Chapter 9

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Sensorless Control of Permanent Magnet Synchronous Machines: Deterministic Methods, Convergence and Robustness ¹

9.1. Introduction

The vector control of permanent magnet synchronous machines (PMSM) requires specific information on the rotor position. This information, necessary for the machine control, is usually supplied by a position sensor set on the machine shaft. There are many disadvantages to the use of this mechanical sensor. First, it increases the volume and the global cost of the system. Moreover, it requires an available shaft extension, which is usually undesirable, particularly in the case of small machines. The installation of this sensor requires a setting relative to the stator (an operation tricky to reproduce in series) and decreases the system reliability. The cost of mechanical sensors and the difficulty of placing them encourage researchers to avoid their use and to study mechanical sensorless control [MAT 92], [AFS 94], [BRU 96], [MAT 96], [BOL 99], [BOD 99] and [HAR 00].

The back-EMF is one of the magnitudes able to supply instantaneous information on the mechanical variables. Its determination requires only the knowledge of some electric variables and of an appropriate model of the machine. This is why a large number of studies carried out in the field of sensorless vector control of the PMSM rely on the estimation of the “back-EMF vector”. There are two categories of methods for estimating the “back-EMF vector”:

- methods based on the estimation of the back-EMF components in a stationary reference frame (in general the equivalent two-phase α-βreference frame). The position and the angular speed of the rotor are then deduced from the polar coordinates of the estimated back-EMF vector [SOL 96], [BOL 99]. These methods have the advantage of suppressing Park transformations for the estimation of the back-EMF vector;

- methods based on the fact that the mean value of the direct component of the back-EMF vector is zero: we estimate the back-EMF components in a hypothetical rotating reference frame. If it coincides with the d-q reference frame linked to the rotor (Park reference frame), the direct component of the back-EMF in this hypothetical reference frame then becomes zero [MAT 92], [ARA 98]. This very important criterion helps to correct the position and the speed of the hypothetical reference frame, so that it synchronizes with the d-q reference frame. The rotor position and speed are then directly deduced from the position and speed of the hypothetical reference frame. The simplicity of the above criterion on the direct back-EMF component is a significant advantage of the methods from this second category. Moreover, the estimated components of the back-EMF vector are constant in steady state, contrary to those estimated in a fixed reference frame. This helps to better adapt the bandwidth of the estimator, and thus to better work at high speed. For these reasons, we concentrate on this approach in this chapter.

The estimation of the back-EMF vector components can be deduced from the electric model of the machine, thanks to a simple calculation or to a state observer. We will thus begin by presenting the electric model of non-salient pole sinusoidal PMSMs for sensorless control applications. Then, by making a usual and non-restrictive hypothesis, we will reduce the order of the obtained model. This model will help us to analyze the convergence of the methods discussed in this chapter.

The methods based on the estimation of the back-EMF vector have generally one common drawback: their convergence domain is limited, even if the system model and its parameters are precise enough. This has significant consequences, notably at start-up. In order to study this problem, a convergence analysis should be made. To facilitate this analysis, we propose first to distinguish the main two tasks to be accomplished by each sensorless method:

- on the one hand, the estimation of the back-EMF vector components;

- and on the other hand, the implementation of a sensorless control law, in order to cancel the error of the estimated rotor position.

Thus, the first task gives the necessary information to the second task, in order to cancel the error of the rotor position. This approach thus helps us to consider the sensorless control as a regulation problem in which the position error should be regulated at zero. Each method of sensorless control thus has its own control law and its own back-EMF estimator. Then, the impact of the back-EMF estimation error on the obtained performances can be studied when the machine model and/or its parameters contain uncertainties.

The reduced order model, developed later in this chapter, helps us to easily analyze the dynamic and static behavior of the existing sensorless control laws. This analysis, carried out in section 9.3, shows us that it is because of the existence of one or several stable undesired equilibrium points that some methods suffer from a limited domain of convergence. To maximize this domain, the only stable equilibrium point of the whole system must be the desired point. We will thus propose a new variable structure sensorless control law, making all undesired equilibrium points unstable. This law is a priori designed for the torque control of a PMSM. But we will show that it is perfectly applicable to the speed regulation. The results obtained by simulation and experimentation will help to test the validity of the model and of the analysis carried out.

The convergence analysis is done for the case where the back-EMF vector components are perfectly correctly estimated. However, in practice, there are always uncertainties in the model, its parameters and measurements. These uncertainties affect the estimated variables according to their sensitivity to uncertainties. It is thus sound to study the robustness of the sensorless control in the presence of the most common uncertainties. This study, done in section 9.5, will complete the convergence analysis already done. We will see that the robustness of the sensorless control at low speed with respect to the stator resistance uncertainty risks being unsatisfactory for some actuators. Indeed, we demonstrate that the uncertainty of the stator resistance must be lower than a maximal threshold (condition [9.52]) in order to operate at low speed. According to this condition, the maximum authorized uncertainty of the resistance depends on the machine parameters, on the maximal value of the stator current module and on the sensorless control parameters. It is thus desirable to either implement an “online” estimator of the stator resistance for reducing this uncertainty [NAH 01b], or a control approach minimizing the sensitivity of the sensorless control with respect to the resistance uncertainty [NAH 07]. The second solution is simpler to carry out than the first, but requires modification of the direct axis current (idref≠0) and thus leads to the increase of losses in the machine. The decrease of the losses as a function of the operating point has been studied for this solution. However, the losses are minimized only if the two solutions are used at the same time. This is presented in section 9.6, concluding with experimental results validating the studied solutions.

9.2. Modeling PMSMs for mechanical sensorless control

We will recall here the model of a non-salient pole sinusoidal PMSM, for sensorless control. The magnetic circuit of the machine is assumed to be non-saturated and the irregularities of the air gap due to the stator slots, as well as the hysteresis phenomena and Eddy currents are neglected.

We consider a reference frame, called δ-γ, localized by the electric angle ϑ, indicating the position of the axis Oδ with respect to the axis Oα, linked to the stator (Figure 9.1). The electric angle 6 indicates the position of the axis Od related to the magnets, and φ indicates the error between Oδ and Od, so that φ = ϑ - θ.

