TIME-VARYING LINEAR OBSERVER FOR TORQUE BALANCING ON A DI ENGINE

Transcription

1 TIME-VARYING LINEAR OBSERVER FOR TORQUE BALANCING ON A DI ENGINE Jonathan Chauvin Gie Corde Phiippe Mouin Miche Castagné Nicoas Petit Pierre Rouchon Centre Automatique et Systèmes, Écoe des Mines de Paris, 6, bd St Miche, Paris, France Institut Français du Pétroe, et 4 Avenue de Bois Préau, Ruei Mamaison, France Abstract: Torque baancing for DI engines aims at eiminating the vibration mode and decreasing the eve of emissions induced by disturbances such as variabiity in injectors efficiency, inertia and geometry between cyinders. The contribution of this artice is to present a rea-time experimenta setup aong simuations and experimenta resuts of estimation of the indicated torque of a DI engine. We buid a time-varying inear mode of the system and design a inear observer to estimate the indicated torque produced by each cyinder. This is achieved the instantaneous engine speed as ony measurement. This work can be used in a next step when unbaance coud be corrected in an adaptive manner by controing the mass injected into each cyinder. Keywords: Time-varying Contro, Observer Design, Engine Modeing, Torque Baancing, Rea-time. INTRODUCTION Torque baancing for diese engines is a major feature of modern engine controers for two main reasons: Diese injection systems introduce a bias in injected masses between cyinders (reative errors can be up to 2% at ow engine speed). Torque production varies from cyinder to cyinder in a monotonic way respect to the injected fue mass. This torque variation is a periodic time function and generates a undesired vibration mode in the engine. This phenomenon is important at ow engine speed and ide mode conditions. To estimate torque production from each cyinder we use the instantaneous engine speed at the crankshaft. Currenty, an engine speed measurement every 45 o of crankshaft rotation seem sufficient to estimate torque baance and to correct it in an adaptive manner by controing the mass injected in each cyinder. Torque baance determination by the measurement of the crankshaft ange has been addressed previousy in the iterature. Most of the soutions proposed have their foundation on a Direct or Indirect Fourier Transform as in (Jianqiu et a., 22a), (Jianqiu et a., 22b) and (Ginoux and Champoussin, 997) or are founded on the derived measures from crankshaft position (van Nieuwstadt and Komanovsky, 997). Our approach use a time-varying mode for the design of an engine torque observer. The goa of our work is to buid a more accurate observer of torque baancing in transient mode (according to torque and engine speed). The paper organizes as foows. In the Section 2, we describe the experimenta setup. We expain the engine dynamics in Section 3 and the approximation we

