Switching Control of Air-Fuel Ratio in Spark Ignition Engines

Transcription

1 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 3-July, FrB.4 Switching Control of Air-Fuel Ratio in Sar Ignition Engines Denis V. Efimov, Member, IEEE, Hosein Javaherian, Vladimir O. Niiforov Abstract he roblem of air-fuel ratio (AFR) stabilization in sar ignition engines is addressed in this aer. he roosed strategy consists of roer switching among two control laws to imrove the quality of the closed-loo system. he first control law is based on the a riori off-line identified engine model and ensures robust and reliable stabilization of the system at large, while the second control law is adative, rovides on-line adative adustment to the current fluctuations and imroves the accuracy of the closed-loo system. he suervisor realizes the switching rule between these control laws roviding better erformance for AFR regulation. Results of alication are reorted and discussed. V I. INRODUCION EHICLE tailie emissions requirement is one of the main restrictions for engine develoment and certification. hree-way catalytic converters (WC) installation in exhaust manifold aims at oxidizing HC and CO and reducing NOx secies. Usually WC ea efficiency is guaranteed if air-fuel ratio (AFR) is close to the stoichiometric value and the conversion efficiencies of WC are significantly reduced away from the stoichiometric value. herefore, the rimary obective of the AFR control system is to maintain the fuel inection in stoichiometric roortion to the ingested air flow (excetion to this occurs in heavy load situations where a rich mixture is required to avoid remature detonation or for more ower). Variations in the air flow affected by the driver serve as the disturbance to the system. Due to its imortance, the roblem of AFR regulation has attracted significant attention during the last few decades [3]. Adative control theory [], [6], [6], [7], robust control aroaches [], fuzzy control systems theory [7], neural networ techniques [], [8] are successfully tested in this articular alication. However, the comlexity of the roblem and growing demands on AFR regulation quality require new solutions, which can combine reliability and erformance of robust control aroaches and the accuracy and he research was suorted by the General Motors Cororation. D.V. Efimov is with Control of Comlex Systems Laboratory, Institute of Problems of Mechanical Engineering, Saint Petersburg, Russia (efde@mail.ru). H. Javaherian. is with Powertrain Systems Research Laboratory, GM Research and Develoment Center, Warren, Michigan, USA (hossein.avaherian@gm.com). V.O. Niiforov is with Deartment of Information and Control Systems, Saint Petersburg State Institute of Information echnology, Mechanics and Otics, Russia (niiforov@mail.ifmo.ru). insensitivity to dynamics changes of adatation methods. Switching control theory gives a solution to this roblem. here exist many good reasons and ractical motivations to use a set of controllers to regulate a single lant as oosed to one controller [], [], [5]. In such a case the natural question arises as how to trade off the advantages and disadvantages of each subsystem for modeling and control. he theory of switched systems addresses this issue regarding the roer switching laws between controllers. Alication of a suervisory (switched) control algorithm may seriously imrove erformance of the system regulation [4]. In this wor we solve the roblem of AFR regulation considering switching between two control laws. he first one is based on robust model-based control algorithm, which ensures stability for all ranges of the system arameters and inuts, but may have shortcomings in satisfying the required accuracy. he second control law is adative and is directed at imroving the quality of transient resonse for the cases of dynamic fluctuation around the reference model (used in the first control). Suervisor erforms activation of the adative control when unsatisfactory quality of the reference model is detected and, hence, imrovement of the robust control is needed. In section detailed roblem statement and some reliminaries are resented. Section 3 contains a descrition of the control algorithms. Suervisor equations are introduced in section 4. Results of the system imlementation are reorted in section 5. II. PROBLEM SAEMEN It is a well-nown fact that an automotive engine is a highly nonlinear multi-variable system and derivation of its recise model is a comlex rocess. his is one reason for the simlified models of engines being very oular in ractice. hese models can tae into account the main features of engine rocesses in the resence of time delays and nonlinearities, which are imortant for controller design or fault detection alications. In this wor nonlinear autoregressive (NARX) model is chosen for AFR dynamics descrition (in this context AFR refers to the non-dimensional engine-out air-fuel ratio sometimes nown as λ): //$6. AACC 5868

