliding mode literature. However, they are not all applicable to the observation problem. For this reason, the ‘super-twisting’ algorithm [21] is chosen and described for the present purpose in the following. Consider a non-linear single-input dynamic system governed by ˙ (t)=h[X(t), t, u(t)] X D21803 © IMechE 2005

(33)

69

where X is the n-dimensional state vector in the real number field, u is the single input control which is bounded, t is the independent variable time and h is a sufficiently smooth vector function. Suppose that the constants s , K , K and C exist such that for X 0 m M 0 with a switching function that satisfies |s(t, X)|s

0

|s|∏s

0 (36)

and the corresponding sufficient conditions for the finite time convergence to the sliding mode are C W> 0 K m 4C K (W+C ) 0 a2 0 M K2 K (W−C ) m m 0 00 and both n(t) and h(t) may be considered as the output of the differentiator. Solutions of the system are understood in the Filippov sense. Parameters may be chosen, for example, in the form m=1.1C, e=1.5C1/2, where C=max |v ¨ (t)|. Differentiator (38) provides for finite-time convergence to the exact derivative of v. The indicated Proc. IMechE. Vol. 219 Part D: J. Automobile Engineering

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torque estimation can be calculated from

C

s Tˆ =I(Q) n(t)−l i |s|+y

4 EXPERIMENTAL RESULTS

P D t

s dt

0 mgr ˆ2 ˆ+ ( f+i )+M(Q)v +Dv i i g g0 T

(39)

where s = v ˆ − v is the switching function, Q y=y +∆ s dt, where y is the initial value of y and l 0 0 0 is a positive gain. Figure 1 shows the block diagram of the SOSM indicated torque observer. The design scheme implies the combination of a second-order sliding mode and a quasi-sliding mode. The latter has a very good tracking performance. The observer is used to force the estimated crankshaft speed to track the measured varying speed and alters the estimated indicated torque in order to reduce the speed estimation error; the convergence is assured by the Lyapunovlike function. A smooth continuous injection signal, −ls/(|s|+y) ∆ t s dt, instead of a discontinuous one, 0 with integrator, is employed to avoid the chattering and to eliminate the steady state error. In general, the crankshaft acceleration information required in the sliding model observer can be calculated by the use of numerical differentiation, but it will introduce errors into the indicated torque estimation. Hence, the SOSM differentiation algorithm is used to differentiate the measured crankshaft speed in real time, which has much higher accuracy and better robustness than the numerical method.

In order to examine the validity of the indicated torque observers, an experimental verification was carried out. A car with a four-cylinder in-line fuelinjected 1.8 litre engine and a five-speed manual transmission was used in the experiment. The instrumentation for model verification includes sensors, signal conditioning circuits, a data acquisition board and a note book PC, etc., which is shown in Fig. 2. The existing crankshaft and camshaft position sensors mounted on the engine are applied to monitoring of all the functions relative to this study by the measurement of the engine crank-angular position. The camshaft sensor is of the Hall effect type, which outputs a signal synchronized with the TDC (top dead centre) position for cylinder one. The crankshaft position sensor is magnetically coupled to the starter ring gear on the flywheel, which provides directly the inexpensive measurement of the crankshaft angular speed and the crank angle. The two sensors form the basis of the experimental validation of the instrumentation. After being isolated, low-pass-filtered and amplified in the signal conditioning circuits, the output signals from the above sensors are passed to the data acquisition board. Data acquisition is a procedure during which the physical signals such as voltage, current, temperature, etc., are transferred into the digital ones so that they can be processed by computer. The programme for the engine test and real-time data process is developed by application of a virtual developing workbench, i.e. LabVIEW.

