Mathematical Problems in Engineering

Volume 2018 (2018), Article ID 3280468, 13 pages

https://doi.org/10.1155/2018/3280468

## Velocity Regulation in Switched Reluctance Motors under Magnetic Flux Saturation Conditions

Correspondence should be addressed to Victor M. Hernández-Guzmán; xm.qau@ghmv

Received 4 July 2017; Revised 8 November 2017; Accepted 10 December 2017; Published 2 January 2018

Academic Editor: Andrés Sáez

Copyright © 2018 Victor M. Hernández-Guzmán et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

It is widely recognized that switched reluctance motors (SRM) have tremendous potential as driving actuators given their unique torque producing characteristics [1–3]. SRM can produce very large torques at relatively slow velocities; they require very low maintenance and produce much higher torques than brushless DC (BLDC) motors [1]. However, the main disadvantages of SRM arise from the fact that they are difficult to control because of their complex nonlinear dynamic model and their multi-input nature. These challenging features of SRM control have motivated several important works on the subject [1–7].

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 [4, 8–10]. In [4] an exact feedback linearizing controller has been presented which is very complex and relies on the exact knowledge of many motor parameters. A backstepping-based controller is introduced in [8] which requires lots of computations deteriorating performance because of numerical errors. A controller focused in optimizing the electric current profile to reduce torque ripple is proposed in [9] but the mechanical subsystem dynamics is not taken into account and any stability analysis explaining the result is not presented. Although control design approach introduced in [10] takes into account flux saturation, an explicit control law and the corresponding simulations results were only presented for the unsaturated model case.

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 [11]. We stress that no work has been presented until now in the literature introducing a stability proof for SRM when hysteresis is used to control electric current at motor.

Some recent theoretical works on SRM control [10, 12–15] assume that only proportional electric current controllers are employed. However, the resulting control laws are very complex because their stability proofs require feedback of many nonlinear terms in order to complete the error equation for electric current. We remark that it is stressed in [16] that complex control laws result in performance deterioration because of numerical errors and actuator saturation. Furthermore, complex control laws also require more powerful hardware because of the large amount of computations to be performed, constraining the use of the motor to high-cost applications [17]. Moreover, it is pointed out in [18] that the electric drives community is not enthusiastic with such complex controllers. Thus, it is important to design controllers that are simple to implement but provided with formal stability proofs resulting in stability conditions useful to understand how the controller works.

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 2 we present the dynamic model that we consider as well as some useful mathematical tools. The torque sharing approach, which is instrumental for our proposal, is presented in Section 3. Our main result is stated in Section 4 and it is proven in Section 5. A numerical example is presented in Section 6 and some concluding remarks are given in Section 7.

Finally, we give some remarks on notation. We use symbol to represent the absolute value of the scalar . The Euclidean norm and the norm of are defined as and , respectively. The spectral norm of an matrix is defined as , where stands for the largest eigenvalue of the symmetric matrix . If is an symmetric matrix then symbol represents the smallest eigenvalue of for all and , where stands for eigenvalues of .

#### 2. SRM Dynamics

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 [19] for further explanation on the working principle of SRM.

For the sake of simplicity and without loss of generality, we will consider a SRM with four rotor poles and three phases (see Figure 1). It is well known that flux saturation is a common feature in SRM. In such a case, [8] suggests modeling this phenomenon in phase as , for , where represents rotor position whereas and stand for flux linkage and electric current in phase . is a function which stands for phase , it is periodic on position and strictly positive, and it is assumed to be given aswhere , is the number of rotor poles, and , , , are real constants. Finally, and are positive constants which have to be obtained experimentally. The dynamical model of such a SRM is given as [13, 20]where and represent electric currents and voltages applied at each stator phase, stands for rotor angular velocity, and generated torque is given aswhereas is the load torque which is assumed to be unknown but constant. Matrix is positive definite, with representing the stator winding resistance, and scalars , represent rotor inertia and viscous friction coefficient, whereas inductance is a positive definite diagonal matrix, where and , where