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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 19, NO. 1, JANUARY 2011
Design and Validation of a Gain-Scheduled Controller for the Electronic Throttle Body in Ride-by-Wire Racing Motorcycles Matteo Corno, Mara Tanelli, Member, IEEE, Sergio M. Savaresi, Member, IEEE, and Luca Fabbri
Abstract—This paper presents the analysis, design and validation of a gain-scheduled controller for an electronic throttle body (ETB) designed for ride-by-wire applications in racing motorcycles. Specifically, the open-loop dynamics of the system are studied in detail discussing the effects of friction based on appropriate experiments. Further, a linear time invariant nominal model of the system to be controlled is experimentally identified via a frequency-domain black box approach, together with the uncertainty bounds on the model parameters. Based on these results a model-based gain-scheduled proportional-integral-differential (PID) controller for throttle position tracking is proposed. The closed-loop stability of the resulting linear parametrically varying (LPV) system is proved by checking the feasibility of an appropriate linear matrix inequality (LMI) problem, and the state space representation of the closed-loop LPV system is experimentally validated. Finally, the performance of the controlled system is compared to the intrinsic limit of the actuator and tested under realistic use, namely both on a test-bench employing as set-point the throttle position recorded during test-track experiments and on an instrumented motorcycle. Index Terms—Electronic throttle body (ETB), gain-scheduled control, linear parameter varying (LPV) model validation, motorcycle dynamics.
I. INTRODUCTION AND MOTIVATION
HE electronic throttle body (ETB) is a mechatronic actuator devoted to the regulation of the air inflow at the engine intake manifold. According to the drive-by-wire paradigm, an accurate control of the ETB dynamics enables a correct and optimized management of the air mass flow rate, which can be managed independently of the rider’s request. The availability of a properly controlled ETB provides several advantages. First
T
Manuscript received May 08, 2009; revised November 26, 2009; accepted July 14, 2010. Manuscript received in final form August 08, 2010. Date of publication September 07, 2010; date of current version December 22, 2010. Recommended by Associate Editor C. Novara. This work was supported in part by MIUR Project “New methods for Identification and Adaptive Control for Industrial 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 Informazione, Politecnico di Milano, 20133 Milano, Italy (e-mail: tanelli@elet. polimi.it;
[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 online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCST.2010.2066565
of all, it can be employed to achieve a regularization of the dynamic relationship between the gas command and the driving torque transmitted to the ground during acceleration maneuvers, thereby offering a smoother vehicle dynamic behavior which can significantly enhance the vehicle handling and driveability. Further, the ETB is also employed as an engine protection mechanism. It ensures that the engine operates within a controlled range, 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 flow rate behavior in the face of a given acceleration command, thus providing a means to customize the vehicle dynamic response to the drivers’ gas request. This feature also allows vehicle manufacturers to personalize the vehicle driving feeling by conferring it either a performance-oriented or a comfort-oriented dynamic behavior, which would be in principle dictated by its mechanical layout, simply via a different tuning of the ETB electronic control system. Finally, of course, an effective ETB control system is a mandatory building block for the design of traction control system 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 by an electrical motor through a reduction system. The throttle dynamic behavior is rendered complex by packaging, cost, and reliability constraints. These constraints often translate into dominant friction and backlash behavior in the transmission, making the control of the valve difficult. In the scientific literature, several control strategies have been proposed for throttle actuation in cars with the common aim of achieving good tracking performance in all working conditions and in the 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 less common than in cars; consequently, little has been published on this topic in the open scientific literature so far. In particular, in [10] a solution for the ETB control of two-wheeled vehicles is proposed employing a variable structure control strategy. It is worth noting that the aforementioned manufacturing constraints become even more strict when the ETB is being designed for two-wheeled vehicles, especially for racing motorcycles. Mass and volumes optimization becomes critical since racing motorcycles are very sensitive even to small changes in the center of mass, see, e.g., [11]–[13]. Furthermore, racing applications
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Fig. 1. Prototype electronic throttle body used in this work. Fig. 2. Schematic representation of the electronic throttle architecture.
