Approximate Optimal Control of Internal Combustion Engine Test Benches

Transcription

1 11 5th IEEE Conference on Deciion and Control and European Control Conference (CDC-ECC) Orlando, FL, USA, December 1-15, 11 Approximate Optimal Control of Internal Combution Engine Tet Benche T. E. Paenbrunner, M. Saano and L. del Re Abtract Thi paper propoe the deign of an optimal imultaneou control of peed and torque for dynamic internal combution engine tet bench by mean of a dynamic extenion to the tate of the ytem which allow to avoid the explicit olution of the Hamilton-Jacobi-Bellman partial differential equation. To thi end, a dynamic control law i firtly deigned baed on a implified and approximate model of the tet bench and then modified in order to cope with non-modeled dynamic and nonlinearitie. The control i deigned on thi bai and it performance of the control law i teted and validated on an accurate imulator of the internal combution engine tet bench. I. INTRODUCTION The purpoe of a dynamic tet bench i the reproduction of the load condition an internal combution engine would undergo in a paenger car, truck or other vehicle without the vehicle. A tet bench guarantee reproducible condition in term of temperature and preure, jut to name a few, and reduce the cot and time required for development and configuration. In a vehicle the velocity and the rotational peed ω E of the engine are a reult of the engine torque T E and the load torque, thi being the direct conequence of the road and vehicle condition. In addition to a peed control loop, thee load condition have to be computed uing model and enforced by a dynamometer connected to the engine crankhaft in a tet bench. In indutrial practice, thi i uually accomplihed by two eparate control loop. In the cae of paenger car imulation, in mot cae the torque i controlled by the accelerator pedal poition α CE, while the rotational peed ω E i controlled by the loading machine. For larger internal combution engine, uually the accelerator pedal poition α CE i directly related to the engine peed while the torque i applied by the load machine. Of coure, the ignificance of the experiment on the tet bench i a direct conequence of the preciion of the control ytem. A a conequence, the ubject ha received attention in different way. For intance, a digital torque controller for a turbocharged dieel engine a well a a direct current dynamometer wa developed in [1] uing a cloed-loop pole aignment technique to tet an internal combution engine with the EPA tranient cycle (ee e. g. []). [3] preent the application of the model referenced adaptive control (MRAC) approach to the engine peed and torque control T. E. Paenbrunner and L. del Re are with the Intitute for Deign and Control of Mechatronical Sytem, Johanne Kepler Univerity Linz, 44 Linz, Autria thoma.paenbrunner@jku.at M. Saano i with the Control and Power Group, Electrical and Electronic Engineering, Imperial College, London, United Kingdom problem, where the Lyapunov tability theory wa ued to derive the parameter update law. More recent approache ue a multi-variable control of the engine-dynamometer ytem. In [4] the cloed loop reference tracking i maximized by balancing the bandwidth of the loop tranfer function to avoid exceive loop interaction in the cloed loop. In [5] a robut invere tracking method i applied to control an internal combution engine tet bench to achieve a high tracking performance. Differently from the invere optimal control problem [6], adopted in [7] which conit in fixing the tructure of the olution of the Hamilton-Jacobi-Bellman PDE and then computing the actual cot that i optimized by the reulting control law in the tandard optimal control problem, which i conidered herein, the oppoite approach i purued, i. e. a reaonable cot functional i decided a priori and then a olution to the partial differential equation i ought [8], [9], [1]. Unfortunately, the explicit olution of the HJB PDE may be hard or even impoible to determine in practical cae, hence everal methodologie to approximate the olution of the HJB partial differential equation in a neighborhood of the origin with a deired degree of accuracy have been propoed, ee for intance [11], [1], [13]. A control deign methodology for an internal combution engine tet bench, within the framework of multi-input