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Zhen Rene-mail: [email protected]

Guoming G. Zhue-mail: [email protected]

Department of Mechanical Engineering,Michigan State University, MI 48824

Integrated System ID and ControlDesign for an IC Engine VariableValve Timing SystemThis paper applies integrated system modeling and control design process to a continu-ously variable valve timing (VVT) actuator system that has different control input andcam position feedback sample rates. Due to high cam shaft torque disturbance and highactuator open-loop gain, it is also difficult to maintain the cam phase at the desiredconstant level with an open-loop controller for system identification. As a result, multi-rate closed-loop system identification becomes necessary. For this study, a multirateclosed-loop system identification method, pseudo-random binary signal q-Markov Cover,was used for obtaining linearized system models of the nonlinear physical system atdifferent engine operational conditions; and output covariance constraint (OCC) control-ler, an H2 controller, was designed based upon the identified nominal model and evalu-ated on the VVT test bench. Performance of the designed OCC controller was comparedwith that of the well-tuned baseline proportional-integral (PI) controller on the testbench. Results show that the OCC controller uses less control effort and has significantlower overshoot than those of PI ones. �DOI: 10.1115/1.4003263�

IntroductionThe continuously variable valve timing �VVT� system was de-

eloped in the early 1990s �1�. The benefits of using VVT fornternal combustion engines include improved fuel economy witheduced emissions at low engine speed, as well as increasedower and torque at high engine speed. The vane-type VVT sys-em �2� is a hydraulic mechanic actuator controlled by a solenoid.lectric motor driven cam phase actuators have become available

ecently due to their fast responses �3�. This paper studies theodeling and control of hydraulic VVT systems.There are two approaches to obtain a control oriented VVT

ystem model for model-based control development and valida-ion. They are physics based system modeling �4� and systemdentification using the system input and output data. In this paper,he closed-loop system identification approach is employed to ob-ain the VVT subsystem model of an internal combustion engine.ystem identification using closed-loop experimental input andutput data was developed in the 1970s �5� and it has been widelysed in engineering practices �6–8�. Closed-loop system identifi-ation can be used to obtain the open-loop system models espe-ially when the open-loop plant cannot be excited at the opera-ional conditions ideal for system identification. For instance,losed-loop system identification is typically applied to identify annstable open-loop plant. In this paper, the closed-loop identifica-ion method was selected due to many factors. The main reason ishat the system open-loop gain of the VVT actuator is fairly highnd the cam shaft has a torque load disturbance, which makes itlmost impossible to maintain the cam phase at a desired opera-ional condition. The other factor is that the VVT system dynamics also a function of engine speed, load, oil pressure, and tempera-ure, which made open-loop system identification difficult. There-ore, open-loop system identification at a desired cam phase is noteasible and the closed-loop identification was selected.

The first portion of this paper describes the process of obtaininginearized system models of the VVT actuator subsystem at vari-us operational conditions using the indirect closed-loop systemdentification approach discussed in Ref. �7�. The q-Markov CO-

Contributed by Dynamic Systems Division of ASME for publication in the JOUR-

AL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received April6, 2010; final manuscript received August 11, 2010; published online March 9,
011. Assoc. Editor: Xubin Song.
ournal of Dynamic Systems, Measurement, and ControlCopyright © 20

om: http://asmedigitalcollection.asme.org/ on 04/02/2015 Terms of Use: ht

Variance equivalent realization �q-Markov Cover� system identi-fication method �9–11� using pseudo-random binary signals�PRBs� was used to obtain the closed-loop system models. Theq-Markov Cover theory was originally developed for model re-duction. It guarantees that the reduced order system model pre-serves the first q-Markov parameters of the original system. Therealization of all q-Markov Covers using input and output data ofa discrete-time system is capable of system identification.q-Markov Cover for system identification uses pulse, white noise,or PRBS as input excitations. It can be used to obtain the linear-ized model representing the same input/output sequence for non-linear systems �11�. It has also been extended to identify multiratediscrete-time systems when input and output sampling rates aredifferent �12�.

