Sliding Mode Control of a Dual-Fuel System Internal Combustion Engine

Transcription

1 Proceeding of the ASME 9 Dynamic Sytem and Control Conference DSCC9 October -4, 9, Hollywood, California, USA DSCC9-59 Control of a Dual-Fuel Sytem Internal Combution Engine Stephen Pace Department of Electrical and Computer Engineering Michigan State Univerity Eat Laning, MI 4884 ABSRAC Air-to-fuel (A/F) ratio i the ma ratio of air-to-fuel mixture trapped inide a cylinder before combution begin, and it affect engine emiion, fuel economy, and other performance Uing an A/F ratio and dual-fuel ratio control oriented engine model, a multi-input-multi-output (MIMO) liding mode control cheme i ued to imultaneouly control the ma flow rate of both port fuel injection () and direct injection () ytem he control target i to regulate the A/F ratio at a deired level (eg, at toichiometric) and fuel ratio (ratio of fueling v fueling) to a given deired level between zero and one A MIMO liding mode controller wa deigned with guaranteed tability to drive the ytem A/F and fuel ratio to the deired target under variou air flow diturbance he performance of the liding mode controller wa compared with a baeline multi-loop (Proportional-Integral-Derivative) controller through imulation and howed improvement over the baeline controller INRODUCION Increaing concern about global climate change and everincreaing demand on foil fuel capacity call for reduced emiion and improved fuel economy Vehicle equipped with fuel ytem have been introduced to market globally In order to improve engine full load performance at high peed, oyota introduced an engine with a toichiometric injection ytem with two fuel injector for each cylinder, ee [] One i a injector generating a dual-fan-haped pray with wide diperion, while the other i a port injector he dual-fuel ytem introduce one additional degree of freedom for engine combution optimization to reduce emiion with improved fuel economy Uing gaoline and ethanol dual-fuel ytem to ubtantially increae gaoline engine efficiency i decribed in [] he main idea i to ue a highly booted mall turbocharged Guoming G Zhu Department of Mechanical Engineering Michigan State Univerity Eat Laning, MI 4884 engine to match the performance of a much larger engine Direct injection of ethanol i ued to uppre engine knock at high engine load due to it ubtantial air charge cooling reulting from it high heat of vaporization he control of A/F ratio i an increaingly important control problem due to federal and tate emiion regulation Operating the park ignited internal combution engine at a deired A/F ratio i due to the fact that the highet converion efficiency of a three-way catalyt occur around toichiometric A/F ratio here have been everal fuel control trategie developed for internal combution engine to improve the efficiency and to reduce exhaut emiion A key development in the evolution wa the introduction of a cloed-loop fuel injection control algorithm [], followed by a linear quadratic control method [4], and an optimal control and Kalman filtering deign [5] Specific application of A/F ratio control baed on oberver meaurement in the intake manifold were developed in 99 [6] Another approach wa baed on meaurement of exhaut gae by an oxygen enor and the throttle poition [7] Hedrick alo developed a nonlinear liding mode control of A/F ratio baed upon the oxygen enor feedback [8] he conventional A/F ratio control for automobile ue both cloed-loop feedback and feed forward control to achieve good teady tate and tranient repone he control trategie, mentioned above, were mainly for ytem Mitubihi began to invetigate combution control technologie for direct injection engine in 996 [9] Due to the introduction of internal combution engine with dual-fuel ytem ( and ), control of both A/F ratio and fuel ratio (ratio of fueling v fueling) became part of a combution optimization problem in [] In thi paper, a control oriented nonlinear model of a dualfuel ytem i developed for control development and evaluation Copyright 9 by ASME Downloaded From: on 5/6/8 erm of Ue:

