Speed Sensor Less Control Of Im Based On A Reduced-order Adaptive Observer

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 15, NO. 6, NOVEMBER 2007

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Speed Sensorless Control of Induction Motors Based on a Reduced-Order Adaptive Observer Marcello Montanari, Sergei M. Peresada, Carlo Rossi, and Andrea Tilli

Abstract—A novel speed sensorless indirect field-oriented control for the full-order model of the induction motor is presented. It provides local exponential tracking of smooth speed and flux amplitude reference signals together with local exponential field orientation, on the basis of stator current measurements only and under assumption of unknown constant load torque. Speed estimation is performed through a reduced-order adaptive observer based on the torque current dynamics, while no flux estimate is required for both observation and control purposes. The absence of the flux model in the proposed algorithm allows for simple and effective time-scale separation between the speed–flux tracking error dynamics (slow subsystem) and the estimation error dynamics (fast subsystem). This property is exploited to obtain a high performance sensorless controller, with features similar to those of standard field-oriented induction motor drives. Moreover, time-scale separation and physically-based decomposition into speed and flux subsystems allow for a simple and constructive tuning procedure. The theoretical analysis based on the singular perturbation method enlightens that a persistency of excitation condition is necessary for the asymptotic stability. From a practical viewpoint, it is related to the well-known observability and instability issues due to a lack of back-emf signal at zero-frequency excitation. A flux reference selection strategy has been developed to guarantee Persistency of excitation in every operating condition. Extensive simulation and experimental tests confirm the effectiveness of the proposed approach. Index Terms—Adaptive observers, indirect field oriented control, induction motor (IM) drives, sensorless, velocity control.

I. INTRODUCTION ECTOR-CONTROLLED induction motors (IMs) [1] without speed sensor have received wide attention in the field of low and medium performance electrical drives. Industrial applications where sensorless IM drives are widely spread are manufacturing machines, belt conveyors, cranes, lifts, compressors, trolleys, electric vehicles, pumps, and fans. With respect to standard vector controlled IM drives, absence of speed/position sensor reduces cost and size and increases the reliability of the overall system. On the other hand, performance degradation strongly limits the safe applicability of sensorless controllers. As a matter of fact, not only reduced

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Manuscript received November 7, 2005; revised August 23, 2006. Manuscript received in final form January 12, 2007. Recommended by Associate Editor A. Bazanella. M. Montanari, C. Rossi, and A. Tilli are with the Center for Research on Complex Automated Systems “G. Evangelisti” (CASY), Department of Electronics, Computer, and System Sciences (DEIS), University of Bologna, 40123 Bologna, Italy (e-mail: [email protected]; [email protected]; atilli @deis.unibo.it). S. M. Peresada is with the Department of Electrical Engineering, National Technical University of Ukraine “Kiev Polytechnic Institute,” Kiev 252056, Ukraine (e-mail: [email protected]). Digital Object Identifier 10.1109/TCST.2007.899714

speed tracking accuracy and load torque rejection capability affect the sensorless approach, but instability phenomena can also occur in the low-speed region and in the regenerating mode when synchronous frequency approaches to zero [2]. Many researchers have focused on the design of sensorless control algorithms for IMs. Nevertheless, no control method has been clearly established as the ultimate solution nor theoretical questions related to the stability of the sensorless controlled IM have been completely and rigorously solved. The reader could refer to [2]–[6] for a detailed overview of various methods. From the control system perspective, the problem is usually faced with field oriented control strategy (direct or indirect) or direct torque control method [4], combined with a speed–flux estimator. Various methodologies have been exploited for speed–flux estimation: MRAS observers [7], [8], extended Kalman filter [9], adaptive observers [10], [11], sliding-mode technique [12]. In more recent papers, the composite analysis of speed estimation combined with the speed–flux controller is carried out. In [13], a sliding mode rotor flux, stator current, and speed observer together with a sliding mode torque-flux controller were designed. Stability analysis of the overall dynamics has been carried out using model-order reduction related to sliding mode technique and applying linearization method. The sensorless controller [14] is based on a velocity observer which exploits a back-emf term in the stator current dynamics. It guarantees semiglobal speed–flux tracking under assumption of known and constant load torque and stator flux measurement (obtained by integration of the stator voltage equations with zero initial conditions). Under the same assumptions a methodology for the combined design of an adaptive flux-speed observer and controller has been presented in [15]. The proposed controller ensures global asymptotic tracking of smooth speed and flux references. In [16], Montanari et al. designed an adaptive speed observer exploiting stator current regulation errors and using rotor fluxes, obtained by stator model integration. Local exponential speed and flux tracking is formally proven under condition of unknown, but constant, load torque. The algorithm in [17] is designed under assumptions of unknown rotor/stator fluxes but with known and smooth load torque. It guarantees local exponential speed–flux tracking, provided that persistency of excitation related to IM observability properties is ensured. The same control problem with explicitly computable domain of attraction has been solved in [18]. These theoretical results represent a significant contribution to the theory of sensorless vector control of IMs. Nevertheless, all of them are based on strong simplifying assumptions, such as required flux measurement and/or known load torque, and therefore, they cannot be considered as “true” speed sensorless solutions suitable for industrial implementation.

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The aim of this paper is to design and test a new speed sensorless controller, which provides local exponential speed/flux tracking and field orientation, on the basis of stator current measurement only, under assumption of unknown constant load torque. The concept of improved indirect field orientation (see [19] and [20]), combined with a novel reduced-order adaptive observer, based on the torque current dynamics (see [21] for a preliminary version with no formal stability proof), are used for the control algorithm development. Speed is estimated through the speed-dependent back-emf signal perturbing the torque current dynamics. The control algorithm generates estimates for time-varying motor speed and constant load torque, that converge to the corresponding true values under persistency of excitation conditions. For controller performances and stability of the overall error dynamics, meaningful properties are a suitable decoupling between the electromechanical, electromagnetic, and estimation subsystems and the possibility to design the dynamics of the speed observer arbitrarily fast with respect to the dynamics of the field-oriented speed–flux controller. On the basis of system structure and fundamental properties of the IM electromagnetic subsystem under persistency of excitation, the stability analysis of the proposed solution is performed using the singular perturbation method [22], [23]. It results that persistency of excitation, which is related to observability properties of the IM model, is satisfied if the dc excitation condition, i.e., null synchronous speed, is avoided. Exploiting the dependence of slip frequency on rotor flux, flux reference may be varied in order to avoid this situation, thus guaranteeing correct speed estimation and stability of the electromagnetic subsystem in every working conditions, defined by the imposed speed reference and applied load torque. In [24], an algorithm for flux reference selection in order to avoid the critical region around zero excitation frequency has been combined with a speed sensorless torque control. In [25], a flux selection strategy is developed and combined with an adaptive full-order speed–flux observer and a field oriented controller. In this paper, a new flux reference selection algorithm is developed, with the aim of imposing the optimal flux which maximizes the modulus of the synchronous speed, considering constraints on the minimum and maximum flux level. Experimental tests demonstrate that the achievable performances are similar to those of standard vector controlled IM drives with speed measurement. The main advantages and novelty of the proposed solution are as follows: • common simplifying assumptions are avoided: no flux measurement or pure integration of stator flux model is required, and load torque is assumed unknown (but constant, according to standard adaptive control approach) for both control and observation purposes; • the controller has a physically based structure and no flux estimation is required; • time-scale separation and system decomposition allows for simple and constructive tuning procedure for controller gains; • observability properties of the IM are guaranteed by the flux reference selection algorithm in any operating conditions.