For the proposed study, the multirate system identification isrequired due to event based cam phase sampling �function of en-gine speed� and time based control sampling. For our test benchsetup, the cam position was sampled four times over an enginecycle. For instance, when the engine is operated at 1500 rpm, thecam position sample period is 20 ms, and the control input isupdated at a fixed sample period of 5 ms. For this study, multiratePRBS q-Markov Cover was used for closed-loop system identifi-cation on the VVT test bench. System models at different engineoperational conditions were identified using closed-loop multirateidentification.

The second portion of the paper presents the control design andvalidation using the output covariance constraint �OCC� controldesign approach �13–15�. The OCC control minimizes systemcontrol effort subject to multiple performance constraints on out-put covariance matrices. An iterative controller design algorithm�15� with guaranteed convergence can be used to find an OCCoptimal controller. Note that an OCC controller is a H2 controllerwith a special output weighting matrix selected by the OCC con-trol design algorithm. The OCC control scheme was applied tomany aerospace control problems due to its minimal control effort�13–15�. In this paper, a nominal model was selected from thefamily of the identified VVT models for the OCC control design.Multiple OCC controllers were designed based upon closed-loopidentified models, and their performances were compared againstthose of the well-tuned baseline PI controller. In order to eliminatesteady-state error, system control input was increased to add anadditional integral input to the system plant used for the OCC
control design. Compared with the PI control, the OCC controllers
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ere able to achieve the similar system settling time to PI onesith significantly less overshoot and control effort.This paper integrates system model identification and controller

esign for the engine VVT subsystem. That is, the control designodel is obtained from closed-loop system identification and the

esigned controller will be evaluated in an actual VVT test bencho show that the integrated system identification and control de-ign provides satisfactory controllers. The system identificationnd control design process demonstrated in this paper will beepeated to improve the closed-loop system performance as ouruture research.

The paper is organized as the following: Section 2 providesramework and formulation of closed-loop system identificationor the VVT actuator system. Section 3 presents the OCC controlroblem and associated design framework. Section 4 describes theest bench setup. Section 5 introduces the closed-loop systemdentification and the OCC controller design results obtained fromhe test bench, along with the discussions of the experiment re-ults. Conclusions are provided in Sec. 6.

System Identification FrameworkConsider a general form of linear time-invariant closed-loop

ystem in Fig. 1, where r is the reference signal, n is the measure-ent noise, and u and y are input and output, respectively.As discussed in the Introduction section, there are many ap-

roaches for the closed-loop identification, which are categorizeds direct, indirect, and joint input-output approaches. In this paper,e utilize the knowledge of the controller to calculate the open-

oop plant model from the identified closed-loop plant model,hich is called indirect approach. To ensure the quality of the

dentified plant, the closed-loop controller in this paper is selectedo be proportional �16,17�.

The input and output relationship of the generalized closed-loopystem, shown in Fig. 1, can be expressed below

y = H · r = GK�I + GK�−1r �1�

et H be identified closed-loop transfer functions from r to y. The

pen-loop system model GID can be calculated using identified H,

ssuming that �I-H�−1 is invertible. The closed-loop controllerransfer function is used to solve for the open-loop system models.

e have

GID = H�I-H�−1K−1 �2�The PRBS signal is used as an input signal for identifying the

losed-loop system model. The most commonly used PRBSs areased on maximum length sequences �called m-sequences� �18�or which the length of the PRBS signals is m=2n−1, where n isn integer �order of PRBS�. Let z−1 represent a delay operator andefine p�z−1� and p�z−1� to be polynomials

p�z−1� = anz−n+1� ¯ � a2z−1

� a1 = p�z−1�z−1� 1 �3�

here ai is either 1 or 0, and � obeys binary addition, i.e.,

1 � 1 = 0 = 0 � 0 and 0 � 1 = 1 = 1 � 0 �4�

nd the nonzero coefficients ai of the polynomial are defined inable 1 and also in Ref. �18�.

