Optimal Profile Tracking for an Electro-Hydraulic Variable ValveActuator using Trajectory Linearization

Huan Li, Guoming G. Zhu, Ying Huang, and Donghao Hao

Abstract— The camless valve is able to provide flexibleengine valve profiles (timing, duration, lift, etc.) to improvethe performance of internal combustion engines. To providea precise valve profile of an electro-hydraulic variable valveactuator (EHVVA) for the desired engine performance, anoptimal tracking controller for the valve rising duration andprofile is designed in this paper. A nonlinear model, elaboratingthe system pressure dynamics determining the valve risingduration, is developed and linearized along the desired risingvalve trajectory. Based on the trajectory linearization, a linearquadratic tracking (LQT) controller is designed with Kalmanoptimal state estimation. The equilibrium control resulted fromthe trajectory linearization is used as the LQT feedforwardcontrol. The control performance is compared with that ofbaseline controllers through both simulation study and benchtests. The transient and steady-state validation results confirmthe effectiveness of proposed control scheme.

I. INTRODUCTION

The increasing concerns of air pollution and energy con-cerns lead to electrification of the vehicle powertrain systemin recent years. For internal combustion engines (ICE), whichhave been the dominant vehicle power source for morethan a century, it becomes nececcary to employ advancedtechnologies to replace traditional mechanical systems withmechatronic systems to meet the ever-increasing demand ofcontinuous improving efficiency and reduced emissions.

The camless VVA (variable valve actuation) is one of thepromising technologies that is able to freely adjust the valveprofile (timing, duration and lift, etc.) and thus is able toprovide additional degree of freedoms to further improvethe engine combustion efficiency [1], [2]. Without the helpof camshaft for valve timing control, the VVA depends onactive electronic control of valve profile to guarantee thedesired engine performance. The valve profile tracking prob-lem includes four basic control objectives for most camlessVVAs: 1) valve timing (opening and closing) control foroptimizing combustion phase and preventing valve collisionwith the engine piston, 2) valve lift control, 3) profilearea (integration of valve displacement over time) controlfor accurate air charge and exhaust quantities, and 4) softseating control for reducing valve noise and improving valvedurability. Note that the overall valve duration control andthe valve transition response (rising/falling) control studied inliterature can be classified into valve timing and profile area

H. Li, Y. Huang and D. Hao are with the School of MechanicalEngineering, Beijing Institute of Technology, Beijing, China. This workwas completed during Li’s visit in MSU. {20081329, hy111,hdh}@bit.edu.cn

G. Zhu is with the Department of Mechanical Engineering, MichiganState University, East Lansing, MI 48824, USA zhug@egr.msu.edu

control, respectively. For the traditional cam-based enginevalves, above properties are automatically guaranteed byproper designing cam profile.

The profile tracking problem for camless VVAs has beenstudied intensively and the four control objectives discussedcan be achieved either individually or simultaniously, de-pending on a specific VVA system design and its application.Adaptive peak lift control was employed in [3] and [4] forrepeatable valve lift. Feedforward control is commonly usedfor valve timing control to compensate for the valve openingor closing delay [5], [6]. Soft seating control is a challengefor the electro-magnetic VVA due to the nonlinear magneticforce and was addressed in [7] with guaranteed valve re-sponse using extrmum seaking control, and reference [8]used the combination of feedforward LQR (linear quadraticregulator) and repetitive learning control to reduce cycle-to-cycle variations. These control designs were conducted forsingle or multiple control objectives and some other studiesdealt with the overall valve profile control as a single trackingproblem. For instance, in [9] the model reference controlcombined with repetitive control was used for the VVA toachieve asymptotic profile tracking performance. The slidingmode control was employed in [10] to achieve repeatabletracking performance with guaranteed seating velocity. Therobust repetitive control in [11] and time-varying internalmodel-based control in [12] were proved to be very effectiveto track the desired flexible valve profile both in steady stateand transient engine operations.