According to the basic sensorless control problem formulation, the controller is designed under assumptions of constant and known motor parameters, while, in real-world applications, these parameters could be affected by significant uncertainties and could vary during motor operations, e.g., thermal drift of stator and rotor resistance. Nevertheless, the proposed solution is suitable for real applications, since its asymptotic stability property ensures intrinsic robustness to parameter uncertainties [23]. Deep analysis of the robustness properties of the proposed controller is beyond the purpose of this paper. By the way, it is well known that the stator resistance is one of the most critical parameters affecting performance of sensorless controllers, mostly in the low speed and regenerating region, when excitation frequency is close to zero [26], [27]. In the proposed controller, the stator resistance variation can be compensated or by measuring the stator temperature, or by means of an online stator resistance estimation algorithm embedded in the sensorless controller, as in [12], [24], [26], and [28]. This paper is organized as follows. The IM model and control problem formulation are given in Section II. The proposed solution is presented in Section III. In Section IV, the stability analysis and the proposed flux selection algorithm are reported. Details on the stability proof based on the singular perturbation technique are given in Appendixes I and II. In Section V, results of simulation and experimental tests are reported. Finally, Section VI draws conclusions. II. INDUCTION MOTOR MODEL AND CONTROL PROBLEM STATEMENT A. Induction Motor Model The equivalent two-phase model of the symmetrical IM, under assumptions of linear magnetic circuits and balanced operating conditions, is expressed in an arbitrary rotating reference frame ( - ) [1] as

(1) where denote stator current, rotor flux, and stator voltage vectors [subscripts and stand for vector components in the ( - ) reference frame], is the rotor speed, is the load torque, and are the angular speed and position of the reference frame ( - ) with respect to a fixed stator reference frame ( - ), where the physical variables are defined. Transformed variables in (1) are given by

with (2) where

stands for any 2-D vector of IM model.

MONTANARI et al.: SPEED SENSORLESS CONTROL OF INDUCTION MOTORS

Positive constants related to electrical and mechanical parameters of the IM are defined as , where is the total rotor inertia, is the are stator/rotor resistances and friction coefficient, is the magnetizing inductance, inductances, respectively, and is the number of pole pairs. B. Control Objectives General specifications for speed-sensorless controlled electric drives require to control the two IM outputs, speed, and rotor flux modulus, defined as

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and -axis current estimate

, estimation errors are defined as and the estimated . The speed tracking error is defined as . slip frequency is

A. Speed-Flux Controller In the framework of improved indirect field oriented control [19], replacing measured speed with the estimated one, the following speed–flux tracking sensorless controller is defined: Flux controller

(6) using the 2-D stator voltage vector , on the basis of measured . Defining , variables vector and are speed and flux reference trajectories, where the speed and flux modulus tracking errors are (3)

Speed controller (7)

Current controller (8)

Following the concept of indirect field orientation [1], [19] the - and -axis flux tracking errors are defined as with (4) is the condition of asymptotic field Note that orientation. From (3) and (4), it follows that the condition implies . The speed–flux tracking control problem is formulated as follows. Consider the IM model (1), (2) and assume the following. A.1. Stator currents are available from measurements. A.2. Motor parameters are exactly known and constant. is unknown, bounded, and constant. A.3. Load torque A.4. Speed and flux reference trajectories are smooth functions with known and bounded first and . second time derivatives, and Under these assumptions, it is required to design an output feedback controller which guarantees local asymptotic rotor speed and flux amplitude tracking together with asymptotic field orientation, i.e., (5)

(9) where are proportional and integral gains of the are proportional gains of the speed controller, are additional correction terms current controller, and designed according to Lyapunov-like technique as shown in is a constant tuning gain. Section IV, and B. Reduced-Order Speed Observer The proposed speed observer is based on the torque current dynamics only, therefore, it is a reduced-order solution with respect to the IM electromechanical model. According to the adaptive control approach, an integral component is adopted to estimate the speed tracking error, while no information on the mechanical model are required. The observer equations are the following:

with all signals bounded. III. SPEED-FLUX SENSORLESS CONTROL ALGORITHM The proposed solution exploits the concept of improved indirect field-oriented control presented by Peresada and Tonielli [19] and an original reduced-order observer based on the torque current dynamics. Controller and observer designs are presented in Sections III-A and III-B, respectively, while properties of the overall controller structure are discussed in Section III-C. as refThe following notations are introduced. Defining currents, respectively, current tracking errors erences for are defined as . Introducing the of load constant , speed estimation , the estimation

(10) where are the observer tuning gains. Remark 1: The main idea for the speed observer design recurrent dylies on the influence of the rotor speed on the , which is estimated namics through the back-emf term as , assuming that the flux vector converges to its reference. Since no flux estimation is performed for the speed reconstruction, the performances are strictly related to flux control properties. This basic consideration will be confirmed by the theoretical analysis in Section IV.

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Fig. 1. Block diagram of the equivalent error dynamics of the mechanical and estimation subsystems.