Fig. 1 Closed-loop identification framework

Then the PRBS can be generated by the following formula

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u�k + 1� = p�z−1�u�k�, k = 0,1,2, . . . �5�

where u�0�=1 and u�−1�= u�−2�= ¯ = u�−n�=0. Defined the fol-lowing sequence

s�k� = �a; If k is even

− a; If k is odd� �6�

Then, the signal

u�k� = s�k� � �− a + 2au�k�� 7�

is called the inverse PRBS, where � obeys

a � a = − a = − a � − a and a � − a = a = − a � a �8�

It is clear after some analysis that u has a period 2m and u�k�=−u�k+m�. The mean of the inverse PRBS is

mu = E2mu�k� �1

2m �k=0

2m−1

u�k� = 0 �9�

and the autocorrelation �Ruu��=E2mu�k+��u�k�� of u is

Ruu�� 1

2m �k=0

2m−1

u�k + ��u�k� = a2 � = 0

− a2 � = m

− a2/m � even

a2/m � odd �10�

Note that the first and second order information of the inversePRBS signal is very close to these of white noise for a largeenough m. The inverse PRBS is used in the q-Markov Coveridentification algorithm. For convenience, in the rest of this paper,the term “PRBS” is used to represent the inverse PRBS.

Consider an unknown �presumed nonlinear� system

x�k + 1� = f�x�k�,w�k��

y�k� = g�x�k�,w�k�� 11�

subjected to an input sequence �w�0� ,w�1� ,w�2� , . . .� generatingthe output sequence �y�0� ,y�1� ,y�2� , . . .�. The unknown system isq-identifiable, if there exists a linear model of the form

x�k + 1� = Ax�k� + Dw�k�

y�k� = Cx�k� + Hw�k� �12�that can reproduce the same output sequence�y�0� ,y�1� ,y�2� , . . . ,y�q−1��, subject to the same input sequence�w�0� ,w�1� ,w�2� , . . . ,w�q−1��. In case that the system is notq-identifiable, it is possible for q-Markov Cover to construct theleast square fit using linear model for the input-output sequence�19,20�.

In this paper, system models were identified in discrete-timedomain using the PRBS graphic user interface �GUI� �12� devel-oped for multirate PRBS q-Markov Cover. The advantage of usingthe PRBS GUI is that the number of Markov parameters and theorder of the identified system model can be easily adjusted basedupon the calculated Markov singular values from the input-output

Table 1 Nonzero coefficients of PRBS polynomial

Polynomial order �n� Period of sequence �m� Nonzero coefficient

2 63 a5, a6

3 127 a4, a7

4 255 a2, a3, a4, a8

5 511 a5, a9

6 1023 a7, a10

7 2047 a9, a11

data.

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OCCConsider the following linear time-invariant system

xp�k + 1� = Apxp�k� + Bpu�k� + Dpwp�k�

yp�k� = Cpxp�k�

z�k� = Mpxp�k� + v�k� �13�

here xp is the state, u is the control, wp represents process noise,nd v is the measurement noise. Vector yp contains all variableshose dynamic responses are of interest. Vector z represents noisyeasurements. Suppose that a strictly proper output feedback sta-

ilizing control law, shown below, is used for plant �13�

xc�k + 1� = Acxc�k� + Fz�k�

u�k� = Gxc�k� �14�

hen the resulting closed-loop system is

x�k + 1� = Ax�k� + Dw�k�

y�k� = yp�k�u�k� � = Cy

Cu�x�k� = Cx�k� �15�

here x= �xpT xc

T�T and w= �wpT vT�T. Formulas for A, C, and D

re easy to obtain from Eqs. �13� and �14�.Consider the closed-loop system �15�. Let Wp and V denote

ositive definite symmetric matrices with dimensions equal to therocess noise wp and noisy measurement vector z, respectively.efine W=block diag�Wp V� and let X denote the closed-loopeighted controllability Gramian from the input W−1/2w. Since A

s stable, X is given by

X = AXAT + DWDT �16�

n this paper, we are interested in finding controllers of the form14� that minimize the �weighted� control energy trace �RCuXCu