In this paper, the profile tracking problem is studiedfor an electro-hydraulic variable valve actuator (EHVVA)system (see [13]) that has dual-lift and build-in soft-landingfeature guaranteed mechanically. As a result, complicatedcontrol scheme is not required for lift and seating velocitycontrol. However, the valve timing and profile area controlare challenging for the studied EHVVA system due to thenonlinear and time-varying nature of the hydraulic system.These nonlinearities include nonlinear flow dynamics andtemperature-sensitive fluid viscosity; see previous publica-tions [14] and [15]. Therefore, this paper studies the profilearea tracking control that is not studied for this EHVVA.

An event-by-event optimal tracking control scheme isproposed for the valve rising duration and profile. A control-oriented valve profile model was constructed based on thesystem supply pressure affected by the fluid supply sys-tem and engine speed. Due to the nonlinear relationshipbetween the valve rising duration and supply pressure, thesystem dynamic model was linearized along the desired valverising trajectory and a discrete-time event-by-event model

2018 Annual American Control Conference (ACC)June 27–29, 2018. Wisconsin Center, Milwaukee, USA

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was developed accordingly. The equilibrium control inputobtained during the model linearization process was usedfor feedforward control and a finite horizon linear quadratictracking (LQT) controller was designed for tracking controlin a closed-loop based on the linearized event-by-eventmodel.

The remaining of the paper is organized as follows.Section II provides the overview and control problem formu-lation for the target EHVVA system. Section III presents thecontrol-oriented model, model linearization along the desiredtrajectory, and the discrete-time event-by-event model. Thefinite horizon LQT controller is designed based on thedecrete-time event-by-event model in Section IV. The simu-lation study is presented in Section V and the experimentalvalidation results are presented in Section VI. Conclusionsare drawn in Section VII.

II. SYSTEM OVERVIEW AND CONTROL OBJECTIVE

A. EHVVA System

Fig. 1 shows the schematic of the studied EHVVA systemwith two discrete lifts. The engine valve is driven by thesupply pressure Ph when the valve actuator is activated andclosed by the return spring force when the valve actuator isdeactivated. The dual-lift (high or low lift) is realized by asolenoid lift valve by opening or closing the lift porter tochange the lift control sleeve position. The system supplypressure Ph is regulated by a hydraulic pump driven usinga DC motor. A predetermined back pressure Pl is set higherthan the fluid tank pressure but lower than Ph to make the liftcontrol sleeve rest steadily on its lower and upper positions,respectively, to guarantee accurate lift control. Precise build-in structure designs including notches, undercuts and orificesinside the actuation cylinder are used to achieve soft landing.The details of the EHVVA system can be found in [14].

B. Control Problem Description

Fig. 2 shows the test data under different supply pressurePh, and illustrates the control objectives of the EHVVA

Lift valve

High

Pressure

Low

pressure

Pl

Return

Spring

Engine

Valve

Actuation

piston

L2 L1

Lift

Control

Sleeve

Valve actuator

Ph

Actuation

cylinder

Motor & PumpM

Fig. 1. EHVVA schematic

Fig. 2. Control problem analysis

system as discussed in the introduction section. With me-chanically guaranteed lift control, soft-seating control, andpreviously addressed valve timing control [15], the remainingtask for the EHVVA system is the tracking control of valveprofile area, which is dominated by the valve rising andfalling durations (defined as the valve transition time between10% and 90% lift), where the dwell and peak lift are deter-mined by valve timing (opening and closing) and lift valve,respectively. From Fig. 2, the falling profile and duration(tvfd) are almost fixed due to the fixed back pressure andreturn spring stiffness, while the rising profile and duration(tvrd) vary significantly with the supply pressure caused byvarying engine speed; see discussions in the Sec. III A.The relationship between the supply pressure Ph and valverising duration tvrd has been studied using bench test dataand a curve-fitted model is developed using a second-orderpolynomial over the typical operational range; see Fig. 3. Itcan be seen from both Figs. 2 and 3 that it becomes verydifficult to increase the valve rising response (i.e. reducethe rising duration) within higher supply pressure region.Therefore, since the valve profile is dominated by the risingduration (supply pressure), a tracking controller can be usedto precisely regulate the supply pressure to achieve desiredvalve rising duration/profile.