2 = i = y( m) = a y( m i) + [ b f( m )] u( m ) + + r d( m ) + v( m), where y R is the regulated outut (in our case we tae fuel-to-air ratio as y ), u [ umin, umax ] is the control inut (fuel ulsewidth in this wor, < umin < umax <+ are n q actuator constraints), d R and f R are the vectors of inuts (may contain hysical engine variables roducts), and are the model olynomials degrees, m is the number of current event (discrete time); v R is a disturbance acting on the system; a = [ a... a ] R, q ( ) B = [ b... b ] R + n ( ) and R = [ r... r ] R + are the model () constant arameters. he advantage of NARX models consists in availability of various methods for their aroximation and simlicity of controls design. It is assumed that a dataset is given, that a riori has collected measured information on y, u and other state variables involved in the vectors d, f for various regimes of engine oeration. hen, alying standard aroaches [4] it is ossible to obtain off-line the vectors of coefficients a, B and R such that the model () reresents dynamics of AFR loo in the engine with sufficient accuracy. he residual error can be assumed bounded and modeled as a art of the exogenous disturbance v. he coefficients a, B can be derived ensuring stability of the model () as well as stability of its inverse with resect to the control inut. Assumtion. Polynomials a and b, have all zeros with norms smaller than one. Under this assumtion, stabilizing controls for the system () can be designed alying simle inversion of its equations (inverse system is stable and, thus, the control algorithm will be realizable). Based on the given dataset, the n q comact sets D R and F R can be comuted which define admissible values for the vectors d and f resectively. he roblem is to design control u( i) [ umin, umax ], i ensuring ractical outut regulation to a given reference yd ( i ), i, i.e., the roerty y( i) yd ( i) Δ should be satisfied for all i and d D, f F for some rescribed Δ> roviding that y() y d () Δ. o this end, recall that a continuous function σ: R+ R+ belongs to class K if it is strictly increasing and σ ( ) = ; additionally it belongs to class K if it is also radially unbounded; and continuous function β : R+ R+ R+ is from class KL, if it is from class K for the first argument for any fixed second one, and it is () strictly decreasing to zero by the second argument for any fixed first one. III. CONROL ALGORIHM In this section descritions of robust model-based and adative controls are resented. A. Model-based control algorithm Under assumtion this algorithm is chosen as a simle inversion of the model () with resect to the control: u( m) = U( m) = b f( m ) yd( m) UPID( m ) ai y( m i) () = r d( m ) = [ b f( m )] u( m ), where due to the resence of the disturbance v (which reflects ossible unmodeled dynamics, measurement noise and aroximated model errors) it is required to use an internal feedbac in the form of the nonlinear PID: m UPID ( m) = e( m) + e( i) + (3) 3 + [ e( m) e( m )] + sign( e( m)) + e( m), where e= yd y is the regulation error,, =, 5 are control arameters, which have to be determined based on real or comuter exeriments. he control () ensures the model inversion and the following closed loo dynamics: y( m) = yd( m) UPID( m ) + v( m). Without U PID the control () forms the so-called feedforward art of the control, which does not contain any deviation errors (it deends on the current and ast values of the inuts and oututs of the engine dynamics and the aroximated coefficients of the model). he following is the condition of the control () alicability. Assumtion. For all f F it holds b f. Since the vector f is comosed of hysical engine variables or their nonlinear functions and roducts, which all have some sets of admissible values, then assumtion can be easily checed for f F and the coefficients b. For instance, f ( i ), i and elements of b can be all ositive (that may be guaranteed by roer aroximation of ()). he control () can not be realized in ractice since there exist constraints on admissible control amlitudes, i.e. it should be within the following bounds: u min u u max. he imlementation of a simle saturation umin if x < umin ; u( m) = sat[ U( m) ], sat( x ) = umax if x > umax ; (4) x otherwise, for stable lants rovides a solution to the roblem. Define the control actuator error as follows 5869