multi-output controller, i propoed in thi work. A novel approach to approximate the olution of the well-known Hamilton-Jacobi-Bellman (HJB) partial differential equation, namely by mean of a dynamic extenion, i teted for thi pecific problem. A modified control law i then implemented on an accurate model of the internal combution engine tet bench, howing good performance in imulation. The ret of the paper i organized a follow: Section II give attention to internal combution engine tet benche, Section III deal with baic definition and notation that will be ued throughout the paper and the dynamic control law. The modified control and imulation reult are preented in Section IV and Section V, repectively. The paper i concluded with comment on the propoed methodology and future outlook in Section VI. II. INTERNAL COMBUSTION ENGINE TEST BENCHES A typical etup of a tet bench i hown in Fig. 1. The engine under tet i connected via a haft with a econd main power unit. Thi may be a purely paive brake wherea on the other hand electric machine, for intance, offer the poibility of an active and tranient operation. The accelerator pedal poition α CE of the internal combution /11/$6. 11 IEEE 85

2 engine and the et value T D, et of the dynamometer torque provide the input to the tet bench. ω E Combution engine α CE Fig. 1. T E Dynamometer T D, et T D ω D Setup of an internal combution engine tet bench. T x denote the meaured or etimated torque, while ω x repreent the meaured rotational peed, where x may be equal to E or D, denoting the internal combution engine and the dynamometer, repectively. The modeling of the tet bench i covered in detail in [5] and [7]. Therefore only the equation neceary for the control deign are ummarized here. To thi end the entire mechanical deign a two-ma ocillator can be decribed by equation of the form ϕ = ω E ω D, θ E ω E = T E c ϕ d (ω E ω D ), θ D ω D = c ϕ + d (ω E ω D ) T D, where ϕ i the torion of the connection haft while θ E and θ D denote the inertia of the internal combution engine and the dynamometer, repectively. The inertia of the adapter flange, the damping element, the haft torque meaurement device and the flywheel are already included in thee value. c characterize the tiffne of the connection haft wherea d decribe the damping. The behavior of an internal combution engine i in general rather complicated. However, a implified torque model of a Dieel engine can be decribed by equation of the form (ee [7] for more detail) (1) Ṫ E = ( c + c 1 ω E + c ω E ) T E + m(ω E,T E,α). () c i >, i = 1,...,3 are parameter to approximate the dynamic behaviour of the internal combution engine, m(ω E, T E, α) i a nonlinear tatic map including all part of the tatic combution engine torque. The employed electric dynamometer i modeled by a econd order low-pa filter, with dynamic ignificantly fater than thoe of the other component of the tet bench, hence they can be neglected in the deign. Within the range of maximum torque and maximum rate of change, the torque of the dynamometer can be decribed by Letting T D = T D, et. (3) v = m(ω E,T E,α), the ytem (1)-() can be rewritten a ẋ = Ax + f (x) + Bu (4) with c A = d, B =, 1 θ E c θ E c θd θ E d θd d θe d θ D 1 θd f (x) = ( ( c 1 x 3 + c x 3 ) T E ) T, the tate x = ( ) T ( ) T x 1 x x 3 x 4 = TE ϕ ω E ω D and the input u = ( ) T v T D, et. The actual control input, i. e. α CE, to apply in order to generate the deired v i then obtained by an (approximative) inverion of the map m( ). The main characteritic of the internal combution engine and the electric machine are ummarized in Table I. While the maximum torque i limited by the electric machine due to a thermal contraint, the maximum peed i given by the internal combution engine. TABLE I CHARACTERISTICS OF MAIN POWER UNITS Characteritic Value Maximum dynamometer torque T D, max 95 Nm Maximum rate of change Ṫ D, max 59 Nm Maximum dynamometer peed ω D, max 1 rpm Maximum engine torque T E, max 34 Nm Maximum engine peed ω E, max 4 rpm III. DYNAMIC CONTROL Conider a nonlinear ytem, affine in the control, decribed by an equation of the form ẋ = f (x) + g(x)u, (5) with f : R n R n and g : R n R n m mooth mapping, where x(t) R n denote the tate of the ytem and u(t) R m the input. The tak of the control i to minimize the cot functional J (x(t),u(t)) = 1 ( q(x) + u