IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 18, NO. 6, NOVEMBER 2010 1325

Synthesis and Experimental Validation of the NovelLQ-NEMCSI Adaptive Strategy on an

Electronic Throttle ValveMario di Bernardo, Senior Member, IEEE, Alessandro di Gaeta, Umberto Montanaro, and Stefania Santini

Abstract—This paper is concerned with the design of a noveladaptive controller, namely the linear quadratic new extendedminimal control synthesis with integral action (LQ-NEMCSI).We present for the first time the analytical proof of asymptoticstability of the controller and experimental evidence of the algo-rithm effectiveness for controlling an electronic throttle body: anelement of any drive-by-wire system in automotive engineering,affected by many nonlinear perturbations.

Index Terms—Adaptive control, automotive control, mecha-tronic, nonlinear control, piecewise smooth systems.

I. INTRODUCTION

I N MANY application areas a control action is aimedat guaranteeing optimality with respect to a certain cost

function subject to some constraints. Optimal control schemesare usually characterized by fixed control gains: a classicalapproach is that of the well-known linear quadratic regulators(LQR)[1]. It has been shown that, typically, LQ schemes lackthe flexibility and the structural stability of other more sophis-ticated control approaches as, for instance, exemplified by thetwo significant cases discussed in [2] and [3].

One way of achieving greater control flexibility is to use adap-tive control schemes, where the control gains are appropriatelyvaried according to the system behavior. To address this issue, anovel family of controllers was presented in [4], where the LQaction is provided via an adaptive control strategy.

Here, we present a novel LQ-adaptive algorithm, namely thelinear quadratic new extended minimal control synthesis withintegral action (LQ-NEMCSI), which relies on minimal knowl-edge of the plant. A simpler version of the algorithm can befound in [4]. The proposed strategy is based on the standardMCS algorithm [5] augmented with integral and robust controlactions, where the reference model is a nominal linear modelcontrolled by a classical LQ optimal strategy. In so doing, the

Manuscript received January 23, 2009; revised June 24, 2009. Manuscriptreceived in final form November 20, 2009. First published January 12, 2010;current version published October 22, 2010. Recommended by Associate EditorA. Giua.

M. di Bernardo, U. Montanaro, and S. Santini are with the Departmentof Systems and Computer Engineering, University of Naples Federico II,Naples 80125, Italy (e-mail: mario.dibernardo@unina.it; umberto.monta-naro@unina.it; stsantin@unina.it).

A. di Gaeta is with the Istituto Motori, National Research Council, Naples80125, Italy (e-mail: a.digaeta@im.cnr.it).

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.2009.2037610

controller has the benefits of the adaptive strategy, while alsomatching the performance of the well known LQ regulator.

We found this novel approach particularly feasible to the con-trol of an electronic throttle body (ETB). The ETB is a mecha-tronic device dedicated to the regulation of the air mass flowrate. In this system a shaped body duct regulates the relation-ship between the angular position of the throttle valve and theincoming air flow into the manifold. The desired plate positionis imposed by a microcomputer in a drive-by-wire configuration(see, for example, [6] and [7]).

From the control perspective, the ETB is a highly nonlinearand uncertain plant, since the transmission friction and thereturn spring limp-home nonlinearity significantly affect thesystem performance (see Section V for further details). Anothercontrol requirement is the simplicity of the strategy that hasto be implemented on a typical low-cost automotive micro-controller. For these reasons, we select the ETB control as asignificant and appropriate test problem to design and vali-date, both numerically and experimentally, the LQ-NEMCSIalgorithm.

In the literature, many control schemes have been proposedto solve the ETB control problem. Typically, they are aimed atachieving a small tracking error with a rapid valve time openingwithout overshoot. To solve the problem, often classicalcontrollers, for example those based on a proportional–inte-gral–derivative (PID) structure [8]–[10], are used, but they areequipped with some feed-forward action to compensate thenonlinearities acting on the ETB. Existing compensators canbe divided in those which are model-based (see, for example,[11], [12], and [13]) and those which are not (see [9]). Furthercontrol techniques are based on constrained optimal control[14] and hybrid approaches [13], [15]–[19], but again they arebased on a good knowledge of the plant dynamics.

With respect to the previous approaches, the adaptive law pro-posed in this paper relies on minimal knowledge of the plant andcan be implemented easily without requiring time consumingexperiments for the precise characterization of the system non-linear dynamics. Moreover, the robustness to unmodeled dy-namics and parameter uncertainties is provided by the adaptivegains of the LQ-NEMCSI strategy.

