Design and Validation Of a Gain-Scheduled Controller for ETC

8/7/2019 Design and Validation Of a Gain-Scheduled Controller for ETC

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18 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 19, NO. 1, JANUARY 2011

Design and Validation of a Gain-Scheduled Controllerfor the Electronic Throttle Body inRide-by-Wire Racing Motorcycles

Matteo Corno, Mara Tanelli, Member, IEEE, Sergio M. Savaresi, Member, IEEE, and Luca Fabbri

AbstractThis paper presents the analysis, design and vali-dation of a gain-scheduled controller for an electronic throttlebody (ETB) designed for ride-by-wire applications in racingmotorcycles. Specifically, the open-loop dynamics of the systemare studied in detail discussing the effects of friction based onappropriate experiments. Further, a linear time invariant nominalmodel of the system to be controlled is experimentally identifiedvia a frequency-domain black box approach, together with the un-certainty bounds on the model parameters. Based on these resultsa model-based gain-scheduled proportional-integral-differential(PID) controller for throttle position tracking is proposed. Theclosed-loop stability of the resulting linear parametrically varying(LPV) system is proved by checking the feasibility of an appro-priate linear matrix inequality (LMI) problem, and the state spacerepresentation of the closed-loop LPV system is experimentallyvalidated. Finally, the performance of the controlled system iscompared to the intrinsic limit of the actuator and tested underrealistic use, namely both on a test-bench employing as set-pointthe throttle position recorded during test-track experiments andon an instrumented motorcycle.

Index TermsElectronic throttle body (ETB), gain-scheduledcontrol, linear parameter varying (LPV) model validation, motor-cycle dynamics.

I. INTRODUCTION AND MOTIVATION

THE electronic throttle body (ETB) is a mechatronic actu-ator devoted to the regulation of the air inflow at the en-

gine intake manifold. According to the drive-by-wire paradigm,an accurate control of the ETB dynamics enables a correct andoptimized management of the air mass flow rate, which can bemanaged independently of the riders request. The availabilityof a properly controlled ETB provides several advantages. First

Manuscript received May 08, 2009; revised November 26, 2009; acceptedJuly 14, 2010. Manuscript received in final form August 08, 2010. Date of pub-lication September 07, 2010; date of current version December 22, 2010. Rec-ommended by Associate Editor C. Novara. This work was supported in part byMIUR Project New methods for Identification and Adaptive Control for Indus-trial Systems and by Piaggio & C. S.p.A., Aprilia Brand.

M. Corno is with the Delft Center for Systems and Control (DCSC),Delft University of Technology, 2628 CD Delft, The Netherlands (e-mail:[email protected]).

M. Tanelli and S.M. Savaresi are with the Dipartimento di Elettronica e In-formazione, Politecnico di Milano, 20133 Milano, Italy (e-mail: [email protected]; [email protected]).

L. Fabbri is with Piaggio & C. S.p.A., Aprilia Brand, 30033 Noale, Venice,Italy.

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

Digital Object Identifier 10.1109/TCST.2010.2066565

of all, it can be employed to achieve a regularization of the dy-namic relationship between the gas command and the drivingtorque transmitted to the ground during acceleration maneuvers,thereby offering a smoother vehicle dynamic behavior whichcan significantly enhance the vehicle handling and driveability.Further, theETBis also employed as an engineprotection mech-anism. It ensures that the engine operates within a controlledrange, for example limiting the engine speed and regulating the

idle speed.From a more advanced vehicle dynamics control perspective,

moreover, the ETB offers a way to differently shape the air flowrate behavior in the face of a given acceleration command, thusproviding a means to customize the vehicle dynamic response tothe drivers gas request. This feature also allows vehicle manu-facturers to personalize the vehicle driving feeling by conferringit either a performance-oriented or a comfort-oriented dynamicbehavior, which would be in principle dictated by its mechan-ical layout, simply via a different tuning of the ETB electroniccontrol system.

Finally, of course, an effective ETB control system is a

mandatory building block for the design of traction controlsystem both for four- and two-wheeled vehicles, e.g., [1][3].Note that, mechanically, a throttle is a simple system; it is

mainly comprised of one or more butterfly valves actuated byan electrical motor through a reduction system. The throttledynamic behavior is rendered complex by packaging, cost, andreliability constraints. These constraints often translate intodominant friction and backlash behavior in the transmission,making the control of the valve difficult. In the scientificliterature, several control strategies have been proposed forthrottle actuation in cars with the common aim of achievinggood tracking performance in all working conditions and inthe face of parametric uncertainties and avoiding overshoots,

which are the main source of discomfort for the driver (see,e.g., [1], [4][9]).

Electronic throttle actuation in motorcycles is far lesscommon than in cars; consequently, little has been publishedon this topic in the open scientific literature so far. In particular,in [10] a solution for the ETB control of two-wheeled vehiclesis proposed employing a variable structure control strategy. It isworth noting that the aforementioned manufacturing constraintsbecome even more strict when the ETB is being designed fortwo-wheeled vehicles, especially for racing motorcycles. Massand volumes optimization becomes critical since racing motor-cycles are very sensitive even to small changes in the center

of mass, see, e.g., [11][13]. Furthermore, racing applications1063-6536/$26.00 2010 IEEE


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CORNO et al.: DESIGN AND VALIDATION OF A GAIN-SCHEDULED CONTROLLER 19

Fig. 1. Prototype electronic throttle body used in this work.

are far more demanding from the performance standpoint thanmarketed solutions.