2 purpoe, and an MIMO liding model controller i developed to regulate the engine A/F and fuel ratio to target level he control trategy i validated in imulation baed upon the developed dual-fuel ytem model hi paper i organized a follow It begin by preenting a implified A/F and fuel ratio control oriented model, followed by a ection that decribe an innovative liding mode controller that regulate both A/F and fuel ratio to deired nonzero target Next, imulation reult are preented, and finally concluion are dicued AN A/F RAIO AND FUEL RAIO MODEL he control problem of thi work i to vary both and fuel ma injection rate ( and ) o that the engine A/F ratio i regulated at a deired level (eg, toichiometric) and the fuel ratio of effective fueling, m &, to fueling, _ E m & E, i maintained at a deired value a hown _ in Figure Note that the effective fueling for i equal to the injected fuel A/F Ratio toich - F/R deired - m & A/F Ratio _ E F/R Controller air A/F Ratio F/R Engine Model FIGURE : AGRAM OF A/F AND FUEL RAIO CONROL PROBLEM A nonlinear model for thi problem, uing implified engine dynamic to model both the engine A/F and fuel ratio, i to be dicued below he air flow, air, i model a air ω ω ω, () where ω i the nominal air flow and ω i the air flow diturbance due to the engine operational condition change he fuel flow for wall-wetting dynamic from the port fuel injector i modeled by the following tranfer function, from [], _ E ( ) α, () ( ) β where α and β are elected to be 5 and 8, repectively, in thi model he fuel flow from the direct injector contain negligible dynamic Due to the three-way catalyt ued for emiion control, mot engine are deign to achieve a target A/F ratio around toichiometry For thi reearch, we ue a normalized target A/F ratio, λ target, which i defined a deired air-to-fuel ratio divided by toichiometric air-to-fuel ratio (46 for gaoline) Note that at toichiometry the normalized target A/F ratio i equal to one he normalized A/F ratio can be expreed a: λ air 46 () Now, the engine equivalence ratio φ i defined a the invere of relative A/F ratio λ and can be approximated uing equation () and () below φ 46 ω, (4) ω ω where equation () i approximated by a firt order aylor expanion For the ret of the paper, we conider only equivalence ratio control intead of A/F ratio he fuel ratio of the dual-fuel ytem i defined a the effective fueling divided by fueling herefore the fuel ratio, R fuel, i calculated a _ E R fuel (5) _ E Similarly, fuel ratio can be approximated by ubtituting equation (4) into (5), replacing the engine equivalence ratio with the target ratio herefore, ω R 46m fuel / φ _ E t arget ω ω & (6) Although fuel ratio i not directly meaureable, the injected fuel ma rate can be etimated by uing both the meaured equivalence ratio (via oxygen enor) φ m and meaured air flow (via ma air flow enor) ince m & _ et air _ m φ /46 m air _ m Hence, the fuel ratio feedback ignal can be obtained uing multiple enor ignal ( φ m and air _ m ), along with the etimated fueling ma rate _ et (a function of injection pule width), that i, _ et _ et R fuel _ et _ et he control oriented equivalence and fuel ratio model, operating at a fixed engine peed of 5RPM, include wall-wetting dynamic of the fuel ytem, average fuel injection delay (5m), exhaut manifold tranport and oxygen (A/F ratio) enor delay (4m), and air flow travel delay from engine throttle to cylinder (m) hee time delay are approximated by unitary gain firt order tranfer function he complete model i divided into three ubytem, a hown in Figure, where the oxygen enor dynamic are denoted a G, the air flow dynamic a G, and the fuel flow dynamic a G he tate pace equation of the three ubytem are hown in equation (7), (8), and (9) x& A x Bu (7) y Cx x& A x Bω (8) y Cx x& A x Bu (9) y C x D u C [ C C ], Copyright 9 by ASME Downloaded From: on 5/6/8 erm of Ue:

3 Air In In Out Out Fuel Flow Dynamic (G) In Out Air Flow Dynamic (G) arget Eq Ratio 4 6 Gain In Out Oxygen Senor Dynamic (G) FIGURE : EQUIVALENCE AND FUEL RAIO MODEL Equivalence Ratio he entire ytem can be expreed by the following tate-pace model A B x& A x CCxx B ω u () A B and the output equation for the equivalence and fuel ratio uing () can be approximated by y EqRatio 46 C C x x () y 46 C x ( C x 65u ) / φ FuelRatio t arg et () he value for the matrice of the engine model are A 5; B 46; C A ; B ; C A - ; B ; -5 C ; D 65 he nonlinear tate pace engine model mut be tranformed into the regular form [] below to apply liding mode control & η f a ( η, δ ( η, ; () z& f ( η, G( η, u, b (4) where the forcing term of equation (), δ(η,, contain the diturbance input ω and f a (η, contain the nonlinear portion of () A change of variable, η x z (5) tranfer the original tate vector, [ x x x x x ] x 5; (6) into the regular form coordinate, where the tranformation matrix i choen uch that η (7) [ x x x 65x ] [ x x (8) z ] hu, the original ytem (repreented in the regular form) i hown in (9) and () It can be een that the model contain two control input correponding to and fueling and one ma air flow diturbance input 5; - 5 η & η -5 η z h( η, - δ - η where f a ( η, -5 z u z& - z u f b ( η, G h ( η, 46η ( η 65z δ ( η, (9) () SLING MODE CONROL SRAEGY Exiting liding mode A/F ratio control application in [] and [4] utilized the binary nature of a HEGO (Heated Exhaut Ga Oxygen) enor to reduce it ocillation reulting from the time delay by uing a dynamic, one dimenional liding urface In thi paper, a two dimenional liding urface i elected to control both equivalence and fuel ratio of the dual-fuel ytem he liding urface i defined a z φ(η), () where the control objective i to regulate to zero by deigning a feedback φ(η) uch that it tabilize equation (9) Chooing z 6η z φ( η) () z -η -η reult in the aymptotic tability at the origin for (9) and decouple the nonlinear dynamic contained in f a (η, from the remaining equation he initial choice of () and conequently () wa to linearize the ytem through feedback and achieve acceptable tabilization of (9) Alo, note that the election of φ(η) doe not contain the tate η and remove it from having any affect on the ytem, which i ignificant ince the tate η contained the input diturbance, δ he election of the control u to cancel the known term of the differentiation of () and the control input v that guarantee aymptotic tability and force toward zero are deigned from [] he reulting control force the ytem tate onto the liding urface in finite time and eventually bring thee tate on the liding urface to zero o achieve the deired nonzero equivalence and fuel ratio, conider η η η ( η η) η η η η () η η z z z ( z z ) z z z z z (4) where and z go to zero, leading η and z converging to the target tate η and z a time goe to infinity ince the liding mode controller regulate the tate,, z, and z to the liding manifold Uing thee target tate, η and z can bring the equivalence ratio and fuel ratio to any deired value o Copyright 9 by ASME Downloaded From: on 5/6/8 erm of Ue:

4 invetigate the tability of the ytem with thee new target tate, φ(η) from () i ued and the final ytem become ( η 65z z ) & -5 - δ η ( η 65z z ) 5η 5η 5 η z (5) -5 z u 5 z z& (6) - z u z It can be een that the ytem i linear with contant matrice plu the forcing δ term For entire ytem tability analyi, the characteritic equation of the linear matrix A * i * det( λi A ) ( λ 5)( λ 5)( λ )( λ 5)( λ ) with all eigenvalue in the left half plane, where ( η 65z ) z -5 A* Note that the target tate η and z can be determined with the given input air diturbance and deired equivalence and fuel ratio he following output equation are the reult of coordinate tranformation of () and () (uing (5)) ~ y EqRatio 5η 5 η( η 65z z ) (7) ~ y 46 5 η (46875z 65u ) φ (8) FuelRatio / t arget Conider the tate equation (9) and () at teady tate by etting derivative