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^{2}

^{3}

^{1}

^{2}

^{3}

We propose a controller for velocity regulation in switched reluctance motors under magnetic flux saturation conditions. Both hysteresis and proportional control are employed in the internal electric current loops. A classical PI velocity controller is employed in the external loop. Our control law is the simplest one proposed in the literature but provided with a formal stability proof. We prove that the state is bounded having an ultimate bound which can be rendered arbitrarily small by a suitable selection of controller gains. Furthermore, this result stands when starting from any initial condition within a radius which can be arbitrarily enlarged using suitable controller gains. We present a simulation study where even convergence to zero of velocity error is observed as well as a good performance when regulating velocity in the presence of unknown step changes in external torque disturbances.

It is widely recognized that switched reluctance motors (SRM) have tremendous potential as driving actuators given their unique torque producing characteristics [

On the other hand, magnetic flux saturation is an undesired phenomenon which appears in normal operation conditions of SRM. This has motivated design of control strategies which take into consideration flux saturation [

Since inductance is not constant in SRM, bandwidth of the electric current dynamics is not constant. This motivates use of both hysteresis control and high-gain proportional control of electric current for SRM control in practice [

Some recent theoretical works on SRM control [

Following the above ideas, the main contribution in the present paper is to introduce a simple control law for velocity regulation in SRM, provided with a stability proof that formally explains how the closed loop system works, when using both hysteresis and proportional control for electric current and taking into account magnetic flux saturation. We prove that the whole state remains bounded and it has an ultimate bound which can be rendered arbitrarily small by suitable selection of controller gains. Moreover, this result stands when starting from any initial condition within a radius which can be arbitrarily enlarged provided that suitable controller gains are chosen. We also show that this stands despite presence of unknown but constant torque disturbances.

This paper is organized as follows. In Section

Finally, we give some remarks on notation. We use symbol

Windings on the stator of a SRM are intended to work as electromagnets. Rotor has neither permanent magnets nor windings and simply consists of a piece of iron provided with several salient teeth or poles. In a SRM torque is generated by reluctance, that is, by means of a torque production mechanism which is identical to that appearing when an electromagnet is placed close to a piece of iron. Electromagnets belonging to one stator phase are activated to attract one pair of rotor poles. Once these electromagnets and rotor poles are aligned this phase is disconnected and the electromagnets belonging to another stator phase are activated to attract another pair of rotor poles. Permanent movement of rotor in any direction is accomplished by activating and disconnecting the stator phases in a suitable sequence. See [

For the sake of simplicity and without loss of generality, we will consider a SRM with four rotor poles and three phases (see Figure

A three-phase and four-pole SRM.

Finally, some important properties of the Euclidean and spectral norms are

According to the working principle description given at the beginning of Section

A graphical example of functions

The following proposition establishes our main result.

Consider model (

Hysteresis nonlinearity. Slop

In Figure

We stress that both hysteresis and proportional controllers are commonly used in practice [

Control scheme in Proposition

In [

Also note that

Approximation of

We obtain the closed loop dynamics in the following. This is achieved by simply replacing controller in Proposition

Now let us obtain the closed loop dynamics of system described in Proposition

In this section we present the proof of Proposition

The scalar function

It is possible to verify, after some algebraic manipulations, that the time derivative of

Expressions in (

Taking into consideration (

Note that, as was explained in paragraph after (

Note that use of the desired electric current

An important difference of our approach with respect to that of other authors [

Note that function

In this section we present a simulation example intended to give some insight into performance achievable with controller in Proposition ^{2}], and

The desired velocity was designed as follows. A ramp takes the desired value from

Simulation results when using controller in Proposition

Motor velocity and desired velocity

Electric current in phase 1

Electric current in phase 2

Electric current in phase 3

The simulation results obtained when using controller in Proposition

Simulation results when using controller in Proposition

Applied voltage in phase 1

Applied voltage in phase 2

Applied voltage in phase 3

In Figures

We observe that large negative voltage spikes appear all the time, especially when load torque is increased. These voltage spikes are naturally produced by the controller as a means of reducing to zero electric current through those stator phases which were required to produce a nonzero torque but, then, they are suddenly required not to produce torque. This happens when sign of

As a second test, in Figures

Additional simulation results when using controller in Proposition

Motor velocity and generated torque

Motor velocity and generated torque (zoom)

Additional simulation results when using controller in Proposition

Electric current in phase 1

Electric current in phase 2

Electric current in phase 3

Additional simulation results when using controller in Proposition

Voltage applied at phase 1

Voltage applied at phase 2

Voltage applied at phase 3

In this paper we have presented a solution for the velocity regulation in SRM when taking into account magnetic flux saturation. The main feature of our proposal is that it includes an internal electric current loop driven by a controller by hysteresis, which is the common scheme in practice to control electric current in SRM. We prove, for the first time, ultimate boundedness of the state. Moreover, the ultimate bound can be rendered arbitrarily small and this result stands when starting from any initial condition within a ball whose radius can be arbitrarily enlarged by choosing suitable controller gains. It is observed in a simulation study that the velocity error converges to zero if the desired torque in steady state is not zero. It is also shown that a good performance is obtained when rejecting unknown step changes in the load torque. We conclude that good performance of the proposed control scheme is achieved if we compare the obtained results with those presented previously in the literature which have been referred to in the simulations section. Moreover, it is important to stress that performance of our proposal has been achieved despite the remarkable simplicity of the control law when compared to previous controllers proposed in the literature.

The authors declare that there are no conflicts of interest regarding the publication of this paper.