Within this context, this work focuses on the controller de-sign for a prototype ETB for a racing motorcycle (see Fig. 1).In particular, the open-loop dynamics of the system are ana-lyzed and the effects of friction are investigated based on ap-propriate experiments. In this respect, dithering is proposed asa simple way to alleviate the problem. Dithering reduces theeffect of friction, thus enabling the identification of a linearmodel of the mechanism. Specifically, a linear time invariantnominal model of the throttle dynamics is experimentally iden-tified via a frequency-domain black box approach. Based on ex-periments carried out in different operating conditions, namelyon a test-bench and on the instrumented motorbike, the un-certainty bounds on the model parameters have also been es-timated. Based on these results a model-based gain-scheduledPID controller for throttle position tracking is proposed to opti-mize the position tracking performance in response to all the dif-

ferent gas request profiles of interest. The closed-loop stabilityof the resulting linear parametrically varying (LPV) system isinvestigated and proven by checking the feasibility of an appro-priate LMI problem, based on the state space representation ofthe closed-loop system.

Further, to investigate the LPV modeling assumptions whichlead to a statement of equivalence between the input/output andstate space representation of the considered LPV system, a val-idation step is carried out. Specifically, by comparing the sim-ulated LPV closed-loop system with experimental data, the va-lidity of the assumptions regarding the parameter-dependent co-ordinate change employed in the state-space realization of the

closed-loop system is assessed. Finally, the gain-scheduled con-troller performance are validated on a realistic input signal andon the instrumented vehicle and compared with the intrinsiclimits of the actuator. It is believed that the proposed controllermatches the performances obtained by more complex controlarchitectures, see, e.g., [10]. Providing a final controller with asimple and manageable structure is crucial in the considered ap-plication, as the target electronic control unit (ECU) on whichit must be implemented offers a limited computing power.

This paper is organized as follows. Section II describes thesystem and the experimental setup. The open-loop systemdynamics are studied in Section III, whereas the effects offriction and the intrinsic performance limits are discussed in

Section IV. The system identification and the model-basedcontrol law design are presented in Section V. The closed-loop

Fig. 2. Schematic representation of the electronic throttle architecture.

stability is proved, via LPV techniques, in Section VI and theexperimental validation of the LPV modeling assumption ispresented. Section VII introduces the experimental results,comparing the closed-loop performance to the intrinsic ETBlimit and testing the system on an instrumented motorbike.

II. SYSTEM DESCRIPTION AND EXPERIMENTAL SETUP

The ETB under analysis, which is a prototype developed forracing application, is depicted in Fig. 1. Thesystem is comprisedof a dc motor, a planetary reduction gear and a linkage thatconnects the shaft of the motor to the shaft of two valves. Thelinkage is required for packaging reasons. In fact, in motorcycleapplications, mass and volumes optimization is critical: buildingthe body so that the motor and the valve were aligned wouldhave affected volumes distribution in a negative manner. Thesystem is equipped with a safety return spring which ensuresthat the engine air is cut off in case of failure of the electronic

system. Two sensors are available for identification and control:an angular potentiometer to measure the throttle plate position,and a Hall effect current sensor measures the motor current.Both sensors have anti-aliasing filters: the potentiometer is fil-tered at 150 Hz, while the current sensor at 500 Hz.

Being the target ECU under development, a National Instru-ments (NI) cRIO real-time controller was used to run experi-ments and for control implementation, while a CAN bus inter-face was employed for data logging. The NI cRIO device is pro-grammable at two different levels; it has an FPGA with a 40MHz clock and a micro controller running at 1 kHz. The motorpulse width modulation (PWM) and the current loop (when

used) are implemented on the field-programmable gate array(FPGA); thanks to this choice, it is possible to have a PWMsignal with a 20 kHz carrier. The 20 kHz frequency was chosenas a trade off between the resolution of the control action andthe satisfaction of the switching hypothesis for the circuit. Theresolution of the PWM is given by the count of FPGA ticks in aperiod of the PWM carrier; thus, the faster the carrier frequency,the lower the final PWM resolution. A carrier frequency of 20kHzyields a resolution of 2000 levels.Trying to increase the res-olution any further would cause audible vibrations in the motor.The FPGA also takes care of data sampling; signals are orig-inally sampled at 20 kHz, and then downsampled to 1 kHz inorder to meet the CAN bus bitrate.

The micro controller is left for higher level control routinesand data processing, namely the actual position control loop and


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Fig. 3. Behavior of the throttle position in open loop in quasi-static tests: (a) opening and (b) closing.

set point generation and filtering. It is anticipated that the targetECU will have a sampling rate of 1 kHz and a 20 kHz PWMcarrier. The design of the final throttle position controller willbe carried out considering this final hardware specification, butfor analysis purposes the full potential of real-time controller(i.e., up to 20 kHz of sampling frequency) can be employed.

Fig. 2 shows a block diagram representation of the throttlecontrol system. As can be seen, the electrical dynamicshave been decoupled from the mechanical ones, which aredescribed by the planetary gear, the return spring, a frictionterm and the LTI throttle dynamics . The interconnectionbetween electrical and mechanical ETB components is due tothe electromotive force (E.M.F.). Finally, the system is com-pleted by the position control loop, (s), which regulates

the throttle position to a desired set-point .