T�ith R�0 and satisfy the constraints

Y = CXCT � Y �17�

here Y�0 are given and X solves Eq. �16�. This problem,hich we call the OCC problem, is defined as finding a full-orderynamic output feedback controller for system �13� to minimizehe OCC cost

JOCC = trace�RCuXCuT�, R � 0 �18�

ubject to Eqs. �16� and �17�.The OCC problem may be given several interesting interpreta-

ions. For instance, assume first that wp and v are uncorrelatedero-mean white noises with intensity matrices Wp�0 and V0. Let E be an expectation operator, and

E�wp�k�� = 0 E�wp�k�wpT�k − n�� = Wp��n�

E�v�k�� = 0 E�v�k�vT�k − n�� = V��n� �19�

efining E�� · �ª limk→�

E� · � and W=block diag�Wp V�, it is easy

o see that the OCC is the problem of minimizing E�uTRu subject

o the OCC constraint YªE�y�k�yT�k�� Y. As it is well known,he constraint may be interpreted as constraint on the variance ofhe performance variables or lower bounds on the residence timein a given ball around the origin of the output space� of theerformance variables �21�.

The OCC problem may also be interpreted from a deterministicoint of view. To see this, define the discrete time domain �� and2 norms

�y�2ª supk�0yT�k�y�k�
�
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�w�22ª �

k=0

�

wT�k�w�k� �20�

and define the �weighted� �2 disturbance set

W ª �w:R → Rnw and �W−1/2w�22 � 1� �21�

where W�0 is a real symmetric matrix. Then, for any w�W, wehave �22,23�

�y��2 � ��Y� and �ui��

2 � �CuXCuT�ii i = 1,2, . . . nu

�22�

where nu is the dimension of u, �� · � denotes the maximum sin-gular value, and � · �ii is the ith diagonal entry. Moreover, Refs.�22,23� show that the bounds in Eq. �22� are the least upperbounds that hold for any signal w�W.

Thus, if we define Yª I�2 in Eq. �17� and R=diag�r1 ,r2 , . . . ,rnu

��0 in Eq. �18�, the OCC problem is theproblem of minimizing the �weighted� sum of worst-case peakvalues on the control signals given by

Jocc = �i=1

nu

ri� supw�W

�ui��2 � �23�

subject to constraints on the worst-case peak values of the perfor-mance variables of the form

supw�W

�y��2 � �2 �24�

This interpretation is important in applications where hard con-straints on responses or actuator signals cannot be ignored, suchas space telescope pointing and machine tool control �14�.

Detailed proof can be found in Ref. �15�. The controller systemmatrices Ac, F, and G can be calculated using an iteration algo-rithm introduced in Refs. �13,15�.

4 VVT System Bench Tests Setup

4.1 System Configuration. Closed-loop system identificationand control design test were conducted on the VVT test bench�Fig. 2�. A Ford 5.4L V8 engine head was modified and mountedon the test bench. The cylinder head has a single cam shaft with aVVT actuator for one exhaust and two intake valves. These valvesintroduce a cyclic torque disturbance to the cam shaft. The camshaft is driven by an electrical motor through a timing belt.

An encoder is installed on the motor shaft �simulated the crankshaft�, which generates crank angle signal with one degree reso-lution, along with a so-called gate signal �one pulse per revolu-tion�. A plate with magnets attached was mounted at the other sideof the extended cam shaft. These magnets pass two hall-effectcam position sensors when the cam shaft rotates, where one camsensor was used to determine engine combustion TDC position�combustion phase�, along with the encoder signals and the otheris used to determine the cam phase. This cam position signalupdates 4 times per cycle.

