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A comparative study of the new LQ-MCS control

on an automotive electro-mechanical system

Mario di Bernardo∗, Alessandro di Gaeta†, Umberto Montanaro∗and Stefania Santini∗

∗University of Naples Federico II, Italy

Email: {mario.dibernardo, umberto.montanaro, stefania.santini}@unina.it

†CNR, National Research Council, Istituto Motori, Naples,Italy

Email: a.digaeta@im.cnr.it

Abstract—This paper is concerned with the design and com-

parison of a new optimal-adaptive control of an electronic throttle

body. Numerical results are complemented by experiments.

I. INTRODUCTION

In automotive systems, the accurate control of the injection

timing and the precise regulation of the air/fuel mixture is

essential in order to guarantee high performance with respect

to traction, emissions, idle speed regime, cold starting manage-

ment, thermal transient and smoother movement during tip/in

tip/out [2]. The main idea is to directly effect the mixture

formulation by the electronic control of the fuel injection and

the exact regulation of the air flow in the manifold by an

Electronic Throttle Body (ETB)(see for example [1]).

In the ETB system, a shaped body duct imposes the rela-

tionship between the throttle valve position and the incoming

air flow into the manifold, while the desired plate position is

imposed by a microcomputer in a drive-by-wire configuration.

The control signal generated by the ECU becomes, by means

of a H-brige power converter, the armature voltage of a DC-

motor. The rotation motion is then transferred from the motor

shaft to the plate shaft through a gear system (some details

are in figure 1).

From the control perspective, the ETB is an highly nonlinear

plant since the transmission friction and the return spring limp-

home nonlinearity significantly affect the system performance.

Moreover the wide variations of process parameters, which

can be caused by production deviations, variations of external

conditions (e.g., temperature) and aging, demand for a con-

troller which is robust against uncertainties. Another control

requirement is the simplicity of the strategy that has to be

implemented on a typical low-cost automotive microcontroller.

Furthermore the feedback loop is closed only on the measure

of the plate angular position via a low resolution sensor.

The aim of this paper is to use the innovative LQ-MCS

(Linear Quadratic Minimal Control Synthesis) control scheme

to control the ETB system. In particular, the dynamics of the

new control are contrasted with those of a more traditional PI

scheme, applied to the problem in [5] where the closed-loop

performances are enhanced by the joint action of a non linear

compensator and a self tuning approach for tuning of the PI

gains.

The robustness of the PI controller is proven by experiments

on the actual plant. The comparative analysis between the two

control philosophies is instead performed by simulation adopt-

ing a mathematical model able to reproduce the experimental

behavior of the plant.

II. MATHEMATICAL MODEL AND IDENTIFICATION

The ETB system, shown in figure 1, is composed by a DC

motor, a reduction gear and a plate equipped with two springs.

These springs, named respectively default and return spring,

are necessary to lead the plate to the home position in case of

failure. Notice that, for security reasons, when a fault happens

and the motor does not generate a driving torque anymore, the

valve has to come back to a default position, called limp home

position, so the driver can limp until to reach the nearest car

service in a safe way. Obviously the valve is not completely

closed at the limp home.

Fig. 1. Throttle Body scheme

By assuming an ideal reduction gear and applying the

Newton’s law to the mechanical subsystem and the Kirchhoff’s

laws to the electrical part related to the armature coil, the

mathematical model of the system can be derived as:

⎧

⎪

All the variable meaning and the related symbols can be found

in Table I.

The main nolinearity in the model is due to the friction

torque depending upon the low quality of the gearbox and

bearings. In engineering, friction plays an important role. It

is the source of self-sustained oscillations that often have

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

⎨

di

dt= −R

dωth

dt

dθ

dt= ωth

Li−Kv

=Kt

LGrωth+1

LGri−Tsp(θ)

Lva

−Tfric(ωth)

JJ

(1)

978-1-4244-1684-4/08/$25.00 ©2008 IEEE 552

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undesired effects in many areas of engineering [3]. The friction

in the ETB is a complex, nonlinear function of throttle angu-

lar velocity which has been modeled by following different

approaches [10], [8], [11], [12]. Since it is important in

applications to conjugate simplicity with accuracy the friction

torque is modelled by assuming a Coulomb viscous Stribeck

function as (for further information see the relevant work of

Hensen [4]):

?