are far more demanding from the performance standpoint than marketed solutions. Within this context, this work focuses on the controller design for a prototype ETB for a racing motorcycle (see Fig. 1). In particular, the open-loop dynamics of the system are analyzed and the effects of friction are investigated based on appropriate experiments. In this respect, dithering is proposed as a simple way to alleviate the problem. Dithering reduces the effect of friction, thus enabling the identification of a linear model of the mechanism. Specifically, a linear time invariant nominal model of the throttle dynamics is experimentally identified via a frequency-domain black box approach. Based on experiments carried out in different operating conditions, namely on a test-bench and on the instrumented motorbike, the uncertainty bounds on the model parameters have also been estimated. Based on these results a model-based gain-scheduled PID controller for throttle position tracking is proposed to optimize the position tracking performance in response to all the different gas request profiles of interest. The closed-loop stability of the resulting linear parametrically varying (LPV) system is investigated and proven by checking the feasibility of an appropriate LMI problem, based on the state space representation of the closed-loop system. Further, to investigate the LPV modeling assumptions which lead to a statement of equivalence between the input/output and state space representation of the considered LPV system, a validation step is carried out. Specifically, by comparing the simulated LPV closed-loop system with experimental data, the validity of the assumptions regarding the parameter-dependent coordinate change employed in the state-space realization of the closed-loop system is assessed. Finally, the gain-scheduled controller performance are validated on a realistic input signal and on the instrumented vehicle and compared with the intrinsic limits of the actuator. It is believed that the proposed controller matches the performances obtained by more complex control architectures, see, e.g., [10]. Providing a final controller with a simple and manageable structure is crucial in the considered application, as the target electronic control unit (ECU) on which it must be implemented offers a limited computing power. This paper is organized as follows. Section II describes the system and the experimental setup. The open-loop system dynamics are studied in Section III, whereas the effects of friction and the intrinsic performance limits are discussed in Section IV. The system identification and the model-based control law design are presented in Section V. The closed-loop
stability is proved, via LPV techniques, in Section VI and the experimental validation of the LPV modeling assumption is presented. Section VII introduces the experimental results, comparing the closed-loop performance to the intrinsic ETB limit and testing the system on an instrumented motorbike. II. SYSTEM DESCRIPTION AND EXPERIMENTAL SETUP The ETB under analysis, which is a prototype developed for racing application, is depicted in Fig. 1. The system is comprised of a dc motor, a planetary reduction gear and a linkage that connects the shaft of the motor to the shaft of two valves. The linkage is required for packaging reasons. In fact, in motorcycle applications, mass and volumes optimization is critical: building the body so that the motor and the valve were aligned would have affected volumes distribution in a negative manner. The system is equipped with a safety return spring which ensures that 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 filtered at 150 Hz, while the current sensor at 500 Hz. Being the target ECU under development, a National Instruments (NI) cRIO real-time controller was used to run experiments and for control implementation, while a CAN bus interface was employed for data logging. The NI cRIO device is programmable at two different levels; it has an FPGA with a 40 MHz clock and a micro controller running at 1 kHz. The motor pulse 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 PWM signal with a 20 kHz carrier. The 20 kHz frequency was chosen as a trade off between the resolution of the control action and the satisfaction of the switching hypothesis for the circuit. The resolution of the PWM is given by the count of FPGA ticks in a period of the PWM carrier; thus, the faster the carrier frequency, the lower the final PWM resolution. A carrier frequency of 20 kHz yields a resolution of 2000 levels. Trying to increase the resolution any further would cause audible vibrations in the motor. The FPGA also takes care of data sampling; signals are originally sampled at 20 kHz, and then downsampled to 1 kHz in order to meet the CAN bus bitrate. The micro controller is left for higher level control routines and 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 target ECU will have a sampling rate of 1 kHz and a 20 kHz PWM carrier. The design of the final throttle position controller will be carried out considering this final hardware specification, but for 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 throttle control system. As can be seen, the electrical dynamics have been decoupled from the mechanical ones, which are described by the planetary gear, the return spring, a friction . The interconnection term and the LTI throttle dynamics between electrical and mechanical ETB components is due to the electromotive force (E.M.F.). Finally, the system is com(s), which regulates pleted by the position control loop, the throttle position to a desired set-point . III. OPEN-LOOP SYSTEM ANALYSIS This section is devoted to analyze the open-loop system behavior, characterized by the electrical dynamics of the dc motor and the mechanical spring characteristic of the throttle body. The electrical dynamics of the dc motor can be described by the following equations: (1) where is the voltage applied to the motor, is the winding and are the dc motor resistance and inductance current, is the electromotive force, which is proportional to the and motor rotational speed . The motor generates a torque which is proportional to the current , whereas the control variable is the applied voltage . System (1) shows that the relation be, and ; tween motor voltage and motor torque depends on this dependency introduces two critical phenomena. First, the strong dependency that the resistance has on the temperature translates into uncertainties on the torque. Second, the electromotive force determines a coupling between the electrical and mechanical dynamics. In mechatronics, these issues are typically solved by designing an inner current control loop to reject these disturbances [14]. This solution yields better results when the inner control loop is run at a higher sampling frequency than
the position one which controls the throttle movement. Unfortunately, this solution could not be adopted because of the limitations in the clock speed of the target ECU and due to the fact that cost constraints prevented the use of additional current sensors. However, in the experimental setup, the current loop option becomes feasible if implemented on the FPGA. As the inner loop control better decouples the mechanical behavior and the electrical behavior, it will be employed to estimate the mechanical dynamics of the return spring, as it makes it easier to isolate and understand the analyzed phenomena. As already mentioned, the current dynamics will be left in open loop in the final controller. It will be seen later that this contributes to increase the uncertainties 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 position is plotted as a function of the input current during opening and closing quasi-static tests, in which the current was increased along a very slow ramp. A clear asymmetry is visible between the opening and closing, the former possessing a fully on/off behavior, whereas the latter shows a sort of staircase descent to the fully closed position. Note, moreover, that the position value obtained for zero current varies significantly (from 0.2 to 0.38) in the different tests and it does not correspond to a fully closed throttle. This clearly confirms the criticalities of the system due to mechanical nonlinearities and friction effects. Further, the spring characteristic has been identified. To this aim, a very low bandwidth proportional-integral (PI) position controller has been designed, so as to stabilize the closed-loop system dynamics and make the controlled ETB able to follow a very slow reference signal constituted by an ascending ramp from 0 to 1 followed by a descending, symmetrical, one. Fig. 4 shows the position-to-current map of the throttle measured in three different tests. By inspecting Fig. 4 it is apparent that the system exhibits a nonlinear hysteretic behavior. Overall, the following three different phases can be outlined in the ascending ramp: • 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. 6. Open-loop valve opening (solid line) and closing (dashed line) when the maximum and the minimum voltage is applied.
