Robust Nonlinear EGR and VGT Control with Integral Action for Diesel Engines

Transcription

1 Robut Nonlinear EGR an VGT Control with Integral Action for Dieel Engine Johan Wahltröm Lar Erikon Vehicular Sytem, Linköping Univerity, SWEDEN Abtract: A robut non-linear multivariable control eign with integral action i propoe an invetigate for control of EGR valve an VGT poition in heavy uty ieel engine. The main control goal i to regulate oxygen/fuel ratio an intake manifol EGR-fraction. Thee are choen a main performance variable ince they are trongly couple to the emiion. A recently evelope non-linear control eign bae on feeback linearization i extene with integral action. The nonlinear controller give an inner loop with goo tability an robutne propertie. It i hown that integral action i neceary to hanle moel error o that the controller can track the performance variable pecifie in the outer loop. In particular the control eign metho utilize a control Lyapunov function an invere optal control, which reult in a control law with robutne propertie interpretable a gain an phae margin. Furthermore, comparion by ulation alo how that the propoe control eign uccefully hanle non-linear effect. Keywor: Lyapunov function; invere optal control. 1. INTRODUCTION Legilate emiion lit for heavy uty truck are contantly reuce. To fulfill the requirement, technologie like Exhaut Ga Recirculation EGR ytem an Variable Geometry Turbocharger VGT have been introuce. The prary emiion reuction mechan utilize to control the emiion are that NO x can be reuce by increaing the intake manifol EGR-fraction x egr an moke can be reuce by increaing the air/fuel ratio Heywoo, Note that exhaut gae, preent in the intake, alo contain oxygen which make it more uitable to efine an ue the oxygen/fuel ratio λ O intea of the traitional air/fuel ratio. The main motive for thi i that it i the oxygen content that i crucial for moke generation. Beie λ O it i natural to ue EGR-fraction x egr a the other main performance variable, but one coul alo ue the burne ga fraction intea of the EGR-fraction. The oxygen/fuel ratio λ O an EGR fraction x egr epen in complicate way on the EGR an VGT actuation. It i therefore neceary to have coorinate control of the EGR an VGT to reach the legilate emiion lit in NO x an moke. Variou approache for coorinate control of the EGR an VGT for emiion abatement have been publihe. Guzzella an Amtutz 1998 preent a goo overview of ifferent control apect of ieel engine with EGR an VGT, an in Nieuwtat et al there i a comparion of ome control approache with ifferent election of performance variable. Other control approache are ecribe in Amtutz an Re 1995, Nieuwtat et al. 1998, Stefanopoulou et al. 2000, Amman et al. 2003, an Rückert et al A robut non-linear multivariable control eign i propoe in Jankovic et al Thi eign inclue contruction of a Lyapunov function an invere optal control which provie a control law that hanle interaction an non-linear propertie in the ytem. However, in thi eign compreor ma flow W c an exhaut manifol preure p em are choen a output an therefore the etpoint for λ O an x egr are tranforme to et-point for W c an p em. Thi tranformation i bae on a plifie thir-orer moel, yieling moel error an conequently inconitent et-point for p em, W c, λ O, an x egr an therefore control error, ee Fig. 3. In orer to ecreae thee control error, thi paper propoe a control eign that exten the one in Jankovic et al with integral action in Sec. 4. The a i alo to illutrate the avantage with integral action an non-linear control in Sec. 6. Further, Sec. 2 ecribe a mean value ieel engine moel