III. OPEN-LOOP SYSTEM ANALYSIS

This section is devoted to analyze the open-loop system be-havior, characterized by the electrical dynamics of the dc motorand the mechanical spring characteristic of the throttle body.The electrical dynamics of the dc motor can be described by thefollowing equations:

(1)

where is the voltage applied to the motor, is the windingcurrent, and are the dc motor resistance and inductanceand is the electromotive force, which is proportional to themotor rotational speed . The motor generates a torque whichis proportional to the current , whereas the control variable isthe applied voltage . System (1) shows that the relation be-tween motor voltage and motor torque depends on , and ;this dependency introduces two critical phenomena. First, thestrong dependency that the resistance has on the temperaturetranslates into uncertainties on the torque. Second, the electro-motive force determines a coupling between the electrical andmechanical dynamics. In mechatronics, these issues are typi-cally solved by designing an inner current control loop to reject

these disturbances [14]. This solution yields better results whenthe inner control loop is run at a higher sampling frequency than

the position one which controls the throttle movement. Unfortu-nately, this solution could not be adopted because of the limita-tions inthe clock speed of the target ECU and due to the fact thatcost constraints prevented the use of additional current sensors.However, in the experimental setup, the current loop option be-comes feasible if implemented on the FPGA. As the inner loopcontrol better decouples the mechanical behavior and the elec-trical behavior, it will be employed to estimate the mechanicaldynamics of the return spring, as it makes it easier to isolate andunderstand the analyzed phenomena. As already mentioned, thecurrent dynamics will be left in open loop in the final controller.It will be seen later that this contributes to increase the uncer-tainties affecting the system dynamical model.

To analyze the nonlinear behavior of the throttle position in

open loop refer to Fig. 3(a) and (b), where the throttle positionis plotted as a function of the input current during opening andclosing quasi-static tests, in which the current was increasedalong a very slow ramp. A clear asymmetry is visible betweenthe opening and closing, the former possessing a fully on/offbehavior, whereas the latter shows a sort ofstaircase descent tothe fully closed position. Note, moreover, that the position valueobtained for zero current varies significantly (from 0.2 to 0.38)in the different tests and it does not correspond to a fully closedthrottle. This clearly confirms the criticalities of the system dueto mechanical nonlinearities and friction effects.

Further, the spring characteristic has been identified. To this

aim, a very low bandwidth proportional-integral (PI) positioncontroller has been designed, so as to stabilize the closed-loopsystem dynamics and make the controlled ETB able to followa very slow reference signal constituted by an ascending rampfrom 0 to 1 followed by a descending, symmetrical, one. Fig. 4shows the position-to-current map of the throttle measured inthree different tests. By inspecting Fig. 4 it is apparent that thesystem exhibits a nonlinear hysteretic behavior. Overall, the fol-lowing three different phases can be outlined in the ascendingramp:

from 0.05 to 0.2 the spring stiffness is constant and approx-imately ;

from 0.2 to 0.9 the spring stiffness decreases to a value of

about ; from 0.9 to 1 the spring stiffness increases to .


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Fig. 4. Identification of the spring characteristic in three quasi-static tests.

Fig. 5. Hysteresis amplitude due to friction when dither of different amplitudesand frequency 75 Hz is added to the current set-point.

In the descending phase, instead, the stiffness remains nearly

constant from 1 to 0.2. Forlowerpositions the stiffness increasesand a negative torque is needed to take the throttle valve to thefully closed position.

The fact that negative currents are needed to fully close thevalve is due to security reasons. As a matter of fact, in com-mercial electronic throttles this feature is explicitly requestedby the law and it is usually realized by employing two differentsprings. The non-zero position corresponding to zero current isthe so-called limp-home position, see, e.g., [1], [5], [6].This fea-ture allows the rider, in case of faults in the dc motor, to safelymove the vehicle off the road. In our case, as the consideredthrottle was designed for racing motorcycles, no explicit speci-fications on the exact value of the limp-home position were fol-lowed, but nonetheless the spring was designed so that in caseof an electric fault the motorcycle engine is guaranteed not tobe instantaneously switched off.

IV. FRICTION EFFECTS AND PERFORMANCE LIMITS

As it emerged in the analysis performed in Section III, thehysteretic spring behavior is due to significant friction effects,mainly due to stiction phenomena. As friction does indeeddegradate the final position controller performance, one maythink of adding a dithering signal to the current input of thedc motor. The chosen dithering signal is a sinusoidal signal,whose frequency is tuned so as to be within the bandwidth of

the electrical dynamics of the dc motor and sufficiently highto not interfere with the regulation of the throttle position.

Fig. 6. Open-loop valve opening (solid line) and closing (dashed line) whenthe maximum and the minimum voltage is applied.

From the final application viewpoint, the tradeoff in designingthe dithering signal is given byon one handthe desire toreduce the stiction effect (the final effect of the dither shouldbe that of keeping the throttle valve excited and just beyond themovement point) andon the otherto choose a dithering am-plitude and frequency which do not cause an excessive powerconsumption. The electric power consumption due to dithering,in fact, can be computed as , whereis the dither amplitude and is the battery voltage (12 V). Due to the system low pass dynamical behavior, thedither amplitude needs to be increased proportionally to itsfrequency; thus, the lowest possible frequency compatible with

the throttle position dynamics should be chosen to minimizepower consumption.

Fig. 5 shows the positive effects of dithering on the hysteresisamplitude due to friction measured in the same quasi-static testsdescribed in Section III. The frequency of the dithering signalhas been set to 75 Hz, while different values of the dither ampli-tude have been tested. As can be seen by inspecting Fig. 5,dithering significantly reduces the hysteresis amplitude, whichdecreases from the original value of approximately 0.5 to 0.05A and the residual hysteresis can be considered negligible forcontrol design purposes. Moreover, it is apparent that increasingthe dither amplitude above 0.5 A does not add signifi-

cant improvements to stiction reduction, while it causes a largerpower consumption. Thus, a dithering signal of amplitude0.5 A seems appropriate for the considered system. Note, fi-nally, that around the limp-home position the ditherhas no effecton the hysteresis. This is due to the mechanical spring layout,whichas discussed aboveis explicitly designed to be stifferaround the limp-home position. It should be noted that the pre-vious analysis was carried out with the help of the inner currentloop; in the final implementation the dither must be applied asa sinusoidal variation of the PWM command.