Here, we present for the first time experimental evidenceshowing clearly that the novel scheme can guarantee excellenttracking performance and transient behavior. All experimentsare performed by using a reliable experimental setup based ona dSPACE control station.

1063-6536/$26.00 © 2010 IEEE

1326 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 18, NO. 6, NOVEMBER 2010

II. BACKGROUND

The MCS algorithm was first introduced in [5] as an exten-sion of the Landau model reference adaptive scheme [20]. It re-lies on minimal knowledge of the plant dynamics. Namely, it isassumed that the controlled system (plant) is controllable and ithas unknown parameters but a known phase canonical structure,as

(1)

where , and

......

.... . .

...

with .Notice that for the sake of clarity we will refer to a single

input system, but the MCS approach was shown to be effectivealso in controlling multiple-input–multiple-output (MIMO) sys-tems [21].

The main aim of MCS control is for the plant states, , totrack asymptotically the states, , of a given asymptoticallystable reference model of the form

(2)

with , being some desired reference signaland (Hurwitz matrix) and given in the same canon-ical form as that of the plant [5]. The MCS control input, say

, consists of a feedforward and a feedback action withtime-varying adaptive gains defined as

(3)

with

(4a)

(4b)

(4c)

(4d)

and and being positive scalar adaptation weights. The outputerror is computed as

(5)

where

(6a)

(6b)

and is the solution of the Lyapunov equation

(7)

Fig. 1. Optimal reference minimal control synthesis.

Note that typically the adaptive gains are started from zero, i.e.,and in (4b) and (4d), respectively.

As shown in [5], the MCS controller can be proven to guar-antee asymptotic stability of the error system and to be robustagainst rapidly varying disturbances and unmodelled nonlinearperturbations [21], [22].

In order to track the behavior of an LQ controlled system, theLQ-MCS scheme has been recently proposed as a viable alter-native in [4]. Essentially, the LQ-MCS consists of implementingthe MCS scheme on the actual plant by selecting, as a referencemodel, a nominal linear model of the plant controlled via a clas-sical LQ strategy (i.e., choosing in Fig. 1 as an optimalcontrol input). In so doing, any mismatch between the nominalmodel and the real plant will be compensated by the adaptiveaction of the MCS, which will also guarantee stability in thosecases where the LQ strategy alone would fail. It is possible toconsider the LQ-MCS (shown in Fig. 1) as a simple way to con-jugate the simplicity and optimality of the LQ action with therobustness of the MCS control.

The steps required to synthesize this control scheme are sum-marized in what follows.

1) Identify a nominal linear model of the plant of interest ofthe form

(8)

The model above represents a rough estimate of the plantmatrices that can be used to synthesize a classical optimalcontrol law.

2) Synthesize a classical LQ optimal controller on the nom-inal plant model selected above in order to minimize thetarget cost function, for example

(9)

with and being appropriate weight matrices.3) Implement the LQ-MCS scheme by using the closed-loop

LQ nominal plant as the reference model for the MCSadaptive controller acting on the real plant.

III. LQ-NEMCSI ALGORITHM

We now improve the previous LQ-MCS scheme by addingtwo further control actions according to the MCS philosophy

DI BERNARDO et al.: SYNTHESIS AND EXPERIMENTAL VALIDATION OF THE NOVEL LQ-NEMCSI ADAPTIVE STRATEGY 1327

[23], [24]. In particular, the general strategy is based on theLQ-MCS control signal augmented by an explicit integral actionand switching action to enhance robustness. Note that the ideaof adding an extra switching action to compensate nonlinear dis-turbances is inspired from variable-structure control approachesfor nonlinear systems as explained in [23], [25].

The control action provided by the new scheme, termed asLQ-NEMCSI, is given as follows:

(10)

where is defined as in (3), whereas the two additionalterms are

(11)

(12)

with

(13)

(14)

(15)

being the output matrix of the plant and the output error asin (5). Notice that the weight has to be a positive scalar con-stant such as the weights and in the classical MCS. Further-more, the additional adaptive gains and are initializedto zero ( and ), such asthe standard gains and . We wish to emphasize that theresulting control law (10) is composed of both continuous (3),(11), and discontinuous (12) actions.

A. Proof of Asymptotic Stability

Asymptotic stability of the LQ-NEMCSI algorithm is provenusing to the passivity theorem of Desoer and Vidyasagar [26].

