profile tracking for an electro-hydraulic variable valve ... · li et al.: profile tracking for an...

338 IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 24, NO. 1, FEBRUARY 2019

Profile Tracking for an Electro-Hydraulic VariableValve Actuator Using Receding Horizon LQT

Huan Li, Ying Huang, Guoming G. Zhu , and Zheng D. Lou

Abstract—The camless valve system is able to provideflexible engine-valve profiles (timing, duration, lift, etc.) tooptimize the performance of internal combustion engines.To provide a precise valve profile of an electro-hydraulicvariable valve actuator and achieve the desired engine per-formance, an optimal tracking controller for the valve-risingduration, a key valve profile parameter, is presented inthis paper. An event-by-event nonlinear model, connectingthe system supply pressure dynamics to the valve-risingduration, is developed and linearized along the desiredvalve-rising trajectory. Based on the trajectory lineariza-tion, a receding horizon linear–quadratic tracking (LQT)controller is designed along with a Kalman optimal stateestimation. The equilibrium control resulted from the modellinearization is used as the LQT feedforward control. Thecontrol performance is compared with that of baselinecontrollers through both simulation study and bench tests.The transient and steady-state validation results confirmthe effectiveness of proposed control scheme.

Index Terms—Linear–quadratic tracking (LQT), optimal,trajectory linearization, valve profile, variable valve actuator(VVA).

I. INTRODUCTION

ON THE one hand, the increasing concerns of air pollu-tion and energy usage has led to the electrification of the

vehicle powertrain system in recent years. On the other hand,internal combustion engines have been the dominant vehiclepower source for more than a century, and it is necessary to em-ploy advanced technologies to replace traditional mechanicalsystems with mechatronic systems to meet the ever-increasingdemand of continuous improving engine efficiency with reducedemissions.

The camless variable valve actuation (VVA) is one of thepromising technologies, which is able to freely adjust the valveprofile (timing, duration, lift, etc.) and, thus, is able to pro-

Manuscript received December 25, 2017; revised April 24, 2018; ac-cepted December 30, 2018. Date of publication January 9, 2019; date ofcurrent version February 14, 2019. Recommended by Technical EditorY. Shi. (Corresponding author: Guoming G. Zhu.)

H. Li and Y. Huang are with the School of Mechanical Engineer-ing, Beijing Institute of Technology, Beijing 100081, China (e-mail:,[email protected]; [email protected]).

G. G. Zhu is with the Department of Mechanical Engineering, Michi-gan State University, East Lansing, MI 48824 USA (e-mail:, [email protected]).

Z. D. Lou is with the LGD Technology, LLC., Plymouth, MI 48170, USA(e-mail:,[email protected]).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMECH.2019.2892038

vide additional degree of freedoms to further improve the en-gine combustion efficiency [1]. Without the help of camshaftfor valve-timing control, the VVA depends on active electroniccontrol of valve profile to guarantee the desired engine perfor-mance. The valve-profile tracking problem includes four basiccontrol objectives for most camless VVAs:

1) valve-timing (opening and closing) control for optimizingcombustion phase and preventing valve collision with the enginepiston;

2) valve-lift control;3) profile-area [integration of valve displacement over time

or crank angle (CA)] control for accurate air charge and exhaustquantities;

4) soft seating for reducing valve noise and improving valvedurability.

Note that the overall valve-opening duration control and thevalve transient response (rising/falling) control studied in liter-ature can be classified into valve-timing control and profile-areacontrol, respectively. For traditional cam-based engine valves,the abovementioned properties are guaranteed by properly de-signed cam profile.

The leading camless VVA technologies include the electro-magnetic VVA (EMVVA), electro-hydraulic VVA (EHVVA),and electropneumatic VVA (EPVVA). The profile tracking prob-lem for those VVAs has been studied intensively. The four con-trol objectives discussed above can be achieved either partiallyor simultaneously, depending on a specific VVA system designand its application. Adaptive peak lift control was employedin [2] for an EMVVA and in [3] for an EPVVA to achieve re-peatable valve lift. Feedforward (FF) control is commonly usedfor valve-timing control to compensate for the valve-openingor valve-closing delays for EHVVA [4] and EPVVA [5]. Soft-seating control is a challenge for the EMVVA because of non-linear magnetic force and was addressed in [6] with guaranteedvalve response using extreme seeking control, and the combi-nation of FF linear–quadratic regulator and repetitive learningcontrol to reduce cycle-to-cycle variations was used in [7]. Thesecontrol designs were conducted for single or multiple controlobjectives, and some other studies dealt with the overall valve-profile control as a single tracking problem. For example, themodel reference control, combined with repetitive control, wasused for the VVA to achieve asymptotic profile tracking per-formance in [8]. The sliding mode control was employed in[9] for an EMVVA to achieve repeatable tracking performancewith guaranteed seating velocity. The robust repetitive controlin [10] and time-varying internal-model-based control in [11]

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LI et al.: PROFILE TRACKING FOR AN ELECTRO-HYDRAULIC VARIABLE VALVE ACTUATOR USING RECEDING HORIZON LQT 339

for an EHVVA were proved to be very effective in trackingthe desired flexible valve profile both under steady-state andtransient-engine operations.