C. Controller and Observer Analysis From IM model (1), control algorithm (6)–(8), and observer (10), the speed–flux tracking and the estimation error dynamics are

(11)

(12)

(13) with (14) (15) The error dynamics (11)–(13) is given by the tracking error dynamics (11) and (12) and the observer error dynamics (13). The former is composed by the nonlinear electromagnetic dynamics in (11) and the linear mechanical dynamics in (12) in feedback with linear/biinterconnection through the coupling term linear properties, while is the flux-error-dependent perturbation for speed estimation. The mechanical and estimation subsystems (12) and (13) are shown in the block diagram of Fig. 1. Two linear second-order dynamics related to PI speed controller and speed estimator can , defined in (14) and be observed, while nonlinear terms (15), establish the interconnection between mechanical/estimation dynamics and electromagnetic subsystem (11). From the scheme reported in Fig. 1, it is natural to define the nominal dynamics of the observer and speed subsystems as the one obtained when no perturbation comes from the electromagnetic subsystem. Hence, the nominal dynamics are given by (12) and and . From its structure, it follows that (13) with of the speed tracking error is the output of the estimation a second-order linear filter with unitary static gain, which can

be viewed as sensor subsystem in the negative-feedback speed control loop. Therefore, it is straightforward imposing the nominal estimation dynamics much faster than the nominal speed subsystem. This is a key element of the controller tuning as explained in Section IV. The high-gain observer design relies on adaptive control theory. As shown in Fig. 1, the integral action with gain in the dynamics in (10) imposes negative feedback with constant static gain to the speed estimation loop, thus achieving linear exponentially stable estimation dynamics (33). In dynamics, as also enlightened by the Appendix I, convergence properties are formally derived. Remark 2: The speed estimation dynamics can be designed , since no flux model arbitrarily fast by selection of gains is embedded in the observer. Nevertheless, independently of how fast the speed observer (13) is, cannot be directly approxowing to the perturbation coming from the flux imated by tracking error dynamics. The flux information obtained by the integration of the stator flux equations with known initial conditions [7], [14]–[16], [26], [27] could be attractive for compensation of the perturbation in the speed observer (13). However, it is well known that this kind of flux estimator is not reliable in actual implementation, owing to effects of imperfect knowledge of stator resistance, nonidealities in stator voltage actuation, and measurement offsets. IV. STABILITY ANALYSIS The stability of the overall error dynamics (11)–(13) is based on structural properties of the feedback interconnected electromagnetic, mechanical, and speed estimation subsystems, generated by the field oriented speed–flux controller together with the speed observer, whose dynamics can be designed arbitrarily fast with respect to the control one. On the basis of considerations in Section III-C, time-scale separation is imposed between the speed–flux tracking error dynamics (slow-subsystem) and the speed observation error dynamics (fast-subsystem). In particular, estimation error dynamics (13), dependent on tuning , is imposed to be faster than electromagnetic (11) gains and mechanical (12) dynamics. The mechanical dynamics can , while current-flux be tuned through selection of gains , due dynamics are characterized by the rotor time-constant to IFO controller structure.

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According to the singular perturbation method [22], [23], the analysis is focused first on the reduced-order system, given by the slow-subsystem with the fast dynamics approximated by its quasi-steady state, as long as exponential stability of the fast-subsystem is guaranteed. Details on the application of the singular perturbation method for systems (11)–(13) are given in Appendix I. The resulting quasi-steady state for the observer error dynamics (13) is the following: (16) Remark 3: From (16), it can be noted the effect of flux amplitude regulation and orientation errors on the speed estimation, as previously mentioned in Remark 1. The reduced-order system is composed by the electromagnetic error dynamics (11) and the mechanical error with their dynamics (12) replacing the fast variables quasi-steady-state expression (16). Defining new state variables , which are proportional to stator flux tracking errors, and substituting (16) into (11) and (12), the reduced-order system expressed with state variables , and is given by

(19) First, local exponential stability of the reduced-order dynamics (17) and (18) is proven, while the full-order system is considered in Appendix II. Stability properties of the electromagnetic (17) and mechanical dynamics (18) are analyzed separately, and then composite reduced-order dynamics is investigated, exploiting the linear-bilinear (and higher order) properties of [see (14)]. In particular, the nominal mechanical subsystem is linear and the stability analysis is straightforward, while the Lyapunov-like method is used for the nonlinear time-varying electromagnetic dynamics. A. Electromagnetic Subsystem In order to analyze the stability properties of electromagnetic dynamics (17) the following candidate Lyapunov function is defined:

(17)

(20) where is a constant positive tuning gain. Its time derivative along trajectories of (17) is

(18)

where are the quasi-steady-state values of , respectively, i.e., obtained from (6) and (9) replacing with from (16), and . Partitioning and into time-dependent and state-dependent parts, their expressions are given by

Selecting (21) and substituting expressions of defined in (9) and evaluated on the slow manifold as in (19), becomes

(22) . with Remark 4: Note that have been suitably selected to compensate for the quadratic cross terms in the time derivative of . No result can be directly obtained from properties of and in (20) and (22), but the previous considerations will be instrumental for the following analysis. Substituting (9) in (17), and defining

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the flux subsystem dynamics can be rewritten as follows:

is differentiable and is bounded. If there exist two pos, such that the persistency of excitation condition itive reals for all

(23)

(27)

is satisfied, then from [30, Lemma 1 (extended persistency of excitation lemma)] it follows that the origin of the linear timevarying system (24), which represents the linearization of (23), is exponentially stable. B. Reduced-Order Dynamics Consider, first, the linearization of the reduced-order system (17) and (18), which is compactly expressed as

where represents the linear part and includes all the higher order terms. , Remark 5: The interconnection matrices between in [see (23)] are not skew-symmetric, owing and to presence of flux tracking errors in the quasi-steady-state estimated speed (16). As a first step, the stability properties of the linearized dynamics of (23)

(28) , and

where

, while

are defined as

(24) with are analyzed. Considering the time-varying coordinate transformation

(25) with

, dynamics (24) in new coordinates become

(26)

where

are defined as

Note that (26) is in standard form of adaptive systems [29, Sec. [defined in 2.8]. Evaluating the time derivative of with ] along the linearized dynamics (26), (20) replacing , hence, are bounded. it follows that From (26), it follows that are bounded, and therefore, are uniformly continuous. Since , it follows that are square integrable signals and, therefore, according to Barbalat’s lemma [23, Lemma 4.2] it results that . Since , and are bounded by assumption, it follows that the dynamical matrix of (26) is continuous and bounded,

Note that the linearized electromagnetic dynamics has been decontains only the rived in Section IV-A, while the matrix reference-dependent parts of . The main feature of the linear approximation (28) is that current-flux and speed dynamics are in series interconnection. is Since the LTV current-flux subsystem globally exponentially stable if PE is satisfied, matrix is and is bounded, it follows that Hurwitz with (28) is GES. Hence, from standard nonlinear control results [23, Th. 3.11], it follows that the origin of the nonlinear reducedorder system (17) and (18) is locally exponentially stable. C. Full-Order Dynamics Standard singular perturbation technique can be directly applied to prove stability of the full-order error dynamics (11)–(13) (see Appendix II). Briefly, the full-order system given by (11)– (13) is presented as feedback interconnected reduced-order system (32) and boundary-layer estimation dynamics (33). The reduced-order system is LES if PE is satisfied, and the boundary layer system (33) is linear and exponentially stable, therefore, the full-order dynamics is LES, provided that parameter is sufficiently small [22, Sec. 7.5]. Hence, it has been proven that local exponential speed–flux tracking together with field orientation are achieved, provided that PE condition (27) is satisfied.

