IDLESPEEDCONTR OLDESIGNAND VERIFICA TIONF ORANA UTOMOTIVEENGINE Andrea Balluc hi Luca Ben en uti MariaDomenicaDiBenedetto Gio anni Girasole AlbertoL
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IDLESPEEDCONTR OLDESIGNAND VERIFICA TIONF ORANA UTOMOTIVEENGINE Andrea Balluc hi Luca Ben en uti MariaDomenicaDiBenedetto Gio anni Girasole AlbertoL

Sangio anniVincen telli ARADES Via San Pantale o 66 00186 R oma Italy al luchi alb ertop ar adesrmcnrit Dip di Informatic a e Sistemistic a Universit adiR oma L Sapienza via Eudossiana 18 00184 R oma Italy envenutidisunir omait EECS Dept University

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IDLESPEEDCONTR OLDESIGNAND VERIFICA TIONF ORANA UTOMOTIVEENGINE Andrea Balluc hi Luca Ben en uti MariaDomenicaDiBenedetto Gio anni Girasole AlbertoL




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Presentation on theme: "IDLESPEEDCONTR OLDESIGNAND VERIFICA TIONF ORANA UTOMOTIVEENGINE Andrea Balluc hi Luca Ben en uti MariaDomenicaDiBenedetto Gio anni Girasole AlbertoL"‚ÄĒ Presentation transcript:


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IDLESPEEDCONTR OLDESIGNAND VERIFICA TIONF ORANA UTOMOTIVEENGINE Andrea Balluc hi ,Luca Ben en uti MariaDomenicaDiBenedetto ,Gio anni Girasole AlbertoL. Sangio anni{Vincen telli ARADES, Via San Pantale o, 66, 00186 R oma, Italy al luchi, alb erto@p ar ades.rm.cnr.it Dip. di Informatic a e Sistemistic a, Universit adiR oma \L Sapienza", via Eudossiana, 18, 00184 R oma, Italy envenuti@dis.unir oma.it EECS Dept., University of California at Berkeley, CA 94720, alb erto@e cs.b erkeley.e du Dip. di Inge gneria Elettric a, Universit a del l'A quila, Po ggio di R oio, 67040 L'A

quila, Italy dib ene de@ing.univaq.it Abstract: The goal of an idle con trol for automotiv e engines is to main tain the engine sp eed within a giv en range, robustly with resp ect to load torque disturbances acting on the crankshaft. Mean v alue mo dels ha e b een used in the past to design idle con trol algorithms. Ho ev er, the b eha vior of the torque generation pro cess and the dynamics of the p o er{train are not captured with enough accuracy to guaran tee that the idle con trol sp eci cations as giv en b y car mak ers are met. W e use a cycle-accurate h ybrid mo del to o ercome these

obstacles. The complexit y of the mo del mak es the problem of synthesizing a feedbac kcon trol la w for the system prohibitiv e. Heuristics ha e b een widely used in the past. In this pap er, w e presen t a divide and conquer approac hto cop e with this problem. The system is decomp osed in three parts. F or eac hpartin isolation, a con trol la wisderiv ed on a simpli ed mo del assuming that the other parts can be con trolled to yield appropriate inputs. The erall con trol strategy is then applied to the system and formal v eri cation is used to ensure that the b eha vior of the con trolled

system meets the sp eci cations. 1. INTR ODUCTION The syn thesis of idle con trol strategy for an in ternal com bustion engine is among the most hallenging problems in engine con trol. The ob- jectiv is main taining the engine sp eed as close as p ossible to the alue that minimizes fuel consumption, while prev en ting the engine from stalling. The di∆cult y lies with the unpredictable load v ariations coming from the in termitten tuse of devices po ered the engine, suc as the air conditioning system and the steering wheel serv o-mec hanism. surv ey on di eren engine mo dels and con trol design

metho dologies for the idle con trol is giv en in Hro at and Sun (1997). Both time{domain (e.g. Butts et al. (1999)) and crank{angle domain (e.g. Y urk vic h and Simpson (1997)) mean{v alue mo dels ha e b een prop osed in the literature. More recen tly , cycle-accurate mo d- els capturing p erio dic engine sp eed v ariations due to torque uctuations, w ere in estigated in Shim et al. (1996). Multiv ariable con trol (Onder and Geering (1993)), con trol (Butts et al. (1999)), con trol (Carnev ale and Mosc hetti (1993)), syn thesis (Hro at and Bo denheimer (1993)), slid- ing mo de con trol

