Robust Control of a Throttle Body for Drive by Wire Operation of Automotive Engines

Transcription

1 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 8, NO. 6, NOVEMBER Robust Control of a Throttle Body for Drive by Wire Operation of Automotive Engines Carlo Rossi, Andrea Tilli, and Alberto Tonielli, Associate Member, IEEE Abstract In recent years, ever more stringent requirements in terms of emissions control, driveability, and safety of automobiles have led to the development of the drive by wire (DBW) concept, a new architecture for engine control systems, with the purpose of managing air, fuel and ignition in an integrated way. The throttle control plays an important role in the development of DBW systems. Despite its apparent simplicity, the position control of the throttle valve is quite a complex problem, due to application constraints and system characteristics. Very high robustness must be linked with limited cost, as required by a mass production device. A cascaded control structure including a nonlinear trajectory generator filter is adopted, allowing each dferent control problem to be solved with the most suitable control algorithm and implementation technology. In this regard, the use of variable structure control techniques is the key element to reaching the solution. Extensive simulation tests are reported to show the performance of the proposed control algorithm. A throttle step from 0.5 to 89.5 indicates good position tracking under realistic operating conditions, with a position error smaller than 1. The same simulation is performed at a battery voltage of 9 V to check the controller robustness. A prototype controller is presented. The experimental implementation of the controller for a step from 2.5 to 85.5 indicates a very smooth position trajectory with a maximum dynamic position error of 7. A small throttle step from 1 to 7 (which contains the nonlinearity of the limp home mode spring) was also tested and resulted in very good position response with the maximum position error of 2.Application specications are fully isfied both in terms of control performance and controller cost. Index Terms Automotive, drive by wire, sliding mode, variable structure. I. INTRODUCTION IN AUTOMOTIVE spark ignition engines the air coming into the intake manold, and therefore the power generated, strongly depends on the angular position of a throttle valve. In traditional systems, the throttle position is actuated by a mechanical link with the accelerator pedal, directly operated by the driver. Automatic air flow regulation requires the introduction of additional actuators. In particular: 1) an actuator for controlling idle speed and release, usually consisting of a bypass of the throttle valve whose section is regulated through a stepper motor and 2) other costly and bulky servomechanisms on the throttle for cruise control and/or traction control functions. In recent years new and increasing requirements in terms of emissions control, driveability, and safety have led to the development of drive by wire (DBW), a new architecture for engine con- Manuscript received April 1, Recommended by Associate Editor, F. Svaricek. This paper was supported in part by Magneti Marelli S.P.A. C. Rossi is with Magneti Marelli S.P.A., Engine Control Division, Bologna, Italy. A. Tilli and A. Tonielli are with the Department of Electronics, Computers and System Science, University of Bologna, Bologna, Italy. Publisher Item Identier S (00) trol systems, with the purpose of integrated air, fuel, and ignition management. The DBW architecture does not require any direct mechanical links between the accelerator pedal and the throttle valve. The throttle actuator is a motorized throttle body (MTB) electrically driven and controlled by an electronic system that mediates between: 1) a driver s request, interpreted by an accelerator pedal position sensor and 2) effective traction possibilities depending upon driveability, safety, and emission limitation constraints. The new architecture does not require additional air actuators, as they are replaced by the electric motor driving the throttle valve, and achieves noticeable improvements in reliability, performance, cost, size, and weight. There are a number of functions that can be achieved or improved with throttle position regulation in a DBW system, which can be divided into two main categories [10], [11]. The first one is related to the thermal engine performance and includes: customized and variable accelerator pedal/throttle position mapping, as a function of operating conditions; idle speed regulation and cold starting management; dashpot management; A/F ratio regulation during transients; engine speed (r/min) limitation; catalyst thermal management (light-off). The second category of functions refers to vehicle behavior and includes: automatic vehicle speed