Fig. 1 Second-order sliding mode observer for indicated torque Proc. IMechE. Vol. 219 Part D: J. Automobile Engineering

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Fig. 2 Instrumentation for model verification

In order to verify the validity of the observers with each other, the measured indicated in-cylinder pressure profile of the engine must be provided in order to obtain the measured, or standard, indicated torque induced by the pressure. Due to limitations of experimental expenses, only one quartz pressure transducer was used in this study, which was mounted into the third cylinder of the engine through a hole drilled on the corresponding cylinder head. The indicated pressure profiles were measured at various speeds and loads using the pressure transducer and its analogue signals were processed in the signal processing conditioning circuits and the data acquisition board. In fact, fluctuations of the indicated in-cylinder pressure inevitably exist in cylinder-by-cylinder or during cycle-by-cycle. For simplicity, these pressure differences are neglected, and the total indicated torque of the multicylinder engine is presumed to be the summation of the torque produced by all individual cylinders according to the firing phase in the firing order, which is seen from equation (3). Figure 3 shows the measured pressure profile of the third cylinder of the testing engine at the crankshaft speed of 2000 r/min and full load. The different observer schemes were used in the experiment to estimate the indicated torque. The

Fig. 3 Measured indicated pressure profile of the third cylinder (2000 r/min, full load)

experimentations were made under various operating conditions of the testing vehicle. It was required that the testing vehicle should carry only two persons on board and be running on good concrete road surfaces without slope and without wind outdoors. Tables 1 and 2 give the indicated torque estimations as well as their errors during one working cycle for steady state operation and under transients. Here, the mean torque estimation is defined as ∆ 4p T (Q) dQ/(4p). The SOSM has the best accuracy of 0 i the estimation. Although it has almost the same upper

Table 1 Indicated torque estimations during one working cycle for three different observer schemes (N m) (41 km/h, third gear) Cylinder 1

Cylinder 2

Cylinder 3

Cylinder 4

Observer

Maximum

Minimum

Maximum

Minimum

Maximum

Minimum

Maximum

Minimum

Mean torque

Mean errors

HG CSM SOSM

316.4 314.5 305.6

−71.3 −69.7 −52.1

322.1 319.3 309.1

−63.2 −65.3 −57.7

−324.7 321.6 310.6

−66.2 −58.7 −59.2

321.6 318.8 304.8

−68.2 −71.2 −53.7

201.2 195.6 181.7

25.5 19.9 6.1

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Table 2 Indicated torque estimations of the engine during one working cycle for three different observer schemes (N m) (the vehicle accelerates slowly from 30 to 60 km/h in 6 seconds in fourth gear) Cylinder 1

Cylinder 2

Cylinder 3

Cylinder 4

Observer

Maximum

Minimum

Maximum

Minimum

Maximum

Minimum

Maximum

Minimum

Mean torque

Mean errors

HG CSM SOSM

352.6 346.2 331.7

−105.0 −98.2 −89.5

357.1 340.2 336.2

−99.3 −101.2 −90.5

361.3 347.4 329.7

−98.3 −99.8 −84.8

359.5 342.8 338.0

−106.2 −89.2 −95.2

221.4 218.7 202.3

31.1 28.4 12.0

bounds of the estimation errors as the high-gain observer (HG), the classical sliding mode observer (CSM) manifests itself by yielding a little higher accuracy because of its good robustness properties with respect to modelling uncertainty and parameter variations. The similarity of the CSM and the HG is that their implementations all require low-pass filters in order to extract estimates of known filter variables. It is known that the steady output of the low-pass filters approximates the average of the system input. It is the averaging of the low-pass filters that induces the intrinsic errors in the CSM and HG. Besides the averaging errors, the chattering effect exists as a number of discontinuous injection in the CSM, but not in the SOSM. Compared with the CSM and HG, the SOSM is robust and exact. Furthermore, Figs 4 to 6 show the indicated torque estimates at low, mid and high speeds of the vehicle. The cylinder firing order of the testing engine is 1–3–4–2 (see Figs 4 to 8). As the running speed of the vehicle increases, the estimation errors for the three observers become large. The three observers all give

Fig. 4 Estimated indicated torques by various observers (20 km/h, second gear) Proc. IMechE. Vol. 219 Part D: J. Automobile Engineering

Fig. 5 Estimated indicated torques by various observers (50 km/h, third gear)

Fig. 6 Estimated indicated torques by various observers (100 km/h, fourth gear) D21803 © IMechE 2005

