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 compared 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 magnitude 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. κ line in Figure 2.2 This is compared with the simulated P signal, Such a function is not (locally) unique when n < p + q. e κ represented by the solid line in the ﬁgure. The observer performs One way to ﬁx a choice of is to use only n of the p + q com- ψ quite well, with relative error3 of less than 0.53% occuring dur- ponents of v and , or some combination thereof. To understand ing transient overshoots. This error is likelydue to nonlinearities this more clearly rearrange (4) as follows

2Physical parameters are scaled such that the figures show fractional change Pa−P0 F G 0 δv from the initial equilibrium value, for example, P = where Pa is the actual Pa δξ + = 0. physical parameter prior to scaling and P0 is the equilibrium value at the start of H 0 −I δψ simulation. 3 4 Relative error is defined as 100 Pe − Pê /Pe. Each matrix element is a calibration parameter. This gives a set of overdetermined equations from which we can influence on the estimate of δx, and specifically δPe, form an estimate of δξ or equivalently κ(v,ψ). With more equations than unknowns, we introducea weight- δ δ ing matrix to allow us to place emphasis on measurements and δ u T u x = M δ = USV δ . states in which we have high confidence, while effectively disre- " y # " y # garding those for which we do not. Specifically, Therefore, since U is orthonormal, F G 0 δv Q δξ + Q = 0, H 0 −I δψ δu U T δx = SV T . (5) δy where " #

Q 0 Examination of the rows of U T and SV T tell us how much Q = 1 0 Q2 information from u and y is transmitted to the linear combination of x. For example, consider the system (1) when the engine is operating at approximately 3000 RPM and 70 Nm, n×n q×q and Q1 ∈ ℜ and Q2 ∈ ℜ . The least squares estimate of δξ is then given by [3] −0.02 0.05 0.99 −0.04 −0.78 −0.16 −0.03 −0.61 δξ T T T T −1 U T =   =[F Q1 Q1F + H Q2 Q2H] −0.15 0.99 −0.05 −0.07 δ  −0.61 −0.04 0.02 0.79  × − T T T T v   [ F Q1 Q1G H Q2 Q2] δψ   " # δv = M δψ . " # 7.220 0 000 0 1.25 0 0 00 S =  0 00.92 0 00  This gives a static estimate of δξ, but as with the observer  0 0 00.0200  design, there are a large number of parameters for on-line im-     plementation and calibration. Therefore, we analyze this static representation of the states to identify which inputs and outputs have the most influence on the estimate. 0.89 −0.06 0.02 −0.13 −0.18 0.39 −0.11 0.01 0 0.03 −0.97 −0.20 Singular Value Decomposition −0.41 0.03 0 0.06 −0.13 0.90  V T = . Consider the turbocharged system defined by (1). We would −0.16 −0.41 0.10 −0.89 −0.01 0    like to construct a static estimate of P from measured variables  0 −0.05 0.99 0.14 0 0  e   and known inputs. Based on our equilibrium analysis, we con-  0 −0.91 −0.10 0.40 0 0    struct M from (2), with Q selected to emphasize the fast, ac-   curate pressure measurements. The wastegate flow input is de- Examining the first row of (5), we see that emphasized via Q, since measurement or estimation of this quan- tity is difficult in practice, −0.02δPi + 0.05δPtip + 0.99δPe − 0.04δNtc = δ δ δ δ 1.00 0 00 0 6.43 ETC − 0.43 VCT + 0.14 Wwg − 0.94 N 0 1.00 00 0 −1.30δPtip + 2.82δPi.  0 00.010 0 0  Q = .  00 01.00 0     0 0 0 01000  Since the contributionsof turbocharger speed and wastegate   δ  0 0 0 0 0100  flow rate are small in a relative sense, we approximate Pe as     δ δ δ δ By analyzing the singular value decomposition (SVD) of M Pe ≈ 2.84 Pi − 0.43 VCT − 0.94 N we can identify a set of inputs and outputs that have significant −1.35δPtip + 6.43δETC. (6) This parameterization aligns well with our physical intuitionthat 1.8 the pressure in the exhaust manifold varies with flow rate and Data restriction. ± 5% error e Specifically, we know from [6] that Pi, N and VCT can be 1.6 used to represent cylinder air charge, and therefore air mass flow rate through the system. In addition, we know that flow rate 1.4 through the turbine is only weakly dependent on turbine speed so that the effective size of the flow restriction due to the turbine 1.2 is approximately constant. So we surmise that the first three pa- Scaled Estimated P rameters in equation (6) represent the effects of air flow rate due to the restriction imposed by the turbine. 1 1 1.2 1.4 1.6 1.8 This implies that the remaining variables, ETC and Ptip, can Scaled Simulated P be used to characterize the effects of the variable flow restriction e due to the wastegate. To relate this to physical behavior, consider Figure 3. STEADY STATE PREDICTION OF Pe USING THE STATIC a steady state operating point defined by air flow rate. At each ESTIMATE. Ptip there is a unique value of ETC that delivers this desired air flow rate. Similarly, there is a unique wastegate position that delivers any given Ptip at this air flow rate. So at steady state, 2 knowledge of ETC and Ptip uniquely defines wastegate position. 0 Thus using our understanding of the physical behavior of Error (%) the system and the variables identified through SVD analysis, we Estimation −2 0 2 4 6 8 10 12 conclude that Wexh, Ptip and ETC can be used to parameterize the 0.02 NL Dynamic Sim Static Estimate equilibrium values of Pe. e 0 Observer P

−0.02 Static Estimate 0 2 4 6 8 10 12 We have identiﬁed a parameterization of exhaust manifold 0.02 pressure through system inputs and outputs. The Implicit Func- TIP 0 tionTheorem [5], applied in steady state to the system model (1), P −0.02 guarantees existence of a functional relationship between these 0 2 4 6 8 10 12 parameters and Pe. A practical way of identifying the relation- 1 ship in a production environment, where engine dynamometer Physical parameters wg 0 scaled relative to initial data is readily available, is through regression. W equilibrium values To generate the same type of data in our modeling environ- −1 0 2 4 6 8 10 12 ment, we simulate the nonlinear model of the turbocharged sys- Time (s) tem (1) [2] over the possible range of Ptip and ETC combina- tions. These data are used to produce the least squares estimate Figure 4. TRANSIENT PREDICTION OF Pe USING THE STATIC ESTI- given by MATE.

in Figure 4. Closer examination shows that Ptip responds rela- Pê = α0 + α1 Ptip + α2 ETC + α3 WexhPtip tively slowly to changes in wastegate, while Pe initially responds α P2 α W 2 (7) + 4 tip + 5 exh very quickly. The relatively slow response of Ptip is due to the fact that the wastegate acts on Ptip via the turbocharger, which has a large inertia. The exhaust manifold, on the other hand, is where Pˆ is the estimate of absolute exhaust manifold pressure. e small, with very fast dynamics and responds quickly to changes The terms of (7) are chosen to achieve the best least squares fit. in wastegate. The estimate relying on P is, therefore, unable to This approach produces an accurate steady state estimate of tip respond as quickly as desired. the simulated exhaust manifold pressure, as demonstrated in Fig- ure 3, which plots the estimated values of Pe versus values gen- erated with the full nonlinear model. Static Estimate with Dynamic Compensation This estimator performs well at steady state but it is also im- Transient analysis shows that our static estimate employing portant to consider the physical aspects of the dynamic environ- Ptip and ETC is insufficient to capture exhaust manifold pressure ment. Transient simulation of the estimator reveals significant dynamics. We now explore lead compensation as a means to error when wastegate is changed quickly. This is demonstrated improve the transient response of the estimator. by example with the step responses shown by the dotted lines For ease of implementation, we consider a simple first order −50 5 −100 0 −150 Error (%) Estimation −5 Magnitude (dB) −200 0 2 4 6 8 10 12 −2 −1 0 1 2 3 4 10 10 10 10 10 10 10 0.1

0 w/ Lead Compensation e 0 −50 P NL Dynamic Sim Static Estimate −100 −0.1 Tranfser function from 0 2 4 6 Static Estimate8 w/ Lead 10 12 W to P −150 wg tip Phase (deg) 1 −200 −2 −1 0 1 2 3 4 10 10 10 10 10 10 10 wg 0 Physical parameters Frequency (rad/sec) W scaled relative to initial −1 Figure 5. BODE DIAGRAM USED FOR LEAD COMPENSATION DE- equilibrium values 0 2 4 6 8 10 12 VELOPMENT. 0.1 0 ETC 2 −0.1 0 2 4 6 8 10 12 1 Time (s)

0 Error (%) Estimation Figure 7. PREDICTION OF Pe USING THE LEAD COMPENSATED −1 0 2 4 6 8 10 12 STATIC ESTIMATE DURING A TRANSIENT DUE TO SIMULTANEOUS NL Dynamic Sim 0.05 Static Estimate THROTTLE AND WASTEGATE COMMANDS. Static Estimate w/ Lead

e 0 P the response of Ptip to wastegate. The throttle, however, also sig- −0.05 nificantly influences Ptip. Initially, Ptip can actually respond very 0 2 4 6 8 10 12 quickly to a throttle command due to the abrupt change in flow 1 rate leaving the volume. Therefore, we need to consider estima-

wg 0 Physical parameters tor performance in the presence of a throttle input as well. The W scaled relative to initial system response to simultaneous inputs of throttleand wastegate equilbrium values −1 isshown in Figure7. Here we see no adverse effects dueto throt- 0 2 4 6 8 10 12 Time (s) tle and in fact there is a substantial reduction in estimation error with lead compensation. Figure 6. TRANSIENT PREDICTION OF Pe USING THE LEAD COM- These results suggest that this simple approach may provide PENSATED STATIC ESTIMATE. a practical strategy for estimating exhaust manifold pressure. Further analysis is required, however, to fully assess robustness filter that can be applied to the outputof (7). Since it is the delay to measurement noise and to determine if the filter coefficients in response of Ptip to wastegate that is problematic for the static must be scheduled with operating condition. estimate, we examine the transfer function from Wwg to Ptip. The bode plot of this SISO system is shown in Figure 5. As a starting point, we choose a lead filter to extend the bandwidth of this CONCLUSIONS AND FUTURE WORK system and then refine our design using nonlinear simulation of Model-based analysis has been used to assess the feasibil- the MIMO system. ity of estimating exhaust manifold pressure for a turbocharged Nonlinear simulation quickly shows that the separation of gasoline engine with an algorithm that is computationally effi- 1 the pole and zero of the lead filter is limited to approximately 2 cient and simple to calibrate. A traditional reduced order linear decade to prevent large overshoot and the associated increase in observer was developed to show that an estimator of this type is estimation error. With this constraint in mind, the zero location indeed feasible. The observer, however, has far too many calibra- of 2.0 and the pole location of 6.25 are subjectively chosen by tion parameters for practical implementation. Therefore, equilib- examining several nonlinear step responses. A small improve- rium analysis was used to find a parameterization of Pe through ment in response for a step in wastegate command is shown in the inputs and outputs of the system. A static estimator based on Figure 6. As expected withlead compensation, the effect of mea- Wexh, Ptip and ETC performed well in steady state but was found surement noise is amplified thus reducing the overall benefit of to be inadequate during transients. Lead compensation improved the compensator. transient performance despite increased sensitivity to measure- Our compensator design is based on the characteristics of ment noise. Further work will include a robustness and calibration as- sessment of lead compensation over the operating range of engine and if necessary, investigation of more sophisticated dynamic compensators for the static estimator. Finally, the results will be validated with engine dynamometer testing.

ACKNOWLEDGMENT The authorsgratefully acknowledge Jeff Cook from the Uni- versity of Michigan and Dave Hagner from Ford Motor Com- pany for many valuable discussions and suggestions.

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