Chapter 8

,

Characterization of Control without a Mechanical Sensor in Permanent Magnet Synchronous Machines ¹

8.1. Introduction

If the industrial development of the permanent magnet synchronous machine (PMSM) is mostly linked to the placing on the market of reliable and efficient position sensors, it is still true that today we are witnessing a second wind for these devices, due to the possibilities of functioning without a mechanical sensor.

These structures help to meet new requirements concerning operation in difficult conditions (temperatures, vibrations, etc.) or else for cost reduction, while maintaining satisfying dynamic performances for a large number of applications. To this effect, many solutions appeared in recent years, in order to rebuild the information “mechanical position of the shaft” necessary for piloting the associated drives. Interested readers can also refer to Chapter 4 of this book or to many other publications such as [CHA 00] or [BOL 99].

Whatever the chosen observation method, the “position” information, rebuilt with a certain precision and above all with certain dynamics, will be injected into the autopilot loop of the machine, in order to ensure drive control. In the search for high dynamic performances, modern drives generally integrate a device that can evaluate the applied load torque, in order to compensate for its harmful effects on the adjustment quality [FOR 10], [BRA 87]. These devices are characterized by certain dynamics, imposed by the designer, in conformity with the temporal constraints of realization. To ensure the stability and the consistency of the functioning on the largest functioning range, we have to consider the different dynamics at stake:

– dynamics of observation of the position;

– dynamics of observation of the disturbance (load torque);

– dynamics of use of the drive in closed-loop, in order to define the pole placement functions of the cutting frequency of the voltage inverter. This is all the more true if we evolve in a non-linear context. In this context, the stability notions must be discussed, in a close proximity to the functioning point perspective, but also in a more general perspective, in order to comprehend regimes of high amplitude fluctuations. This raises an additional difficulty inherent in the analysis tools to be used and particularly in their handling, requiring a good knowledge of the electromechanical phenomena controlling the speed and/or position drive.

8.1.1. State observation and disturbance observer

Since the emergence in the 1960s of the notion of state, the observer theory has achieved important progress and above all a certain maturity authorizing efficient industrial applications [LUE 71]. In a linear, non-linear, deterministic or else in a stochastic context, many examples of use have shown the efficiency of such procedures. In the case of the PMSM, we have to consider the speed and position of the revolving mechanical shaft as two observable state variables. This observation will be directly led by the measurement of the absorbed currents and the knowledge of the voltages imposed by the voltage inverter or indirectly by the reconstruction of the electromotive forces. For the search of performances, we can also consider the load torque that can appear as an additional state variable or as an exogenous disturbance. Very generally, the efficient solutions are based on the establishment of an extended state observer leading to the reconstruction of the speed, the position and the load torque. The observer in its form of full order is of dimension 5:

[8.1]

This solution requires the use of an evolution model of the load torque, difficult to formulate given the exogenous characteristic of this magnitude, but whose bypass is simple by choosing a null evolution on the scale of the sampling period:

[8.2]

However, reinjection of this magnitude in the feedback loop of the speed and/or the position must be discussed, with regard to the observation dynamics imposed, which, if they are too significant, can destabilize the adjustment device.

8.1.2. Interaction of the dynamics of control and observation

Setting and using a mechanical magnitude observer comes down to using rebuilt pieces of information, thus replacing the direct measurements. The dynamics restriction leads to a filtering role for the observer. This can interact with the control law. Let us study the set system-observer-control in the dynamic interactions angle.

8.1.2.1. Linear case

For the linear case, we can consider the following equations, for a full order observer:

[8.3]

If we assume that the system has controllability and observability properties, this system is in closed-loop with 2xn dimensions and can be represented by the state equation [8.4], where L represents the observer gain and K represents the adjustment gain of the control:

[8.4]

The dynamic matrix can be factorized as follows:

[8.5]

We will note that the properties of matrix Λ^' help to express the eigenvalue by:

[8.6]

which shows the non-interaction between the observer dynamics and the dynamics of the looped system.

Thus, the dynamics placement is easy, because we have to freely define the observation dynamics, compared to the dynamics desired in closed-loop. Of course, the presence of noises imposes a restriction on the observation dynamics, that we generally limit to three or four times the natural dynamics of the open loop system.