control and intelligent cruise control; effective torque control, including traction control and ABS management; vehicle speed and performance limitation; smoother movement during acceleration/deceleration (driveability control); integration with electronic, hydraulic and mechanical gear box; vehicle dynamic control; integration with anticollision and guidance systems. The basic function of a DBW system consists clearly in good control of the throttle position. The requirements to be achieved in terms of accuracy, response time, and robustness are severe, particularly when mass production constraints are imposed. MTB dynamical performance to large signal variation has to guarantee at least the same behavior of a traditional pedal/throttle body system. At the same time, the response to small signal variations should be equally fast, especially during idle speed regulation, where also accuracy becomes a critical factor. From the control viewpoint this problem might appear to be one of simple position tracking control under variable load torque. Nevertheless, system operation close to the car /00$ IEEE

2 994 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 8, NO. 6, NOVEMBER 2000 Fig. 1. Throttle body functional scheme. Fig. 2. Safety spring torque (T ) as a function of throttle position (#). engine and constraints on the cost imposed by mass production transform this apparently simple control problem into quite a complex one. Joint methodological and technological design must be performed to ensure the best tradeoff between control complexity, technological feasibility, robustness, and cost limitation. The control algorithm adopted must be extremely robust although suitable for simple and inexpensive implementation. Among all the robust control techniques proposed in the literature a variable structure (VS) approach has been considered in this project [1]. A three-level cascaded control structure is used [2]. The inner current controller is designed using hysteresis control techniques [3]. The intermediate velocity controller is designed as a VS controller with integral action [6]. The outer position controller is designed as a digital linear controller. The position reference signal is given by a smooth trajectory generator (STG) able to generate position trajectories with variable bounds on velocity and acceleration [7], [12]. This paper is organized as follows. Section II contains the description of the problem. Section III describes the control architecture and the design of the dferent controllers. Simulation results are reported in Section IV, while the experimental setup and results are given in Section V. The theoretical stability proof for the velocity controller is reported in the Appendix. II. PROBLEM DESCRIPTION The motorized throttle body (MTB) used is manufactured by Magneti Marelli, Bologna, Italy. A schematic representation is given in Fig. 1. The main parts are: the throttle valve, driven by a permanent magnet dc motor (this motor is fed by a controlled four-quadrant dc/dc converter connected to the vehicle battery); the body duct, suitably shaped to ensurethe desired relationship between the air flow and the throttle valve position; a set of loaded springs ensuring a safety recovery position [referred to as limp home (LH) position] for the throttle when no driving torque is generated by the dc motor. The resulting torque of the springs is reported in Fig. 2, which shows that the LH position (5 ) lies inside the normal operating range ( ) of the throttle. The main functional specication for a DBW system is a fast throttle positioning with accurate reference tracking. The system equations are as follows. 1) Motor Model: (1) (2) (3) TABLE I SYSTEM PARAMETERS FOR THE MTB CONSIDERED where throttle position and velocity; armature current and voltage of DC motor; (both are limited for technological reasons: ); motor torque coefficient; motor resistance and inductance; overall inertia (throttle motor); overall load torque. 2) Load Torque Model: (4) where load torque generated by the safety springs, whose nonlinear characteristic is reported in Fig. 2; friction torque, nonlinearly depending on the throttle velocity; aerodynamic torque due to the engine air flow whose value is assumed to be unpredictable; bounded values are assumed for the aerodynamic torque and its derivative. Owing to manufacturing tolerances (low-cost mass production), variable operating conditions and wear of mechanical parts, nominal values of some system parameters are known with uncertainty while others may have a large variance around their nominal value. Table I reports the nominal values of parameters and their expected range for the MTB considered. The most important system parameters are strongly variable. For example: The electrical time constant can vary by a factor larger than 5 (from 0.25 to 1.33 ms).