that i ue in ulation for tuning an valiation of the evelope controller. The control eign metho ue in thi paper i ecribe in Sec. 3. Finally, for efficient calibration an automatic controller tuning metho i evelope in Sec DIESEL ENGINE MODEL A moel for a heavy uty ieel engine i ue in ulation for tuning an valiation of the evelope controller. Thi ieel engine moel i focue on the ga flow, ee Fig. 1, an it i a mean value moel with even tate: intake an exhaut manifol preure p an p em, oxygen ma fraction in the intake an exhaut manifol X O an X Oem, turbocharger pee ω t, an two tate ecribing the actuator ynamic for the two control ignal ũ egr an ũ vgt. Thee tate are collecte in a tate vector x x = [p p em X O X Oem ω t ũ egr ũ vgt ] T

2 W egr EGR cooler u δ p W ei X O Intake manifol Cyliner Intercooler u egr EGR valve W eo p em X Oem Exhaut manifol W c u vgt Compreor Turbine Fig. 1. Sketch of the ieel engine moel ue for ulation, control eign, an tuning. It ha five tate relate to the engine p, p em, X O, X Oem, an ω t an two for actuator ynamic ũ egr an ũ vgt. There are no tate equation for the manifol temperature, ince the preure an the turbocharger pee govern the mot portant ytem propertie, uch a nonminum phae behavior, overhoot, an ign reveral, while the temperature tate have only minor effect on thee ytem propertie. The reulting moel i expree in tate pace form a ẋ = fx,u,n e where the engine pee n e i coniere a an input to the moel, an u i the control input vector u = [u δ u egr u vgt ] T which contain ma of injecte fuel u δ, EGR-valve poition u egr, an VGT actuator poition u vgt. A etaile ecription an erivation of the moel together with a moel tuning an a valiation againt tet cell meaurement i given in Wahltröm The erivative of the engine tate variable are given by 1, an the oxygen concentration in the exhaut ga i calculate in 2. Further, the main performance variable are efine by 3. t ω t = f 2 x,u t p em =f 1 x,u, 1a t p = R a T W c W egr W ei V 1b t X O = R a T X Oem X O W egr p V X Oc X O W c 1c t X Oem = R e T em p em V em X Oe X Oem W f W ei ω t W t 1 X Oe = W ei X O W f O/F 2 W f W ei W egr x egr =, λ O = W ei X O 3 W c W egr W f O/F 3. ROBUST NONLINEAR CONTROL The control eign metho ue in thi paper i bae on a non-linear multivariable metho propoe in Sepulchre et al an Jankovic et al It inclue contruction of a Control Lyapunov Function CLF an invere optal control that guarantee robutne of optal controller. The control eign metho i briefly reviewe below. In invere optal control, the goal i to firt eign a control law an then etermine which cot function it minize. However, in orer to obtain a ple relation between the control law an the cot function, the cot function i choen quaratic in u accoring to 0 lx u T Rxut, lx > 0, Rx > 0 4 In tanar optal control Bryon an Ho, 1975, the control law that minize 4 for the ytem ẋ = fx gxu y = hx jxu, x R n, u,y R m i foun by olving the Hamilton-Jacobi-Bellman equation 0 = min[lx u T Rxu L f V x L g V xu] 5 u that ha the olution ux = 2 R1 L g V T x 6 where V x i a Control Lyapunov Function CLF an the notation L q V x enote the Lie erivate of V x along the vector fiel qx. Conequently, the goal i to firt eign V x an the matrix Rx, an then etermine lx in orer to ee which cot function the control law 6 minize. The optal control law 6 guarantee tability an robutne propertie. In particular, if R 1 = iag{γ 1,,γ m } an γ i > 2max px 7 x where L f V L f V 2 L g V 4 px = L g V 2, if L g V 0 0, if L g V = 0 then the control law 6 give aymptotic tability an it i robut to input uncertaintie that have 1/2, gain margin an ±60 phae margin