The evolution of the tracking error defined as in (6a) can bederived from the plant and model reference dynamics in (1) and(2) under the switching control action in (10), as

(16)

where models rapidly varying dis-turbances and unmodelled nonlinear perturbations acting on theplant. We remark that the matrix is the closed-loop dynamicmatrix of the LTI reference model (8) under the action of the LQstrategy. After simple algebraic manipulations, given the canon-ical form of matrices and , we can rewrite (16) as

(17)

where

(18)

Fig. 2. Closed-loop error dynamics represented as an equivalent feedbacksystem.

(19)

(20)

(21)

(22)

(23)

The error dynamics (17) can be seen as a nonlinear system witha linear feedforward block and a nonlinear feedback block ac-cording with the block scheme in Fig. 2.

B. Passivity of the Forward Dynamics

Consider now the feedforward path (see (17) and Fig. 2)

(24)

Since the dynamical system (24) has got a linear structure, toprove its passivity it is sufficient to choose a positive definitequadratic storage function such as

with (25)

Using (7), (5), (6b), and (18), the time derivative of the storagefunction can be now computed as

(26)

as the matrix is positive definite. In so doing, passivity ofthe feedforward system is proven [27]. Notice that, sinceis asymptotically stable according to the classical theory on LQcontrol theory, the Lyapunov (7) always has a solution [1].

C. Passivity of the Feedback Dynamics

To prove that the feedback block will only ever produce afinite amount of energy, the Popov’s inequality must be satisfied.Specifically, the following integral inequality must hold [26]:

(27)

for all where is a finite constant independent of. Decomposing the Popov’s integral (27) as

(28)

1328 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 18, NO. 6, NOVEMBER 2010

where

(29)

(30)

then condition (27) is verified if each of the integral terms,and , are greater than some finite negative quantities, sayand .

Consider now the first integral term (29). The MCS adapta-tion law is (see (19))

(31)

with

(32)

where the integral and proportional adaptive components are,respectively

(33)

Thus, by using the above expressions, in (29) can be decom-posed as

(34)

which is greater than a negative definite constant.Consider now the term (30), where . We

have now to prove that

(35)

This is verified if , and thisis trivially satisfied if

(36)

From the definition of the adaptation law for the in (15),it is apparent that the gain evolution is monotonously increasing.Thus, for any given constant value , there always exists afinite time instant, say , such that for all . Ifwe now set and , we can conclude that,for all , the passivity condition (36) is verified together

Fig. 3. (a) Picture of motorized throttle body (DV-E5, Bosch). (b) Schematicof the electronic throttle body.

with the inequality (35). In so doing, passivity of the feedbackpath is proven.

Remarks:• For any time instant the bounded input-output prop-

erty of the feedback passive systems guarantees bounded-ness of the error dynamics [21].

• The convergence of the adaptive gain to a finite valuein the absence of persistent perturbations can be easilyshown by following the approach in [25]. In practice,might keep growing if persistent disturbances are present.In this case, a simple implementation solution is to lock theevolution of over a certain threshold.

IV. EXPERIMENTAL CASE OF STUDY: THE ELECTRONIC

THROTTLE BODY

The ETB [see Fig. 3(a)] is a mechatronic device dedicated tothe regulation of the air mass flow rate of an internal combus-tion engine. It is located between the air filter box and the in-take manifold. When the throttle plate opens an airflow sensordetects this change and communicates it to the electronic con-trol unit (ECU). As a consequence, ECU varies the amount ofinjected fuel in order to maintain the desired air-fuel ratio.

The reference signal is the solution of a trade off betweenthe driver request (acceleration pedal position) and the effec-tive traction possibilities depending upon driveability, safetyand emission constraints. The control signal generated by theECU becomes, by means of an H-bridge power converter, thearmature voltage of a dc-motor. The rotation motion is thentransferred from the motor shaft to the plate shaft through a gearsystem. A schematic of the ETB is shown in Fig. 3(b). Despiteits apparent simplicity, the system behavior is affected by manynonlinearities which can dramatically alter its dynamics. Theycan be briefly summarized as follows.

DI BERNARDO et al.: SYNTHESIS AND EXPERIMENTAL VALIDATION OF THE NOVEL LQ-NEMCSI ADAPTIVE STRATEGY 1329

Fig. 4. Experimental stick and slip phenomena: armature voltage (dashed line)and plate position (solid line). Notice that, for the comparison of the time historyof both the variables on the same plot, the armature voltage is multiplied by 10).