Fig. 3. Correlation between supply pressure and valve rising duration

III. SYSTEM MODELLING

A. System Dynamics

The system supply pressure is determined by the systeminlet flow rate regulated by the DC motor (see Fig. 1)using an analog voltage reference signal Vm (0∼5V ) andthe system outlet flow rate is affected by the periodic valve

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event as a function of engine speed Ne due to the pulse-flowloss. The system supply pressure can be modelled as follows.

Ph = k1Ne1

1 + τ1s+ k2Vm

1

1 + τ2s+ c0 (1)

where k1, k2, and c0 are calibration coefficients; τ1 and τ2are the time constants for the outlet and inlet flow rates,respectively. The model is calibrated by comparing the testdata and the model results in Simulink.

Let x1 = k1Ne1

1+τ1sand x2 = Ph − c0 denote the

pressure component determined by engine speed and thesupply pressure with an offset c0, respectively. Then the2nd order polynomial in Fig. 3 can be expressed as y =ax22 + bx2 + c, where y denotes the valve rising durationtvrd; and a, b, and c are identified calibration coefficients.This leads to a 2nd order nonlinear model for the system.

x1 = − 1

τ1x1 +

k1τ1d

x2 = (1

τ2− 1

τ1)x1 −

1

τ2x2 +

k2τ2u+

k1τ1d

y = ax22 + bx2 + c

(2)

where x = [x1, x2]T represents the system state vector andy is the output; d represents the engine speed Ne consideredas an exogenous input to the system; and u represents thecontrol input Vm.

B. Trajectory Linearization

It can be seen that (2) is a nonlinear model due tothe exogenous input d and the nonlinear output equationy. A linearized model is needed for the optimal linearquadratic tracking control design. At each operation pointalong the desired tracking trajectory, the system model canbe linearized using the classical Jacobian linearization, i.e.,model (2) can be transformed into the linearized form:{

∆x = A∆x+Bu∆u+Bd∆d

∆y = C∆x(3)

where ∆x = x − x0, ∆y = y − y0, ∆u = u − u0, and∆d = d − d0 are the variation variables and the systemcoefficient matrices are determined as follows.

A =∂f

∂x

∣∣∣∣(x0,u0,d0)

Bu =∂f

∂u

∣∣∣∣(x0,u0,d0)

Bd =∂f

∂d

∣∣∣∣(x0,u0,d0)

C =∂g

∂x

∣∣∣∣(x0,u0,d0)

Note that (x0, y0, u0, d0) denotes the equilibrium point thatcan be solved by (2) at the steady state (x1 = 0, x2 = 0)along the given tracking trajectory r over the valve rising du-ration (i.e., y = y0 = r) and given the measurable exogenousinput d (i.e., d = d0, ∆d = 0). The determined equilibriumpoint will be used to calculate the system coefficient matricesand the solved equilibrium point control u0 will be used asa nominal (feedforward) control. A closed-loop controllerfor the variation control input ∆u need to be designed tocompensate for the deviation from the desired trajectory with

the presence of system noise and disturbance. Therefore, thecontrol input for the system can be written as

u = ∆u+ u0 (4)

C. Discrete-Time Event-by-Event Model

The linearized continuous-time model (3) with ∆d = 0can be discretized using the forward Euler approximation.Considering the system input noise w and output mea-surement noise v, the discrete-time event-by-event model isobtained as:{

∆x(k + 1) = A(k)∆x(k) +B(k)∆u(k) + w(k)

∆y(k) = C(k)∆x(k) + v(k)(5)

where,

A(k) =

1− Ts

τ10

( 1τ2− 1

τ1)Ts 1− Ts

τ2

, B(k) =

0

Tsk4τ2

,C(k) =

[0 2ax20 + b

],

and Ts is the valve event period. Note that the input noise wand measurement noise v are assumed to be zero mean andindependent random vectors such that

E {w(k)} = 0, W = E{w(k)wT (k)

}> 0

E {v(k)} = 0, V = E{v(k)vT (k)

}> 0

(6)

where W and V are corresponding covariance matrices.Table I shows the identified parameters for the event-by-eventmodel.