Jankovic et al., CONTROL DESIGN WITH INTEGRAL ACTION A control eign without integral action of a ieel engine uing the metho in Sec. 3 i propoe in Jankovic et al Thi give an inner loop that hanle nonlinearitie an ecouple the ytem, in aition it ha goo tability an robutne propertie. However, a will be hown in Sec. 6, integral action i neceary to hanle moel error o that the controller can track the performance variable λ O an x egr pecifie in an outer loop. Therefore, the propoe eign i extene with integral action, reulting in the propoe cloe-loop ytem with integral action hown in Fig Control eign moel In orer to get a ple control law, the eventh-orer moel in Sec. 2 i plifie to a moel with three tate:

3 λ O x egr Setpoint from ma balance W c W egr Setpoint from 3:r orer moel p em Integral action y 1 y 2 CLF an invere optal control W c W f v 2 u 2 v 1 u 1 Wegr Flow to opening inverion u vgt u egr ENGINE W c p em Fig. 2. Block iagram of the cloe-loop ytem, howing; et-point calculation, integral action on the flow W c, nonlinear control of the flow u 1 an u 2, an flow to opening inverion. p, p em, an the compreor power P c. Thi moel i bae on the control eign moel evelope in Jankovic et al ṗ = k W c u 1 k e p ṗ em = k em k e p u 1 u 2 W f P c = 1 τ η m P t P c η c P c W c = T amb c pa p / µa P t = η t c pe T em 1 /p em µe u 2 The variable k em = k em T em, W f = W f u δ,n e, an k e = k e n e are treate a external lowly varying ignal an k, τ, η m, η c, T amb, c pa,, µ a, η t, c pe, an µ e are contant. 4.2 Output an et-point The eign objective i to regulate λ O an x egr to their et-point λ O an x egr. However, λ O can not be calculate from the control eign moel in Sec Further, p em ha to be choen a one output in orer to get table zero ynamic Jankovic et al., Therefore, the following output are choen y 1 = W c Wc, y 2 = p em p em 9 The et-point λ O an x egr are tranforme to the etpoint Wc an p em in two tep. Firtly, the equilibrium for W c an W egr of the ma balance 1b 1 are calculate from λ O an x egr Wc = W f β β 2X 2 4λ O O/F 1 x egrx Oc Oc x egr Wegr = 1 x egr W c 8 10 where β = λ O O/F X Oc 1 x egr O/F x egr, X Oc i the contant oxygen concentration in air paing the compreor, an O/F i the toichiometric relation between oxygen an fuel mae. Seconly, the equilibrium for p an p em of the thir-orer moel in Sec. 4.1 are calculate from W c an W egr p = W c W egr k e p em = 1 where η cmt = η c η m η t. c pa p µa T amb W c c pe η cmt T em W c W f µe Integral action Since the et-point 11 are bae on the plifie thirorer moel, moel error lea to that the value for p an p em are inconitent with the et-point for W c an W egr in 10. A a conequence, control error will appear in 9, yieling error in λ O an x egr thi will be illutrate in Fig. 3. In orer to ecreae thee control error, the following approxate integral action i ue ė = δ e KW c Wc = δ e K y 1 12 where δ i mall, but mut be poitive to enure table zero ynamic. The choice of y 1 a input to the integral action enure that the et-point Wc i achieve. The next tep i then to enure that Wegr i achieve an thi i one by feeing the tate e into p em. The et-point p em epen nonlinearly on Wc an Wegr it i therefore natural to utilize 11 when etermining the gain from e to p em. A a reult the following et-point for p em i receive p e = W c Wegr e k e p eme = p c e µa pa T amb W µe c e 1 c pe η cmt T em Wc W f e To plify the control eign, the Taylor expanion of p eme i ue accoring to p eme p em0 e p em0e = p em e p em0e 13 where e p em0 = 1 / 2 with 1 = [ µ a W c W c W f W c W egr µa1 W c W egr µa k e µ a µ a Wf ] cpa p em µe1 T amb 2 = c pe η cmt k e µ a µ e µ aµ e T em W c W f 2 Uing the et-point 13 for p em, the output become y 1 = W c W c, y 2 = p em p em 1 e/ 2 14 Note that integral action can not be performe on λ O ince λ O can not be calculate from the control eign moel. Further, if integral action i performe on x egr, table zero ynamic can not be guarantee. 