Estimation of Exhaust Manifold Pressure in Turbocharged Gasoline Engines with Variable Valve Timing

ESTIMATION OF EXHAUST MANIFOLD PRESSURE IN TURBOCHARGED GASOLINE ENGINES WITH VARIABLE VALVE TIMING Julia H. Buckland J. W. Grizzle Mrdjan Jankovic J. S. Freudenberg Research and Advanced Engineering Dept. of Electrical Engineering and Computer Science Ford Motor Company University of Michigan Dearborn, Michigan 48124 Ann Arbor, Michigan 48109 Email: [email protected] ABSTRACT estimate of Pe based on commonly measured signals. Feedforward A/F control in turbocharged gasoline engines Most turbocharged gasoline applications employ turbines with variable valve timing requires knowledge of exhaust mani- equipped with wastegates that open to divert flow around the tur- fold pressure, Pe. Physical conditionsin the manifoldmake mea- bine to control boost. Both the turbine and an open wastegate surement costly, compelling manufacturers to implement some can be considered flow restrictions in the exhaust path of the en- form of on-line estimation. Processor limitations and the cali- gine. This interpretation of the physical system implies that the bration process, however, put constraints on estimator complex- pressure in the exhaust manifold varies with mass flow rate and ity. This paper assesses the feasibilityof estimatingPe withanal- the effective size of the restrictions. gorithm that is computationally efficient and relatively simple to The total effective restriction varies greatly with wastegate calibrate. A traditionalreduced order linear observer is found to opening. For a given command from the powertrain control mod- perform well but has too many calibration parameters for prac- ule (PCM), the wastegate positionmay vary from its minimum to tical implementation. Using the performance of the observer as maximum deflection depending on the operating condition. This a benchmark, static estimation is explored by parameterizing the positionis not typically measured. Therefore the size of the flow equilibrium values of Pe with both the inputs and the outputs of restriction in the exhaust path is not accurately known, making the system. This nonlinear static estimate, combined with simple estimation of exhaust manifold pressure quite difficult. This is lead compensation, yields a practical observer implementation. in contrast to modern diesel applications which employ variable geometry turbines where vane position is readily available. In addition, cost and durability concerns also discourage INTRODUCTION measurement of temperature or other useful properties down- Modern automotive emission control systems for gasoline stream of the engine. Thus, Pe must be inferred from measure- engines rely heavily on feedforward air-fuel ratio (A/F) control ments of signals significantly distant and often related through to meet strict emissions regulations. For turbocharged applica- dynamic elements. tions, it has been shown [1] that knowledge of exhaust manifold Nonetheless, cost and time to market pressures dictate a pressure (Pe) is helpful in meeting the stringent accuracy require- simple and efficient estimation algorithm. Online computing ments of the feedforward controller. When variable valve timing resources are limited in automotive-quality processors, while is added, the significance of Pe rises dramatically, since consider- growing hardware complexity and regulatory requirements con- able valve overlap may occur over a large portion of the operating tinue to increase the number of computations and memory re- envelope. Measuring Pe can be quite problematic, however, due quirements of the software. Not coincidentally, calibration effort to the harsh environment in the manifold. System cost and dura- is also increasing, with each additional task lengthening devel- bility considerations drive a strong desire by automakers for an opment time and increasing cost. As such, strategies are highly scrutinized prior to implementation and those with added states Throttle or complex calibration procedures must demonstrate a clear and significant advantage to gain acceptance. Here we use model-based analysis to determine the feasibil- Intercooler ity of estimating Pe with a simple, efficient algorithm that can be easily calibrated. First we consider a traditional reduced order linear observer. Although nonlinearities in the system lead to a large number of calibration parameters, this approach es- tablishes feasibility of a solution and provides a benchmark for Wastegate Actuator comparison. Static estimation is then explored. Analysis of a linearized, steady state model of the system leads to a static lin- Compressor ear estimate of exhaust manifold pressure based on conditions Turbine in the intake. This estimator provides excellent accuracy when Figure 1. SYSTEM SCHEMATIC. applied to the nonlinear model in steady state. Transient performance, however, is shown to be poor due to the slow dynamics connecting the intake and exhaust. Simple lead compensation OBSERVER DEVELOPMENT of the nonlinear static estimate produces a practical implementa- The system (1) has four states, two of which are measured, tion with excellent steady state accuracy and improved transient Pi and Ptip. Therefore, a reduced order linear observer is explored performance. for estimation of Pe. Consider a linear representation of (1) given by SYSTEM DESCRIPTION δx˙ = Aδx + Bδu (2) The system under consideration is an I-3 turbocharged en- δ δ gine equipped with variable intake cam timing, a conventional y = C x pneumatically operated wastegate to control boost and an intercooler to increase charge density and reduce tendency for engine where δ indicates deviation from the equilibrium point about knock. A schematic of the system is shown in Figure 1. which the system was linearized. In order to compare the relative The model used for concept development is described in [2]. influence of system parameters, (2) represents a system scaled This model includes most ofthe major components of the desired relative to this equilibriumpoint such that deviation is defined in system, with a notable exception being the pneumatically actu- terms of fractional change.1 ated wastegate. The effects of wastegate are incorporated in the Following the development described in [3], the linear sys- model through a virtual actuator, wastegate flow rate, by assum- tem is partitioned by grouping measured and unmeasured states ingflow throughthe valve is a knowninput. This enables investi- as follows gation of the estimation problem, for this and future applications, without the limitations imposed by current actuator technology. T A four state representation of the modeled system is given x1 =[Pi,Ptip] T by x2 =[Pe,Ntc] , x˙ = f (x,u) such that T x =[Pi,Ptip,Pe,Ntc] (1) T u ETC VCT W N δx˙1 A11 A12 δx1 B1 =[ , , wg, ] δx˙ = = + δu T δx˙2 A21 A22 δx2 B2 y =[Pi,Ptip] δx1 δy = δx1 =[I2x2 02x2] . δx2 where Pi is intake manifold pressure, Ptip is throttle inlet pressure, Pe is exhaust manifold pressure, Ntc is turbocharger shaft speed, ETC is throttleangle, VCT is variable cam timing, Wwg is The estimatex ˆ2 is defined by flow rate through the wastegate and N is engine speed. Since engine speed is measured, this model representation z˙ = Arz + Brw uses N as an input; and since temperature changes slowly com- pared to the remaining states, manifold temperature dynamics xˆ2 = Crz + Drw 1 are ignored. This simplified representation of the turbocharged For example, when δWwg = 0.1 wastegate flow is perturbed by +10% from system facilitates formal analysis of the estimation problem. the equilibrium value for Wwg used for linearization. in the system that are not captured by the linear model. 0.5 0 ESTIMATOR DEVELOPMENT Error (%) Estimation The reduced order observer is quite effective, but it has two −0.5 0 2 4 6 8 10 12 states and twenty parameters that must be calibrated,4 many of 0.02 which must be scheduled with operating condition for this non- NL Dynamic Sim 0.01 Observer linear system. Given the processor and calibration complexity e P 0 limitations discussed in the Introduction, a simpler approach is preferred. Therefore we pursue a static representation of exhaust −0.01 0 2 4 6 8 10 12 manifold pressure. 1 wg 0 Physical parameters Equilibrium Analysis W scaled relative to initial equilibrium values Consider a continously differentiable nonlinear system −1 0 2 4 6 8 10 12 Time (s) ξ˙ = g(ξ,v) (3) Figure 2. TRANSIENT PREDICTION OF Pe USING A TRADITIONAL ψ = h(ξ) REDUCED ORDER OBSERVER. where ξ ∈ ℜn denotes the states of the nonlinear system, v ∈ ℜp where represents the control inputs and the system outputs are given by ψ ∈ ℜq. It’s equilibriumpoints are given by the solutions of [4] w =[u y]T − Ar =[A22 LA12] g(ξ,v) = 0 − − − Br = [(B2 LB1) (A22 LA12)L + (A21 LA11 )] ψ = h(ξ). Cr =[I2x2] Dr =[02x2 L]. Let g(ξ0,v0) = 0 and ψ = h(ξ0) be one such solution and let It can be shown that since (A,C) is an observable pair, (A ,A ) 22 12 0 = Fδξ + Gδv (4) is an observable pair. It can also be shown that the estimation δψ δξ error, (x2 − xˆ2), goes to zero when the observer gain matrix L is = H , chosen such that (A22 − LA12 ) is asymptotically stable. For this − application L is chosen such that the eigenvalues of (A22 LA12) be the linearization of (3) about the equilibrium point. By the are faster than the eigenvalues of A22 . Implicit Function Theorem [5], there exists a continously differ- Observer performance is evaluated using a nonlinear simu- entiable function κ(v,ψ), defined in a neighborhood of the equi- lation of the model described in [2]. The simulation model is librium point, such that modified by adding measurement noise with a maximum magni- tude of approximately ±1 kPa to both Pi and Ptip. These noisy raw measurement signals are then filtered at 30 rad/sec to rep- 0 = g(κ(v,ψ),v) resent the effects of digital signal processing typically employed ψ = h(κ(v,ψ)) by a PCM. Simulation results showing the transient response of the ob- F server to step changes in wastegate are shown by the dot-dashed as long as rank H = n.

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