In this paper, the profile tracking problem is studied for anEHVVA system described in [12] with dual-lift and built-in soft-landing mechanism. For the current cam-based VVA systems,one trend is to adopt cost effective two-step lift (or two-lift)designs. Note that since a camless VVA design with continu-ous variable valve timing and duration is able to meet most ofthe engine’s operational needs with only one lift, a second dis-crete lift with a substantial height shall add additional controlflexibility and benefits [13]. For the EHVVA system studied,complicated control scheme is not required for lift control andseating-velocity control. However, the valve-timing and profile-area controls are challenging for the studied EHVVA systembecause of the nonlinear and time-varying nature of the hy-draulic system. These nonlinearities include nonlinear flow dy-namics and temperature-sensitive fluid viscosity; see previouspublications [14] and [15]. Therefore, this paper studies theprofile-area tracking control that has not been investigated yetfor this EHVVA [12].

As combustion control updated cycle-by-cycle and/or withthe engine cycle is becoming more and more important for ad-vanced combustion technologies to satisfy the ever-increasingdemand for high efficiency with low emissions, including thecombustion mode transition between spark ignition (SI) andhomogeneous charge compression ignition (HCCI) combustion[16], the stratified lean combustion [17], and the turbulent-jet-ignition [18]. The engine cycle-by-cycle air management usingengine-valve control is becoming a necessity for those new com-bustion technologies in addition to conventional ignition andfuel controls. Therefore, in this study, an event-by-event optimaltracking control scheme is proposed for the valve-rising durationand opening profile based on the control-oriented valve-profilemodel as a function of the system supply pressure and enginespeed. Because of the nonlinear relationship between the valve-rising duration and supply pressure, the system dynamic modelwas linearized along the desired valve-rising trajectory and adiscrete-time, event-by-event model was developed accordingly.The equilibrium control input obtained during the model lin-earization process was used for FF control and a receding hori-zon linear–quadratic tracking (LQT) controller for valve-profiletracking was designed based on the linearized event-by-eventmodel.

The main contribution of this paper is the development ofevent-by-event valve model of opening duration and the utiliza-tion of receding horizon LQT control based on moving opti-mization of model predictive control (MPC).

The rest of this paper is organized as follows. Section IIprovides the overview and control problem formulation for thetarget EHVVA system. Section III presents the control-orientedmodel, model linearization along the desired trajectory, and thediscrete-time, event-by-event model. The receding horizon LQTcontroller is designed on the basis of the discrete-time, event-by-event model in Section IV. The simulation study is presentedin Section V, and experimental validation results are presentedin Section VI. Conclusions are drawn in Section VII.

Fig. 1. EHVVA schematic.

II. SYSTEM OVERVIEW AND CONTROL OBJECTIVE

A. EHVVA System

Fig. 1 shows the schematic of the studied EHVVA system withtwo discrete lifts. The engine valve is driven by the pressurized(Ph ) hydraulic fluid when the valve actuator is activated. Whenthe valve actuator is deactivated, the engine valve is closed by thereturn spring force and pushes the high-pressure fluid back intothe low-pressure (Pl) side. The dual-lift (low lift: L1 or high lift:L2) is realized using a solenoid lift valve by opening or closingthe lift porter to change the lift control sleeve position. Thesystem supply pressure, Ph , is regulated by a hydraulic pumpdriven by a dc motor and maintained by a linear accumulator.The predetermined low (back) pressure, Pl , is set higher thanthe fluid tank pressure but lower than Ph to make the lift controlsleeve rest steadily on its lower or upper position to ensureaccurate lift control. A top snubber, which consists of a checkvalve, a variable orifice, an undercut Ls1 connected to the topport, and a top notch on the top edge of the actuation piston, isprecisely designed to achieve soft landing during valve closing;while a bottom snubber, which consists of a bottom notch on thebottom edge of the actuation piston and a dent Ls2 on the top ofthe lift control sleeve, is designed to realize soft landing duringvalve opening. The bottom snubber is simpler than the top onebecause there is a squeeze fluid film between the piston and thelift control sleeve, and, thus, no direct metal-to-metal impact isobserved during the valve opening. The details of the EHVVAsystem can be found in [14].

B. Control Problem Description

Fig. 2 shows the test data under different supply pressure,Ph , and illustrates the control objectives of the EHVVA sys-tem discussed in the introduction section. With mechanicallyguaranteed lift control, soft-seating control, and previously


Fig. 2. Control problem analysis.

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

addressed valve-timing control [15], one of the remaining tasksfor the EHVVA system is the tracking control of valve profilearea, which is dominated by the valve-rising and valve-fallingdurations (defined as the valve transition time between 10%and 90% lift), where the dwell and peak lift are determinedby valve timing (opening and closing) and lift, respectively.From Fig. 2, the falling profile and falling duration, tvfd , arealmost fixed because of the fixed back pressure, Pl , and re-turn spring stiffness, while the rising profile and rising duration,tvrd , vary significantly as a function of the supply pressure dueto the varying engine speed; see discussions in Section III-A.The relationship between supply pressure, Ph , and valve-risingduration, tvrd, has been studied using bench test data, and acurve-fitted model is developed using a second-order polyno-mial over the typical operational range; see Fig. 3. It can beseen from both Figs. 2 and 3 that it becomes very difficult toreduce the valve-rising duration by increasing the supply pres-sure when it is close to 2 ms. This nonlinear nature leads toa variable-acceleration motion due to the time-varying springforce, where the transition time for a specific distance is a non-linear function of the driving force (acceleration). It is veryimportant to compensate the nonlinearity and to have accuratevalve-rising duration control because a small deviation in thetime domain could bring significant deviation in the CA domain[for example, 1 ms is equivalent to 30 ◦CA at 5000 r/min)],leading to significant engine performance deterioration. Sincethe valve profile is dominated by the rising duration (supplypressure), a tracking controller can be used to precisely regulatethe supply pressure to achieve the desired valve-rising duration(or profile).