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D. Persistency of Excitation Condition and Flux Reference Selection It is important to get a physically meaningful comprehension of the persistency of excitation result (27), in order to understand in which IM operating conditions the proposed controller fails to be asymptotically stable due to lack of PE. Then, in order to avoid these conditions, an algorithm for the reference flux selection is proposed. A theoretical analysis of PE condition (27) is complex considering generic operating conditions of the IM. Assuming conand load torque , it stant speed and flux references in (19) is constant, hence, . When follows that , i.e., with dc excitation, it results

hence, the matrix in (27) is positive semi-definite and the PE , choosing , it condition fails. When results

Fig. 2. Selected flux reference estimated torque J 1 T^ .

as a function of speed reference

!

and

TABLE I INDUCTION MOTOR DATA

for all Hence, it follows that the PE condition is satisfied under all constant operating conditions apart from zero frequency. depends on , and , the flux reference can Since be used as an extra degree of freedom to avoid lack of PE, i.e., . To this purpose a simple strategy for the flux reference selection which maximizes synchronous frequency taking into account physical flux bounds is proposed. In particular, considering the steady-state contributions only, the following auxiliary synchronous frequency can be defined: (29) Hence, for given the flux reference can be selected is maximum, while satisfying the constraint such that . Flux bound represents the minimum admissible flux not impairing the IM performances, while is defined according to IM magnetic saturation. The following flux reference selection algorithm solves the proposed maximization problem: if

if

(30) with . In Fig. 2 it is shown how the flux reference is selected depending on and . Notice that the minimum flux level is selected during motoring mode, while the maximum flux level is imposed only in some regenerating conditions. Solution (30) guarantees that dc-excitation is always avoided.

The flux selection strategy is enabled only at low excitation frequency and with high torque current. In fact, at high excitation frequency, no observability issues arise, hence, the flux reference is imposed to the rated value, or a different flux selection technique, e.g., for power efficiency optimization or flux weakening, can be utilized. Moreover, operation with low torque current and low excitation frequency is not critical from a practical viewpoint, since the small motor torque is comparable to friction torque; on the other hand, at low motor torque, the variation of excitation frequency achieved by changing the flux level is less relevant [see (29)]. Additionally, in order to reduce sensitivity with respect to measurement noise and estimation errors, signal filtering and hysteresis are introduced to avoid spurious commutations of the flux reference. Moreover, in the implementation the flux reference (30) is filtered, in order to provide bounded time-derivatives and to avoid quick flux changing, which may induce large flux errors and torque oscillations. V. SIMULATION AND EXPERIMENTAL RESULTS The proposed speed sensorless control algorithm has been tested by means of simulations and experiments using a 1.1-kW induction motor whose rated data are reported in Table I.

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Fig. 3. Block diagram of the simulated IM model with the speed sensorless controller.

A. Controller Tuning Controller parameters have been selected according to the following tuning rules, which derive from the structure of the error dynamics (11)–(13) and according to the analysis reported in Appendix I. Some implementation aspects, such as robustness with respect to parameter uncertainties and unmodelled dynamics, sensitivity with respect to measurement noise, issues related to discrete-time implementation of the algorithm, have been taken into account in a heuristic way, performing some simulations and/or experiments in order to refine the controller tuning. Selecting gains and , the speed controller is tuned imposing a time constant and damping factor for the secondorder nominal mechanical error dynamics (12) (see Fig. 1). The second-order speed observer dynamics (13) is designed imposing a time constant and damping factor by selection of and . From a practical viewpoint, time-scale separation between dynamics (12) and (13) is achieved with . The bandwidth of the current error dynamics is imposed by selecting the proportional -axis and -axis current controller gains . Selection of large gains , which leads to the current-fed condition, ensures fast current dynamics, and reduced sensitivity to imperfect knowledge of IM electrical parameters, such as stator resistance. Nevertheless, large gain implies a high value of gain , which is computed according to (21), thus, leading to high sensitivity to current measurement noise superimposed on .

The following controller gains have been chosen for all simulations and experiments. The speed controller gains are set at 120 s 7440 s , imposing a mechanical time constant 11.6 ms. The tuning gains of the speed observer are selected as 240 s 93400 s , imposing an 3.3 ms. Current controller gains observer time constant have been set at 150 s 300 s . B. Simulation and Experimental Set-Up The continuous-time version of the algorithm is considered in the simulations. Ideal stator voltage actuation is assumed, i.e., no pulse-width modulation (PWM) switching of the inverter is considered and no noise is superimposed on the stator currents. A detailed block diagram of the controller (6)–(10) together with the IM model (1) is shown in Fig. 3. The IM model is expressed in the stationary reference frame ( ), while the controller is implemented in the rotating reference frame ( ). In the controller, direct Park transformation (2) is utilized to transform stator currents into , while the inverse Park transformation is utilized to compute stator voltages from . In the controller equations, are analytically computed from expressions of in (6) and (7). The experimental tests have been carried out using a rapid prototyping station, which includes the following: • a personal computer acting as the operator interface; • a custom floating-point digital signal processor (DSP) board (based on TMS320C32) connected to the ISA PC bus; the DSP board performs data acquisition (eight 12-bit A/D data channels and two interfaces for incremental

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Fig. 4. Speed, flux references, and load torque profile (dashed line).