Fig. 5. Hysteresis amplitude due to friction when dither of different amplitudes and 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. For lower positions the stiffness increases and a negative torque is needed to take the throttle valve to the fully closed position. The fact that negative currents are needed to fully close the valve is due to security reasons. As a matter of fact, in commercial electronic throttles this feature is explicitly requested by the law and it is usually realized by employing two different springs. The non-zero position corresponding to zero current is the so-called limp-home position, see, e.g., [1], [5], [6]. This feature allows the rider, in case of faults in the dc motor, to safely move the vehicle off the road. In our case, as the considered throttle was designed for racing motorcycles, no explicit specifications on the exact value of the limp-home position were followed, but nonetheless the spring was designed so that in case of an electric fault the motorcycle engine is guaranteed not to be instantaneously switched off. IV. FRICTION EFFECTS AND PERFORMANCE LIMITS As it emerged in the analysis performed in Section III, the hysteretic spring behavior is due to significant friction effects, mainly due to stiction phenomena. As friction does indeed degradate the final position controller performance, one may think of adding a dithering signal to the current input of the dc 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 high to not interfere with the regulation of the throttle position.
From the final application viewpoint, the tradeoff in designing the dithering signal is given by—on one hand—the desire to reduce the stiction effect (the final effect of the dither should be that of keeping the throttle valve excited and just beyond the movement point) and—on the other—to choose a dithering amplitude and frequency which do not cause an excessive power consumption. The electric power consumption due to dithering, , where in fact, can be computed as is the dither amplitude and is the battery voltage ( 12 V). Due to the system low pass dynamical behavior, the needs to be increased proportionally to its dither amplitude frequency; thus, the lowest possible frequency compatible with the throttle position dynamics should be chosen to minimize power consumption. Fig. 5 shows the positive effects of dithering on the hysteresis amplitude due to friction measured in the same quasi-static tests described in Section III. The frequency of the dithering signal has been set to 75 Hz, while different values of the dither amplihave been tested. As can be seen by inspecting Fig. 5, tude dithering significantly reduces the hysteresis amplitude, which decreases from the original value of approximately 0.5 to 0.05 A and the residual hysteresis can be considered negligible for control design purposes. Moreover, it is apparent that increasing 0.5 A does not add signifithe dither amplitude above cant improvements to stiction reduction, while it causes a larger power consumption. Thus, a dithering signal of amplitude 0.5 A seems appropriate for the considered system. Note, finally, that around the limp-home position the dither has no effect on the hysteresis. This is due to the mechanical spring layout, which—as discussed above—is explicitly designed to be stiffer around the limp-home position. It should be noted that the previous analysis was carried out with the help of the inner current loop; in the final implementation the dither must be applied as a sinusoidal variation of the PWM command. Before addressing the position control design, it is interesting to investigate the intrinsic limits of the considered electronic throttle, in order to have a benchmark for the performance evaluation of the final closed-loop system. These limits were tested both 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.
V. IDENTIFICATION OF THE THROTTLE DYNAMICS AND CONTROLLER DESIGN
G s
j!
Fig. 8. Estimated frequency responses from measured data ^ ( ) in three ( ) and the associdifferent experiments, the nominal parametric model ated uncertainty bounds.