III. SYSTEM MODELING

A. System Dynamics

The system supply pressure is determined by the entire hy-draulic system of the high-pressure route including the dc motorin torque mode, where the shaft torque, T , is proportional to thecontrol voltage, Vm (0–5 V) (i.e., T = a1Vm ), the fixed dis-placement pump, the relief valve with a constant mean reliefflow rate of Qre , and the actuation cylinder that acts as a high-pressure fluid consumer with a constant stroke volume Vc andan out-flow rate (Qout = VcNe/120) as a function of enginespeed Ne . According to the technique specification of the fixeddisplacement pump [19], the operating (outlet) pressure, Pout ,and flow rate, Qpump , can be derived as follows:

Pout =ηT

0.0159Vp+ Patm =

ηa1V m

0.0159Vp+ Patm (1)

Qpump = a2Pout + a3 (2)

where η is the hydraulic mechanical efficiency; Vp is the fixedpump displacement; Patm is the pump inlet (atmosphere) pres-sure; a1, a2, and a3 are constants.

Under steady-state operations, the flow in the high-pressureroute can be assumed to be steady Hagen-Poiseuille flow [20],where the pressure drop along the flow path, ΔP , can be deter-mined by the net flow rate as follows:

ΔP =2L

R

(Qpump − Qout − Qre

)(3)

where R is the Reynolds number related constant; L is the lengthof the flow path. Then, the supply pressure, Ph , can be derivedas follows:

Ph = Pout + ΔP. (4)

Combining (1)–(4), the following static supply pressuremodel can be obtained:

Ph = k1Ne + k2Vm + c0 (5)

where k1, k2, and c0 are lumped coefficients that can be cali-brated within the typical operational range. Since the outlet flowrate under low- and high-lift operations is different because ofthe difference in stroke volume, Vc , the coefficients in (5) forlow- and high-lift operations are different. The calibration sur-faces of the linear model for the two discrete lifts are shown inFig. 4.

Because of the use of linear accumulator [14], the event-by-event supply pressure change can be approximated using twofirst-order transfer functions of the net flow rate change deter-mined by the inlet and outlet flow rates, and they are linear func-tions of the dc motor control voltage, Vm , and engine speed, Ne

(see Fig. 4), respectively. Therefore, the system supply pressuredynamics can be modeled as follows:

Ph = k1Ne1

1 + τ1s︸︷︷︸outlet flow

+ k2Vm1

1 + τ2s︸︷︷︸inlet flow

+ c0 (6)


Fig. 4. Static pressure model validation.

Fig. 5. Dynamic pressure model validation.

where τ1 and τ2 are the time constants for the outlet and inletflow rates, respectively, and are calibrated by comparing the testdata and model simulation results; see Fig. 5.

Let x1 = k1Ne1

1+τ1sand x2 = Ph − c0 denote the pressure

components determined by engine speed and the supply pressurewith an offset c0, respectively. Then the second-order polyno-mial in Fig. 3 can be expressed as y = ax2

2 + bx2 + c, where ydenotes the valve-rising duration, tvrd; a, b, and c are calibrationcoefficients. This leads to the following second-order nonlinearmodel for the system:

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

x1 = − 1τ1

x1 +k1

τ1d

x2 =(

1τ2

− 1τ1

)x1 − 1

τ2x2 +

k2

τ2u +

k1

τ1d

y = ax22 + bx2 + c

(7)

where x = [x1, x2]T represents the system state vector, and yis the output; d is engine speed, Ne , which can be measured inreal-time and is considered a known exogenous system input; atlast, u represents the control input, Vm .

B. Model Linearization Along the Desired Trajectory

It can be seen that (7) is a nonlinear model due to the nonlinearoutput equation for y. A linearized model is needed for theoptimal LQT control design. At each operational point along thedesired tracking trajectory, the system model can be linearizedusing the classical Jacobian linearization, i.e., model (7) can betransformed into the following linearized form:

{Δx = AΔx + BuΔu + BdΔd

Δy = CΔx(8)

where

Δx = x − x0, Δy = y − y0

Δu = u − u0, Δd = d − d0

are the variation variables, and the system coefficient matricesare determined by the following:

A =∂f

∂x

∣∣∣∣(x0,u0,d0)

Bu =∂f

∂u

∣∣∣∣(x0,u0,d0)

Bd =∂f

∂d

∣∣∣∣(x0,u0,d0)

C =∂g

∂x

∣∣∣∣(x0,u0,d0)

where x = f(x, u, d) and y = g(x) are the associated nonlinearsystem dynamic equations. Note that (x0, y0, u0, d0) denotes theequilibrium point solved by (7) at steady-state (x1 = 0, x2 = 0)for the given tracking trajectory, r, and measured exogenousinput, d. That means for any given reference, r, (to be tracked bythe system output, y) and real-time measured exogenous input,d, an equilibrium point can be found with y0 = r and d0 = d(i.e., Δd = d − d0 = 0). Then, x10 can be solved directly bygiven d0, i.e., x10 = k1d0; x20 can be solved by y0 using thesecond-order order polynomial in (7), and the solution withinthe physical operational range should be taken; u0 can be solvedsubsequently by given x10, x20, and d0.