encoder), implements the control algorithm and generates the PWM signals (two symmetrical three-phase PWM modulators with programmable dead-time); • a 50 A/400 V three-phase inverter, operated at 10-kHz switching frequency; dead-times of the inverter have been set to 1.5 s; • a 4-pole, 50-Hz, 1.1-kW standard induction motor, whose data are listed in Table I; • a vector-controlled permanent magnet synchronous motor used to provide the load torque. In order to filter out the modulation ripple, two stator phase currents, measured by Hall-effect zero-field sensors, are simultaneously and synchronously sampled at the symmetry point of the PWM signals. Only for monitoring purposes, the motor speed is measured by means of a 512 pulse/revolution incremental encoder. Dead-time effects have been compensated with a simple method based on current sign [5], [31]. The simple Euler method has been used in order to get the discrete time version of the control algorithm, with 200- s sampling time. C. Operating Sequences The following flux and speed references and load torque profiles are used during tests, as shown in Fig. 4. 1) The machine is excited during the initial time interval 0–0.12 s applying a flux reference trajectory starting at 0.01 Wb and reaching the motor rated value of 0.86 Wb with maximum first and second time derivatives equal to 8 Wb/s and 500 Wb/s , correspondingly. 2) The unloaded motor is required to track the speed reference trajectory characterized by the following phases: starting at 0.4 s from zero initial value, speed reference trajectory reaches 100 rad/s at 0.45 s; from this time to 1.3 s 1.3 s to 1.35 s a constant speed is imposed; from the motor is required to stop at zero speed reference. Maximum absolute values of the first and second derivatives of the speed reference trajectory are equal to 2200 rad/s and 20 000 rad/s , correspondingly. Speed tracking during reference trajectory variations requires a dynamic torque equal to the motor rated one. 3) From time 0.7 s to 1.0 s a constant load torque, equal to 100% of the motor rated value (7.0 Nm), is applied. D. Simulation and Experimental Results Dynamic performance of the controller during speed tracking and load torque rejection has been tested in various speed-load operating conditions.

Fig. 5. Simulation results: dynamic behavior of the sensorless controller: Speed errors and q -axis current estimation error; stator currents; stator voltages; rotor flux errors.

In the first test, simulations and experiments during transient with 100-rad/s maximum speed and 7.0-Nm constant load torque have been performed. In Fig. 5 simulation results are shown. Speed estimation and tracking errors are negligibly small during speed trajectory tracking, while a maximum speed tracking error of about 14 rad/s is present during load torque rejection. Time-scale separation between speed estimation and regulation dynamics guarantees accurate speed estimation with maximum estimation error of about 6 rad/s. As a result, the speed, current, and flux tracking errors caused by the unknown load torque perturbation are compensated by the load estimation mechanism based on the estimated speed tracking error. Results related to experimental tests performed under the same operating conditions of Fig. 4 are shown in Fig. 6. From Figs. 5 and 6, it can be noted good similarity between simulation and experimental results. Zero steady-state speed estimation

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Fig. 7. Experimental results: dynamic behavior of the improved IFO controller.

Fig. 6. Experimental results: dynamic behavior of the sensorless controller with maximum speed reference equal to 100 rad/s and applied load torque equal to 7.0 Nm: Speed errors and q-axis current estimation error; stator currents; stator voltages.

and tracking errors are achieved when constant speed reference and load torque are applied. Similar dynamic performance and amplitude of speed tracking and estimation errors are obtained during load torque rejection, with maximum speed tracking error of about 13 rad/s. It is worth noting that, in actual implementation, maximum speed tracking and estimation errors are about 4 rad/s during speed reference variation. This is mainly due to the imposed fast speed transient (50 ms), IM parameter uncertainties and nonidealities such as dead time effects and

stator voltage inverter distortion. Stator currents are quite close to the reference ones. The proposed sensorless controller is compared with the improved indirect field oriented (IFO) control algorithm [19], [20], which is based on speed measurement, under the same operating conditions of Fig. 4. IFO controller parameters are selected as follows. The proportional and integral speed controller gains have been set as in the sensorless controller, i.e., , proportional and integral current controller gains are , and . The experimental results reported in Fig. 7 demonstrate the dynamic performance of the improved IFO controller. Maximum amplitude of the speed tracking error is 11 rad/s during load torque rejection transient and it is almost null during speed tracking. The settling time achieved during load torque rejection is the same of the sensorless controller case. Comparison of the experiments performed with the sensorless controller (see Fig. 6) and with the improved IFO controller (see Fig. 7) demonstrates that the achievable performances are similar to those obtained with speed feedback, under the tested operating conditions. The second set of experiments have been performed to test the performance of the proposed controller under low-speed operating conditions. These regimes are very critical since they are characterized by reduced index of PE ( can be almost zero in regenerating mode), increasing sensitivity to IM parameter uncertainties (e.g., stator and rotor resistance variations) and to inverter nonidealities (e.g., dead-time and resistive losses) [26]. The same sequence of motor operations (see Fig. 4) with maximum speed of 10 rad/s and maximum acceleration of 550 rad/s is used to test the dynamic behavior of the controller under constant rated load (7 Nm) and regenerating ( 7 Nm) applied torques. Note that in the regenerating case synchronous speed is 8.5 rad/s. Transients reported in Figs. 8 and 9 confirm that no speed tracking performance degradation is present with respect to the high-speed case for both motoring and regenerating modes. In Fig. 10 rejection of constant rated load torque with zero speed reference is considered. During load torque application from 0.7 s to 1.0 s synchronous speed is 11.5 rad/s, as confirmed also from the stator current profiles, hence, PE condition is verified. An important observation can be made from all previous experiments (see Figs. 6 and 8–10): the final test conditions (from 1.35 s) are characterized by lack of PE, in fact , and . Nevertheless, the proposed solution guarantees negligible speed error for this particular condition. The third set of experiments was carried out to test the behavior of the speed controller when PE condition is not satisfied. The performances of controller with and without the flux reference selection algorithm of Section IV-D are

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Fig. 8. Experimental results: dynamic behavior of the sensorless controller with maximum speed reference equal to 10 rad/s and applied load torque equal to 7.0 Nm: speed errors and stator currents.

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Fig. 10. Experimental results: dynamic behavior of the sensorless controller with speed reference equal to 0 rad/s and applied load torque equal to 7.0 Nm: speed errors and stator currents.

Fig. 11. Speed reference (solid line) and load torque profile (dashed line).

Fig. 9. Experimental results: dynamic behavior of the sensorless controller with maximum speed reference equal to 10 rad/s and applied regenerating torque equal to 7.0 Nm: speed errors and stator currents.