3 ROSSI et al.: ROBUST CONTROL OF A THROTTLE BODY FOR DRIVE BY WIRE OPERATION OF AUTOMOTIVE ENGINES 995 Fig. 3. Control architecture adopted. The supply voltage, affecting the control gain, can vary by a factor of 3 (from 6 to 18 V). The safety spring torque is discontinuous at the LH position; the LH position is defined with more than 10 of uncertainty; the torque discontinuity is relevant and corresponds to more than 40% of the motor nominal torque. the friction torque has strong variations in the specied temperature range ( C C) III. CONTROLLER DESIGN A. Control Specications The main topic of this paper is the design of a position-tracking controller for the throttle valve, which is robust to MTB parameter variation and suitable for low cost mass production. More specically the requirements are as follows. The controller is not based on a detailed model of the motor-driven valve characteristics. The position reference is generated by the standard engine control unit (ECU), remotely located with respect to MTB, where the expected sampling time for digital algorithms implemented on ECU is about 4 ms. The computational load added to ECU is very low. Besides, the control system must ensure the following. The position regulation error (static condition) is less than 0.1. The position tracking error (dynamic condition) is less than 7. The valve opening time (0 to 90 with 5% tolerance) is less than 130 ms with a supply voltage higher than 9 V. With between 6 9 V there are no specications on opening time. B. Control Architecture A simple linear controller (analog or digital) cannot be adopted because of unknown or strongly time-variable system parameters. A digital implementation can achieve a high level of robustness through sophisticated self-tuning or on-line adaptation algorithms, at the expense of heavy computational load. Since ECU cannot handle signicant added computational load, another powerful microprocessor or DSP would be required which would exceed cost constraints. Therefore, a special-purpose solution integrating robust control requirements and technological implementation constraints is required. To get high performance at low cost, control algorithm and implementation technology cannot be considered independently. Strong interaction between the two design phases is required in order to maximize the performance/cost ratio. The authors have shown in [2] that a cascade of VS controllers can lead to a simple and robust hardware implementation of the motor control system. Moreover, thanks to the flexibility ensured by the cascaded control structure, the three control loops can be designed using, for each one, the best combination of control method and implementation technology. Applying these concepts to the MTB control, the cascaded architecture shown in Fig. 3 has been defined. There are three separate sections: 1) the position controller, implemented at discrete-time in the ECU; 2) the ECU/MTB interface; 3) the two fast-dynamics velocity and current controller, implemented at continuous-time in the MTB. A linear discrete-time feedback position controller with feedforward action is used. To ensure that parameters of the reference trajectories (velocity, acceleration) are always set according to the time variable operating conditions of MTB, a nonlinear smooth trajectory generator (STG) operating with variable velocity and acceleration bounds is added at this control level [7], [12]. According to specication on the position dynamics, the sampling time limitation on the algorithms implemented on ECU does not represent a serious problem. On the other hand, direct implementation in ECU software permits strong integration of the position controller into the complete DBW system. A second-order filter is introduced in the ECU/MTB interface to smooth the digital to analog converter output (velocity set point). This is required to isfy constraints on the set-point derivative of the VS velocity controller. The velocity loop is based on a VS velocity controller with integral action. This ensures robustness to system parameter variation and external load. A velocity estimator is included to reconstruct the velocity from the valve position measurement.