Before addressing the position control design, it is interestingto investigate the intrinsic limits of the considered electronicthrottle, in order to have a benchmark for the performance eval-

uation of the final closed-loop system. These limits were testedboth for the opening and closing dynamics by applying either


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Fig. 7. Open-loop valve position (solid line) and normalized current (dashed line) for a maximum voltage opening (left) and closing (right) test.

Fig. 8. Estimatedfrequencyresponsesfrom measured data G ( j ! ) in threedifferent experiments, the nominal parametric model

G ( s )

and the associ-ated uncertainty bounds.

the maximum or the minimum voltage to the dc motor. The re-

sults are shown in Fig. 6: as can be seen, the 01 opening occursin 87 ms, while the closing in 73 ms.More details are shown in Fig. 7, where the position is plotted

along with the normalized current. Analyzing Fig. 7, the effectsof the electromotive force and the stiction can be noted. Fo-cusing on the opening dynamics, in the first phase the throttle isnot moving and the current reaches its peak; once the initial fric-tion is broken the throttle starts moving thus generating a elec-tromotive force that the battery cannot overcome and thereforea drop in the current is observed. The same behavior is mirroredin the closing dynamics.

These actuator limits are appropriate for racing applications:as a matter of fact, as it will be shown in Section VII, a profes-

sional driver requests a full-open/full-close throttle variation inat most 100 ms.

V. IDENTIFICATION OF THE THROTTLE DYNAMICS AND

CONTROLLER DESIGN

For the design of the throttle position control loop, a classicalmodel-based indirect design approach has been used (see, e.g.,[15]). The first step of this approach is to derive a model ofthe controlled system. Classical black-box open loop modelidentification requires to excite the system with an input signalwhose frequency components span the frequency range ofinterest. This approach could not be applied to the system athand because of the two-state behavior of the open-loop throttleshown in Fig. 3(a) and (b). Specifically, note that as soon as theexcitation signal reaches an amplitude large enough to breakthe static friction, the throttle plate immediately gets to a fully

open (or closed) configuration.This problem can be solved carrying out the identification in

closedloop. To this end, a low-bandwidth position controller hasbeen designed and a frequency-domain identification procedureimplemented. Specifically, the position controller was fed witha reference signal constituted by a multi-frequency sinusoidalsignal (from 0.01 to 20 Hz) of amplitude 0.05 centered aroundthe nominal position .

Then, in order to estimate a non parametric model of thefrequency response of the overall system the inter-mediate PWM signal and the output position have been em-ployed. Namely, the frequency response estimate

has been computed according to the following expression [16]:

(2)

where denotes the cross spectrum of . Note that theadopted identification procedure yields an unbiased frequencyresponse estimate also in the case of closed-loop identification[16].

Fig. 8 shows the comparison of the estimated frequency re-sponses obtained from three different experiments: two testsperformed on the test bench and one test performed on the in-strumented bike. As can be seen, the identified models exhibit a

certain degree of variability, which will be thoroughly addressedin the next section.


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Fig. 9. Controlled system responses to a 0.1 and a 0.6 position reference stepcommands. The responses have been normalized to improve readability.

Fig. 10. Architecture of the gain-scheduled PID controller.

Finally, using the obtained non-parametric frequency re-sponse estimate in the first test bench test (dashed-dotted linein Fig. 8), a transfer function model for the system hasbeen obtained, by solving a nonlinear weighted least-squaresfitting problem. Namely, the parameter vector is the mini-mizer of the following cost function:

(3)

where the weights have been tuned to privilege the fittingwithin the frequency range [5, 15] Hz, are the sam-ples of the frequency response estimate (i.e., that obtained in

the first test bench test). In this work the optimization problem(3) has been solved via an iterative approach based on thedamped Gauss-Newton method [17]. The model order has beendetermined trying to obtain a good tradeoff between modelcomplexity and accuracy. A better fitting could have beenobtained by increasing the model order, but this would leadto the inevitable risk of over-fitting. It has been found moreuseful to focus the optimization procedure on the frequencyrange [515] Hz, which is the interval within which the desiredcutoff frequency of the closed-loop system is expected to be,and therefore the frequency range where a more accurate modelis needed.

Note that in order to avoid numerical issues potentially asso-

ciated with frequency-domain polynomial fitting, the data havebeen rescaled by normalizing the frequency range of interest.

The final expression of the identified frequency-response ofthe nominal system has the form

(4)

Based on the parametric nominal model (4), a fixed-structure

two-degrees of freedom PID controller with anti-windup andset-point weighting, [18], [19], has been implementedin ve-locity formwith the aim of achieving a cutoff frequency of 10Hz and a phase margin of 70 , needed to minimize closed-looposcillations and overshoots, which have to be avoided as muchas possible as they are felt by the rider and limit his/her confi-dence in the vehicle.

Note that, by analyzing a professional driver request (see,e.g., Fig. 19), onenotices that thefastest full-open/full-close ma-neuver lasts approximately 100 ms. Thus, to mimic real inputs,from here on over the employed set point signal is filtered witha 20 Hz low-pass filter. The continuous time transfer functionof the regulator can be written as

(5)

where are the ideal PID tuning parameters, andis the set-point weight of the derivative term. The model uncer-tainties, previously mentioned and shown in Fig. 8, are also con-firmed by the closed-loop validation of the controlled system.Fig. 9 shows the normalized responses of the controlled systemto two different position step commands: a 0.10 and a 0.6 step.Inspecting Fig. 9, it is apparent that the two responses are quali-

tatively different. Namely, the closed-loop response to the smallamplitude step exhibits an overshoot, whereas the large ampli-tude step response is well damped.

