Chapter 4

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Drive Controls with Synchronous Motors ¹

4.1. Introduction

The speed (and/or position) control of an electric motor is often the ultimate aim of its control. The industrial applications of this device are countless: for machine tools, for robots and for special machines. Each time a production machine works, the motor is fulfilling this task. We talk about an “axis” for these applications, because a robot has “six axes”, when it can set a tool in a certain point of the space (three dimensions) with a certain direction (three other dimensions). One motor per axis is necessary. A special machine can gather tens of axes, and a production factory, hundreds or even thousands: each operation has its axis. The motors in charge of this operation must quickly and efficiently move the tools. Figure 4.1 shows an example of a “multi-axis” control system. In this chapter, we will mainly take a look at the speed control, with a short presentation of the position control.

Indeed, the industry of equipment goods‘ production supplies “electronic speed variators”, comprising the motor, its supply (the static converter) and its speed control: traditionally, with the help of a high level decision-making element, the user supplies the speed reference, frequently coming from a position control.

The speed control of an electric motor is located at the border of several domains: electricity (even better, electronics), mechanics, automatics. Figure 4.2 gives the schematic block diagram of the speed control of an electric motor (here with AC, hence the self-control presence), but a control by a DC motor would be similar.

The load (the carried mechanical part) can be simple, purely inertial, with for example a constant load torque. This is usually the case considered by classical books and this will also be the case here. However, the users often encounter much more complex loads. We will only quote two really representative loads:

– a mechanical load with variable inertia, as robots or recoilers-spoolers;

– a load with elastic links and oscillating modes, such as rolling mills.

But with these examples, we leave here the domains related to the electric motor‘s control, seen by the power electrical engineering specialists who estimate having fulfilled their task when they have carried out good torque control. The “axis control” in the more complex cases that we just quoted comes within the field of general control engineering science, applied to complex mechanical systems. A book of the present treatise considers these questions, [HUS 09].

We will thus only consider a few simple cases, in order to examine various problems of the speed control of a synchronous motor directed on the obtained performances with the two main torque controls already studied in this book: control in the a-b-c reference frame (Chapter 2) and control in the d-q reference frame (Chapter 3).

We start from an already carried out torque control (for example by the methods seen previously) and thus with a “decoupling” between the d and q axes. Let us recall that this decoupling is often not very efficient in a-b-c and that it can be very powerful in d-q. We will propose a speed control approach (P, IP controllers and introduction to the load observers) and we will examine some robustness aspects.

The reader should know that a complete study of these problems would require several complete books specialized on these questions. These problems have been the subject of intense work by the entire scientific and technical community from the years 1985-2000 ([GRA 86, BOS 86, BERG 87, CHA 88, BUY 89, FAD 89, LEO 90, LEP 90, LEP 91, LOU 92, LOU 95, ROB 95, HAU 97, GRE 97, LOU 99, STU 00]), from which the control structures summarized here were created.

The control has an outside environment: the speed references are generated by a “high level” calculating and decision-making unit that takes into account the characteristics of the production system. This field is the mechanical engineering field, with knowledge of the tools‘ trajectories that would give the position references θ . The motor drives the mechanical loads that can have very different characteristics. We will only study here the simple, but generic, case previously defined. Indeed, in this case, the mechanical load is described by an inertia, a viscous friction coefficient and a constant load torque, as seen in equation [4.1]:

[4.1]

The speed controller aims to supply the desired value for the electromagnetic torque. This will be the torque reference, whose value of currents for the motor is to be deduced; this is the role of the studies on torque controls. Therefore, we have devoted two chapters to the supply determination, either in currents in the natural a- b-c reference frame (Chapter 2), or in currents in the d-q reference frame (Chapter 3).

This chapter thus assumes knowledge of the supply of the synchronous motors studied in Chapters 2 and 3. In these chapters, we are at the core of the subject of electric motor control.

4.2. Principles adopted for speed controls: case of IP controllers

We will examine the properties of the speed controls in two main cases: when the torque control is implanted in the a-b-c reference frame (as seen in Chapter 2) and when it is implanted in the d-q reference frame (as seen in Chapter 3). We will adopt a not very detailed common approach (there are indeed too many details to be able to discuss them in this chapter).

For example, in order not to weigh down the discussion, we will assume that the current dynamics are very fast compared to the speed dynamics. This property must be concretely verified in each specific case. If we impose very fast speed dynamics (case of the motors leading to a very low inertia), whose order of magnitude is comparable to the current dynamics, we should examine the torque effect between these dynamics (because it is destabilizing, it creates oscillations, etc.).

With the hypothesis of the very fast current dynamics, the current noted I_x very quickly reaches its reference value, noted I_{x _ ref} in this general presentation. The electromagnetic torque appearing in [4.1] verifies an equation of the form:

[4.2]

where the coefficients and variables noted K_x and I_x depend on the case discussed. A typical control example is given in Figure 4.3: this is a representative speed control with an integral-proportional (IP) controller, completed by an anti-windup device [AST 97], used to compensate for the effects due to the current limit. This limit is necessary for safety reasons: we need to limit the current amplitude to avoid exceeding the supply capacities, to avoid converter and the motor accidents and the accidents relative to overcurrents (see section 4.4.5).

The functional diagram, representing the design model of the regulation is given in Figure 4.4.