2 made to obtain a inear equation in Section 4. We then reca the notion of residue in Section 5. In Section 6, we describe our observer design. We concude in Section 7 by concusions and the future work 2. EXPERIMENTAL SETUP FOR CONTROL DESIGN The synthesis of the state based observer-controer was first vaidated on a 4 cyinder DI engine mode. This reference mode is based on the Chmea combustion modeing (Chmea and Orthaber, 999). The reference mode incuded dynamics of the combustion and the transformation into indicated torque and after into effective torque. The transmission is represented as in (Chmea and Orthaber, 999). A more reaistic mode non inearities and effects due to the imperfection of fywhee tooth wi be integrated in a second time. The observer has been tested instantaneous engine speed measurements coming from the reference mode and aso tested on the test bench. The synopsis of the compete simuation is described on the figure beow. We impemented the simuation mode (reference mode + observer + controer) on a rea time simuator based on xpc- Target (Support, n.d.). The rea time ratio is not achieved due to the compexity of the reference mode. With a GHz Pentium based computer, second of engine simuation is computed in 3 seconds. This HiL patform was transferred to a fast prototyping system by adding specific input/ouputs boards. The same code was kept and impemented in the contro system in order to be tested on the test bench. 3. ENGINE DYNAMICS We want to reconstruct the combustion torque crankshaft veocity as ony measurement. Foowing (Kiencke and Niesen, 2) we write the dynamics equation. The torque baance on the crankshaft is given by d dα ( 2 J(α) α2 ) = T comb (α) T oad (α) T fric (α) () Let T mass = T comb T oad T fric denote the mass torque. The combustion torque is aso referred to as indicated torque. We consider that the oad torque T oad is a ow frequency signa and that friction torque is a second order poynomia in α (Cavina and Ponti, 999). We can spit the mass torque into two terms T mass = J(α) α + dj 2 dα α2 (2) The first term represents the osciating masses and the second one represents the rotating masses. We use a mode of two massed connected by a singe rod (Fehrenbach, 99) (see aso (Kiencke and Niesen, 2)). The overa rod mass m rod is an osciating part m rod,osc = m rod osc a rotating part m rod,rot = m rod rot The two engths rot and osc rot + osc = are defined by the position of the center of gravity of the connecting rod. The osciating mass is : m osc = m piston + m rod osc and the rotationa mass on each cyinder is: m rot CY L = m crank CY L + m rot rod The crankshaft mass is deduced from the moment of inertia m crank = J Crank r 2 After further cacuations we have T mass = J α + dj 2 dα α2 J(α) = m rot r 2 + m osc CY L dj dα = 2m osc CY L j= ds j d 2 s j dα dα 2 j= ( ds j dα )2 4. NOTION OF RESIDUES AND USE FOR TORQUE BALANCING The residue is the reative error of the effective work. It is a cassica notion for torque baancing as expained in (Kiencke and Niesen, 2). It can be written R i = CY L 4π αi + 2π CY L t=α i 2π CY L 4π T ind dt 4π t= T inddt 4π 4π t= T inddt (3) where the ange on the crankshaft α i represents the midde point between the TDC and the BDC of the cyinder i. By definition CY L i= R i = The residues wi aow us to know the equiibrium of the stroke. When the system is perfecty baanced the residues are. A positive residue in cyinder i reveas that this cyinder generates more torque than the average of the 4 cyinders. This unbaance can be corrected by injecting ess fue into cyinder i. If convergence time is not an issue, a simpe integra controer can be used to correct cyinder i. More sophisticated adaptation aws can be deveoped and seem efficient in practice (see (?))

3 Figure. Goba Scheme 5. DISCRETE TIME-VARYING LINEAR APPROXIMATION For further cacuations, the torque baance is an ange-dependent differentia equation timederivatives J(α) α = T comb (α) T oad(α) f(α) α 2 (4) f = dj dα T oad = T oad + T fric refers to the extended torque. We can reformuate the second derivative of the crankshaft ange to have Toad α d α dα = J(α) (T comb(α) T oad(α) f(α) α 2 ) (5) Using a first order approximation on the integration of the previous equation, we can break the dependence on time and on crankshaft ange and ony save a dependence on the square of the crankshaft ange speed. α 2 (n + ) α 2 (n) 2 α J(n) (T comb(n) Toad(n) f(n) α 2 (n)) (6) We use as anguar path α = 6 o. The mode is then discretized respect to ange steps instead of time steps. Using the square of the crankshaft ange speed α 2 as the first state variabe x, we get the inear equation x (n+) = ( 2 α J(n) f(n))x (n)+ 2 α J(n) x 2(n) (7) where { x (n) = α 2 (n) x 2 (n) = T comb (n) T oad(n) (8) This formuation of the probem wi aow us to use cassica methods of contro to estimate the indicated torque. 6. OBSERVATION OF THE INDICATED TORQUE From the previous dynamics we ook for an observer of the system to estimate the indicated torque. The system has a two-dimensiona state and a singe measurement (crankshaft speed). It must be compemented another equation refecting the evoution of T comb (n) T oad (n). 6. x 2 considered as an unknown constant As a first try we consider that the variation of the indicated torque during one anguar step is sma Observation Mode We use as second equation dx 2 dα = (9)