(Kjergaard et al. (1994)) and
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LQ-based optimization (Abate and Di Nunzio (1990)) ha b een applied to idle con trol on ariet of mo dels. Ho ev er, fully satisfactory solution has still to emerge. In this pap er, w euse ybrid mo del to describ e the cyclic b eha vior of the engine, th us capturing the e ect of eac spark ignition on the generated torque and the in teraction b et een the discrete torque generation and the con tin uous p o er-train and air dynamics. consider traditional spark ignition engine without Gas Direct Injection (GDI). The torque generated eac cylinder and

applied to the engine crankshaft can b e assumed to b e a function of the spark ignition time, and of the air-fuel mixture mass loaded in the cylinder during the in tak e phase. Since the air-to-fuel ratio is assumed to b e constan t (at the stoic hiometric v alue), then the mixture mass is con trolled the throttle plate p osition and is sub ject to the dynamics of the cylinder lling. Hence, the ailable con- trols for the idle problem are: the spark ignition time and the p osition of the throttle v alv e, whic regulates the air in o ybrid mo del of the plan is obtained from the general mo del

of an in ternal com bustion engine presen ted in refer- ence Balluc hi et al. (2000b), using nonlinear expressions and mo del parameters iden ti ed on a commercial car in collab oration with our indus- trial partner, Magneti Marelli. In Balluc hi et al. (2000b) and Balluc hi et al. (2000a), the problem of main taining the crankshaft sp eed within giv en range as formalized as \safet y" sp eci cation for the ybrid closed{lo op system or sim- pli ed engine ybrid mo del, where some nonlin- ear expressions w ere linearized and no actuation dela as considered, the idle con trol problem as solv ed

computing analytically the set of all ybrid states for whic there exists ybrid con trol strategy meeting the sp eci cation. The class of all \safe" con trollers obtained b y the pro- cedure w as called the maximal c ontr ol ler . Despite the simpli cation of the mo del, the expression of the maximal con troller obtained in Balluc hi et al. (2000b), Balluc hi et al. (2000a) is quite complex. Consequen tly , the implemen tation of a particular con troller extracted from the maximal con troller is quite exp ensiv e in terms of computing time and memory osolv e the problem in w ys that are

industrially feasible, eha eto tak ein to accoun nonlinear- ities, actuation dela ys and et ha to eep The e ect of a spark command on the torque generation is \stronger" than the one of a throttle plate command, since air in o w is sub ject to b oth manifold dynamics and dela y due to mix compression. Hence, sudden loads can b e uc h b etter comp ensated with spark ignition than with air in o w, while air in o w can b e used to con trol the engine in steady state. A safet y sp eci cation requires the system to sta y within a set of sp eci ed safe states. close lo ok at implemen tation costs.

The ap- proac hw e prop ose in this pap er is to select semi- heuristically an easy{to{implemen tcon troller and then v erify that it satis es the sp eci cations. erifying the system b ysim ulation and protot yp- ing is certainly p ossible but w e cannot guaran tee that the system will satisfy the sp eci cations in all op erating conditions. In fact, it is often the case that idle con trol needs extensiv empirical adjustmen ts in the car. In this pap er, w e presen t a rst cut for a metho d- ology based on divide and conquer approac to obtain the con troller and formal tec hniques to guaran

tee that the con troller is con tained in the maximal con troller and, hence, it satis es the sp eci cations. In our case, the closed{lo op system is view ed as comp osed of three sub-systems (in tak e manifold, cylinders and p o er{train sub-systems). The con- trol la is deriv ed simplifying substan tially eac h of the sub-systems so that the h ybrid nature of the mo del is ignored and the sub-systems are linearized and discretized with xed sampling time. The con trol la is syn thesized so that the constrain ts are satis ed in this simpli ed domain. Then, this con trol la w is applied to the

full- edged system. Of course, at this p oin t, there is no guar- ante that the con trol la w will yield a closed{lo op system that satis es the constrain ts ev en though the decomp osition and the corresp onding simpli - cations ha ebeen made so that w edonotw ander far from the original mo del. ormal eri cation to ols ha b een considered possible solution to this problem. Che ckmate an automatic to ol for sim ulation and formal eri cation of ybrid systems dev elop ed at Carnegie Mellon Univ ersit (see Ch utinan and Krogh (1998, 2000)) has b een tried, but the complexit of the mo del of the