TABLE IMODEL PARAMETERS

k1 k2 τ1 τ2 c0 a b c

-1.2× 10−3 2.05 0.19 0.5 -3.49 0.18 -4.7 32.41

IV. LINEAR QUADRATIC TRACKING CONTROL

In this section, a finite horizon LQT controller is designedto make the system output y(k) track the reference r(k).More specifically, since the control design will be basedon the variation model (5) to provide a variation controlinput ∆u(k) at each operation point, the control target is tomake the variation output ∆y(k) track the variation reference∆r(k). Since the state ∆x1 used for state-feedback cannot bemeasured, a Kalman filter is used as optimal states observerwith the presence of system noise.

A. Finite Horizon LQT Control

The control objective of the finite horizon LQT controlis to minimize the tracking error e(k) defined in (7) withthe feasible control effort Vm along a predefined trackingtrajectory. Note that under transient engine operations, thetracking reference of the valve profile is often predefined ordetermined for a few engine cycles to achieve desired perfor-mance, e.g. a fast and smooth valve time and lift transitionis required for the SI-HCCI (spark ignition-homogeneous

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charge compression ignition) combustion mode transitioncontrol [16]. Tracking error e(k) is defined as

e(k) = ∆y(k)−∆r(k) = C(k)∆x(k)−∆r(k) (7)

To simplify notations, ∆xk, ∆yk, ∆uk, ∆rk, ek, Ak, Bkand Ck are used to denote ∆x(k), ∆y(k), ∆u(k), ∆r(k),e(k), A(k), B(k), and C(k) at current time step k in therest of the paper. The performance cost function of the LQTcontroller at each control step (valve event) is defined as

J(k) =1

2eTkfFekf +

1

2

∑kf−1

k=k0[eTkQek + ∆uTkR∆uk] (8)

where F = FT ≥ 0, Q = QT ≥ 0, and R = RT > 0are given weighing matrices. k0 ∼ kf is a finite horizonmoving window at each control step k for predefined N -step(N = kf−k0) tracking reference. That is, an N -step optimalcontroller is designed at each control step for the given N -step tracking trajectory and only the first control step will beused, which is the so-called moving optimization problem inMPC (model predictive control). Since the LQT controlleris designed based on the variation model (5) linearized atthe equilibrium point (current operation point) k = k0, thetracking reference for the variation output ∆yk is

∆rk = r(k)− r(k0), k = k0, ..., kf .

The optimal tracking problem can be solved by followingthe minimum principle approach in [17]. The variationcontrol ∆uk can be obtained as

∆uk = −∆uFBk + ∆uFFrk = −LFBk∆xk + LFFkgk+1

(9)LFBk = [R+BTk Kk+1Bk]−1BTk Kk+1Ak

LFFk = [R+BTk Kk+1Bk]−1BTk(10)

where ∆uFB and ∆uFFr are the state feedback and ref-erence feedforward control, respectively. The matrix K inthe control gains LFBk and LFFr and the vector g canbe obtained by backwards solving the Riccati equation (11)offline and vector equation (12) online using the boundaryconditions in (13), respectively.

Kk = ATkKk+1[I+BkR−1BTk Kk+1]−1Ak+CTk QCk (11)

gk =ATk {I −Kk+1[I +BkR−1BTk Kk+1]−1

BkR−1BTk }gk+1 + CTk Q∆rk

(12)

Kkf = CTk FCk, gkf = CTk F∆rk (13)

Note that in (9) the system state ∆x1k in ∆xk used forfeedback control cannot be measured and is estimated onlineusing the Kalman filter addressed in the next subsection.

B. Kalman State Estimation

The Kalman state estimation is a stochastic filter thatminimizes the covariance of the estimation error to providean optimal state estimation subject to Gaussian noise inputs.For a given initial state ∆x0, it uses the current outputmeasurement ∆yk and control ∆uk to estimate the next stepstate ∆xk+1 in the following form.

∆xk+1 = Ak∆xk +BK∆uk +Hk(∆yk − Ck∆xk) (14)

Note that the initial state is calculated as follows

∆x0 = xk−1 − xk,0 (15)

The subscript ”k, 0” denotes the equilibrium point at valveevent k. It shows that the initial state is updated oncethe equilibrium point xk,0 is switched, i.e., a new trackingreference rk or exogenous input dk is observed. The Kalmanfilter gain Hk is obtained as follows

Hk = AkΣkCTk (CkΣkC

Tk + V )−1 (16)

where the estimation error covariance matrix Σk is calculatedrecursively by the following difference Riccati equation usingthe initial condition Σ0 = E(∆x0∆xT0 ).

Σk+1 = AkΣkATk +W −HkCkΣkA

Tk (17)

Using the estimated state vector in (9) and substituting it into(4), the system control input can be obtained as

uk = ∆uk + uk,0 = −LFBk∆xk +LFFkgk+1 + uk,0 (18)

V. SIMULATION VALIDATION

The discrete-time event-by-event model and the LQTcontroller based on the trajectory linearization (LQTFF) areimplemented in Simulink and validated through simulationstudy. For the purpose of comparison, the open-loop controlthat only uses the equilibrium point control u0 as feed-forward (FF) control, and the PID (proportional-integral-derivative) control with and without the feedforward u0 (PIDand PIDFF) are also implemented. Fig.4 illustrates the fourcontrol schemes.

Trajectory

LinearizationPlant

PID

(a) Open-loop Feedforward (FF) (b) PID

Plant

Plant

PID

(c) PID with Feedforward (PIDFF)

Trajectory

Linearization

PlantLQT

(d) LQT with Feedforward (LQTFF)

Trajectory

Linearization

Kalman Filter

Fig. 4. Implemented control schemes

TABLE IICONTROL PARAMETERS

Scheme u0 kp kil kih kd F Q R N

FF√

- - - - - - - -

PID(SmG) - 18.4 6 6 3 - - - -

PID(LaG) - 18.4 12 12 3 - - - -

PID(ScG) - 18.4 6 12 3 - - - -

PIDFF(ScG)√

10 2 3 0.9 - - - -

LQTFF√

- - - - 103 100 1 6

The control parameters, the weighting matrices (F , Q,and R) for the LQT controller and the control gains forthe PID controllers, are tuned in simulations based upon

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the transient tracking performance and system stability. ThePID controller are tuned with fixed small ki gain (SmG),fixed large ki gain (LaG), and ki gain scheduled (ScG) withsupply pressure to show the effect of system nonlinearity.The control parameters for each control scheme are listedin Table II. Note that kil and kih denote the scheduled kigain at the lower pressure bound (6 MPa) and upper pressurebound (9 MPa) of the fluid supply system, respectively.

Fig. 5. Simulation results of step reference tracking at 1000 rpm

Fig. 5 shows the simulation results of step reference tack-ing for the valve rising duration at 1000 rpm engine speed.The tracking performances, corresponding supply pressureresponses and control inputs u for six control schemes arepresented in the upper three plots of Fig. 5, respectively. Thecontrol input components (u0, ∆uFB , ∆uFFr) of the LQTcontroller are analyzed in the bottom plot. It can be seenfrom the first plot that the open-loop control with equilibriumpoint feedforward (FF) can have good transient response andsteady-state accuracy if the model (2) used for u0 calculationis accurate, otherwise steady-state error may exist. Comparedwith the FF scheme, the PID control with fixed small gain(SmG) has similar response performance in low pressureregion while poor performance in high pressure region. Usinglarge gain (LaG) can increase the transient performancewith large overshoot in low pressure region. This trade-offrelationship coincides with the system nonlinearity (recallFig. 3) that it consumes more control effort in the highpressure region to achieve the same rising duration incrementthan that consumed in the low pressure region. Therefore,control gain scheduled with supply pressure is necessaryfor the PID controller to achieve good transient responsein both low and high pressure regions as shown by the