The cam phase actuator system consists of a solenoid drivercircuit, a solenoid actuator, and a hydraulic cam actuator. The

Fig. 2 VVT phase actuator test bench

solenoid driver is controlled by a pulse-width modulation �PWM�

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ignal, where its duty cycle is proportional to the direct currentdc� voltage command. An electrical oil pump was used to supplyressurized oil for both lubrication and as hydraulic actuatinguid of the cam phase actuator. The cam actuator command volt-ge signal is generated by the Opal-RT prototype controller andent to the solenoid driver. The PWM duty cycle is linearly pro-ortional to input voltage with maximum duty cycle �99%� corre-ponding to 5 V. The solenoid actuator controls the hydraulic flu-ds �engine oil� flow and changes the cam phase. The cam positionensor signal is sampled by the Open-RT prototype controller andhe corresponding cam phase is calculated within the Opal-RTeal-time controller.

Within an engine cycle, the cam position sensor generates fouram position pulses sampled by the Opal-RT real-time controller.y comparing these pulse locations with respect to the encoderate signal, the Opal-RT controller calculates the cam phase withne crank degree resolution. After the error between the calculatedam phase and the cam reference signal is obtained, the cam ac-uator control command is obtained from the controller K. Figure

shows the signal diagram of the VVT control system. Referenceignal r and the measured cam phase signal y can be recorded inhe Opal-RT controller for system identification. In this paper,system model” refers to the transfer function between the controlnput u �voltage� and calculated cam phase signal y �degree�;controller” refers to transfer function K between the error andutput.

4.2 VVT Open-Loop Properties. The cam phase actuatoras an output range of 30 crank degrees. Figure 4 shows anpen-loop step response of the VVT phaser. Input to the system isstep between 0 V �1% duty cycle� and 5 V �99% duty cycle�. It

an be found that the cam phase system has a settling time about.5 s for advancing �rising� and 1.0 s for retard �falling�, demon-trating its nonlinear characteristics of the VVT system. This isainly due to the fact that the VVT actuator has different dynam-

cs for advancing and retarding. For advancing, the actuatingorque generated by the oil pressure overcomes the cam load

Fig. 3 VVT system diagram

Fig. 4 Cam phase actuator open-loop step response

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torque and moves cam phase forward; for retarding, the oiltrapped in the actuator bleeds back to the oil reserve when the camphase is pushed back by the cam shaft load. This difference leadsto the response characteristics difference for advance and retardoperations, which makes the system nonlinear. This phenomenonwill be discussed in Sec. 5.

Figure 5 shows the VVT system steady-state responses viaopen-loop constant inputs with a 0.1V interval �2% duty cycle�between 0 and 5 V. It can be observed that for open-loop control,the cam phase actuator behaves almost like a binary state and it isvery difficult for the VVT actuator to maintain a desired nonsat-urated cam timing position due to the actuator hysteresis charac-teristics, cam load, and engine oil pressure variations. This indi-cates that open-loop system identification, which requires to holdthe actuator operate at a desired location during the system iden-tification process, is almost impossible. Therefore, closed-loopsystem identification is adopted in this research. A proportionalcontroller is selected for the closed-loop system identification inorder to ensure good closed-loop system identification accuracy�16,17�.

5 Bench Test Results

5.1 Closed-Loop Identification. The operating point andcontroller gain need to be selected carefully due to the systemproperty. The solenoid drive circuit has an operational range be-tween 0 and 5 V that corresponds to 1–99% of the solenoid PWMduty cycle. Therefore, in order to avoid saturation, we have toselect the phase actuator operation condition carefully; otherwise,the control input might be saturated, leading to high system iden-tification error. Therefore, the PRBS signal magnitude was se-lected to be 12 deg, nominal operational condition was centered at14 deg cam phase, and the controller proportional gain was 0.1�V/deg�. To obtain a family of system transfer functions, the sys-tem identification bench tests were conducted at different enginespeeds and oil pressures. For demonstration, we selected two en-gine speeds at 1000 rpm and 1500 rpm and a constant oil pressureof 60 psi. Recorded reference signals and system response datawere processed using MATLAB PRBS-GUI �12�.