po er train is to o high for the to ol to complete the analysis in reasonable time. The basic idea of our metho dology is to erify eac sub-system in isolation assuming that the b eha vior of the other sub-systems satis es an appropriate set of condi- tions. Then, the consistency of the assumptions on the b eha viors is v eri ed, kno wing that eac hof the sub-systems w orks correctly in isolation. The sub-systems are the same as the ones iden ti ed for the deriv ation of the con trol la w. Ho ev er, the beha vior of the closed-lo op po er-train sub- system is still to o complex to v erify

formally , since it con tains the h ybrid p o er{train mo del. Hence, the h ybrid p o er{train mo del is decomp osed itself in the con tin uous part and the discrete part and eri ed follo wing the same divide and conquer approac h. Note that this approac h is a particular case of the assume{guaran tee paradigm widely used in formal eri cation. While our metho dol- ogy has b een fully de ned, w e still ha e some di∆-
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culties with the application of the formal v eri ca- tion to ols ev en with the decomp osed sub-systems. In this pap er, rep ort the ybrid mo del, the con trol la

w deriv ation pro cedure and the parts of formal v eri cation w eha e b een able to complete. o summarize, our metho dology consists of divid- ing the erall system in set of in terconnected sub-systems, simplifying the mo del for eac h, de- riv ea con trol la w for the decomp osed, simpli ed system and nally formally erify that the con- trol la (p ossibly mo di ed to tak in to accoun some of the simpli cations made) satis es the constrain ts of the full edged mo del. T o the b est of our kno wledge, this approac his no el in h ybrid con trol. The pap er is organized as follo ws: in Section 2,

description of the ybrid mo del of the engine in the idle region of op eration is recalled. In Section 3, an idle sp eed con troller is prop osed and in Section 4, the eri cation results on the beha vior of the closed{lo op ybrid system are rep orted. 2. PLANT HYBRID MODEL In this section w e brie y describ e a h ybrid mo del of 4{strok in ternal com bustion engine. This mo del has b een obtained from the ybrid mo del presen ted in reference Balluc hi et al. (2000b), b sp ecifying nonlinear expressions and mo del pa- rameters on the bases of the exp erimen tal data obtained from a commercial

car at idle sp eed pro- videdusb ythe er-train Division of Magneti Marelli in Bologna (Italy). Accuracy of this y- brid mo del has b een tested on a di eren tcon trol problem b y exp erimen tal v alidation (see Balluc hi et al. (1999)). The mo del is comp osed of four in teracting blo c ks, namely the in tak manifold, the cylinders, the po er-train and the actuators, as sho wn in g- ure 1. The in tak manifold pressure dynamics is con tin uous-time pro cess con trolled b y the throttle- alv p osition Denoting the pressure, manifold dynamics is mo delled as )= )) )+ )) )) (1) where is the

quivalent thr ottle ar giv en in terms of throttle angle arameters and dep end in strongly nonlinear fashion on the geometric c haracteristics of the manifold, on the ph ysical c haracteristics of the gas and atmo- sphere, and on the curren tv alue of the pressure While in traditional engines the throttle v alv eis directly connected to the gas p edal, in electronic{ throttle systems, it is con trolled the engine con trol system to ac hiev e b etter p erformance. The dynamics of actuation of the throttle v alv e(usu- ally DC motor) is mo delled linear rst{ order dynamical system: )= )+ (2)

where is the DC motor input oltage whic is assumed to b e a discrete time signal pro duced with a sampling p erio d The cylinders blo c mo dels the torque genera- tion. The torque generated eac piston at eac h cycle dep ends on the thermo dynamics of the air{fuel mixture com bustion pro cess. The pro le of dep ends on the phase of the cylinder, the piston p osition, the mass of air and of fuel loaded during the in tak e phase, and on the spark ignition timing. F or idle sp eed v alues, the quan tit of air loaded in to eac cylinder at the end of the in tak e run can b e assumed to dep end only

on the alue of the in tak manifold pressure at the in tak e end time int as int )) int (3) Assuming that fuel injection is regulated an 15 10 10 15 20 Fig. 2. Ignition e∆ciency function at lo engine sp eed. inner con trol lo op that main tains the air-to-fuel ratio to stoic hiometry , the pro le of the torque generated b y eac h cylinder can b e describ ed b ya piecewise constan t function that is assumed to b e zero ev erywhere except in the expansion phase in whic )= Gm (4) where the gain is constan parameter, is the spark adv ance and the ignition e∆ciency function ) has, in general, the