• Piece-Wise Linear Restoring Torque. When a failure of thedc motor occurs, for safety reasons it is necessary to en-sure that the valve comes back to a default position (calledlimp-home position) [28]. To guarantee the limp-home,two additional springs are used. The resulting elastic torqueis a piece-wise linear function of all the admissible angles.

• Friction. The friction is mainly due to the low quality ofthe bearings.

• Impacts. Impacts occur when the plate hits the mechanicalconstraints.

• Backlash. This effect is due to a spacing between the gearteeth.

Another important issue is the uncertainty on the system param-eter values due to manufacturing tolerances, variable operatingconditions or mechanical wear [7].

A. Experimental Nonlinear Behavior

Using the setup described in appendix A, it has been possibleto capture and confirm experimentally the nonlinear behaviorexhibited by the throttle body.

A notable consequence of friction is the presence of stick-slipbehavior. When this kind of unwanted dynamics appear, rigidbody elements alternatively stick and slip with respect to eachother.

The experimental stick-slip oscillations exhibited by thethrottle valve are shown in Fig. 4 under an open-loop slowlyvarying armature voltage.

The combined action of friction and spring torques causes anhysteretic behavior in the system response (see Fig. 5) [11], [29].

When the throttle body dynamics are controlled via the feed-back of the angular position of the plate, other relevant nonlineareffects appear. For instance, it is well known that in systems withfriction under an integral control action (like PID controller),an equilibrium (set point) can be easily transformed into a limitcycle when the parameters change. This undesired phenomenon

Fig. 5. Experimental hysteresis phenomenon: experimental data from the plant(dotted line) and average data (solid line).

Fig. 6. Experimental phase plot showing the hunting phenomenon under a PIcontrol action (the integral gain is 0.05, the proportional gain is 0.1 and referencesignal is 45 [deg]).

is known in the literature as hunting [11], [30], [31]. Experi-mental results show that this is an important nonlinear phenom-enon since the controlled angular plate position exhibits an un-wanted limit cycle in the presence of a constant reference signal(see Fig. 6).

Notice that, since tuning of the controller is a crucial aspectof the design, an adaptive strategy is worthwhile exploring.

B. Performances of the ETB via Classical Controller

In this section, we experimentally investigate the perfor-mance achievable via a properly tuned PI controller equippedwith a model based feed-forward action for the compensationof nonlinear dynamics. A filter is added on the reference signalto limit the tracking error during rapid tip-in/tip-out maneuvers.Further details on control design can be found in [32].

1330 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 18, NO. 6, NOVEMBER 2010

Fig. 7. Experimental results, PI with a model-based feed-forward action: �reference plate position (dashdotted blue line); � filtered reference signal tobe tracked (dashed red line); � angular plate position (solid black line), whenthe reference is a: (a) square wave with period 4 [s] and amplitude of 5 [deg]around 27 [deg]; (b) a step of 55 [deg]; (c) constant at 45 [deg] for long time; (d)a sinusoidal wave with period 3 [s] and amplitude of 25 [deg] around 50 [deg].

Fig. 7 highlights that the PI is not effective in some cases dueto the presence of nonlinearities. In particular: 1) the friction

torque heavily increases the tracking error when the referencesignal is a square wave [see Fig. 7(a) and (b)], especially in thecase of small amplitude signals as in Fig. 7(a); 2) in the caseof low velocities the plate gets stuck because of friction, as inFig. 7(d); 3) the presence of hunting phenomenon is unavoid-able, as in Fig. 7(c).

V. SYNTHESIS OF THE LQ-NEMCSI FOR ETB CONTROL

A. Control Specifications

The achievable control specifications obviously dependon the admissible performance of the adopted hardware (seeAppendix A). Essentially they are related to requirements onthe system dynamic response and they are set as follows:

1) the settling time is required to be 130 [ms] for a valveopening [7] (notice that the settling time is defined as theminimum time after which the throttle plate angle remainswithin 5% of its steady-state value);

2) no overshoot should be present in the step response (fur-thermore, the throttle plate shall never hit the mechanicalend-stroke);

3) the maximum value of the steady state tracking error is setto be 0.1 [deg] [9];

4) the maximum absolute dynamic error should be 7 [deg] [7].Furthermore the control system should be robust to variationsof the process parameters, and the controller synthesis shouldbe simple and automated as much as possible [15].

B. Design of the LQ Controlled Reference Model

As only minimal knowledge of the plant is required for thecontrol synthesis, the first step is the choice of an approximateLTI model of the ETB. This can be found mainly in two differentways: 1) using a standard linear identification procedure and 2)simplifying a more complete nonlinear model of the plant (whenit is available).