A number of Markov parameters to be matched by system iden-tification were used to optimize the identification accuracy �seeFig. 6�, and the identified system model order is determined by thedominant dynamics of PRBS response data �see Fig. 7�. Figure 7shows the order selection diagram produced during PRBS systemID at 1500 rpm. It shows the diagonal elements of Schur decom-position of the system response data matrix. The diagonal ele-ments of the Schur decomposition were plotted in a decreased

Fig. 5 Cam phase actuator open-loop steady-state responses

order. Each dot in the plot corresponds to one state of the identi-


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Fig. 8 Bode diagram of ope

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fied model. Detailed algorithm can be found in Ref. �12�. The plotshows a dominant first order dynamic because the order indexchart has the largest gap between the first and second dots. Thegap between the fourth and fifth dots is larger than the gap be-tween the second and third order gaps. Therefore, the order of theidentified model was selected to be 4 in order to keep the modelorder low without losing major system dynamics. The rest of sys-tem identification parameters are shown in Table 2.

Using the identified closed-loop model and Eq. �2�, a fourthorder open-loop plant model �see Appendix� at 1500 rpm is ob-tained. The corresponding open-loop Bode diagram �Fig. 8� showsthat there exists a dynamic mode at around 12.5 Hz, which isequal to the engine cycle frequency of 12.5 Hz �80ms/cycle� at1500 rpm. It is believed that the resonance observed was not thesystem dynamics of the cam phaser system but rather the externaldisturbance due to the cyclic dynamics of timing belt and camshaft torque disturbance due to valve actuations. Therefore, wedecided to exclude it from the identified model to be used forcontrol design. A second order model is obtained by selecting theidentified close-loop model to be 2. The plant model calculatedfrom the identified second order model has an almost identicalbehavior to the fourth order model without the 12.5 Hz mode �seeFig. 8 and Appendix�.

A fourth order closed-loop model was identified at 1000 rpm.Similar to the case at 1500 rpm, the identified model has a dy-namic mode at about 8 Hz, which corresponds to the engine cycleperiod �8.3 Hz, 120 ms/cycle�. However, in this case, a 2nd order

hysical responses
Fig. 6 Identified and p
Fig. 7 Identified model order selection

Table 2 System identification parameters

ngine speed �rpm� 1000 1500nput sample rate �ms� 5 5utput sample rate �ms� 30 20utput/input sample ratio 0.167 0.25RBS order 13 13ignal length �s� 81.88 81.88arkov parameter number 90 60

D open-loop model order 4 2 and 4

n-loop plant at 1500 rpm

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pen-loop model was not obtained directly from system identifi-ation. Note that in this case there exist a pair of nonminimalhase zeros shown in the root locus �Fig. 9� at the frequency closeo engine cycle frequency. To eliminate the dynamics at this fre-uency, a second order model was obtained by removing the pairsf pole-zero from the fourth order plant model �see Appendix�.he second order plant model has very similar frequency responses the fourth order plant model except without the dynamics in-roduced by the cyclic engine cam load �Fig. 10�.

5.2 Validation of Identified Model. To evaluate the accuracyf these identified models, their step responses were comparedith these from the bench tests. Since the open-loop step response

annot be obtained for the VVT actuator, their closed-loop re-ponses were compared in this study. The same proportional con-

Fig. 9 Root locus of the identifi

Fig. 10 Bode diagram of ope

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trol gain of 0.1 V/deg was used for the step responses. A step inputof 12 crank degrees was used. For the identified models, simula-tions were conducted in SIMULINK under the same conditions. Thenormalized step responses are compared in Fig. 11 at 1000 rpmand Fig. 12 at 1500 rpm. Note that the oscillations in the recordedresponses are mainly due to the cyclic valve torque load distur-bance and low cam phase sampling resolution, which demon-strates the difficulty for a proportional controller to maintain thecam phase at a desired level.