pro le sho wn in gure 2. The spark ignition time is commonly de ned in terms of the spark adv ance that denotes the di erence b et een the angle of the crankshaft when the cylinder is at the end of the compression strok e and the one at the time of ignition.
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er-train Crankshaft clutch ge ar Driveline Prima ry Seconda ry Driveline Intak eManifold spark spark actuato rs Cylinders Ignition Fig. 1. Mo del of the engine. When neutral gear selection is engaged and the clutc h is released, the secondary driv e-line is dis- connected from the engine so that the po er- train is describ

ed only in terms of the crankshaft sp eed and position the con tin uous time system )= )+ )) (5) )= (6) where are constan parameters, is the torque pro duced b y the engine, and represen ts the load torque acting on the crankshaft. The ignition actuators ust pro duce the spark ignition at ev ery cycle, according to the curren con trol algorithm decision, and sync hronously with the crankshaft p osition . Since the ignition con trol tak es time to actuate, then, it has to be decided su∆cien tly in adv ance to mak esurethat it is prop erly deliv ered to the plan t. The spark is in general

ignited with a di eren spark adv ance at ev ery cycle. The v alue of the spark adv ance m ust then b e computed at the end of the in tak e strok e, so that the ignition subsystem can b e programmed to ignite the spark at the prop er time. Fig. 3. Engine h ybrid mo del at idle sp eed. In conclusion, the b eha vior of a four{cylinder in- line engine and po er-train can be represen ted using only one discrete state, as sho wn in gure 3. In fact, the engine kinematics are suc h that, at an time, eac h cylinder is in a di eren strok e of the engine cycle and only one cylinder is generating the

torque .Ev ery half of a crankshaft run, i.e., when = 180, the con trol la ) is computed to obtain the spark ignition to b e applied to the cylinder that is en tering the compression strok eat the curren t dead cen ter. The throttle v alv e con trol ) is computed at xed frequency 1 = and will a ect the amoun of air loaded the cylinder that is p erforming the in tak e strok eatthe curren sampling time. The torque is the total torque acting on the crankshaft. 3. IDLE SPEED CONTR OL DESIGN The sp eci cation for idle sp eed con trol is to main- tain the crankshaft rev olution sp eed around

nominal alue so that it nev er exits range with 0. This sp eci cation has to be ac hiev ed for an alue of the load torque disturbance within a lo er b ound zero and an upp er b ound 0. The prop osed con trol is a state feedbac k con trol. While the spark adv ance feedbac is com- puted at dead{cen ter times, the throttle feedbac ) is set at xed{frequency sampling times. Let denote the sequence of dead{cen ter times and let denote the curren t dead{cen ter time. According to the engine ybrid mo del describ ed in the previous section and sho wn in gure 3, the spark adv ance con trol ), hosen at

time will a ect the alue of the torque +1 that driv es the crankshaft from time +1 to time +2 The amoun of driving torque +1 dep ends also on the alue of the load disturbance ), acting on the same time in terv al [ +1 ;t +2 ). The result of the action of this torque will b e a giv en alue of crankshaft sp eed +2 at time +2 Let denote the sequence of sampling times of the throttle alv con trol feedbac ). The con trol actions ) for ;t +1 ) driv
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Fig. 4. Con trol sc heme. the manifold dynamics during an in tak strok starting from the dead{cen ter time to the dead{ cen ter time +1

The amoun tof mass loaded b the cylinder at time +1 dep ends on these con trol actions. Note that the corresp onding torque will b e pro duced only from time +2 to time +3 ,due to the dela yin tro duced b y the compression strok e. Due to the complexit of the system to con trol, our strategy is to decomp ose the system in to three nested sub-systems and to devise an appropriate con trol for eac of them in isolation, assuming that the v ariables that connect the sub-system to the others satisfy an appropriate assumption. The in terconnect v ariables are sub ject to con trol and the assumption

is hosen so that the con troller can indeed mak e them true when applied to the appropriate sub-system. Ev en with this divide and conquer approac h, the con trol problem for eac of the sub-system is highly complicated. Hence, decided to eliminate most of the sources of di∆cult so that an easy{to{implemen con trol la w can b e deriv ed. Of course, there is no a priori guaran tee that the con trol la w deriv ed with these heuristics satisfy the constrain ts when applied to the ybrid mo del. The goal of the eri cation step is to pro that the strategy pa ys o : the constrain ts are indeed satis

ed. 3.1 Contr ol ler structur The con troller is comp osed of three nested lo ops: an engine sp eed con trol in the outer lo op, whic is resp onsible for the generation of torque alues the cylinders are requested to pro duce; torque con trol in the middle lo op, whic regulates the torque pro duced b y the engine to the desired alue and is implemen ted the spark adv ance feedbac ); a manifold con trol in the inner lo op, whic his resp onsible for regulation of the mass loaded the cylinders and implemen ted the throttle v alv e feedbac ). The o erall con trol sc heme is sho wn in gure 4. The

