Fuzzy Adaptive Control of a Variable Geometry Turbocharged Diesel Engine

Transcription

1 Fuzzy Adaptive Control of a Variable Geometry Turbocharged Diesel Engine Mariagrazia DOTOLI, MEMBER, IEEE, and Paolo LINO, MEMBER, IEEE Dipartimento di Elettrotecnica ed Elettronica, Politecnico di Bari, Via Re David, 75 Bari, Italy, dotoli@poliba.it Dipartimento Elettrico, Elettronico e Sistemistico, Università degli Studi di Catania, V.le Andrea Doria 6, 955 Catania, Italy, lino@de .poliba.it Abstract -- A fuzzy control approach for the adjustment of the boost pressure of a Variable Geometry Turbine (VGT) supercharged Diesel engine is proposed. The VGT adapts the boost pressure to the target reference for different engine speeds by adjusting the turbine blades, resulting in a reduction of both fuel consumption and gas emissions, while preserving efficiency. We design an adaptive fuzzy control law according to the following steps: first, a standard PI controller is devised, then an equivalent fuzzy controller is built, finally the fuzzy controller is made non linear by tuning its input/output parameters using an optimization algorithm. Further, modification of the membership functions is investigated. A large number of simulations on a zero-dimensional model of the engine prove the effectiveness of the proposed control strategy with reference to stability and transient performance in comparison with standard PI techniques. I. INTRODUCTION Diesel engines are widely employed in transport applications due to their reliability, low fuel consumption and high efficiency. The main disadvantages of Diesel engines are their low power density, due to higher stress and compression ratio than in spark ignition engines, and the noxiousness of exhaust gases. One of the most common strategies to overcome such inconveniences is to supercharge the engine. In fact, while in normally aspirated motors the combustion air flow is directly introduced in the cylinders at atmospheric pressure, in supercharged engines air is previously compressed, allowing a larger mass to be introduced into the cylinders. Therefore, more fuel burns in the same volume, while keeping the same speed and piston displacement. In mechanical superchargers the compressor is driven by the engine, thus a fraction of its energy is lost. In turbocharged Diesel engines the exhaust gases energy is exploited to drive the supercharger. This results in an increase in efficiency, an improvement in torque response at low engine speeds and, on account of a lower equivalence ratio, a reduced amount of particulate emissions [8]. Since the turbine performances strongly depend on the mass outflow from the cylinders, in fixed geometry supercharged engines the compressor pressure ratio at low engine speeds is lower than needed. A variable geometry turbocharger can provide different values of the boost pressure for any engine operating condition by modifying the orientation of the turbine blades; hence, the Variable Geometry Turbine (VGT) regulates the power transmission from the rotor to the shaft. The control action is accomplished by an electronic control unit by way of a pneumatic actuator, to which the error between the actual boost pressure and the target reference is fed. The pressure reference is stored as a table in a memory with values depending on the engine speed and the driver s pedal position. Classical control methods, such as Proportional Integral Derivative (PID) control, do not guarantee a satisfactory behavior at each operating point of a VGT supercharged engine, due to high system non-linearity, ageing of mechanical parts and environmental conditions [5]. Some multivariable PID model based controllers are compared in [7]: the authors show that more complex control structures than PID guarantee higher performances and robustness. In [] the use of two separate adaptive PID regulators, one working at steady state and the other during fast transients, results in good dynamic responses. Clearly, non linear approaches can cope effectively with large system non linearities, often present in Diesel systems [3]: in [] the authors show how employing fuzzy control results in an improvement of the airflow response at low engine speeds and different loads, with considerable reductions in the design and implementation efforts. In this paper a fuzzy control approach for the adjustment of the boost pressure of a VGT supercharged Diesel engine is proposed and applied to a