4 996 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 8, NO. 6, NOVEMBER 2000 Fig. 4. Typical current tracking. The current loop is based on a simple fixed-frequency hysteresis controller, suitable for direct integration into the power converter electronics. C. Current Controller It is well known from the literature (see, for example, [2] [5]) that current controlled pulse width modulators are simple VS controllers ensuring fast dynamic response, high robustness and low cost implementation. Among the dferent implementations a fixed-frequency one is adopted to isfy implementation constraints imposed by the manufacturer of the power electronics circuit. As explained in Fig. 4, at the beginning of each modulation period the control voltage sign is applied to the motor according to the sign of the current error. Once zero error is reached the control voltage is applied and maintained up to the end of the modulation period. After defining the current reference and the current error, the dynamical model (1) of the current can be rewritten in error form as or sign (5) It is easy to demonstrate (see, for example, [2]) that sliding mode (SM) existence conditions (infinite switching frequency) can be used also to establish the stability of current controlled PWM modulators operating at finite switching frequency. The well-known condition for SM existence on surface is [1] sign (6) Substituting (5) in (6), after some simple computations it follows that the stability of the proposed controller is ensured (7) D. Velocity Controller Owing to fast dynamics and robustness ensured by the adopted current controller, the assumption of the current-source converter can be made and the velocity controller design can be based on a first order model. This model is obtained joining equation (2) with (4). In order to analyze and design the controller structure, the velocity-tracking control problem is transformed into the equivalent stability problem for the system in error form. After defining the velocity reference and the velocity error, the error model is or, equivalently where disturbance and input coefficient are given by (8) (9) (10) Under some hypotheses on the bounds of parameter, disturbance and its derivative, the stability of system (7) can be ensured by the following three-term controller: where sign (11) (12) Control law (11) and (12) is a modication of a VS control law with integral action originally proposed in [6]. By defining a positive velocity error bound, it can be proved that: 1) the system (9) is stabilized and, consequently, the velocity tracking is guaranteed; 2) the velocity error reaches the boundary layer in a finite time and lies inside ; the controller parameters ( ) in (11) and (12) isfy the following inequalities: (13) Both system parameters and the external reference affect the system stability. Robustness is achieved enough control voltage is applied to compene for internal and external disturbances. The control voltage is variable, depending on the voltage level of the battery (see Table I). As a consequence, the current reference (the value and its derivative) must be carefully bounded to ensure the current tracking (i.e., current error stability) under all operating conditions. The smooth trajectory generator and the velocity controller design impose these bounds. where (14) (15) reflect the assumptions that there are finite bounds on the derivative of the disturbance and on the coefficient.

5 ROSSI et al.: ROBUST CONTROL OF A THROTTLE BODY FOR DRIVE BY WIRE OPERATION OF AUTOMOTIVE ENGINES 997 The proof of the previous statement is quite complicated. The interested reader can refer to the Appendix, [2], [6] and [9]. Here some general considerations are made to better clary its use. The characteristics of MTB, as well as the external load torque and the external reference are all considered as bounded disturbances in the design of the proposed controller [see (9)]. The velocity tracking is achieved without an exact knowledge of system and load parameters. Only bounds on parameter values and their derivatives are used in the design. Moreover, design inequalities (14) ensure some degrees of freedom in the selection of control coefficients and, since they are not a strictly necessary condition to ensure stability, they could even be violated, checking stability by simulations. It must be pointed out that the finite bound on the total disturbance derivative [see (15)] is a strictly necessary condition for system stability. This means: 1) the velocity reference must have bounded first and second derivatives; 2) the load torque and its derivative must be limited. The first condition is ensured by the joint action of the STG in the position controller and the second-order filter that reconstructs a continuous velocity reference from the discrete one (see Fig. 3). Referring to the second condition, there are two structural discontinuities in the load torque: 1) at LH position ( has a discontinuity) as shown in Fig. 2; 2) for ( has a discontinuity due to static friction). In these two isolated operating points the disturbance derivative is not limited and the velocity tracking is lost. Since in all other operating points is limited, the robustness of the proposed controller ensures a fast tracking recovery. With respect to the controller originally proposed in [6], this modied solution does not produce an SM motion with since the discontinuous action in the control law has been replaced with a high-gain urated action. A limit cycle arises whose amplitude is always smaller than the design parameter. This limit cycle is the equivalent of the chattering caused by operation at finite switching frequency in VS controllers. The control law (11) adopted has some signicant advantages it is compared with classical VS control laws, since: the maximum amplitude of the residual tracking error is a