We write the equations of the proportional and integral loops of the device:

[4.3]

where:

[4.4]

with:

[4.5]

and:

[4.6]

We can deduce, from the previous equations (or from the diagram in Figure 4.4), the transfer functions (in feedback and in regulation) in closed-loop limited to the only dynamics of the slow mechanical variables:

[4.7]

where we have set out:

[4.8]

The two parameters intervening in the denominator in [4.7] (the time-constants Ƭ_yΩ and Ƭ_bΩx ) are adjustable, in particular by the gain (see [4.11]). We can thus choose them to impose the dynamics.

We adopt the following adjustment method: first, we choose a “reference time- constant”, which will be the electromechanical time-constant of the motor in open- loop, determined from the model in the d-q reference frame, similar to the model of the DC machine:

[4.9]

Then, we choose to impose on the closed-loop system two equal time-constants. This is the “critical damping”. We note this time-constant Ƭ_m :

[4.10]

It is then sufficient to choose the single parameter λ_n. It is necessary to take a high parameter to impose fast dynamics (but with high current peaks), or a low parameter to limit the dynamics and the current peaks. Let us note that λ_n=1 is a “medium” value. Indeed, we neglect the viscous friction coefficient (we then set out f= 0) and the adjustment (the controller design) is given by:

[4.11]

In the following sections, we will see how the controls behave, according to whether the design occurs in the a-b-c reference frame (where we use several almost mandatory simplifications) or in the d-q reference frame (where we could avoid these approximations).

4.3. Speed controls designed in the a-b-c reference frame (application to a non- salient pole machine)

4.3.1. General information

The controls in the a-b-c reference frame are very widespread. Chapter 2 has shown that their precise models are complex and that their performances are limited, but that the simplicity of the implementations makes them attractive. We can consider that they are interesting when simplifying hypotheses are acceptable: for example, non-salient pole machine, very fast dynamics of the electrical variables compared to the dynamics of the mechanical variables.

In this account, limited to the essential properties, we assume in addition that the f coefficient is very small. The digital machine values were given in Chapter 1 (appendix section 1.7.1). As we criticize the controls designed in the a-b-c reference frame several times, let us indicate one advantage: we regulate the currents measured (and not mathematical reconstructions), which is an advantage from the safety point of view (protection against the current peaks in case of an unexpected accident). In some applications, this solution must be absolutely necessary.

4.3.2. IP speed controller with an IP current controller in the a-b-c reference frame

The hypothesis modeled by [4.2] is particularly well-adapted to control in the a-b-c reference frame. Indeed, because modeling in the a-b-c plane is complicated, it is indeed practical to admit the fast currents hypothesis. Thus, for the speed controller design, we assume that there is a perfect decoupling between the speed dynamics, assumed to be slow, and the currents dynamics, assumed to be infinitely fast. Then, we can adapt model [4.2] to the a-b-c case:

[4.12]

thus:

[4.13]

Figure 4.3 has given a typical example of speed control with an IP controller, complemented by an anti-windup device. In this section, we will only examine the essential properties and we will only consider an IP speed controller without any anti-windup device, with “reasonably fast” speed dynamics, given by λ_Ω= 2.5 (see [4.10]).

We will consider two cases: a case where the IP current loops are “slow”, with a “small gain” on the current loops (let us specify that this adjustment is not randomly chosen and that it is an adjustment that would be quite suitable when it is used for control in the d-q reference frame), and a case where current loops are “fast” (with “large gains”).

We can observe the speed response at two intervals: there is first a “large movement” defined by a start at zero speed up to 90% of the rated speed; then, at t= 1 s , there is a “small movement”, from 90 to 100% of the rated speed.

Figure 4.5 gives an example of the speed response with slow current loops. We knew that these small gain loops (see Chapter 2, section 2.4.2.2) had a mediocre behavior, with a strong static error, as clearly observed on the curves. The device is unable to regulate the speed (top curves), because it cannot supply the requested torque (bottom curves).

We can however observe that the performances are not perfect, as seen on the electromagnetic torque during the interval in which the torque reference limit is active (necessary constraint to impose current limits for the device safety, when the desired torque is higher than the maximal authorized torque C_M ). The effectively obtained torque has a static error increasing with the speed. The speed loop corrects this error (thanks to its integral effect), by increasing the torque reference, but this correction is done with slower dynamics. We observe that the integrating circuit drifts during the torque limit, which leads to an important speed overshoot (top curves). The anti-windup device role is to eliminate this effect, but we will only study it in the case of the regulations designed in the d-q reference frame (see section 4.4.5).

The dynamic performances are thus worse than with a control structure actually knowing how to quickly eliminate static errors. In addition, this “large gain” solution can be non-implantable for technological reasons. This solution (control in the a-b-c reference frame) was the first to be established because of its simplicity and it has really helped. But, when it was time to increase the performances, thanks to the improvement of the digital components performances and the power electronics, several improvements were proposed: “advanced” current controllers in the a-b-c reference frame, or control structures in the d-q reference frame. We will devote a brief section to the examination of the improvements brought about by an advanced controller, the resonant controller, and we will then thoroughly study controls in the d-q reference frame.

