Marise Bafleur
email: marise@laas.fr

Fabrice Caignet

Nicolas Nolhier

David Trémouilles

Tackling the challenges of System level ESD: from efficient ICs ESD protection to system level predictive modeling

Keywords: system level ESD, Human Metal Model, behavioral modeling, VHDL-AMS, ESD protection

This paper provides a review of the main tools that will allow developing a system efficient ESD design (SEED) approach. It starts with copying with the challenges of the narrowing of the ESD design window with temperature and high-voltage I/Os. We then present our behavioral modeling approach using VHDL-AMS and characterization tools such as TLP testing. Finally, the efficiency of the modeling methodology is illustrated with three case studies.

INTRODUCTION

Protection against ESD at system level is a main challenge for embedded electronic systems. It is particularly true in automotive applications where the electronic equipment in vehicles is continuously increasing over the years. More and more complex electronic modules have to work together without disturbing or being disturbed by electromagnetic interferences and fast transient events. To address these requirements, several electromagnetic standards such as IEC61000-4-2 [START_REF]-2: testing and measurement techniqueselectrostatic discharge immunity test[END_REF] or ISO10605 [START_REF]Road vehicles -Test methods for electrical disturbances from electrostatic discharge[END_REF] have been defined to assess the immunity of systems to both direct and indirect ESD stress during operation.

The IEC standard is designed for completed systems. However original equipment manufacturers (OEMs) are requesting integrated circuits manufacturers to also test individual components to this standard. The Human Metal Model (HMM) standard test method [START_REF]Electrostatic Discharge Sensitivity Testing -Human Metal Model (HMM) -Component Level[END_REF] is intended to define a test method for evaluating components using the IEC-61000-4-2 waveform [START_REF]-2: testing and measurement techniqueselectrostatic discharge immunity test[END_REF].

It has to be underlined that the ESD robustness at system level does not depend only on the ESD robustness of the standalone components but also on the complex interactions between the various elements (passive, active, PCB traces) implemented on the printed circuit board (PCB).

In addition, automotive electronics is required to operate at higher temperatures than consumer products. This trend will be accelerated with the advent of electric vehicles and the introduction of wide band gap semiconductors such as GaN and SiC that require silicon drivers operating at high temperature (T≥200°C). It has to be reminded that system level ESD standards require tests for both unpowered and powered systems. As a result, it is important to take into account the ambient temperature for the design of ESD protections.

Given these various requirements and constraints, to perform System Efficient ESD Design (SEED) [START_REF]White Paper 3 : "System Level ESD Part I: Common Misconceptions and Recommended Basic Approaches[END_REF], it is essential to develop high-robustness ESD protections but also to be able to simulate/predict the real behavior of the system under such stress. The main challenge for OEMs to perform predictive modeling is the lack of information regarding the internal ESD strategy of the ICs. A very similar problem arose a few years ago for IC signal integrity issues and gave birth to the IBIS standard [START_REF]IBIS (Input Output Buffer Information Specification)[END_REF].

In this paper, we review the main challenges that need to be tackled to answer requirements for system-level-ESD robustness. In section 2, we first present ESD design protection approaches to cope with narrowed ESD design window in high voltage technologies as well as high temperature issues. In section 3, we present our behavioral ESD predictive modeling based on using ESD-upgraded IBIS models. In section 4, we will detail the characterization methodology we developed as an alternative to using an ESD gun. After presenting the SEED approach in section 5, we will describe in section 6 three case studies involving on the one hand two digital circuits and on the other an analog one. Finally we will give some perspectives of this work.

HIGH-ROBUSTNESS ESD PROTECTION

In automotive applications, providing ESD protection for all the different I/Os from low to high voltages is a real challenge, some particular pins of the ICs being required to withstand very high HBM and IEC robustness (≥ 8kV).

The ESD design window is defined on the one hand by the operation voltage of the I/O to be protected (VDD) and on the other hand by its breakdown voltage (BV) (Fig. 1). In smart power technologies, this window is narrowed by two constraints: -the increase of VDD whereas the oxide breakdown voltage remains the same for high voltage I/Os, -and the detrimental effect of temperature that tends to increase the on-resistance of protection devices.

To cope with the first issue, one solution consists in using ESD protection devices with no snapback. In general, NPN transistors are preferentially used as ESD protections for their very low on-resistance (R ON ). However, their high gain results in a strong snapback that is a major drawback to meet high-voltage specifications.

One way, to solve this problem, is to replace the NPN by a PNP. Indeed, PNP transistors exhibit a weak or no snapback. However, they have a serious handicap lying in their very much higher R ON related to the low mobility of holes carriers. Using careful design guidelines (minimum emitter length, low base doping, abrupt collector doping, base contact suppression, matrix design to maximize emitter perimeter), it is possible to achieve area-efficient PNP-based ESD protections [START_REF] Renaud | High robustness PNP-based structure for the ESD protection of high voltage I/Os in an advanced smart power technology[END_REF]. An additional improvement consists in taking advantage of the vertical parasitic PNP of the technology that operates much like a diode (Fig. 2). The assisting diode can be used to set the clamping voltage of the structure. As shown in Fig. 2, using such an implementation, a reduction of R ON by a factor of seven is achieved: 0.1 mΩ.cm 2 compared to the initial PNP device. Thanks to this low on-resistance, the measured HBM robustness is 8kV for a 100x100µm 2 silicon area.

Regarding high temperature operation, SOI smart power technologies qualified up to 200°C or even more are now proposed [START_REF]Smartis1 Technology[END_REF]. A classical ESD protection solution is to use a central MOS power clamp. It has the advantage to use devices of the technology library that can be simulated by the designers but it generally requires a large silicon area. In addition, its on-resistance is very sensitive to temperature, thus degrading the ESD robustness (It 2 ) at high temperature that can drop from 1.74A at 25°C to 1.34A at 200°C.

To counterbalance this detrimental effect, an interesting solution consists in combining MOS and bipolar effects in the same structure and also taking advantage of the parasitic SCR triggering to improve the ESD robustness [START_REF] Arbess | Combined MOS-IGBT-SCR structure for a compact high-robustness ESD power clamp in smart power SOI technology[END_REF], as shown in Fig. 3.

Such IGBT-MOS-SCR structures provide a high ESD robustness (It 2 > 5.5 A) with a ten times smaller silicon area compared to the initial LDMOS-based power Given the strong snapback related to the SCR triggering, original design solutions to improve the immunity to latch-up of these structures consist in engineering the source side (Fig. 3), such as locally reducing the channel, and applying a gate voltage to increase the holding current, I H . The best results are obtained with a reduction of 20% of the channel and a voltage bias on the gate allowing increasing I H from 7 mA with the initial design to 40 mA with the optimized design. To even more increase I H , several cells can be arranged in parallel to reach the required 100 mA (Fig. 4, bottom)..

SYSTEM EFFICIENT ESD DESIGN (SEED)

As previously mentioned, IEC61000-4-2 standard requires testing the system both under powered and unpowered conditions. The ESD protection scheme of an IC is generally designed for withstanding HBM stress under unpowered conditions. This is one of the main reasons why there is no possible direct correlation between the ESD IC robustness and the ESD system robustness as shown by Fig. 5 [START_REF]System Level ESD Part II: Implementation of Effective ESD Robust Designs[END_REF].

A possible misunderstanding may also have its origin in the fact that device (HBM) and system (IEC) standards look like being similar tests although they are very different. The model in both cases is made up by a capacitor and a resistor but their values are very different: device HBM: R=1500 Ω, C=100 pF; IEC 61000-4-2: R=330 Ω, C=150 pF. As a result, 8kV HBM corresponds to 5.2A current whereas 8kV IEC to 24A. This would mean that, if a pin of an IC is required to withstand 8kV IEC, the surface of the ESD protection would have to be almost five times the one required for the same HBM robustness. Increasing the size of the protections is not acceptable and the only way to fulfill such harsh requirements is to adopt co-design approaches. This is the purpose of the SEED approach proposed by the Industry Council [START_REF]White Paper 3 : "System Level ESD Part I: Common Misconceptions and Recommended Basic Approaches[END_REF].

One important point highlighted by the SEED approach is to better understand ESD events within systems. It is possible to predict the behavior of the ESD protection scheme of an unpowered IC under an HBM stress but this is much more complex when the IC is submitted to an ESD stress while operating. To validate that the implemented protection strategy will provide the targeted robustness under system level ESD stress, behavioral modeling and simulation is then essential.

PREDICTIVE BEHAVIORAL MODELING

For system designers, predicting the impact of ESD stress requires having access to information about the ICs including their internal ESD protection strategy. This latter information is proprietary and generally not available. The same problem occurred years ago in the field of Signal Integrity (SI) and was solved via the proposal by Intel of the IBIS standard that was first standardized in 1995 (ANSI/EIA-656) [START_REF]IBIS (Input Output Buffer Information Specification)[END_REF]. The main advantage of IBIS models is to provide exchangeable information useful for the system designer while preserving proprietary information.