design parameter; under the above-mentioned smoothness conditions on, the velocity regulator produces a continuous output (current reference). This condition is required in (7) to ensure SM existence on the current controller. In order to bound the current reference, uration is added to the controller integral action. Moreover, to limit the current reference derivative, parameters of the velocity controller must isfy the added condition (16) where is the maximum achievable current derivative. It is quite dficult to strictly very (16) for each operating condition, since the current derivative must be bounded according to actual value and MTB state [see (1)]. As explained in more detail in the next section, STG produces a position reference complying with energetic constraints with dferent. This means that the current reference, generated by the VS velocity controller, will be automatically near to isfying (16) for each normal operating condition. Therefore, it is admissible to select controller parameters,, and in order to isfy (16) with reference only to the nominal case. A direct measurement of the throttle velocity is not available. The velocity must be estimated from the position measurement made by a linear potentiometer. Some care must be taken in performing the signal derivative in such a noisy environment. Tradeoff between noise sensibility and bandwidth must be carefully considered. The proposed velocity controller is suitable for inexpensive analog implementation. E. Reference Generation and Position Controller Good position tracking can be achieved the reference is always consistent with MTB energetic limitations. The MTB model can be used to get the maximum positive and negative throttle accelerations for every mechanical state of the valve and power supply value, under the assumption of the worst case external load. Neglecting some small terms, the results are shown in (17) and (18) at the bottom of the page. The maximum positive throttle acceleration as a function of and is shown in Fig. 5, with V. These results are computed using typical values of MTB parameters. The fastest position and velocity reference consistent with the current MTB state, are on-line generated by the trajectory generator, built as a state-variable filter controlled by a VS controller, as shown in Fig. 6. To explain the trajectory generator design, let us assume that and are the input position reference and its derivative, and are the corresponding filtered output, and are the variable positive bounds imposed on the output velocity and acceleration modules. Let us define the position and velocity errors as (17) (18)

6 998 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 8, NO. 6, NOVEMBER 2000 Fig. 5. Maximum positive acceleration as a function of # and!. Fig. 6. Block diagram of trajectory generator. The following discrete-time smooth trajectory generator (STG) is used to produce the position reference whose velocity and acceleration are bounded at and, respectively. sign sign (19) (20) (21) (22) (23) Equations (19) (23) are the discrete-time version of the STG proposed in [7]. They represent the cascade of two integrators (adopted to reconstruct the velocity feedforward signal) feedback controlled by a VS controller derived from bang-bang control. Bounds on acceleration and velocity ( and ) can be arbitrarily imposed at run-time through dedicated controller parameters always keeping the filter stability. The main characteristic of this trajectory generator is that the output trajectory tracks the input reference without overshoot, in almost the minimum time consistent with the selected bounds on acceleration and velocity. Further details on STG characteristics and a discussion on the adopted discrete-time implementation are given in [12]. In the MTB application considered, the velocity bounds are fixed to a value depending on physical constraints, while the acceleration bounds are variable. Acceleration limits could be computed as a function of and according to (17) and (18). This solution was not adopted since values given by these expressions are not very reliable, owing to uncertain and variable MTB parameters. A simpler approach is proposed. Three ranges are considered: for each range, acceleration bounds are fixed to minimum values obtainable by (17) and (18) with the minimum supply voltage in the range. This is sufficient to isfy requirements on the throttle opening time and guarantees the controllers a regulation energetic margin. The position controller implemented at the ECU level is designed as a digital PI controller with velocity feedforward action. IV. SIMULATION EXPERIMENTS Extensive simulation experiments were made to test the controller performance. The first set of experiments was made to check control performances of the two inner analog loops, while the second set was intended for the overall position controller. In the first set of experiments all system parameters are kept constant at their nominal value, as reported in Table I. This is not a limitation since the values of system parameters are not used to tune the current and velocity controllers. Besides, constant system parameters in (4) mean variable load torque. A. Current Controller Two dferent simulations at the two dferent switching frequencies of 10 KHz and 20 KHz are reported in Fig. 7(a) and (b), respectively. Good tracking is achieved and little chattering is observed when the highest switching frequency (20 KHz) is considered. B. Velocity Controller A smooth trapezoidal reference is used, similar to the one generated by the position controller. The controller parameters adopted are A A s A/(rad/s) A s A/(rad/s) A/s s rad/s