4.3.3. IP speed controller with a resonant current controller

We will consider the same speed control as in the previous section, the current loops being regulated by the resonant controllers (see Chapter 2, section 2.4.4). Figure 4.7 confirms what was stated at the end of the previous section: the use of resonant controllers (being naturally of infinite gain in sinusoidal steady state) eliminates the static torque error with the (fast) current loop dynamics. This is clear, on the bottom curves, during the interval where the torque is limited to the C_M value: the electromagnetic torque is indeed maintained at this value, without any drop. Therefore, the transient in speed is faster than with the IP current controllers. We note that there is, once again, a speeding that could be eliminated by an anti- windup effect. Figure 4.8 gives an enlargement of the phase current a, compared to its reference, on both sides of the instant when the current stops being limited to a maximum amplitude (concretely: five times the efficient nominal amplitude); up to t = 0.37 s, the current is lower than the ideal reference and then follows the reference exactly. The reference is always (in this example) in phase with the flux derivative, created by excitation (noted ψ'_af or ψ_praf ). We thus find the optimality criterion again, see Chapter 2.

This solution (resonant controller) and others (regulation by hysteresis) are competing with a different approach that we will now examine: controls designed in the d-q reference frame.

4.4. Determination of the speed controls designed in the d-q reference frame (application to a salient pole machine)

4.4.1. General information

A priori, the controls in the d-q reference frame have the best dynamic performances, thanks to the torque controls laws presented in Chapter 3. They can be applied to salient pole synchronous machines, which we will indeed do in the following examples. The machine that will be used for our examples has numerical values compatible to those of the non-salient pole machine, that was used for the examples of the previous section (section 4.3); the numerical values were given in the appendix section of Chapter 1 (section 1.7.1). We will examine the main properties of these controls in the generic example that we have already defined (a mainly inertial load). In the d-q reference frame, the equations are apparently simple. This simplicity is formal because we had to establish current controls, which are relatively complex to carry out (self-control and state reconstruction). But this design leads to a formalism that is simple for the modeling of speed control and sophisticated and efficient laws. We will only mention a few of them, in a voluntarily simplified framework: input-output linearization, load torque observation, a robustness example. The control in d-q will lead to discussions, that would have been unnecessarily complicated to do in the case of controls in the a-b-c reference frame.

4.4.2. Introductory example: speed control with compensation or decoupling

To determine a simple control, we adopt the “decoupling” hypothesis between the “electric dynamics”, assumed to be fast, and the “mechanical dynamics” assumed to be slow. Thus, to design a speed controller, we assume that the current loops are very fast. From the speed loop point of view, the current instantaneously reaches its steady state. Thus, the dynamic equations of the regulated current are approximated by:

[4.14]

These equations replace [4.2] and we have:

[4.15]

In this account, we have chosen the “simple” version for the current reference of the d axis: I_{d _ ref} = 0 , because it authorizes comparisons with the different approaches, including with those concerning the non-salient pole machines (see the control in the a-b-c reference frame, in section 4.3).The control model comes from the dynamic model limited to the “mechanical part” of the motor (see [4.1] and [4.12]) and adapted to our discussion:

[4.16]

with:

[4.17]

This writing details the questions often implied in the conventional accounts, but we would like to clarify them here. The indices “est” recall that we can only use estimated parameters in the controls, that the index “ref” specifies that we seek to determine the references for the regulations; the index “des” indicates the desired values, that we wish to impose on the motor and that will give, in practice, the reference values.

An important remark: in this account, we strictly distinguish the “control model” and the “dynamic model”. The control model is only used to design the controllers and can admit approximations (see the simplified equations such as [4.14]). Thus, in

these controls in d-q, we assume that the current regulations are the best regulations seen in Chapter 3 and we ignore the current of the d axis. Therefore, for the design of the speed controller, we assume:

[4.18]

[4.19]

However, the dynamic model (detailed in Chapter 1) takes into account all the transients: i_d is not equal to zero and the torque is given by:

[4.20]

It is necessary to note that, the current control being very efficient in the examples presented below (and determined with the complete dynamic model), the d axis current seems always equal to zero on the curves that we examine. An approach for an efficient control is given by the “input-output linearization with state feedback” ([LEP 90, LEP 91]). In this approach, we impose that the speed must have an arbitrary, but physically feasible dynamics. Since we admit the 1^st order model [4.16], we can impose on the system to also have 1^st order dynamics defined by [4.21]:

[4.21]

where Ƭ_2Ω is chosen by the designer. The control law is immediately deduced from [4.16] and [4.17]:

[4.22]

We observe that it is a proportional controller (term in (Ω_ref -Ω)) associated with compensation terms (terms in Ωand C_{ch_est}(t)). We immediately notice that the load torque C_{ch_est} (t) is, by nature, imposed by the environment and that it is rarely known. The control law [4.22] is thus generally impossible to carry out under this form. A classical solution consists of ignoring this term, thus assumed to be equal to zero:

[4.23]

The elimination of the effect of this “disturbance” is then entrusted to an integrating effect in the controller, hence the popular success of the PI controller. We will examine its variant, called IP which has two advantages, from our point of view:

– this controller does not introduce a zero in the transfer function, and there is thus no reason to worry about the overshoots introduced by the zero: the dynamic performances are only imposed by the poles of the transfer function in closed loop, poles that can be imposed;

– and it can be seen as an extension of the previously presented “input-output linearization” method: we write two equations that we associate with [4.16], by taking into account [4.23]. The first equation [4.24] defines the desired dynamics of the speed, by analogy with [4.21], where the term y_Ω plays the role of the speed reference:

[4.24]

The other equation is known: [4.6] models the integral effect of the controller and defines the dynamics of y_Ω , as previously shown in section 4.2.