4.4 Feeback linearization The firt tep in the control eign metho i to contruct a CLF V x, which i one uing feeback linearization. For the fourth orer moel 8 an 12, an the output 14, the relative egree become r 1 = 1 an r 2 = 1. Conequently, ẏ can be formulate a ẏ = Gxu Fx 15

4 where the matrix i invertible with an where b = a = Gx = [ ] a b k em k em µa1 k µ p a p Wc amb µa p p amb c pe η cmt 1 pamb p em µe T em c pa p µa τ T amb ak e p aw c W c Fx = τ 1 δ e KW c Wc k em k e p W f 2 Then by applying the feeback an input tranformation u = G 1 xα y Fx w 16 an the change of coorinate x [y,z], the ytem 8 an 12 i tranforme into the ytem ẏ = α y w 17 ż = f 0 y,z g 0 y,zw 18 where w i the new input, α i a poitive calar contant, an z = [p,e] T. 4.5 Stability of the zero ynamic When feeback linearization i ue it i neceary to invetigate the tability of the zero ynamic, which are efine by ż = f 0 0,z For 18 the zero ynamic become ṗ = q 1p q 2 p,e pµa q 3e 19 ė = δ e where q 1 p = c pa k p T amb Wc µ q 2 p,e = c pa k µ a p a T amb τwc 1e µe µ c pe η cmt p a amb 1 p 2 p T em em µ p a amb 1 δe q 3 e = c pa k em T amb Wc k em Wc W f 2 µe 1 1e c 2 p pe η cmt T em c pa k em T amb Wc em To analyze the tability of the zero ynamic 19, the Lyapunov function V = 1 2 c z1z c ee 2 i ue where z 1 = p µa q 3e. The zero ynamic 19 are aymptotically table if V =c z1 z 2 1 µ a p µa1 q 1 c e δ e c z1 q 2c 3ez 1 e δ c z1q 3e 2 q 2 4c e 2 < 0 for all [z 1,e] 0 which i true if δ > 0 an c z1 4 < c e δq 3 µ e2 a p µa1 q 1 q 2 Thi i poible if q 1 /q 2 < 0, which i true if 1 e/ 2 p em. Thi relation i achieve by plementing the integral action 12 in icrete form with anti-winup accoring to ǫ n = e n1 T δ e n1 K y 1 { ǫn, if ǫ e n = n e min e min, if ǫ n < e min where T i the ample te an 4.6 Contruction of a CLF e min = 2 1 p em In orer to obtain a ple control law, a quaratic Lyapunov function V = c 1 y 2 1 c 2 y i contructe, which i a CLF for the ytem 17, ince V = 2c 1 α y 2 1 2c 2 α y 2 2 < 0 for w = 0 an all y Control law Uing the CLF 20, the control law 6 can be applie to the ytem 17. However, thi control law will contain the non-linear cancellation 16 that i enitive to moel error. In orer to obtain a robut control law, thi cancellation i avoie by applying the control law 6 to the ytem 15 intea, uing the ame Lyapunov function V. Since V i a CLF for the ytem 17, it i a CLF for all ytem that can be obtaine from 17 by feeback, coorinate, an input tranformation which inclue the ytem 15. To apply the control law 6, the control input are centere at their et-point value v 1 = u 1 Wegr, v 2 = u 2 Wc W f Thi lea to that v = 0 rener the equilibrium point y = 0. A ple control law for the ytem 15 i achieve by chooing R 1 = iag{γ 1,γ 2 }, yieling [ ] v1 = [ ] v 2 2 R1 L G V T γ1 c = 1 ay 1 c 2 k em y 2 γ 2 c 1 by 1 c 2 k em y 2 Thi conclue the ecription of the control ytem in Fig AUTOMATIC CONTROLLER TUNING In the propoe control eign there are five tuning parameter: γ 1, γ 2, c 1, c 2, an K. The tuning objective are to minize the error between λ O an λ O an between x egr an x egr, an to achieve the inequality 7. However, it i ifficult to achieve thee objective by manually tuning, epecially the lat objective. Therefore an efficient metho for automatic tuning of the parameter i evelope.