7 ROSSI et al.: ROBUST CONTROL OF A THROTTLE BODY FOR DRIVE BY WIRE OPERATION OF AUTOMOTIVE ENGINES 999 Fig. 7. Simulation of current control with continuous reference at dferent switching frequencies: (a) 10 KHz and (b) 20 KHz. Fig. 9. Smooth trajectory generator outputs: (a) position, (b) velocity, and (c) acceleration. Fig. 8. Simulation of velocity control with filtered trapezoidal set-point: (a) set-point and actual velocity and (b) velocity error. Very good tracking is reported in Fig. 8(a), confirmed by the small velocity tracking error reported in Fig. 8(b). Confirming the theoretical analysis, the velocity tracking is lost both at the starting time s and at the time s, when the position crosses the LH. As expected the tracking is quickly recovered after the load discontinuity. C. Position Controller A step command from to is assumed. Throttle velocity and acceleration are bounded to 2000 /s (35 rad/s) and /s (1047 rad/s ). The response of the STG is shown in Fig. 9, reporting filtered position, bounded velocity, and acceleration. A first set of experiments consists in testing the MTB controller under real operating conditions. A sampling time of 4 ms is assumed, as imposed by time constraints on the ECU controller. A switching frequency of 10KHz is assumed in the current controller. Throttle velocity and acceleration are bounded to 2000 /s (35 rad/s) and /s (1047 rad/s ). A good response is given in Fig. 10. The effects of the velocity feedforward action are clearly reported in Fig. 10(a), where the actual position seems to precede its reference. The discontinuous velocity reference, generated by the discrete controller, is filtered with a second-order filter as reported in Section III-E. The second set of simulations consists in testing the operation under low battery voltage operating conditions. The same operating conditions as in the first set of simulations are consid- Fig. 10. Simulation of MDTB controller (discrete time T = 4 ms): (a) position tracking and (b) velocity tracking. Fig. 11. Simulation of MDTB controller when operating at 9 V (discrete time. T =4ms): (a) position tracking and (b) velocity tracking. ered, but a supply voltage of 9V is assumed. The trajectory parameters are (automatically) changed accordingly. The throttle velocity is still bounded to 2000 /s (35 rad/s) while the acceleration is limited to /s (700 rad/s ). Tracking performance comparable to that in the previous experiment is shown in Fig. 11. In particular the effect of a dferent bound on acceleration can be observed in Fig. 11(b).

8 1000 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 8, NO. 6, NOVEMBER 2000 Fig. 12. Simplied schematic diagram of current and velocity controller. Fig. 14. First experiment on MDTB controller: (a) velocity reference and (b) actual velocity estimation. Fig. 13. First experiment on MDTB controller: (a) real position and (b) position error. V. EXPERIMENTAL SETUP AND RESULTS A prototype system was designed to perform experiments on an actual MTB. A custom analog board, equipped with MOS-FET power transistors and power supply, was designed for current and velocity controllers. A simplied schematic diagram is shown in Fig. 12. To speed up the experiments, standard analog components were adopted. In the following figures some offset and noise effects are still present. Better results would be obtained in terms of residual offset and noise an optimized board, based on hybrid electronics, was adopted. The digital position controller and the STG were implemented on a rapid prototyping station developed by the authors at the University of Bologna, Italy [8]. Powerful DSP and input output (I/O) (analog and digital) boards located on the bus of a standard personal computer, joined to a special purpose operating system handling all the real-time and I/O drivers, simplies the development of digital controllers. Built-in oscilloscope functions are useful to check performance and save the most relevant waveforms. A sampling time of 4 ms was adopted for the digital part and a switching frequency of 10 KHz was set in the current controller. Two dferent experiments were performed. The first experiment corresponds to the simulation shown in Fig. 10, with a small dference in the two extreme positions, set to 2.5 and 85.5, respectively, for safety reasons. A very smooth throttle position trajectory is shown in Fig. 13(a). The position tracking error (within the specication) is given in Fig. 13(b). The rather long tail on the position error is due to some residual offset in the experimental analog Fig. 15. Second experiment on MDTB controller: (a) real position and (b) position error. hardware that prevents the use of the best integral action gain in the position controller. The velocity reference and the actual velocity estimation are given in Fig. 14. The second experiment shows the effect of LH position crossing ( ). A small position step from 1 to 7 is applied to the STG. Still very good position response is shown in Fig. 15. The position scale has been selected to show the small residual effect on the actual position trajectory caused by the LH crossing. VI. CONCLUSION A robust position controller for motorized throttle body in automotive applications is presented. Complexity of the control problem is explained and a control architecture is presented, ensuring very high robustness at limited cost. The cascaded control structure adopted is illustrated. It enables the solution of the control problem with the most suitable control algorithm and implementation technology. The use of variable structure control techniques represents the key element to achieving the solution. The controllers designed at the dferent cascaded level are presented and discussed. The extensive set of simulation tests reported shows good performance for the proposed control algorithm. The implemented prototype controller is also presented. The experimental results confirm the feasibility of the proposed approach. Application specications are fully isfied both in terms of control performance and controller cost.