task the engine sp eed con troller is pro vid- ing reference torque that, when pro duced, main tains the engine sp eed inside the sp eci ed range robustly with resp ect to the action of load torque disturbances. A feedbac k con trol that ac hiev es this task is designed assuming that the sp ark advanc e midd le lo op wil l e able to pr duc the queste tor que The feedbac con trol la is designed considering the sub-system with the po er{train blo c konly . The con troller is a PI con- troller that maximizes the stabilit of the closed lo op where the con tin uous dynamics of the p o er{ train is

discretized, w e assume that b et een and there is a simple dela The torque con trol has to regulate the spark adv ance so that it pro duces the reference torque requested b y the outer lo op. A feedbac k con trol ac hieving this task is designed assuming that the thr ottle valve inner lo op wil l able to pr ovide suitable se quenc of masses of air lo ade into the cylinders during the intake str okes . This feedbac kcon trol is designed considering the sub- system with the cylinders blo c konly . This con trol la has no dynamics and pro duces the spark adv ance and In this mo dels the self{lo

op transition is triggered b y the dead{cen ter ev en t. The reference torque and the spark adv ance feedbac k con trol ) are set at eac h dead{cen ter for the cylinder that is en tering the compression strok e. As a result of the design of the torque con trol feed- bac k, obtain sp eci cation for the manifold con trol inner lo op. This sp eci cation is expressed in terms of lo er and an upp er bound on the alue of mass of air that should ha b een loaded in to the cylinders in the previous in tak strok e. The ob jectiv e of the manifold con trol is th us pro- viding an amoun t of mass loaded b

y the cylinders within the range sp eci ed the middle lo op torque con trol. The comp onen ts of the plan tto be tak en in to consideration are the in tak manifold and cylinder lling dynamics, and the throttle alv e actuator. T o simplify the con trol la w deriv a- tion, the manifold dynamic is linearized ab out the equilibrium pressure, i.e., the co e∆cien ts of the dynamic equations are computed with )= eq corresp onding to )= and = 0. In addi- tion, the dynamics are discretized with sampling time . The con troller is a discrete-time PID. In this mo del the self{lo op transition is go erned

xed frequency trigger with period eac h transition the throttle v alv e feedbac )is computed and applied. Note that, while the throt- tle alv signal is triggered xed frequency sampling, the sp eci cation for the in tak e manifold con trol is giv en in terms of the loaded air masses
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whose time base is the sequence of dead{cen ter times. This async hronism bet een the feedbac lo op and the sp eci cations will be considered in the v eri cation of the p erformances of the closed lo op system. 3.2 The design of the fe db ack c ontr ol lo ops 3.2.1. Sp d and tor que c ontr ol lo

ops. ollo wing the ideas presen ted in the previous section, deriv e the con trol la ws b y simplifying the h ybrid mo del. In this section w e detail the simpli cations and sho ho the con trol la is deriv ed. The ybrid nature of torque generation pro cess is appro ximated b y means of a xed frequency 30 dicrete time pro cess and the driv eline dynamics is discretized with sampling period so that it reduces to: k )= k )+ k k )) with and 1)( =a ). Moreo er, from the assumption that the middle lo op will b e able to pro duce the requested torque it follo ws that k )= k (7) where k is the

driving torque during the expansion phase from time k to k and k is the reference torque computed at the end of the in tak e phase b y the sp eed con troller. This torque dep ends on the mass of air k )= k )) k loaded in the cylinder en tering the compression phase and on the spark adv ance k )that will b e applied to the same cylinder. e design a feedbac kcon trol suc h that the torque k )em ulates that of a PI con troller, giv en y: )= 1+ 1)  (8) This con troller is able to asymptotically comp en- sate constan torque disturbances. The gain and the time constan are selected as to

stabi- lize the system dynamics k )= k )+ k (9) in the absence of the disturbance torque k ). The spark adv ance is then computed simply in erting equation k )= Gm k )) k )) taking in to accoun the saturation min max ]. If the computed spark adv ance is not saturated, then it will be able to pro duce the desired torque, as in (7), on the basis of the mass of air loaded at time k urthermore, the middle{lo op con trol has to gen- erate the sequence of reference v alues k that is giv en as input to the manifold con trol inner lo op. This reference alue, when pro duced, will a ect the