4 during α. Between two anguar steps we get x (n + ) = ( 2 α J(n) f(n))x (n) + 2 α J(n) x 2(n) x 2 (n + ) = x 2 (n) () We have the system { X(n + ) = A(n)X(n) y(n) = CX(n) X(n) = [ α 2 (n) T comb (n) T 2 α 2 α A(n) = ( f(n)) J(n) J(n) C = [ ] oad(n) ] T We buid the observer by inear feedback on the crankshaft anguar speed. { ˆX(n + ) = A(n) ˆX(n) L(n)(ŷ(n) y(n)) Indicated Torque [Nm] Indicated Torque : ref(bod bue), est(red) Figure 2. Indicated torque (rpm, 2bar) Residues in % : ref(bod bue), est(red) ŷ(n) = C ˆX(n) so the error satisfies { X(n + ) = (A(n) L(n)C) X(n) ỹ(n) = C X(n) X(n) = X(n) ˆX(n). Residues [%] Ordor of the cyinders Observer convergence The dynamics of our system is periodic, so we just have to check stabiity for T k= (A k L k C). We numericay investigate the convergence of the system. The eigenvaues of T k= (A k L k C) are [, 4 ] T which proves the convergence of the observer Resuts and Comments Figure 3. Residues (rpm, 2bar) 96 Engine Speed : ref(bod bue), est(red) Simuation resuts In Figures 2, 3, and 4 we have ref : reference pp : resut of the first observer poe pacement dx 2 dα = The simuation is at rpm an BMEP of 2bar. For the simuation part, we introduced offsets on the mass injected in each cyinder. Cyinder : 2% of the reference mass Cyinder 2: % of the reference mass Cyinder 3: -2% of the reference mass Cyinder 4: 5% of the reference mass We introduced aso some noise for the simuation which reproduces the imperfection of the fywhee. Ne [rpm] Figure 4. Engine Speed (rpm, 2bar) the bench the in-cyinder pressure and we test the observer on the fywhee veocity measurement. The setting point is not the same as the simuation one Resuts from the bench Figures 5 and 6 dispay the resut of the estimator on experimenta data. We reconstruct the indicated torque from Comments The simuation and the resut on the bench are reativey satisfactory. Quaitativey, the residues are we estimated. On the

5 Indicated Torque [Nm] Indicated Torque : ref(bod bue), est(red) ( 2 α 2 α 2 α f(n)) A(n) = J(n) J(n) J(n) sin(n α) C = [ ] Again, we buid the predictor by inear feedback on the crankshaft anguar speed. The observer dynamics reads { ˆX(n + ) = A(n) ˆX(n) L(n)(ŷ(n) y(n)) Figure 5. Indicated torque on the bench Residus [%] Residus %: ref(bod bue), est(red) Order of the cyinders Figure 6. Residues on the bench other hand the quantitative prediction is not accurate. In order to drive the residues to zeros, ony the quaitative information matters: simpe PID woud work on this fuy actuated system provided the sign of the gains are known. The idea behind this observer is the simpest as we introduced no more knowedge about the system. Using more physics on the second equation, we now aim at increasing the quaity of the observation. ŷ(n) = C ˆX(n) so the error satisfies { X(n + ) = (A(n) L(n)C) X(n) ỹ(n) = C X(n) X(n) = X(n) ˆX(n) Observer convergence As we said above, the dynamics is periodic so we just have to check stabiity for T k= (A k L k C). The eigenvaues of T k= (A k L k C) are [.67, 3.5 7, ] T which proves the convergence of the observer. These ast eigenvaues correspond to an average mode Ā eigenvaues having the foowing moduus [.9933,.538,.886] T. A question of particuar interest is the design of the gain scheduing that woud give an improved convergence. This is a point we are focusing on Resuts and Comments Resuts from the simuation Figures 7 and 8 show the comparaison of the two observers and the better resuts of the second one. 4 Indicated Torque : ref(bod bue), est pp (dash red) est pp2 (green) 6.2 x 2 modeed as a stabe second order fiter known parameters Observation Mode The second observer is based on the fact that the indicated torque is periodic, so we consider x 2 as a stabe second order fiter known parameters. We use for second equation x 2 (n) = a(n) + b(n) sin(n α) () As the indicated torque varies in transient mode (according to torque and engine speed), the parameters a and b must be adapted. Both a and b are now incuded in the state. The dynamics reads { X(n + ) = A(n)X(n) y(n) = CX(n) X(n) = [ α 2 (n) a(n) b(n) ] T Indicated Torque [Nm] Figure 7. Indicated torque the second observer Comments Accuracy is improved but remains an issue. Convergence is obtained in practice but its interpretation is tricky. Indeed, for a given n, the pair (A(n), C) is not observabe. One eigenvaue of the observabiity matrix is aways.