Each three-phase signal can be represented by a vector g_abc=(g_a g_b g_c). The zero sequence component of the current being zero, the zero sequence components of the other signals (flux and voltage) do not contribute to the electromechanical energy conversion.

We thus neglect all zero sequence components. Only the projection of the vector g_abc in a normal plane is considered. This plane may be characterized by α-β, d-q, or 8-γ coordinates, represented in Figure 9.1. Every signal g can thus be represented by its components α-β: g_αβ = (g_α g_β), or d-q: g_dq = (g_d g_q), or δ-γ: g_δγ = (g_δ g_γ). The relations between these different representations are:

[9.1]

where P(θ) is the conventional Park transformation defined by:

[9.2]

The electric equations of a non-salient pole PMSM in the fixed α-βreference frame are described by [LEO 84]:

[9.3]

where i_α, i_β, v_αand v_βare respectively the α-βcomponents of the current vector and of the voltage vector; and Ω,is the angular rotor speed. and L_s are respectively the stator resistance and the stator inductance. K_f = Pψ_f is the back-EMF coefficient with ψ_f being the magnet flux through the direct equivalent circuit and P the number of poles pairs.

From [9.1] and [9.3], the electric equations of non-salient pole sinusoidal PMSMs with a non-salient rotor in the δ-γreference frame are written as follows:

[9.4]

with:

[9.5]

Ω_s indicates the angular speed of the δ-γreference frame. e_δand e_γare the back-EMF components in the δ-γreference frame, defined by the following relation:

[9.6]

The motor torque is a function of i_δ, i_γand the position error φ between the d and δ axes:

[9.7]

The evolution of the rotor angular speed of the Ωrotor is thus described by:

[9.8]

where Γ_ch is the load torque, assumed to be a monotonic function of the angular speed.

In the case of mechanical sensorless control of PMSMs, the rotor position 6 is not measured and the d-q reference frame cannot be located. Consequently, the d-q components of the electric signals and thus the motor torque are unknown.

A very common solution to this problem consists of defining a δ-γreference frame, whose position ϑand speed Ω_s are known (undefined until then). From the relation [9.5] and by assuming that the initial value ϑ₀ is arbitrarily chosen, the following equation describes the evolution of ϑas a function of Ω_s:

[9.9]

It is thus necessary to define Ω_s, so that the position error φ= ϑ- θ vanishes; this is achieved using the only available information on the system: the stator currents i_δand i_γobtained from the measured currents, and the measured or estimated stator voltages v_δand v_γ.

It is important to note that the reference torque is only imposed if ϑconverges to 6; in other words, if the position error φconverges to 2kπ.

9.2.1. State model

Let us consider the state vector X, the input vector U and the output vector Y:

[9.10]

The dynamic behavior of PMSMs without mechanical sensor is described by the following non-linear model:

[9.11a]

with:

[9.11b]

The problem of sensorless control of PMSMs then comes down to the research of an appropriate control law for the input vector U=[Ω_s v_δv_γ]^t, using linear or nonlinear control design techniques. The control objectives are the regulation of φto 0 (or to 2kπ), and the control of i_δand i_γat their setpoints i_δref and i_γref, using only the electric measures i_δand i_γ(Figure 9.2) and independently of the initial state of the system.

NOTE.- We can easily show that system [9.11] is locally controllable, but its observability depends on the angular speed Q, [NAH 01c]. Indeed, if the speed is zero, system [9.11] is not observable. This explains why conventional sensorless control of non-salient pole PMSMs fails at standstill and at low speed.

9.2.2. Reduced-order model

For simplicity reasons, we make the following common hypothesis.

HYPOTHESIS.- The mechanical variables are clearly slower than the electric variables. System [9.11] is then a non-linear system with two time-scales, with a mechanical dynamics much slower than the electric dynamics. So, we can decompose it into two sub-systems of reduced order: a fast (electrical) sub-system and a slow (mechanical) sub-system.

For instance, x=[φ Ω]^t the state vector of the mechanical variables and z=[i_δ i_γ]^t the one of the electric variables. We can then reformulate system [9.11] as follows:

[9.12]

with:

This separation of the electrical and mechanical modes helps us to implement a two loop control strategy [KHA 96]: a faster loop to control the electric currents and a second loop for the mechanical variables. According to relation [9.7], in order to control the motor torque, we need to control the electric currents i_δand i_γand to regulate the position error φ.

The first two variables (i_δ and i_γ) are controlled by current regulators. The model of the electrical sub-system for the design of these regulators can be considered as linear in the fast time-scale τ=t/ε:

[9.13]

where d_e(x,z) = f₃(x)+εd₂(z)u_m plays the role of an external disturbance. The electric control u_e should thus ensure the current regulation, whatever the disturbance d_e. In order to do that, we can use any suitable type of regulators. The objective is to obtain a short response time, a zero steady state error and a good disturbance rejection. Here, the design of the current regulators is not discussed and we only suppose that the δ-γ components of the current vector are well controlled at their references by sufficiently efficient regulators. This enables us to reduce the system order in the mechanical time-scale, by neglecting the electric dynamics:

[9.14]

where the reference currents I_d and I_q can be constant (torque control) or functions of the estimated angular speed (speed control). First, we consider them as constant and the speed regulation effect on the performances of sensorless control will be studied later in section 9.3. The interested reader can refer to [NAH 01c], for more details on the decomposition of the model [9.11] in two sub-systems of reduced order.

Taking into account the development above, the boundary layer model of the machine (model of the mechanical sub-system) is as follows:

[9.15a]

where:

[9.15b]

[9.15c]

The slow control Ω_s, obtained from a sensorless control law must ensure the regulation of the position error φ to 2kπ;which requires knowledge of φ, when it is not measurable. Thus, we need to use the state observers, in order to estimate the position error φ.