9 ROSSI et al.: ROBUST CONTROL OF A THROTTLE BODY FOR DRIVE BY WIRE OPERATION OF AUTOMOTIVE ENGINES 1001 APPENDIX The velocity tracking problem in Section III-D has been transformed into an equivalent stability problem for the system in error form (A1) If the three-term VS stabilizing controller (11) (12) is considered, the resulting closed-loop second-order system is the following: sign (A2) where. Hypotheses: Based on physical bounds on unknown or time-varying coefficients, the following mathematical hypotheses can be formulated. H1) The velocity error is a bounded variable and the state dependent disturbance is a continuous function, with piecewise continuous derivative. H2) The time derivative of is bounded where is a known positive constant. H3) The coefficient is positive and bounded. Then the following theorem can be stated: Theorem: Considering the closed-loop system (A2), hypotheses H1, H2, H3 and design inequalities (11) (12) are isfied then for every initial state there is a finite time such that where is arbitrary. Proof: System (A2) can be rewritten in state-space form as sign (A3) From H2 and (14) it follows that the sign of in (A3) does not depend on the term. The following four regions can be determined according to the sign of and : where and where and where and where and (A4) For every initial state and for each consistent with H1 and H2, the system trajectories will always evolve from a region to region, owing to state derivatives signs. Thus, each trajectory intersects the axis on a succession of points individuated by ( ). In Fig. 16 a typical trajectory is given. Fig. 16. VS controller typical trajectory in phase plane As shown in [9], (13) and (14) guarantee that, with the worst disturbance sign, a single asymptotically stable limit cycle is present in region (A5) It will be proved that there are no other stable limit cycles for the system (A3) with the worst disturbance, under hypotheses H1, H2, H3 and (14). This implies that closed-loop system trajectories will converge to in a finite time, for each initial state and for each consistent with H1 and H2. Defining the region, contained by, as (A6) the proof proceeds in two steps. Step 1: In this part, initial states belonging to are considered. For each initial state in, with worst case disturbance, the system behavior is the same as that analyzed in [9], thus system trajectories will converge to the unique asymptotically stable limit cycle contained in. Step 2: In this part initial states not belonging to are considered. As previously stated, each system trajectory intersects axis on a succession of points ( ). Thus, with the worst case disturbance sign, the following is necessary and sufficient condition to ensure that there are not other limit cycles out of : (A7) With the worst disturbance, the system trajectories evolve in state-space according to: (A8) sign sign Let assume, without loss of generality, that at time system state is ( ), with [the case follows by symmetry]. Starting from, the system trajectory evolves in and, at, reaches the surface in. There are two cases. If, the trajectory will evolve as shown in [9] and it will converge to the sole limit cycle contained in ; in fact, is smaller than.if [see (A1)] the trajectory will evolve in, it will intersect the axis at

10 1002 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 8, NO. 6, NOVEMBER 2000,in[ ], where is the solution of the following equation: (A9) Subsequently, the system trajectory will reach the axis, in ( ), where is the solution of the following equation: Due to hypotheses (14) From (A9), (A11) and (14) it follows that (A10) (A11) [3] J. Holtz, Pulsewidth modulation A survey, IEEE Trans. Ind. Electron., vol. 39, pp , Oct [4] D. M. Brod and D. W. Novotny, Current control of VSI-PWM inverters, IEEE Trans. Ind. Applicat., vol. IA-21, pp , May/June [5] C. Rossi and A. Tonielli, Robust current controller for a three-phase inverter using finite-state automaton, IEEE Trans. Ind. Electron., vol. 42, Feb [6] R. Zanasi, Sliding mode using discontinuous control algorithm of integral type, Int. J. Contr., vol. 57, no. 5, pp , [7] C. G. Lo Bianco, A. Tonielli, and R. Zanasi, Nonlinear trajectory generator, in Proc. IEEE IECON 96 Int. Conf., Taipei, Taiwan, R.O.C., Aug [8] R. Morici, C. Rossi, and A. Tonielli, Fast prototyping of nonlinear controllers for electric motor drives, in IFAC World Congr. 