G
the maximum or the minimum voltage to the dc motor. The results are shown in Fig. 6: as can be seen, the 0–1 opening occurs in 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 effects of the electromotive force and the stiction can be noted. Focusing on the opening dynamics, in the first phase the throttle is not moving and the current reaches its peak; once the initial friction is broken the throttle starts moving thus generating a electromotive force that the battery cannot overcome and therefore a drop in the current is observed. The same behavior is mirrored in 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 professional driver requests a full-open/full-close throttle variation in at most 100 ms.
For the design of the throttle position control loop, a classical model-based indirect design approach has been used (see, e.g., [15]). The first step of this approach is to derive a model of the controlled system. Classical black-box open loop model identification requires to excite the system with an input signal whose frequency components span the frequency range of interest. This approach could not be applied to the system at hand because of the two-state behavior of the open-loop throttle shown in Fig. 3(a) and (b). Specifically, note that as soon as the excitation signal reaches an amplitude large enough to break the static friction, the throttle plate immediately gets to a fully open (or closed) configuration. This problem can be solved carrying out the identification in closed loop. To this end, a low-bandwidth position controller has been designed and a frequency-domain identification procedure implemented. Specifically, the position controller was fed with a reference signal constituted by a multi-frequency sinusoidal signal (from 0.01 to 20 Hz) of amplitude 0.05 centered around the nominal position . Then, in order to estimate a non parametric model of the the interfrequency response of the overall system mediate PWM signal and the output position have been employed. Namely, the frequency response estimate has been computed according to the following expression [16]: (2) where denotes the cross spectrum of . Note that the adopted identification procedure yields an unbiased frequency response estimate also in the case of closed-loop identification [16]. Fig. 8 shows the comparison of the estimated frequency responses obtained from three different experiments: two tests performed on the test bench and one test performed on the instrumented bike. As can be seen, the identified models exhibit a certain degree of variability, which will be thoroughly addressed in the next section.
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The final expression of the identified frequency-response of has the form the nominal system (4)
Fig. 9. Controlled system responses to a 0.1 and a 0.6 position reference step commands. The responses have been normalized to improve readability.
Based on the parametric nominal model (4), a fixed-structure two-degrees of freedom PID controller with anti-windup and set-point weighting, [18], [19], has been implemented—in velocity form—with the aim of achieving a cutoff frequency of 10 Hz and a phase margin of 70 , needed to minimize closed-loop oscillations and overshoots, which have to be avoided as much as possible as they are felt by the rider and limit his/her confidence in the vehicle. Note that, by analyzing a professional driver request (see, e.g., Fig. 19), one notices that the fastest full-open/full-close maneuver lasts approximately 100 ms. Thus, to mimic real inputs, from here on over the employed set point signal is filtered with a 20 Hz low-pass filter. The continuous time transfer function of the regulator can be written as
(5)
Fig. 10. Architecture of the gain-scheduled PID controller.
Finally, using the obtained non-parametric frequency response estimate in the first test bench test (dashed-dotted line for the system has in Fig. 8), a transfer function model been obtained, by solving a nonlinear weighted least-squares fitting problem. Namely, the parameter vector is the minimizer of the following cost function: (3) have been tuned to privilege the fitting where the weights within the frequency range [5, 15] Hz, are the samples 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 the damped Gauss-Newton method [17]. The model order has been determined trying to obtain a good tradeoff between model complexity and accuracy. A better fitting could have been obtained by increasing the model order, but this would lead to the inevitable risk of over-fitting. It has been found more useful to focus the optimization procedure on the frequency range [5–15] Hz, which is the interval within which the desired cutoff frequency of the closed-loop system is expected to be, and therefore the frequency range where a more accurate model is needed. Note that in order to avoid numerical issues potentially associated with frequency-domain polynomial fitting, the data have been rescaled by normalizing the frequency range of interest.