Taking into account the fact that the reduced-order model [9.15] and the full-order model [9.11] are both non-linear, the estimation of φrequires a non-linear state observer. This approach is studied in [NAH 01c]. There is another solution: the δ-γcomponents of the back-EMF vector, functions of the position error φ, appear as external disturbances in the linear model of the electrical sub-system [9.13]. This leads to estimating them using a disturbance observer of this reduced-order linear system. Then, knowing that the 8 component of the back-EMF (e_δ) converges to zero, when φ converges on 2kπ (relation [9.6]), we can replace the regulation problem of φ at 2kπ with the regulation of e_δat zero. This approach is discussed in the following section.

9.3. Convergence analysis of mechanical sensorless control laws

The δ-γcomponents of the back-EMF vector are not directly measurable and thus, we have to estimate them. This estimation can be done using electric equation [9.4]. This will be studied in section 9.4. Here, we will only discuss the convergence analysis of the sensorless control laws and we will thus assume that the estimation of e_δand e_γis perfect whatever the estimator; the robustness with respect to the estimation error of e_δand e_γ(error due to the uncertainties on the model and on the measurements) is studied in section 9.5 and in [NAH 00]. We thus have:

[9.16]

The convergence analysis of the different methods of sensorless control is carried out with the help of Lyapunov's first method, which consists of studying the local stability of the equilibrium points of the closed-loop system. First we study a linear control law [MAT 96], and then a variable structure law [NAH 04].

9.3.1. Proportional-type control law

This approach results from Matsui's second method, called the current model-based control algorithm. It has a digital formulation for an implementation in discrete time controllers [MAT 96], but here we present it in its equivalent continuous form. Figure 9.3 shows the block diagram of this control law, whose expression is given in [9.17].

[9.17]

where α and β are the parameters of the sensorless control to be determined so that the position error φ converges to 2kπ, whatever the initial conditions and the operating point. In order to do this, it is of course necessary for (φ=2kπ (k=0, ±1, ±2, …) to be the stable equilibrium points of system [9.15] with the control law [9.17]. But this is not enough, because other stable solutions imply that the convergence on the desired solution (φ=2kπ) is not global. In the following, we study the stability of the system equilibrium points [9.15]- [9.17]. To simplify the study, we assume at first that the control strategy of the motor torque is the one usually applied to non-salient pole PMSMs to maximize the output, i.e.:

[9.18]

We also assume that the reference torque is constant (torque control); the case of a speed regulated motor be studied later in section 9.3.

In order to obtain the model of the slow system in closed loop, we replace [9.16] in [9.17] and then in [9.15]. Taking into account [9.18], we have:

[9.19]

So that (φ=2kπ is a system equilibrium [9.19], whatever the rotor speed (Ω), if a is the following:

[9.20]

The choice of β will determine the stability of the desired solution (φ=2kπ). Here, we will make a local study. Indeed, if φ=2kπ is the only stable solution and in the absence of any other attractor (limit cycle on the φ-Ωplane), the domain of convergence of this stable solution is extended to the validity domain of the model [9.19], which covers the whole φ-Ωplane in the ideal case (global convergence).

System [9.19] has four equilibrium points. They are as follows (provided that Γ_ch(0) = 0):

[9.21a]

where:

[9.21b]

with:

[9.21c]

NOTE.- If Γ_ch(0) ≠ 0, the points p₁ and p₂ shift along the ϕ axis.

The study of the local stability of these equilibrium points shows that the desirable solution (p₃) is stable if:

[9.22]

where sgn indicates the sign function and b is a normalized gain (without unit), representing the only degree of freedom that we have for adjusting the convergence speed of the position error to 2kπ. This parameter also plays an important role in the local stability of the four equilibrium points. Figure 9.4 shows the impact of the parameter b on the stability of points p₁ to p₄ for a positive I_q. It should be noted that the location of p₄ depends on b (φ_e4 = -2tan^-1(b)), as shown in Figure 9.4.

We can notice in this figure, that we have to choose a positive b for the desirable solution (p₃) to be stable. With this choice, p ₁ is inevitably unstable. But the stability of p₂ and p₄ depends on the value chosen for b: if b > 1, p₄ is stable and p₂ unstable; and the opposite if b < 1. In these two cases, there is always an awkward stable equilibrium point preventing some state trajectories of the system to converge on p₃. Consequently, the convergence domain of p₃ is not global.

We will note that a high gain b > 1 (fast correction of the position error), p₃ and p₄ are stable, and p₄ is located in a sector where φ_e4 is between -π and -π/2, and the equilibrium speed of the motor (Ω*) is negative for a positive I_q (opposite rotation direction).

Figure 9.5 shows the trajectories of system [9.19] in the ϕ-Ω phase plane for b = 2, Ω⁺ = 100 rd/s and Ω* = -60 rd/s. This figure shows the convergence problem of the control law [9.17]: the trajectories coming from a zero speed initial point lead to p₃, only if the initial position error is limited (|ϕ₀|< π/2).

Speed inversion is not possible either: all the trajectories starting at a “sufficiently negative” initial speed converge on p₄. However, it is shown in [NAH 01c] that this last problem can be easily solved by replacing sgn(I_q) in [9.22] by sgn(ê_γ) (see [9.23]).

The new phase plane with βin [9.23] is illustrated in Figure 9.6. The convergence problem mentioned above still remains: at Ω= 0, all the trajectories for which φ₀ is between –π/2 and +π/2 lead to p₃; otherwise, they converge on p₄. The experimental results presented later confirm this conclusion:

[9.23]

In the case where 0 < b < 1, the equilibrium point p₄ is unstable (a saddle point), but p₂ becomes a stable fixed point and prevents some trajectories from converging on p₃. This is demonstrated in Figure 9.7, where b =1/2. We notice in this figure that p₄ has moved in the phase plane and is not stable, contrary to the previous case (see Figure 9.6 with b = 2). However, the start-up with a strongly incorrect initial position (| φ₀ | > π/2) is doomed to fail, because of the attraction of the state trajectories by p₂.