0

comparatively evaluated when motor operates in regenerating mode. In these tests, parameters of the flux reference selection algorithm have been chosen as follows: minimum and maximum fluxes are 10% of the rated flux, i.e., 0.77 Wb, 0.95 Wb, while the resulting flux reference is filtered by the nonlinear filter [32] with maximum first and second time derivatives equal to 2 Wb/s and 50 Wb/s . According to Fig. 11, a speed reference profile with steady-state value 7.5 rad/s is imposed and constant rated regenerating torque ( 7 Nm) is applied from 0.6 s to 1.3 s. In the first test, reported in Fig. 12, a constant flux reference is adopted, leading to (i.e., lack of PE) when the regenerating

torque is applied. From this test it is clear that speed is not correctly estimated, hence, nonnull speed tracking error is obtained. Differently, in the second test, reported in Fig. 13, the flux reference is selected according to (30). In this way, the synchronous speed is not equal to zero and, as a result, correct speed estimation and almost null speed tracking error are achieved. In order to test the behavior of the controller during motoring, plugging, regenerating, zero speed, and zero frequency conditions, a slow speed reference from 20 to 20 rad/s and back with acceleration equal to 10 rad/s is imposed to the IM, with constant rated torque equal to 7.0 Nm. During tests, minimum and maximum flux levels are imposed at 0.74 Wb and 1.11 Wb and flux reference is filtered, such that maximum time derivative of the flux reference is equal to 10 Wb/s. The flux selection strategy (30) is enabled only if excitation frequency becomes lower than 15 rad/s (in modulus) and estimated torque is greater than 2.2 Nm, while it is disabled for becoming greater than 25 rad/s. In other operating conditions, flux reference is imposed to 0.86 Wb. Experimental results are reported in Fig. 14. Flux reference is maintained at its constant rated value 0.86 Wb till 2.45 s (when

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0

Fig. 14. Experimental results: speed transient from 20 to 20 rad/s and back, with applied torque equal to 7 Nm, with optimal flux reference selection: speed, synchronous speed, and flux reference; stator currents, speed estimation error.

Fig. 12. Experimental results: dynamic behavior of the sensorless controller with speed reference equal to 7.5 rad/s and applied regenerating torque equal 7.0 Nm, with constant flux reference: speed errors, flux reference, and to synchronous speed; stator currents.

0

15 rad/s); then, flux level is decreased at 0.74 Wb, hence, -axis current decreases, while excitation frequency and torque current increase. Induction motor works in motoring mode till 2.4 s ( 0 rad/s), then it starts operating in plugging mode . At 3.2 s, the flux level is increased at 1.11 Wb, thus avoiding operation close to zero excitation frequency. The -axis current increases, while slip frequency dechanges sign, thus IM operates in regenerating creases and mode. At 4.2 s, when 25 rad/s, the flux level is imposed to its rated value. Speed transient from 20 to 20 rad/s is performed in a similar way. During operations, estimation and tracking error with maximum amplitude of about 3 rad/s are present during flux change, due to magnetic saturation effect. The presented tests confirm that the proposed flux reference selection algorithm combined with the speed sensorless control are profitable to overcome the problem of zero frequency excitation, when PE is not satisfied or is close to fail. VI. CONCLUSION

Fig. 13. Experimental results: dynamic behavior of the sensorless controller with speed reference equal to 7.5 rad/s and applied regenerating torque equal to 7.0 Nm, with optimal flux reference selection: speed errors, flux reference, and synchronous speed; stator currents.

0

In this paper, a novel speed sensorless controller for IM, which guarantees local exponential tracking of smooth speed– flux references together with field orientation, has been presented. In the proposed method, no flux estimation or reconstruction through integration of the stator voltage model is needed, while load torque is assumed constant but unknown. A persistency of excitation condition, which fails to be guaranteed at dc excitation, is required for stability. Physical understanding of such result has been exploited to develop an effective flux reference selection procedure, in order to avoid instability issues. Experiments and simulations show that the proposed controller combined with the flux selection algorithm is suitable for high-performance sensorless controlled IM drives. The proposed controller has been designed under assumption of known and constant motor parameters; nevertheless, electrical parameters may vary during motor operations and it is

MONTANARI et al.: SPEED SENSORLESS CONTROL OF INDUCTION MOTORS

well known that parameter uncertainties affect the stability and transient performance of sensorless controllers. The design of adaptive version of the controller with respect to motor parameters will be the subject of further works. APPENDIX I ANALYSIS OF THE FULL-ORDER ERROR DYNAMICS USING SINGULAR PERTURBATION TECHNIQUE Methodology and formalism of singular perturbation technique [22], [23] have been exploited in order to give a formal approach to time-scale separation of the full-order error model and to perform the stability analysis. Introducing the following , and state variables: , selecting tuning parameters according to (31) with

and defining the small parameter , error dynamics (11)–(13) is expressed using as as state variables

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, the quasi-steady state From (33) with is obtained as the unique isolated solution of the alge. It is given by braic equation

(36) Note that from definition of , the quasi-steady state for corresponds to the quasi-steady state for and given in (16). Defining the boundary-layer variable and , the so-called boundary layer dynamics is defined as (37) where and are considered as fixed parameters [see (33)]. According to Tikhonov’s Theorem for nonlinear time-varying systems, if the origin of the boundary layer system is asymptotically stable and the small parameter is sufficiently small, the behavior of the full-order dynamics (32) and (33) can be approximated by the so-called reduced-order system

APPENDIX II STABILITY OF THE FULL-ORDER DYNAMICS (32)

Stability proof is carried out transforming the full-order error dynamics (11)–(13) into standard form for the singular perturbation method and exploiting the following properties: series interconnection of the linearized reduced-order flux and speed subsystems, exponential stability of the flux dynamics provided that PE condition is satisfied, and exponential stability of the speed and estimation isolated subsystems. Imposing tuning relations given in Appendix I, from (32) and (33) the full-order error dynamics is rewritten with state variables [see (9), (21), and Appendix I for definitions] as

(33) (38) Systems (32) and (33) are in standard singular perturbation form (34) (35) where state vector, vector, parameter, and that

denotes the “slow” denotes the “fast” state represents the small positive perturbation are smooth and bounded functions, such .