From both Figs. 11 and 12, it can be observed that the systemdc gains of both actual system and identified model are fairlyclose; for the transient response, the step down responses are veryclose for both model and actual system at both engine speeds, butthe step up responses of the identified model at 1500 rpm is faster

fourth order plant at 1000 rpm

n-loop plant at 1000 rpm


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han the actual system. This is mainly due to the nonlinear char-cteristics of the VVT actuator discussed in the VVT open-looproperty section. When cam phase is advanced, the VVT actuators driven by the engine oil pressure to overcome the cam shaftorque load, whereas when cam phase is retarding, the VVT ac-uator is pushed by cam load torque and returns freely. This is whyhe system has a different step up and step down responses. Iturns out that the identified mode approximates the step downesponse.

A family of system models was obtained from bench tests atifferent engine speeds and oil pressures �Fig. 13�. A second orderodel �25� was selected as the nominal model for control design

elow

G =0.0003s2 − 0.06s + 647.2

s2 + 7.615s + 20.67�25�

Note that this system plant has a pair of nonminimal phaseeros, indicating that high control gain will destabilize the closed-oop system.

5.3 OCC Controller With Single Input. In this section, con-rollers were designed and validated on the test bench. Step inputas used as reference signal and varies between 20 and 0 de-rees. A PI controller was well tuned for the VVT system on theest bench for comparison purpose. The PI tuning process wasompleted at different engine speeds and oil pressures. The tunedI controller shown in Eq. �26� achieves good balance betweenast response time and little oscillations at different conditions

Fig. 11 Closed-loop step res

Fig. 12 Closed-loop step respo



Kbase�s� = 0.2 +0.1

s�26�

For the OCC design, system plant matrices of nominal modelwere obtained from Eq. �25�

Ap = A = − 7.62 − 20.68

1 0� Bp = Dp = B = 1

0�

Cp = Mp = C = �− 0.063 647.39 � D � 0 �27�

Controller design parameters were selected as

Wp = 1 V = 0.01 R = �1� �28�Using the OCC iterative control design algorithm in Ref. �15�,

an OCC controller was obtained

KOCC�s� =194.8 s + 2701

s2 + 131s + 8582�29�

However, the controller was not able to maintain the cam phase atthe desired level, and it has a large steady-state error �see Fig. 14�.To improve the performance, an integrator was added to the plantto eliminate steady-state error �see Fig. 15�. A fourth order con-troller �the third order OCC controller plus the first order integra-tor� was obtained below

nse comparison at 1000 rpm

nse comparison at 1500 rpm

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KOCC−i =− 239.9s2 − 2751s − 1.1 � 104

s�s3 + 51.3s2 + 1305s + 1.97 � 104��30�

otice that the order of plant used for controller design is in-reased by one and as a result the order of the full-order controllers also increase by one. After combining the full-order controllerith the added integrator, the order of the new controller is in-

reased by two compared with the original controller.The OCC control with integrator has a large overshoot with

scillations �Fig. 14�. In order to eliminate steady-state error andeduce response time, a multi-input control design with propor-ional and integral inputs was proposed.

Fig. 13 Family o

Fig. 14 Step response for OCC controllers

Fig. 15 OCC design framework with an integrator

21012-8 / Vol. 133, MARCH 2011


5.4 OCC Controller Design With Multi-Input. For thedual-input control design, the controller has an additional integra-tor input to the plant �Fig. 16�. Noise intensity matrices Wp and Vwere the same as Eq. �28�. The input weighting matrix dimensionincreased to two due to additional integral input and it was se-lected as R=diag�1 20�. Note that in this case the input effort costratio between direct control and integral control is 1 to �20. Thedual-input controller was designed and shown in Eq. �31�, and itsperformance at 900 rpm with 45 psi oil pressure is compared withthe base PI controller in Eq. �26�. Figure 17 shows that bothcontrollers have very similar response times and steady-state er-rors and OCC controller has significantly less overshoot. The ma-

Fig. 16 Multi-input OCC design framework

entified models
f id
Fig. 17 Step response comparison


tp://asme.org/terms

jooP

slom5pmcaudssttOso

dw

E

Fp

J

Downloaded Fr

or reason for the OCC controller �with an integrator� to have lessvershoot than that of the PI one is due to the fact that the full-rder OCC controller contains more dynamics �full order� than theI controller.