are the ideal PID tuning parameters, and where is the set-point weight of the derivative term. The model uncertainties, previously mentioned and shown in Fig. 8, are also confirmed by the closed-loop validation of the controlled system. Fig. 9 shows the normalized responses of the controlled system to two different position step commands: a 0.10 and a 0.6 step. Inspecting Fig. 9, it is apparent that the two responses are qualitatively different. Namely, the closed-loop response to the small amplitude step exhibits an overshoot, whereas the large amplitude step response is well damped. As the final controller must achieve good performance and absence of overshoots in the face of all possible reference signals compatible with a driver’s gas request, the controller parameters must be properly tuned to improve the closed-loop performance in the case of small amplitude set-point variations. To this aim, a scheduling strategy for the PID parameters as functions of the requested position variation seems a promising choice. Note that, in principle, when a single control system must be designed in order to guarantee the satisfactory closedloop operation of a given plant in many different operating conditions a genuine gain scheduling approach can be followed, see, e.g., [12], [20], [21], [22]. This framework asks to find one or more scheduling variables which completely parameterize the operating space of interest and to define a parametric family of linearized models for the plant associated with the set of operating points of interest. Finally, a parametric controller can be designed to ensure the fulfillment of the desired control objectives 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 the ETB. In fact, the underlying assumption of LPV identification techniques (see [29]–[32]) is that the identification procedure can rely on one global identification experiment in which both the 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 the ETB where genuine open-loop identification is unfeasible. Thus it has been decided to experimentally determine (by trial-and-error) an adaptation rule of the PID controller parameters and then prove its stability a posteriori. This approach has another significant advantage in the context of the considered application, which relates to computational complexity. As a matter of fact, besides the need to store the lookup tables with the adaptation functions, the controller order and its structure are unaltered, whereas genuine LPV controllers have in general complex and high-order structures and are computationally very intensive. Thus, the proposed solution is particularly suitable for being implemented on motorcycle ECUs, which offer a limited computing capability. Hence, several experiments have been performed to estimate and the the static maps used to schedule the integral time as static functions of the requested posiderivative constant . The proportional gain and the set-point tion variation weight have proved to be effective when proper constant values are chosen. The final controller architecture is shown in Fig. 10. Note that, as the chosen scheduling variable, i.e., , is anti-causal, it is in fact the requested set-point variation computed based on the set-point derivative as follows: Fig. 12. Step responses of the closed-loop system: fixed-structure PID controller (dashed line) and gain-scheduled PID controller (solid line) for small step responses (top plot) and big step responses (bottom plot).
(6) Equation (6) shows that is computed by considering (at the current time instant ) the averaged value of the set-point —and propderivative—averaged over a time window by assuming that it remains agating it forward over constant over the latter time interval. The values of and have been experimentally tuned to 7 and 100 ms, respectively. Fig. 11 shows the adaptation laws of the integral time and derivative constant of the PID controller. As can be seen, the adaptation law is simply a linear one, whereas the scheduling of the integral time , which experiments have shown to be the most influential parameter, is more elaborate. The shape of the curve has been derived pointwise and then interpolated with continuous functions.
Fig. 12 assesses the effectiveness of the proposed gain-scheduled PID controller, showing a comparison of the step responses obtained with the fixed structure and the gain-scheduled PID controller on a 0.1 and 0.7 reference step commands. As can be seen, the gain-scheduled PID controller renders the response better damped than the fixed structure one and—most importantly—the closed-loop performance is consistent for all setpoint amplitudes. VI. STABILITY ANALYSIS AND LPV MODEL VALIDATION In the previous section, a gain-scheduled PID controller for the throttle position tracking has been proposed. Specifically, the scheduling laws for the PID parameters have been experimentally determined by optimizing the closed-loop system response 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 derivaof the controller as static functions of the expected tive time comvariation of the reference position signal puted via (6). Once the scheduling law is implemented, the whole controlled system can be seen as an LPV system. Furthermore, the LPV framework also accounts for structured uncertainties in the ETB dynamic model. As a matter of fact, the analysis of the identified model obtained in different working conditions—see Fig. 8—has highlighted that the system model is subject to a certain amount of uncertainty which can be accounted for by allowing a variability in the position of the first zero and in the . transfer constant of the nominal transfer function Fig. 8 also shows the uncertainty boundaries when the position is moved within the of the lower frequency zero of interval Hz and the transfer constant varies in the . As can be appreciated from Fig. 8, interval the structured uncertainty describes the variability of the system in the frequency range of interest. According to the adopted black-box approach, the choice of the uncertainty-modeling parameters has been driven by complexity considerations. Specifically, we have looked for the smallest set of parameters that could account for the whole variability shown by the experimental data in the frequency range of interest. Three time-varying parameters are therefore identified, so that the . resulting 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 time variability accounts for dynamic variations both of the system uncertainties, which are expected to vary as functions of the specific ETB, of the engine temperature, of the lubrication conditions, and of the set-point, i.e., the gas request command. have been set Specifically, the time derivatives bounds on . The bound on the time to derivative of the set-point variation has been determined by analyzing several gas request profiles commanded by a professional rider on race circuits tests, while the velocity bounds on the variation of the zero and of the transfer constant have been
chosen with dynamics compatible with thermal and lubrication effects. The variability ranges of the parameters vector can be described, in the parameter space, by a 3-D polytope. By plugging the corresponding value of the parameters in and in and closing the loop, a transfer function of the closed-loop system for each point in the parameter space can be obtained. However, to resort to LPV techniques for the closed-loop stability analysis, one needs to obtain a state space representation of the closed-loop system. This step gives rise to two different issues. Specifically, as we start from local models of the closed-loop system obtained by evaluating the parameter vector at fixed points of the polytope, one needs to interpolate the local models and this would in principle ask that all the state space realizations are in the same coordinate basis, [33]. Second, in LPV systems—which are a special class of time-varying systems—the usual notions of equivalence between input/output (I/O) and state space representations which hold for LTI systems are not valid anymore, unless a dynamic variation of the parameters is permitted (see [34], [35]). In general, the interpolation problem can be dealt with by resorting to balanced realizations [33], [36]. In the considered case, however, as both the system and the controller structure were known, an analytical state-space model both for the uncertain ETB dynamics and for the gain-scheduled PID controller has been obtained by performing a symbolic realization of both and . Based on such model, it is possible to write the LPV closed-loop system as