Figure 4.9a defines the speed control structure with the help of the “2^nd order control model” (thus voluntarily simplified: we ignore the current dynamics). We can observe in particular the control equation:

[4.25]

We find again the transfer function in closed-loop (already written in [4.7]), but limited to the only speed reference:

[4.26]

We observe that this transfer function is imposed by the designer; therefore this function is exact and it is useless to set out f = 0 . The effect of f is compensated for by the control (see [4.27] and Figure 4.9a), which is physically feasible, since we measure the speed. The control law (determination of the reference current in the q axis) is limited to:

[4.27]

There is no longer any need to know an estimate of the load torque. The integral variable y (given by [4.6]) will be responsible for generating the compensation of all the unknown disturbances: first, the disturbance created by the load torque, but also the disturbances created by a bad knowledge of all the parameters intervening in the models (in particular in [4.27]).

Then, we choose the poles of the transfer function. For example, we impose that the system has two equal time-constant, that we note Ƭ_mq (“critical aperiodic” case).

Thus, we must have:

[4.28]

from which we find the design rules of the IP controller:

[4.29]

[4.30]

4.4.3. Discussion on the speed controls

The choice by the designer of the Ƭ_mq time-constant imposing the dynamics is arbitrary. We propose a “robust” choice: Ƭ_mq must be small enough, but not too small, so that the design respects various constraints (also see section 4.4.4):

– Ƭ_mq small enough to impose fast enough dynamics, since the rapidity is a quality criterion of the motors used in industries where productivity is essential;

– but not too small, because too fast dynamics could “excite” the hidden dynamics, in particular those of the current loops that we neglected in the design.

However, this last limit can be overcome by resuming the study with a complete model and by imposing the position of all the poles, which is physically possible, since all the state variables are measurable (or known) and there are enough adjustment parameters to adjust all the poles. Thus, if we apply the adjustment method by compensation, Chapter 3 has shown that we can impose on the current of the q axis, an arbitrary 2^nd order dynamics:

[4.31]

where the time-constants ( Ƭ_yq and Ƭ_aq ) are chosen by the designer. Figure 4.9b gives the 4^th order control model, relative to the q axis; this model takes into account the current dynamics.

Then, the transfer function in speed is given by equations [4.1], [4.6] and [4.27], from which we deduce the complete expression of the transfer functions of feedback and regulation:

[4.32]

where Ƭ_μ and the denominator coefficients D(p) have as expressions:

[4.33]

[4.34]

[4.35]

and the relation between the time-constant Ƭ_2Ω and the gain k_Ω (see [4.25] and [4.24]):

[4.36]

This model can be used in several ways:

– by setting out p = 0, we can verify that the static error (on constant entries) is equal to zero, even if the load torque is unknown (but slowly variable, as it is conventional), and also, even if the model is inaccurate (the estimated values of K_est , f_est and J_est can be different from the instantaneous values: K, f and J); in a similar way, the transfer function of the current loop can be inaccurate. This is a conventional property due to the integrating effect;

– the complete model [4.32]-[4.35] helps to reach the 4 poles of the system (roots of the characteristic equation: D(p)=0 ):

-  either by strictly imposing them, since they are all adjustable (in particular see Chapter 3 for the adjustment of the current loops),

-  or by verifying, after a rough adjustment (such as the one presented here), that the system is sufficiently stable and has a response time compatible with the specifications;

– it also verifies the robustness of the system: we can compute the value of the poles, when the estimate values of the parameters move away from the exact values. We will not develop these questions, because there are too many elements to examine for a reasonable length account. However, we see that we have indeed the suitable tool to do these studies;

– finally, the adjustment depends on several “dimensioning” parameters. Therefore, the discussion quickly becomes complex, if we examine the effect of each of them. Thus, if the dynamics is very fast, during a high speed transient desired by the speed reference, the control will impose a high amplitude torque and thus will generate a very high q axis current, and the Joule losses will be high. The high speed fluctuations are thus potentially dangerous. The classical solution consists of putting limits on the current references. We will distinguish I '_{q _ ref} , the “desired reference” and I_{q _ ref} , the reference actually applied to the input of the regulation and which takes into account the authorized current maximums for the motor safety. The “sat” function ([4.37]) is clarified by Figure 4.10 and by the constraint [4.39]:

[4.37]

There is thus a compromise to be found between the dynamic performances, because the rapidity requires high currents, and the security requires current limits. The dimensioning problems interfere with the regulation problems. They are thus very complex and relative to each application: dimensioning of the motor, of its supply, of its mechanical load and of the functioning cycles [LOU 09]. For the recognition of all these parameters, the industrial designers of the automated systems exploit specialized software. We will examine some aspects of these questions.