The determined equilibrium point (x0, y0, u0, d0) is used tocalculate the system coefficient matrices. The solved equilib-rium point control, u0, will be used as a nominal (FF) controlto regulate the valve-rising duration, y, to track the desiredtrajectory, r. Since there are always inevitable calibration andmodeling errors, noise, and disturbances for the actual systemleading to deviations of the valve profile, y, from its target rif the predetermined nominal control u0 is used, therefore, aclosed-loop controller for the variation control input, Δu, needsto be designed to compensate for the deviation from the desiredtrajectory, when modeling errors, system noise, and disturbanceare presented. Therefore, the control input for the system can bewritten as follows:

u = Δu + u0. (9)

C. Discrete-Time, Event-by-Event Model

The linearized continuous-time model in (8) with Δd =d − d0 = 0 can be discretized using the forward Euler approx-imation. Considering system input noise, w, and output mea-surement noise, v, the discrete-time, event-by-event model isobtained as follows:

{Δx(k + 1) = A(k)Δx(k) + B(k)Δu(k) + w(k)

Δy(k) = C(k)Δx(k) + v(k)(10)

where

A(k) =

⎡

⎣1 − Ts

τ10

(1τ2

− 1τ1

)Ts 1 − Ts

τ2

⎤

⎦ , B(k) =[

0Ts k4τ2

]

C(k) =[

0 2ax20 + b]


TABLE IMODEL PARAMETERS

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

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

}> 0

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

}> 0 (11)

where E{·} is an expectation operator; W and V are the cor-responding covariance matrices. Table I shows the identifiedparameters for the event-by-event model.

IV. LQT CONTROL

In this section, a receding horizon LQT controller is designedto make the system output, y(k), track reference r(k). To bemore specific, since the control design will be based on thevariation model in (10) to provide a variation control input,Δu(k), at each operational point, the control target is to makethe variation output, Δy(k), track variation reference, Δr(k).Since the state Δx1 used for state-feedback cannot be measured,a Kalman filter is used as optimal state observer.

A. Receding Horizon LQT Control

The control objective of the receding horizon LQT controlis to minimize the tracking error, e(k), defined in (12) with thefeasible control effort, Vm , along a predefined finite horizontracking trajectory. Tracking error, e(k), is defined as follows:

e(k) = Δy(k) − Δr(k) = C(k)Δx(k) − Δr(k). (12)

To simplify notations, Δxk , Δyk , Δuk , Δrk , ek , Ak , Bk ,and Ck are used to denote Δx(k), Δy(k), Δu(k), Δr(k), e(k),A(k), B(k), and C(k) at current time step, k, in the rest ofthis paper, respectively. The performance cost function of thereceding horizon LQT controller is calculated at each controlstep (valve event) over a finite horizon as follows:

J(k) =12eTkf

Fekf+

12

kf −1∑

k=k0

[eTk Qek + ΔuT

k RΔuk ] (13)

where F = FT ≥ 0, Q = QT ≥ 0, and R = RT > 0 are thegiven weighting matrices. k0 and kf define the beginning andthe end of the finite horizon moving window at each controlstep, k, for predefined upcoming N -step (N = kf − k0) track-ing reference. That is, an N -step optimal controller is designedat each control step for the given N -step tracking trajectory, andonly the first control step is used. This is the so-called movingoptimization problem in MPC. Note that under transient-engineoperations, the tracking reference of the valve profile is oftenpredefined or determined a few engine cycles ahead to havethe best performance possible. For example, a predefined 8-step(cycle) valve transition is required for the SI–HCCI combustionmode transition control [16]. It is also worth mentioning that

there are many predefined trajectory tracking problems for manymechatronic systems, such as the desired trajectory trackingduring clutch or gear engagement for automotive gear shiftingprocess [21], and trajectory generation (planning) and trackingcontrol for machining tools [22].

Since the receding horizon LQT controller is designed on thebasis of the variation model in (10) linearized at the equilib-rium point (or current operational point), k = k0, the trackingreference for variation output, Δyk , is given as follows:

Δrk = r(k) − r(k0), k = k0, ..., kf .

The optimal tracking problem can be solved by following theminimum principle in [23]. The variation control, Δuk , can beobtained as follows:

Δuk = −ΔuFBk + ΔuFFrk = −LFBkΔxk + LFFk gk+1 (14)

LFBk = [R + BTk Kk+1Bk ]−1Kk+1Ak

LFFk = [R + BTk Kk+1Bk ]−1 (15)

where ΔuFB and ΔuFFr are the state feedback and reference FFcontrol, respectively. Matrix K in the control gains LFBk andLFFr and vector g can be obtained by solving Riccati equation(16) offline and vector equation (17) online using the boundaryconditions in (18), respectively. We have the following:

Kk = ATk Kk+1[I − BkR−1BT

k Kk+1]−1Ak + CTk QCk

(16)

gk = ATk {I − Kk+1[I + BkR−1BT

k Kk+1]−1

× BkR−1BTk }gk+1 + CT

k QΔrk (17)

Kkf = CTk FCk , gkf = CT

k FΔrk . (18)

Note that in (14) the system state Δx1k in Δxk used forfeedback control cannot be measured and is estimated onlineusing the Kalman filter addressed in the next section.

B. Kalman State Estimation

The Kalman state estimation is a stochastic filter that mini-mizes the estimation error covariance to provide an optimal stateestimation subject to Gaussian noise input. For a given initialstate, Δx0, it uses the current output measurement, Δyk , andcontrol, Δuk , to estimate the next state, Δxk+1, in the followingform:

Δxk+1 = AkΔxk + BK Δuk + Hk (Δyk − CkΔxk ). (19)

Note that the initial state is calculated as follows:

Δx0 = xk−1 − xk,0. (20)

The subscript “k, 0” denotes the equilibrium point at valve eventk. It shows that the initial state is updated once the equilibriumpoint, xk,0, is switched, i.e., when a new tracking reference, rk ,or exogenous input, dk , is observed. The Kalman filter gain, Hk ,is obtained as follows:

Hk = AkΣkCTk (CkΣkCT

k + V )−1 (21)

where the estimation error covariance matrix, Σk , is calculatedrecursively by the following difference Riccati equation using


the initial condition Σ0 = E(Δx0ΔxT0 ):

Σk+1 = AkΣkATk + W − HkCkΣkAT

k . (22)

Using the estimated state vector in (14) and substituting itinto (9), the system control input can be obtained as follows:

uk = Δuk + uk,0 = −LFBkΔxk + LFFk gk+1 + uk,0. (23)

Note that this is possible because of the nature of the forwardcalculation of Σk .