(7) is the position set-point and is where the measured throttle position. As for the equivalence notion between I/O and state space realizations of LPV systems, it should be pointed out that the standard LTI realization is only an approximation in the LPV case. It is well known that for LTI syscorresponds to the tems, if the state space model then all the state transfer function
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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 is square and nonsingular, are equivalent to the original one, in the sense that they give rise to the same input-output behavior. In the LPV case, however, the above notion of equivalence class does not hold anymore. This concept is better illustrated by considering the state space representation (7) and the parameter-de. If the coordinate pendent coordinate change transformation is applied to system (7) one obtains
(8) From the above relations, it can be seen that the obtained realization (7) can be regarded as a good approximation of the LPV system, i.e., the state-space model can be considered sufficiently close to its I/O representation, only if the time variation of the underlying coordinate transformation, i.e., the term is negligible, which corresponds to accounting for a static parametric dependence only in the I/O-tostate-space transformation. Unfortunately, a formal expression for the approximation error as a function of the problem data is very difficult to achieve, and this constitutes a challenging open problem in the LPV modeling and identification context. However, it is possible to perform a validation step to experimentally validate the LPV model (7). Note that this validation issue is rarely addressed in the LPV modeling and control literature, even though it constitutes a crucial part in assessing the soundness of any LPV model and controller which is derived based on local models. Here, the validity of the state space closed-loop system (7) has been checked by simulating the LPV system (7) and comparing the results with experiments carried out on the instrumented motorbike with the ETB controlled via the proposed gain-scheduled PID controller. To account for the parametric uncertainties in the ETB dynamics, in the simulations the two uncertain parameters and were varied by applying sinusoidal
perturbations with amplitude and frequency tuned according to their respective magnitude and velocity bounds. As for the set , it was computed via the measured set-point point variation position according to (6) and used as input for the LPV model simulation. The results of this validation step are shown in Figs. 13 and 14, where the simulated and measured closed-loop throttle positions are compared, using a highly exciting sine sweep reference signal spanning the frequency range from 0.5 to 15 Hz. Specifically, to increase readability, Figs. 13 and 14 show four details of 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 response shows very good agreement with the measured one, thereby confirming the validity of the state-space LPV model (7) obtained for the closed-loop system. Once the LPV state space representation has been validated, it is possible to apply LPV stability analysis techniques. Here reference is made to the following result [23], [24]. matrixTheorem 6.1: The system (7) is stable if there is a valued function satisfying
for all , where and is the bound on the time derivative of the vector . The notation indicates that every combination of and should be included in the inequality. The above problem is an infinite dimensional one. In particular, the infinite dimensionality comes from the fact that is a function of and that the above conditions must hold for . Several techniques are available in the literature to all reduce the problem to a finite dimensional one. In this context the parameter space gridding (see, e.g., [12]) has been preferred. Namely, the following steps are performed: 1) grid the set ;
CORNO et al.: DESIGN AND VALIDATION OF A GAIN-SCHEDULED CONTROLLER
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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 employed: (9) and a 512 points grid has been chosen, constituted by the 8 points for each polytope coordinate. The problem is finally translated into a system of 4608 LMIs, whose feasibility is succesfully checked via YALMIP [37] and SeDuMi. Remark 6.1: It should be noted that the gridding approach does not formally guarantee stability unless certain conditions on the gridding density are satisfied [23]. In principle, one may think of avoiding the gridding procedure, as there exist approaches in the LPV literature which formulate the stability analysis problem as a feasibility LMI problem of finite dimension, see, e.g., [38]–[41]. To do this, however, the LPV system must be written either as a linear fractional representation (LFR) or as an affine LPV system. Unfortunately an LFR representation could not be derived for the system at hand and an affine system structure could not be forced without adding too much conservativeness. Specifically, to write the system closed-loop dynamic matrix in such an affine form via a set of time varying parameters, the parameter space has to be significantly enlarged and, most importantly, it would loose its physical meaning and would not contain the set-point . For the above reasons, the gridding approach was variation preferred. Note, however, that the gridding approach has the advantage of being readily applicable in the case one should decide to model the uncertainty in a different way from that considered herein. Remark 6.2: Numerically, the LMI feasibility problem is very sensitive to the condition number of the involved matrices. Of course, the realization problem to be solved influences the condition number of the system matrices and thus the LMI problem itself. In this respect, the most sound numerical approach is that of using a balanced realization, while avoiding controllability and observability canonical forms which are known 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 the scheduling parameters was known, so that a symbolic realization could be performed. In this case, the final state space model, which was then evaluated at the different points of the grid, did not present critical numerical issues but for the presence of the controller integrator. To alleviate this problem, the controller integrator has been approximated with a low frequency pole. As shown in Fig. 15, this approximation does not alter the transfer function of the controller around the cutoff frequency of 10 Hz. Remark 6.3: The LPV validation and stability analyses have been carried out in continuous time. As the final implementation of the control algorithm is done in discrete time, it is important to verify that the stability properties are not lost in the discretization process. The discretization of the controller has been done according to the Euler’s forward method, which does not always
Fig. 15. Original controller transfer function (solid line) and low frequency pole approximation (dashed line).