4.4.4. Examples of regulation choices. The interest of an IP controller: its limits

The results, as they appear in [4.29], leave the choice of the ?_mq time-constant free: if it is small, the system will be fast, but the desired current will be high (and thus dangerous); and vice versa, if ?_mq is large, the desired current will be low (safe), but the system will be of course slow. The saturation of the current amplitude (Figure 4.10) is the one protecting the system, at the expense of the dynamics. To quantify this remark, (what does “large” or “small” mean?), we take as a “reference value”: ?_em , the “electromechanical time-constant”¹ coming from the DC motor (whose armature is very similar to the “q axis” of the synchronous motor) and whose expression is given by [4.9]. We will first examine the motor behavior for a “small time-constant” ?_mq and a reasonable value of the maximal current I_{q _ ref} :

[4.38]

[4.39]

where I_qN is the rated value of the q axis current. We assume that the thermal

inertia is enough to protect the motor during current peak periods at I_{q_max} = 5 . I_qN ,

periods that are “not too long” during large transients (but the duration of a large transient is shorter when I_{q_max} is larger; therefore, the global optimization is

complex).

In the following sections, we will examine several transient types: first, a setting speed at no-load with a large step (from 0 to 90% of the rated speed Ω_N), then, a small step (from 90% to 100% of the rated speedΩ_N), and finally a phase loading with a step of the load torque, the latter being equal to the nominal torque C_N.

Figure 4.11 shows the corresponding transients, in reduced values (in particular the currents are compared to the rated current (I_N = I_qN)). The system behavior is

mainly determined by the maximum values accepted by the current: it is constant during large intervals and thus the speed fluctuations are (approximately) formed of line segments. But, from the point of view of the speed controller containing an integrating circuit, the durations of the saturation transfers are very long; on these intervals, the current actually applied is much lower than the desired current (by

1. Often called the “mechanical time-constant” in many technical documents.

examining the reduced values I_{q_ref} I_N : they are very high). These durations are seen as delays and the “integrating circuit integrates too much”, which is destabilizing: it is thus logical to observe speed oscillations of high amplitudes.

4.4.5. Examples of the regulation choices: IP controller with an anti-windup device

A classical solution to this well-known problem consists of limiting the integrating effect of the speed controller, thanks to an anti-windup device. Anti- windup devices present numerous variants. A classic example – adapted by us to the IP integrating circuit case – is given in Figure 4.12, where we observe that we inject, again at the input of the integrating circuit, a z signal proportional to the difference between the desired current reference and the reference applied, this has a stabilizing effect.

In our diagram, the generated signal has the dimension of a speed, which helps to define an λ_AW adjustment parameter without dimension. The z signal expression is as follows:

[4.40]

The integrating circuit (whose y_? output also has the dimension of a speed) is described by the equation:

[4.41]

Figure 4.13 shows the same transients as those in Figure 4.11 ( ?_mq = ?_em /10) in the case where the adjustment coefficient of the anti-windup device is high (λ_AW = 25). We observe a true stabilization created by the z signal.

The fast adjustments with integrating circuits stabilized by an anti-windup device are the non-linear controls, similar to variable structure controls, since there are several operating modes. Their performances depend on various factors:

- parameters of the original system;

-  desired dynamic performances (imposed by the controller designed in linear operation, i.e. without the anti-windup device);

-  amplitude of the considered disturbances. Here, we have chosen λ_AW, large enough to limit the inrush current of the q axis at an amplitude lower than the I_{q_max} limit, during the second speed transient (10% of the nominal value at t = 1s), so that this “small movement” has a linear functioning;

- and, obviously, an acceptable amplitude for the current peaks.

4.4.6. Examples of regulation choices: IP controller with limited dynamics

Consequently, the detailed analysis of all the factors intervening in such a device is complex and will not be part of this discussion, which is limited to the fundamental properties of some classical linear regulations.

This is why we will now examine some properties of quite fast systems, but systems that do not have an anti-windup device (thus λ_AW= 0).

The dynamics is limited during the large transients (typically at start, from zero speed towards the rated speed) by the protection in current and there should not be too important an overshoot when we get to the end of the large transient; it is also necessary that the current does not have any current peak with saturation during a typical small amplitude transient (here a small reference step with an amplitude equal to 10% of the rated speed).

In the example in Figure 4.14, we have chosen T_mq = T_em /2.5 . The rise in speed is linear during the start, because of the saturation in current.

If the dynamics is not too fast, “the integrating circuit does not integrate too much” and the first overshoot is not excessive.

The response to the small final step does not have any current saturation.

Figure 4.15 details (enlargement) the transient responses during small transients (linear behavior): first, a response to a small speed step, followed by a phase loading (slot of a load torque equal to the nominal torque). We observe, from top to bottom:

– the speed references and the torque, and the response in torque;

- an enlargement of the speed response. On the small speed step, this is a 4^th order dynamics (of the q axis: two orders for the current loop and two orders for the speed regulation, see model [4.32]), but the time-constants relative to the mechanical variables are much more important than the time-constants relative to electric variables (choice of the controllers so that ?_mq / ?, = 32.6 and ?_mg = 0.056 s). Thus, the response has two “very dominant poles”. The speed response is indeed of the “damped 2^nd order” type. Limited to the 2^nd order (the simplified model is given by [4.28]), the expression of the response of a system with two time-constants equal to ?_mq, is given by the formula:

[4.42]

With this formula, we can verify that the response time at 95% really corresponds to a 0.265 s duration, as we can indeed observe in Figure 4.15.