C. Online Algorithm Implementation

The receding horizon LQT controller based on trajectory lin-earization and the Kalman filter is implemented online using thefollowing steps.

Step 1: For the current operation point, k, given rk and dk

(Nek ), solve for equilibrium point (xk,0, yk,0, uk,0,dk,0) by letting y = r, x1 = 0, and x2 = 0 in (7).

Step 2: Using (xk,0, yk,0, uk,0, dk,0), solve for the systemmatrices (Ak ,Bk , Ck ) of the discrete-time, event-by-event model in (10), where uk,0 will be used asFF control.

Step 3: Solve Ricatti equation (16) backward offline forKk+1 using the boundary condition, Kkf , in (18);calculate the state feedback control gain, LFBk , andreference FF control gain, LFFk , offline by (15) usingKk+1.

Step 4: Define an N -step finite horizon window fromk0 to kf for the predefined tracking referencesrk , ..., rk+N −1, i.e., k0 = k, kf = k + N − 1; solvethe vector equation in (17) backward online for gk+1

using the N -step references, Kk+1 solved in Step 3,and the boundary condition gkf in (18).

Step 5: Calculate the Kalman filter gain, Hk−1, using (21),the error covariance matrix, Σk , using (22) (for Hk

used in the next control step), and the state estimate,Δxk , using (19) using the initial condition givenin (20), the measurement, Δyk−1 = yk−1 − yk−1,0,and the variation control input, Δuk−1, from the lastcontrol step.

Step 6: Calculate the system control input, uk , by (23) us-ing the FF control, uk,0, solved in Step 2 and thevariation control, Δuk , (the first control in the finitehorizon window) calculated by (14) using the con-trol gains obtained from Step 3, the vector solved inStep 4, and the states estimated in Step 5.

Step 7: Go to Step 1.Note that the stability analysis of the receding LQT control

for linear systems can be found in [24]. Since the EHVVAsystem is nonlinear, the stability analysis of the EHVVA systemis challenging. The stability analysis of the proposed recedingLQT will be part of future work.

V. SIMULATION VALIDATION

The discrete-time, event-by-event model and the LQT con-troller based on the trajectory linearization (LQTFF) are im-plemented in Simulink and validated using simulation study.

Fig. 6. Implemented control schemes. (a) Open-loop Feedforward(FF), (b) PID, (c) PID with Feedforward (PIDFF), (d) LQT with Feed-forward (LQTFF).

TABLE IICONTROL PARAMETERS

For the purpose of comparison, the open-loop control that usesonly the equilibrium point control, u0, as FF control, and theproportional-integral-derivative (PID) control with and withoutthe FF u0 (PID and PIDFF) are also implemented. Fig. 6 illus-trates the four control schemes.

The control parameters, including weighting matrices (F ,Q, and R) for the LQT controller and the control gains forthe PID controllers, are tuned in simulations based upon thetransient tracking performance and system stability. The finitehorizon window size, N , for the tracking reference in this studyis selected as small as possible with acceptable tracking per-formance. Note that increasing the value of N improves thetransient tracking performance since more reference informa-tion is available during controller design, while it may becomeless practical to predefine a very long trajectory for the engine-valve control and more difficult for trajectory prediction if anupper level prediction algorithm is used for the desired trajec-tory. The PID controller is tuned with fixed small gain (SmG),ki , fixed large gain (LaG), ki , and scheduled gain (ScG), ki ,by supply pressure to show the effect of system nonlinearity.The control parameters for each control scheme are presentedin Table II. Note that kil and kih denote the scheduled ki gain atthe lower pressure bound (6 MPa) and the upper pressure bound(9 MPa) of the supply fluid, respectively. Two simulation sce-narios, the step reference tracking with constant engine speedand the steady reference tracking with transient-engine speed,are studied for each control scheme.

A. Step Reference Tracking With Constant EngineSpeed

Fig. 7 shows the simulation results of the step reference track-ing for the valve-rising duration at 1000 r/min engine speed. Theinitial condition used for the simulation is [x1in i , x2in i , dini ] =[−1.2, 9.49, 1000] calculated based on equilibrium point


Fig. 7. Simulation results of step reference tracking at 1000 r/min.

relationships, x1in i = k1dini and rini = ax22i n i

+ bx2i n i + c, withthe known initial tracking reference, rini = 4, and initial exoge-nous input, dini = 1000 (i.e., engine speed 1000 r/min). Thetracking performances, the corresponding supply pressure re-sponses, and control inputs, u, for six control schemes are pre-sented in the upper three plots in Fig. 7, respectively. The controlinput components, u0, ΔuFB, ΔuFFr , of the LQT controller areanalyzed in the bottom plot. It can be seen from the first plotthat the open-loop control with equilibrium point FF has goodtransient response and steady-state accuracy if the model (7),used for u0 calculation, is accurate; otherwise, steady-state er-ror may exist. Compared with the FF scheme, the PID controlwith a fixed SmG has similar response performance in low-pressure region (reference step-up) and poor performance inhigh-pressure region (reference step-down). Using a LaG im-proves the transient performance but with large overshoot inthe low-pressure region. This tradeoff relationship coincideswith the system nonlinearity (recall Fig. 3), which needs morecontrol effort in the high-pressure region to achieve the samerising duration increment than that needed in the low-pressureregion. Therefore, scheduling control gain with supply pressureis necessary for the PID controller to achieve good transient re-sponse both in low- and high-pressure regions, as shown by thePID control scheme with ScG. Combining the equilibrium pointFF and gain scheduling PID control, the PIDFF(ScG) schemeleads to improved transient performance. The responses of sup-ply pressure, Ph , and control input, u, are consistent with thetacking performance for each control scheme. It can be noticedthat both the open-loop and the PID control schemes result ina delayed response to the tracking reference signal. The LQTcontroller (LQTFF), however, can track the desired trajectoryvery closely with the minimum transient tracking error withoutany tracking delay mainly because of the reference feedfowardcontrol, ΔuFFr , obtained from the predefined upcoming N -step

