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`
`A Performance Improvement in Idle-Speed
`Control System with Feedforward Compensation
`for the Alternator Load Current
`Hitoshi lnoue and Shoichi Washino
`Industrial Electronics & Systems Development Lab.
`Mitsubishi Electric Corp.
`
`MAR 11990
`SAE
`L18RARY
`
`International Congress and Exposition
`Detroit, Michigan
`February 26 - March 2,1990
`
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`A Performance Improvement in Idlespeed
`Control System with Feedforward Compensation
`for the Alternator Load Current
`Hitoshi lnoue and Shoichi Washino
`Industrial Electronics & Systems Development Lab.
`
`Mitsubishi Electric Corp.
`
`ABSTRACT
`
`idle
`It is well-known that decreasing
`speeds is one of the ways to reduce fuel con-
`sumption. On the other hand, it is also well-
`known that even slight fluctuations of the idle
`speed cause unpleasant vibrations of the vehicle
`when this speed is set at low values. Therefore,
`it is important to create idle-speed control
`(ISC) systems that undergo less idle-speed fluc-
`tuations with respect to various load disturb-
`ances, in order to reduce fuel consumption with-
`out giving rise to unpleasant vibrations.
`The first topic of this paper is a line-
`arized model which derives a new feedforward
`compensation method for reducing idle-speed
`fluctuations caused by load distrubances for
`specific electric loads. The second concerns
`the control results of our ISC system using this
`feedforward compensation. The final topic is a
`discussion of the validity of the parameter
`values used in the feedforward compensation.
`
`FOR IMPROVEMENTS OF AUTOMOTIVE ENGINE
`PERFORMANCE, the following items immediately
`suggest themselves : high power, high fuel
`economy, high response, and low vibration of the
`vehicle. Many investigators have been engaged
`in such improvements. Recently, of all these
`improvements, fuel economy has again attracted
`attention, because it will also be able to
`reduce the greenhouse effects of carbon dioxide.
`For improving fuel economy, there are
`several approaches: (1) decreasing frictional
`losses in engines, (2) improving combustion in
`engines, and (3) decreasing fuel consumption in
`a given driving mode. Reducing idle speed are
`obvious means of decreasing fuel consumption,
`because the fuel consumed while idling makes no
`contribution to work done by the engine. On the
`other hand, it will be readily recalled that
`decreasing idle speeds often causes not only
`degradation of idling stability, but also un-
`pleasant vibrations of the vehicle caused by
`
`large fluctuations of idle speed under slight
`load disturbances. Therefore, the technical
`issues of decreasing idle speed flucturations
`are important to satisfy the requirements of
`both fuel economy and good driveability. As
`candidates to solve this issue, idle-speed
`control (ISC) systems are appropriate.
`Unfortunately, the degree of control
`achieved by current commercially-available ISC
`systems is still unsatisfactory. Therefore,
`many investigators have been making efforts to
`improve the control provided by ISC systems.
`For examples, Takahashi et al. have discussed
`the application of modern control theory to ISC
`systems.
`Osawa et al. have also proposed
`the app i ation of an adaptive control scheme
`to ISC. 125 Despite these efforts, it seems that
`no remarkable progress in the control provided
`by ISC systems is to be found in the literature.
`Quite apayt- from the issue of fuel economy,
`there is another important reason for reducing
`the excursions of idle speed from the nominal
`value. Recently, automobiles have been equipped
`with many kinds of electrical equipment and as a
`result, every electrical load variation can
`cause idle-speed fluctuations. As noted pre-
`viously, such electrical load variations will
`cause unpleasant vibrations for those in the
`vehicle.
`In this paper, we propose an ISC system
`which can greatly reduce idle-speed fluctuations
`caused by various electrical load variations.
`Oui ISC system manipulates air-flow rates both
`proportionally to the load currents of alter-
`riators and their derivatives. We show theore-
`tically that this feedforward manipulation of
`air-flow rates results in a significant reduc-
`tion of transient idle-speed fluctuations. Then,
`we confirm experimentally that this ISC system
`shows the expected characteristics. We finally
`
`* Number in parentheses designate references at
`the end of the paper
`
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`
`discuss the agreement of both proportional and
`derivative gains between theory and experiment.
`
`shown in Eq.(5).
`
`DERIVATION OF IDLE SPEED DYNAMICS, INCLUDING
`THE ALTERNATOR
`
`Despite the great influence that alterna-
`tors can have on idle-speed dynamics, there have
`been few papers discussing alternator dynamics
`and linking them with engine-speed dynamics
`while idling.
`Therefore, we start our discus-
`sion with a review of alternator and engine-
`speed dynamics. For simplicity, we confine our
`discussion to the frequency domain.
`First we derive the dynamics of an alter-
`nator. Figure 1 shows the equivalent circuit
`of an alternator. The instantaneous relations
`load current Ia (A],
`among field current I~[A),
`induced voltage E [v] and required torque T 'Nm]
`are represented by the following equations.(3)
`
`E = P M y, If
`
`(V]
`
`T = P M If 1,
`
`[Nm)
`
`(1)
`
`(2)
`
`where P is the polar number, M is mutual in-
`ductance (H] , N is engine speed [rpml , up is
`pulley ratio, and wm is mechanical angular speed
`( = 7~ N a p / 3 0 ) . By taking deviations of various
`physical quantities from an equilibrium state
`in Eqs. (1) and (2) and normalizing by equilib-
`rium value, we obtain the follwoing equations.
`
`where A means deviations from their equilibrium
`values, subscript 0 their equilibrium values,
`and superscript * the values normalized by them,
`respectively.
`The field circuit is represented by a
`series connected resistance and inductance.
`The relation between excited voltage Er [v] and
`If is obtained by Laplace transformation as
`
`I Battery L Electric
`
`Lwd
`
`L -- -- --- - - - - - - - - - - - - - - - - - -J
`Alternator
`
`Fig. 1 Alternator Circui
`
`equivalent inductance : Lf CH)
`equivalent resistance : rf (i2)
`
`The relation among E, I, and load voltage
`also represented by a series connected
`V [V)is
`resistance and inductance. But the equivalent
`inductance of stators can be neglected. There-
`fore, we derive the following simple equation.
`
`Generally, the terminal voltage V is regu-
`lated to a constant voltage, typically 14(V),
`by controlling the field current. As is well-
`known, this control of the field current is
`performed by modulating the pulse width of the
`current. Therefore, strictly speaking, the
`action of the voltage regulator cannot be ex-
`pressed by a block diagram. However, for
`simplification, it is assumed that terminal
`voltage AV is in proportion to field current AIf
`and that an effective gain Kf is so defined. We
`have also approximated the control action by a
`simple feedback loop, because our primary object
`is to derive the overall dynamics of the alter-
`nator, not to derive the strict dynamics of the
`voltage regulator itself. From the above equa-
`tions, the alternator dynamics shown in Fig. 2
`can be easily derived.
`Next, we derive the dynamics of engines
`We have already described this derivation. (4-6j
`Therefore, only the results of our papers are
`cited here.
`
`AT, = K~ exp (- SL ) A P ~ *
`
`(8)
`
`where ia is air-flow rate [kg/s) through a
`throttle valve, Pb manifold absolute pressure
`N engine speed [rpm] , Te brake torque
`(pa),
`(Nm) , L dead time (s),
`and Tj torque disturb-
`ance (~m] for engines, respectively. The time
`constants Za and zd and other related parameters
`are defined by the following equations:
`
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`
`Engine
`Dynamics
`
`Inertio
`
`~ d ( l + sTd)
`
`Field
`
`Effect~ve Garn
`
`Voltage Regulator
`
`Fig. 2 Idle-Speed Dynamics, Including
`Alternator
`
`where Vm is manifold volume [m3] , Vh stroke
`volume (m3) , nv volumetric efficiency, 3 mement
`of inertia (kgm2] , and c resistance coefficient
`(kgm] , respectively.
`Equations (7)- (9) give the engine-speed
`dynamics as shown in Fig. 2 (the upper parts).
`Therefore, the overall dynamics of idle-speed
`including an alternator are given by the block
`diagram in Fig. 2.
`Figure 2 readily suggests the mutual inter-
`actions between an engine and an alternator.
`Suppose that the engine speed is decreased from
`its original equilibrium value by any load
`(AN* < 0).
`This means a reduction of the
`terminal voltage (AV* < 0).
`Theref ore, the
`regulator will increase its field current so as
`to keep the terminal voltage constant ( A I ~ ~ > o ) .
`This causes a torque increase (AT) and it
`reduces the engine speed. That is, there is
`qualitatively a visible reduction in engine
`friction effect when using an alternator. Beside
`this qualitative understanding of the alternator
`action, we can derive a control strategy for
`imrpoving control results of ISC systems as
`shown below. For this9; purpose, we derive a
`Assuming that
`relation among AN*, AGa and AI,'.
`the effective gain Kf is large enough for
`practical use and the deadtime L small enough,
`we can easily obtain the following relation
`using the block diagram in Fig. 2.
`
`0
`
`~ a o Gal
`Air-flow Rate
`Fig. 3 Relation between Engine Speed
`and Air-Flow Rate
`
`loads, we must control intake air flow rates
`according to both the proportional and deriva-
`tive of the currents. This feedforward compen-
`sation is the theoretical basis of our ISC
`system.
`(2) In Eq.(lO) the torque To caused by the
`current of the alternator always appears in the
`form of the difference Kd-To.
`This implies
`the alternator effectively decreases the fric-
`Therefore, alternators
`tion Kd of engine.
`degrade idle stability.
`In other words, idle
`stability will be different according to alter-
`nator characteristics for a given engine. As
`a special case, the variation of idle speed-AN*
`is equal to the variation of air-flow rate A G ~ * ,
`when To is equal to Kd. Then we obtain the
`relation as shown in Fig. 3. It should be noted
`that the relation between engine speeds and air
`flow rates is represented by a straight line
`which passes through the origin.
`
`DETERMINATION OF CONTROL PARAMETERS FOR ISC
`SYSTEM
`
`In order to design ICS system with the
`feedforward compensation formulated in Eq.(ll),
`we must: determine the values of parameters To,
`
`the following useful results
`
`From Eq.(lO),
`are obtained.
`(1) If we can control air-flow rates so as to
`maintain the following equation, we can always
`make AN* = O for any variation of load current
`AI,*.
`
`Manifold
`
`t
`i
`Sensor
`
`mtric Ti
`: e ~ ~
`
`That is, in order to reduce idle-speed fluctua-
`tions by torque disturbances due to electric
`
`Fig. 4 Experimental Installation
`
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`
`Kp and za, and then determine the value of both
`proportional gain ( 2 T o / ~ ) and derivative gain
`(
`2
`~
`~in the controleer of the I S C system. ~ )
`
`
`
`
`Figure 4 shows our experimental equipment
`for identifying the parameters in the idle-speed
`dynamics (Fig. 2).
`As shown in Fig. 4, the
`apparatus comprises an automotive engine equipped
`with a pressure sensor for each cylinder, a
`panifold absolute pressure (MAP) sensor, a load-
`current sensor, and an air-flow sensor. A 2.0-
`lit. 4-cylinder spark-ignition engine fitted
`with an electronically controlled fuel-injection
`system was used in these experiments. The
`indicated torque Ti of each cylinder was deter-
`mined from the cylinder pressure sensors, the
`manifold absolute pressure Pb by the MAP sensor,
`the load current of alternator, Ia. by the
`current sensor, and the air-flow rate Ga by the
`air-flow sensor, respectively. All experiments
`were performed under quasi-static conditions in
`while idling.
`The load-current sensor used in this ex-
`periment is composed of a hall element which has
`linear characteristics between input currents
`and output voltages as shown in Fig. 5. The ISC
`valve is a linear solenoid valve with high
`response. The indicated torque was calculated
`with a data analyzer in which both the cylinder-
`pressure signal and the crank-angle signal were
`processed.
`Table 1 shows the engine operating condi-
`tions, i.e. the equilibrium values of several
`parameters, in the experimevt. The relations
`among the air-flow rate A G ~ * , the manifold
`absolute pressure Apb*, the indicated torque AT;
`and the load current AI,"
`were measured, when
`the load current A 1 2 was increased gradually so
`speed constant ( ANk = 0 ) .
`as to keep the equilibrium value ,of the engine
`The experimental
`results are shown in Fig. 6 - 8 . Here it is
`assumed that the mechanical torque deviation can
`
`be neglected, that is, the deviation of indi-
`cated torque is equal to that of brake torque
`from the equilibrium value. Thus, the relations
`among AT;, ATe and AT are represented by the
`following equations.
`
`ATi (indicated torque) = ATe (brake torque)
`
`= A T (in Fig. 2) (12)
`
`We can determine the values of some para-
`meters from experimental results. Obviously the
`value of 2To in Eq.(ll) is obtained by taking
`gradient in Fig. 6. As previously mentioned,
`value of Kp in Eq. (11) can be determined by
`taking the gradient in Fig. 7 according to the
`idle - speed dynamics in Fig. 2, because A P ~ " = A ~ ~ *
`is valid of there are no variations in volu-
`
`Table 1 Engine operating condition
`(equilibrium values)
`
`Engine speed
`
`No = 7 5 0 lrprnl
`
`Air-f low rate
`
`Gao = 2.72 [ g / s l
`
`Load current
`
`= 14 IAI
`IaO
`
`Manifold absolute
`pressure
`
`Pbo =30.53 [ kPal
`
`Indicated torque
`
`T
`
`= 15.0 [Nrnl
`
`-
`10
`E z
`% *-
`a g 6 -
`e 3
`G 4 -
`a
`
`0
`
`Current A
`
`2To = 4.8 Nm
`
`,o.O/
`
`/ O
`
`P 0
`
`I
`
`I
`
`1.5
`0.5
`1.0
`~ o a d Current n1aX
`
`I
`
`2.0
`
`Fig. 5 Output Characteristics of
`Current Sensor
`
`Fig. 6 Relation between Torque Disturbance
`and Load Current
`
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`
`metric efficiency. According to the experimental
`results of Fig. 8, however, it can be seen that
`.
`This means we cannot neglect
`AFbyc = 0.77 A G ~
`
`the variations in volumetric efficientcy. Thus,
`in order to determine the value of Kp, it is
`necessary to take into account the effect of the
`volumetric efficiency on Kp. Theoretical treat-
`ment of the effect is mentioned in Appendix in
`detail : here, we cite the result. When we
`define the gradient in Fig. 7 as K;
`and the
`gradient in Fig. 8 as Kab, the value of Kp is
`given by the following equation.
`
`where Kb is the ratio of the indicated torque
`variation generated to that of pumping loss
`caused by the variation of a manifold absolute
`pressure and the value of the Kb is determined
`Since Kb = 4.85
`theoretically (see Appendis).
`[~m] and the values of To, KP' and Kab are
`determined by taking gradients in Fig. 6, Fig. 7
`and Fig. 8, respectively, we obtain the value of
`the proportional gain P in Eq.(ll).
`
`Next we determine the derivative gain in
`E q l l The value of Za is determined accord-
`ing to the definition as previously mentioned.
`In the case of the engine used in our experiment,
`the relation between the volumes of a manifold
`and a stroke is represented by Vm= Vh. And the
`value of the volumetric efficiency is estimated
`By using the above results,
`as qvo = 0.6.(~)
`the following value is obtained.
`
`According to previously obtained results, we
`derive the value of derivative gain D in Eq.(ll).
`
`Air -flow Rate a ~ a '
`
`Thus, the control parameters P and D have
`been determined. In the following, we will
`discuss the control provided by the ISC system
`implementing the above feedforward logic.
`
`Fig. 7 Relation between Brake Torque and
`Air-Flow Rate
`
`EXPERIMENTAL RESULTS AND DISCUSSION
`
`Me 0.2
`
`Q
`
`Kab = 0.77
`
`Fig. 8 Relation between Manifold Absolute
`Pressure and Air-Flow Rate
`
`We would like to discuss the control re-
`sults for our ISC system with proportional gain
`P and the derivative gain D, whose values were
`determined experimentally as described above.
`Before starting to explain the control
`results, we would like to comment on the
`differences between analogue and digital control.
`As is well-known, actual control of ISC is
`performed based upon sampled values in an
`electronic control unit (ECU).
`Therefore, the
`feedforward algorithm in ECU is given by the
`following equation.
`
`where Ts is the interval [s] of control action,
`Ke is the conversion coefficient as implemented,
`AI,(~)
`is the kth value of the 1?ad current of
`the alternator in the ECU, and AGa,(k) is the
`command for the ISC valve.
`Typical experimental results for the degree
`of control are shown in Fig. 9 and Fig. 10.
`Figure 9 shows the control of idle speeds both
`with and without feedforward in the case for
`
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`
`discontinuous changes in the load current of the
`alternator. Figure 10 shows this for another
`variation of the load current of the alternator.
`These patterns of the load currents, which are
`often encountered in practical vehicles, are
`achieved by using the electric accessories; for
`example, headlights, and power windows. It is
`evident from Fig. 9 and Fig. 10 that our ISC
`system has superior ability to minimize idle-
`speed variations despite torque variations.
`Finally, we discuss the agreement between
`the control gains obtained experimentally and
`theoretically. In principle, this ISC system
`must be able to eliminate the fluctuations of
`idle speed completely except for some approxi-
`mations. First of all, we can conclude good
`agreement of the proportional gains between ex-
`perimental and theoretical. There are no dis-
`crepancies of the averaged idle speed before and
`after the load application, as shown in Fig. 9
`and 10. Before we discuss the agreement of
`derivative gains, some considerations should be
`
`P
`W 650
`
`With Feed forward
`
`Time s
`
`Fig. 9 Experimental Result No. 1
`
`Fig. 11 Definition of N1, N 2
`
`0, " w" 6 5 0
`
`-.
`-
`a 700
`c .-
`o,
`5 650
`
`With Feedforward
`
`2 17rpm
`
`800
`Without Feedforward
`g 750
`-90rpm
`
`g 700
`.-
`
`'0 ; ; ; ; 6 7 8 b i b
`
`Derivative Gain
`
`Fig. 10 Experimental Results No. 2
`
`Fig. 12 Relation between Variation of
`Engine Speed and Derivative Gain
`
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`
`mentioned. As shown in Fig. 9 and Fig. 10, we
`can see that both an undershoot and an overshoot
`of idle speed persist with feedforward, when the
`alternator's electric load is disturbed. This
`situation is shown in Fig. 11 schematically.
`There may be several reasons why both the under-
`shoot and the overshoot of idle speed exist in-
`cluding : (1) the deadtime before generating
`torque, and (2) the essentially discrete action
`in engines. Now we define an optimal experimen-
`tal derivative gain Ke D/Ts corresponding to the
`theoretical derivative gain D in Eq.(14) as
`follows. As shown in Fig. 11, there are both
`undershoot N1 and overshoot N2 if the feedfoward
`algorithm is implemented. It may be supposed
`that both N1 and N2 vary with increases in the
`experimental derivative gain Ke D/Ts in Eq.(15).
`Figure 12 shows the experimental results for
`behaviors of both N1 and N2 with increases in
`experimental derivative gain. In the case of
`Fig. 12, it would be natural to define the
`optimal gain as Ke D/Ts = 5. On the other hand,
`using the theoretical derivative gain D in Eq.
`(14), the value of the derivative gain Ke D/TS
`is theoretically determined from the following
`equation.
`
`Because, in the case of our experiment, Ke =
`5.23, Ts = 0.028 [s) .
`This values gives the
`nearest value to the experimental derivative
`gain Ke D/Ts. Therefore, we consider the control
`parameter P and D obtained experimentally show
`good agreement with those obtained theoretically.
`
`CONCLUSION
`
`(1) We have developed an ISC system which
`strongly suppresses idle-speed
`fluctuations
`under load disturbances, particularly electrical
`loads.
`(2) Our model of idle-speed dynamics including
`alternator dynamics gives clearly not only the
`basis of the control procesdure for our ISC
`system but also the physical aspects of engine
`id1 ing speed.
`
`ACKNOWLEDGMENTS
`
`The authors would like to acknowledge the
`members of the Himeji Works of Mitsubishi Elec-
`tric Corp. for achieving this work.