T u ) dt, (6) where q : R n R + i poitive emi-definite, ubject to the dynamical contraint (5), the initial condition x(t ) = x and the requirement that the zero equilibrium of the cloed-loop ytem be locally aymptotically table. Herein we conider formally the problem of approximating the olution of the regional dynamic optimal control problem, the definition of which i given in the following tatement. Problem 1: Conider ytem (5) and the cot functional (6). The regional dynamic optimal control problem conit in determining an integer ñ, a dynamic control law of the form ξ = α (x,ξ ), (7) u = β (x,ξ ), with ξ (t) Rñ, α : R n Rñ Rñ, β : R n Rñ R m and a et Ω R n Rñ containing the origin of R n Rñ uch that 851

3 the cloed-loop ytem ẋ = f (x) + g(x)β (x,ξ ), ξ = α (x,ξ ), ha the following propertie: (i) The zero equilibrium of the ytem (8) i aymptotically table with region of attraction containing Ω. (ii) For any ū(t) and any (x,ξ ) uch that the trajectory of the ytem (8) remain in Ω J((x,ξ ),β) J((x,ξ ),ū). It i well-known that if the calar function V : R n R + i a olution of the Hamilton-Jacobi-Bellman (HJB) partial differential equation (8) V x f (x) 1 V xg(x)g(x) T Vx T + 1 q(x) =, (9) then the tatic tate feedback u o = g(x) T Vx T olve the regional dynamic optimal control with ñ =. Unfortunately, the explicit olution of the HJB PDE may be hard or impoible to find in pecific ituation. Therefore, we conider herein a different notion of the olution of (9), a detailed in the following definition. Definition 1: Conider ytem (5). A C 1 mapping P(x) : R n R 1 n, zero at zero, i aid to be an algebraic P olution of (9) if there exit σ(x) x T Σ(x) x >, for all x R n \ {}, with Σ(x) : R n R n n, Σ() =, uch that P(x) f (x) + 1 q(x) 1 P(x)g(x)g(x)T P(x) T + σ (x), (1) and P(x) i tangent at x = to the ymmetric poitive definite olution of the algebraic Riccati equation aociated with the linearized problem, i. e. P(x) T = P. x x= Propoition 1: [14] Let P(x) be an algebraic P olution with Σ() >. Then, there exit a matrix R >, a neighborhood of the origin Ω R n and k uch that for all k k the function V (x,ξ ) = P(ξ )x + 1 x ξ R, (11) i poitive definite and atifie the partial differential inequality V x f (x) +V ξ ξ + 1 q(x) 1 V xg(x)g(x) T V T x, (1) with ξ = kvξ T, for all (x,ξ ) Ω. A dicued in the previou ection, a implified model of the tet bench can be decribed by equation a in (4) with f (x) = [ f 1 ]. To begin with, without lo of generality, let the algebraic P olution be of the form P(x) = x T P + Q(x), where Q : R 4 R 1 4 contain higherorder polynomial of the tate variable x. In other word, the linear olution i modified, adding the term Q(x), in order to dominate the nonlinear term of the algebraic inequality (1) aociated with the ytem (4). In the following, exploiting the pecific tructure of the term f in ytem (4), we let Q(x) be defined a Q(x) = [ Q1 (x) ]. Let q(x) = x T Qx in the cot functional (6), where the poitive definite matrix Q weight the relative error for the different component of the tate. Then, the algebraic Hamilton-Jacobi-Bellman inequality (1) i olved with repect to the unknown Q 1 (x) obtaining a olution of the form Q 1 (x) = N (x) D(x). It can be hown that the reulting olution i indeed welldefined for all the value of interet of the tate variable, exploiting in particular phyical contraint on x, namely x 1 mut be poitive, the minimum value for the peed of the engine ω E and of the dynamometer ω D i defined by the idle peed of the internal combution engine and mut be greater than min{ω E } = 63 rpm = 65.9 rad, i. e. x , x , and finally the torion of the haft i coniderably maller than the value of the other component, x x i, i = 1,3,4. Finally, a tated in Propoition 1, the computation of an algebraic P olution i enough to obtain the dynamic control law ξ = kvξ T (e,ξ ), u = B T V T x (e,ξ ), (13) where e i denote the tracking error for the variable x i with repect to the correponding reference value, namely e i = x i x i,et, i = 1,...,4. The function V i defined a in (11) with P(x) introduced above. The control law (13) approximate the olution of the regional dynamic optimal control problem for the ytem (4) with repect to the intantaneou cot q(x) = x T Qx. IV. MODIFIED CONTROL The dynamic control law (13) propoed in the previou ection i deigned