Figure 9.8 shows the simulation and experimental results obtained during startup with this sensorless control law. The parameters of the machine are given in the appendix. The δ-γcomponents of the stator currents, being the only available ones (see Figure 9.2), are well controlled with PI regulators. The reference currents are set at i_δref=0 and i_γref=3 A. The proportional control law [9.17] with a in [9.20], βin [9.23] and b = 2, has been used to obtain Ω_s. Then, the estimated position (ϑ) is obtained from [9.9] whereϑ₀ is set at zero. We can notice that despite an initial error in the position (φ₀ ≅ -π/3 rd), the machine correctly starts up and the position error (φ) converges to zero. The d-q components of the current also converge to the δ-γcomponents and the torque control thus comes to control the current i_γ

The same test has been carried out with a more important initial error (φ₀ ≅ -5π/4 rd). The obtained results are given in Figure 9.9. As expected (see Figure 9.6), the position error (φ) does not vanish in this case, and converges on -2ρ (ρ= tan^-1(b) ≅ 0,35π rd). The angular speed converges to a negative value (despite a positive i_γ) and the torque is no longer controlled.

All these experimental results confirm the validity of the presented model (relationship [9.11]) and validate the results of the convergence analysis of the proportional control law [9.17].

We can thus conclude that control law [9.17] does not guarantee the global convergence of the estimated position. Indeed, as demonstrated by our analysis, this convergence depends on the initial error on the estimated position, and the domain of attraction of the desired solution (zero position error) is limited, because of the presence of another stable, but not desired solution. This phenomenon is present in most sensorless methods existing in the literature [NAH 01c].

In the following, we will propose a variable structure control law, with a domain of attraction much larger than the previous one, covering all the entire studied domain (global convergence).

9.3.2. Variable structure control law

As previously seen, the presence of an undesired stable equilibrium point limits the convergence domain of the desired solution. It can be easily shown that if it is desirable to keep a simple expression like that of the law [9.17], it is necessary to vary the (3 coefficient in order to make the awkward equilibrium point (p₄ or p₂) unstable when the state trajectory is within close proximity.

It is obvious that α must remain the same as in [9.20], so that the desired solution (φ=2kπ) remains an equilibrium point of system [9.19].

In order to reach this objective, we will introduce into the control law new nonlinear terms that can only be functions of the components ê_δand ê_γ. For reasons of notation simplicity, we keep (3 as in [9.23] and we add the coefficient K = K(ê_δ,ê_γ) in [9.17], as follows:

[9.24]

According to the bifurcation diagram (Figure 9.4), K must be positive, in order not to modify the stability of the desired solution (p₃). In addition, the product b^.K determines the stability of the equilibrium points (see Figure 9.4): if b K > 1, p₄ is stable and is located in zone III^- (Figure 9.10) for a positive I_q; and if b K < 1, p₄ is unstable and is located in zone IV⁺ (φ-Ω plane in Figures 9.6 and 9.7).

Thanks to a good choice for K, we can make the point p₄ unstable by imposing b K< 1 when the state vector of the system is in zone III^- indicated on the ϕ-Ω plane in Figure 9.10. The p₄ point then moves in zone IV⁺ and p₂ becomes a stable spiral. The system state thus leaves zone III^- and converges on p₂, by inevitably going through zone IV⁺ (p₂ being a spiral, as shown in Figure 9.7). In order to prevent the system trajectory from converging on p₂, as soon as the system state is in zone IV⁺, we now impose b K > 1 : first, this makes p₂ unstable and on the other hand, it leads to the displacement of p₄ in zone III^-. Even if p₄ becomes stable during this operation, it is now sufficiently far away from the system state to no longer be able to absorb it. The K coefficient must thus meet the following conditions:

[9.25]

Consequently, a good choice among others for K is the following:

[9.26]

In order to respect conditions [9.25] with the choice of K in [9.26], the b coefficient must meet the following conditions:

[9.27]

Thus, the proposed variable structure control law is given under the following form:

[9.28a]

with:

[9.28b]

Figure 9.11 shows the block diagram of the proposed control law. It should be noted that the additional calculation load of this control law compared to [9.17] is only limited to a test on the sign of ê_δ . Consequently, the proposed solution does not have any (material or algorithmic) difficulty for a DSP.

Figure 9.12 shows the trajectories of the closed-loop system for b = 2, Ç= 0.75 and Ω⁺= 100 rd /s. As we can notice, all the system trajectories converge on p₃, whatever the initial point. This confirms the global convergence of the system, ensured by the proposed variable structure control law, and the good practical behavior of the control. A start-up test without any mechanical sensor with law [9.28] has been carried out on the experimental bench, in the same conditions as those of the test in Figure 9.9. PI regulators have been used to control i_δand i_γ(see Figure 9.2) with the reference currents set at i_δref = 0 and i_γref = 3 A. The sensorless control parameters are the same as above (b = 2 and Ç = 0.75) and the initial position is highly erroneous (φ₀ ≅ -3π/2 rd).

We notice in Figure 9.13 that, despite this important initial error, control law [9.28] manages to make the position error vanish, where proportional law [9.17] failed (see Figure 9.9). Indeed, the position error (φ) quickly converges to zero, and the d-q current components converge to the controlled currents (i_δ and i_γ). We also note that the motor torque (proportional to i_q) is first negative because of the important initial position error, which leads to starting up the machine in the wrong direction. But the torque and thus the speed are corrected as soon as φbecomes weak. To evaluate the performances during the torque inversion and thus the transit through Ω=0, another test is performed and the obtained results are illustrated in Figure 9.14. We notice that the position error remains small and that the machine easily changes rotation direction.

However, a small deviation from zero on ϕ is noticed when the speed and thus the back-EMF are low. This was expected, knowing that all the back-EMF based sensorless techniques have difficulties operating at standstill or at low speed. This is because of the fact that back-EMF is weak at low speed and consequently reliable information on mechanical signals is not available (see the remark on the observability of the system in section 9.2.1).

NOTE.– The machine is controlled in current (torque control) in the tests presented above. In [NAH 01c], the convergence of the sensorless control in presence of a speed control loop is analyzed. The block diagram of the closed-loop system is illustrated in Figure 9.15. It is shown that this new loop does not change the conclusions on the convergence of the proportional and variable structure control laws. Here, we will only present the experimental results when the initial position is highly wrong (φ₀ ≌ -4π/3 rd) and Ω_ref = 1,000 rpm and i_δref = 0. Figures 9.16 and 9.17 respectively show the results obtained with laws [9.17] (βin [9.23] and b = 2) and [9.28] (b = 2 and Ç = 0.75).