preserve stability. Thus, to assess the validity of the obtained results in a discrete time setting, the approach presented in [42] is considered. Specifically, we focus on evaluating two important , the former being the upper bound on quantities , and the sampling time that guarantees numerical convergence of the discretization algorithm and stability preservation, and the latter being the maximum local discretization error. Considering the closed loop continuous time LPV system , one has with dynamic matrix (10) where indicates an eigenvalue and is the spectral abscissa. Equation (10) shows that the chosen sampling period of 1 ms ensures that the desired properties hold. Further, given a which can be maximum relative local discretization error tolerated, an upperbound on the required discretization time can be computed as (11) where
where is the state space and is the control space. In (10) and (11) the maximization and minimization over have been computed using the same grid employed in the stability analysis. The state space has been estimated by simulating the continuous time closed-loop system with different throttle position references recorded during test tracks for all the parameter values in the above mentioned grid. The control space is defined by the upper and lower limits of the dc motor voltage. Using the
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Fig. 17. Plot of the driver request measured in a test track lap (dotted line) and the measured throttle position (solid line).
Fig. 16. Comparison between the measured closed-loop system behavior (solid line) and the intrinsic open-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 smaller 1.1 ms. This upperbound is satisfied by the chosen that sampling frequency of 1 kHz. VII. EXPERIMENTAL RESULTS Before turning to test the closed-loop system performance against a real driver request signal, it is worth comparing the achieved closed-loop performance of the throttle position control with the actuator intrinsic limits discussed in Section IV. To this aim, let us refer to Fig. 16, which shows a comparison between the measured closed-loop system behavior and the intrinsic open-loop performance limits both for the output position and the requested dc motor current. As can be seen, the closed-loop system provides performances which are indeed quite close to the system limits, hence exploiting the full actuator capability. Note, moreover, that these tests have been performed with no dithering applied to the real system; the measured current shown in Fig. 16 proves that the measurement noise present in the real system when a genuine dynamic excitation is applied (recall the the need for dithering signal arose in face of quasi-static tests to estimate the return spring characteristic) provides the requested degree of dithering by itself. Further, the system was tested on the instrumented motorbike on a test track with a professional rider. The complete results on the 60s-long lap measurements are shown in Fig. 17.
Fig. 18. Detail of the closed-loop behavior in response to intermediate variations of the throttle position: set-point (dotted line) and measured (solid line) throttle position.
To better analyze the system performance in response to different types of driver’s 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 throttle position in the range 0.45–0.7 with opening and closing ramps. Further, Fig. 18(b) shows a detail of the maneuver where a very fine-grain modulation is performed by the rider around small throttle openings 0.07–0.15, which are the most critical as they continuously cross the position range where there is a significant change in the return spring stiffness (see Fig. 4). As can be seen, the system response is very accurate and it follows the driver’s request 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 driver’s request. As can be seen, the fastest driver’s request imposes on
CORNO et al.: DESIGN AND VALIDATION OF A GAIN-SCHEDULED CONTROLLER
Fig. 19. Detail of the closed-loop behavior in response to intermediate variations of the throttle position: set-point (dashed line) and measured (solid line) throttle position.