In industrial applications, the first approach can be preferable.Specifically, it is assumed that the throttle body is described bya classical rotational mass-spring-damping model, as

(37)

where ; [deg] and [deg/s] are the angularplate position and velocity, respectively. Classical identificationmethods, e.g., LS algorithms, can then be used to tune the modelparameters , , and .

Obviously, if a detailed nonlinear model is available, itssimplified version can be used for model reference design.Here, according with a nonlinear plant model previously de-rived for analysis and numerical simulation purposes [32] (seeAppendix B for further details), the following simplified modelhas been used for the control synthesis:

where , [V] is the armature voltage of thedc motor, [Kgm ] is the total moment of inertia, [Nm/A]is the dc motor torque constant, [Nm/deg] is the equivalent

DI BERNARDO et al.: SYNTHESIS AND EXPERIMENTAL VALIDATION OF THE NOVEL LQ-NEMCSI ADAPTIVE STRATEGY 1331

stiffness due to the restoring springs and [deg/rad]is a dimensional constant.

Once the nominal model has been selected, it is necessary tosynthesize a classical LQ optimal controller for the referencemodel. The weight matrices, and

, have been found heuristically, so as to achieve the desiredsettling time of 130 [ms].

Note that the presence of bounded uncertainties in themodel parameters and unmodelled nonlinear dynamics can beregarded as time varying disturbances acting on the plant (seeSection IV and [21]). This term has been checked to satisfythe matching conditions requirements of the MCS approach asshown in Appendix B.

C. Details on the Implementation of the LQ-NEMCSI Strategy

In implementing the control action (12), the following issueshave to be considered.

1) The presence of the switching function in (12) introduceschattering (high frequency switching of the control signal).In order to avoid this unwanted phenomena, the discontin-uous control action can be smoothed as [23]

(38)

where is a sufficiently small positive constant to be ap-propriately chosen.

2) A smooth trajectory reference (STR) first-order filter isintroduced both to limit the tracking error during tip-in/tip-out conditions and to reduce the unavoidable noise onthe external reference [7].

3) As the velocity of the plate is not available, a proper deriva-tive filter has to be used to reconstruct the plate velocity.In so doing, unavoidable noise and delays are introducedin the closed-loop system along the velocity channel wors-ening the overall performances. A way to overcome thisproblem is to use state observers for the velocity estima-tion according with some approaches that can be found inthe ETB literature (see, for example, [16], [19], [28], [33]).Since one of the most strict industrial requirements is thereduction of the processing time and memory needed forthe implementation of controllers on an ECU [15], here wetest the control scheme assuming in (4a) equal to zero,removing in this way the need for an online estimation ofthe angular velocity. The experimental results confirm thatsuch an “unorthodox” choice does not affect the overallperformance of the controller.

4) As it usually happens when implementing the MCSstrategy [21], the scalar quantities and which modu-late the adaptive gains in (4a) and (4c), have to be chosenheuristically as a trade off between convergence time andreactivity of the control action. Our choice for the weightsof the adaptive strategy is , , ,and .

5) Locking of the adaptive gains was implemented in the mi-crocontroller. In particular, the adaptive gain is frozenwhen it exceeds 25% of the maximum allowable voltagevalue, i.e., battery voltage.

Fig. 8. Experimental results. Square wave with period 4 [s] and amplitude 50[deg]. Angular plate position: � plate position reference (dashdotted blue line);� plate position reference model to be tracked (dashed red line); � plateposition (solid black line). (a) begin of the manoeuvre � � ���� ��� [s]; (b) endof the manoeuvre � � ������� ����� [s].

VI. EXPERIMENTAL RESULTS

The proposed adaptive LQ-NEMCSI controller has beenwidely tested over a long reference signal of the throttle po-sition composed by a mixed sequence of canonical signals,comprising square, sinusoidal, and step functions as well asfree driver commands.

To illustrate the effects of the gain adaptation on the trackingerror, we show in Fig. 8(a) and (b) the reference position andthe valve position close to the beginning of the test and afterabout 100 [s]. We can clearly see that, as time increases, thecontrol gains evolve causing a better and better tracking of thereference model state. Specifically, the undershoot present atthe beginning of the maneuver disappears as the controller gainadapt.