5 5.1 Cot function for tuning The automatic tuning metho i obtaine by formulating a non-linear leat quare problem min V θ θ > 0 where θ i the tuning parameter θ = [γ 1,γ 2, c 1, c 2, K] T 21 The cot function V θ i calculate a N 2 2 eλo t i exegr t i V θ = γ λo γ egr i=1 λ ONorm x egrnorm 2 uegr t i u egr t i1 uvgt t i u vgt t i1 u egrnorm γ p max 2pxti /minγ 1,γ 2,0 p Norm u vgtnorm where t i i the te at ample number i. The motive for the ifferent term in the cot function are: Term 1: Minize λ O error e λo = λ O λ O Term 2: Minize EGR error e xegr = x egr x egr. Term 3 an 4: Minize ocillation in the EGR valve an VGT control ignal. The term have equal weight. Term 5: If the inequality 7 i not atifie, thi term minize the ifference between the right an left han ie of 7 uing a high penalty, γ p = 100. However, thi oe not guarantee that the inequality 7 i atifie for all operating point. A een in 22 all the term are normalize in orer to get the ame orer of magnitue for the five term. The weighting factor γ λo an γ egr are et to 1 an the contant δ i et to Optization A olver ha been evelope for the optization problem tate in the previou ection, an it conit of three phae. Firtly, the tuning parameter are initialize manually. Seconly, a globalization heuritic metho i ue to can a large region aroun the initial value in orer to avoi ening up in a ba local minum. Thirly, a tanar non-linear local leat quare olver i ue. A etaile ecription of the heuritic metho i given in Wahltröm CONTROLLER EVALUATION The performance of the control ytem in Fig. 2 i evaluate by comparing three ifferent control ytem: CLF with integral action: The control ytem in Fig. 2 where the controller tuning metho in Sec. 5 i applie to the tranient cycle efine by Tab. 1. CLF without integral action: Same a above, but K i et to zero an remove from θ in 21. PID: The control tructure with PID controller an automatic tuning metho propoe in Wahltröm It ha the following main feeback loop u egr = pi 1 e λo 23 u vgt = pi 2 e xegr λ O [] EGRfraction [%] EGRvalve [%] VGTpoition [%] Table 1. A efinition of the tranient cycle that i ue for tuning an conit of tep between 17 ifferent operating point. The cycle tart at 1 an en at 17, an pen 10 in each point. The value for λ O an x egr are obtaine by calculating the tationary point of the eventh orer moel for the 17 point below. u δ [mg/cycle] u egr [%] Operating point CLF with integral action CLF without integral action u vgt [%] n e [rpm] Setpoint Te [] Fig. 3. Illutrative example ulating CLF with an without integral action howing that the control error are reuce if an integral action i ue. where e λo = λ O λ O an e xegr = x egr x egr. The full eventh orer moel, ecribe in Sec. 2, i ue a plant moel to evaluate the three control ytem above. All their control parameter are hel contant for the entire operating region, i.e. no gain cheuling i ue. In aition, a low pa filter i applie to all variable aume to come from an oberver. 6.1 Benefit with integral action In Fig. 3, CLF with integral action an CLF without integral action are ulate. Step in λ O an x egr are performe at n e = 1200 rpm an u δ = 130 mg/cycle. The reult i that the control error in λ O an x egr are reuce if an integral action i ue. 6.2 Benefit with non-linear control In Fig. 4, CLF with integral action an PID are ulate. The ame tep in λ O an x egr are performe a in Fig. 3. The reult i that PID give lower control compare to CLF. In Fig. 5, CLF with integral action an PID are ulate at two other tep in λ O an x egr compare to Fig. 4.