KOCC−2in =84.5s3 + 935.2s2 + 1164.5s + 220

s4 + 122.2s3 + 7464.7s2 + 3022s�31�

5.5 Controller Performance Comparison. Table 3 showsystem response comparison of PI and multi-input OCC closed-oop systems. Both controllers have zero steady-state error, withscillation magnitude of 1 deg �lowest possible and limited byeasurement resolution�. Both controllers have almost identical

% �within 1 deg� settling time and 10–90% rising time. Com-ared with the base line PI controller, the OCC controller hasuch lower overshoot. In some operational conditions, the OCC

ontroller reduces the PI controller’s overshoot by 50%. For thedvance step �from 20 to 0 deg�, the multi-input OCC controllerses less control effort than PI. In the retard step, the control effortifference is smaller �Fig. 18�. The reason is that in the advancetep, all the control effort is created by the actuator; in the retardtep, engine oil pressure is working with actuator. At steady-state,he dual-input OCC controller shows a larger oscillation magni-ude than the PI controller. This is due to the fact that the designedCC controller has a higher gain than PI and therefore is more

ensitive to the change in error signal, which has the resolution ofne crank degree in the experiment.

As part of our future research, robust gain scheduling controlesign using identified models at different operating conditionsill be studied.

Table 3 Controller p

ngine speed�rpm�

Oil pressure�psi�

Advanced performanc

Overshoot�deg�

Settling time�s�

PI OCC PI OCC

900 45 7 3 2.16 2.241200 5 4 2.39 2.201500 5 4 2.01 2.261800 6 4 2.10 1.98900 60 6 3 2.26 2.011200 6 3 2.49 2.001500 6 3 1.71 1.971800 5 3 1.61 1.84

ig. 18 Control effort comparison at 900 rpm with 45 psi oil
ressure


6 ConclusionThis paper applies integrated system modeling and control de-

sign process to a continuously VVT actuator system. Constrainedby the sample rate of the crank-based cam position sensor �a func-tion of engine speed� and time based control scheme, the actuatorcontrol sample rate is different from the cam position sensor one.Due to cam shaft torque load disturbance and high actuator open-loop gain, it is also difficult to maintain the cam phase at thedesired level with an open-loop controller. The closed-loop mul-tirate system identification is required. Closed-loop system iden-tification using PRBS q-Markov Cover was applied to obtainopen-loop system models of a VVT cam actuator system. Theproposed closed-loop system identification approach providesmodels whose time responses are fairly close to bench responses.An output covariance constraint controller was designed based onthe identified model and tested on the test bench. The controllerhas an extra integrator for better response performance. Comparedwith PI controller, the multi-input OCC controller uses less energyand has similar closed-loop response time. OCC controller alsoreduces overshoot up to 50%.

AcknowledgmentThis work was partially supported by the U.S. Department of

Energy under Grant No. DE-FC26-07NT43275.

Appendix

G1000 rpm_4�s�

=3.6 � 10−4s4 − 5.3s3 + 586s2 − 1.8 � 104s + 2.1 � 106

s4 + 11.63s3 + 2780s2 + 3.21 � 104s + 4.6 � 104

G1000 rpm_2�s� =3.56 � 10−4s2 − 5.27s + 592.2

s2 + 11.63s + 16.7

G1500 rpm_4�s� =0.012s4 − 3.1s3 + 1354s2 − 2.9 � 104s + 9.0 � 106

s4 + 14.54s3 + 5971s2 + 8.54 � 104s + 2.38 � 105

G1500 rpm_2�s� =0.0124s2 − 2.04s + 1582

s2 + 16.78s + 34.82

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