Fig. 8. Simulation results of steady reference tracking at transient-engine speed.

tracking references, which dominates the control when the ref-erence changes. Note that the equilibrium FF control is used tocompensate the system nonlinearity, which reduces the sensi-tivity of LQTFF control gains to the operational condition (e.g.,supply pressure) and makes it possible to use fixed control gainsto achieve similar tracking performance under different supplypressure. This significantly reduces the effort of gain tuning overthe whole system operational range.

B. Steady Reference Tracking With Transient-EngineSpeed

Fig. 8 shows the simulation results of the steady referencetracking for the valve-rising duration under transient-engineoperation, where the engine speed increases from 1000 to1500 r/min within 1 s (10 engine cycles or valve events) andthen recovers at the same rate. Note that among the PID con-trol schemes, only the scheme with ScG [PID (ScG)] is usedin the transient-engine speed test. The initial condition for thesimulation is [x1in i , x2in i , dini ] = [−1.2, 10.13, 1000]. It can beseen that for each control scheme there are spikes in valve-rising duration because of the corresponding spikes in sup-ply pressure during engine acceleration and deceleration pro-cesses, respectively. This is because the change in the speedof the engine leads to the variation in valve-event frequency,which subsequently results in changes in the pulse outflow rateand supply pressure (see Section III-A). Without the equilib-rium point FF control, u0, obtained from the measured ex-ogenous input, d, (engine speed) and tracking reference, r,the PID (ScG) control has the slowest control response (seetrace u in Fig. 8), thus with the largest transient tracking er-ror of rising duration, while the other three control schemeswith FF u0, FF, PIDFF(ScG), and LQTFF, are able to suppress


Fig. 9. Experimental setup and control system configuration.

the variation in the supply pressure more quickly, thereby lead-ing to smaller transient tracking errors. With the state feedbackcontrol, ΔuFB, (see the bottom plot in Fig. 8) obtained from themoving optimization at each control step, the LQTFF controlhas the fastest response and the minimum tracking error dur-ing the transient operation. Note that the reference FF control,ΔuFFr , remains at zero because of the steady reference input,i.e., Δr = 0.

VI. EXPERIMENTAL VALIDATION

The experimental validation was conducted on the EHVVAprototype [12] designed for a 2.0L Ford Duratec engine. Fig. 9shows the test bench experimental setup and the associatedcontrol system configuration. The proposed LQT controllerwith Kalman filter is implemented into a microcontroller (NI-myRIO) that detects the valve-rising duration using a laser dis-placement sensor (Micro-Epsilon) and generates the analog con-trol signal, Vm , for the dc motor to regulate the system supplypressure. The supply pressure was measured by pressure sensorfor model calibration and the validation of Kalman state estima-tion. A simulated engine speed (frequency) signal generated bythe microcontroller was used to trigger the valve event by valve-timing (opening and closing) control [15]. A low side drivecircuit was used to drive the lift valve solenoids for lift control.The open-loop FF, PID, and FF PID control schemes presentedin Table II were also implemented in the microcontroller for thepurpose of comparison. Note that the control gains for the PIDand LQT control schemes were retuned in tests based upon theoriginal values used in simulations because of the system noisesintroduced in the test bench.

The transient operational conditions used in the simulationstudies (see Figs. 7 and 8) are also used for experimental val-idation. Additionally, since the repeatable valve profile is veryimportant for repeatable air charge and exhaust quantities foreach engine cycle, 200-cycle tests under both steady-state and

Fig. 10. Test results of step reference tracking at 1000 r/min.

the engine speed ramp-up transient operational condition areconducted to validate the repeatability of the valve-rising dura-tion tracking.

A. Step Reference Tracking With Constant EngineSpeed

Fig. 10 illustrates the test results of the step reference trackingfor the valve-rising duration at 1000 r/min, and the results agreewith the simulation results (see Fig. 7) well with several excep-tions. First, a small, steady-state tracking error appears in theopen-loop FF control at the low-pressure region because of themodeling error and is eliminated by the closed-loop controllers.Second, compared with the FF control scheme, the PID con-trol scheme (ScG) has the slowest closed-loop response and theFF PID control [PIDFF(ScG)] only shows slight improvement.This is mainly due to the fact that PID gains are limited by thesystem noise for guaranteed closed-loop stability. The trackingperformance including the 5% settling time, mean absolute per-centage error (MAPE), and the maximal error during the stepreference tracking are presented in Table III. With the help ofthe Kalman state estimation, the LQT controller (LQTFF) isable to track the transient trajectory with the fastest responseand the minimum transient tracking error (both MAPE and MaxError) among all the control schemes. The bottom plot in Fig. 10shows the Kalman state estimation results. Note that the state,Δx1(x1), is not measurable and remains at zero (x1 keeps un-changed) because x1 is excited by the engine speed, which isunchanged under this condition.