In the first case, the position error (φ) converges to -2ρ (with ρ = tan-¹(b)) and the angular speed Ωas well as Ω_s converge to a negative value. Consequently, the speed regulation error (ε_Ω= Ω_ref-Ω_s) is not cancelled and the integral action of the speed controller goes to saturation. This forces i_γrefat the saturation without being able however to correct the position error. On the other hand, the variable structure law [9.28] presents satisfactory results. The presence of a speed controller thus does not modify the results of the convergence analysis above.

These experimental results validate the convergence analysis and the machine modeling carried out for sensorless control presented in section 9.2. These results also confirm the good performances of the proposed variable structure control law, notably at start-up with any initial position error. However, good operating of the studied control laws depends on the quality of the estimation of the back-EMF vector components. This will be presented in the following section.

9.4. Estimation of the back-EMF vector

The δ-γ components of the back-EMF are not directly measurable. This is why we have to estimate them. There are different estimators of the back-EMF vector, among which we can distinguish several observation methods: linear observers [NAH 04], sliding mode observers [NAH 07], Kalman filter [ZED 07], etc. They are all based on a model representative of the machine and of the available electric measurements. The voltage equations of the machine in their different forms are the most frequently used models. Some authors use the model with the α-βcomponents of the electric signals (model [9.3]), whereas others prefer the model linked to the revolving reference frame (δ-γor d-q). The first has the advantage of being independent of any rotation, but its disadvantage lies in the alternating nature of the processed signals, deteriorating the estimation quality at high speed. Moreover, depending on the chosen sensorless control law, a rotation a posteriori is often necessary to find the δ-γcomponents of the back-EMF vector again, from the α-βcomponents. We also have to note that the use of the model linked to the d-q reference frame is not advisable, because it does not reflect the true machine behavior seen by the control, when the latter does not have direct access to the rotor position.

Here, we propose a linear observer based on model [9.4] linked to the δ-γreference frame, in order to estimate the back-EMF vector. Depending on the model, the δ-γcomponents of the back-EMF can be considered as slowly variable disturbances in the electric equations, which helps us to estimate them with the help of the following state and disturbance observer:

[9.29]

with and The estimated components of the back-EMF (ê_δand ê_γ) asymptotically converge on e_δand e_γ, in the absence of uncertainty in the model, its parameters and its measures. The observer gains are determined by a pole placement method. Indeed, the observer [9.29] leads to a linear dynamics of the estimation error given by the following relation:

[9.30]

with and The characteristic equation of this system is as follows:

[9.31]

We can then impose the estimation error cancellation dynamics with an appropriate choice of the coefficients k_d1, k_d2, k_q1 and k_q2. This dynamics should be sufficiently fast compared to the dynamics of the state variables, and it should be sufficiently slow in order to attenuate the measurement noise and rapid dynamics.

Whatever the method chosen for the estimation of the back-EMF vector, the estimated components risk being more or less precise, because of the uncertainties related to the model, its parameters and its sensors. In practice, the most common uncertainties in this field are related to the parameters of the machine electric model. These uncertainties and their consequences are studied in the following section. Interested readers can refer to [AKR 09] or [NAH 07] for other uncertainty sources.

9.5. Robustness of sensorless control of PMSM with respect to parameter uncertainties

In the following we will study the robustness of the proposed sensorless control method with respect to the machine parameters. Because control law [9.28] depends only on electric variables and parameters, it is very robust with respect to the uncertainties on the mechanical parameters. Consequently, we will only study the robustness with respect to the electric parameter uncertainties, which are:

[9.32]

where the index 0 indicates the nominal value of the parameter. Let us note that in practice, the lower and upper limits of the parameters are known. Even if the nominal model of the machine corresponds to a non-salient pole PMSM (L_d0=L_q0=L₀), L_q can be different from L_d in the general case (AL = L_q-L_d ≠ 0). The machine, supposed to be non-salient, can thus be more or less with salient poles. Taking into account the PMSM model with salient pole and assuming I_d = 0, we can show that the estimate of the δ-γback-EMF components is wrong; because of the parameter uncertainties (see [NAH 01a]):

[9.33]

To simplify the notation, we set:

[9.34]

Without loss of generalities, we assume that I_q is positive and that Ω_s is given by the following expression:

[9.35]

with:

From [9.33], [9.34] and [9.35], the control signal Ω_s and the motor torque Γ_m are written as follows:

[9.36]

[9.37]

with:

By replacing [9.36] and [9.37] in [9.15], the model of the mechanical sub-system is then described by the following equations:

[9.38a]

with:

[9.38b]

[9.38c]

In the following, we will study the influence of these parameter uncertainties defined in [9.32] on the performances of system [9.38].

9.5.1. Uncertainty on the stator inductances

Let us assume that and K_f are exactly known ( =0 and κ =1). Therefore:

[9.39a]

[9.39b]

The new stable equilibrium point of the system is thus shifted compared to the ideal case (φ= 0), as shown in Figure 9.18, for = 0.5L₀ and = 0.75L₀. From relation [9.39a], we can easily notice that this shift is vigorously due to the error on (D_q≠0). Consequently, only the uncertainty of leads to a static error on the rotor position.

The sensitivity of the system with respect to the uncertainty of L_d, is strongly reduced thanks to the choice of the reference current I_d = 0. The proposed sensorless method thus works well for salient pole PMSM, if the following condition is met [NAH 01a]:

[9.40]

where D_max is the maximal value of |D|. We must then choose b < b_max, where b_max is obtained from [9.40] for a given D_max. Figure 9.19 shows b_max as a function of D_max for c = 1.75 (worst case scenario).

However, we notice that the displacement of the desired equilibrium point due to the uncertainties on the stator inductances, remains rather limited thanks to the low value of these inductances in PMSM.