conidering a rather implified model of the internal combution engine tet bench. Therefore the control action need to be modified, a explained in the following, in order to cope with ome of the nonlinearitie of the internal combution engine which are not included in the implified model, uch a aturation and hyterei, to name jut a few. In particular, an integral action i added to the control law deigned in the previou ection. More pecifically, letting ν i denote the i-th component of the dynamic control law (13), i = 1,, we define the actual control input implemented on the accurate internal combution engine model a u i = ν i (e,ξ ) + k i ν i (e,ξ )dt, (14) for i = 1,. Additionally we let the gain k be a function of the derivative which i implemented in the imulation a η+1 85

4 ω E, et T E, et Dynamic control Integral action Integral action with variable gain Modified control α T D, et Engine tet bench ω E ω D T E T D Fig.. Modified internal combution engine tet bench control. with η > of the tracking error for the peed of the engine, namely k (ė 3 ). In particular, the gain i defined uch that, when the error on the tracking for the engine peed change too fat, the integral action i negligible compared to the other component of the control ignal. Thi choice i reaonable, a hown in imulation, and needed in order to avoid an exceively aggreive reference profile for the torque of the dynamometer. Large error between the deired and the meaured engine peed and engine torque, due to change in the reference, are mainly, and relatively rapidly, compenated by the action provided by the dynamic control developed in the previou ection. Therefore, the effect of the integrator i limited due to the hort tranient repone of the tate variable when the reference value i modified. In fact, roughly peaking, thi implie that the tracking error belong to a neighborhood of the origin after a relatively hort amount of time and conequently the integral of the error are mall compared to the action of the dynamic control law. However, the integral action are more effective in the neighborhood of the origin mentioned above, i. e. cloe to zero tracking error, where they are actually needed in order to avoid a contant error in the teady-tate repone to the deired reference. Note additionally that, locally around the origin, the linear part of the dynamic control law i dominant with repect to higher-order term, hence the integral provide a tandard integral action. To ummarize the deign, the dynamic control law contain nonlinear term that allow to obtain an extremely fat repone to reference change, wherea the linear part of the dynamic control law itelf together with the integral action, with contant gain, provide accurate tracking preciion cloe to zero tracking error. Fig. how the cloed loop compoed of the internal combution engine tet bench and the modified control. The reference for engine peed ω E, et and engine torque T E, et are given. While the meaurement of engine peed ω E, dynamometer peed ω D and torque T D of an electric machine i relatively eay, the meaurement of the engine torque T E i unfortunately very difficult. Shaft torque meaurement device are expenive, at tet benche for development of tet bench control they are ometime preent. At mot tet benche a haft torque meaurement i not available and ha to be etimated uing an oberver. A compariion between an Extended Kalman Filter (EKF), a High Gain Oberver and a Sliding Mode Oberver (SMO) to etimate the engine torque T E i given in [15]. However, in imulation the engine torque T E can be acceed directly. V. RESULTS The developed controller ha been teted and validated on a high quality imulator of a tet bench. Thi imulator ha already been ued to deign a robut invere control (ee [5]) or model baed oberver for torque etimation (ee [15]). In addition to the dynamic of the entire mechanical decription the imulator alo include a more precie, nonlinear databaed model of the internal combution engine, the already mentioned dynamic and limitation of the dynamometer a well a diturbance effect. For example, the internal combution engine model take the dynamic of the accelerator pedal and combution ocillation into account. Additive white Gauian noie well reproducing the meaurement noie known from our tet bench i uperimpoed to the imulated rotational