Keeping the concept of IBIS description, we proposed to build IC models based on IBIS information and including ESD protections models that can be extracted without knowing the proprietary details of the protection strategy. Such model could be used into any simulator tool.

The simulation is set with the assembly of building blocks following the hierarchy of the system composed of multiple elements: ICs, PCB traces, passive components and the test bench environment, as illustrated in Fig. 6 [START_REF] Caignet | Behavioral ESD protection modeling to perform system level ESD efficient design[END_REF][START_REF] Monnereau | Building-up of system level ESD modeling: Impact of a decoupling capacitance on ESD propagation[END_REF].

As previously mentioned, IBIS files provide a lot of information that can be useful to build the IC model. IBIS files include ESD protection models, but they are not sufficient to predict faults [START_REF] Caignet | Investigating the probability of susceptibility failure within ESD system level consideration[END_REF]. IBIS simply describes the ESD protection scheme only with diodes between Input/Output, VDD and VSS. These diodes are static ones and their model are only valid from -VDD to 2*VDD, which is a range that is too small for ESD events simulation. Moreover, IBIS suffers from a lack of information concerning the central Power Clamp (PC) protection between VDD and VSS, which is crucial for the ESD strategy, as it drives the current from VDD to the IC ground.

For accurate ESD predictive modeling, IBIS models information need then to be upgraded with a full ESD protection strategy description of the IC. To keep the IBIS concept, no prior knowledge of IC internal structures should be needed, which is often the case for system designers at OEM level.

Measurement data are needed to extract information about on-chip ESD protections. TLP tester and curve tracer are used to extract pulsed (quasi-static) and static characteristics between each pair of IC pins.

From this measurement, we proposed a behavioral description of the ESD protection scheme that has the advantage to protect proprietary information. In CMOS technologies, ESD power clamp based on a big MOS transistor driven by a more or less complex triggering circuit provides a central protection between VDD and VSS. The proposed behavioral model is a two-state machine like a diode. As the actual RC-triggering delay is not known, the triggering condition is set on a dV/dt.

For protection structures exhibiting a snapback behavior, four states need to be defined as shown in Fig. 7 for an SCR. Six voltage and current couples (Vx, Ix) define the inflection points of the SCR and the equivalent equations for states 0, 1, 2 and 3. To avoid convergence issues, the equivalent parasitic capacitance (often given in IBIS) is added in parallel to the SCR.

It is necessary to validate the models on the different discharge paths that involve more than one protection element. Fig. 8 illustrates a comparison between TLP measurement and simulation for a power clamp. Globally, the discrepancy is lower than 20%. The largest one observed for the path Out-to-Vss may be due to serial resistances that are not properly extracted when characterizing individual protection devices.

ESD SYSTEM LEVEL CHARACTERIZATION

The ESDA Human Metal Model standard practice provides IC manufacturers and customers with a Fig. 6 ESD system level modeling methodology. Fig. 7 Behavioral description of a SCR using a state machine. testing method applicable to devices that utilize the current waveform of IEC 61000-4-2. It establishes the procedure for testing, and characterizing the ESD sensitivity of component pins that will be directly connected to external connectors or ports on a completed system both under un-powered and powered states. The ESD pulse source is generally an ESD gun. However, for characterization purpose and modeling calibration, this source is too noisy and it is preferable to use a 50Ω coaxial source such as a TLP (Transmission Line Pulse) tester.

The TLP tester as previously mentioned is used to characterize the protection devices at each pin of the IC. It can also be used to inject the ESD stress on specific pins. A "one-ohm measurement" setup as defined by the EMC standard IEC61967 [START_REF]Integrated circuits, Measurement of electromagnetic emissions, 150kHz to 1GHz -Part 4: Measurement of conducted emissions -1W/150W direct coupling method[END_REF], allows measuring the current flowing into the circuit.

For susceptibility studies, the ESD TLP stress is injected into the VDD pin using the DPI (Direct Power Injection) technique [START_REF]Integrated circuits, Measurement of electromagnetic immunity -150 kHz to 1 GHz -Part 4: Direct RF power injection method[END_REF] through a DC block composed of a 6.8nF capacitor.

Another interesting injection method is based on using a Very Fast TLP tester coupled to a near-field injection probe. The injection probe is moved above the surface of the circuit to produce cartography of the sensitive zones of the circuit [START_REF] Lacrampe | Characterization And Modeling Methodology For IC's ESD Susceptibility at System Level Using VF-TLP Tester[END_REF].

Reversely, the scan probe can be used to monitor the propagation of an ESD stress within a system. The magnetic probe allows extracting the current waveforms using integral and fast Fourier Transform methods [START_REF] Caignet | Non-invasive system level ESD current measurement using magnetic field probe[END_REF].

CASE STUDIES

In the following, we present several examples where behavioral simulation allows getting a better understanding of the observed behavior of the system under an IEC stress. Three cases are discussed: two digital circuits -a simple one based on a CMOS inverter [START_REF] Caignet | Behavioral ESD protection modeling to perform system level ESD efficient design[END_REF] and a more complex one based on a 16-bit microcontroller [START_REF] Vrignon | Investigation of effects of an ESD pulse injected by a near-field probe into an oscillator block of a 16-bit microcontroller[END_REF] -and finally the case of an analog circuit, namely an audio amplifier IC [START_REF] Giraldo | Impact of the Power Supply on the ESD System Level Robustness[END_REF].

CMOS commercial inverter

The studied system is built around a simple inverter in 0.25um CMOS technology. As Fig. 9 shows, an external decoupling capacitor is placed as close as possible to the IC, between the VDD pin and the ground plane. Another capacitance is placed close to the input as an Electromagnetic Interference (EMI) filter or external ESD protection. TLP tester is used to inject a 100ns rectangular pulse with 1 ns rise and fall times. Two injection cases are investigated: into the input to ground and into the output to ground. The equivalent model of the IC extracted from the available IBIS file is reported in Fig. 10 (a). To ESD upgrade this model for ESD simulations, we performed TLP measurement on the pins of the circuits and identified a power clamp between VDD and VSS and an ESD protection exhibiting a strong snapback (SCR) at the input. According to this measurement, we upgraded the IBIS model according to Fig. 10 (b). parasitics of the external capacitor. The simulations show that both physical and behavioral models of the protections give the same results. This is mainly due to the fact that the parasitic elements play a more determinant role than the protections that have a very low intrinsic resistance (lower than 1Ω).

This behavior was also validated on the output pin of the system under study using an alternative near-field measurement technique. Fig. 12 presents different cartographies acquired at different times of the ESD stress. It clearly shows that current injection is first absorbed by the decoupling capacitor and later on split between the ESD protection scheme of the circuit and the capacitor.

16-bit microcontroller

The near-field probe can also be used to inject the ESD stress by scanning it above the surface of the IC.

It was successfully used to study the ESD sensitivity of a commercial 16-bit microcontroller in combination with a Very Fast TLP tester [START_REF] Vrignon | Investigation of effects of an ESD pulse injected by a near-field probe into an oscillator block of a 16-bit microcontroller[END_REF].

In this experiment, we have flashed in the circuit a simple program to deliver a square signal period of 40 MHz, which is used as a reference to identify if the circuit runs correctly. This program uses the oscillator and PLL blocks to create the output signal. The near-field probe is moved above the chip to extract the ESD immunity cartography as shown in Fig. 13. The susceptibility criterion in measurement consists in finding, for different ESD pulse shapes, the amplitude required to stop the program or activate the clock monitor failure. Two zones are more susceptible to ESD: RESET and PLL/oscillator. The susceptibility of the RESET block is related to the exceeding of the input threshold voltage. However, more investigation was needed to understand the behavior of the PLL/oscillator block that induces clock losses (Fig. 13).

We performed simulations to validate that the microcontroller clock losses are actually induced by the ESD stress injection. For the simulation, only the following blocks are modeled: reduced model of the microcontroller including the oscillator block with its power supply, package model for the four involved pins, injection probe model and mutual coefficient coupling the loop probe with the package inductances of the pins. The simulation allowed reproducing the same faults with a good correlation with measurement.

Audio amplifier

We present hereafter a case study illustrating a strong miscorrelation between the HBM robustness and the IEC one and analyze the reasons of this behavior [START_REF] Giraldo | Impact of the Power Supply on the ESD System Level Robustness[END_REF]. The circuit is an audio amplifier IC realized in a 0.25µm CMOS technology. The main blocks of the IC are represented in Fig. 14, together with the external devices (decoupling capacitor C DEC and charge pump capacitor C OUT ) required on the testing board. The audio amplifier is composed of two symmetrical amplifiers (AOP) that provide the right and left headset sound outputs (OUTR and OUTL). The power supply VBAT can vary from 2.7V to 5.2V and an internal charge pump generates a symmetrical negative power supply -VBAT. The LDO regulators provide ±2.2V to the AOP. This IC was not originally designed to withstand IEC gun testing. A study was then carried out to assess its robustness under such stress both in powered and unpowered conditions. The gun stress is applied to the amplifier outputs (OUTR or OUTL) with its ground (GND) connected to the ground plane of the set-up. Both positive and negative stresses were applied as defined by the specifications. The results are summarized in Table 1.