3 δ ( m) = b f ( m)[ u( m) U( m)], then the closed loo dynamics of (), (4) taes the form: y( m) = y ( m) U ( m ) + v ( m), (5) d PID v ( m) = v( m) +δ( m). Proosition. Under assumtions and there exist constants, =, 5 such that for any solutions of the system (), (4) with d D, f F for all i : ei ( ) β ( e(), i) + γ ( v ), [, i] i [, i ] v = su v ( ), β KL, γ K. All roofs are omitted due to sace limitations. Additionally adusting values of the coefficients =, 5 one can imrove the erformance of the control (4). For examle, coefficient rovides insensitivity to static errors (integral art of PID), coefficient 4 cancels disturbances with amlitudes less than 4, and coefficient 5 may hel on large deviations of the error. he estimate obtained in roosition is close to inutto-state stability roerty introduced in the aer [8]. B. Adative control algorithm Desite the fact that the control (4) has feedbacs aimed at disturbances, aroximation error and measurement noise attenuation, in some cases an additional adatation of the control is further needed. An imortant issue is that the model () has been aroximated on a large a riori collected dataset, and the coefficients a, B and R suit well for all d D, f F, but for some oerating conditions, which were not well resented in the dataset, there exists another set of coefficients a, B and R which reresents dynamics of AFR more accurately. In fact, for almost all modes of engine oeration there exist such coefficients locally woring better than global ones a, B and R. hus, the roblem of the coefficients a, B and R identification with osterior udate of the control can be osed. o solve the roblem it is roosed to use the conventional identification algorithm [5] denoting y( m) =ω( m) θ, (6) where θ= [ a b... b r... r ] and ω ( m) = [ y( m )... y( m ) f( m) u( m)... f( m ) u( m ) d( m)... d( m ) ] is the vector of regressors. hen we obtain the following arameterization for the identification error ε ( m) = y( m) y( m) = ω( m)[ θ θ ( m)], where θ= [ a b... b r... r ] is the adustable vector of estimates for θ and y( m) is the outut of the adative observer: y( m) = a ( m) y( m i) + r ( m) d( m ) + + [ b ( m) f( m )] u( m ). = i = Note that the model (6) has a form similar to (), however, for the coefficients a, B and R it is assumed that vi ( ) =, i (the coefficients locally aroximate the system dynamics exactly). he observer (7) also has the form () under θ substitution instead of θ. o design the adatation algorithm for θ let us choose the conventional quadratic error functional, Qε ( m) =.5 ε ( m), whose minimization is ensured by the following gradient adatation algorithm θ ( m) =θ( m ) +γ( m) ω( m ) ε( m ), (8) where γ ( m ) > is a design arameter. o secify conditions of the algorithm (8) alicability we imose the following restrictions on the engine dynamics. Assumtion 3. For all d D, f F it holds: there exist series a, B and (7) R and Δ, such that the model (6) is valid with a, B and R for all i + for all with + = +Δ, = ; ω( i ), i ; for any i there exist K and < ρ < such that i+ K = i P P ρ I, P = I ω( ) ω( ) ω( ), where I is the identity matrix of corresonding dimension. his assumtion has three arts. First, it is assumed that the time range of the system oeration can be decomosed on subintervals i +,, where the model (6) is valid for some a, B and R. Secondly, it is assumed that the regressor ω norm differs from zero (i.e. yi ( ) for all i ). he last art is a variant of ersistency of excitation condition required for the convergence of the adusted arameters to θ = [ a b,... b, r,... r, ]. Proosition. Let assumtion 3 hold, then observer (7) with adatation algorithm (8) for γ ( m) = ω( m ) has the following estimate on the arameters identification error θ ( m) = θ( m) θ convergence θ ( i) θ ( ) ex[ ln( ρ) ( i mod K ) ], i +,. Under conditions of roosition the arametric error θ ( m ) asymtotically converges to zero, then taing control 587

4 um ( ) = satum ( ( )), Um ( ) = b f( m) yd( m) UPID( m ) ai y( m i) = r d( m ) = [ b f( m )] u( m ) () it is ossible to ensure the model (6) stabilization, where UPID ( m ) is defined by (3). Proosition 3. Under assumtion 3 there exist constants, =, 5 such that for any solutions of the system (), (7), (8), () with d D, f F for all i +, ( β KL, γ K ): ei ( ) β( e( ), i ) +γ( θ ( ) ). he advantage of the control () is that ( ) i according to roosition, therefore, if the adative algorithm is active for sufficiently long time (constants Δ from assumtion 3 are large enough), then the adative control (7), (8) and () ensure exact regulation of AFR dynamics at the stoichiometric value. hus, in this section two control algorithms are roosed which ensure inut-to-state stabilization of AFR dynamics. Both controls are based on the measurement information (the first one designed for the aroximated off-line model, the second one for the on-line model). he issue of the imrovement of the closed-loo system quality with the use of secial switching between these