6 λ O [] EGRfraction [%] CLF with integral action PID 1.94 EGRvalve [%] VGTpoition [%] Setpoint Te [] Fig. 4. Illutrative example ulating CLF with integral action an PID howing that PID give lower control compare to CLF. λ O [] EGRfraction [%] CLF with integral action PID Setpoint EGRvalve [%] VGTpoition [%] Te [] Fig. 5. Illutrative example ulating CLF with integral action an PID at two other tep in λ O an x egr compare to Fig. 4 howing that PID give ocillation an CLF give no ocillation. The operating point in Fig. 5 rener higher DC-gain in u egr λ O an u vgt x egr the loop that are ue a feeback in 23 compare to the operating point in Fig. 4. The reult i that PID give ocillation, while CLF give no ocillation. Since the ame control parameter are ue in Fig. 4 an 5 for CLF repectively PID, the concluion i that CLF hanle the non-linear effect in the ieel engine, uch a non-linear gain. 7. CONCLUSIONS A robut non-linear multivariable control eign with integral action ha been propoe an invetigate for control of oxygen/fuel ratio an EGR fraction. The control eign metho inclue contruction of a CLF by uing exact linearization an invere optal control, which provie a control law that i robut to tatic input uncertaintie that have 1/2, gain margin an ±60 phae margin. Thi metho i applie to a fourth orer moel incluing integral action on the compreor ma flow. Further, ulation of the propoe control eign how that control error are reuce if an integral action i ue compare to a control eign without integral action. The ulation alo how that the propoe control eign hanle the non-linear effect in the ieel engine. ACKNOWLEDGEMENTS The Sweih Energy Agency an Scania CV AB are gratefully acknowlege for their upport. REFERENCES M. Amman, N.P. Fekete, L. Guzzella, an A.H. Glattfeler. Moel-bae Control of the VGT an EGR in a Turbocharge Common-Rail Dieel Engine: Theory an Paenger Car Implementation. SAE Technical paper , January A. Amtutz an L. Del Re. EGO enor bae robut output control of EGR in ieel engine. IEEE Tranaction on Control Sytem Technology, page 37 48, Arthur E. Bryon an Yu-Chi Ho. Applie Optal Control. Taylor an Franchi, L. Guzzella an A. Amtutz. Control of ieel engine. IEEE Control Sytem Magazine, 18:53 71, J.B. Heywoo. Internal Combution Engine Funamental. McGraw-Hill Book Co, M. Jankovic, M. Jankovic, an I.V. Kolmanovky. Contructive lyapunov control eign for turbocharge ieel engine. IEEE Tranaction on Control Sytem Technology, Mrjan Jankovic, Roolphe Sepulchre, an Petar Kokotovic. CLF bae eign with robutne to ynamic input uncertaintie. Sytem an Control Letter, M. Nieuwtat, P.E. Moraal, I.V. Kolmanovky, A. Stefanopoulou, P. Woo, an M. Wile. Decentralize an multivariable eign for EGR VGT control of a ieel engine. In IFAC Workhop, Avance in Automotive Control, M.J. Nieuwtat, I.V. Kolmanovky, P.E. Moraal, A.G. Stefanopoulou, an M. Jankovic. EGR VGT control cheme: Experental comparion for a high-pee ieel engine. IEEE Control Sytem Magazine, J. Rückert, A. Schloßer, H. Rake, B. Kinoo, M. Krüger, an S. Pichinger. Moel bae boot preure an exhaut ga recirculation rate control for a ieel engine with variable turbine geometry. In IFAC Workhop: Avance in Automotive Control, Roolphe Sepulchre, Mrjan Jankovic, an Petar Kokotovic. Contructive Nonlinear Control. Springer-Verlag, A.G. Stefanopoulou, I.V. Kolmanovky, an J.S. Freuenberg. Control of variable geometry turbocharge ieel engine for reuce emiion. IEEE Tranaction on Control Sytem Technology, 84, July Johan Wahltröm. Control of EGR an VGT for emiion control an pumping work minization in ieel engine. Technical report, Linköping Univerity, 2006.