`
`REFERENCES
`
`1. Takahashi, T., Ueno, T., Yamamoto, A., and
`Sabuichi, H., "A Simple Engine Model for
`Idle Speed Control" SAE Paper 850291 (1985)
`2. Osawa, M., Ban, H., and Miyashita, M.,
`"Stochastic Control For Idle Speed Stability"
`FISITA 885066 (1988)
`
`3. Miyairi, S., "New Technology of Electrical
`Machinery and ~ ~ ~ a r a t u s " (19751, P. 10,
`Maruzen (in Japanese)
`4. Washino, S., Nishiyama, R., and Ohkubo, S.,
`"A Fundamental Study for the Control of
`Periodic Oscillation of SI Engine Revolu-
`tions" SAE Paper 860411 (1986)
`5. Washino, S. and Inoue, H., "A Study of Idle
`Speed Control for the relation between PI
`gain and time lag of Idle Speed Dynamics1'
`JSAE 882085 (1988) (in Japanese)
`6. Washino, S., "Automobile Electronics1' (1989),
`P.34, Gordon and Breach Science Publishers
`7. Awano, S., "Technology of Internal Combus-
`tion Engines" (1978), P.73, Sankaidoo Pub-
`lishing Co. Ltd. (in Japanese)
`
`Appendix
`
`Strictly speaking, according to.Fig. 8, the
`deviation of the air-flow rate A G ~ was not
`equal to that of the manifold absolufe pressure
`with increasing load current AI,* . This
`AP;
`is caused by the variation of volumetric effi-
`ciency qv* which is not equal to zero. There-
`fore, we consider the idle-speed dynamics in
`case of Anlr* f 0.
`.. .
`First we derive the idle-speed dynamics of
`the engine for AqV* # 0. The following equations
`are derived from mass-flow continuity of intake
`air, the state equation of gas in the manifold
`and the definition of volumetric efficiency.
`
`ia : Air-flow rate through throttle valve [kg/s]
`: Air mass in manifold volume [kg)
`Ge : Air-flow rate through intake valve [kg/s)
`Rm : Gas constant of air [pam3/kgK]
`Tm : Air temperature in manifold [K]
`(Other notations are the same as the body of
`this paper)
`
`Here, we define the absolute pressure PC which
`is the product of the manifold absolute pressure
`Pb and the volumetric efficiency nv. As in the
`body of this paper, we take the deviation from
`equilibrium value and normalize the deviation by
`the equilibrium value.
`Then we obtain the
`following equations.
`
`By taking the deviation from equilibrium value
`in Eq.(~3), we also obtain the following equa-
`tion.
`
`9
`
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`and
`By b o t h d i f f e r e n t i a t i n g Eq.(A2)
`L a p l a c e t r a n s f o r m , Eq.(A6)
`i s o b t a i n e d .
`
`t a k i n g
`
`S u b s t i t u t i n g Eq.(AlO)
`f o l l o w i n g e q u a t i o n :
`
`i n ( ~ 8 ) , we o b t a i n t h e
`
`i n
`and Eq.(A6)
`S u b s t i t u t i n Eq.(A4), Eq.(A5)
`E q . ( A l ) , we f i n a l l y g e t t h e f o l l o w i n g e q u a t i o n :
`
`where,
`
`It i s assumed t h a t t h e v a r i a t i o n of both a i r - f u e l
`r a t i o and f r i c t i o n c o u l d be n e g l e c t e d and
`t h a t
`Eq.(A12)
`i n t h e body of t h i s paper i s a l s o v a l i d .
`Thus t h e d e v i a t i o n of b r a k e t o r q u e
`i s g i v e n a s
`t h e f o l l o w i n g e q u a t i o n .
`
`ATe = ( Kc APC* + K c Apb*) exp (-sL)
`
`i n ( ~ l l ) , we o b t a i n t h e
`S u b s t i t u t i n g Eq.(A9)
`f o l l o w i n g e q u a t i o n , which i s o u r propose:
`
`Here, we a r e a b l e t o d e t e r m i n e t h e v a l u e s of K p '
`and Kab by u s i n g F i g . 7 and F i g . 8 , r e s p e c t i v e l y .