9.5.2. Uncertainty on the torque coefficient

Let us assume that the only uncertain parameter is K_f. In this case, = △L = ==0, h_R=D_q=D=0 and h(φ,ρ) becomes zero. The new expression of

g(φ,ρ) is:

[9.41]

Taking into account the fact that the equilibrium points are solutions of g(ϕ,ρ) = 0, the existence condition of the equilibrium points is as follows:

[9.42]

This condition determines a lower limit on b as a function of κ_max. Figure 9.20 shows b_min as a function of κ_max for c = 0.25 (worst case scenario).

For all these values of κ≠1, the new equilibrium point is shifted compared to φ= 0, as illustrated in Figure 9.21 for an overestimation of 20% on K_f.

9.5.3. Uncertainty on the stator resistance

Let us now assume that the resistance is the only uncertain parameter: ≠ 0, ΔL= ==0 and K=1. Consequently, h_R≠0, D = D_q = 0, and system [9.38] is described by the following equations:

[9.43a]

with:

[9.43b]

We can then notice that the dynamic behavior of the system at low speed, where lg(φ,ρ).Ω| remains lower than |h_R|, is completely disturbed. Indeed, as long as | h_R | > | g(φ,ρ).Ω|, the position error φ cannot settle down and continues to increase (or decrease) if h_R is negative (or positive). Consequently, the motor torque (K_fI_qcosφ>) oscillates with a null mean value, as well as the rotor speed. This can be interpreted by a closed orbit (limit cycle) on the surface of a cylindrical phase plane, as illustrated in Figure 9.23 [NAH 04].

In the following, we first seek a sufficient condition of the existence of this limit cycle, when the uncertainty of the resistance is important. Then, we deduce the necessary condition of operating at low speed with sensorless control, from the sufficient condition of the existence of the limit cycle.

Let us consider the following system:

[9.44]

Let us assume that f1(φ>,Ω) does not change sign in the domain E, representing the low speed region defined by:

[9.45]

where Ω_t is a threshold speed described as follows:

[9.46]

with g_m the absolute maximal value of g(φ,ρ):

[9.47]

If (φ(t),Ω(t))∈ for every t>t₀ and if there is no stable equilibrium point in , a stable limit cycle is then the only possible solution. Thus, all the system trajectories converge on a closed orbit on the cylindrical surface of the phase plane (Figure 9.23).

Let us consider the case of start-up. We want to know if the absolute value |Ω | can reach Ω_t during a “long enough time” T_t. If the answer is negative, the trajectory is absorbed by a closed orbit. T_t is the maximum time where the sign of the motor torque Γ_m=K_fI_q cosy does not change. In order to maximize T_t whereas Ω(0)=0<Ω_t, we take the case where |f₁|is minimum (to limit the dynamics of y) and |f₂|is maximum (to maximize the acceleration of Ω). To do so, we can write:

[9.48]

It is also necessary to note f_1min> 0 in . By replacing f₁ and f₂ in [9.44] respectively with f _1min and f_2max, we obtain the following system:

[9.49]

This system has the following solutions for t > 0:

[9.50]

If T_r is the necessary time for Ωto reach Ω_t, we have:

[9.51]

In the case where the motor torque Γ_m = K_fI_q cos φchanges sign before Ω reaches Ω_t , the speed will not be able to leave . It is thus sufficient that φ>(T_r) - φ(0) > π, so that the limit cycle appears. This gives:

[9.52]

This condition is sufficient for the existence of the limit cycle. By replacing h_R with I_q/K_f (relation [9.34]), we can deduce the necessary condition of sensorless low speed operating in the presence of the stator resistance uncertainty:

[9.53]

We need to note that this condition is necessary but not sufficient. Thus, in practice, to have a good low speed operation, || must be much lower than _max. Figures 9.24 and 9.25 show the trajectories of system [9.43], respectively with h_R = -22 and h_R = -56 when h_max = 50. As we can notice in Figure 9.24, the sensorless control works relatively well thanks to |h_R|<h_max, and the system trajectories end at an equilibrium point close to the ideal equilibrium point, with a small steady state error on the rotor position. In the second case where |h_R| > h_max (Figure 9.25), the low speed operating of the system (at start-up or during the speed inversion) is completely disturbed and the sensorless control fails. We can notice the divergence of the position error φand the oscillation of the angular speed Ω.

NOTE.– The problem of the appearance of the limit cycle is present in all the methods that can be reformulated by equations [9.43] in the presence of an uncertainty on the stator resistance (law [9.17] for example). In that case, the necessary condition [9.53] on the maximal acceptable error in the stator resistance must be respected. However, the limit cycle disappears in the methods where the sensorless control law contains an integrator on ê_δ[MAT 96]. In that case, the dynamics of the position error is governed by:

[9.54]

The last term, representing the integral action on ê_δ, compensates for the term h(φ,ρ) little by little, hence the disappearance of the limit cycle. However, in the case where condition [9.53] is not satisfied, the system starts up correctly only after a suitable compensation of h(φ>,ρ). And yet, because ê_δand thus its integral are weak at low speed, the compensation of h(φ>,ρ) is not immediate and the angular speed oscillates at low speed, as shown by the phase plane in Figure 9.26. Let us note that the counterpart of this advantage lies in the fact that an undesired stable equilibrium point (p₄) reappears because of the integral term in the sensorless control law (see Figure 9.25). So, the convergence to the desired solution (p₃) is no longer global [NAH 03].

In order to verify the validity of the obtained analytical results, we did a series of tests in the case of sensorless control with uncertain resistance. A start-up test from an initial position error of-4π/3 rd is performed when the resistance is underestimated by 20%. Taking into account the machine parameters and those chosen for the control _max = 0.45R_s0 (||< _max ). Figure 9.26 shows the results of this simulation. As we can see, the motor torque is well controlled and the speed reaches its reference. However, because of the error on the resistance, a static error appears on the estimated position (φ_e ≠ 0).