TABLE IIITRACKING PERFORMANCE OF STEP REFERENCE

Fig. 11. Test results of steady reference tracking at transient-enginespeed.

TABLE IVTRACKING PERFORMANCE IN TRANSIENT-ENGINE SPEED

B. Steady Reference Tracking With Transient-EngineSpeed

Fig. 11 and Table IV show the test results of the steady ref-erence tracking for the valve-rising duration under transient-engine speed. The test results also agree with the simulationresults (see Fig. 8) well except that the FF PID [PIDFF(ScG)]control does not show significant transient performance im-provement compared with the open-loop FF control because ofthe system noise that limits the control gains for closed-loop sta-

Fig. 12. Steady-state tracking in 1000 r/min within 200 engine cycles.(a) Valve-rising duration distribution. (b) Histogram of tracking deviationin steady-state.

bility. This is consistent with the test results for step referencetracking (see Fig. 10). However, the closed-loop PIDFF(ScG)control scheme has potential advantage over the open-loop FFcontrol scheme for steady-state performance, considering themodeling error, disturbance, system aging, etc. The FF LQT(LQTFF) control, with the shortest settling time and the smallesttransient error, also has the best transient performance becauseof the dynamic output feedback control, ΔuFB.

C. Repeatability Validation in 200 Engine Cycles

1) Steady-State Operations: The steady-state (when bothtracking reference and engine speed are constants) tracking per-formance for the valve-rising duration is evaluated for 200 en-gine cycles at 1000 r/min, and the valve-rising duration distribu-tion and statistic distribution of the tracking error are illustratedin Fig. 12. Since it is very important to keep the valve-risingduration within a small (CA) region to guarantee repeatable aircharge and exhaust quantities for each engine cycle, in Fig. 12(b)the tracking error in time domain is converted into the CA do-main at an engine speed of 5000 r/min to evaluate the worst-casetracking performance.

The probabilities of deviation distribution within ±1 ◦CAand the absolute mean and maximal tracking error for eachcontrol scheme are presented in Table V. It can be seen thatthe probabilities of ±1 ◦CA error distribution for the open-loop FF, gain-scheduled PID [PID(ScG)], and gain-scheduledPID with FF [PIDFF(ScG)] control schemes are very close,and the performance (±1 ◦CA error distribution, mean error,and maximal error) of the FF scheme is slightly better thanthat of the PID(ScG) control scheme, indicating that the open-loop control can achieve comparable steady-state performance,


TABLE V200-CYCLE STEADY-STATE TRACKING PERFORMANCE

Fig. 13. Engine speed ramp-up test from 1000 to 5000 r/min within200 engine cycles. (a) Valve-rising duration distribution. (b) Histogram oftracking deviation in engine speed transient.

TABLE VI200-CYCLE TRANSIENT-ENGINE SPEED TRACKING PERFORMANCE

provided that an accurate equilibrium point control can be ob-tained from the system model in (7). The FF LQT (LQTFF)control scheme has the best steady-state tracking performance,with 84% probability of ±1 ◦CA error distribution that is morethan 20% higher than that of the FF, PID(ScG), and PIDFF(ScG)control schemes, indicating that fairly good valve repeatabil-ity has been achieved. The corresponding improvements of themean and maximal tracking errors for the LQTFF control alsocan be found in Table V.

2) Engine Speed Transient Operations: The test results ofthe engine speed transient operations (engine speed ramps uplinearly from 1000 to 5000 r/min within 200 engine cycleswith the constant tracking reference) are shown in Fig. 13 andTable VI. It can be seen from Table VI that the performance(±1 ◦CA error distribution, absolute mean and maximal errors)

of the FF scheme deteriorates in transient operations, comparedwith that in steady-state because of the reduced model accuracyover the wide engine speed operational range. It also can beseen by the histogram of the FF control scheme in Fig. 13(b)that the mean rising-duration error deviates away from 0. How-ever, the closed-loop control schemes, PID(ScG), PIDFF(ScG),and LQTFF, are able to compensate for the modeling error andachieve similar tracking performance including the ±1 ◦CA er-ror distribution, mean error, and maximal error, compared withthe steady-state tracking (see Table V). The LQTFF controlscheme still shows more than 20% improvement with respectto the repeatability performance (±1 ◦CA error distribution),compared with other controllers. It can be seen that among theclosed-loop controllers, PID(ScG), PIDFF(ScG), and LQTFF,only the PID(ScG) control scheme has performance deteriora-tion for the ±1 ◦CA error distribution during the engine speedramp-up. This shows the significance of the equilibrium pointFF control that adapts to the engine-speed change by consideringit as an exogenous input in the system model in (7).

From both 200-cycle repeatability tests with and without theengine-speed variation, the proposed LQTFF controller showssignificant improvement on repeatability, compared with theother controllers, i.e., FF, PID, and PIDFF. That is mainly dueto the employment of the Kalman filter providing the optimalstate estimation for feedback control. It can be noticed from theopen-loop FF control that the noise level of the system output yis around ±0.2 ms (5% of the tracking reference). Note that PIDand PIDFF controllers use the measured y directly for feedbackcontrol, while the LQTFF controller uses the state feedbackbased on states optimally estimated by the Kalman filter withminimized estimation error in presence of system noise, leadingto improved system repeatability. However, the Kalman state es-timation allows the LQTFF controller to use large control gainsfor the state feedback control, especially under transient-enginespeed or supply pressure operations, whereas gains of PID con-trollers need to be carefully tuned to balance both stability andperformance because of high measurement noise, leading toconservative performance.