-  we can observe (which is natural) that the current reference of the q axis exactly follows the torque reference;

- and the current loop is perfectly compensated and quickly reacts: the current of the d axis remains practically at zero and the current of the q axis follows very closely its reference: these controls indeed verify what we call the “decoupling of the d and q axes”.

The speed response is very well approximated by a 2^nd order, because the dynamics of the currents and the speed are decoupled (let us recall that we have chosen _mq/_iq= 32.6). We could bring these dynamics closer (by accelerating the dynamics of the speed loop) and impose the dynamics of the two regulations. Indeed, we have seen that the system is of the 4^th order, but the four state variables (i_d ,y_d , Ω , y_n) are all measurable and there are indeed four adjustable parameters. This means that the 4 poles of the system can all be set without restriction. But we will not develop here, discussions on the purely “automatic” aspects of these questions.

Conclusion: with these adjustments, the IP regulation functions ideally. But we have observed that it was necessary to limit the bandwidth of the speed loop. In particular, let us observe the elimination of the load torque effect, in the enlargement in Figure 4.16. During the phase loading (nominal torque), the speed (in reduced value) decreases to the minimum value 0.842 for t = 1.652 s. There is thus a speed drop of 16%.

We can recall here that we had to install an integral effect on the speed controller to eliminate the static errors due to the disturbances (here the load torque). The system that was of 1^st order was then put up to the 2^nd order (let us clarify that we limit ourselves in this analysis to the “slow mechanical” modes). We could wish to decouple the two problems: the adjustment of the speed regulation on one side, and the elimination of the disturbance on another side; this is what we will present in the following section.

4.4.7. Example of an advanced regulation: P controller associated with an integral observer

4.4.7.1. General information, hypotheses, modeling

We have just seen that to eliminate an unknown disturbance (here the load torque), it is common to install an integrating effect at the level of the controller (PI or IP type). It then appears that that the order increases by one unit for the transfer feedback function (i.e.:Ω(p)/Ω_ref), while the problem concerns the regulation transfer function (i.e.: Ω(p)/C_load).

And yet there is a technique to solving this problem by separating these two aspects: this is the use of a load torque observer. A previous book discusses observation questions [FOR 10], and one chapter is entirely devoted to the load torque observation [FAD 10] with excellent recognition of the various problems involved in this question. We limit ourselves here to a simple example, inspired by a structure proposed by researchers from the catholic university of Louvain [ROB 95]. We present this method, and in the discussion of the results, we will see that we will be able to accelerate the observation dynamics (and consequently the load torque compensation) independently from the dynamics adjustment of the speed regulation.

Therefore, without harming the safety rules (limit of the current peaks), we accelerate the dynamics of the set and we improve the regulation. Figure 4.17 gives the basic diagram used for the design.

We consider here the following hypotheses:

– the motor dynamics is represented by the equation of a purely inertial load:

[4.43]

We also again find the hypothesis of the previous studies, where the load torque is assumed to be constant at intervals:

[4.44]

But in practice, the load torque often has a more complex expression, and a first approach (still very simple) of the robustness of this method will require consideration of the effect of a viscous friction term:

[4.45]

We also assume that the currents are perfectly regulated and the axes are perfectly decoupled. We then have:

[4.46]

There again, this is a simplifying hypothesis, because even by assuming that the decoupling is perfect, the currents each have a 2^nd order dynamics (see Chapter 3), as specified by the diagram in Figure 4.18.

Validity tests of this method will take into account the complete dynamics of the system, including the effects of a non-decoupling of the d and q axes, if need be.

4.4.7.2. Design of the controller alone

In the approach with an observer, we design the controller by ignoring the load and its observer. Under these conditions, the dynamic model of the feedback in closed-loop is written:

[4.47]

The model is of the 1^st order:

[4.48]

And we see that we can impose the time-constant ?_mq of the closed-loop system (method of the input-output linearization) by adjusting the gain k₃ :

[4.49]

4.4.7.3. Design of the observer

The goal must be defined: we must determine a time-constant ?₃ (see Figure 4.17), so that the I₀ signal (which has the dimension of a current) could physically be interpreted as a load torque image. Thus, we should be able to obtain a transfer function of the form:

[4.50]

where the time-constant ?₀ could be imposed by the designer by adjusting the gain k₀. The model assumed in the formula [4.50] is of the 1^st order: this hypothesis must be verified.

Moreover, we observe that if we have C₀=K-I₀ (which is implied by the formula [4.50] in steady state), the load torque is perfectly compensated and the control, designed in section 4.4.7.2, becomes exact.

Then the observer equation is deduced from the following properties and hypotheses:

- we admit that the current dynamics is much faster than the speed dynamics (let us recall that these dynamics are imposed by the designer). Therefore, we assume that the current (here of the q axis) is instantaneously equal to its reference. Consequently, the speed equation can be simply written:

[4.51]

– the structure of the observer is based on principles appearing with precision in Figure 4.17: the reference current I_{q _ ref} is the sum of two terms:

- one is the signal I₁ , coming from the controller. It is the main control;

- the other is the signal I₀ , coming from the observer (to design) and designed to compensate for the load torque effect C₀ . In total, we have the relation:

[4.52]

– the signal I₀ is obtained from an error made by the observer: if the estimate speed is different from the measured speed, this is because there is a load torque disturbing the speed trajectory:

[4.53]

?ˆ – finally, the “estimate speed” ? is given by a simulation in open loop:

[4.54]

Let us note here that we can anticipate a result: formula [4.54] is in practice a simulation of the no-load model [4.43] (C₀=0). The fact that it is a pure integrating circuit (hence the hypothesis f= 0) will give a property to the global structure that is comparable to the controllers with integral effect.

Equations [4.51] to [4.54] help to obtain a relation between I₀ and C₀ (as desired in [4.50]), also involving the speed ?, but the latter will disappear thanks to a sound choice of ?₃ . Thus, it is necessary to eliminate I₁, I_{q_ref} and Ωˆ in the previous equations (from [4.51] to [4.54]).

We write the relations in the Laplace transformation and we obtain:

[4.55]

We immediately see that the term ? disappears if:

[4.56]

Indeed, we find the assumed result: the observer is a simulation of the motor in open loop. We obtain the desired model [4.50] with:

[4.57]

4.4.7.4. Global transfer function

We have studied the regulation (section 4.4.7.2) and the observation (section 4.4.7.3) separately. It is natural to seek the global transfer function without this separation. A priori, we can start from the equation:

[4.58]

We make the time-constants chosen by the designer appear: ?₀ for the observer (see [4.57]) and ?_mq for the regulation (see [4.49]).

We can then write the following result:

[4.59]

We observe that the observer choice (see [4.57]) leads to a simplification:

[4.60]

We obtain the expected results:

- the transfer function in feedback is of the 1^st order (however, let us recall that we have neglected the current dynamics) and the time-constant ?_mq is adjustable (see [4.49]);

- the transfer function in regulation is of the 2^nd order, which is logical: the load torque observer is of the 1^st order (as the “mechanical” equation of the system) and it calculates the load torque effect on the speed, which is also of the 1^st order. The observer is adjustable by [4.57].

4.4.7.5. Examples of transients and performances

We have seen that the examination and the comparison of the performances is difficult because of the interaction of the dynamic properties (that the designer can choose with great freedom) and of the dimensioning constraints (it is necessary to limit the current peaks).

Therefore, we will limit ourselves here to a few remarks.

To compare the responses of the system to steps, such as those presented in section 4.4.6, we consider an adjustment with limited feedback dynamics, so that the response time in speed control is comparable to the response time of the adjustment with the IP controller: we choose _mq = 0,8 .?_em.

But we can accelerate the transfer function of the regulation by choosing 10.

Figure 4.20 gives transients that we can compare to those in Figure 4.14.

The limit in current creates a response to the large speed step, which is very comparable, with an overshoot (due to the limit effect in current on the integrating circuit) of the same order of magnitude.

Figure 4.19 clarifies the transients in the linear case (without transfer in current saturation): responses at a small speed step and at a load torque, such as those in Figure 4.15. Indeed, we observe the similarity of the response time: the similarity was predicted to be 3 ._mq = 0.335s and this exact value is observed (in the case of an IP controller, the response time was 0.266 s). The current peak with the P controller is indeed that of a 1^st order system (almost): the peak is higher than with the IP controller, but this peak lasts for a shorter time. As we could expect, the torque step is eliminated quicker and the speed drop is thus reduced.

Figure 4.21 gives an enlargement of the response after the load torque step. This result is to be compared with the result in Figure 4.16 for the IP controller). We observe at the instant t = 1.626 s a speed minimum with a drop of 8.5% (to be compared with 16% in the case with IP, however obtained with a little faster speed control). This is accompanied by a response in torque (thus in current) of moderate amplitude: we observe that the adjustment separated from the feedback and the regulation really brings the desired improvements.

The discussion on the performances obtained with such a controller could be very long. We will only consider a robustness test on a delicate point: indeed, the method has assumed that the model of the mechanical load was a pure integrating circuit, which allows us to design a purely integral observer that gave to this control, properties comparable to those of a controller with an integral effect. And yet, this is an approximation, because the presence of a viscous friction (see Figure 4.18) is frequent.

Figure 4.22 gives the responses in torque and in speed in the case where f is high (f = 0.01 in SI units, thus f-Ω_max = 3.14 N.m, a value to be compared with the steps of the nominal torque: C_N = 2.3 N.m). We observe that the speed is correctly regulated. The enlargement of the response to a load step in Figure 4.23 confirms the good robustness: the presence of an additional term in the torque expression (see [4.45]) does not prevent the integral observer from compensating the torques. The dynamics is slowed down, but the operation is still correct.

Here we stop the discussion on the speed regulations of the axes operated by a synchronous motor. This outline of a few properties of the “axis control” shows the extreme richness of this problem.

4.5. Note on position regulations

We have centered this account on the speed controls of the motorized axes by a synchronous motor. Indeed, the common “speed regulators” carry out this function, the position control being carried out by a higher hierarchical element.

But we observe that more and more designers of industrial production materials propose this function in the axis control. We can thus make a few remarks on this aspect. Figure 4.24 gives the principle of a purely proportional control, sufficient because with the position being the speed integral, there will not be any static error.

Figure 4.25 gives the “control diagram” of the position regulation. As we neglect the (very fast) currents dynamics, this is a 3^rd order system.