1993, Sydney, Australia, Aug [9] C. Bonivento and R. Zanasi, Discontinuous integral control applied to the orientation of a spacecraft, in IFAC Symp. Robust Control Design, Rio de Janeiro, Brazil, Sept [10] R. J. Tudor, Electronic throttle control as an emission reduction device, in Annual SAE Congr., 1996, Paper [11] H. M. Streib and H. Bischof, Electronic throttle control (ETC): A cost effective system for improved emissions, fuel economy, and drivability, in Annu. SAE Congr., 1996, Paper number: [12] R. Zanasi, C. G. Lo Bianco, and A. Tonielli, Nonlinear filters for the generation of smooth trajectories, Automatica, vol. 36, no. 3, Mar Hence, from (14), it follows that Consequently (A12) (A13) Carlo Rossi was born in Fabriano, Ancona, Italy, on January 3, He received the Dr. Ing. degree in electronic engineering and the Ph.D. degree in system science and engineering from the University of Bologna, Italy, in 1989 and 1993, respectively. In 1989, he joined the Department of Electronics, Computer, and System Science (DEIS) of the University of Bologna. In 1993, he spent six months at the University of Calornia, Santa Barbara, as a Visiting Researcher from the Italian National Council of Research. In 1995, he joined the Engine Control System Division of Magneti Marelli S.p.A., Bologna, where he is currently System Analysis and Simulation Manager. His research interests include electric motor drives, geometric approach to nonlinear control, nonlinear observers, variable structure systems, and hybrid control applied to automotive systems. and, due to (14) and (A11) (A14) Andrea Tilli was born in Bologna, Italy, on April 4, He received the Dr. Ing. degree in electronic engineering and the Ph.D. degree in system science and engineering from the University of Bologna, Italy, in 1996 and 2000, respectively. Since 1997, he has been at the Department of Electronics, Computer, and System Science (DEIS) of the University of Bologna. His current research interests include nonlinear control techniques, variable structure systems, electric drives, active power filters, and DSP-based control architectures. Thus (A15) (A16) This means that design inequalities (14) isfy condition (A7). REFERENCES [1] V. I. Utkin, Variable structure systems with sliding modes, IEEE Trans. Automat. Contr., vol. AC-22, pp , Feb [2] C. Rossi and A. Tonielli, Robust control of permanent magnet motors: VSS techniques lead to simple hardware implementations, IEEE Trans. Ind. Electron., vol. 41, Apr Alberto Tonielli (A 92) was born in Tossignano, Bologna, Italy, on April 1, He received the Dr. Ing. degree in electronic engineering from the University of Bologna, Italy, in In 1975, he joined the Department of Electronics, Computer and System Science (DEIS) of the University of Bologna, with a grant from the Ministry of Public Instruction. In 1979, he started teaching as an Assistant Professor. In 1980, he became Permanent Researcher. In 1981, he spent two quarters at the University of Florida, Gainesville, as Visiting Associate Professor. In 1985, he became Associate Professor of Control System Technologies at the University of Bologna. Currently, he is Full Professor of Automatic Control at the same university. His current research interests are in the fields of nonlinear and sliding mode control for electric motors, nonlinear observers, robotics, and DSP-based control architectures.