(39)

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(40) Decomposing signals into time-varying state-independent, fast-variable-independent, and fast-variable-dependent terms, respectively, i.e., expressing the generic signal as , the full-order error dynamics can be rewritten enlightening decomposition into “nominal” dynamics, slow-variable-dependent and fastvariable-dependent coupling terms as in (41)–(43), shown at the bottom of the page. Additional signals are defined in (19) and

The following properties of (41)–(43) are meaningful for the stability analysis. is 1) If PE condition is satisfied, system exponentially stable, hence, from the converse Lyapunov function theorem for LTV systems [23, Th. 3.10] there exist a continuously differentiable symmetric matrix and a continuous symmetric matrix such that . , and as in Appendix I, matrices 2) Selecting are Hurwitz, hence, there exist symmetric matrices such that and . 3) Interconnection terms in (41)–(43) can be expressed as linear and higher order polynomial functions of state variables. Bearing in mind that the smallest powers are dominant nearby the origin, interconnection terms can be bounded by linear or bilinear bounds, which hold uniformly in time within given compact sets of state variables. , Assuming that

(41)

(42)

(43)

MONTANARI et al.: SPEED SENSORLESS CONTROL OF INDUCTION MOTORS

with compact sets containing the origin, denoting , the following holds. the Euclidean norm with • There exist bounded functions (where dependence on time is due to reference signals , and load torque ), and there exist positive , such that constants

. • There exist bounded functions and there exists a positive constant

, , such that

. • There exist bounded functions and there exists a positive constant

, , such that

. • There exist bounded functions and there exist a positive constant that

, , such

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Remark 6: It is worth noting that since the existence of the Lyapunov function for the flux subsystem relies on converse is not Lyapunov theorem and PE condition, expression of explicitly known. Moreover, intensive use of Young inequalities and comparison functions lead to conservative evaluation of the domain of attraction and tuning rules. Additionally, choice of comparison functions is not unique, therefore, different evaluations are possible. Anyhow, performed stability result is significant from the theoretical viewpoint. for the Considering the Lyapunov function boundary layer dynamics, its time derivative along trajectories of (43) is

where Young inequalities have been applied to terms dependent , and . Introducing on the composite Lyapunov function and applying Young inequalities to bilinear terms, it holds , with

. • There exist bounded functions

, . It is possible to select coefficients with sufficient degrees of freedom and (enlightening to choose sufficiently small parameter and dynamics), in time-scale separation between , thus obtaining order to impose . Therefore, the of the full-order equilibrium point error dynamics is locally exponentially stable. and

there exist positive constants

and , such that

. contains linear terms, Previous inequalities enlighten that while does not. Consider first the dynamics. Exploiting properties stated before, the time derivative of the (with Lyapunov function ) along trajectories of (41) and (42) is

where the Young inequality has been applied to the term dependent on . Assuming and selecting sufficiently small such , that , in the preliminary case of and it holds , thus proving local exponential stability of the reduced-order error dynamics (41) and (42) (see also results in Section IV-B).

ACKNOWLEDGMENT The authors would like to thank Prof. A. Tonielli from CASYDEIS, University of Bologna, for his help and support during the preparation of this paper. REFERENCES [1] W. Leonhard, Control of Electrical Drives, 3rd ed. Berlin, Germany: Springer-Verlag, 2001. [2] J. Holtz, “State of the art of controlled ac drives without speed sensor,” Int. J. Electron., vol. 80, no. 2, pp. 249–263, 1996. [3] K. Rajashekara, A. Kawamura, and K. Matsuse, Sensorless Control of AC Motor Drives. New York: IEEE Press, 1996. [4] P. Vas, Sensorless Vector and Direct Torque Control. London, U.K.: Oxford Univ. Press, 1998. [5] J. Holtz, “Sensorless control of induction motor drives,” Proc. IEEE, vol. 90, no. 8, pp. 1359–1394, Aug. 2002. [6] J. Holtz, “Sensorless control of induction machines—With or without signal injection?,” IEEE Trans. Ind. Electron., vol. 53, no. 1, pp. 7–30, Feb. 2006. [7] C. Schauder, “Adaptive speed identification for vector control of induction motors without rotational transducers,” IEEE Trans. Ind. Appl., vol. 28, no. 5, pp. 1054–1061, Sep./Oct. 1992. [8] F. Peng and T. Fukao, “Robust speed identification for speed-sensorless vector control of induction motors,” IEEE Trans. Ind. Appl., vol. 30, no. 5, pp. 1234–1240, Sep. 1994. [9] Y. Kim, S. Sul, and M. Park, “Speed sensorless vector control of induction motor using extended Kalman filter,” IEEE Trans. Ind. Appl., vol. 30, no. 5, pp. 1225–1233, Sep. 1994. [10] H. Kubota and K. Matsuse, “Speed sensorless field-oriented control of induction motor with rotor resistance adaptation,” IEEE Trans. Ind. Appl., vol. 30, no. 5, pp. 1219–1224, Sep. 1994. [11] M. Montanari, S. Peresada, A. Tilli, and A. Tonielli, “Speed sensorless control of induction motor based on indirect field-orientation,” in Proc. 35th IEEE Conf. Ind. Appl., 2000, pp. 1858–1865.