particular engine. The resulting design criteria are independent from the engine model and allow to quickly tune the controller parameters with partial knowledge of the system characteristics. More in detail, we design the controller according to the following steps: first, a PI regulator is devised using the classic Ziegler-Nichols method; then, an equivalent fuzzy controller is built, by means of a suitable choice of input/output (I/O) scaling gains, membership functions and rules, as described in [6]; finally, the control surface is made non linear by modifying the controller input membership functions (MFs) and applying a gain scheduling technique to the I/O gains, which are tuned by means of the Nelder-Mead optimization algorithm. In order to test the effectiveness of the proposed control strategy, a zerodimensional model of the case study is developed via the so-called filling and emptying technique, that makes it possible to simplify the model structure while keeping a good level of accuracy. A large number of simulations provide evidence of the superiority of the illustrated regulator with respect to PI controllers, with particular reference to stability and transient performances. The paper is organized as follows. First, we briefly describe the case study and derive a formal model of the engine. Then, in section III we describe in detail the proposed regulator. In the subsequent section we highlight the effectiveness of the proposed control technique by way of simulations; furthermore, comparisons with linear PI control laws are carried out. Finally, we discuss the main conclusions reached. II. CASE STUDY MODEL DESCRIPTION The engine under investigation is a four cylinders VGT supercharged Diesel engine. We assume that the working fluid is a mixture of ideal gases always in equilibrium for all chemical compositions and pressure-temperature conditions. In order to simplify the fluid dynamics description, a model based on the filling and emptying technique is set up [4]: assuming

2 small enough pipe dimensions, we adopt a lumped capacities representation, in which the fluid thermodynamic properties are spatially constant but time variant. In particular, the zerodimensional approach describes the engine by means of five elements: the turbine and the compressor, the intake and exhaust manifolds and the cylinders (see Fig.). Each component is characterized by a different set of thermodynamic state variables and may be described by the ideal gas law and conservation of the mass, conservation of energy and dynamic equilibrium equations. The application of the principle of conservation of mass to the intake and exhaust manifolds yields the following equation: p d V T = min mout R dt () where p and T are the gas pressure and temperature in the manifold volume V, respectively, R is the universal gas constant, m in is the mass inflow and m out the mass outflow. Neglecting heat transfers through walls, the conservation of energy equation for the intake and exhaust manifolds can be written as follows: V dp cv = min hin mout hout R dt () where c v is the specific heat at constant volume, calculated for the manifold under study and assumed as a time invariant, h in is the input fluid enthalpy content, h out is the enthalpy content of the fluid in the manifold. The application of the conservation of mass principle to the cylinders yields the following equation: dmcyl = ma + µ f ms dt (3) where M cyl is the overall mass in each cylinder, m a is the mass inflow from the intake valve, m s is the mass outflow to the exhaust valve and µ f is the apparent burned fuel rate. The latter is calculated as the sum of two contributions, under the assumption that combustion develops in three sequential steps, namely combustion delay, premixed combustion and diffusive combustion []. The gas pressure p cyl in the cylinder is obtained by applying the ideal gas law: pcyl Vcyl = Mcyl Rcyl Tcyl. (4) Temperature T cyl is calculated by means of the energy conservation equation: dtcyl dq dvcyl Mcyl cvcyl = pcyl + dt dt dt ( ) ( ) ( h u ) + ma hva ucyl ms hvs ucyl + + µ f f cyl where c vcyl is the specific heat at constant volume, Q is the heat transfer in the cylinders, V cyl is the instantaneous cylinder volume, depending on the piston position, h f is the fuel low calorific value, h va is the input fluid enthalpy content, h vs is the exhaust fluid enthalpy content and u cyl is the fluid internal energy. Neglecting the conductive