the system a full-open/full-close maneuver which lasts approximately 100 ms, hence within the range of the throttle intrinsic dynamic limits. The system response is accurate also in this critical case, and the maximum delay in the response is of 10 ms, which is considered well beyond the limit of driver’s perception. Finally, it is worth pointing out that, in the whole test, the steady-state error is of the same order of magnitude as the analog-to-digital (A/D) converter quantization, i.e., of approximately 0.0015. VIII. CONCLUDING REMARKS This paper presented a complete analysis of an electronic throttle system for ride-by-wire application in racing motorcycles. The electrical and mechanical dynamics of the system have been studied and the effects of friction based on appropriate experiments analyzed. Further, a model-based gain-scheduled position control system for throttle position tracking has been proposed. The stability of the closed-loop system has been proved via LPV techniques by solving an appropriate LMI feasibility problem and the LPV modeling assumptions employed in deriving a state-space model of the closed-loop system have been experimentally validated. Finally, the performances of the controlled system have been shown to be very close to the intrinsic limit of the actuator and the overall gain-scheduled controller effectiveness has been assessed on an instrumented test vehicle. ACKNOWLEDGMENT The authors would like to thank Prof. M. Lovera for the many fruitful discussions and suggestions on LPV modeling and validation issues. They would also like to thank the reviewers for their comments and suggestions. REFERENCES [1] C. Rossi, A. Tilli, and A. Tonielli, “Robust control of a throttle body for drive by wire operation of automotive engines,” IEEE Trans. Control Syst. Technol., vol. 8, no. 6, pp. 993–1002, Nov. 2000. [2] M. Tanelli, C. Vecchio, M. Corno, A. Ferrara, and S. Savaresi, “Traction control for ride-by-wire sport motorcycles: A second order sliding mode approach,” IEEE Trans. Ind. Electron., vol. 56, no. 9, pp. 3347–3356, Sep. 2009. [3] M. Corno and S. Savaresi, “Experimental identification of engine-toslip dynamics for traction control applications,” Euro. J. Control, vol. 16, pp. 88–108, 2010.
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Matteo Corno jointly received the Master of Science in computer and electrical engineering from the University of Illinois, Chicago, and the “Laurea” Degree cum laude and the Ph.D. degree cum laude with a thesis on active stability control of two-wheeled vehicles from the Politecnico di Milano, Milan, Italy, in 2005 and 2009, respectively. During his Ph.D., he had a six-month internship at Alenia Spazio (now Thales Alenia Space). In 2008, he had been a Visiting Scholar with the University of Minnesota, Minneapolis. In 2009, After a joint post-doc position at Politecnico di Milano and Johannes Kepler University, Linz, he joined Delft University of Technology, Delft, The Netherlands, as an Assistant Professor with the Delft Center for System and Control. His current research interests include dynamics and control of two and four wheeled vehicles, nonlinear estimation techniques, and LPV control.
Mara Tanelli (M’05) was born in Lodi, Italy, in 1978. She received the Laurea degree in computer science engineering and the Ph.D. degree in information engineering with a thesis on active braking control systems design for road vehicles from the Politecnico di Milano, in 2003 and 2007, respectively, and the Master of Science degree in computer science from the University of Illinois, Chicago, in 2003. She is currently an Assistant Professor of automatic control with the Dipartimento di Elettronica e Informazione, Politecnico di Milano. She is also currently with the Dipartimento di Ingegneria dell’Informazione e Metodi Matematici, Università degli studi di Bergamo, Dalmine, Italy. Her main research interests focus on control systems design for ground vehicles, estimation, and identification for automotive systems, control systems design for agricultural tractors, and identification and control for active energy management of data centers. Dr. Tanelli was a recipient of the Dimitri N. Chorafas Ph.D. Thesis Award and the Claudio Maffezzoni Ph.D. Thesis Award for her Ph.D. thesis. In 2008, she and her coauthors received the Rudolf Kalman Best Paper Award for the best paper published in 2007 in the ASME Journal of Dynamic Systems Measurement and Control.
Sergio M. Savaresi (M’00) was born in Manerbio, Italy, on 1968. He received the M.Sc. degree in electrical engineering and the Ph.D. degree in systems and control engineering from the Politecnico di Milano, Milan, Italy, in 1992 and 1996, respectively, and the M.Sc. degree in applied mathematics from the Catholic University, Brescia, Italy, in 2000. After the Ph.D., he was a Management Consultant with McKinsey & Company, Milan, Italy. He was a Visiting Researcher with Lund University, Lund, Sweden; University of Twente, Ensende, The Netherlands; 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 the Politecnico 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 International 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 control applications. Dr. Savaresi is an Associate Editor of the IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, the European Journal of Control, and the IET Control Theory and Applications. He is a member of the Editorial Board of the IEEE Control Systems Society.
Luca Fabbri was born in Owo, Nigeria, in 1963. He received the M.Sc. degree in mechanical engineering from the University of Padova, Padova, Italy, in 1990. From 1990 to 1993, he was with Aprilia, working as a mechanical designer in the racing unit. From 1993 to 2006, he was responsible for vehicle development in the racing unit, where he led the design and development of racing motorcycles for the categories 125cc, 250cc, 500cc, and SuperBike. Currently, he is the Innovation Manager of the Motorcycle Engineering section for the brand Units Aprilia e Derbi within the Piaggio Group.