TABLE I

IEC gun tests results in function of the power supply VBAT 0V 3.8V 5.2V

IEC failure level 7kV 2kV 1kV

As shown by these results, the IEC robustness decreases from 7kV to as low as 1kV when the IC is powered. The failure analysis revealed that the device that is failing is the n-MOS output transistor M 0 , the physical damage being a drain-source melt filament. Such failure signature indicates that the parasitic bipolar transistor was triggered on.

To understand why the ESD protection strategy is not efficient when the IC is powered, we performed behavioral simulations. The protection strategy is based on four MOS Power Clamp and the body diodes of the output MOS transistors M 0 , M 1 , M 2 , and M 4 .

We carried out simulations for a positive IEC stress applied to OUTR(L) when the IC is not powered and it resulted that the Power clamps are not involved at all and the discharge current is entirely absorbed by the external capacitors C DEC following IEC(1) path on The results of the simulations are summarized in Fig. 16 that plots the drain to source voltage of transistor M 0 . When the IC is unpowered, this voltage is below 1V and then no current flows in the transistor since its bipolar trigger voltage is around 10V. However, when powered up, the gate of transistor M 0 is biased thus significantly lowering the bipolar trigger voltage and favoring its activation.

This example shows that under IEC stress, the powered active circuit can interact with the ESD protection strategy and make it inefficient. Only a global protection strategy approach taking into account the final system and its operation would allow achieving system efficient ESD design.

CONCLUSION

Designing ESD-robust integrated protections provides HBM safe ICs but does not guarantee they will survive in their application. Field returns clearly showed that there is not a straight correlation between HBM robustness and the system level one. Behavioral modeling associated to appropriate characterization tools allows better understanding the behavior of a system during ESD events such as the IEC stress. The

Fig. 1

 1 Fig. 1 ESD design window showing the detrimental effect of temperature.

Fig. 2

 2 Fig. 2 View of the improved PNP ESD protection (top) and TLP characteristics of two optimized PNP-diode structures (30V and 60V clamping voltages) and of a standalone optimized PNP (bottom).

Fig. 3

 3 Fig. 3 IGBT-MOS-SCR structure exhibiting 50% IGBT/MOS ratio (top) and proposed source engineering solutions to increase SCR holding current (bottom).

Fig. 4

 4 Fig. 4 TLP characteristics of an IGBT-MOS-SCR structure with 80% IGBT/MOS ratio for different temperatures (top) and holding current results for 1P1N structures (50% IGBT) in parallel (bottom).

Fig. 5

 5 Fig. 5 Comparison of IC level and system level ESD failure threshold of various systems (A-J) showing that HBM protection is not related to System level ESD robustness [9].

  Diodes are good examples to present the philosophy. Only two parameters are needed: the triggering voltage, Vt 1 , and the on-state resistance, R ON . This defines a very simple two-state machine diagram. When the voltage across the diode is below Vt 1 , the diode is off (State 0), no current flows into the protection. Otherwise we are in state 1 with the equation V= R ON .I.

Fig. 8

 8 Fig. 8 TLP measurement vs simulation using VHDL-AMS model of a Power clamp for two stress modes: Vdd to Vss (left) and Out to Vss (right).

Fig. 11

 11 Fig. 11 illustrates the case of the injection of an ESD stress into the input pin IN, with and without an external decoupling capacitance. Without capacitance, the TLP current flows as expected through the ESD structure. However, when added, the decoupling capacitor diverts all the current until its voltage reaches the triggering voltage of the SCR. After snapback, the ESD structure conducts the current from the decoupling capacitor to the ESD discharge current creating a strong current peak. Its amplitude is determined by the package inductance and the

Fig. 9 Fig. 10 Equivalent

 910 Fig. 9 Simplified schematic of the inverter circuit implemented to study the ESD stress propagation.

Fig. 11

 11 Fig. 11 Measurement vs simulation comparison of a TLP stress injection between IN and VSS of the inverter circuit with and without 6.8nF decoupling capacitor. Behavioral IBIS-based models are compared to semi-physical models.

Fig. 13 ESD

 13 Fig. 13 ESD pulse injection system with a near-field probe (left), ESD immunity cartography (right) and resulting clock losses (bottom).

Fig. 15 .

 15 These capacitors are large enough (1µF) to sink the ESD current without a voltage fluctuation on VBAT large enough to trigger the Power Clamp. When the IC is powered up, there is an additional discharge path, IEC(2) in Fig.15, involving transistor M 0 .

Fig. 14

 14 Fig. 14 Overview of the audio amplifier in IEC test conditions. C DEC and C OUT are external capacitors.

Fig. 15

 15 Fig. 15 Current discharge paths for a positive IEC stress applied to the output under powered conditions.

Fig. 16

 16 Fig. 16 Simulated drain-to-source voltage of M 0 as a function of power supply for 1kV IEC stress.

proposed approach is based on upgrading IBIS models with behavioral ESD relevant models and using VHDL-AMS for their description. This approach is now being discussed within ESDA standard working group WP14, System level ESD, to standardize it in the same way as IBIS.


R Delpoux
email: romain.delpoux@ec-lille.fr

T Floquet

High order sliding mode control for sensorless trajectory tracking of a PMSM

Keywords: PMSM, sensorless, sliding modes, reference rotating frame

The paper presents a new sensorless approach for permanent magnet synchronous motor (PMSM). Current sensors are assumed available, but position and velocity sensors are not. Based on the electrical equations, sliding mode observers are designed to estimate the back-EMF of the motor. These estimations are used to reconstruct the position and the velocity. From this estimation, a robust sliding mode control is developed which ensures the position tracking of the motor. A new reference frame is used that presents advantages similar to the standard (dq) frame, but without the need for a position sensor. The efficiency of the algorithm is shown through experimental results. The approach is potentially applicable to other types of synchronous motors as well.

Introduction

Permanent Magnet Stepper Motors (PMSM's) are widely used in industry for position control, especially in manufacturing applications. PMSM's are more robust than brush DC motors and produce high torque per volume. They are often controlled in open-loop, although the potential loss of synchronism limits operation away from resonances and from high acceleration trajectories. Using closed-loop control methods with position sensors of sufficient accuracy can solve these problems. Recent research has focussed on whether the performance of closed-loop control methods could be achieved using sensorless systems. In this case, sensorless refers to systems that do not have position sensor nor velocity, although current sensors are still assumed to be available.

One can find work in the literature which deals with the design of a control supposing that the rotor position is known [START_REF] Chiasson | Nonlinear speed observer for the PM stepper motor[END_REF][START_REF] Nollet | Observer-based second order sliding mode control laws for stepper motors[END_REF][START_REF] Defoort | A Third-Order Sliding-Mode Controller for a Stepper Motor[END_REF]. Concerning "sensorless" based control, different approaches have been treated. One can refer to the overviews of [START_REF] Johnson | Review of sensorless methods for brushless DC[END_REF][START_REF] Schroedl | Sensorless control of permanent-magnet synchronous machines: An overview[END_REF]) treating brushless DC motor and PMSM, respectively. Among the different approaches to treat this problem, the most current are the high frequency injection method [START_REF] Jang | Analysis of permanent-magnet machine for sensorless control based on high-frequency signal injection[END_REF][START_REF] Zhu | A simplified high frequency injection method for PMSM sensorless control[END_REF], observer based approach such as Extended Kalman Filter [START_REF] Bolognani | Extended-range PMSM sensorless speed drive based on stochastic filtering[END_REF][START_REF] Bendjedia | Position Control of a Sensorless Stepper Motor[END_REF], linear observer [START_REF] Son | Sensorless operation of permanent magnet motor using direct voltage sensing circuit[END_REF], nonlinear observer [START_REF] Ortega | Estimation of Rotor Position and Speed of Permanent Magnet Synchronous Motors With Guaranteed Stability[END_REF][START_REF] Shah | Sensorless Speed Control of Nonsalient Permanent Magnet Synchronous Motors[END_REF][START_REF] Khlaief | A Nonlinear Observer for High Performance Sensorless Speed Control of IPMSM Drive[END_REF][START_REF] Tomei | Observer-Based Speed Tracking Control for Sensorless Permanent Magnet Synchronous Motors With Unknown Load Torque[END_REF], adaptive interconnected observer [START_REF] Hamida | An Adaptive Interconnected Observer for Sensorless Control of PM Synchronous Motors With Online Parameter Identification[END_REF]) and sliding mode observers [START_REF] Kim | A High-Speed Sliding-Mode Observer for the Sensorless REFERENCES Speed Control of a PMSM[END_REF][START_REF] Fiter | Sensorless Control of a Stepper Motor Based on Higher Order Sliding Modes[END_REF][START_REF] Zribi | Static and dynamic sliding mode control schemes for a permanent magnet stepper motor[END_REF]. Although not exhaustive, this list gives an idea on the issues.