On the basis of both simulation and experimental results, itcan be concluded that the proposed FF LQT (LQTFF) controlis able to track the desired valve profile (rising duration) withsatisfactory tracking performance both under steady-state andtransient operational conditions.

VII. CONCLUSION

In this paper, an LQT controller with the Kalman state estima-tion is designed based on trajectory linearization for a developednonlinear model of the EHVVA system. The equilibrium pointcontrol obtained from the trajectory linearization is used as theFF control for the receding horizon LQT. Both simulation andexperimental validations are conducted, and the results matchvery well. In the experimental study, the proposed LQT con-troller shows significant improvement both under transient andsteady-state operations, compared with the open-loop FF, PID,and FF PID controllers. The LQT controller is able to track thedesired valve trajectory with minimal tracking error within five


valve events for the step reference tracking and within eightvalve events for the engine speed transient test, respectively.Both are around twice faster than the responses of the FF PIDcontroller (the best controller among the controllers for compar-ison study). The repeatability validation tests over 200 enginecycles show around 20% improvements for the LQT controllerfor the ±1 ◦CA tracking error distribution both in steady-stateand transient-engine speed operational conditions, comparedwith other studied controllers. The improvements of the pro-posed FF LQT controller are mainly due to the combination ofequilibrium point FF, state feedback based on estimated states,and reference FF control, along with the utilization of Kalmanfilter for optimal state estimation in the presence of systemnoise.

Currently, the proposed algorithm is validated for one VVAon a test bench. For practical applications, multiple VVAs needto be installed on the target engine. Since it is very important tohave cylinder-to-cylinder consistency, a variable displacementpump driven by the engine itself shall be used to supply pres-surized fluid for all valves [12], where the fluid pressure can beregulated by controlling the pump displacement. In this case,only the inlet flow model in (6) needs to be modified for theproposed control algorithm to have the desired cycle-by-cycleconsistency and valve profile tracking performance.

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Huan Li received the B.S. and Ph.D. degreesin mechanical engineering from the Beijing Insti-tute of Technology, Beijing, China, in 2012 and2018, respectively.

From 2015 to 2017, he was a Visiting Studentwith the Department of Mechanical Engineer-ing, Michigan State University, East Lansing, MI,USA, where he was involved in the control ofelectro-hydraulic variable valve actuators for au-tomotive engines. He is currently a Staff Engi-neer with the GAC R&D Center, Guangzhou,

China, where he is involved in the control of (hybrid) electric vehicles.

Ying Huang received the B.S. degree in elec-trical engineering from the Beijing University ofAeronautics and Astronautics, Beijing, China, in1989, and the M.S. and Ph.D. degrees in me-chanical engineering from the Beijing Instituteof Technology, Beijing, China, in 1992 and 2003,respectively.

She was a Visiting Researcher with the De-partment of Mechanical Engineering, WayneState University, Detroit, MI, USA, in 1999. Sheis currently a Professor with the School of Me-

chanical Engineering, Beijing Institute of Technology, where she is alsothe Vice Director of the Laboratory of Power System Engineering. Hercurrent research interests include the modeling and control of engines,automatic transmissions, hybrid powertrains, integrated powertrain con-trol, and vibration control of engines and powertrains.

http://www.secofluid.fr/uploads/catalogue/produits/notice/Boschrexroth-pgf-en.pdf

http://www.secofluid.fr/uploads/catalogue/produits/notice/Boschrexroth-pgf-en.pdf


Guoming G. Zhu received the B.S. degree inmechanical engineering and the M.S. degree inelectrical engineering from the Beijing Universityof Aeronautics and Astronautics, Beijing, China,in 1982 and 1984, respectively, and the Ph.D.degree in aerospace engineering from PurdueUniversity, West Lafayette, IN, USA, in 1992.

He was a Technical Fellow in advanced pow-ertrain systems with Visteon Corporation, anda Technical Advisor with Cummins Engine Co.,Ltd. He is currently a Professor with the Depart-

ment of Mechanical Engineering, Michigan State University, East Lans-ing, MI, USA. He has more than 30 years of experience related to controltheory, engine diagnostics and control, and vibration control. He has au-thored or co-authored more than 230 refereed technical papers and twobooks, and he holds more than 40 U.S. patents. His current researchinterests include closed-loop combustion control of internal combustionengines, engine system modeling and identification, hybrid powertraincontrol and optimization, and vibration suppression of aero-structuralsystems.

Dr. Zhu is Fellow of the SAE and the ASME. He is an Editorial BoardMember of the International Journal of Powertrains. He was the ProgramChair of the 2018 ASME Dynamic Systems and Control Conference andan Associate Editor for the ASME Journal of Dynamic Systems, Mea-surement, and Control.

Zheng D. Lou received the B.S. degree in me-chanical engineering from the Zhejiang Univer-sity, Hangzhou, China, in 1982, and the M.S. andPh.D. degrees in mechanical engineering fromthe University of Michigan, Ann Abor, MI, USA,in 1986 and 1990, respectively.

He is currently the founder and president ofLGD Technology, LLC. He was a Technical Spe-cialist and Fellow in manufacturing and thermal-fluid systems with Ford Motor Company and Vis-teon Corporation. He was also a Senior Engi-

neer with the Schaeffler Group USA, Inc. He was a Founder of theJiangsu Gongda Power Technologies Co., Ltd., Changshu, China. Hehas more than 27 years of experience related to product developmentand manufacturing in powertrains, climate control, thermal and fluid sys-tems, camless engine-valve systems, suspension systems. He has au-thored or co-authored more than 20 refereed technical papers and holds45 patents.

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