8.1.2.2. Non-linear case

In non-linear, the state observation is a bit more delicate, because the observability notion in itself is more difficult, since it depends on the initial conditions and on the system inputs [ZEM 07]. Nowadays, there is still no universal method for the synthesis of the observers in this framework. The approaches used are either an extension of the linear methods after linearization of the model around an equilibrium or after non-linear transformations, thus producing a linear system; or they use specific algorithms updated in recent years, such as the high gain observers and the other observers in sliding regime, built from the non-linear model. Anyway, observation and adjustment dynamics strongly interfere and the global stability is no longer guaranteed by separating the observation gains and the adjustment gains in closed-loop. Many studies confirm and illustrate this situation [MIS 89], [RAJ 98] in the general case, as well as in the particular case of the magnet synchronous machine [GAS 04], [LEP 93], [MUR 93].

The general model of the permanent magnets synchronous machine with salient poles, considered in a revolving reference mark with the rotor, with an exogenous load torque, appears as a non-linear system closely connected in the control:

[8.7]

The appearing non-linearities are of two types, with first a mechanical-electric interaction (products ω.i_d and ω.i_q) and with then, a production of the electromagnetic torque, the result of a crossed term i_d.i_q. However, if we consider that the speed is in very slow evolution compared to the dynamics of the electric part, the mechanical-electric decoupling authorizes the writing of the following linear system:

[8.8]

with:

This second writing, if legitimized by the value of the parameters, opens the scope of the linear systems and of the associated properties.

Let us note that the electromagnetic torque involves two current components. This gives the possibility of optimizing the drive performances, by choosing the best distribution in the sense of one criterion (functioning at unit power factor, obtaining a maximal torque for a given current, etc.) [LAJ 91]. In the framework of the research of high dynamic performances, the mechanical-electric decoupling hypothesis is often modified and we can usually favor model [8.7].

8.1.3. Poles placement for control and observation

The notion of a pole is only meaningful for a linear model. We can extend this notion to the systems whose dynamic evolution of the speed remains low, compared to the evolution of the electric or magnetic magnitudes. Therefore, the use of model [8.8] leads to an analysis of the dynamics and to a fast dimensioning of the different adjustment gains, for the observer as well as for the control.

The exploitation of the dynamic matrix leads to the localization of the poles’ evolution in the complex plan, in function of the rotation speed (Figure 8.2). At low speed, the maximum dynamics is limited to the ratio R_s / L_d , and at high speed, the imaginary part is given by ω_h and the poles’ module by ω_m . Depending on the machine dimensioning and thus on the function to carry out, the maximum dynamic will be given by the ratio R_s / L_d if the maximal speed ω remains low in front of R_s / L_d , or by ω if the speed is high compared to R_s / L_d .

Generally, the ω_m parameter is the dimensioning parameter, for conventional industry applications and all the more for aeronautics applications, where high speeds are favored for reasons of machine volume. It is only for the positioning or for the low speeds that the ratio R_s / L_d remains predominant, for the design of the control laws, with:

[8.9]

and the module of the complex poles:

[8.10]

We still have to note that the new magnets machines have increasingly low inductances and that the monitoring of the current ripples requires an increase in the cutting frequency or resorting to multi-level supply topologies, resulting in a virtual multiplication of the cutting frequency. This is how the band-pass of the observers of the mechanical magnitudes (speed or position), ω_bpo is to be put in relation with the cutting angular frequency, which itself is linked to the sampling angular frequency ω_e ( ω_e = ω_d / k with k varying between 1 and 3) and the module of the complex poles ω_m . More reasonably, it is necessary to consider the band-pass of the current loop ω_bpc. This angular frequency is itself limited by the sampling angular frequency ω_e, with at least one half-decade of margin and must remain higher than the characteristic angular frequency ω_m of at least one decade, i.e.

For a stable and satisfying functioning in terms of dynamic response, the bandpass of the observers of the mechanical magnitudes ω_bpo should remain lower than ω_e by at least one half-decade, i.e.

In addition, the band-pass of use of the drive (speed or position) ω_u often remains much lower than the band-pass of the current loop for obvious reasons of overcurrent restrictions. Thus, we can consider the hierarchical organization of the angular frequencies or frequencies according to Figure 8.3

These considerations rely on a linear approach with an environment not taking account of the parametric uncertainties and of the presence of state and measurements noises, which limits its impact. It is clear that in the framework of a wider industrial approach, these notions must be considered and solved with the use of a possibly extended Kalman filter for example.

Thus, the notion of the observation dynamics is no longer this formally imposed, but it is controlled via the minimization of the trace of the covariance matrix of the state vector. The gains are directly adjusted as a function of the quality of the prediction model. When the prediction error is low, which results in the relevance of the model, it is possible to increase the gains and thus the observation dynamics; a contrario, when the prediction error increases, testifying to a divergence between the measurements reality and the prediction of the model, it is necessary to reduce the gains to limit the impact of the introduction of wrong (or at least hardly relevant) information.

The problem of the choice of the dynamics is thus carried forward on the calibration of the covariance matrices of the state and measurement noise. This aspect has been the focus of much work in recent years, without however giving a formal answer.

The state of the art in the domain shows methods adapted to a group of problems. Concerning synchronous machine control without a mechanical position sensor, we will keep the methods establishing a link between a covariance matrix and the poles’ placement in closed-loop.

This new approach artfully uses the advantages of maintaining the dynamic performances defined thanks to a quadratic linear criterion, using pole placement, a method very frequently used nowadays. The main advantage is to know at the nominal point the form of the response, in order to accelerate the dynamics imposed by the control. This method, whose theory is developed in [AND 89] has been the subject of more recent interesting applications in the frame of a positioning system in [FER 98].

8.2. Sensorless control of PMSM, thanks to an extended Kalman filter

As already mentioned, the extended Kalman filter (EKF) is an interesting candidate solution for the observation of the position and the speed of the synchronous machine, in the context of noises and uncertainties of the model. Therefore, it has been the subject of many academic and industrial applications, since its creation in 1958 [KAL 60].

8.2.1. A brief reminder on the Kalman filter (KF)

It is not useful to detail the KF theory here, because there are many books mentioning its different aspects with relevance. We can quote for example the following references [WEL 06], [BEL 84], [SAY 94]. Let us however specify here the formulations used for the resolution of our problem, by mentioning the different necessary stages.

From the equations of the standard observer:

[8.11]

we will note the aprioriestimate of the vector [8.12], an estimate carried out from pieces of information known at the k instant. The implementation of the discrete KF is thus broken down into two stages. First, we have a prediction stage, where we estimate the state at the k.Te instant as a function of the state and the measurements carried out at the (k +1).Te instant. The recurrent equations helping to carry out this prediction are the equations of the deterministic model.

The second stage, the correction phase, consists of updating the state estimate from the new measurement at this instant and from the estimate apriori :

[8.12]

This stage requires the use of a correction gain K_k . This is precisely the calculation of this gain that makes the originality of KF. Indeed, in the traditional observers functioning according to the same principle, the gain is in general fixed. Here the gain value will be calculated as a function of the prediction quality. If the prediction error is small, this testifies to the model validity, of the noises of measurement and of the incident state. We can grant an important trust to the model. Thus, the observation will be of quality and we can increase its dynamics through this K_k gain. On the contrary, if the prediction error is important, we have to give a low trust level to the model and thus strongly decrease the gain in the correction process. The calculation of this gain must thus integrate an evaluation of the prediction error.

We also define the following estimate errors:

[8.13]

And the covariance matrices of the observation errors:

[8.14]

The matrices P_k^- and P_kare defined positive matrices. They give an indication of the precision of the estimates:

[8.15]

with the P_k^- matrix that we can express as:

[8.16]

The gain K must seek to minimize the variance of the a priori estimate error; the criterion is then given:

[8.17]

The optimal gain is given by:

[8.18]

i.e.:

[8.19]

The covariance matrix must also be updated. We thus find:

[8.20]

To summarize, here are the necessary calculations, from a general initial model:

[8.21]

First, it is necessary to define the four following matrices:

[8.22]

and then to carry out the following calculations: P_k ^-=F_k. P_k-₁.F_k^t+W_k.Q_k-1W_k^t

[8.23]

in order to obtain this state vector.