contribution, e.g. assuming constant cylinder walls temperature T w, we may compute the heat transfer Q as the sum of the convective and radiative contributions. The convective term is: ( ) Qc = hc Asc Tw T cyl (6) where A sc is the instantaneous heat transfer surface, depending on the crankshaft angle, and h sc is a convective heat transfer coefficient, depending on the fluid working conditions. The radiative term is: (5) 4 4 ( ) Qr = β σ Asc Tcyl Tw where σ is the Stephan-Boltzman constant and β is a coefficient computed by way of a polynomial fit of experimental data, depending on the engine speed and the equivalence ratio. The cylinders intake and exhaust valves are represented as converging nozzles. Assuming stationary flow, we distinguish between two different gas conditions, described by the following dynamic equations, depending on the output/input pressure ratio p out /p in : k+ p = in k p out k p k m A out eff in k pin pin if (7) RT (8) k pout k > p + in k (subsonic flow condition in the outlet section) k + p k m = in Aeff k in k + if RT (9) k pout k p + in k (sonic flow condition in the outlet section), where k is the gas elastic constant, referred to the intake air flow, A eff is the equivalent outlet section surface, evaluated by means of the valve displacement curve, and depends on the crankshaft angle. The turbocharger is equipped with a centripetal VGT and a centrifugal compressor. By interpolating data stored in experimental maps it is possible to calculate the turbine and compressor efficiencies and the output air flow from the compressor, corresponding to different turbocharger speeds and pressure ratios. Evaluating the power supplied by the turbine and received by the compressor is straightforward. For the compressor we obtain: k p k P = out C mout cp Tin in C p η () where c p is the specific heat at constant pressure, assumed invariant during the transformation and referred to the intake mass flow, T in is the environmental temperature and η C is the compressor efficiency. For the turbine we obtain: k p k P = out T min cp Tin ηt pin () where η T is the turbine efficiency. The VGT flow conditions are determined by modeling the turbine as an adiabatic converging nozzle with variable output section, therefore we adopt dynamic equations analogous to Eq.8 and Eq.9. More in detail, in these expressions p in represents the exhaust manifold pressure, p out the environmental pressure and A eff the nozzle throat section, which is adjusted by the controller in order to regulate the boost pressure. Finally, the turbocharger speed is calculated by applying the dynamic equilibrium momentum equation: dω T P + = T P J C T ν ωt dt ω () T

3 where J T is the turbocharger moment of inertia, ν is the shaft viscous friction coefficient and ω T is the turbocharger speed. Similarly, by applying the dynamic equilibrium momentum equation to the driving shaft, we can evaluate the engine speed ω e : R p + dω J e = e M mmt C p Cr dt where J e and ( / ) p πτ (3) M R πτ are the crankshaft and the vehicle moment of inertia, respectively, M is the overall vehicle mass, R p the tires circumference, τ the gear transmission ratio, mmt the driving torque, C p the organic friction torque and C r the load torque, taking into account the road and viscous friction. Torques C p and C r are computed by means of polynomial fits of experimental data at different engine speeds and fuel mass introduction values. When the controller is working, the pneumatic actuator adjusts the turbine blades orientation and, consequently, the flow section by means of a plunger, resulting in a variation in turbocharger speed and, as a consequence, in boost pressure. The actuator dynamic behavior involves a time delay between the control decision and its accomplishment. We approximate the actuator dynamics by means of a first order system: daeff ( AS Aeff ) = τ A dt (4) where τ A is the actuator time constant and A S represents the actuation signal generated by the control system. The above model is validated by comparing simulation and experimental results, see section IV for details. III. THE PROPOSED REGULATOR A simplified representation of the control system is shown in Fig., including the engine, two control units, the fuel injection controller and the boost pressure controller. The latter regulates the turbine blades orientation by varying the duty cycle applied to the pneumatic actuator s electrovalve that adapts the boost pressure to the set point. The pressure reference is read from a map and depends on the engine speed and the fuel injected during the thermodynamic cycle. As mentioned previously, the commonly adopted strategy for the boost pressure regulation employs PID controllers, thanks to their simple structure and tuning rules. In general, such control laws do not lead to high performances, due to the high system non-linearity. We show in the following how to design an adaptive fuzzy regulator overcoming such inconveniences. Since the relation of the fuzzy controller parameters with the system response is not evident, we follow some criteria that simplify the design of the regulator. In the following the main control design steps are highlighted. The first step is to devise a standard PI controller based on the block diagram in Fig.. The discrete time PI control action u at the current step n is described by the following equation: n upin = K p en ej T + c τ i j= (5) where K p is the proportional gain, τ i the integral time constant, T c the sampling time, e j the error at step j. The PI controller parameters are settled according to the Ziegler-Nichols rules and manually tuned to improve performances. Then, in order to obtain a quick reaction to set-point variations, gain scheduling of the fuzzy regulator parameters is performed depending on the error; more in detail, a different set of manually tuned parameters, namely K p and τ i, is employed above a fixed threshold of the error, selected on account of simulation results. The second step consists in building an equivalent fuzzy controller to the previous one. The fuzzy regulator must be linear so the control surface must be flat; these conditions are satisfied by a suitable choice of input variables, I/O gains, MFs, rules and implication operators [6]. Input variables are the boost pressure error and the change of error, the output variable is the change in control action. As a consequence, we obtain an incremental PI fuzzy regulator: the control action is computed by integrating the output variable. The input MFs are equally spaced symmetric triangles and do not overlap more than two at a time; the sum of membership degrees of two overlapped MFs is always equal to. The output MFs are equally spaced singletons. The number of the input MFs determines the number of rules and output MFs; this is not a restrictive choice and is related to the requested precision depending on the input ADC converter resolution and the width of input variables universes of discourse: by fixing seven MFs for both the input variables, we obtain thirteen output MFs and fortynine rules, see Fig.3 and Tab.. Finally, the adopted fuzzy operators are: product as AND operator, bounded sum as OR operator, min as implication method, center of gravity (COG) as defuzzification method. The described fuzzy regulator is linear and can be made equivalent to a particular PI controller by calculating its I/O gains as a function of the PI parameters. In the discrete time domain the control action u FUZn is computed as follows: ( ) (6) ufuzn = CU j GCU Tc j where CU j is the normalized output at step j and GCU the output scaling factor. Since the linearity condition is satisfied, we may express CU j as the sum of normalized inputs E j and CE j : ( ). (7) ufuzn = Ej + CEj GCU Tc j If we take into account the input scaling factors GE and GCE we have:.(8) ej ej ufuzn = GCU GE ej + GCE Tc T j c In few steps it is straightforward to obtain: GE ufuzn = GCE GCU ej Tc + e n GCE j. (9) By comparing Eq.5 and Eq.9, we find the relationships between the PI and the fuzzy equivalent controllers parameters: GCE GCU = Kp GE GCU = τ. () i The previous two equations with three unknowns admit an infinite number of solutions; we exploit the remaining degree of freedom in order to keep the fuzzy I/O variables within the respective universes of discourse; the process is repeated for each parameters set. 3

4 The final step in the control design is to make the regulator nonlinear, with the aim of achieving good performances at different operating conditions. In order to grant smooth dynamics of the controlled variable in the vicinities of the set point, we modify the input MFs by investigating two different options: it is possible to shift the triangles upper vertices to the universes of discourse bounds or to replace the central MFs with trapezoidal ones, resulting in a large operating field of the main diagonal central fuzzy rules. The latter choice can be easily implemented, demands lesser changes and leads to better outcomes, as proved by simulation tests; the trapezoidal MFs shapes are set by using the trial and error procedure, see Fig.3 for details. Once the fuzzy regulator structure is designed, the I/O scaling factors are tuned by means of the Nelder-Mead optimization method, a simple and fast geometric algorithm that does not compute derivatives: at each step a 4-simplex is generated from the one in the previous step by way of geometrical transformations, namely by displacing the vertex in which the function to minimize displays a local maximum, then satisfaction of the stop criteria is verified [9]. As regards the system under study, the main control objective is a fast response with low oscillations, therefore the settling time is a good choice as performance index to minimize. IV. SIMULATION RESULTS In order to demonstrate both the model features and the effectiveness of the proposed controller, a large number of simulations is carried out by employing the MATLAB-Simulink software package. A set of preliminary tests investigates stationary conditions, with constant engine speed and fueling input as well as a fixed position of the turbine blades. In Fig.4 and Fig.5 the corresponding cylinders pressure and intake mass airflow in one thermodynamic cycle are reported respectively. The resulting pressure in the cylinders increases just after the intake valve is closed, i.e. during the compression phase, until the expansion phase starts shortly after the top dead center is reached; the pressure peak corresponds to the beginning of ignition. The intake mass airflow displays two negative peaks, due to the advance time in the intake valve opening and the delay time in the intake valve closing, corresponding to gas refluxes into the intake manifold. The resulting dynamics are in accordance with the engine expected behavior, therefore a primary check of the model capabilities is accomplished. A second set of tests investigates transient conditions. The working conditions are determined by the introduced fuel rate, corresponding to a third gear acceleration profile; the boost pressure reference is read from a map. The experimental set-up employs a PI regulator with the following manually tuned parameters set: K p = {.5,.4} τi = {.,.8} ; the error switch threshold is set to 5 mbar. The experimental results are initially compared with the simulation ones using the same regulator, showing how the model is able to predict the system behavior. Fig.6 shows the simulation engine speed; this is exactly equivalent to the experimental one, which is not displayed. In Fig.7 the experimental and simulation boost pressure diagrams are compared: when the pressure reference settles to a constant value, some differences in the controlled variable oscillations are noticeable. The amplitude of such oscillations is bigger in the real case than in simulation: this can be explained by considering the fluid dynamics simplification introduced with the filling and emptying technique and the low accuracy in the electromechanical actuator model. This inconsistency is acceptable and balanced by the reduced computational effort deriving from the model simplification. Figs.8, 9 and display the simulation boost pressure diagram, the electrovalve input duty cycle and the turbine equivalent area (or variable nozzle throat section) for the fuzzy controlled system, respectively. Tuning of the I/O gains is accomplished by employing the fminsearch MATLAB Optimization toolbox function, leading to the following parameters set: GE = {.96,.698} GCE = {.4,.38}. GCU = {.76,.399} The resulting control surface is plotted in Fig.9. Compared with the PI controller, the proposed regulator grants a lesser rising time and a lower oscillation amplitude; moreover, the boost pressure overshoot is remarkably reduced. It must be noticed that the PI parameters are optimized for a narrow working range, while further simulations may show that the dynamic performances of the proposed fuzzy regulator are independent from the set-point, demonstrating the superiority of such control approach on the traditional PI approach. CONCLUSIONS In this paper we proposed a novel fuzzy control approach for the boost pressure adjustment of a VGT turbocharged Diesel engine. The design criteria are independent from the system model, and the regulator may be built in few steps, namely: setting up of a PI controller, establishing an equivalent fuzzy controller and making it non linear by tuning the input/output parameters using an optimization algorithm. moreover, the simplified controller structure allows easy implementation on the control unit. A large number of simulations on a zero-dimensional model of the system showed the effectiveness of the illustrated regulator with respect to standard PI control laws, with reference to stability and transient performances. The adopted model is able to predict the system behavior and is simple enough to request a small computational effort. ACKNOWLEDGMENTS This work was supported by the MURST 4% funds. The authors wish to thank Prof. Bruno Maione from DEE-Politecnico di Bari for his valuable assistance and Mr. Marco Lombardo for his kind help in the collection and evaluation of the results. APPENDIX In the following the main engine technical characteristics are reported. V c = [m 3 ] V a =.5 [m 3 ] V s =.3 [m 3 ] J e =.5 [Kg m ] J T =.84 [Kg m ] M = 3 [Kg] R p =.954 [m] τ = 3. single cylinder volume intake manifold volume exhaust manifold volume crankshaft moment of inertia turbocharger moment of inertia vehicle mass tires circumference third gear transmission ratio 4

5 REFERENCES [] R. Buratti, A. Carlo, E. Lanfranco and A. Pisoni, DI Diesel Engine with Variable Geometry Turbocharger (VGT): A Model Based Boost Pressure Control Strategy, Meccanica, no. 3, 997, pp [] G. Ferrari, Motori a Combustione Interna, Il Capitello, Torino,. [3] L. Guzzella and A. Amstutz, Control of Diesel Engines, IEEE Control Systems Magazine, vol.8, no. 5, October, 998, pp [4] J.B. Heywood, Internal Combustion Engine Fundamentals, McGraw-Hill International Editions, 988. [5] I. Kolmanovsky, P. Moraal., M.J. van Nieuwstadt, and A.G. Stefanopoulou, Issues in Modelling and Control of Intake Flow in Variable Geometry Turbocharged Engines, in Proc. of the 8 th IFIP Conference on System Modeling and Optimization, Detroit, 997. [6] M. Mizumoto, Realization of PID Controls by Fuzzy Control Methods, in Proc. of the IEEE Int. Conf. on Fuzzy Systems, 99, pp [7] M.J. van Nieuwstadt, I.V. Kolmanovsky, P.E. Moraal, A. Stefanopoulou, M. Jankovic, EGR-VGT Control Schemes: Experimental Comparison for a High-Speed Diesel Engine, IEEE Control Systems Magazine, vol., no. 3, June,, pp [8] A. Stefanopoulou, I. Kolmanovsky, J.S. Freudenberg, Control of Variable Geometry Turbocharged Diesel Engines for Reduced Emissions, IEEE Trans. on Control Systems Technology, Vol.8, No.4, July, pp [9] The MathWorks, Optimization Toolbox User s Guide,. [] R.S.Wijetunge, C.J.Brace, J.G.Hawley, N.D.Vaughan, Fuzzy Logic Control of Diesel Engine Turbocharging and Exhaust Gas Recirculation, in Proc. of the Control UKACC Int Conf. on Control, Mini-Symposium on Engine Control Systems, University of Cambridge, UK, September. engine speed Pset fuel REFERENCE injection PRESSURE MAP 3 Pboost Degree of membership Degree of membership z- z Kp Ki*Tc PRESSURE LIMITER z min LIMITED INTEGRATOR Fig. PI controller block diagram. duty cycle d.c. Aeff PNEUMATIC ACTUATOR E, CE NS NZS NZ Z PZ PZS PS NS NZS NZ Z PZ PZS PS NB NMB NM PM PMB PB CU Aeff Fig.3 Fuzzy controller input/output membership functions. ELECTROVALVE 8 Pee, Tee,φee ELECTRONIC CONTROL UNIT Pee, Tt,φs 6 4 PNEUMATIC ACTUATOR C P a, T u,φee INJECTION SYSTEM T Ps, Ts,φs p cyl [MPa] 8 6 INTAKE MANIFOLD (pa,ta,φa) 4 CYLINDERS DIESEL ENGINE (pcyl,tcyl,φcyl) EXHAUST MANIFOLD (ps,ts,φs) crankshaft angle [grad] LOADS Fig.4 Stationary cylinders pressure in one thermodynamic cycle. Fig. Four cylinders Diesel engine layout.. CE E Tab. Fuzzy controller rule table. NS NZS NZ Z PZ PZS PS NS NB NMB NM NS NZS NZ Z NZS NMB NM NS NZS NZ Z PZ NZ NM NS NZS NZ Z PZ PZS Z NS NZS NZ Z PZ PZS PS PZ NZS NZ Z PZ PZS PS PM PZS NZ Z PZ PZS PS PM PMB PS Z PZ PZS PS PM PMB PB m a [Kg/sec] crankshaft angle [grad] Fig.5 Stationary intake mass air flow in one thermodynamic cycle. 5

6 8 duty cycle % Fig.6 Engine speed for a third gear acceleration profile. Fig.9 Electrovalve input duty cycle with fuzzy regulator. simulation experimental 8 7 boost pressure [mbar] set-point A e ff [cm ] Fig.7 Comparison of experimental and simulation boost pressure for a third gear acceleration profile with PI regulator. Fig. Turbine equivalent area with fuzzy regulator. set-point system response boost pressure [mbar] Fig.8 Simulation boost pressure diagram for the fuzzy controlled system Fig. Adaptive fuzzy regulator control surface. 6