peed. The developed controller i compared with the controller developed in [7]. Thi controller ha already a much better performance than exiting tandard implementation with two eparate control loop. The output of the engine the engine torque T E and the engine peed ω E i depicted in Fig. 3 and Fig. 6 for two different, rather imple tet cae. In the firt imulation the engine torque T E i increaed from T E = 1 Nm to T E = Nm, while the engine peed ω E i kept contant. Both the engine torque T E and the engine peed ω E are reduced at the ame time in a further imulation. The engine torque T E i decreaed from T E = 15 Nm to T E = 1 Nm, the engine peed ω E from 5 rad to rad. However, it i poible to define a number of imilar engine operation reproducing real engine tranient appearing in a vehicle. The ue of the controller developed herein lead to a ignificant reduced under- or overhooting repectively of the engine torque T E when a change of the operation point occur. The engine torque T E how a lightly reduced rie time, however the final value i reached about 4 time a 853

5 Engine torque T E in Nm Reference 1 Preent MIMO Controller Developed Controller 8 Not only Fig. 3 how the difference between the compared controller. Difference are alo viible in the input ignal of the tet bench (ee Fig. 4). While both controller increae the accelerator pedal poition α CE quickly at t = 1, the developed controller avoid an overhoot of the accelerator pedal poition α CE. Furthermore, uch overhoot can increae the polluting emiion, ee e. g. [16]. The lower part of Fig. 4 how the dynamometer torque T D, et during the firt imulation. While the controller tuned in [7] a well a a decoupled ingle-input ingleoutput controller increae the torque immediately and thu caue the before mentioned drop in the peed, the propoed controller prevent thi drop by a lower increae of the dynamometer torque et value T D, et. 5 4 ξ 1 Engine peed ω E in rad/ ξ 5.. Fig. 3. Reference and imulated engine torque T E (top) a well a engine peed ω E (bottom) for firt imulation. ξ 3 5 fat. Uing the controller tuned in [7] the engine peed ω E drop down by around 1 % due to the udden increae of the engine torque T E and the coupling. In contrat, even a mall increae of engine peed ω E reult by applying the developed controller. In both cae the peed i uperimpoed by an additive diturbance caued by the reolution of the haft encoder and combution ocillation. Dyno torque T D, et in Nm Accelerator pedal poition α in % Preent MIMO Controller Developed Controller Fig. 4. Accelerator pedal poition α C E (top) and dynamometer torque et value T D, et (bottom) for firt imulation. ξ Fig Development of the tate ξ during the firt imulation. Fig. 5 how the time hitorie of the tate ξ i, i = 1,...,4 of the dynamic extenion. A expected, thee tate converge to the origin. However, the change of the operation point at t = 1 can clearly be een. Furthermore, the ize of the individual tate can be recognized in thee difference. Compared to the engine torque T E, the engine and the dynamometer peed ω E and ω D repectively (tate x i, i = 1,3,4), the torion of the haft ϕ (tate x ) i alway mall. Thi alo jutifie the aumption at the end of Section III. The tate ξ 3 and ξ 4 how a very imilar behaviour in the ame range of value, which i not upriing, ince thee tate are aociated with the engine peed ω E and the dynamometer peed ω D. The ame hold for the econd imulation. Again, the tationary final value of the engine torque T E i achieved fater and without an underhoot. The deired engine peed ω E i achieved in a much horter time by uing the developed controller. However, in contrat to the controller developed in [7] the engine peed ω E how a light underhoot limited to about.5 %. Although the cot function J TE = N T E T E, et k=1 T E, et J ωe = N ω E ω E, et ω E, et k=1 854