PID control scheme with scheduled gain (ScG). Combiningthe equilibrium point feedforward and gain scheduling PIDcontrol, the PIDFF(ScG) scheme leads improved transientperformance. The supply pressure and control input re-sponses are consistent with the tacking performance for eachcontrol scheme. It can be noticed that both the open-loop andPID control schemes result a delayed response to the trackingreference signal. The LQT controller (LQTFF), however,can track the desired trajectory very closely with minimumtransient tracking error without tracking delay mainly dueto the referene feedfoward control ∆uFFr obtained fromthe predefined upcoming N -step tracking references anddominates the control when the reference changes.

VI. EXPERIMENTAL VALIDATION

The experimental validation was conducted on theEHVVA prototype [14]. Fig. 6 illustrates the test resultsof the step reference tracking for the valve rising durationat 1000 rpm, and the results agree with the simulationresults well with several exceptions. First, small steady-statetracking error appears in the open-loop feedforward (FF)control at low pressure region due to the modeling errorand is eliminated by the closed-loop controllers. Second,the responses of PID control schemes (SmG, LaG, andScG) are very slow, compared with the FF control schememainly due to the presented system noise limited PID gainsfor closed-loop stability. The control performance can beimproved using the feedforward PID control (PIDFF(ScG)).

Fig. 6. Experiment results of step reference tracking at 1000 rpm

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With the help of the Kalman state estimation, the LQTcontroller (LQTFF) can track the transient trajectory in 5events with minimum transient tracking error among allcontrol schemes. The transient response is 4 events fasterthan that of the PIDFF controller which is the best amongthe other controllers. The bottom plot in Fig. 6 showsthe Kalman state estimation results. Note that the state∆x1/x1 is not measurable and it remains at zero (x1 keepsunchanged) because x1 is excited by the engine speed whichis unchanged under this condition.

The steady-state tracking performance (constant r) for thevalve rising duration is evaluated for 200 engine cycles at1000 rpm and the statistic distribution of the tracking erroris illustrated in Fig. 7. Since it is very important to regulatethe valve rising duration within a small crank angel (CA)deviation from the target to guarantee repeatable air chargeand exhaust quantity for each engine cycle, the trackingerror in time domain is converted into the crank angledomain at engine speed of 5000 rpm to evaluate the worst-case tracking performance. The probabilities of deviationdistribution within ±1◦CA for the FF, PID, PIDFF, andLQTFF control schemes are 44%, 58.5%, 65.5%, and 74.5%,respectively.

Fig. 7. Histogram of steady-state tracking deviation in 200 engine cycles

VII. CONCLUSIONS

In this paper, an LQT (linear quadratic tracking) con-troller with Kalman state estimation is designed based ontrajectory linearization for a developed nonlinear modelof the EHVVA (electro-hydraulic variable valve actuator)system. The equilibrium point control obtained from thetrajectory linearization is used as the feedforward for thedesigned LQT controller. Both simulation and experimentalvalidations are conducted and the results match well. Inthe experimental study, the proposed LQT controller showssignificant improvement under both transient and steady-stateoperations, compared with the open-loop feedforward, PID,and feedforward PID controllers. The LQT controller is ableto track the transient trajectory of the valve rising durationin 5 valve events with minimal tracking error, which is 4events faster than the feedforward PID controller (the bestcontroller among the studied controllers). The steady-statetracking tests show 30%, 16%, and 9% improvements for the

±1◦CA tracking error distribution, compared with the open-loop feedforward, PID, and feedforward PID controllers,respectively. Future work is to study and validate the valveprofile tracking control under transient engine speed.

ACKNOWLEDGMENT

The authors would like to thank the China ScholarshipCouncil (CSC) for supporting Huan Li’s study at MichiganState University as a joint PhD student.

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