We present next a set of wide opening manoeuvres wherethe valve is required to open and close over a position rangeof about 70 [deg]. This is a particularly challenging problemas the restoring force is, in practice, characterized by differentelastic coefficients depending on the actual position of the plate(see Appendix B). Despite the lack of any explicit modellingof such spring behavior, experiments confirm that the adaptivecontroller does indeed guarantee good tracking performance inthese operating conditions, as shown in Fig. 9. Note that themaximum transient error is below 7 [deg] as required and goesdown to less than 0.1 [deg] at steady-state. The correspondingcontrol signal and gain evolution are within the acceptable ex-perimental bounds.

1332 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 18, NO. 6, NOVEMBER 2010

Fig. 9. Experimental results. Square wave with period 4 [s] and amplitude70 [deg]. (a) Angular plate position: � plate position reference (dashdottedblue line); � plate position reference model to be tracked (dashed red line);� plate position (solid black line). (b) Position tracking error. (c) Controlleradaptive gains: � (solid line), � (dashed-dotted line). (d) Controller outputvoltage.

We move now to the case of small amplitude reference signals.In this scenario the presence of stick-slip motion due to frictionbecomes particularly relevant as the position and velocity vari-ations are relatively low. As shown in Fig. 10 our novel control

Fig. 10. Experimental results. Sinusoidal wave with period 4 [s] and ampli-tude 5 [deg]. (a) Angular plate position: � plate position reference (dashdottedblue line); � plate position reference model to be tracked (dashed red line);� plate position (solid black line). (b) Position tracking error. (c) Controlleradaptive gains: � (solid line), � (dashed-dotted line). (d) Controller outputvoltage.

schemecopeseffectivelywithsuchunwantednonlinearperturba-tions (which again are not modelled explicitly for control systemdesign). Similar results were obtained for different small ampli-tude signals and are not reported here for the sake of brevity.

DI BERNARDO et al.: SYNTHESIS AND EXPERIMENTAL VALIDATION OF THE NOVEL LQ-NEMCSI ADAPTIVE STRATEGY 1333

Fig. 11. Experimental results. Free driver command. (a) Angular plate posi-tion: � plate position reference (dashdotted blue line); � plate position ref-erence model to be tracked (dashed red line); � plate position (solid blackline). (b) Position tracking error. (c) Controller adaptive gains: � (solid line),� (dashed-dotted line). (d) Controller output voltage.

Further validation of the controller was carried out on aset of free manoeuvres as shown in Fig. 11. We observe

Fig. 12. Experimental results. Sequence of steps. (a) Angular plate position:� plate position reference (dashdotted blue line); � plate position referencemodel to be tracked (dashed red line); � plate position (solid black line).(b) Position tracking error. (c) Controller adaptive gains: � (solid line), �(dashed-dotted line). (d) Controller output voltage.

that excellent tracking performance is achieved when morerealistic free driver commands are chosen as reference signals.In particular, for the manoeuvre Fig. 11(a) details on the

1334 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 18, NO. 6, NOVEMBER 2010

evolution of absolute tracking error, adaptive gains and controlaction are traced in Fig. 11(b)–(d), respectively. The trackingerror during all the manoeuvre is less than the one prescribedby automotive requirements (7 [deg]) remaining lower thanabout 2.6 [deg].

Finally, we consider the case of a multistep reference signalapplied to the valve towards the end of the control horizon.Such signal is shown in Fig. 12. Here, again the LQ-NEMCSIcontroller guarantees satisfactory performance over the entiremanoeuvre.

All the validation tests described above confirm that indeedthe new adaptive control scheme presented in this paper is a vi-able and effective strategy to be implemented experimentallyshowing excellent performance when applied to the ETB setupof interest. Note that for the sake of brevity, we omit the evolu-tion of the adaptive gains and determining the amplitudeof the integral and switching actions in the LQ-NEMCSI con-troller for all the experimental tests which have been reported. Ingeneral, we observe the switching action to have small boundedamplitude and similarly for the integral action.

It is worth mentioning here that performing a long ma-noeuvre can be useful to implement an automatic procedurefor the tuning of the controller gains. Namely, the gains couldbe set to the final steady-state values obtained at the end ofthe manoeuvre and then adapted about those values to copewith uncertainties and external disturbances. Despite not beinga requirement of the algorithm we presented in the paper,this might represent a practical low-cost alternative to moreclassical gain tuning schemes.