6 Engine torque T E in Nm Engine peed ω E in rad/ Reference Preent MIMO Controller Developed Controller Fig. 6. Reference and imulated engine torque T E (top) a well a engine peed ω E (bottom) for econd imulation. have not been ued to deign the propoed controller, they allow an aement of the propoed controller and controller tuned without any or uing an other cot function. N i the number of meaurement. All meaurement were recorded with a contant ampling time of 1 m. Table 3 ummarize the reult for both imulation. The calculated cot are caled to allow a better comparion. TABLE II COMPARISION BETWEEN EXISTING MIMO CONTROLLER AND DEVELOPED CONTROLLER 1 t imulation nd imulation J TE J ωe J TE J ωe Exiting MIMO controller Propoed controller While the cot according to an error in the engine torque T E are about the ame, a ignificant improvement in the peed characteritic can be determined. It hould alo not be forgotten that the compariion wa not done with a tandard tet bench controller, intead a multi-input multioutput controller wa ued. VI. CONCLUSION AND OUTLOOK The optimal control of an internal combution engine tet bench i dicued in thi paper. The tate of the ytem i extended to avoid the calculation of the explicit olution of the correponding Hamilton-Jacobi-Bellmann partial differential equation. The dynamic control law i afterward modified to cope with non-modeled dynamic and nonlinearitie. The propoed controller how a good tracking performance compared with other control: The engine torque i achieved quickly without overhoot, the engine peed how light, but almot limited overhoot. The effect of coupling i limited. In contrat to other multi-input multi-output control, the calculation and the tuning of the propoed controller i imple, eaily adopted and extended to other tet benche and tet benche with different etup. Meaurement applying the developed multi-input multioutput controller will be performed on the tet bench for which the imulator wa created. Such a controller will alo be deigned for a tet bench with a truck engine with about 175 kw loaded by a hydrodynamic dynamometer. An ongoing project motivate the extenion of the developed optimal control to internal combution engine tet benche with multiple dynamometer, namely tet benche with an electric machine and a hydrodynamic dynamometer. VII. ACKNOWLEDGMENTS The author gratefully acknowledge the ponoring of thi work by the COMET K Center Autrian Center of Competence in Mechatronic (ACCM). REFERENCES [1] T. Tuken, R. R. Fullmer, and J. VanGerpen, Modeling, identification, and torque control of a dieel engine for tranient tet cycle, in SAE Technical Paper Serie, February paper 935. [] DieelNet, Emiion tet cycle. March 11. [3] D. Yanakiev, Adaptive control of dieel engine-dynamometer ytem, in 37th IEEE Conference on Deciion and Control, December Tampa, Florida, USA. [4] B. J. Bunker, M. A. Franchek, and B. E. Thomaon, Robut multivariable control of an engine-dynamometer ytem, IEEE Tranaction on Control Sytem Technology, March [5] E. Gruenbacher and L. del Re, Robut invere control for combution engine tet benche, in 8 American Control Conference, June Seattle, Wahington, USA. [6] R. J. Freeman and P. V. Kokotovic, Invere optimality in robut tabilization, SIAM Journal of Control and Optimization, vol. 34, July [7] E. Gruenbacher, Robut Invere Control of a Cla of Nonlinear Sytem. PhD thei, Intitute for Deign and Control of mechatronical Sytem, Johanne Kepler Univerity Linz, Autria, 5. [8] B. D. O. Anderon and J. B. Moore, Optimal Control: Linear Quadratic Method. Prentice Hall, [9] D. P. Berteka, Dynamic Programming and Optimal Control Volume I, third edition. Athena Scientific, 5. [1] A. E. Bryon and Y. C. Ho, Applied Optimal Control: optimization, etimation, and control. Taylor and Franci, New York, [11] W. M. McEneaney, A cure-of-dimenionality-free numerical method for olution of certain HJB pde, SIAM Journal on Control and Optimization, vol. 46, no. 4, pp , 7. [1] D. L. Luke, Optimal regulation of nonlinear dynamical ytem, SIAM J. Control, vol. 7, no. 1, pp. 75 1, [13] T. Hunt and A. J. Krener, Improved patchy olution to the Hamilton- Jacobi-Bellman equation, in 49th IEEE Conference on Deciion and Control, December Atlanta, Georgia, USA. [14] M. Saano and A. Atolfi, Dynamic olution of the HJB equation and the optimal control of nonlinear ytem, in 49th IEEE Conference on Deciion and Control, December Atlanta, Georgia, USA. [15] P. Ortner, E. Gruenbacher, and L. del Re, Model baed nonlinear oberver for torque etimation on a combution engine tet bench, in 8 IEEE Conference on Control Application, Augut Hagen, Germany. [16] J. R. Hagena, Z. S. Filipi, and D. N. Aani, Tranient dieel emiion: Analyi of engine operation during a tip-in, in 6 SAE World Congre, April Detroit, Michigan, USA. 855