This article presents a solution for mechanical sensorless control of a PMSM based on high order sliding modes. Sliding mode theory is commonly used for the design of robust nonlinear observers or control laws. Indeed, sliding modes provide very good properties with respect to perturbations and uncertainties. Another interesting property of the sliding modes is the finite time convergence, which unlike the asymptotic convergence, do not requires the use of separation principle theorem to prove the convergence of an observer based control. The use of second order sliding modes reduces the high frequency commutations known as Chattering. In the literature, a wide range of applications using sliding modes can be found : (Bartolini et al. 2003, Bartolini and Pisano 2003[START_REF] Butt | Estimation of Gasoline-Engine Parameters Using Higher Order Sliding Mode[END_REF][START_REF] Canale | Vehicle Yaw Control via Second-Order Sliding-Mode Technique[END_REF][START_REF] Davila | Finite-time state observation for nonlinear uncertain systems via higher-order sliding modes[END_REF][START_REF] Defoort | Sliding-Mode Formation Control for Cooperative Autonomous Mobile Robots[END_REF][START_REF] Drakunov | Stabilization and tracking control for an extended Heisenberg system with a drift[END_REF][START_REF] Floquet | Higher-order sliding mode stabilization for a class of nonholonomic perturbed systems[END_REF][START_REF] Fridman | High-order sliding-mode observation and fault detection[END_REF][START_REF] Martinez | Hybrid Sliding-Mode-Based Control of Underactuated Systems With Dry Friction[END_REF][START_REF] Pisano | Cascade control of PM-DC drives via second-order sliding mode technique[END_REF][START_REF] Riachy | Second order sliding mode control of underactuated Mechanical systems I: Local stabilization with application to an inverted pendulum[END_REF].

From a control engineer perspective, the most used control laws are designed in the field oriented frame, referred as d -q frame. It offers several advantages since it provides a simplified structure for the control, by avoiding sinusoidal functions. This frame from a sensorless point of view is not usable since it relies on the use of the position sensor. The proposed approach is based on the use of a different reference frame which is obtained from a reference position instead of a measured position. This frame permits to have the same properties than the d -q frame, with the advantage that it does not require the actual motor position. In the literature, this frame was introduced for sensorless control and parameter identification in [START_REF] Morimoto | Sensorless control strategy for salient-pole PMSM based on extended EMF in rotating reference frame[END_REF][START_REF] Morimoto | Mechanical Sensorless Drives of IPMSM With Online Parameter Identification[END_REF][START_REF] Zheng | Sensorless control of PMSM based on extended kalman filter[END_REF][START_REF] Shi | Online Identification of Permanent Magnet Flux Based on Extended Kalman Filter for IPMSM Drive With Position Sensorless Control[END_REF][START_REF] Delpoux | Parameter estimation of permanent magnet stepper motors without position or velocity sensors[END_REF].

The main contribution of the article is the use of second order sliding modes observers, based on Super Twisting Algorithms (STA) for the estimation of the position and the velocity of the motor. The observers use the input voltages and the measured currents only, to estimate the motor back-EMF. From this estimation, both position and velocity can be reconstructed. These estimations are used for the control. Here, we propose an observer based sliding mode control for the motor position trajectory tracking. This control is based on the flatness property of the PMSM [START_REF] Sira-Ramírez | A Passivity plus Flatness Controller for Permanent Magnet Stepper Motor[END_REF]. The theory is validated through experiments that were performed using a test bench available at the LAGIS laboratory at the École Centrale de Lille.

This article is divided into four parts. The first section presents the model of the PMSM in the three different frames, particularly the reference rotating frame. The higher order sliding modes as well as the control strategy are also presented in this section. Then second order sliding mode observers are developed to estimate the position, velocity and acceleration of the PMSM in the section 3. The control law is derived Section 4 together with the stability of the observer based control. The last section is devoted to experimental results.

Problem Statement

PMSM Model

The model of the PMSM is given in three different frames (see Fig. 1). After a description of the model in the variables (a -b), the model in the rotating frame (d -q) is presented. This model is useful for the control law design. However this model is obtained from the measured position. In sensorless application, the use of this model is not suitable. Hence a reference rotational frame based on the reference trajectory, called (f -g) frame is introduced.

Model in the phase variables (a -b)

Equations ( 1) give the standard PMSM model in the phase (or winding) variables where v a and v b are the voltages applied to the two phases of the PMSM, i a and i b are the two phase currents, L is the inductance of a phase winding, R is the resistance of a phase winding, K is the back-EMF constant, θ is the angular position of the rotor, ω is the angular velocity of the rotor, n p is the number of pole pairs (or rotor teeth), J is the moment of inertia of the rotor (including the load), f v is the coefficient of viscous friction and C r is the load torque, which may vary as a function of the time.

                 L di a dt = v a -Ri a + Kω sin(n p θ), L di b dt = v b -Ri b -Kω cos(n p θ), J dω dt = K (i b cos(n p θ) -i a sin(n p θ)) -f v ω -C r , dθ dt = ω, (1)

Model in the rotating frame (d -q)

The phase model can be transformed using Park's transformation [START_REF] Park | Two-reaction theory of synchronous machines generalized method of analysis-part I[END_REF]):

[i d , i q ] T = U (θ) [i a , i b ] T , (2)
[v d , v q ] T = U (θ) [v a , v b ] T , (3)
where

U (θ) = cos(n p θ) sin(n p θ) -sin(n p θ) cos(n p θ) . (4)
Using this change of coordinates, the system (1) is transformed into the so-called (d -q) model

                 L di d dt = v d -Ri d + n p Lωi q , L di q dt = v q -Ri q -n p Lωi d -Kω, J dω dt = Ki q -f v ω -C r , dθ dt = ω.
(5)

The (d -q) transformation is commonly used for PMSM's (and synchronous motors in general), because it results in constant voltages and currents at constant speed (instead of the highfrequency phase variables). Also, the model highlights the role of the quadrature current i q in determining the torque. However, the (d -q) transformation is based on the position θ, which is not directly available in sensorless applications.

Model in the rotating reference frame (f -g)

To overcome the problems caused by the use of the position in the d -q frame, a different reference frame that uses a reference position instead of the real position is proposed in this work. The transformation is expressed in matrix form as

[i f , i g ] T = U (θ r ) [i a , i b ] T , (6) [v f , v g ] T = U (θ r ) [v a , v b ] T , (7)
where

U (θ r ) = cos(n p θ r ) sin(n p θ r ) -sin(n p θ r ) cos(n p θ r ) . ( 8
)
and θ r is an arbitrary reference position.

The PMSM model in the transformed variables is

                         L di f dt = v f -Ri f -Kω sin(n p ∆θ) + Ln p ω r i g , L di g dt = v g -Ri g -Kω cos(n p ∆θ) -Ln p ω r i f , J dω dt = K(i f sin(n p ∆θ) + i g cos(n p ∆θ)) -f v ω -C r , dθ dt = ω, (9a) (9b) (9c) (9d)
where ∆θ = θ -θ r and ω r = dθ r /dt. The (f -g) frame is potentially useful as θ r may be defined as the reference position that the motor is supposed to track. Then, the (f -g) model approximates the (d -q) model, with the advantage that it is valid and computable even if θ r is not exactly equal to θ which cannot be used in the control law.

High order Sliding mode

The principle of higher order sliding mode control is to constrain the system trajectories to reach and stay, after a finite time, on a given sliding manifold S r in the state space (Emel'yanov et al. 1986, Perruquetti and[START_REF] Perruquetti | Sliding mode control in engineering[END_REF]). Consider a system whose dynamics is given by:

x = f (t, x) + g(t, x)u, (10)
where x ∈ R n is the system state, u ∈ R is the control and f , g are sufficiently smooth vector fields. The sliding manifold is defined by the vanishing of a corresponding sliding variable S : R + × R n → R and its successive time derivatives up to a certain order. In the literature, the "sliding" algorithms are mainly of second order. The second order sliding set is written:

S 2 = {(t, x) ∈ R + × R n : S(t, x) = Ṡ(t, x) = 0}.
The second derivative w.r.t the time of the sliding surface can be written:

S = φ(t, S, Ṡ) + ϕ(t, S, Ṡ)U, (11)
where U = u (respectively U = u) for systems of relative degree 2 (relative degree 1) w.r.t S.

Under the assumption that there exists positive constants S 0 , k m , K M , C 0 such that ∀x ∈ R n and |S(t, x)| < S 0 , the system satisfies the following conditions:

0 < k m ≤ |ϕ(t, S, Ṡ| ≤ K M , |φ(t, S, Ṡ)| < C 0 . (12)
Different sliding mode algorithms can be found in the literature, here are recalled only the algorithms used in the article: the Super Twisting Algorithm and the Twisting Algorithm.