VII. CONCLUSION

We have presented the synthesis and validation of a novelclass of adaptive controllers, the LQ-NEMCSI. The main fea-tures of these controllers are to require minimal knowledge ofthe plant and to be robust to nonlinear perturbations by meansof an adaptively estimated switching action. The controller isbased on the family of MCS model reference adaptive con-trollers. The idea is to use an optimally controlled referencemodel and implement the control action onto the plant by meansof an adaptive controller consisting of state feedback and feed-forward actions enhanced by means of a switching action andan integral action. Analytical proof of asymptotic stability wasgiven. Then, the controller was validated experimentally on ahighly nonlinear plant of relevance in applications, the ETB.The experimental validation confirms the effectiveness of thenovel scheme to achieve the desired performance in a numberof different situations.

Note that the control design and validation reported in thispaper strongly indicates the feasibility of the new controlstrategy we presented for the control of ETB and other non-linear plants in commercial applications. Future work willaddress this issue.

APPENDIX AEXPERIMENTAL SETUP

The experimental setup consists of: 1) an ETB (DV-E5,Bosch), embedding a dc motor; 2) two dual resistive angular

position sensors; 3) a battery voltage sensor; 4) an H-bridgepower circuit (to drive the DC motor); 5) an Hall effect currentsensor (LTA 50P/SP1, LEM); 6) signal conditioning circuits;7) a station for rapid control prototyping (RCP).

The open-loop response of the ETB plant can be summarizedthrough two characteristic times. Namely, the time necessaryto open wide the valve under a battery voltage step (12 [V]) 100 [ms], and the current-less return time,namely the time necessary to close the valve in free evolution

350 [ms].The RCP is a dSPACE-based Multiprocessor System

equipped with the DS1003 (DSP TMS320C40, 60 Mflops)and DS1004 (DEC Alpha AXP 21164, 600 Mflops) processorboards. An analog DS2201 (20 ch., 12 bit, 30 kHz) and adigital DS4002 (8 ch. CAP/CMP res. 30 bit/200 ns, freq.max. 833 kHz) board allow the I/O handling. The DSP isprogrammed in MATLAB/Simulink (MathWorks) environment[34] and the experiments are managed and instrumented by aControlDesk application [35]. Furthermore, an oscilloscope(TDS-3014, Tektronix) is used to perform high frequencymeasurements.

Note that, using the hardware described above, the executionof our control task is performed in 10 [ s] with a sample timeof 1 [ms].

APPENDIX BMATHEMATICAL MODEL OF THE THROTTLE BODY

The ETB plant (shown in Fig. 3) mainly consists of a dcmotor, a reduction gear and a plate where the springs necessaryto lead the plate in the limp-home position are also located.

The mathematical model of the plant can be constructedstarting from the simple models that describe each part of thesystem and then considering the interaction between them. Theoverall model can then be derived as (see [36] for details on themodel derivation)

(39)

where, given the minimum and the maximum allowed angles,respectively and , [deg] is the plateposition; [deg/s] is the velocity of the plate; is the currentacross the coil of the armature; [V] is thevoltage source across the coil of the armature (being thebattery voltage); [H] is the equivalent inductance of the ar-mature coil; [ ] is the equivalent resistance of the armaturecoil; [Vs/rad] is the velocity constant determined by the fluxof the permanent magnets; [Nm/A] is the torque constant;[Kgm ] is the equivalent moment of inertia; is the transmis-sion ratio due to the gear; [Nm] is the torque due to the pres-ence of the springs which gives the restoring torque; [Nm]represents all friction torques. Parameter values for our experi-mental setup are reported in Table I.

Nonlinear Restoring Torque and Friction: The elastictorque in (39), is not simply a linear function of all the

DI BERNARDO et al.: SYNTHESIS AND EXPERIMENTAL VALIDATION OF THE NOVEL LQ-NEMCSI ADAPTIVE STRATEGY 1335

TABLE IETB MODEL PARAMETERS

admissible angles, but is a PWL (piece-wise linear) functiongiven by

ifififif

(40)

(41a)

(41b)

(41c)

where , , [Nm/deg] are the stiffness coefficients ineach region of interest; [Nm] is the minimum torque nec-essary to close the valve; [Nm] is the minimum torquenecessary to open the valve; [deg] is the limp-home an-gular position; [deg] is the clearance between the teeth ofthe gear; [deg] is the discontinuity point of the slope of theelastic torque. Further details on the experimental derivation ofexpression (40) in the range of interest is reported in [36], whileall parameter values are given in Table I. The identified elastictorque model is shown in Fig. 13.

The model of the friction torque is based on a static Coulombmodel modified in order to include the Stribeck effect as

where [Nm] is the Coulomb friction torque; [Nm] is thestiction friction torque; [deg/s] is the Stribeck velocity[Nms/rad] is the equivalent linear damping coefficient.