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[12] T. Chern, J. Chang, and K. Tsai, “Integral-variable-structure-controlbased adaptive speed estimator and resistance identifier for induction motor,” Int. J. Control, vol. 69, no. 1, pp. 31–47, 1998. [13] Z. Yan, C. Jin, and V. I. Utkin, “Sensorless sliding-mode control of induction motors,” IEEE Trans. Ind. Electron., vol. 47, no. 6, pp. 1286–1297, Dec. 2000. [14] M. Feemster, P. Aquino, D. M. Dawson, and A. Behal, “Sensorless rotor velocity tracking control for induction motors,” IEEE Trans. Control Syst. Technol., vol. 9, no. 4, pp. 645–653, Jul. 2001. [15] R. Marino, P. Tomei, and C. M. Verrelli, “A global tracking control for speed-sensorless induction motors,” Automatica, vol. 40, no. 6, pp. 1071–1077, Jun. 2004. [16] M. Montanari, S. Peresada, and A. Tilli, “Sensorless control of induction motors with exponential stability property,” presented at the Eur. Control Conf., Cambridge, U.K., 2003. [17] M. Montanari, S. Peresada, and A. Tilli, “Sensorless control of induction motor with adaptive speed–flux observer,” in Proc. 43rd IEEE Conf. Dec. Control, 2004, pp. 201–206. [18] R. Marino, P. Tomei, and C. M. Verrelli, “Nonlinear tracking control for Sensorless induction motors,” in Proc. 43rd IEEE Conf. Dec. Control, 2004, pp. 4423–4428. [19] S. Peresada and A. Tonielli, “High-performance robust speed–flux tracking controller for induction motor,” Int. J. Adapt. Control Signal Process., vol. 14, pp. 177–200, 2000. [20] S. Peresada, A. Tilli, and A. Tonielli, “Theoretical and experimental comparison of indirect field-oriented controllers for induction motors,” IEEE Trans. Power Electron., vol. 18, no. 1, pp. 151–163, Jan. 2003. [21] S. Peresada, M. Montanari, A. Tilli, and S. Kovbasa, “Sensorless indirect field-oriented control of induction motors, based on high gain speed estimation,” in Proc. 28th IEEE Conf. Ind. Electron., 2002, pp. 1702–1709. [22] P. V. Kokotovic, H. K. Khalil, and J. O’Reilly, Singular Perturbation Methods in Control: Analysis and Design. London, U.K.: Academic Press, 1986. [23] H. K. Khalil, Nonlinear Systems, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1996. [24] M. Depenbrock, C. Foerth, and S. Koch, “Speed sensorless control of induction motors at very low stator frequencies,” presented at the 8th Eur. Conf. Power Electron. Appl., Lausanne, Switzerland, 1999. [25] H. Kubota, I. Sato, Y. Tamura, K. Matsuse, H. Ohta, and Y. Hori, “Regenerating-mode low-speed operation of sensorless induction motor drive with adaptive observer,” IEEE Trans. Ind. Appl., vol. 38, no. 4, pp. 1081–1086, Jul./Aug. 2002. [26] J. Holtz and J. Quan, “Sensorless vector control of induction motors at very low speed using a nonlinear inverter model and parameter identification,” IEEE Trans. Ind. Appl., vol. 38, no. 4, pp. 1087–1095, Jul./Aug. 2002. [27] K. D. Hurst, T. G. Habetler, G. Griva, and F. Profumo, “Zero-speed tacholess IM torque control: Simply a matter of stator voltage integration,” IEEE Trans. Ind. Appl., vol. 34, no. 4, pp. 790–795, Jul./Aug. 1998. [28] M. Montanari and A. Tilli, “Sensorless control of induction motors based on high-gain speed estimation and on-line stator resistance adaptation,” in Proc. 32nd Annu. Conf. IEEE Ind. Electron. Soc., 2006, pp. 1263–1268. [29] K. S. Narendra and A. M. Annaswamy, Stable Adaptive Systems. Englewood Cliffs, NJ: Prentice-Hall, 1989. [30] R. Marino, G. L. Santosuosso, and P. Tomei, “Robust adaptive observers for nonlinear systems with bounded disturbances,” IEEE Trans. Autom. Control, vol. 46, no. 6, pp. 967–972, Jun. 2001. [31] S. G. Jeong and M. H. Park, “The analysis and compensation of deadtime effects in PWM inverters,” IEEE Trans. Ind. Electron., vol. 38, no. 2, pp. 108–114, Mar./Apr. 1991. [32] R. Zanasi, C. Guarino Lo Bianco, and A. Tonielli, “Nonlinear filters for the generation of smooth trajectories,” Automatica, vol. 36, no. 3, pp. 439–448, 2000.

Marcello Montanari was born in Ravenna, Italy, in 1974. He received the Dr. Ing. degree in computer science engineering and the Ph.D. degree in automatic control from the University of Bologna, Bologna, Italy, in 1999 and 2003, respectively. Since 2003, he has held a Postdoctoral position with the Department of Electronics, Computer, and System Science, University of Bologna. He is a cofounder of ARCA Tecnologie srl, Bologna, Italy, a spin-off company of the University of Bologna, started in 2004. His current research interests include applied nonlinear and adaptive control, in the field of electric drives, automotive applications, and electro-mechanical and electro-hydraulic systems.

Sergei M. Peresada was born in 1952, in Donetsk, USSR. He received the Diploma of electrical engineering from the Donetsk Polytechnic Institute, Donetsk, Ukraine, in 1974 and the Candidate of Technical Sciences degree (corresponding to the Ph.D. degree) in control and automation from the Kiev Polytechnic Institute, Kiev, Ukraine, in 1983. Since 1977, he has been with the Department of Electrical Engineering and Automation, National Technical University of Ukraine “Kiev Polytechnic Institute,” Kiev, Ukraine, where he is currently a Professor of control and automation. He has been a Visiting Professor with the University of Illinois (Urbana-Champaign), Urbana, Department of Electronics, Computer, and System Science (DEIS), University of Rome Tor Vergata, Rome, Italy, and the Institute of Advanced Study University of Bologna, Bologna, Italy. He is the author of about 150 scientific publications and a co-author of the volumes: Theory and Control of Electrical Drives and Control of Electromechanical Systems. His research interests include nonlinear and adaptive control of electromechanical systems based on ac motors, control of power converters. Dr. Peresada currently serves as an Associate Editor of the IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY.

Carlo Rossi received the Dr.Ing. degree in electronic engineering and the Ph.D. degree in system science and engineering from the University of Bologna, 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. From 1994 to 2001, he was with Magneti Marelli Powertrain, Bologna, Italy, where he had the responsibility of the Control Design and System Analysis Groups. Since 2001, he has been an Associate Professor with the Automatic Control Department, University of Bologna. He is a cofounder and CEO of ARCA Tecnologie srl, Bologna, Italy, a spin-off company of the University of Bologna started in 2004. He is also a member of the executive board of the Centre for Research on Complex Automated Systems (CASY), University of Bologna. His research interests include electric motor drives, nonlinear control, internal combustion engines, and vehicle powertrain modeling and control. Dr. Rossi was a recipient of the Best Paper Award from the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS in 1994.

Andrea Tilli was born in Bologna, Italy, on April 4, 1971. He received the Dr.Ing. degree in electronic engineering and the Ph.D. degree in system science and engineering from the University of Bologna, Bologna, Italy, in 1996 and 2000, respectively, where his thesis was based on nonlinear control of standard and special asynchronous electric machines. Since 1997, he has been with the Department of Electronics, Computer, and System Science (DEIS), University of Bologna, where, in 2001, he became a Research Associate. His current research interests include applied nonlinear control techniques, adaptive observers, electric drives, automotive systems, power electronics equipments, active power filters, and DSP-based control architectures. Dr. Tilli was a recipient of a research grant from DEIS on modeling and control of complex electromechanical systems in 2000.