control laws is discussed in the following section. IV. HE SUPERVISOR Either of the roosed algorithms from section 3 ossess its own advantages. he control algorithm designed off-line is rather reliable (it ensures stability for all oerating modes of the engine) and robust (it is not sensitive to disturbances and unmodeled dynamics), but it may fail to ensure good accuracy over the entire range of oerating conditions. he adative control has some transients after which it is tuned to comensate for all the errors at a articular engine oerating condition. he switching algorithm executed in the suervisor has to combine the advantages of these controls neglecting their shortcomings and rovides the closed loo control with an imroved erformance. For this urose, note that the main difference between these controls consists in the models on which they are based. he following erformance functionals evaluate the models accuracy on the last L stes: m m ( ) = q= m L ( ), q= m L ( ) = ( ) i ( ) [ b f( q )] u( q ) r d( q ), = = J m L e q e q y q a y q i J ( m ) = L e ( q ), () e ( q) y( q) a ( m) y( q i) = i = = [ b ( m) f( q )] u( q ) r ( m) d( q ). In this case activation of the control with the most accurate model loos reasonable, that is the idea of suervision algorithm in this wor, but the switching among nonlinear systems is not so trivial. Even if the systems are asymtotically stable or inut-to-state stable as in our case, an inaroriate switching strategy may lead to instability []. he roblem of switching among inut-to-state systems has been addressed in the revious wors [4], [9], []. he main idea there consists in the dwell-time mechanism alication. Dwell-time constant restricts the rate of switching between the controls and for a sufficiently slow rate the stability of the closed loo system is guaranteed. For the rest of the section let u ( m ) be defined by () and u ( m ) be given as in (). heorem. Let assumtions 3 hold and there exist dwell-time constant τ D > such that βi( r, τd ) λ r, r for some <λ<. If sw+ sw τ D, w, where s w is the instant of w th switch, then in the system () with control u( m) = uz( s w )( m), z( sw ) {,} for any solutions the following estimate is satisfied: ei ( ) β ( e(), i) + γ ( l ), β KL, γ K, [, i ] vi ( ) if i sw = ; l( i) = θ ( sw) if i sw =. In the case of theorem, switching of the controls results in changing of the disturbance. he dwell-time switching algorithm ensures boundedness of the system traectories and the theorem resents worst-case estimate on the closed loo error behavior. Dwell-time switching algorithm still leaves room to further suervisor rule design focusing on the imrovement of transient AFR behavior. Keeing in mind the erformance functionals () and dwell-time rule from theorem, the following suervision algorithm is roosed: u( m) = u ( m), z( s ) {,}, w, () z( s w ) w arg inf m s { ( ) ( )} if ; w+τ J m < J m s D w = sw+ = arg inf m s { J( m) J( m)} if s, w+τ < D w = z( sw+ ) = 3 z( sw), z( s ) =, s =, where s w, w determines the time instant of the last switch; τ D > is the dwell-time that revents chattering (high frequency switching between the control algorithms), and from theorem ensures the closed-loo stability. Since s = the system starts with model-based control (), then after dwell-time if accuracy of the adatation model (7) imroves ( J( m) < J( m) ) the adative control () has to be 587

5 activated. If after dwell-time the accuracy of the model () again becomes better ( J ( m) < J ( m) ) the control () would be switched on. V. APPLICAION RESULS he roosed switching control has been tested in two vehicles with V8 engines: one with 5.7l engine and another with a 5.3l engine. he schedule of testing was as follows. At the first ste based on the database of revious exeriments the model () was derived for both vehicles (the coefficients a, B and R satisfy assumtion ). he results of model () tests are shown in Fig..a,.c and Fig..b,.d for 5.7l and 5.3l engines, resectively (the figures.c and.d demonstrate zoomed lots of the figures.a and.b). As we can deduce from these lots the quality of both models are of sufficient accuracy for the control () design. a. b. c.. AFR.9 x 4 4 FPW x 4 J J a c..4. y J x 4 AFR b. AFR x x 4 d ŷ y x x 4 Fig.. AFR model accuracy verification ŷ Fig.. raectories for the vehicle with a 5.7l engine a.. AFR.9 x 4 6 b. 5 FPW 4 3 At the second ste the controls () and () are calculated. Assumtion is verified for the control () on the a riori collected dataset. Coefficients, =, 5 are assigned as zero initially, and after some exerimentations they are tuned to some values roviding accetable erformance. For the control () the same values of coefficients, =, 5 are chosen. he assumtion 3 is taen to be valid (the first art and the last two arts can be verified on the dataset). After that, the system is ready for exerimental testing. c. x 4 x -4 J J J x 4 Fig. 3. raectories for the vehicle with a 5.3l engine he results of the tests are shown in figures and 3 for the vehicles with the 5.7l and 5.3l engines, resectively. As we can conclude from these results the adative model rovides better quality of aroximation of AFR dynamics very frequently, but changes in modes of engines oeration results in bacward activation of the model () based controls. For both vehicles, exeriments confirm alicability of the 587