Fig. 14 shows the shape of the friction nonlinear term whenthe friction parameters have been set according to those inTable I. Further details can be found in [32], [36], while theparameters values are reported in Table I.

Fig. 13. Experimental restoring torque: (a) plate position greater than the limp-home; (b) plate position lower than the limp home.

Rearranging the Model Equations: We now rearrange (39)in order to put the system in the form prescribed to show therobustness of the MCS approach [25] (see Section IV).

Since the electrical dynamics are faster than the mechanicalones, we can neglect the dynamic of the current, thus yielding

(42)

where the current is a static function of the armature voltageand angular velocity of the plate as

(43)

1336 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 18, NO. 6, NOVEMBER 2010

Fig. 14. Nonlinear term of the friction torque based on identified plant param-eters.

Now we rewrite the spring torque as

(44)

if ,ifif ,if

(45)

where

(46a)

(46b)

(46c)

(46d)

and the friction torque as

(47)

where

(48)

Substituting expressions (44), (47) in the model (42), weobtain

(49)

being .

Define now the state and the input vectors as, , and the parameters vector.

System (49) can be written as

(50)

where

(51)

Considering now the parameter uncertainties , assumed to bebounded, system (50) can be recast as

(52)

with

(53)

where and. Note that, since the parameter uncertainties

and the values of the physical parameter are bounded, thefunction is also be bounded as required in [25].

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[36] A. di Gaeta, U. Montanaro, S. Santini, and M. di Bernardo,“Modeling and identification of an electronic throttle body,” Isti-tuto Motori, National Research Council, Naples, Italy, Tech. Rep.2009RR1887, 2009.

Mario di Bernardo (S’06) received the Ph.D. degreein engineering mathematics from the University ofBristol, Bristol, U.K., in 1998.

He is currently an Associate Professor with theDepartment of Automatic Control, University ofNaples Federico II, Naples, Italy. He was appointedto a Lectureship with the Department of EngineeringMathematics, University of Bristol, in 1997, and thenpromoted to a Readership and a Full Professorship.From 2001 to 2003, he was an Assistant Professorwith the Department of Automatic Control, Univer-

sity of Sannio, Sannio, Italy. His research interests include the broad area ofnonlinear systems, on both dynamics and control. He authored and coauthoredmore than 150 international scientific publications.

Dr. Bernardo is currently Associate Editor of the IEEE TRANSACTIONS ON

CIRCUITS AND SYSTEMS—I: REGULAR PAPERS. He is a member of the orga-nizing committees of the IEEE Symposia on Circuits and Systems and has beenchair or co-chair of many scientific events. In 2004, he was elected to the gov-erning board of the Italian Society for Chaos and Complexity and in 2006 and2009 to the Board of Governors of the IEEE Circuits and Systems Society. He re-ceived funding from major funding bodies and industries including the EPSRC,the European Union, the Italian Ministry of Research and University, Jaguar En-gineering Centre, QinetiQ. On the 28th February 2007, he was honoured withthe title of ’Cavaliere della Repubblica Italiana’ (equivalent to a British OBE)for scientific merits by the President of the Italian Republic.

Alessandro di Gaeta received the “Laurea” degree(M.Sc.) in computer science engineering and thePh.D. degree in automatic control from the Univer-sity of Naples “Federico II”, Naples, Italy, in 1999and 2002, respectively.

Since 2003, he has been a Researcher with theIstituto Motori of the National Research Council ofItaly, Naples, Italy, where he is presently with theHigh Efficiency Spark Ignition Engines Division.His research interests include modeling and controlof internal combustion engines and of mechatronic

systems.

Umberto Montanaro received the “Laurea” degree(M.Sc.) in computer science engineering and thePh.D. degree in automatic control from the Univer-sity of Naples Federico II, Naples, Italy, in 2005 and2009, respectively.

His research interests include the modeling of non-smooth systems, adaptive, and switching control.

Stefania Santini received the “Laurea” degree(M.Sc.) in control engineering and the Ph.D. degree inautomatic control from the University of Naples Fed-erico II, Naples, Italy, in 1996 and 1999, respectively.

In 1999, she was a Visiting Researcher withthe Measurement and Control Laboratory (ETHZuerich), Svizzera. In 2001, she became AssistantProfessor of Automatic Control with the Departmentof Systems and Computer Engineering, Universityof Naples Federico II. Her current research interestsinclude the analysis and control of nonsmooth

dynamical systems and automotive control.Dr. Santini is a member of the Technical Committee on Automotive Controls

of the IEEE Control System Society.