The results of the same test in the case where the error on the resistance is -50% (|| > _max) are shown in Figure 9.27. We notice that the start-up attempt fails and a limit cycle appears, as expected. The position error φcontinually increases (h_R<0) and the angular speed oscillates. The control system losses control the motor torque, and the estimation of the δ-γcomponents of the back-EMF is entirely incorrect.

To validate these simulation results, we have carried out two series of experiments using the simulation conditions previously quoted. The first series (Figure 9.28) concerns the start-up with the sensorless control for a resistance error much lower than the authorized limit . The second series (Figure 9.29) highlights the appearance of the limit cycle when the uncertainty on the resistance exceeds its limit .

NOTE.- Taking into account the study above, we can conclude that the uncertainty on the stator resistance is the most harmful for sensorless control of PMSMs. Two solutions can be considered: reduce this uncertainty by estimating the stator resistance or make the sensorless control unaffected by this uncertainty. These solutions are discussed in the following section.

NOTE.- The measurements are assumed to be correct in this chapter. The influence of the measurement error of the DC link voltage (if applied) and of the currents on the performances of the sensorless control, has been studied in [NAH 07].

9.6. Sensorless control of PMSMs in the presence of uncertainties on the resistance

Despite the simplicity and the efficiency of the sensorless methods based on the back-EMF estimation, their robustness at low speed with respect to the uncertainty on the stator resistance is not satisfactory. Indeed, we have demonstrated that the uncertainty on the stator resistance must meet the condition [9.53], in order to operate at low speed. According to this condition, _max depends on the machine parameters (K_f, P and J), on the maximal value of the current module (I_{s max}) and on the sensorless control parameters, appearing in the expression of g_m (relation [9.47]). Evidently, for the safe operation of sensorless control at low speed, it is necessary for _max to be as large as possible. But in practice, we cannot freely increase _max. It is thus desirable to implement, either an “on-line” estimator of the stator resistance that maintains close to zero [NAH 04], or a control approach helping us to minimize the effect of on the sensorless method.

9.6.1. Online estimation of the resistance

The proposed identification method is based on the cancellation of the error between the effective output and the output estimated from a reference model. It consists of estimating a current, called i_γe, from the following model, using the estimated resistance :

[9.55]

We notice that this is the model of the machine, when the δ-γ reference frame converges to the d-q reference frame (φ = 2kπ). In this model, we assume that the “off-line” identification of K_f and L_s is sufficiently precise. In practice, the inductance can vary with the machine saturation, but we have shown in the previous section, that the uncertainty on the inductance leads only to a limited error on the estimated position. We also assume that the measured currents do not include any uncertainties (offset or gain) and that the control voltages are correctly applied to the machine. The impact of the control voltages was studied in [NAH 07].

With the above hypotheses, it is clear that the error between the estimated current i_γe and the measured current i_γ is inevitably due to the error on the estimated resistance.

We thus propose the following integral estimator to correct :

[9.56]

where is the estimation error of the current i_γ. The η coefficient must thus be defined so that the estimation error of the resistance = - converges to zero. Because the variations of are slow, we choose η, so that the dynamic of is slow. The estimation error of the resistance is thus part of the slow variables of the system (ϕ and Ω in the model [9.12]). From relations [9.43], [9.55] and [9.56], the new model of the slow sub-system with the estimator of the resistance is written as follows:

[9.57]

This system has the four equilibrium points:

[9.58]

with Ω⁺ =Γ_ch^-1(K_fI_s) and Ω^- =Γ_ch^-1(-K_fI_s). The study of the local stability of system [9.57], around the desired equilibrium point p₃ gives the following sufficient conditions:

[9.59]

We notice that the first two conditions are the same as in relation [9.29] for the convergence on the position error, and the third condition guarantees the convergence of the resistance estimation error. We set out:

[9.60]

With the help of Lyapunov's first method, we can demonstrate that the only stable equilibrium point of the system is the one desired (p₃) with the choice of [9.59]. Let us note that this result is valid even if Ç = 0, i.e. for the proportional sensorless control law (section 9.3.1). Figure 9.31 shows the trajectories of the system [9.57] for b = 2, Ç= 0.75, η₀ = 5 and Ω⁺ = -Ω2^-=100 rd/s. The initial error on the estimated resistance is -50% ( R_s=-0.40 Ω). The same trajectories on the space φ>-Ω-R_s are illustrated in Figure 9.31.

As we can notice, all the trajectories of the system converge on p₃, whatever the initial point. These results show the efficiency of the sensorless control with the proposed resistance estimator. Indeed, without this estimator and with the same initial error on the resistance, the sensorless control could not operate at low speed (Figure 9.24).

The experimental results, illustrated in Figure 9.32, show the starting of the machine when the initial estimated resistance is inevitably incorrect, as in Figure 9.29.

We notice that after a few flickers, the machine successfully starts up and the estimated resistance is slowly corrected (depending on the adjustment carried out for η_o), before reaching its final value that can seem very high (≅10Ω).

This value is explained by the fact that the resistance seen by the control is not only the stator resistance, but also the set of all the resistances due to the connector technologies, to the cables and above all to the voltage drops on the semiconductor components of the voltage source inverter supplying the machine. They are particularly non-negligible at low voltage. It is thus not surprising to obtain such a high estimated resistance, because it leads to a good estimation of the back-EMF, and thus helps to have a successful sensorless start-up of the machine.

NOTE.- The analysis carried out on the sensorless control with the resistance estimator neglects the error on other parameters of the machine. In practice, all the uncertainties (of measurements or model) result in a voltage error that will be “absorbed” by the estimated resistance.

9.6.2. Minimization of the sensitivity of the sensorless control with respect to the resistance

In order to obtain a good functioning of the sensorless control in presence of uncertainties on the resistance, we present another solution in this section.

It consists of modifying the reference of the direct component of the current vector (I_d), in order to minimize the sensitivity of the sensorless control to the resistance uncertainties. We take again model [9.15] with I_d≠0 and we obtain the new estimated components of the back-EMF vector for non-salient pole sinusoidal PMSMs with parameter uncertainties:

[9.61]

In this case, the evolution of the position error (ϕ) with the control law [9.28] is described by the following equation:

[9.62a]

with:

[9.62b]

[9.62c]

[9.62d]

[9.62e]

As in section 9.5, we notice that the dominant uncertainty at low speed and at standstill in equation [9.62] is the term related to the error on the resistance (h_R).