6 roosed aroach. VI. CONCLUSION Switching control algorithm for air-fuel ratio regulation is develoed and ractically tested for two vehicles. he controller contains three arts: robust model-based control, adative control and the suervisor. he first control is designed for aroximated off-line model using a riori available exerimental dataset, the adative control is based on the adusted (i.e. tuned) real-time model. Both models and controls have similar structure and the only difference is the tye of information used for their design (off-line or on-line measurements). he suervisor rovides switching between these controls taing into account the current accuracy of the models. If off-line aroximated model has better quality, then the robust control is active. In situations when the adatively adusted model has better accuracy of AFR dynamics reresentation during some number of revious events, the adative control is switched on. Such an scheme allows the controller designer to combine the reliability of robust control (which was intensively tested and it ensures admissible quality of AFR regulation for all oerating regimes) and the flexibility of the adative control (which can imrove the erformance due to the higher local accuracy of the AFR dynamics aroximation). Practical imlementation and intensive tests has demonstrated the alicability of the aroach. [9] Hesanha J.P., Liberzon D., Morse A.S. (). Suervision of integral-inut-to-state stabilizing controllers. Automatica, 38(8), [] Hesanha J.P., Morse A.S. (999). Certainty equivalence imlies detectability. Syst. Controls Lett., 36,. 3. [] Huang., Liu D., Javaherian H. and Sin N. (8). Neural sliding mode control of engine torque, Proc. 8 IFAC riennial World Congress, Seoul, South Korea. [] Liberzon D. (3). Switching in Systems and Control. Systems and Control: Foundations and Alications. Boston, MA: Birhauser. [3] Liu D., Javaherian H., Kovaleno O. and Huang. (8). Adative critic learning techniques for engine torque and air-fuel ratio control, IEEE ransactions on Systems, Man and Cybernetics, Part B: Cybernetics, 38(4), [4] Lung L. (999). System Identification: heory for the User (second ed.). Prentice-Hall, Englewood Cliffs, NJ. [5] Morse A.S. (995). Control using logic-based switching. In: rends in control (A. Isidori (Ed.)), Sringer-Verlag, [6] Powell J.D., Feete N.P., Chang C.-F., (998). Observer-based air fuel ratio control. IEEE Control Systems Magazine, 8(5), [7] urin R.C., Geering H.P. (995). Model-reference adative A/F-ratio control in an SI engine based on Kalman-filtering techniques. Proc. American Control Conference, [8] Zhai Y., Yu D. (7). RBF based feedforward feedbac control for air-fuel ratio of SI engines. Proc. IFAC Worsho on Advanced Fuzzy and Neural Control. ACKNOWLEDGMEN We would lie to than Dr. Man-Feng Chang of GM R&D for his continued suort of the roect and many useful discussions during the course of this study. REFERENCES [] Ault B.A., Jones V.K., Powell J.D., Franlin G.F. (993). Adative air-fuel ratio control of a sar ignition engine. SAE aer No [] Brandstetter M. (996). Robust Air-Fuel Ratio Control For Combustion Engines. Cambridge, UK, Ph.D. thesis. [3] Coo J.A., Kolmanovsy I.V., McNamara D., Nelson E.C., Prasad K.V. (7). Control, comuting and communications: echnologies for the wenty-first century Model. Proc. of the IEEE, 95, [4] Efimov D., Loria A., Panteley E. (9). Multigoal outut regulation via suervisory control: alication to stabilization of a unicycle. Proc. American Control Conference 9. [5] Fradov A.L., Miroshni I.V., Niiforov V.O. Nonlinear and Adative Control of Comlex Systems. Dordrecht, Boston, London: Kluwer Academic Publishers, 999. [6] Franceschi E.M., Muse K.R., Peyton-Jones J.C., Mai I.H. (7). An Adative Delay-Comensated PID Air Fuel Ratio Controller. Proc. SAE-7--34, World Congress & Exhibition, Detroit. [7] Ghaffari A., Shamehi A.H., Sai A., Kamrani E. (8). Adative Fuzzy Control for Air-Fuel Ratio of Automobile Sar Ignition Engine. Proc. World Acad. Science, Engineering, echnology,, [8] Jiang Z-P., E.D. Sontag and Wang Y. (999). Inut-to-state stability for discrete-time nonlinear systems. Proc. 4th IFAC World Congress (Beiing), Vol. E,