To work with parameters without dimension, we set out:

[4.61]

The transfer function in position in closed-loop is deduced from [4.26] and the control laws:

[4.62]

[4.63]

We obtain:

[4.64]

We observe that there will not be any static error on a constant input. We can design this regulation with two different strategies:

– the “speed regulator” is autonomous with its own adjustments, we can only adjust the gain k_θ and we will obtain results that will not be very efficient. Indeed, we cannot simultaneously adjust the response time and the overshoot with only one adjustment;

– but if we have the possibility to simultaneously design the speed controller and the position controller, we can completely impose the dynamics. We consider an example where we choose that the system has three time-constants equal to one arbitrary value ?_mq . This strategy is obviously not the only possible one, but it is simple and allows us to reason with the only ?_mq parameter. It is necessary that the denominator of [4.64] is identical to:

[4.65]

Which leads to the adjustment:

[4.66]

The choice of τ_mqis arbitrary. We present the example of a motor that does first a half-turn (starting from 0 to τ), followed by a complete turn in the other direction (transient from -k to --k ). The choice τ_mq = τ_m /5 leads to a first completely linear transient (without transfer in current limit) and to a second transient with short transfers in the current limit. In Figure 4.26, we observe a “critical” damping behavior, as expected.

4.6. Conclusion

This chapter was devoted to questions relative to the drive control driven by a synchronous motor. This is a very rich subject, because it mixes problems associating power electrical engineering, mechanics, control and manufacturing. We have limited the discussion to some aspects, primarily power electrical engineering aspects: the torque controls designed in the natural a-b-c reference frame do not have a priori the same performances as the torque controls designed in the rotor d-q reference frame.

The simplicity of establishment in the a-b-c reference frame would require more complex and more efficient currents and speed controllers (we have presented the example of the resonant current controller).

The installation in the d-q reference frame is more complex (the modern components however propose integrated solutions), but it is possible to easily determine very efficient speed controls. We have presented a design coming from control science: conventional P or IP controllers associated with improvements that are easy to design and to establish: a controller with an anti-windup device and a controller associated with a load torque observer.

The design methods of the axis controls are various and numerous. The method that we used in this brief account can be summarized in a few words: placement of the robust poles. But there are other approaches, as we can see in general books ([HAU 97], [GRE 97]).

The conventional methods set with the control of axes operated by DC motors ([LOR 97], [LOU 02]) can easily be transposed to the control of the q axis of the synchronous motor.

Control science has developed general methods that can be applied to the axis control to obtain high performances and robustness, especially when the mechanical load is complex, as mentioned in the introduction.

Let us limit ourselves to quoting a few of them: optimal control [ABO 04], [BON 02]; robust controls [BER 02], including the CRONE method [MAT 99], the H_∞ method [DUC 99] and the poles placement [LAR 93]; non-linear controls [LEP 90], [LEP 91], [FOS 95]; control by sliding mode [GLU 93]; RST controllers [BOU 10]; feed-forward controls [BOU 06]; and flat control [DEL 04].

We have presented continuous versions of the control models, but we can, of course, digitize the equations and algorithms [SEV 69], [LAN 02].

4.7. Appendices

4.7.1. Numerical values of the parameters

The examples of the dynamics presented in this chapter concern two motors: a non-salient pole motor (control in a-b-c, section 4.3) and a salient pole motor (control in d-q, section 4.4). The numerical values of their parameters have been given in Chapter 1 (appendix, section 1.8.1).

4.7.2. Nomenclature and notations

We have given in Chapter 1 (section 1.8.2) the notations relative to the definitions of models for the modeling in Chapters 2 and 3, and the notations relative to the torque and currents regulations. We complete this with notations specific to this chapter.

4.7.2.1. Regulations in the a-b-c reference frame

Definition of the parameters for the design of the controllers designed in the a-b-c reference frame:

-  k_Ω, τ_yΩ: gain and time-constant of the speed controller;

-  τ_em =R_s.J / K_q² : electromechanical time-constant;

-  k_b = (k_t /k_i) .K_abc .k_Ω, τ_bΩ= J/k loop gain and time-constant associated with the speed loop.

4.7.2.2. Regulations in the d-q reference frame

Definition of the parameters for the design of the controllers and of the observer designed in the d-q reference frame:

-  τ_2n: time-constant of adjustment for the design of the speed regulation;

-  k_Ω2, τ_yΩ: proportional gain and time-constant of the integrating circuit of the speed controller;

-  T_mq: adjustment parameter of the speed controllers (IP and with an observer);

– I'_{q_ref} , I_{q_ref} ,I_{q_max} : desired value for the reference of the q axis current (will

be saturated for safety reasons), value effectively applied, maximal authorized value for the current of the q axis;

– λ_AW : adjustment parameter of the anti-windup device;

– k₃ : proportional gain of the speed controller with observer;

– τ₃ : time-constant of the estimating model of the speed by the observer;

– k₀ : gain of the load torque observer;

– C₀ : amplitude of the load torque, assumed to be piecewise constant;

– τ₀ : time-constant adjustment parameter of the load torque observer;

– I₁ , I₀ : current reference coming from the load torque observer;

4.8. Bibliography

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1 Chapter written by Jean-Paul LOIUS, Damien FLIELLER, Ngac Ky NGUYEN and Guy STURTZER.