As the final controller must achieve good performance andabsence of overshoots in the face of all possible reference sig-nals compatible with a drivers gas request, the controller pa-rameters must be properly tuned to improve the closed-loop per-formance in the case of small amplitude set-point variations.To this aim, a scheduling strategy for the PID parameters asfunctions of the requested position variation seems a promisingchoice. Note that, in principle, when a single control systemmust be designed in order to guarantee the satisfactory closed-

loop operation of a given plant in many different operating con-ditions a genuine gain scheduling approach canbe followed, see,e.g., [12], [20], [21], [22]. This framework asks to find one ormore scheduling variables which completely parameterize theoperating space of interest and to define a parametric family oflinearized models for the plant associated with the set of oper-ating points of interest. Finally, a parametric controller can bedesigned to ensure the fulfillment of the desired control objec-tives in each operating point (see, e.g., [23][28]).

In the present case, though, the linear parameter-varying(LPV) identification step is nearly impossible to perform on theETB. In fact, the underlying assumption of LPV identificationtechniques (see [29][32]) is that the identification procedure

can rely on one global identification experiment in which boththe control input and the scheduling variables are (persistently)


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Fig. 11. Adaptation laws of the integral time T and derivative constant T of the PID controller.

excited in a simultaneous way. This cannot be done on theETB where genuine open-loop identification is unfeasible.Thus it has been decided to experimentally determine (bytrial-and-error) an adaptation rule of the PID controller param-eters and then prove its stability a posteriori. This approach hasanother significant advantage in the context of the consideredapplication, which relates to computational complexity. As amatter of fact, besides the need to store the lookup tables withthe adaptation functions, the controller order and its structureare unaltered, whereas genuine LPV controllers have in generalcomplex and high-order structures and are computationallyvery intensive. Thus, the proposed solution is particularlysuitable for being implemented on motorcycle ECUs, whichoffer a limited computing capability.

Hence, several experiments have been performed to estimate

the static maps used to schedule the integral time and thederivative constant as static functions of the requested posi-tion variation . The proportional gain and the set-pointweight have proved to be effective when proper constantvalues are chosen. The final controller architecture is shownin Fig. 10. Note that, as the chosen scheduling variable, i.e.,the requested set-point variation , is anti-causal, it is in factcomputed based on the set-point derivative as follows:

(6)

Equation (6) shows that is computed by considering (atthe current time instant ) the averaged value of the set-pointderivativeaveraged over a time window and prop-agating it forward over by assuming that it remainsconstant over the latter time interval. The values of and havebeen experimentally tuned to 7 and 100 ms, respectively.

Fig. 11 shows the adaptation laws of the integral time andderivative constant of the PID controller. As can be seen, the

adaptation law is simply a linear one, whereas the schedulingof the integral time , which experiments have shown to bethe most influential parameter, is more elaborate. The shape of

the curve has been derived pointwise and then interpolated withcontinuous functions.

Fig. 12. Step responses of the closed-loop system: fixed-structure PID con-troller (dashed line) and gain-scheduled PID controller (solid line) for smallstep responses (top plot) and big step responses (bottom plot).

Fig. 12 assesses the effectiveness of the proposed gain-sched-uled PID controller, showing a comparison of the step responsesobtained with the fixed structure and the gain-scheduled PIDcontroller on a 0.1 and 0.7 reference step commands. As canbe seen, the gain-scheduled PID controller renders the responsebetter damped than the fixed structure one andmost impor-tantlythe closed-loop performance is consistent for all set-point amplitudes.

VI. STABILITY ANALYSIS AND LPV MODEL VALIDATION

In the previous section, a gain-scheduled PID controller forthe throttle position tracking has been proposed. Specifically,the scheduling laws for the PID parameters have been experi-

mentally determined by optimizing the closed-loop system re-sponse to different reference inputs. The proposed solution is


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Fig. 13. Time domain validation results of the LPV model at a frequency of 1.4 Hz (upper plot) and of 6.25 Hz (lower plot).

based on the scheduling of the integral time and the deriva-tive time of the controller as static functions of the expectedvariation of the reference position signal com-puted via (6).

Once the scheduling law is implemented, the whole con-trolled system can be seen as an LPV system. Furthermore, theLPV framework also accounts for structured uncertainties inthe ETB dynamic model. As a matter of fact, the analysis of theidentified model obtained in different working conditionsseeFig. 8has highlighted that the system model is subject to acertain amount of uncertainty which can be accounted for by

allowing a variability in the position of the first zero and in thetransfer constant of the nominal transfer function .Fig. 8 also shows the uncertainty boundaries when the positionof the lower frequency zero of is moved within theinterval Hz and the transfer constant varies in theinterval . As can be appreciated from Fig. 8,the structured uncertainty describes the variability of the systemin the frequency range of interest. According to the adoptedblack-box approach, the choice of the uncertainty-modelingparameters has been driven by complexity considerations.Specifically, we have looked for the smallest set of parametersthat could account for the whole variability shown by theexperimental data in the frequency range of interest. Threetime-varying parameters are therefore identified, so that theresulting parameter vector can be defined as .Further, note that all the parameters vary with respect to time,with bounds on the velocity of their time variation. The timevariability accounts for dynamic variations both of the systemuncertainties, which are expected to vary as functions of thespecific ETB, of the engine temperature, of the lubricationconditions, and of the set-point, i.e., the gas request command.Specifically, the time derivatives bounds on have been setto . The bound on the timederivative of the set-point variation has been determined byanalyzing several gas request profiles commanded by a profes-

sional rider on race circuits tests, while the velocity bounds onthe variation of the zero and of the transfer constant have been

chosen with dynamics compatible with thermal and lubricationeffects. The variability ranges of the parameters vector can bedescribed, in the parameter space, by a 3-D polytope.