6 Residues [%] Residues in % : ref(bod bue), est pp (dash red) est pp2 (green) Ordor of the cyinders Figure 8. Residues the second observer Thus the gain scheduing design is somewhat more difficut to achieve. Today it seems this formuation of the probem coud be improved further, by adding more harmonics to the approximation for instance. 8. NOMENCLATURE Symbo Units Physica Variabes BMEP [bar] Break Mean Effective Pressure BTC Bottom-Top Center CYL Number of cyinders J [kgm 2 ] Moment of inertia J crank [kgm 2 ] Crankshaft moment of inertia [m] Connecting rod enght m crank [kg] Crankshaft mass m osc [kg] Osciating mass m rod [kg] Rod mass m rod,osc [kg] Osciating rod mass m rod,rot [kg] Rotationa rod mass r [m] Radius of the crankshaft R i Residue of cyinder i s [m] Piston stroke ṡ [m/s] Piston veocity s j [m] Piston veocity of cyinder j TDC Top-Down Center T comb [Nm] Combustion torque T fric [Nm] Friction torque T ind [Nm] Indicated torque T oad [Nm] Load torque Toad [Nm] Extended oad torque 7. CONCLUSION AND FUTURE WORK This artice summarizes the evoution of the observer design at IFP for torque baancing on DI engine. After expaining the engine dynamics in Section 3 and the mode we use in Section 4, we expored two ways to describe the evoution of the indicated torque and showed the resuts on simuation and on the bench (Section 6). So far we have some good resuts on the estimation of the torque baance, though their quantitative accuracy can be improved. By using the periodicity of the indicated torque and a first harmonic deveopment, we have better resuts but have not finished the exporation of this method yet. The quaity of the observation was improved and a proof of the convergence of this type of observer is in progress. Future directions seem cear. For the contro design method, we have to evauate the strengths and weaknesses of the previous methods by increasing the size of the decomposition (by adding terms in cos(α), sin(2α), cos(2α),... ). Concerning physica modeing, an other estimation of the oad torque, not depending on the singe time estimation of the indicated torque at TDC and BDC. We wi have to introduce a more accurate mode of transmission and its non inearities. This part wi be required for use on a vehice whose transmission is ess stiff. Moreover, the probem of torque baancing can be incuded in the more goba probem of cyinder baancing which incude torque baancing but aso air fue ratio reguation. 9. REFERENCES Cavina, N. and F. Ponti (999). Fast agorithm for on-board torque estimation. In: Proc. of SAE Conference. Chmea, F. and G. Orthaber (999). Rate of heat reease prediction for direct injection diese engines based on purey mixing controed combustion. In: Proc. of SAE Conference. number Fehrenbach, H. (99). Mode-based combustion pressure computation through crankshaft anguar acceerartion anaysis. Proceedings of 22 nd Internationa Symposium on Automotive Technoogy. Ginoux, S. and J.C. Champoussin (997). Engine torque determination by crankange measurments: State of art, future prospects. In: Proc. of SAE Conference. number Jianqiu, L., Y. Minggao, Z. Ming and L. Xihao (22a). Advanced torque estimation and contro agorithm of diese engines. In: Proc. of SAE Conference. Jianqiu, L., Y. Minggao, Z. Ming and L. Xihao (22b). Individua cyinder contro of diese engines. In: Proc. of SAE Conference. number Kiencke, U. and L. Niesen (2). Automotive Contro Systems For Engine, Driveine, and Vehice. SAE Internationna. Support, Mathworks XPC Target 2 (n.d.). van Nieuwstadt, M.J. and I. Komanovsky (997). Cyinder baancing of direct injection engines. In: Proc. of the IEEE Conf. American Contro Conference.