`i s determined t h e o r e t i c a l l y a s
`The v a l u e of Kb
`Eq. (A131 .('I
`
`Kb : i n d i c a t e d t o r q u e c o e f f i c i e n t g e n e r a t e d
`by pumping l o s s caused by v a r i a t i o n of
`a m a n i f o l d a b s o l u t e p r e s s u r e
`
`t h e i d l e - s p e e d dynam-
`The b l o c k diagram of
`i c s i n c a s e of Anv* # 0
`i s o b t a i n e d a s shown i n
`F i g . A l , which i s d e r i v e d from Eq. (A4), Eq. (671,
`Eq, (A8) and Eq. ( 9 )
`i n t h e body of
`t h i s p a p e r .
`When we compare F i g . A1 w i t h F i g . 2 , we o b t a i n
`t h e f o l l o w i n g e q u a t i o n f o r t h e r e l a t i o n between
`AP$
`and ATe.
`
`By t h e d e f i n i t i o n of PC, we c a n d e r i v e Eq.(A10).
`
`t h e Kp v a l u e w i t h v a r i a -
`
`Thus we have o b t a i n e d
`t i o n of q$.
`As mentioned above, we cannot n e g l e c t t h e
`, when we
`i n £ l u e n c e of
`t h e v a r i a t i o n of
`t h e v a l u e of K p . We, however,
`can
`i d e n t i f y
`t h e v a r i a t i o n of q$ on
`n e g l e c t
`t h e e f f e c t of
`t h e m a n i p u l a t e d a i r - f l o w r a t e i n a c t u a l c o n t y o l ,
`The r e l a t i o n among AN*, AG,* ,
`a s shown below.
`Anv*
`and A I ~ * i s d e r i v e d from F i g . A l ,
`i n t h e
`same manner a s t h e body of t h i s p a p e r .
`
`t h e a i r - f l o w r a t e t h a t should b e
`From Eq.(A14),
`t o g e t A ~ = o
` f o r an
`m a n i p u l a t e d
`i n o r d e r
`l o a d c u r r e n t AIL'(
`i s
`a r b i t r a r y v a r i a t i o n of
`g i v e n by Eq . (A151 .
`
`t h e v a r i a t i o n
`C o n s i d e r i n g t h e i n f l u e n c e of
`of *
`on
`t h e m e n i p u l a t e d a i r - f l o w r a t e , we
`compare
`t h e v a l u e of
`t h e f i r s t t e r m w i t h
`t h e
`v a l u e of
`t h e second t e r m i n t h e r i g h t - h a n d s i d e
`of Eq.(A15). By u s i n g t h e r e s u l t s among F i g . 6 ,
`7 , and 8
`i n t h e body of
`t h i s p a p e r , we o b t a i n
`t h e r e l a t i o n betwee APb* and AIa*
`a s shonwn i n
`By u s i n g Eq.(A10) and
`F i g . A2.
`t h e r e s u l t s of
`F i g . A2 and F i g . 8 ,
`t h e d e v i a t i o n of t h e volu-
`m e t r i c e f f i c i e n c y nv*
`w i t h
`i n c r e a s i n g
`l o a d
`c u r r e n t A I ~ *
`i s o b t a i n e d . Using
`t h e above
`r e s u l t s , t h e r e l a t i o n between b o t h v a l u e s of t h e
`f i r s t t e r m and t h e second t e r m i n t h e r i g h t - h a n d
`
`Engine
`Dynamics
`
`Alternator
`
`- - - - - - - t
`
`AN*
`
`I
`I
`( K ~ ( I +
` ST^)
`- - _ - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
`I
`
`F i g . A1 Engine Dynamics i n c a s e of Anv* = 0
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`side of Eq.(A15) is calculated as the following.
`
`(the second term)/(the first term) = 0 . 0 2 8
`
`Evidently, the influence of the variation of the
`volumetric efficiency on the manipulated air-
`flow rate can be neglected in actual control.
`Thus the influence of the variation of rlv*
`on the manipulated air-flow rate may be neg-
`lected, but not, of course, the influence on the
`identification of the parameter Kp.
`
`Load Current a l a X
`
`Fig. A2 Relation between Manifold Absolute
`Pressure and Load Current
`
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`
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