By plugging the corresponding value of the parameters inand in and closing the loop, a transfer func-

tion of the closed-loop system for each point in the parameterspace can be obtained. However, to resort to LPV techniquesfor the closed-loop stability analysis, one needs to obtain a statespace representation of the closed-loop system. This step givesrise to two different issues. Specifically, as we start from localmodels of the closed-loop system obtained by evaluating the

parameter vector at fixed points of the polytope, one needs tointerpolate the local models and this would in principle askthat all the state space realizations are in the same coordinatebasis, [33]. Second, in LPV systemswhich are a special classof time-varying systemsthe usual notions of equivalence be-tween input/output (I/O) and state space representations whichhold for LTI systems are not valid anymore, unless a dynamicvariation of the parameters is permitted (see [34], [35]).

In general, the interpolation problem can be dealt with byresorting to balanced realizations [33], [36]. In the consideredcase, however, as both the system and the controller structurewere known, an analytical state-space model both for the uncer-tain ETB dynamics and for the gain-scheduled PID controllerhas been obtained by performing a symbolic realization of both

and . Based on such model, it is possible towrite the LPV closed-loop system as

(7)

where is the position set-point and isthe measured throttle position. As for the equivalence notion be-tween I/O and state space realizations of LPV systems, it shouldbe pointed out that the standard LTI realization is only an ap-proximation in the LPV case. It is well known that for LTI sys-

tems, if the state space model corresponds to thetransfer function then all the state


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Fig. 14. Time domain validation results of the LPV model at a frequency of 8.5 Hz (upper plot) and of 12.5 Hz (lower plot).

space models defined by , where issquare and nonsingular, are equivalent to the original one, in thesense that they give rise to the same input-outputbehavior. In theLPV case, however, the above notion of equivalence class doesnot hold anymore. This concept is better illustrated by consid-ering the state space representation (7) and the parameter-de-pendent coordinate change . If the coordinatetransformation is applied to system (7) one obtains

(8)

From the above relations, it can be seen that the obtained re-alization (7) can be regarded as a good approximation of theLPV system, i.e., the state-space model can be considered suf-ficiently close to its I/O representation, only if the time varia-tion of the underlying coordinate transformation, i.e., the term

is negligible, which corresponds toaccounting for a staticparametric dependence only in theI/O-to-state-space transformation. Unfortunately, a formal expressionfor the approximation error as a function of the problem data is

very difficult to achieve, and this constitutes a challenging openproblem in the LPV modeling and identification context. How-ever, it is possible to perform a validation step to experimen-tally validate the LPV model (7). Note that this validation issueis rarely addressed in the LPV modeling and control literature,even though it constitutes a crucial part in assessing the sound-ness of any LPV model and controller which is derived basedon local models.

Here, the validity of the state space closed-loop system (7)has been checked by simulating the LPV system (7) and com-paring the results with experiments carried out on the instru-mented motorbike with the ETB controlled via the proposedgain-scheduled PID controller. To account for the parametric

uncertainties in the ETB dynamics, in the simulations the twouncertain parameters and were varied by applying sinusoidal

perturbations with amplitude and frequency tuned according totheir respective magnitude and velocity bounds. As for the setpoint variation , it was computed via the measured set-pointposition according to (6) and used as input for the LPV modelsimulation.

The results of this validation step are shown in Figs. 13 and14, where the simulated and measured closed-loop throttle posi-tions are compared, using a highly exciting sine sweep referencesignal spanning the frequency range from 0.5 to 15 Hz. Specif-ically, to increase readability, Figs. 13 and 14 show four detailsof the validation results, at four different frequencies within the

whole frequency span of the experiment.As can be seen in Figs. 13 and 14 the simulated responseshows very good agreement with the measured one, therebyconfirming the validity of the state-space LPV model (7) ob-tained for the closed-loop system. Once the LPV state spacerepresentation has been validated, it is possible to apply LPVstability analysis techniques. Here reference is made to the fol-lowing result [23], [24].

Theorem 6.1: The system (7) is stable if there is a matrix-valued function satisfying

for all , where and is the boundon the time derivative of the vector . The notationindicates that every combination of and should beincluded in the inequality.

The above problem is an infinite dimensional one. In partic-ular, the infinite dimensionality comes from the fact thatis a function of and that the above conditions must hold forall . Several techniques are available in the literature toreduce the problem to a finite dimensional one. In this contextthe parameter space gridding (see, e.g., [12]) has been preferred.

Namely, the following steps are performed:1) grid the set ;


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2) pick a basis for so that ;3) check the conditions of Theorem 6.1 for each point of the

grid.For the case under study, the following basis has been em-

ployed:

(9)

and a 512 points grid has been chosen, constituted by the 8points for each polytope coordinate. The problem is finallytranslated into a system of 4608 LMIs, whose feasibility issuccesfully checked via YALMIP [37] and SeDuMi.