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mean indicated torque for the kth cylinder during the power stroke in the crank angle domain is defined as 1 T˜ (k)= i p

Fig. 7 Estimated mean indicated torques by various observers at steady state

P

TDC(k+1)

T (Q) dQ i TDC(k) where the integration angle interval, p, is selected from the TDC position of the kth cylinder to the TDC position of the next cylinder, i.e. the (k+1)th cylinder in the firing order. The smallest mean indicated torque estimations apparently indicate a possible misfire in the fourth cylinder. Figure 8 gives torque estimations under the condition of misfire in the fourth cylinder during acceleration (from 30 to 60 km/h in 6 seconds). This clearly indicates the faulty cylinder. The non-steady-state test clearly indicates that the SOSM has much better torque estimations than the other methods.

5 CONCLUDING REMARKS

Fig. 8 Estimated indicated torques by various observers during transients

better indicated torque estimations at low and mid speeds than at high speed. The SOSM exhibits the best estimation results at all speeds. It should be noted that the SOSM algorithm has a large computational burden and needs a large memory size of the computer. In order to further validate the feasibility of the indicated torque estimation based on the nonlinear observers, in-cylinder misfire phenomena were simulated for the test engine by pulling up the ignition cables of individual cylinders(s) or stopping the supply of fuel for the cylinder(s). Figure 7 shows the estimated mean indicated torques for individual cylinders by various observers in the case of misfire at an engine speed of 2000 r/min in third gear. The D21803 © IMechE 2005

Non-linear observers represent a set of powerful approaches that significantly improve the performance of the estimation used in automotive diagnosis and control. This paper has described how to design indicated torque observers that have good accuracy and robustness. These observers have their own characteristics and advantages, and can be properly chosen in practice. Experimental validations show that these observers may make more accurate estimations for indicated torque at a low or mid speed of the vehicle. The second-order sliding mode observers especially have good robustness and accuracy, not only for steady state operation but also under transients. It is feasible and promising for a real-time torque estimation to find future applications such as those for on-board vehicle diagnosis, for engine misfire detections or automotive control, etc. Further study is focused on the improvement of modelling accuracy, the reduction of measurement noise and the enhancement of computational efficiency.

ACKNOWLEDGEMENTS The work described in this paper was supported by the Trans-Century Training Programme Foundation for the Talents by the Ministry of Education, China and Schandong Natural Science Foundation (Grant Y2002F17). Proc. IMechE. Vol. 219 Part D: J. Automobile Engineering

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Yunsong Wang and Fulei Chu

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APPENDIX Notation A A p C D D f r g i i g i 0 I J k m m

1

m 2 M N P r R c s sign T i T r T fp T i v v v˙ v d g

T

frontal area (m2) area of piston top (m2) aerodynamic drag coefficient damping coefficient road rolling resistance coefficient gravity acceleration (m/s2) road slope resistance coefficient transmission rate rate of main reducing gear logogram of a polynomial equivalent lumped inertia (kg m2) sequence number of the cylinder relative to the first cylinder vehicle mass (kg) equivalent reciprocating lumped mass (kg) equivalent rotating lumped mass (kg) logogram of a polynomial number of cylinders cylinder indicated pressure (Pa) wheel radius (m) crank radius (m) Laplace variable sign function the indicated torque (N.m) the reciprocating torque (N.m) the friction and pumping loss torque (N.m) the load torque (N.m) longitudinal vehicle speed (m/s) vehicle acceleration (m/s2) conversion coefficient of rotating mass transmission efficiency D21803 © IMechE 2005

On-line estimation of indicated torque in automotive engines

l r s w Q Q˙ or v Q¨

ratio of crank radius and connecting rod length air density (kg/m3) switching function firing phase (rad) crank angle (rad) angular speed (rad/s) angular acceleration (rad/s2)

D21803 © IMechE 2005

75

Acronyms CSM HG IC PC SOSM TDC

classical sliding mode observer high gain observer internal combustion personal computer second-order sliding mode observer top dead centre

Proc. IMechE. Vol. 219 Part D: J. Automobile Engineering