This is all the more awkward as during start-up or the speed inversion, the equivalent resistance of the system seen by the control can strongly vary, as previously noticed.

However, the expression of h_R given in [9.62c] offers the possibility of canceling this term, independently of the value of R, by choosing:

[9.63]

or under another form:

[9.64]

This relation shows that the choice of an I_d proportional to I_q is really interesting for the good operation of sensorless control at low speed. Moreover, the coefficient of this proportional link (tan ρ =b ^.c) is perfectly known by the control and does not have any uncertainty. Figure 9.33 illustrates the (φ-Ω phase plane of the system with a strong uncertainty on the resistance (the same than in Figure 9.24), when I_d is set according to [9.63].

As we can notice, the limit cycle (see Figure 9.24) has disappeared and all the system trajectories converge on the desired equilibrium point (p₃). Figure 9.34 shows the block diagram of the system with a variable I_d.

However, let us note two important elements:

- for ê_δor ê_γclose to zero, I_d is subjected to strong fluctuations because of the sign function [9.62e]. Then, the i_δcurrent will try to follow these fluctuations with the dynamics imposed by the current regulator. This will disturb the motor torque according to [9.11], as long as φ ≠ kπ. It is thus desirable for I_d not to depend on sgn(ê_s) and sgn(ê_γ), which leads us to average these terms. Consequently, imposing an average I in practice could be sufficient, i.e. :

[9.65]

- for a non-salient pole PMSM, the direct current component only gives losses. This is why its reference is generally fixed at zero. And yet, with the method proposed here, I_d is different from zero and we have additional losses. But those do not disappear with I_d = 0, because of the steady state error on φ, which will imply a stator current of higher amplitude, in order to obtain the same torque and thus to reach the same functioning point. This steady state error on the estimated position (φ_e) depends of course on , but also on b, on the torque and on the speed at the operating point. We can show that if the speed is important enough, so that:

[9.66]

the losses with I_d = 0 will be lower than those with I_d given in [9.65]. In that case, it is better to switch at I_d = 0, in order to minimize the losses. We can also plan to use an on-line resistance estimator (see section 9.6.1) to decrease and thus φ_e and the losses.

Figure 9.35 shows the start-up test with variable I_d (equation [9.65]), when the resistance is highly erroneous (||> _max , see Figure 9.29). The control parameters are the same as previously (b = 2, ζ = 0.75). We notice that, despite an important initial error on the estimated position (ϕ₀ ≅ -3π/2 rd), the sensorless control corrects the estimated position, and this in presence of an important uncertainty on the resistance. We can easily show that the speed regulation does not modify this result.

9.7. Conclusion

In this chapter, we first presented a model for sensorless control of non-salient pole sinusoidal PMSMs. Then, we showed that the problem of sensorless control can be interpreted as a simple regulation problem at zero of a state variable (the position error in our specific case): a sensorless control method is thus nothing more than a control law regulating this state variable. The behavior of the closed-loop system thus depends on the sensorless control law, used when the model, and thus the back-EMF estimation are assumed to be precise. By applying the method of singular perturbations, the developed model has been broken down into two models of reduced order, corresponding to two electrical and mechanical sub-systems; thus facilitating the convergence analysis of the existing sensorless control laws. It has been shown that even in the case of a perfect estimation of the back-EMF vector, the convergence domain of the sensorless controls could be limited, because of the existence of several stable equilibrium points that can prevent the convergence on the desired equilibrium point (zero position error).

Then, a variable structure control law, enabling the “global convergence” has been analyzed and its characteristics have been particularly studied in the presence of model parameter uncertainties. At first, the influence of the parameter uncertainties on the estimated back-EMF components was evaluated. Taking into account the fact that these estimations are used by the sensorless control law, their uncertainties disturb the behavior of the closed-loop system. We analyzed it in presence of parameter uncertainties and showed that the uncertainty on the stator resistance has the most important influence on the system operating at low speed and at start-up. Indeed, a limit cycle appears when the stator resistance is too erroneous. Thus, the sufficient condition of the existence of this limit cycle has been given and checked by the study of the state trajectories of the system in the phase plane. The experimental and simulated results have confirmed the obtained analytical results.

In order to make the limit cycle disappear, two solutions have been proposed: the “online” identification of the resistance and the modification of the direct axis current (variable Id). The first helps to reduce the uncertainty of the resistance and leads to a relatively robust sensorless control with respect to the parameter uncertainties. The second modifies the control strategy of the stator currents, in order to minimize the sensitivity of the sensorless control with respect to the uncertainties on the resistance. The presented experimental results have shown that the second is more efficient, but increases the losses in the machine. Ideally, we should use the two solutions at the same time, in order to take advantage of each of them.

9.8. Appendix 1

The machine used is a non-salient pole sinusoidal PMSM, with the following main characteristics:

Rated power: 2 kW

Rated voltage: 200 V

Stator resistance: 0.8 Ω

Direct inductance: 10 mH

Moment of inertia: 3×10^-3 kg.m²

Viscous friction coefficient: 24×10^-3 kg.m²/s

Rated speed: 4000 tr/min

Nominal current: 5 A

Number of poles pairs: 7

Inductance in quadrature: 10 mH

9.9. Appendix 2

In the case of a salient pole PMSM ( ≠ ) controlled without a mechanical sensor, the electric model of the machine is as follows [NAH 01c]:

Taking into account the hypothesis on the separation of the time scales (section 9.2.2) and of equation [9.18], this model is simplified in electric steady state in the following equations:

The voltages v_δ and v_γ play a crucial role in the estimation of the δ-γ components of the back-EMF. In the presence of parameter uncertainties, the estimated back-EMF components are as follows:

We use these expressions in section 9.5, in order to analyze the sensitivity of sensorless control methods.

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1 Chapter written by Farid MEIBODY-TABAR and Babak NAHID-MOBARAKEH.