Another nonlinear effect also arises because the stiffness of

the default spring is always greater than stiffness of the return

spring. For this reason, the elastic torque is not simply a linear

function of all the admissible angles, but is a PWL (piecewise

linear) function given by:

Tfric(ωth) =

Tc+(Ts−Tc)e−|ωth

ωs|?

sign(ωth)+βωth.

(2)

Tsp(θ) =

⎧

⎪

⎪

⎪

⎪

⎩

⎨

Ks3

0

Ks1

Ks2

?θ−?θLH−Δθ

?θ−?θLH+Δθ

2

??−Tclose

??+Topen

if θ < θLH−Δθ

2≤ θ ≤ θLH+Δθ

if θLH+Δ

??+Topen

2

if θLH−Δθ

?ˆθ−?θLH+Δθ

2

22≤ˆθ

?θ−ˆθ?+Ks1

2

if θ >ˆθ

(3)

The parametric identification of the system model (1) on the

base of the experimental data has been performed by following

different approaches. In particular the parameters R, L, Kv, Kt

have been identified according to the procedures in the work

of Pavkvic et al. [13]. Values of the parameters J, β, Tc, ωs

have been estimated applying an optimization technique based

on a least-square algorithm. All the remaining parameters have

been derived by the knowledge of the experimental set-up and

geometrical considerations.

The effectiveness of the identification procedure is shown

from the validation results in figure 2, where it is evident that

the model is able to capture the system behavior not only in

steady-state conditions, but also during the valve opening and

closing.

Fig. 2.

experimental data(dotted line) and model predictions (solid line). Results refer

to a triangular shaped input voltage.

Time history of the plate angular position. Comparison between

III. PI CONTROL

The experimental setup is composed by the throttle body, the

DC motor, the micro-controller and the DC power converter.

The DC power converter is based on the integrated circuit

TC4422, a diode RHRP15120, a mosfet IRFBF4710 (rise time

of 30[ns]) and a 4700[pF] capacitor. The control law has

been discretized and then implemented in a low cost micro-

controller PIC PIC16F877A (RAM 368 byte, EEPROM 256

byte, maximum clock frequency 20[MHz]).

The design of the control scheme is based on a simplified

linearized version of model (1) derived under the following

assumptions:

1) the plateposition usually

?

simply reduces to Ks1

θ−

2) the friction action is given only by the term βωth(see

eq. (2))

The control loop, shown in figure 3, is composed by a

PI controller acting on the error between desiderated and

actual plate position and a model based feed-forward action

to compensate nonlinear effects. The presence of a reference

governor, which is essentially a Smooth Trajectory Generator,

helps to reduce errors during rapid transients, while a filter

on the output reduce the noise on the feedback signal. The

worksintherange

θLH+Δθ

2,ˆθ

?

, thus the elastic force in equation (3)

??

θLH+Δθ

2

??

+Topen

Fig. 3. Control scheme

PI gains are tuned in order to ensure a phase margin for

the control system so that it is robust with respect to a

delay of 4ms determined by the ECU sampling time. The

parameters of the transfer functions of the output filter and

the STR filter are tuned heuristically by tacking into account

the main frequencies of the signals in the close loop system,

the necessary degree of attenuation of noise in the measured

signal, and the desiderate response of the closed-loop plant.

The feed-forward action is designed, starting form the friction

and the elastic torque models in eqs. (3)-(2) and the effective

reference signal θrif, as:

Af f=?Tsp(θrif)+Tssign?˙θrif

The effectiveness of the proposed control scheme is verified

via experiments. Performances are summarized in figures 4

and 5 where the controlled signal well tracks the reference,

respectively a piecewise constant and a sawtooth signal. Notice

the presence of the delay in figure 5.

?+β?˙θrif

??

R

KtGr.

(4)

IV. LQ-MCS CONTROL SCHEME

The main disadvantages of the PI control are the time con-

suming tuning of the control parameters and the impossibility

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Fig. 4.

results: plate angular position (blue solid line) against the PWC (piecewise

constant) reference signal.

Time history of the closed-loop plate angular position. Experimental

Fig. 5.

results: plate angular position (blue solid line) against the sawtooth reference

signal.

Time history of the closed-loop plate angular position. Experimental

to guarantee some optimality in the closed-loop performances.

In order to overcome the drawbacks of the PI controller, we

propose an optimal-adaptive model reference approach. This

control algorithm extends the family of the Minimal Control

Synthesis schemes [14], [15], [16].