Remark 6.1: It should be noted that the gridding approachdoes not formally guarantee stability unless certain conditionson the gridding density are satisfied [23]. In principle, onemay think of avoiding the gridding procedure, as there existapproaches in the LPV literature which formulate the stabilityanalysis problem as a feasibility LMI problem of finite dimen-sion, see, e.g., [38][41]. To do this, however, the LPV systemmust be written either as a linear fractional representation(LFR) or as an affine LPV system. Unfortunately an LFRrepresentation could not be derived for the system at handand an affine system structure could not be forced withoutadding too much conservativeness. Specifically, to write thesystem closed-loop dynamic matrix in such an affine formvia a set of time varying parameters, the parameter space hasto be significantly enlarged and, most importantly, it wouldloose its physical meaning and would not contain the set-pointvariation . For the above reasons, the gridding approach waspreferred. Note, however, that the gridding approach has the

advantage of being readily applicable in the case one shoulddecide to model the uncertainty in a different way from thatconsidered herein.

Remark 6.2: Numerically, the LMI feasibility problem isvery sensitive to the condition number of the involved matrices.Of course, the realization problem to be solved influencesthe condition number of the system matrices and thus theLMI problem itself. In this respect, the most sound numericalapproach is that of using a balanced realization, while avoidingcontrollability and observability canonical forms which areknown to be ill-conditioned, [33], [36].

For the problem at hand, however, we took advantage of the

fact that the dependence of the controller parameters on thescheduling parameters was known, so that a symbolic realiza-tion could be performed. In this case, the final state space model,which was then evaluated at the different points of the grid, didnot present critical numerical issues but for the presence of thecontroller integrator. To alleviate this problem, the controller in-tegrator has been approximated with a low frequency pole. Asshown in Fig. 15, this approximation does not alter the transferfunction of the controller around the cutoff frequency of 10 Hz.

Remark 6.3: The LPV validation and stability analyses havebeen carried out in continuous time. As the final implementationof the control algorithm is done in discrete time, it is importantto verify that the stability properties are not lost in the discretiza-

tion process. The discretization of the controller has been doneaccording to the Eulers forward method, which does not always

Fig. 15. Original controller transfer function (solid line) and low frequencypole approximation (dashed line).

preserve stability. Thus, to assess the validity of the obtained re-sults in a discrete time setting, the approach presented in [42] isconsidered. Specifically, we focus on evaluating two importantquantities , and , the former being the upper bound onthe sampling time that guarantees numerical convergence of thediscretization algorithm and stability preservation, and the latterbeing the maximum local discretization error.

Considering the closed loop continuous time LPV system

with dynamic matrix , one has

(10)

where indicates an eigenvalue and is the spectral ab-scissa. Equation (10) shows that the chosen sampling period of1 ms ensures that the desired properties hold. Further, given amaximum relative local discretization error which can betolerated, an upperbound on the required discretization time canbe computed as

(11)

where

where isthe state space and is the control space.In (10) and(11) the maximization and minimization over have been com-puted using the same grid employed in the stability analysis.Thestate space has been estimated by simulating the continuoustime closed-loop system with different throttle position refer-ences recorded during test tracks for all the parameter values

in the above mentioned grid. The control space is defined bythe upper and lower limits of the dc motor voltage. Using the


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Fig. 16. Comparisonbetween themeasured closed-loop systembehavior (solidline) and the intrinsicopen-loop performance limits(dashed line): position (top)and current (bottom).

above method, it is found that a maximum relative local error of

discretization of 4% is guaranteed with a sampling time smallerthat 1.1 ms. This upperbound is satisfied by the chosensampling frequency of 1 kHz.

VII. EXPERIMENTAL RESULTS

Before turning to test the closed-loop system performanceagainst a real driver request signal, it is worth comparing theachieved closed-loop performance of the throttle position con-trol with the actuator intrinsic limits discussed in Section IV.To this aim, let us refer to Fig. 16, which shows a comparisonbetween the measured closed-loop system behavior and the in-

trinsic open-loopperformance limits both for the outputpositionand the requested dc motor current.As can be seen, the closed-loop system provides perfor-

mances which are indeed quite close to the system limits, henceexploiting the full actuator capability. Note, moreover, thatthese tests have been performed with no dithering applied tothe real system; the measured current shown in Fig. 16 provesthat the measurement noise present in the real system when agenuine dynamic excitation is applied (recall the the need fordithering signal arose in face of quasi-static tests to estimatethe return spring characteristic) provides the requested degreeof dithering by itself.

Further, the system was tested on the instrumented motorbike

on a test track with a professional rider. The complete results onthe 60s-long lap measurements are shown in Fig. 17.

Fig. 17. Plot of the driver request measured in a test track lap (dotted line) andthe measured throttle position (solid line).

Fig. 18. Detail of the closed-loop behavior in response to intermediate varia-tions of the throttle position: set-point (dotted line) and measured (solid line)throttle position.

To better analyze the system performance in response to dif-ferent types of drivers gas modulations, Figs. 18(a) and (b) and

19 show different details of the complete lap, which correspond,respectively to the solid, dotted, and dashed boxes in Fig. 17.Specifically, Fig. 18(a) shows a detail of the time interval

s, where the rider requests variations of the throttleposition in the range 0.450.7 with opening and closing ramps.Further, Fig. 18(b) shows a detail of the maneuver where a veryfine-grain modulation is performed by the rider around smallthrottle openings 0.070.15, which are the most critical as theycontinuouslycross the position range where there is a significantchange in the return spring stiffness (see Fig. 4). As can be seen,the system response is very accurate and it follows the driversrequest with minimal overshoot in both situations.