torque generated in the time in terv al k ;k +2 and is computed simply in erting equation k )= Gm k where the parameter , with min <' <' max is the nominal spark adv ance to be used during idle con trol. The tuning of the con trol parameter to v alue smaller than max allo ws some degree of reacting to fast positiv e torque requests using the spark adv ance. Similarly ,setting to a v alue greater than min ,allo ws to react to fast negativ torque requests. Note that the mass of air loaded in to the cylinder at time k i.e., at the end of the in tak phase, dep ends on the throttle con trol

applied on the time in terv al [ k ;k ]. 3.2.2. Intake manifold ontr ol lo op. The ob jec- tiv eofthein tak e manifold con trol is making the cylinders load the amoun tof mass sp eci ed b the torque con trol. or the design of the in tak e manifold con trol, w assume that the reference signal is pro duced at xed frequency = and is suc that for an appropriate in teger The robustness of the closed{lo op system with resp ect to this assumption will b e c hec ed in the eri cation phase. The in tak e manifold and cylinder lling dynamics are discretized with sampling p erio d equal to the throttle

v alv econ trol sampling time . Hence, a PID con troller that pro duces con ergence of the manifold pressure to the target setp oin t~ (whic is a piecewise constan tv alue of p erio d obtained from in erting equation (3)) is used for the throttle piecewise{constan con trol feedbac ): )= 1+ 1) 1) )) (10) Sim ulation results are rep orted in gure 5. The crankshaft rev olution sp eed has to b e main tained around the nominal alue 800 rpm with maxim um excursion of 50 rpm so that 30 =n =37 5 msec. The alue of the po ertrain parameters are 531 and 95 49; those of the in tak manifold are: 4, 20 94,

1821, 10 3, and 15. Finally the PID con trollers are de ned b y the follo wing time constan ts and gains: 0781, 234 msec, 0613, =135 7 msec, ^ =29 5 msec. Figure 5 sho ws
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3.2 3.4 3.6 3.8 4.2 4.4 4.6 4.8 12 13 14 15 16 17 18 19 3.2 3.4 3.6 3.8 4.2 4.4 4.6 4.8 750 760 770 780 790 800 810 3.2 3.4 3.6 3.8 4.2 4.4 4.6 4.8 −17 −16 −15 −14 −13 −12 −11 −10 −9 −8 −7 3.2 3.4 3.6 3.8 4.2 4.4 4.6 4.8 4.2 4.4 4.6 4.8 5.2 5.4 5.6 3.2 3.4 3.6 3.8 4.2 4.4 4.6 4.8 4.5 5.5 6.5 7.5 Fig. 5. Sim ulation results. relev an t signals

in a sim ulation of the closed{lo op system, obtained with a load torque that is a step of amplitude 5 Nm applied at time = 5 sec. 4. CLOSED{LOOP SYSTEM BEHA VIOR VERIFICA TION In this section, w need to sho wthat the h ybrid feedbac ks describ ed in sections 3.2.2, 3.2.1 and applied to the full h ybrid system mo del describ ed in section 2, ac hiev ethe task of main taining the engine sp eed within the sp eci ed range, for an action of the load disturbance, pro vided that the initial condition of the ybrid system b elongs to sp eci ed set. In other ords, need to sho that giv en set of initial

conditions is robust in arian t set for the closed lo op system obtained y applying the prop osed idle sp eed con trol to the ybrid engine mo del. This result guaran tees that, if the ybrid system state is steered inside this set, then the prop osed idle feedbac con trol can b e activ ated and the idle sp eed sp eci cation will b e met under an y load disturbance. Note that the set is not a maximal in arian t set, but a subset. In Balluc hi et al. (2000a), the maximal robust in arian t set for an engine at idle sp eed w as com- puted analytically for simpli ed engine ybrid mo del, where the

nonlinear expressions w ere lin- earized and no actuation dela ys w ere considered. The main ob jectiv there as to establish the best p erformances ac hiev able giv en engine without adding an y detail and/or constrain ts on the idle sp eed con troller. The appro ximated linear expressions w ere used to mak e the analytical com- putation feasible. In that case, the largest set of initial conditions for whic h at least an idle sp eed con troller exists w as deriv ed. In this pap er, added details and constrain ts on the con troller, sp ecifying actuator dela ys and dynamics in the ybrid mo del