Super Twisting Algorithm

For systems of relative degree 1, the Super Twisting Algorithm will be used [START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF][START_REF] Levant | Universal single-input-single-output (SISO) sliding-mode controllers with finite-time convergence[END_REF]. It is a second order sliding mode defined by:

u st (S) = u 1 (S) + u 2 (S), (13)
with:

u1 (S) = -αsgn(S), u 2 (S) = -λ|S| 1 2 sgn(S). ( 14
)
The finite time stability of the algorithm is proved using a Lyapunov function. With relative degree equal to 1, one has:

Ṡ = u st + Π. ( 15
)
where Π is a bounded perturbation, whose time derivative is also bounded. From the control law defined equation ( 13), this equation can be written:

Ṡ = ξ -λ|S| 1/2 sgn(S), ξ = Π -αsgn(S), (16)
where α and λ are positives gains to be defined. The finite time stability of the algorithm is derived from [START_REF] Barbot | Iterative higher order sliding mode observer for nonlinear systems with unknown inputs[END_REF].

Consider the equations of the system (16) and note :

ψ = ψ 1 ψ 2 = |S| 1/2 sgn(S) ξ .
Leading to:

ψ = |ψ 1 | -1 -λ 1 -α 0 ψ + 0 Π = |ψ 1 | -1 M ψ + 0 |ψ 1 |Π . (17)
Define the Lyapunov candidate function:

V = ψ T P ψ, (18)
where P = p 1 p 3 p 3 p 2 is a symmetric positive definite matrix. The time derivative of V along the solution of ( 17) is given by:

V = |ψ 1 | -1 ψ T (M T P + P M )ψ + 2ψ T P 0 |ψ 1 |Π . ( 19
)
where:

2ψ T P 0 |ψ 1 |Π ≤ k 1 ψ 2 1 + k 2 ψ 2 2 , with k 1 = Π(2|p 3 | + ǫ) and k 2 = Π p 2 2
ǫ , for all ǫ > 0. Thus:

V ≤ |ψ 1 | -1 ψ T M T P + P M + k 1 0 0 k 2 ψ. ( 20
)
Because M is a Hurwitz matrix, the observer gains α and λ can be chosen such that the matrix

-Q = M T P + P M + k 1 0 0 k 2 ,
is negative definite. By application of LaSalle theorem, the finite time convergence ψ toward zero is proven [START_REF] Barbot | Iterative higher order sliding mode observer for nonlinear systems with unknown inputs[END_REF], i.e. the finite time convergence of S and ξ toward zero. Thus after finite time ξ = 0 leading to Ṡ = 0.

In some cases, more particularly the observers, we resort a linear stabilizing term. This additive term aims to reduce the noise and accelerating the convergence towards the sliding surface. The approach given in [START_REF] Shen | Uniformly Observable and Globally Lipschitzian Nonlinear Systems Admit Global Finite-Time Observers[END_REF] extends the preceding results to this case.

Twisting Algorithm

For systems of relative degree 2, the Twisting algorithm can be used. This algorithm can be written has follow:

w w t (S, Ṡ) = -λ M sgn(S) if S Ṡ > 0, -λ m sgn(S) if S Ṡ ≥ 0. (21)
From [START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF] one can find the sufficient conditions which show the finite time convergence onto the real second order sliding set:

λ m > 4 KM S0 , λ m > C0 km , λ M > KM λm km + 2 C0 km , (22)
where:

0 < k m ≤ |ϕ(t, S, Ṡ)| ≤ K M and |φ(t, S, Ṡ)| < C 0 .
The finite time convergence of this algorithm based on a Lyapunov function can be found in [START_REF] Orlov | Discontinuous Systems[END_REF]).

Control strategy

Previously, three different frames to express the model of the PMSM have been presented. It will be shown that the f -g frame is suitable for sensorless control of PMSM. The control law is proposed for the position tracking of PMSM. The control is designed from the flatness property of the motor. Indeed, in (Sira-Ramírez 2000) it was shown that PMSM were flat systems, where the flat outputs are the direct currents and the position. For this system to be controlled, it is necessary to design a control loop for the direct currents and another for the position. For the current loop, the sliding variable is chosen so that the system has relative degree equal to 1, a Super Twisting algorithm can be used. The sliding variable measurement only is needed and this algorithm reduces the chattering phenomena. For the position control loop, the system is of third degree. A judicious choice of the sliding variable lead to a system with relative degree equal to 2 with respect to the sliding variable which enables the use of a Twisting algorithm, for which the position, the velocity and the acceleration are needed.

For a mechanical sensorless control, based on this described control law, the position and the velocity must be estimated but also the acceleration of the motor. Here, the proposed approach is based on the back-EMF estimation. The electrical equations (9a), (9b) are used to estimate the back-EMF using input voltages and measured currents only. From this estimation, the position and the velocity can be reconstructed for the control. For this purpose, two sliding mode observers are designed. These observers are advantageous in the presence of the nonlinearities of the back-EMF. The observers convergence is ensured under the assumption that the time derivatives of these terms are bounded. Moreover these observers ensure the finite time convergence of the estimate that facilitates the proof of the closed-loop stability of the whole control-observer. The proposed control law also needs the estimation of the acceleration to be designed. It is estimated from the mechanical equation (9c) using the estimated position and velocity.

Moreover, the PMSM is not observable at zero velocity, as it is proven in [START_REF] Ezzat | Observer-controller scheme using high order sliding mode techniques for sensorless speed control of permanent magnet synchronous motor[END_REF]). To overcome the unobservability case, in the sequel, a low-speed strategy to bring the motor states in the observable domain is proposed. Thus, the main objective of this work is the robust position trajectory tracking without any mechanical sensors.

The control scheme is shown Fig. 2.

i f i g θ ω ω |ω lim | i f i g v a v b θ Ω i a i b v f v g i f,ref θ ref x ref v ref x Figure 2
. System global scheme.

Mechanical sensorless sliding mode observers

In this paragraph, sliding mode observers are designed for the estimation of the position, the velocity and the acceleration. The observers are designed in the (f -g) frame. The position and the velocity are estimated using the electrical equations. These estimations are then used for the acceleration estimation based on the mechanical equation.

Electrical equations based observers

Observer design

The electrical equations of the motor in the (f -g) frame are given by ( 9a) and (9b). Taylor & Francis and I.T. Consultant

These equations can be represented under the form:

ẋfg = A f g x f g + u f g + d f g , y f g = C f g x f g , (23)
where

x f g = [i f i g ] T , u f g = 1 L (v f -Ri f ) 1 L (v g -Ri g )
T and,

A f g = 0 n p ω r -n p ω r 0 , d f g = d f d g = K L ω sin(n p ∆θ) -K L ω cos(n p ∆θ)
and C f g = 1 0 0 1 , the disturbance d f g are introduced by the permanent magnets. A second order sliding mode observer is designed. Given that is acts on the sliding variable and its first derivative, the model ( 23) is rewritten under an augmented form, where the variables d f and d g are the augmented state variables to be estimated:

ẋfg = A f g x f g + B f g u f g + d f g , y f g = C f g x f g , (24)
where

x f g = [i f i g d f d g ] T and, A f g = A I 2×2 0 2×2 0 2×2 , B f g = I 2×2 0 2×2 , C f g = I 2×2 0 2×2 and d f g =   0 2×1 K L ( ω sin(n p ∆θ) + n p ω∆ω cos(n p ∆θ)) -K L ( ω cos(n p ∆θ) -n p ω∆ω sin(n p ∆θ))   ,
where ∆ω = ω -ω r . The sliding mode observer is constructed by replacing the disturbance voltages by the output injection matrix χ f g (y f g -ŷfg ) yet to be defined. A stabilizing linear part L f g (y f g -ŷfg ) is also introduced. The observer is defined by:

ẋfg = A f g xfg + B f g u f g -χ f g (y f g -ŷfg ) -L f g (y f g -ŷfg ), ŷfg = C f g xfg . ( 25
)
The output injection matrix χ f g (y f g -ŷ f g ) is defined by the Super Twisting algorithm (relative degree equal to 1 w.r.t. the sliding variable):

χ f g (y f g -ŷfg ) =     λ f |i f -îf | 1 2 sgn(i f -îf ) λ g |i g -îg | 1 2 sgn(i g -îg ) α f sgn(i f -îf ) α g sgn(i g -îg )     .
The linear term is defined by:

L f g =   l f 0 0 l g 0 2×2   ,
where l f and l g are adjustable gains yet to be defined.

The observation errors are defined by the vector:

ǫ f g = x f g -xfg =     ǫ f ǫ g ǫ df ǫ dg     =     i f -îf i g -îg d f -df d g -dg     .
From equations ( 23) and ( 25), the errors dynamics are given by:

ǫfg = ẋfg -ẋfg = (A f g -L f g C f g )ǫ f g + d f g -χ(ǫ f g ). ( 26
)
Under physical assumption, one can consider that the velocity and the acceleration of the motor are bounded. Thus, the disturbances d f and d g are upperbounded by Π d = K L (| ω| max + n p |ω∆ω| max ). The finite time convergence conditions of the Super Twisting algorithms are satisfied choosing the parameters α, λ and the gain matrix L f g such that they satisfy the conditions given by the Lyapunov function introduced in (18).

After a finite time, a second order sliding motion occurs on the sliding manifold {ǫ f g = ǫfg = 0} implying:

       ǫf = 0, ǫg = 0, ǫdf = 0 = K L ( ω sin(n p ∆θ) + n p ω∆ω cos(n p ∆θ)) -α f sgn(ǫ f ), ǫdg = 0 = -K L ( ω cos(n p ∆θ) -n p ω∆ω sin(n p ∆θ)) -α g sgn(ǫ g ), (27)
whether:

                         df = K L ω sin(n p ∆θ), dg = - K L ω cos(n p ∆θ), α f sgn eq (ǫ f ) = ω ω df -n p dg ∆ω, α g sgn eq (ǫ g ) = ω ω dg + n p df ∆ω, (28a) (28b) (28c) (28d)
where α * sgn eq (ǫ * ) with * ∈ {f, g} represents the equivalent output injections obtained after filtering.

Position and velocity reconstruction

Position estimation:

From equations ( 28a) and (28b), the estimation ∆θ est of ∆θ is given by:

∆θ est =                              undefined if ω = 0, 1 np arctan df dg if ĉ > 0, π/2n p if ŝ = 1, -π/2n p if ŝ = -1, 1 np arctan df dg -π if ĉ < 0,
and ŝ > 0,

1 np arctan df dg + π if ĉ < 0, and ŝ < 0, (29)
where ĉ = dg / d2 f + d2 g , ŝ = df / d2 f + d2 g are the estimations of cos(n p ∆θ) and sin(n p ∆θ), respectively. The output value of the function arctan computed using (29) takes values in the interval -π np , π np , leading to discontinuous output. Velocity estimation: From equations (28a) and (28b) the estimation of the velocity module could be reconstructed as:

|ω est | = d2 f + d2 g . ( 30
)
This expression gives only the absolute value of the velocity. However, substituting ω and ω in equations ( 28c) and ( 28d), one has:

∆ω est = 1 n p α g sgn eq (ǫ g ) df -α f sgn eq (ǫ f ) dg d2 f + d2 g . (31)
From equations ( 30) and (31), one deduces the velocity estimation:

ω est = ω r + ∆ω est . ( 32
)
Remark 3.1 Note that the acceleration could also be obtained from equations (28c) and (28d).

The discontinuities of the Super Twisting algorithm acting on these equations make them more sensitive to chattering phenomenon, the resulting expression for the acceleration is ill conditioned. An alternative method is discussed in the next section.

Mechanical equation based observer

In order to apply a second order the sliding mode control, one proposes to design an observer to estimate the acceleration. Therefore the mechanical equation in the (f -g) frame is rewritten replacing the position and the velocity by their estimates

ω = 1 J (K(i f sin(n p ∆θ est ) + i g cos(n p ∆θ est ) -f v ω est -C r ) . ( 33
)
The load torque is considered as a perturbation denoted:

d ωest = - C r J .
The augmented system is defined by:

ẋωest = A ωest x ωest + B ωest u ωest + d ωest , y ωest = C ωest x ωest , (34)
where x ωest = ω est d ωest T , u ωest = K J (i f sin(n p ∆θ est ) + i g cos(n p ∆θ est )) and,

A ωest = -fv J 1 0 0 , B ωest = 1 0 , d ωest = 0 -1 J dCr dt
and C ωest = 1 0 .

The observer under the augmented form becomes:

ẋωest = A ωest xωest + B ωest u ωest + χ ωest (y ωest -ŷωest ) + l ωest (y ωest -ŷωest ), ŷωest = C ωest xωest , (35)
where

χ ωest (y ωest -ŷωest ) = λ ωest |y ωest -ŷωest | 1 2 sgn(y ωest -ŷωest ) α ωest sgn(y ωest -ŷωest ) , l ωest (y ωest -ŷωest ) = l ω 0 (y ωest -ŷωest ).
The observation error is defined by:

ǫ ωest = x ωest -xωest = ǫ y ω est ǫ dω est = y ωest -ŷωest d ωest -dωest , (36)
leading to the dynamics:

ǫωest = ẋωest -ẋωest = (A ωest -l ωest C ωest ) ǫ ωest + d ωest -χ ωest (ǫ y ω est ). ( 37
)
Under the assumption that the coefficient of Coulomb friction is differentiable, and that the derivative of C r is bounded i.e. dCr dt max < Π Cr where Π Cr is a positive constant , the finite time convergence of the observer is guaranteed choosing the gains α ωest , λ ωest , and l ωest satisfying the conditions of the Lyapunov function (20). Thus after finite time, one has ǫωest = ǫ ωest = 0, leading to the estimation of the acceleration ωest . Note that at the same time the load torque is also estimated.

This section has shown a method for the estimation of the position, the velocity and the acceleration. The next section is devoted to the control law design.

Sliding mode control for the PMSM

The control law is designed considering that the PMSM model is flat. The flatness theory was introduced in [START_REF] Fliess | Flatness and defect of nonlinear systems: Introductory theory and examples[END_REF]. A system is said flat if the states and the inputs of the system can be expressed from the flat outputs and a finite number of their time derivatives only. The flatness of the PMSM was shown in [START_REF] Sira-Ramírez | A Passivity plus Flatness Controller for Permanent Magnet Stepper Motor[END_REF] in the rotating (d -q) frame, without Coulomb friction (i.e. unperturbed), with the flat outputs θ and i d . To be transformed into the (f -g) frame, it suffices to apply the transformation (4) using U (∆θ) instead of U (θ), leading to the flat outputs in the (f -g) frame Y f g = (y f g,1 , y f g,2 ) = (θ, i f cos(n p ∆θ) + i g sin(n p ∆θ))

Flatness property allows defining easily a reference trajectory (denoted Γ r ), satisfying the system dynamics. Considering that θ r is the reference position to be tracked, the reference model of the motor in the (f -g) frame is obtained, replacing θ by θ r in the model and is given by:

                 L di f,r dt = v f,r -Ri f,r + n p Lω r i g,r , L di g,r dt = v g,r -Ri g,r -n p Lω r i f,r -Kω r , J dω r dt = Ki g,r -f v ω r , dθ r dt = ω r . (38)
The flat outputs reference trajectories are defined by θ r and i f,r . From these outputs, all the reference variables can be defined using the equations:

ω r = dθr dt , i g,r = 1 K J d 2 θr dt 2 + f v dθr dt , v f,r = L dif,r dt + Ri f,r -N L K dθr dt J d 2 θr dt 2 + f v dθr dt , v g,r = JL K d 3 θr dt 3 + 1 K (Lf v + RJ) d 2 θr dt 2 + Rfv K + K + N Li f,r dθr dt . ( 39
)
The dynamics of the tracking error:

e f g = [e f , e g , e ω , e θ ] T =     i f cos(n p e θ ) + i g sin(n p e θ ) -i f r -i f sin(n p e θ ) + i g cos(n p e θ ) -i gr ∆ω ∆θ     . (40)
are given by:

       ėf = 1 L (v f -Re f + n p L(e ω e g + e ω i g,r + e g ω r )) , ėg = 1 L (v g -Re g + n p L(e ω e f + e ω i f,r + e f ω r ) -Ke ω ) , ėω = 1 J (Ke g -f v e ω -C r ), ėθ = e ω , (41)
with:

v f v g T = U (e θ ) v f v g T - v f,r v g,r T .
The flat variables can be expressed as a function of the inputs:

       ėf = 1 L v f + µ 1 (e f g ), e (3)
θ = K JL v g + µ 2 (e f g ) + f v J 2 C r - 1 J dC r dt , (42a) (42b)
where:

µ 1 (e f g ) = 1 L (-Re f + N L(e ω e g + e ω i g,r + e g ω r )) , µ 2 (e f g ) = -K JL (Re g + N L(e ω e f + e ω i f,r + e f ω r ) + Ke ω ) -fv J 2 (Ke f -f v e ω ).

Observation based state feedback control

The control law is designed using the state estimation obtained Section 3. The error vector is defined between the estimated variables and the reference trajectories by:

ξ = [ξ f , ξ g , ξ ω , ξ θ ] T , (43)
with

    ξ f ξ g ξ ω ξ θ     =     i f cos(N (θ est -θ r )) + i g sin(N (θ est -θ r )) -i f,r -i f sin(N (θ est -θ r )) + i g cos(N (θ est -θ r )) -i g,r ω est -ω r θ est -θ r     , (44)
Leading to:

       ξf = 1 L v f + µ 1 (ξ f g ), ξ (3)
θ = K JL v g + µ 2 (ξ f g ) + f v J 2 C r - 1 J dC r dt , (45a) (45b)
where:

µ 1 (ξ f g ) = 1 L (-Rξ f + N L(ξ ω ξ g + ξ ω i g,r + ξ g ω r )) , µ 2 (ξ f g ) = -K JL (Rξ g + N L(e ω ξ f + ξ ω i f,r + ξ f ω r ) + Kξ ω ) -fv J 2 (Kξ f -f v ξ ω ).
It is desired to ensure the convergence to the origin of the system (45). In this section, the perturbations are unknown, but supposed bounded as well as their time derivatives. It is then important to propose a control law which guarantee the convergence despite the perturbations. From equation ( 45), one has to design a control law for the i f current tracking and a law for the position tracking.

i f current tracking

In order to ensure the current tracking, the following sliding variable is chosen:

S f = ξ f . (46)
From equation (45a), the time derivative of this variable is given by:

Ṡf = ξf = 1 L v f + µ 1 (ξ f g ). ( 47
)
Since the control appears in the first derivative, the system has a relative degree equal to 1 w.r.t. the sliding variable. A first order sliding mode algorithm would have been sufficient in this case. However the chattering phenomena is more important if the discontinuous control is applied directly on the time derivative of the current. It is proposed here to use a second order algorithm so that the discontinuous action is applied to the second derivative. The Super Twisting algorithm which require the knowledge of ξ f , only, is used. Define first a static state feedback v f :

1 L v f = -µ 1 (ξ f g) + u st (S f ), ( 48
)
where u st (S f ) is the Super Twisting algorithm defined in ( 13). The gains α and λ are determined considering the system (16) without perturbations.

The time derivative of the Lyapunov function ( 20) is given by:

V = |ψ| -1 ψ T (M T P + P M )ψ, ( 49
)
with P a symmetric positive definite and M = -λ 1 -α 0 .

The gains α and λ must be chosen such that:

-Q 1 = M T P + P M, (50)
with Q 1 positive definite. This condition is satisfied if the matrix M is Hurwitz, whether:

tr(M ) = -λ, det(M ) = α. ( 51
)
Choosing α and λ strictly positive, this condition is satisfied.

Position tracking

In order to obtain a system with relative degree equal to 2, the sliding variable is defined by:

S θ = kξ θ + ξθ , (52)
with k > 0. Moreover, the sliding variable depends on the position and the velocity only, and do not depends on the motor parameters. The successive time derivatives of S θ are:

Ṡθ = k ξθ + ξθ , Sθ = k ξω + ξω = k J (Kξ g -f v e ω ) + K JL v g + µ 2 (ξ f g ) -k J -fv J 2 C r -1 J dCr dt . (53)
In [START_REF] Nollet | Observer-based second order sliding mode control laws for stepper motors[END_REF], the Sampled Twisting algorithm was used. This algorithm does not require the time derivative of the sliding variable, but it uses the difference over a sampling period of the variable, which is very sensitive to measurement noise. Thus in the section 2.2, we have introduced the Twisting algorithm to treat systems of relative degree equal to 2. Choosing the control input v q as follow:

K JL v g = - k J (Kξ g -f v ξ ω ) -µ 2 (ξ f g ) + w t (S θ , Ṡθ ), ( 54
)
where w t (S θ , Ṡθ ) is the twisting algorithm defined by equation ( 21).

The second time derivative of the sliding variable ( 52) can be written as:

Sθ = φ 1 (t) + w t (S θ , Ṡθ ), ( 55
)
where:

φ 1 (t) = - k J - f v J 2 C r - 1 J dC r dt . ( 56
)
Equation ( 55) is expressed with the form given in equation ( 11). Thus the finite time convergence towards the surface S θ = 0 is guaranteed if the conditions ( 22) for the controller gains λ m and λ M :

λ m > k J -fv J 2 C r -1 J dCr dt max , λ M > λ m + 2 k J -fv J 2 C r -1 J dCr dt max , (57)
holds.

Therefore, the presented control law guarantees exponential convergence to 0 of the tracking error, in the case where the estimation from the observers are equal to the real variables. Note that this control law is robust with respect to disturbances and parameter uncertainties. However the system is nonlinear. The separation principle is then not valid. In the next subsection we will study the convergence of the control law and the observers together.

Closed loop stability

The state dynamics of the complete system including the tracking errors as well as the estimation errors Ξ [e f , e g , e ω , e θ , ǫ f , ǫ g , ǫ df , ǫ dg ǫ y ω est , ǫ dω est ] T , (58) are given by:

Ξ =                 -R L e f + n p (e ω e g + e ω i g,r + e g ω r ) -R L e g + n p (e ω e f + e ω i f,r + e f ω r ) -K L e ω 1 J (Ke g -f v e ω -C r ) e 3 -l f ǫ f + n p ω r ǫ g -n p ω r ǫ f -l g ǫ g K L ( ω sin(n p e θ ) + n p (e ω + ω r )e ω cos(n p e θ )) -K L ( ω cos(n p e θ ) -n p (e ω + ω r )e ω sin(n p e θ )) (-fv J -l ωest )ǫ y ω est + ǫ dg -1 J dCr dt                 +                1 L v f 1 L v g 0 0 w st,1 (ǫ f ) w st,1 (ǫ g ) ẇst,2 (ǫ f ) ẇst,2 (ǫ g ) w st,1 (ǫ y ω est ) ẇst,2 (ǫ y ω est )                , ( 59
)
where:

ω = 1 J (K(i f sin(-N e θ ) + i g cos(-N e θ )) -f v (e ω + ω r ) -C r ) .
To prove the exponential convergence of the system (59), one has to show that the trajectories of the complete system (59) remain bounded on a finite time interval. Indeed, if the system is bounded, the trajectories are maintained in a compact until the observers converge. Then since the designed observers converge in finite time, after this finite time the system will behave exactly like the system controlled with the state feedback described previously. To prove that the system is bounded the estimated variables ξ have to be written as a function of the system states Ξ, the reference trajectory Γ r and the control inputs.

ξ f ξ g = U (∆θ est ) îf îg - i f,r i g,r , (60)
Taylor & Francis and I.T. Consultant with:

     ∆θ est = -1 np arctan df dg , îf = ǫ f + i f = ǫ f + e f + i f,r , îg = ǫ g + i g = ǫ g + e g + i g,r .
(61) Similarly:

ξ ω ξ θ = ω est θ est - ω r θ r =   ± L K d2 f + d2 g -1 N arctan df dg   - ω r θ r . ( 62
)
After substitution in the system (59), of the control equations v f and v g with the expressions ( 60) and ( 62) of the errors between the estimated variables and the reference variables, one obtain a model depending on the states, the desired trajectories and the control inputs:

Ξ = F (Ξ, Γ r , w st,1 (ǫ f ), w st,1 (ǫ g ), ẇst,2 (ǫ f ), ẇst,2 (ǫ g ), w st,1 (ǫ y ω est ), ẇst,2 (ǫ y ω est )). ( 63
)
Without loss of generalities, one can assume that the chosen reference trajectory is bounded. One has then, using (60):

ξ f = O(ǫ f + e f ), ξ g = O(ǫ g + e g ). (64)
Since the current i f and i g are saturated, as well as their estimates values îf and îg , the tracking errors of the currents e f and e g and estimation errors ǫ f and ǫ g are bounded. Thus, the error variables ξ f and ξ g are bounded. Moreover, the Super Twisting and the Twisting algorithms are bounded.

The system can be written as Ξ = f (Ξ) + g. Thereby, including all the dominations in the complete system, one obtain the inequalities:

|| Ξ|| ≤ Q||Ξ|| + g, (65)
where Q and g are positive constants.

Integrating (65) one obtain:

||Ξ(t)|| ≤ ||Ξ(0)|| + t 0 (Q||Ξ(τ )|| + g)dτ. (66)
Applying the Gronwall lemma, one has:

||Ξ(t)|| ≤ ||Ξ(0)|| exp(Ct) + g Q exp(Ct -1), ( 67
)
where C is a positive constant. Therefore, the complete state system Ξ is bounded. This proves the exponential convergence of the mechanical sensorless control law.

5 Experimental results

Open-loop strategy

As mentioned Section 3, the position is not identifiable at standstill. To overcome the problem, a low speed scenario is described. If the motor velocity is lower than a certain velocity |ω lim |, the motor is driven in open loop. The following control law in the rotating reference frame is used:

v = R 2 + (N ω ref L) 2 I 2 max + (Kω ref ) 2 , if v < V max , V max , else. ( 68
)
I max is the current limit, V max is the voltage limit. v is applied to the motor using the transformation v a = v cos(N θ ref ) and v b = v sin(N θ ref ). Because this method does not rely on position and velocity sensors or on estimates of these variables, we will refer to the control algorithm as the open-loop controller.

When the PMSM is controlled in open-loop, it can be assumed that Ω r ≃ Ω as long as the motor keeps synchronism. This condition can be verified without a sensor because loss of synchronism results in a stalled motor and/or considerable vibrations.

The open-loop controller is useful at low speed, but falls short of what would be desired from a sensorless control law. In particular, this strategy is not able to achieve great acceleration profiles and the motor cannot operate in certain resonant region.

Reference trajectory

In order to shown the performances of the proposed algorithm, a reference trajectory has been designed. Thanks to the flatness properties, the reference trajectory is completely determined from the flat outputs. It is then sufficient to determine θ r and i f,r . Form these variables, all the remaining reference variables are computable.

The objective is to bring the motor position from a position θ r (t i ) = θ i to a position θ r (t f ) = θ f using smooth dynamics without discontinuities. The following constraints are chosen:

       θ r (t i ) = θ ri , θ r (t f ) = θ rf , θr (t i ) = 0, θr (t f ) = 0, θr (t i ) = 0, θr (t f ) = 0, θ (3) r (t i ) = 0, θ(3) r (t f ) = 0. ( 69
)
Having 8 constraints, the minimal polynomial must be of degree 7:

θ r (t) = θ ri + (θ rf -θ ri )( a 0 ∆(t) 7 + a 6 ∆(t) 6 + a 2 ∆(t) 5 + a 3 ∆(t) 4 +a 4 ∆(t) 3 + a 5 ∆(t) 2 + a 6 ∆(t) + a 7 ), (70)
with:

∆(t) = t -t i t f -t i ,
and:

a 0 = 20, a 1 = -70, a 2 = 84, a 3 = -35, a 4 = 0, a 5 = 0, a 6 = 0, a 7 = 0.

The direct reference current is defined to be null, in order to minimize the Joule losses and maximize the motor torque [START_REF] Bodson | High performance nonlinear feedback control of a permanent magnet stepper motor[END_REF], it leads to i f,r = 0A. The trajectory of the current could also have been chosen as in [START_REF] Verl | Torque Maximization for Permanent Magnet Synchronous Motors[END_REF], but the optimization is not the goal of this article. From these two variables, ω r , ωr and i g,r are determined from (38). The experimentations are realized using a stepper motor bench developed in LAGIS at École Centrale de Lille (see Fig. 3). The motor parameters have been identified using a mechanical sensorless procedure described in [START_REF] Delpoux | Parameter estimation of permanent magnet stepper motors without position or velocity sensors[END_REF] It is important to note that number of pole pairs is higher than conventional two-phase permanent magnet synchronous motors. Such a rotor design has a low moment of inertia, resulting in outstanding acceleration behavior. However, this large number of poles produces currents and voltages in the fix frame at higher frequencies, which vary at n p = 50 times the frequency of the motor rotation. The computer hardware on the test-bench is a dSpace 1104, with Con-trolDesk software as interface. Using the library RTlib developed by dSpace, the algorithms are implemented in C language. The input voltages v a and v b of each coils are delivered by two D/A outputs of the dSpace and amplified by two linear amplifiers. The currents i a and i b are measured using hall effect sensors with a precision of 1% of the nominal current I n = 3A. The power supply provides a maximum voltage v max = 30V and i max = 3A. Fig. 2, show the whole control scheme. The sampling frequency for the experiments is constant and equal to 10 -4 s.

Test-Bench Description

Experimental results

The performances of the approach are demonstrated by choosing a particular reference trajectory. The objective is to show that the motor is able, starting from a null velocity, to track the reference trajectory even for negative velocities. As mentioned before, the open loop control is used for velocities lower than |ω lim |, where ω lim is chosen equal to 3rad.s -1 . The switching instants are represented in the following figures by a magenta line. The desired trajectory is defined applying (70) twice, the first time starts with a position equal to zero to reach an intermediate position, from 0 to 18rad in 2seconds. Then equation ( 70) is applied from the intermediate position until a final position equal to 0. This leads to a velocity profile which is positive during the first 2 seconds, and negative the 2 last ones.

Convergence of the observer based on electrical equations

The position and the velocity are estimated from the observer (26). In order to be well reconstructed, it must be shown that the observers converge correctly. In Fig. 4 are plotted the currents i f and i g and their estimates îf and îg . The figure shows that the estimated currents are close to the measured ones. The estimations errors ǫ f and ǫ g are less that 0.01A. The measured disturbance voltages d f = K L ω sin(n p ∆θ) and d g = -K L ω cos(n p ∆θ) are drawn Fig. 5 with their estimates df and dg and appear to be closely related. The unknown inputs of the observer are well reconstructed. This shows the convergence of the observer, from these results, one can be able to reconstruct the estimated position and velocity.

Position and velocity reconstruction

The position is reconstructed from the estimation of cos(n p ∆θ) and sin(n p ∆θ), which enables to estimate the error between the desired trajectory and the motor position. The variable ∆θ est is then computed modulo 2π np . One has to verify that the error between the position and the reference position remains in this interval. In the case where the error leaves the interval, one has to add ±2 π np to compensate for the error. A procedure is implemented to count these "jumps". The integer k is incremented if the estimation leaves the interval by the upper bound and decremented conversely. Such an algorithm is sensitive to measurement noise: a "jump" caused by the noise could be interpreted by the passage to another interval. However, the Super Twisting algorithm plays the role of a filter and provides a continuous estimation ∆θ est . From this estimate, the position can be reconstruct as: Fig. 6 shows the estimates of these variables, which are plotted while the motor is controlled in closed loop. Firstly it should be noted that the position error is always less than the distance between two poles. The maximum error is around 0.02rad while the angle between two poles is equal to 2π 50 = 0.126rad. The estimated position is reconstruct using (71) where k is equal to zero. The resulting estimation is plotted Fig. 7. The figure shows the closed loop position tracking. Again the estimated position is plotted when controlled in closed loop. In open loop, the position is equal to the integral of the reference velocity plus the initial position, which is zero at the beginning of the experimentation. Thereafter, the position takes the last estimated value. The estimated position is plotted in green on the Fig. 7. The second subplot shows the error between the measured and estimated position. The error is less than 0.01rad. The blue curve represents the closed loop performance, and it can be seen that the tracking error in closed loop is less than 0.02rad. This means that over three rounds, the position is tracked without mechanical sensor with a precision less than one degree. The velocity estimate is plotted Fig. 8. This figure presents the reference (red), the measured (blue) and the estimated (green) position in the first subplot and in the second subplot the error ǫ ω between the measured and the estimated position (green) and the tracking position error e ω . The experiment shows that the motor reaches 20rad.s -1 with a precision around 1rad.s -1 .

θ est = θ r + ∆θ est + 2k π N . (71)
Finally, the acceleration is low pass filtered using a discrete third-order filter (Butterworth). The resulting estimation is plotted with the reference acceleration Fig. 9. Remark 5.1 It is important recall that the motor used in the experiment has a large number of pole n p = 50, which limits the speed. In the literature, for example, in [START_REF] Ortega | Estimation of Rotor Position and Speed of Permanent Magnet Synchronous Motors With Guaranteed Stability[END_REF]) n p = 4, in [START_REF] Khlaief | A Nonlinear Observer for High Performance Sensorless Speed Control of IPMSM Drive[END_REF]) n p = 3. The results presented in this section prove the effectiveness of the proposed approach.

Conclusion

There has been considerable interest in developing sensorless control methods for synchronous motors, and permanent magnet stepper motors in particular. The objective is to replace position and velocity sensors with less costly and more reliable current sensors (which are often present anyway).

In this paper, a new approach was proposed for mechanical sensorless control of PMSMs using current sensors only. The model is expressed in a frame, which is obtained from the reference position instead of the measured position. The frame has the advantages of the (d -q) frame without necessity of the measured position and moreover it has the advantage to be valid even when the reference position is different from the measured one. Using a second order sliding mode observer the position, the velocity and the acceleration were estimated. Experimental results that show a very good estimation of this variables. From the observed variables, a trajectory tracking was designed using sliding mode control. This control strategy has the advantage to be robust with respect to external perturbations.

Using an initial scenario, which is used to control the motor when the position is not identifiable, the experimental results show that we are able to realize a position tracking in a large range of velocities, even for negative velocities. The position tracking error is less than 0.02rad which seems to be really reasonable in sensorless applications.

The low speed sensorless control and observability is still an open problem. In many articles, the experimental results are shown at non-null speed only. In this article was described an open loop procedure to overcome this problem. The experimental results have shown very good results with this strategy. However, the theoretical proof is missing. We did not talk neither about the switch between the open loop and the closed loop strategy.

Figure 1 .

 1 Figure 1. Variables in the different frames.

Figure 3 .

 3 Figure 3. Stepper-motor test bench.

  . The motor characteristics, with coils in series are • Inductance L = 9mH, • Resistance R = 3.01Ω, • Back-EMF constant K = 0.27N.m/A, • Moment of inertia J = 3.18.10 -4 kg.m 2 , • Coefficient of viscous friction f v = 2.37.10 -3 N.m.S/rad, • Coefficient of Coulomb friction C r = 0.0752N.m, • Number of pole pairs n p = 50.

Figure 4 .

 4 Figure 4. (top) Currents and their estimations, (bottom) observations errors.

Figure 5 .

 5 Figure 5. Disturbance voltages and their estimations.

Figure 6 .

 6 Figure 6. Position error and its estimate.

  Figure 7. (top) Reference position, measured position and estimated position, (bottom) position tracking error and position observation error.

)

  Figure 8. (top) Reference velocity, measured velocity, estimated velocity, (bottom) velocity tracking error, velocity observation error.

)Figure 9 .

 9 Figure 9. Reference acceleration and observed acceleration.

Taylor & Francis and I.T. Consultant