The main idea is to integrate the classical LQ optimal

control approach with the MCS (Minimal Control Synthesis)

algorithm. In so doing we try to achieve the optimal perfor-

mances of classical LQ control schemes while maintaining the

simplicity of use of the MCS algorithm and its benefits. The

MCS scheme is implemented on the actual plant by selecting,

as a reference model, the nominal model of the plant controlled

via a classical LQ optimal strategy, i.e. choosing the input to

reference model (um(t) in figure 6) as an optimal control input.

In so doing any mismatch between the nominal model and the

actual plant will be compensated by the adaptive action of the

MCS, which will also guarantee stability in those cases where

the LQ strategy alone would fail.

Figure 6 describes the LQ-MCS scheme. Further details,

proof of asymptotic stability based on the passivity theorem,

and performances of the controller on a representative case of

study can be found in [17], [9].

For the design of the ETB control via LQ-MCS, a linear

model is selected as nominal plant. Notice that it is the

same derived in section III for the PI synthesis. The weight

Fig. 6.LQ-MCS control scheme.

matrices Q and R are chosen to guarantee that bandwidth of

the reference model is equal to the closed-loop plant controlled

via PI. Since the LQ-MCS control is a full state control, but

the velocity measure is not available in the actual plant, it is

necessary to derive the velocity information from the angular

potion measurement. To this aim a third-order Butterworth

filter is used, that ensures a zero phase offset in the useful

band and the maximum roll-off outside. The bandwidth of

the filter is tuned taking in account the tradeoff between the

accuracy of the reconstruction and the constraint imposed by

its numerical implementation.

Figure 7 shows the good matching between the LQ refer-

ence model and the nonlinear model plant when the LQ-MCS

control is active.

Fig. 7.

signal; dotted line: output of the optimal reference model; solid line: plant

output

Closed-loop system under LQ-MCS control. Dashed line: reference

The comparison results between the LQ-MCS and the PI

performances are reported in figure 8. It is noteworthy to high-

light the importance of the feed-forward action that improves

the tracking of the reference signal when the PI controller

is active. Notice that the exact compensation of all the plant

nonlinearities is impossible, since the parameters of the plant

change due to mechanical wear. for this reason, results suggest

the adaptive approach can be extremely useful to obtain zero

error in steady state regime even without compensation.

V. CONCLUSION

In this paper we have highlighted the advantages provided

by ETB control and related problems. In order to merge

optimality with robustness the use of a optimal adaptive

scheme, named LQ-MCS, is proposed. The effectiveness of

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Fig. 8. Comparison between PI control and LQ-MCS algorithm. Time history

of the reference signal (black dotted line) v.s. the plant output in the case of

the PI control without feedforward action (green dotted dashed line), the PI

control strategy with feedforward action (red dashed line) and the LQ-MCS

without feedforward action (blue solid line).

TABLE I

NOMENCLATURE

symbol

i[A]

ωth[rad/s]

θ[rad]

[θmin,θmax]

va[V]

R[Ω]

L[H]

Kv[Vs/rad]

description

armature current of the DC motor

plate velocity

plate angular position

minimum and the maximum plate position angles allowed

source voltage across the coil of the armature

equivalent armature coil resistance of the DC motor

equivalent armature coil inductance of the DC motor

velocity constant determined by the flux of the permanent

magnets into the DC motor

torque constant of the DC motor

gear ratio of the reduction gear

moment of inertia of the plate and motor

friction torque

Coulomb friction torque

stiction friction torque

Stribeck velocity

damping coefficient

elastic torque

stiffness coefficients in each region of interest

minimum torque to close the valve

minimum torque to open the valve

limp home position

clearance between the teeth of the gear

discontinuity point of the slope of the elastic torque

Kt[Nm/A]

Gr

J[Kgm2]

Tfric[Nm]

Tc[Nm]

Ts[Nm]

ωs[rad/s]

β[Nms/rad]

Tsp[Nm]

[Ks1,Ks2,Ks3]

Tclose[Nm]

Topen[Nm]

θLH[rad]

Δθ[rad]

ˆθ[rad]

the control action is verified by simulation results. Moreover

the performance of the proposed strategy has been compared

to the one achievable through a simple PI regulator.

VI. ACKNOWLEDGEMENTS

A special thank to Mario Vaccaro, a recent graduated in

Control Engineering at the Universit` a degli Studi di Napoli

Federico II, for his contribution in setting-up the PI controller.

The next step in the future will be the experimental

verification of the LQ-MCS on the ETB problem.

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