8.2.2. Application to the PMSM case

In the reference frame directed by the position of the rotor flux, the voltage equations are as follows:

[8.24]

The load torque being considered as an exogenous magnitude and the speed with slow evolution in front of the electric magnitudes, we can use a simple model of the speed evolution, characteristic of the fact that its evolution is negligible on the scale of a sampling period [ZHE 08]:

[8.25]

We can thus form a state observation model with the currents, the speed and the position:

[8.26]

These equations do not depend on the mechanical parameters or on the load torque. Its use is simple. We can write this system under the following generic form, characteristic of a non-linear discretized system:

[8.27]

We use as inputs the direct currents and the quadrature assumed to be correctly rebuilt from the measurement of two line currents. These variables are marked by the y vector:

[8.28]

and with:

[8.29]

This is of course the state vector that we seek to estimate. By considering the measurement and state noises, the equations discretized in the 1^st order become:

[8.30]

[8.31]

w represents the state noise and v represents the measurement noise. We assume that these noises have good properties, i.e. that they are comparable to non-correlated white Gaussian noises, characterized by a null average and by Q and R covariance matrices. The determination of the Q and R matrices is always tricky, since the stochastic characteristics of the noises are not generally well known.

[8.32]

For the discretization of the state system, we can use a simple method, such as the Euler method, ensuring a good compromise between precision and the calculation load [8.32]. We can thus define the F matrix, so that:

[8.33]

The observer is formed in the reference frame directed by the position of the rotor flux. It can be used for non-salient pole machines, as well as for salient pole machines.

To limit the number of measurements, we can also use the reference voltages instead of the voltages measured by phase, which remain tricky to measure. The structure of the control without sensor thus becomes:

8.2.3. Simulation results

We will now present the simulation results and show the incidence of the adjustment of the Q and R matrices. The trials are carried out with a magnet synchronous machine with non-salient poles, with the following characteristics: R_s = 97 mΩ ; L_d = L_q = 16 mH ; f = 0.089 N.m.rd.s^-1 ; J = 0.004 Kg.m² ; P = 4 ;E = 33 v ; F_d = 20 kHz.

In the 4^th order EKF, the equation for the speed observation is expressed by:

[8.34]

This equation is the most important of the four state equations. We analyze the parameter effects on this equation by digital simulation.

The initial adjustments of the covariance matrices of the state and measurement noises have been defined by the following values.

The observation results on the speed loop for case number 1 are shown in Figure 8.5

The same trial is used for case no. 2 in Figure 8.6, then for case no. 3 in Figure 8.7 and finally for case no. 4 in Figure 8.8

We will note that the observer gain depends on the choice of the Q and R matrices. If the Q and R matrices vary in the same proportions (case no. 1 and no. 4), the K_k feedback gains remain identical and thus the convergence dynamics will be preserved.

The choice of the value will depend on the context of use and will be kept after different trials, leading to an adjustment that is considered satisfying. For the following, the curves presented in this document are obtained by choosing the adjustment of case no. 4.

8.3. Comparison with the MRAS (model reference adaptive system) method

There are other observation methods using the variable gains and which are adaptable to the context. Among them, there is the MRAS approach, based on the reference model and on the cancellation of an estimate error with the help of a variable gain calculated from a Lyapunov function [LIA 03].

The observer model is of the 2^nd order. Here, we are in the case of a non-salient poles machine, i.e. L_d = L_q = L :

[8.35]

The estimate errors are expressed by:

[8.36]

where:

and by choosing the following Lyapunov function:

[8.37]

using the stability analysis, we can express the observed speed:

[8.38]

or by setting out:

we can express the evolution of the observed speed by:

[8.39]

The observation structure is given in Figure 8.9.

By simulation, the gains of the MRAS method and of the Kalman filter are compared. It is clear that the methods have some common points and comparable performances.

8.4. Experimental results comparison

Experiments are carried out on a device using the DSP C6711 and a FPGA for the digital piloting.

The device is represented in Figure 8.11, with its supply device, its load and its control. In Figures 8.12 and 8.13, we show the effective evolutions of the effective speed and position, rebuilt during a low speed functioning (4Hz) followed by an acceleration.

In the same way, we can visualize the reconstruction of the mechanical position information in Figure 8.13.

Speed and position monitoring is completely satisfactory compared to the predictions and in any case is sufficient for a large number of applications. In Figure 8.14, we show a functioning with a lower speed (2 Hz) and in the presence of a load surge. The functioning remains very satisfactory, even if the lowering speed is not negligible and linked to a non-support in the control loop of this type of disturbance. To this effect, we will see in the following section how it is possible to reduce this disturbance effect and how we can improve the global performances of the application. In addition, a functioning with an even lower speed (< 1 Hz), makes ripples appear, notably for start-up in load and for important load surges. This aspect is prone to open research with an industrial expectation.

8.5. Control without sensor of the PMSM with load torque observation

Load torque is an external magnitude, strongly influencing the dynamic behavior of the speed drive. Thus, to improve the performance, it becomes interesting to estimate this magnitude and to compensate for its effects in the control loop. There are several solutions for estimating this magnitude: either the load torque is independently rebuilt or in a coupled way at the speed and/or position observer [FOR 10]. Here, we will mention the EKF extension to an additional magnitude: the load torque. To estimate the load torque and use the complete equation of the speed, the 4^th order Kalman filter is increased to the 5^th order. The load torque is then considered as an additional state variable. The observed torque will then be used in a compensation loop.

The complete equation of the electric speed then becomes:

[8.40]

Thus, the equations of the new EKF are modified and written:

[8.41]

After 1^st order discretization we have [MAR 65]:

[8.42]

It is now necessary to define the dynamic matrix, and thus we obtain:

[8.43]

This structure is capable of rebuilding the speed, the position and the load torque. Thus, the control without mechanical sensor can be defined according to its structure represented in Figure 8.15. The magnitudes at the input are the line currents and the voltages that we can replace with their references. The observed torque is used in compensation on the reference torque, to improve the control performances during load impacts.

This compensation makes a low-pass filter appear, which is crucial for the good functioning of the set. Indeed, too quick a compensation of the load torque can create ripples and destabilize the speed loop. The low-pass filter, wedged a half-decade lower than the utilized band-pass (see Figure 8.3), ensures satisfying functioning of the set. The experimental results made under the same conditions as the previous section are presented in Figures 8.16, 8.17 and 8.18 below. All the advantages of the compensation for the rejection of a disturbance clearly appear.

The observed load torque is reliably rebuilt. There seems to be a small delay, due to the adjustment of the Q and R matrices of the EKF.

This observer can also be integrated into a feedback position loop by using, for example, a control by state feedback. In Figure 8.20, the whole device is represented with an integral action and an anticipation term helping to follow a reference trajectory.

The experimental results are shown in the figures below (Figure 8.21, 8.22, 8.23), during a trajectory follow-up, making speed levels of different values appear. The anticipation gain K_θ is calculated in order to compensate for a pole of the closed-loop, which ensures a response without overrun for a step request. On the other hand, for a slope response a speed error appears, which is visible in Figure 8.22. By choosing a K_θ gain cancelling the X_rmagnitude in steady state, during a step request, we will have a null decay error, if the request is in keyway [VOR 95].

8.5.1. Control by state feedback on the currents

During conventional control, the current measurement carries the noises and disturbances, inherent in the physical reality of the experimental assembly.

[8.44]

If the measured currents are used in the regulator feedback loops and if the voltage set point appears at the output of the regulators and can be imposed without major distortion, the currents errors will be null with a delay linked to the adjustment band-pass.

[8.45]

Under this condition, the effective currents in the motor can be expressed by:

[8.46]

There is thus a component corresponding to the noises and disturbances in the effective currents, which can affect the drive performances. The Kalman filter can eliminate the noises and disturbances based on the motor model and on the covariance matrices. Thus, we can use the filtered currents to replace the measured currents, in order to avoid the reinjection of noises in the regulation loops [MAX 88].

The simulation results shown below testify to the advantage of this simple and inexpensive modification, in terms of quality on the current evolution.

In experiments, the effective currents cannot be known, and we cannot compare them. The experimental results show the measured and observed currents in Figures 8.27 and 8.28.

The extended Kalman filter, chosen to estimate the speed, the position and the torque of the motor is an efficient solution in the field of sensorless control. The observed torque used in compensation, improves the control performances in terms of disturbance rejection. Replacing the measured currents with the observed currents contributes to the adjustment quality, on the current loop and thus on the torque loop, as well as on the global quality of the positioning or of the speed control.

8.6. Starting the PMSM without a mechanical sensor

Even if the Kalman filter is an interesting solution, there is still a major problem with control without a mechanical sensor: start-up. Indeed, it is necessary to know the rotor position at start-up, in order to generate the desired torque, in amplitude as well as in direction. There are two different ways consisting of locating the rotor or of pre-locking it. The conventional methods to estimate the initial position are applied in general to the rotors with field poles, i.e. to salient pole machines. The pre-locking method before start-up of the motor is a solution strongly affected by the load characteristic and cannot be applied in all cases.

In this section, we propose a method to directly start the motor, without any knowledge of the initial position. This method can be applied to a large number of machines. It relies on modeling in a two-phase reference frame, revolving at the rotor speed.

8.6.1. Equilibriums of the system without a mechanical sensor

Functioning of the synchronous machine without a mechanical sensor is done from any configuration of the rotor and leads to non-unique equilibrium states. Thus, we can define the error of the position estimate:

[8.47]

The reference frames directed by the effective and the estimated position, are represented below in Figure 8.29.

The equations of the motor in the reference frame directed by the effective position are:

[8.48]

where L = L_d = L_q , which characterizes the non-salient poles machine. Thus:

[8.49]

The input variables in these two reference frames are:

[8.50]

Thus the equations in the reference frame directed by the estimated position become:

[8.51]

The speed equation is given by:

[8.52]

and the electromagnetic torque by:

[8.53]

The electric speed of the rotor:

[8.54]

In practice, the effective position can never be known. Thus, the observer equations can be given by:

[8.55]

The estimated position is calculated by integration of the estimated speed:

[8.56]

The errors between the effective and the estimated variables are:

[8.57]

We thus obtain the equations for estimate errors:

[8.58]

[8.59]

The system must converge on an equilibrium state that we will study. The equilibriums must satisfy:

[8.60]

By assuming that the estimated currents are equal to the effective currents; a situation reinforced by the presence of current regulators.

[8.61]

Thus the equations of the equilibrium points become:

[8.62]

We can obtain the following solution:

[8.63]

[8.64]

Particularly, if the motor load is null, the second equilibrium becomes [MOB 00], [FAD 07]:

[8.65]

8.6.2. Analysis by simulation

By using the 4^th order Kalman filter, we generally consider that the initial position is known and thus that the value of the initial estimated position is always null for the observer.

In the figures below, we consider the motor start-up from different initial positions, so that the rotor is in advance or delayed compared to the requested rotation direction.

We observe behaviors dependent on the value of this error. Thus, in Figure 8.30, we show different scenarios as a function of several initial conditions (Figures 8.30a to 8.30j).

We notice an evolution systematically striving to cancel the position error, even if the rotor can evolve in the opposite direction: cases a, b, c and d.

If the initial position error is more important, we notice important rotor ripples (Figure 8.30e to 8.30g) resulting in a non-receivable behavior.

For higher values, the motor does not start (Figure 8.30h and 8.30i).

When the initial position error is equal to-π / 2 (Figure 8.31), we can notice two starting situations:

– the torque at the stop is important enough to avoid any displacement and the rotor remains at standstill.

– or the motor is at vacuum with a low dry friction torque and the rotor can ripple.

At the proximity of this value, two different behaviors are observed depending on whether the value is approached by an upper or lower value.

When the system converges on the value γ = ±π / 2 (Figure 8.32), the estimated speed and the effective speed are identical. Thus, the motor currents reach a maximum as a function of the current regulator, but the electromagnetic torque still remains null. Thus, the motor rotor is maintained in this equilibrium state [8.66].