of the engine. Hence, the v eri cation task is more complex and cannot be carried out analytically need to sho wthat, from a giv en set of initial conditions, the con troller syn thesized in the previous section ac hiev es the idle sp eed sp eci cation. Due to the added details on actuations haracteristics and due to the c hoice of a particular con troller, this set of initial conditions will b e necessarily con tained in the maximal robust in arian t set for idle sp eed con trol computed for the nonlinear mo del. The eri cation is carried out using existing au- tomatic to ols (Matlab and Chec

kmate). Ho w- ev er, in tro ducing the en tire closed{lo op system in the automatic to ols is prohibitiv ely complex. Hence, w e decided to apply an assume{guaran tee paradigm to erify the correctness of the con- troller. The eri cation is based on the same decomp osition of the system used in the design of the feedbac con trols. Ev en with the assume{ guaran tee approac hw eha e not b een able to com- plete the formal v eri cation pro cess. W e rep ort in this section the basic ideas and the partial results eha e b een able to obtain. As in the case of the con trol syn thesis pro cess, w

consider rst the p o ertrain closed{lo op system, then the cylinders closed{lo op system, and nally the in tak e manifold closed{lo op system.
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Applying the assume{guaran tee paradigm, the correctness of the b eha vior of the sp eed con trol outer lo op is v eri ed assuming that the tor que c on- tr ol midd le lo op wil l pr duc the queste values of tor que with a single phase delay First, w ev erify that, if, at eac h dead cen ter, the con guration ( n;T ) is inside the set represen ted in gure 6, then the con tin uous ev olution of will satisfy the system sp eci cation ;n ],

during the curren t strok e and under an y action of the torque disturbance .Theset is obtained ybac kw ard in tegration of equations (5)-(6) o er a strok efor [0 5] Nm . Figure 6 sho ws the set 10 12 14 16 18 20 750 760 770 780 790 800 810 820 830 840 850 Fig. 6. The set Hence, the sp eed con troller pro duces correct beha vior if, for an ev olution of the closed{lo op system, at eac h dead cen ter the engine torque and the engine sp eed b elong to the set In order to erify this, assume that the torque con trol middle lo op will pro duce the requested alues of torque with a single phase dela

. Under this assumption, if the initial con guration of the PI con troller is c hosen suc h that the con gura- tion ( ;T ;z ) of the closed{lo op system +1 1) ( )(11) +1 (12) +1 (13) b elongs to the maximal in arian set con- tained in then the correct beha vior of The index refers to the -th dead cen ter. the sp eed con troller is eri ed. In (11), de- notes the dead-cen ter time that is mo deled as b ounded unkno wn disturbance in the range [1 +) )]. Then, to complete the eri cation of the sp eed con troller w eha e to com- pute a robust in arian t set for dynamics (11),(12) and (13), con

tained in .W e are curren tly in- estigan ting the use of p olyhedral appro ximations of the reac h sets to carry out this computation. Notice that, since the assumed beha vior for the torque closed{lo op sub-system is that of pro duc- ing the desired torque with a single phase dela the sp eci cation to b e v eri ed for the middle sub- system do es not dep end on the ev olution of the outer con trol lo op. This allo ws us to formally erify that if the in tak manifold con trol inner lo op will exhibit an input-output beha vior that is con tained in the stream depicted in gure 7, in terms of

requested mass 1) and loaded mass during the in tak strok es ), then the torque con trol middle lo op is able to pro duce the requested b eha vior b et een the desired torque and the engine torque Finally in the last step of the assume-guaran tee approac h, w e will consider the v eri cation of the in tak manifold con trol inner lo op to sho that the beha vior of the throttle alv con trol is cor- rect, in the sense that it pro vides the assumed beha vior sp eci ed in gure 7. An ob jectiv e of this eri cation is to sho w that the discrete represen ta- tion of the pressure dynamics, used in the

design of the throttle alv con trol, mo dels su∆cien tly ell the con tin uous ev olution of the manifold pres- sure. A second issue is v erifying that the in terac- tion b et een the t o lo ops with di eren t triggers (the throttle con trol running at xed frequency = and torque con trol sync hronized with the engine dead cen ters) pro duces a correct b eha vior. will erify the correct b eha vior of the in tak manifold inner lo op using existing automatic to ols (Matlab and Chec kmate). 5. CONCLUSIONS The idle sp eed con trol problem has b een formal- ized as safet sp eci cation for ybrid mo

del of the engine and the po er-train. The ybrid mo del describ es the in tak e manifold and cylinder lling dynamics, the torque generation pro cess and the p o er{train, as w ell as throttle and spark ignition actuators. An idle sp eed con troller has b een designed exploiting the decomp osition of the system in three comp onen ts: the engine sp eed con troller, the torque con troller and the in tak manifold con troller. or the design of eac con- troller, simpli ed mo del of the plan is used. Eac con troller is syn thesized assuming that the
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100 150 200 250 300 350 400 450