It is also interesting to analyze Fig. 19, which shows the

system response to a very sharp full-close/full-open drivers re-quest. As can be seen, the fastest drivers request imposes on


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Fig. 19. Detail of the closed-loop behavior in response to intermediate varia-tions of the throttle position: set-point (dashed line) and measured (solid line)throttle position.

the system a full-open/full-close maneuver which lasts approx-

imately 100 ms, hence within the range of the throttle intrinsicdynamic limits. The system response is accurate also in thiscritical case, and the maximum delay in the response is of 10ms, which is considered well beyond the limit of drivers per-ception. Finally, it is worth pointing out that, in the whole test,the steady-state error is of the same order of magnitude as theanalog-to-digital (A/D) converter quantization, i.e., of approxi-mately 0.0015.

VIII. CONCLUDING REMARKS

This paper presented a complete analysis of an electronicthrottle system for ride-by-wire application in racing motorcy-cles. Theelectrical and mechanical dynamics of the system have

been studied and the effects of friction based on appropriate ex-periments analyzed. Further, a model-based gain-scheduled po-sition control system for throttle position tracking has been pro-posed. The stability of the closed-loop system has been provedvia LPV techniques by solving an appropriate LMI feasibilityproblem and the LPV modeling assumptions employed in de-riving a state-space model of the closed-loop system have beenexperimentally validated. Finally, the performances of the con-trolled system have been shown to be very close to the intrinsiclimit of the actuator and the overall gain-scheduled controllereffectiveness has been assessed on an instrumented test vehicle.

ACKNOWLEDGMENT

The authors would like to thank Prof. M. Lovera for the manyfruitful discussions and suggestions on LPV modeling and val-idation issues. They would also like to thank the reviewers fortheir comments and suggestions.

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Matteo Corno jointly receivedthe Masterof Sciencein computer and electrical engineering from the Uni-versity of Illinois, Chicago, and the Laurea Degreecum laude and the Ph.D. degree cum laude with athesis on active stability control of two-wheeled ve-hicles from the Politecnico di Milano, Milan, Italy, in2005 and 2009, respectively.

During his Ph.D., he had a six-month internship atAlenia Spazio (now Thales Alenia Space). In 2008,

he had been a Visiting Scholar with the Universityof Minnesota, Minneapolis. In 2009, After a jointpost-doc position at Politecnico di Milano and Johannes Kepler University,Linz, he joined Delft University of Technology, Delft, The Netherlands, as anAssistant Professor with the Delft Center for System and Control. His currentresearch interests include dynamics and control of two and four wheeledvehicles, nonlinear estimation techniques, and LPV control.

Mara Tanelli (M05) was born in Lodi, Italy, in1978. She received the Laurea degree in computerscience engineering and the Ph.D. degree in infor-mation engineering with a thesis on active brakingcontrol systems design for road vehicles from thePolitecnico di Milano, in 2003 and 2007, respec-tively, and the Master of Science degree in computerscience from the University of Illinois, Chicago, in

2003.She is currently an Assistant Professor of auto-matic control with the Dipartimento di Elettronica

e Informazione, Politecnico di Milano. She is also currently with the Dipar-timento di Ingegneria dellInformazione e Metodi Matematici, Universitdegli studi di Bergamo, Dalmine, Italy. Her main research interests focuson control systems design for ground vehicles, estimation, and identificationfor automotive systems, control systems design for agricultural tractors, andidentification and control for active energy management of data centers.

Dr.Tanelli wasa recipientof theDimitri N. Chorafas Ph.D. ThesisAwardandthe Claudio Maffezzoni Ph.D. Thesis Award for her Ph.D. thesis. In 2008, sheand her coauthors received the Rudolf Kalman Best Paper Award for the bestpaper publishedin 2007in the ASMEJournal of Dynamic Systems Measurementand Control.

Sergio M. Savaresi (M00) was born in Manerbio,Italy, on 1968. He received the M.Sc. degree in elec-trical engineering and the Ph.D. degree in systemsand control engineering from the Politecnico di Mi-lano, Milan,Italy, in 1992 and1996, respectively, andthe M.Sc. degree in applied mathematics from theCatholic University, Brescia, Italy, in 2000.

After the Ph.D., he was a Management Consultantwith McKinsey & Company, Milan, Italy. He wasa Visiting Researcher with Lund University, Lund,Sweden; University of Twente, Ensende, The Nether-

lands; Canberra National University, Australia; Stanford University, Stanford,CA; Minnesota University, Minneapolis; and Johannes Kepler University, Linz,Austria. Since 2006, he has been a Full Professor in automatic control with thePolitecnico di Milano and is currently the Head of the mOve research team(http://move.dei.polimi.it). He is an author of six patents, over 60 papers on In-

ternational Journals, and 150 papers on international conferences proceedings.His main interests include the areas of vehicles control, automotive systems,data analysis and system identification, nonlinear control theory, and controlapplications.

Dr. Savaresi is an Associate Editor of the IEEE TRANSACTIONS ON CONTROLSYSTEMS TECHNOLOGY, the European Journal of Control, and the IET ControlTheory and Applications. He is a member of the Editorial Board of the IEEEControl Systems Society.

Luca Fabbri was born in Owo, Nigeria, in 1963. Hereceived the M.Sc. degree in mechanical engineeringfrom theUniversity of Padova, Padova, Italy, in 1990.

From 1990 to 1993, he was with Aprilia, workingas a mechanical designer in the racing unit. From

1993 to 2006, he was responsible for vehicle devel-opment in the racing unit, where he led the designand development of racing motorcycles for thecategories 125cc, 250cc, 500cc, and SuperBike.Currently, he is the Innovation Manager of theMotorcycle Engineering section for the brand Units

Aprilia e Derbi within the Piaggio Group.

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