The simulation shows us (Figure 8.32) that if there is static friction on the motor rotor, the equilibrium states occupy a more important domain. This domain is predetermined by the relation:

[8.66]

with i_qmax , the maximum current limited by the control law or the inverter.

By the simulation shown in Figure 8.34, we can obtain the convergence zones from the different initial errors. For a positive reference current i_q>0 , we see in the top figures that in domain A, the system directly converges at the equilibrium γ= 0. In the B domain, the estimate error slightly increases, then decreases and converges at zero. In domain C, the error increases and goes through the convergence domain B, then finally converges at zero. For the equilibrium γ = -π / 2, the convergence domain is determined by the static friction torque C_fs in accordance with relation [8.66]. For a negative reference current i_q <0 , we notice a similar behavior for the rotation direction.

8.6.3. Modification of the control law for a global convergence

The previous study showed that the equilibrium γ= 0 is to be favored, because the other two equilibriums are produced by the current restriction and by the static friction torque. We can seek a method to avoid these undesired equilibriums. For this, we add a compensation term to the q axis equation, with an adjustable gain for value k .

[8.67]

k is a coefficient independent of the initial position error and must be chosen between 0.2 and 0.8, depending on the context. This coefficient can modify the observation quality after start-up and it is desirable to make it strive for 0, after the obtained start-up. The simulation results for a vacuum start-up are shown in Figures 8.35 and 8.36.

For a functioning in load (50% of C_n ), the behavior is also satisfactory (Figure 8.36).

In Figure 8.38, we visualize experimental results obtained with the extended 5^th order Kalman filter and for a load worth 50% of C_n and k = 0.3.

8.7. Conclusion

For control without a mechanical sensor, the synchronous machine opens new perspectives for specific applications needing low costs or reliability levels adapted as a function of the context. The processes of position estimate are associated with a start-up protocol, such as the one presented here and consist of compensating a term on the torque equation. These processes have reached full growth levels, so that nowadays the industrial diffusion is operational. This phenomenon is accelerated by the progress in launched micro-informatics, measurable by the large number of technological solutions available on the market.

Let us note that today the usable algorithms lead to start-ups with significant loads, getting closer to the maximal load, for machines with non-salient poles as well as salient poles.

Digital implementation of these processes requires a certain know-how for the placement of the different dynamics. This chapter shines some light on this difficult problem, linked to the machines’ characteristics, to the performances of the means of control, to the function to be carried out and to the environment in which the “actuator” will evolve.

8.8. Bibliography

[AND 89] ANDERSON B.D.O, Optimal Control, Linear Quadratic Methods, Prentice Hall, Englewood Cliffs, NJ, USA, 1989.

[BEL 84] BELLANGER M., Analyse des signaux et filtrage numérique adaptatif, Dunod, Paris, 1984.

[BOL 99] BOLOGNANI S., OBOE R., ZIGLIOTTO M., “Sensorless full-digital PMSM drive with EFK estimation of speed and rotor position”, Transactions IEEE Industrial Electronics, vol. 46, no. 1, p. 184-191, February 1999.

[BRA 87] BRANDENGURG G., SCHÉFER U., “Influence and partial compensation of backlash and Coulomb friction for a position-controlled elastic two-mass system”, Proceedings of EPE. Grenoble, p. 1041-1048, 1988.

[CHA 00] CHABOT F., Contribution à la conception d’un entrainement basé sur une machine à aimants permanents fonctionnant sans capteur sur une large plage de fonctionnement, PhD Thesis, INP, Toulouse, 2000.

[FAD 07] FADEL M., ZHENG Z., LI Y., “Globally Converging Observers for SPMSM Sensorless Control”, IECON 2007, 33^rd Annual Conference of IEEE Industrial Electronics Society, p. 968-973.

[FER 98] FERRETTI G., MAGNANI G.A., ROCCO P., “LQG control of elastic servomechanism based on motor position measurements”, AMC, Advanced Motion Control, Coimbra, Portugal, p. 617-622, 1998.

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1 Chapter written by Maurice FADEL.