500 100 200 300 400 500 600 Fig. 7. Assumed b eha vior for the in tak e manifold closed{lo op subsystem. other sub-systems can be con trolled to giv the appropriate input-output b eha vior. The con trol la so obtained is then applied to the ybrid system mo deled in its en tiret and sim ulation results are obtained. Since there is no guaran tee that the b eha vior of the closed{lo op system will satisfy the sp eci cations, apply formal v eri cation tec hniques to demonstrate that the con trol la w do es indeed meet the constrain ts. Since formal eri cation is ery complex task, applied an

assume{guaran tee approac that views the system as comp osed of the three sub- systems for whic the con trol la as deriv ed. The assume{guaran tee principle allo ws us to ob- tain set of formal eri cation problems that are solv able within the domain of presen t formal eri cation to ols. References M. Abate and V. Di Nunzio. Idle sp eed con trol using optimal regulation. ec hnical Rep ort 905008, SAE, 1990. A. Balluc hi, L. Ben en uti, M. D. Di Benedetto, G. M. Miconi, U. ozzi, T. Villa, H. ong- oi, and A. L. Sangio anni-Vincen telli. Max- imal safe set computation for idle sp eed con- trol of

an automotiv e engine. In Nancy Lync and Bruce H. Krogh, editors, Hybrid Systems: Computation and Contr ol ,v olume 1790 of c- tur Notes in Computer Scienc pages 32{44. Springer-V erlag, New Y ork, U.S.A., 2000a. A. Balluc hi, L. Ben en uti, M. D. Di Benedetto, C. Pinello, and A. L. Sangio anni-Vincen telli. Automotiv e engine con trol and h ybrid systems: Challenges and opp ortunities. Pr dings of the IEEE 88, "Sp ecial Issue on Hybrid Sys- tems" (in vited pap er)(7):888{912, July 2000b. A. Balluc hi, M. D. Di Benedetto, C. Pinello, C. Rossi, and A. L. Sangio anni-Vincen telli. Hybrid con

trol in automotiv e applications: the cut-o con trol. utomatic , 35, (in vited pap er) Sp ecial Issue on Hybrid Systems(3):519{535, Marc h1999. K. R. Butts, N. Siv ashank ar, and J. Sun. Appli- cation of optimal con trol to the engine idle sp eed con trol problem. IEEE T ans. on Contr ol Systems T chnolo gy , 7(2):258{270, Marc h 1999. C. Carnev ale and A. Mosc hetti. Idle sp eed con trol with tec hnique. ec hnical Rep ort 930770, SAE, 1993. A. Ch utinan and B.H. Krogh. Computing p olyhe- dral appro ximations to dynamic o w pip es. In Hybrid Systems V , Lecture Notes in Computer Science.

Springer-V erlag, New ork, U.S.A., 1998. A. Ch utinan and B.H. Krogh. Appro ximate quo- tien t transition systems for h ybrid systems. In Pr c. of the 2000 A meric an Contr ol Confer enc Chicago, IL, June 2000. D. Hro at and B. Bo denheimer. Robust auto- motiv idle sp eed con trol design based on syn thesis. In Pr c. IEEE meric an Contr ol Confer enc , pages 1778{1783, S. F rancisco, CA, 1993. D. Hro at and J. Sun. Mo dels and con trol metho dologies for IC engine idle sp eed con trol design. Contr ol Engine ering Pr actic , 5(8), Au- gust 1997. L. Kjergaard, S. Nielsen, T. esterholm, and E.

Hendric ks. Adv anced nonlinear engine idle sp eed con trol systems. ec hnical Rep ort 940974, SAE, 1994. C. H. Onder and H. Geering. Mo del-based ultiv ariable sp eed and air-to-fuel ratio con trol of a SI engine. ec hnical Rep ort 930859, SAE, 1993. D. Shim, J. P ark, P .P . Khargonek ar, and W. B. Ribb ens. Reducing automotiv engine sp eed uctuation at idle. IEEE ans. on Contr ol Systems T chnolo gy , 4(4):404{410, July 1996. Stephen Y urk vic h and Melinda Simpson. Crank- angle domain mo deling and con trol for idle sp eed. SAE Journal of Engines 106(970027): 34{41, 1997.