term equal to zero, leading to the ytem teady-tate tate η and z expreed a a function of the deired output ratio and input diturbance he target tate η and z can be obtained with given target ratio and diturbance he zero target tate liding mode control trategy i modified uch that the cloed-loop ytem converge to the deired target tate, ee Figure air η z Figure : Schematic diagram of liding mode control trategy o improve the performance of the liding mode controller, the tabilizing function φ(η) can be generalized a follow 6 ε z φ( ), (9) - - where ε i a poitive contant Subtituting (9) into (9) yield, - 5 d & η - 5 η ε () ε where d 95 Equation () can be viewed a & η A( ε) g(, ε) () where, - 5 d A( ε ) - 5, ( η, ε ) g ε ε It can been een that A(ε) i Hurwitz if ε > and alo d ε d ε () Let Q I and define X(ε) > a the olution of the Lyapunov equation auming ε > A ( ε ) X ( ε ) X ( ε ) A( ε ) Q he maximum and minimum eigenvalue of X(ε) are plotted a function of ε in Figure ε > 5 5 λ min of X(ε) λ max of X(ε) ε > FIGURE 4: MAXIMUM AND MINIMUM EIGENVALUES OF X(ε) Define the Lyapunov function V() X(ε), leading to the following propertie: -λ λmin ( X ( ε )) V ( ) λmax ( X ( ε )), V A - Q -λmin ( Q), V X X λmax ( X ) he derivative of V() atifie V& ( ) ( A ( ε ) X ( ε ) X ( ε ) A( ε )) εd ( ε ) X min ( Q) λ herefore the origin i exponentially table if max ( X ( ε )) εd () 4 Copyright 9 by ASME Downloaded From: on 5/6/8 erm of Ue:

5 V & ( ) < or ε < λmax ( X ( ε )) d (4) he tability condition will hold auming η η η γ for all time, where γ i a poitive contant, and can be retated a: ε < λmax ( X ( ε ))d γ (5) Figure 4 how that λ max (X(ε)) for all ε >, thu ε < < (6) d γ γ 85 γ Equation (6) how that if γ i mall, then there exit an ε > with guaranteed tability SIMULAION RESULS A MIMO controller wa deigned for comparion purpoe to the nonlinear MIMO liding mode controller hi controller cacaded two controller uch that the firt i ued to control the equivalence ratio and the econd PI i ued to reduce the fuel ratio error, ee Figure 5 he controller gain for both and PI controller are hown in able Eq Ratio error Kd Ki Kp Kd Ki Kp Controller error Ki Kp PI Controller FIGURE 5: SCHEMAIC AGRAM OF CONROLLER x Simulation # he imulation ue a contant input diturbance ω of plu 5 percent noie, adding a tep input of 5 to the contant diturbance at the 6 th econd It alo ha a fuel ratio reduction from 6 to 4 at the 9 th econd, and how that the controller reject the diturbance quickly Figure 6 alo how the and fuel control input for both and liding mode controller It can be oberved that the liding mode controller provide quick fueling input of both and ytem, leading to better diturbance rejection and tranient repone over the repone able ummarize the overhoot and ettling time for both and liding mode controller ABLE : COMPARISON OF CONROLLERS FOR SIMULAION # % Overhoot (after 6 th econd) Settling ime (after 9 th econd) e (after th econd) Eq Ratio 59% 44% 6 ec < 5 ec 6 % 6% 9 ec ec Equivalence Ratio All controller gain parameter were tuned uch that the repone of the cloed-loop ytem wa table and had a fat a repone a poible Simulation of the liding mode controller under different air flow input diturbance were conducted and compared to the controller ABLE : AND PI CONROLLER PARAMEERS K p K p 5 K p 5 K i K i K i 5 K p K p 7 he gain matrix β in the liding mode controller for each imulation wa tuned uch that the tranient repone wa acceptable and it wa elected a 6 β 7 (7) A unitary target equivalence ratio and 6% (6) target fuel ratio were choen for each imulation Figure 6 how the cloed-loop repone of the controller and liding mode controller for Fueling Fueling FIGURE 6: CLOSED LOOP RESPONSE OF SIMULAION # 5 Copyright 9 by ASME Downloaded From: on 5/6/8 erm of Ue:

6 Equivalence Ratio Fueling ABLE : COMPARISON OF CONROLLERS FOR SIMULAION # % Overhoot (after 6 th econd) Settling ime (after 9 th econd) e (after th econd) Eq Ratio 99% 5% 45 ec 9 ec 8 7 9% 7% 5 ec < ec 4 5 Figure 8 how the cloed-loop repone of both and liding mode controller for Simulation # hi imulation ha an airflow diturbance of zero and ω i equal to unity, which contitute the wide open throttle (WO) cae he imulation decreae the target fuel ratio from 5 to zero (% fueling) Since the engine peed i fixed, WO implie high engine load and the imulation how that complete direct injection fueling can be achieved by the controller, and therefore can be ued to uppre engine knock at high engine load Equivalence Ratio Fueling FIGURE 7: CLOSED LOOP RESPONSE OF SIMULAION # Figure 7 how the cloed-loop repone of both and liding mode controller for Simulation # hi imulation decreae the target equivalence ratio from to 9 at the 6 th econd, and increae it back to unity at the 9 th econd Again, the contant input diturbance ω wa plu 5 percent noie Figure 7 alo diplay both fueling input of and liding mode controller Although the controller ha a fater equivalence ratio repone than the liding mode one, it i under the penalty of the fuel ratio control accuracy (huge pike during the tranition) On the other hand, the liding mode controller provide mooth tranition for both equivalence and fuel ratio herefore, the liding mode control repone are more favorable over the one able ummarize the overhoot and ettling time for both and liding mode controller for Simulation # It i worth to noting 9% fuel ratio overhoot for the controller while the liding mode controller only ha % Fueling Fueling FIGURE 8: CLOSED LOOP RESPONSE OF SIMULAION # 6 Copyright 9 by ASME Downloaded From: on 5/6/8 erm of Ue:

7 CONCLUSION A multi-input-multi-output liding mode controller wa developed baed upon the A/F and fuel ratio model to achieve nonzero deired equivalence and fuel ratio target he liding mode controller wa validated with the developed imple nonlinear dual-fuel ytem model and compared to the multi-loop controller he future work i to implement it into a real-time controller REFERENCES [] akuya Ikoma, Shizuo Abe, ukihiro Sonoda, Hiao Suzuki, Yuichi Suzuki, and Maatohi Baaki, Development of V-6 5-liter Engine Adopting New Direct Injection Sytem, SAE [] Lelie Bromberg, Daniel Cohn, and John Heywood, Optimized Fuel Management Sytem for Direct Injection Ethanol Enhancement of Gaoline Engine, US patent application 6/6 [] J G Rivard, "Cloed-loop Electronic Fuel Injection Control of the IC Engine," in Society of Automotive Engineer, 97 [4] J F Caidy, et al, "On the Deign of Electronic Automotive Engine Control uing Linear Quadratic Control heory," IEEE ran on Control Sytem, vol AC-5, October 98 [5] W E Power, "Application of Optimal Control and Kalman Filtering to Automotive Sytem," International Journal of Vehicle Deign, vol Application of Control heory in the Automotive Indutry, 98 [6] N F Benninger, et al, "Requirement and Performance of Engine Management Sytem under ranient Condition," in Society of Automotive Engineer, 99 [7] C H Onder, et al, "Model-Baed Multivariable Speed and Air-to- Control of an SI Engine," in Society of Automotive Engineer, 99 [8] S B Choi, et al, "An Oberver-baed Controller Deign Method for Automotive Fuel-Injection Sytem," in American Control Conference, 99, pp [9] Kume, et al, "Combution echnologie for Direct Injection SI Engine," in Society of Automotive Engineer, 996 [] G Zhu, et al, Combution Characteritic of a Single- Cylinder Engine Equipped with Gaoline and Ethanol Dual- Fuel Sytem, SAE [] L Guzzella, Introduction to Modeling and Control of Internal Combution Engine Sytem: Springer-Verlag Berlin Heidelberg, 4 [] H Khalil, Nonlinear Sytem, ed: Prentice Hall, [] M Won, et al, Air to Control of Spark Ignition Engine Uing Dynamic Control and Gauian Neural Network, in American Control Conference, 995, pp [4] J Souder, et al, Adaptive Control of Air-Fuel Ratio in Internal Combution Engine, International Journal of Robut and Nonlinear Control, vol 4, 4 7 Copyright 9 by ASME Downloaded From: on 5/6/8 erm of Ue: