LPV Decoupling and Input Shaping for Control of Diesel Engines

American Control Conference Marriott Waterfront, Baltimore, MD, USA June -July, WeB9.6 LPV Decoupling and Input Shaping for Control of Diesel Engines Javad Mohammadpour, Karolos Grigoriadis, Matthew Franchek, Yue-Yun Wang, and Ibrahim Haskara Abstract The paper presents the results of application of linear parameter varying LPV decoupling control and a prefilter to improve the tracking performance in the air path of Diesel engines modeled as a quasi-lpv system. The proposed decoupling method benefits the multi-variable control of multiinput multi-output MIMO systems with variable operating conditions, variable parameters and nonlinear behavior. The results of this paper illustrate the reduced variability and performance enhancement of the two inputs EGR valve effective area and VGT effective area and two outputs and mass air flow dynamic system of the air path of Diesel engines, where there is a significant coupling in the system dynamics. The proposed design method combines a prefilter used to shape the input with the LPV feedback control based on an LPV decoupling method proposed here to achieve the tracking with desired transient performance specifications. The prefilter is designed based on the closed-loop dynamics resulting from the LPV design, and a systematic input shaping prefilter design process is developed. The designed prefilter successfully extends the closed-loop bandwidth. Simulation results demonstrate the effectiveness of the input shaping prefilter. Moreover, the designed prefilter is structurally simple and computationally efficient. I. INTRODUCTION Modern diesel engines equipped with exhaust gas recirculation EGR system and variable geometry turbochargers VGT introduce inherent feedback loops from the exhaust manifold to the intake manifold, which due to the highly nonlinear dynamic behavior of the system results in a multivariable nonlinear control problem. Among the first control methodologies applied to the multivariable system of diesel engines were robust control [, minimum-time optimal control [, Lyapunov-based nonlinear control [, [, and gain-scheduled PI control with directional compensation [8. A performance comparison between various control design methods applied to a hardware setup was reported in [. Next series of work in this area focused on capturing the system dynamics in a linear parameter varying LPV framework, where based on a quasi-lpv model of the air path system of diesel engine, Jung [ designed and experimentally validated a gain scheduled robust controller using an LMI-based formulation associated with the H control design problem. More recently, Wei and del Re [ designed gain-scheduled controllers based on the LPV models they obtained using black-box system identification. The advantage of the approaches proposed in [ and [ is the reduced calibration effort and online tuning at the This work was supported by General Motors Company, Warren, MI. Department of Mechanical Engineering, University of Houston, Houston, TX 77. Powertrain Systems Research Lab, General Motors, Warren, MI 89. Corresponding author, email: jmohammadpour@uh.edu. expense of higher computational load due to the complex structure of the control law. An attractive feature of the PID controllers is their simple structure; however, they are unable to cope with the nonlinearities and hence have to be gain scheduled extensively. The present paper proposes a method that enjoys the advantages of both simplicity of a PI controller and adaptation of an LPV controller. The authors previously developed an intuitive method for linear parameter varying LPV decoupling control [6. The proposed approach sought to benefit the multi-variable control of multi-input multi-output MIMO systems with variable operating conditions, variable parameters or nonlinear behavior. The method was shown to improve the closedloop performance and reduce the variability of MIMO systems with significant coupling in the system dynamics. The MIMO decoupled feedback LPV controllers were designed to reduce the coupling effects. In particular, the method used a parameter-dependent static and dynamic inversion and SVD decomposition of the system transfer function to minimize the effects of the off-diagonal terms in the MIMO system transfer function matrix. The parameter-dependent decoupling matrices were selected along with the appropriate LPV controller design to guarantee the closed-loop performance specifications. In this paper, the simulation results will be shown to illustrate that the proposed LPV decoupling control design method provides significant advantages for control of Diesel engines. For this purpose, the coupling issue in the air handling system including two inputs and two outputs subject to nonlinear behavior and variability in the operating conditions with different engine speed and fueling will be addressed. In particular, the proposed decoupled LPV control method results in: i a systematic methodology for on-line adaptation of the decoupling matrices to guarantee a desired decoupling in the presence of, engine speed and fuel variability, and ii a combined LPV control design that uses the decoupled system to guarantee improved closed-loop performance. In the proposed design method, the overall LPV controller, i.e., the LPV decoupling transformation matrix combined with the controller, is scheduled based on and engine speed to accommodate the operating conditions variability. Based on the simulation results presented in this paper, it is observed that the proposed LPV design method works well for the full range of engine operation and reduces the variability of the closed-loop system significantly. Moreover, the designed LPV controller provides robustness against the variation of engine operating conditions, nonlinearities, and model uncertainties. However, the 978---7-7//$6. AACC 77

commanded and air mass flow cannot be tracked satisfactorily during transient engine operations due to the bandwidth limitation dictated by the feedback control. Therefore, a second degree of freedom, i.e., an input shaping prefilter, is employed here based on the closed-loop dynamics to achieve the desired tracking performance. The systematic input shaping prefilter design process previously developed by the authors in [ is used in this paper to demonstrate that the designed prefilter successfully extends the closed-loop tracking bandwidth for the controlled air handling system. It will be seen that the input shaping does not affect the stability of the closed-loop system; rather it simply modifies the input signal. The simulation results for the twoinput two-output model of the Diesel engine air handling system demonstrate the improved tracking performance using the input shaping prefilter. II. LPV MODELING OF AIR PATH IN DIESEL ENGINES In this section, we review a simplified model of the air handling system in Diesel engines represented by a third order nonlinear differential equation and show how to reformulate the system dynamics in the quasi-lpv form. In this paper, we present a quasi-lpv representation of the air handling system that is appropriate for the design purposes. In the simplified nonlinear model of the air path system, the manifold dynamics is described only by differentiating the ideal gas law resulting in one differential equation for each one of the intake and the exhaust manifold pressure. The turbocharger dynamics is also approximated by the power transfer with a known time constant. The resulting model is [ ṗ i = RT i V i W ci + W xi W ie ṗ x = RT x W ie + W f W xi W xt = τ + η m P t where the mass flows and the turbine power are represented by the following equations. W ci = η c c p T a pi µ p a W xi = A rζ egr p x RTx N V d W ie = η v RT i 6 W xt = A vgtζ vgt RTx P t = W xt c p T x η t p a p i pa p a µ The above simplified third-order model was used by Jung [ to derive a quasi-lpv model. It was shown [, [ that the simplified model results in higher steady-state offset for open-loop simulation compared to the full-order model corresponding to the air handling system. It is not however a concern since a well-designed feedback controller should be able to take care of the offset. The simplified model is also able to capture the non-minimum phase behavior of the transfer function from VGT area to air flow. In the above equations, N, T, W ij, V, W f, p, A r, A vgt, ζ egr, ζ vgt, P, γ, c p, c v, R, η, η v and η m represent engine speed in rpm, temperature in K, mass flow rate from i to j in kg/sec, manifold volume in m, fuel rate injected to cylinder in kg/sec, manifold pressure in kpa, EGR valve effective area in m, VGT nozzle effective area in m, EGR valve opening percentage, VGT nozzle opening percentage, engine power in W, specific heat capacity ratio, specific heat capacity at constant pressure in J/kg/K, specific heat capacity at constant volume in J/kg/K, universal gas constant in J/kg/K, isentropic efficiency, volumetric efficiency, and turbocharger mechanical efficiency, respectively. We also refer to [ for the values of the engine parameters. For simplicity of the equations, the case where the flow becomes choked is not considered here. Combining the equations of the simplified nonlinear model and the flow equations above, the following nonlinear differential equations are obtained. ṗ i = RT i V i + RT i V i η c c p T a pi p x RTx ṗ x = η v T x T i V d N + RT x RT x RT x V d µ η v p a V i N p x RTx p x p a Tx P c = τ + η mη t c p T x τ p a p a p i A r W f p a A vgt A r A vgt pa µ px Tx The above equations can be put in the quasi-lpv form using the LPV parameter vector ρ = [,,. Reformulating the above nonlinear system into the quasi-lpv form becomes possible when the information from the variables in the vector ρ are available in real-time. The equations of the quasi-lpv representation of the nonlinear system read ṗ i ṗ x = Aρt Ar + Bρt[ A P vgt c 78

where Aρ = Bρ = a ρ a q ρ b ρ b ρ ρ c a q ρ,ρ ρ b q ρ,ρ ρ b q ρ ρ c c q ρ q ρ ρ where a i, b j and c k are constant values, and q = q = q = ρ µ p a ρ ρ q = ρ p a ρ p a ρ ρ µ pa are dependent on the LPV parameter vector ρ. In the statespace representation, A r and A vgt are the manipulated variables for the quasi-lpv model, and N and W f are external inputs that can be measured but not manipulated. It is noted that no approximation is involved in deriving the quasi-lpv model from the original nonlinear model. Nonuniqueness of the quasi-lpv representation for a nonlinear system can make the design conservative. However, we have previously shown [ that the developed quasi-lpv model has sufficient accuracy to be employed for control design purposes. Considering the and mass air flow as the outputs of interest, we have the following as the output equation: [ Wci = Cρt where Cρt = ρ [ d q ρ. III. CLOSED-LOOP CONTROL DESIGN PROCEDURE The proposed LPV decoupling control design method for the air handling system demonstrates significant advantages in terms of dealing with the coupling issue in the system of two inputs and two outputs represented in the previous section. The air handling system is naturally subject to nonlinear behavior and variability in the operating conditions with different engine speed and fueling. Analytical framework of the LPV decoupling method is described in [6, and simulation results presented in [6 use SVD-based and inversion-based LPV decoupling. It is noted that the simulation results shown in [6 are based on a parameter-dependent static decoupling, whereas the results of the present paper compare the static and dynamic decoupling. Figure shows the block diagram of the closed-loop system formed by augmenting the air handling system including two inputs EGR valve effective area and VGT nozzle area and two outputs Fig.. Block diagram of the closed-loonterconnection of the air handling system, LPV decoupling matrix and the controller Engine speed rpm 8 6 6 8 Injected fuel kg/h 6 8 Fig.. Profiles of engine speed N and the injected fuel W f and mass air flow, the parameter-dependent decoupling matrix Ws,ρ, and a diagonal PI controller, whose design is performed based on the performance specifications. It is noted that the notation Ws,ρ represents the compensator transfer function matrix associated with the frozen LPV parameter vector ρ. A. Simulation Results Presented in this section are the results of application of the developed LPV decoupling control strategy to control the air handling system. The control objective is to achieve an acceptable tracking performance with a low overshoot and settling time. The designed controller is expected to be able to adapt to different operating conditions, i.e.,, engine speed and fuel. In this section, we first provide a comparison between the closed-loop performance achieved from the fixed decoupling and that achieved from the proposed LPV decoupling method using different combination of the LPV parameters including, engine speed and injected fuel. The objective of such a comparison is to show that for the highly nonlinear air handling system, the standard decoupling methods would fail, whereas the proposed decoupling method combined with the standard control design is able to provide a satisfactory closed-loop performance. Next, we show the closed-loop simulation results achieved using a dynamic LPV decoupling. For the simulation purposes, we consider that the engine speed and fuel are measurable in real-time as depicted in Figure. We also take into account limits on the control outputs, i.e., EGR 79

Fresh air flow kg/h 8 6 6 8 Intake manifold pressure kpa output 6 8 EGR valve effective area mm VGT area mm 6 7 8 9 6 7 8 9 Fig.. Profiles of the inputs and the closed-loop system outputs fresh air flow and for the static decoupling case valve effective area and the VGT area, to be within a desired range. First stes to obtain an explicit characterization of a parameter-varying matrix resulting in the decoupling in the system transfer function matrix. The decoupling matrix has been calculated as Ws,ρ = [ β s a ρ ρ q a q ρ + s k s c β q q q ρ + s k where β and β are constant parameters depending only on the fixed parameters of the system representation, and k and k are the tuning parameters. We first apply a static LPV decoupling matrix calculated by setting s = in. The results of using this decoupling matrix along with two simple PI controllers tuned using the standard methods in the literature are shown in Figures and. The figures illustrate the tracking performance of the closed-loop system using static decoupling and the control inputs required to achieve the tracking, respectively. It is noted that, in this case, we have scheduled the decoupling matrix based on a measurement of engine speed, fuel and. Scheduling the decoupling matrix based on all elements of the LPV parameter vector ρ is not desirable. Therefore, we investigated the effect of the different LPV parameters on the closed-loop system performance. To this purpose, we compared three scenarios in adapting the decoupling matrix W as follows: i the decoupling matrix is, i.e., it is not adapted in real-time. ii the decoupling matrix is scheduled only based on the engine speed and fuel as shown in Figure, while the intake and exhaust manifold pressures are considered fixed. iii the decoupling matrix W is scheduled only based on, while the other components of the LPV parameter vector ρ are considered fixed. The simulation results considering the above cases are demonstrated in Figures and 6. The results suggest that Fig.. Profiles of the EGR valve actuator input and the VGT actuator signal for the static decoupling case the decoupling based on the has a significant impact on the closed-loop tracking performance. This can be clearly seen in Figure at t = sec, where the change in the command input is compensated quickly in the fresh air flow output when the information from boost pressure is used in the decoupling matrix. Surprisingly, there is not much difference between the closed-loop performance achieved from the case where a fixed decoupling matrix is used and the case where real-time information from both fuel and engine speed is also used. Due to the weak transient performance shown by the large spikes achieved using a static decoupling, we will next apply a dynamic LPV decoupling. Illustrated in Figure 7 is the tracking performance of the closed-loop system using the decoupling matrix adapted based on only. Figure 8 shows the control effort actuator signals required to track the desired amount of air to flow to the cylinders as well as the intake manifold pressure. It is noted that scheduling of the decoupling matrix is done only based on for the illustrated simulation results of the next section. IV. DESIGN OF INPUT SHAPING FILTER In this section, we show how to extend the closed-loop bandwidth using the input shaping method presented in [. Using the dynamic decoupling transformation in and the open-loop transfer function Gs, the decoupled open-loop transfer function matrix becomes G d s = GsWs = [ + s k a s d s a ρ s+k + s k. As observed from the coefficients of the above transfer function matrix, the decoupled open-loop system only depends on engine speed N which implies that considering the quasi-lpv representation, if an adaptive strategy for the controller gains is sought, it will be enough to schedule them only based on the engine speed. This would suffice to capture the variability of the system operating conditions. To close the loop, we used the same PI controllers of the fixed 8

Fresh air flow kg/h 8 6 EGR valve effective area mm Intake manifold pressure kpa 6 8 6 8 Fig.. Profiles of the inputs and the closed-loop system outputs: fresh air flow top plot and bottom plot for the static decoupling case VGT area mm 6 8 9 8 7 6 6 8 Fig. 6. Profiles of the EGR valve actuator input and the VGT actuator signal for the static decoupling case gains designed in the previous section for individual loops. The frequency responses obtained illustrate that closed-loop bandwidth of the first input channel is less than.hz, which can deteriorate the transient performance. Next, we design a prefilter to shape the inputs and improve the transient response of the outputs of interest, i.e., boost pressure and mass air flow. Using the input shaping method proposed in [ and considering a bandwidth approximately equivalent to Hz for the first loop, the following prefilter is designed: s + P f s = diag[.86s +.9s +.8s +,.s +..s +.s. +.s +. We shaped the second input channel to change the bandwidth associated with this channel. Shaping the input commands using the designed prefilter results in a significant increase in the closed-loop bandwidth. Figure 9 shows the tracking performance of the twodegree-of-freedom closed-loop feedback configuration. Both and mass air flow present an improvement as far as the transient response. It is readily observed from the plots that the settling time has significantly decreased due to the increased bandwidth of the closed-loop system. Illustrated in Figure are the control input signals, i.e., EGR valve effective area and VGT nozzle area. The latter plot shows a decrease in the energy of the input signals compared to the case where no prefilter is used. Also, the large spikes clearly seen in Figure 8 have been removed in Figure that demonstrates the improved transient performance obtained from taking advantage of a two-degree-of-freedom design. V. CONCLUSION In this paper, we illustrate closed-loop tracking performance enhancement in the air handling system of Diesel engines achieved using an input shaping method combined with a parameter-dependent decoupling design method. The proposed design method addresses nonlinearities and changes in the operating conditions, and the input shaping method is used to improve the closed-loop tracking performance during engine transient operations based on the closedloop dynamics resulting from the LPV design. The LPV control design method for the system uses a two-step procedure, where first a parameter-dependent transformation is employed and adapted in real-time to reduce the coupling between the undesired set of inputs and outputs, and then for the decoupled system, SISO controllers are designed using standard methods in the literature. Simulation results show the significant improvement achieved by using the prefilter that results in an extension in the tracking bandwidth without affecting the stability of the closed-loop system. 8

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