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

Authors: Victor M. Hernández-Guzmán; Jorge Orrante-Sakanassi; Fortino Mendoza-Mondragón
Journal: Mathematical Problems in Engineering
 (2018)

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## 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.


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## Body

## 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 Section2 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 symbolζ to represent the absolute value of the scalar ζ. The Euclidean norm and the 1-norm of x=[x1,…,xn]T are defined as x=xTx and x1=∑i=1nxi, respectively. The spectral norm of an n×n matrix A(x) is defined as A=λM{ATA}, where λM{ATA} stands for the largest eigenvalue of the symmetric matrix ATA. If A(x) is an n×n symmetric matrix then symbol λm{A} represents the smallest eigenvalue of A for all x∈Rn and A=maxi|λi{A}|, where λi{A} stands for eigenvalues of A(x).


## 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 Figure1). It is well known that flux saturation is a common feature in SRM. In such a case, [8] suggests modeling this phenomenon in phase i as ψi(q,Ii)=ψsarctan⁡(βLi(q)Ii), for i=1,2,3, where q represents rotor position whereas ψi(q,Ii) and Ii stand for flux linkage and electric current in phase i. Li(q) is a function which stands for phase i, it is periodic on position q and strictly positive, and it is assumed to be given as(1)Liq=l0+∑n=1∞lncos⁡nNrq-i-12π3+cnsin⁡nNrq-i-12π3,where i=1,2,3, Nr is the number of rotor poles, and l0, ln, cn, are real constants. Finally, ψs and β are positive constants which have to be obtained experimentally. The dynamical model of such a SRM is given as [13, 20](2)Dq,II˙+Cq,IIω+RI=U,(3)Jω˙+bω=τ-τL,where I=[I1,I2,I3]T and U=[U1,U2,U3]T represent electric currents and voltages applied at each stator phase, ω=q˙ stands for rotor angular velocity, and generated torque is given as(4)τ=∑i=13ψs2βLi2qdLiqdqln⁡1+β2Li2qIi2,whereas τL is the load torque which is assumed to be unknown but constant. Matrix R=diag⁡{r,r,r} is positive definite, with r>0 representing the stator winding resistance, and scalars J>0, b>0 represent rotor inertia and viscous friction coefficient, whereas inductance D(q,I)=diag⁡{D1(q,I1),D2(q,I2),D3(q,I3)} is a positive definite diagonal matrix, where(5)Diq,Ii=ψsβLiq1+β2Li2qIi2,i=1,2,3, and C(q,I)=diag⁡{C1(q,I1),C2(q,I2),C3(q,I3)}, where(6)Ciq,Ii=ψsβ1+β2Li2qIi2dLiqdq,i=1,2,3.Figure 1
A three-phase and four-pole SRM.Finally, some important properties of the Euclidean and spectral norms are(7)±yTDxw≤Dxyw,yTExy≥λmExy2,yTExy≤λMExy2,Dxy≤Dxy,yTw≤yw,DxGx≤DxGx,for all x,w,y∈Rn and n×n matrices D(x), G(x), and E(x) with E(x) being a symmetric matrix.


## 3. Torque Sharing Approach

According to the working principle description given at the beginning of Section2, it is clear that such a motor operation results in a large torque ripple due to instantaneous switching from one to another phase. In order to reduce these torque variations, in this article we resort to the torque sharing approach (see [2, 4, 10, 19]). Given a desired torque τ∗, this torque has to be generated by the contribution of the three stator phases of the motor; that is,(8)τ∗=m1qτ∗+m2qτ∗+m3qτ∗.The scalar function, mi(q), is referred to as the sharing function of phase i,i=1,2,3, and represents the fraction of the desired torque to be contributed by phase i. The sharing functions are designed as follows. Consider two sets for each stator phase i [2, 4, 10]:(9)Θi+=q:dLiqdq≥0,Θi-=q:dLiqdq<0.Define mi+(q) and mi-(q) as bounded functions with bounded and continuous first derivatives such that(10)mi+q>0,∀q∈Θi+mi+q=0,∀q∈Θi-;∑i=13mi+q=1,∀q∈R,mi-q>0,∀q∈Θi-mi-q=0,∀q∈Θi+;∑i=13mi-q=1,∀q∈R,and assign(11)miq=mi+q,τ∗≥0mi-q,τ∗<0.Note that mi(q) and dLi(q)/dq become zero simultaneously. See Figure 2 for a graphical example of functions mi(q) and dLi(q)/dq. Furthermore, we additionally require functions mi+(q) and mi-(q) to be chosen such that mi(q)→0 as fast as (q-q0)ρ→0 where q0 is the rotor position such that dLi(q0)/dq=0 and for some integer ρ>0 which depends on the particular Li(q). This is to ensure that Ii∗ and I˙i∗ are continuous and bounded as q→q0 [19]. See Section 6, for example.Figure 2
A graphical example of functionsmi(q) and dLi(q)/dq for a three-phase and four-pole SRM. The latter are represented by sinusoidal waveform signals and the former are the other waveform signals.


## 4. Main Result

The following proposition establishes our main result.Proposition 1.
Consider model (2) and (3), in closed loop with the following controller:(12)U=HI∗-I-αξ-k1ωξ+Cq,II∗ω,I∗=I1∗,I2∗,I3∗T,(13)Ii∗=β1ζi+β2fζi,ifdLiqdq≠00,ifdLiqdq=0,i=1,2,3,(14)ζi=exp⁡2βLi2qmiqτ∗/ψsdLiq/dq-1β2Li2q,k1>ψsβ2dLqdqM,(15)τ∗=-Kpω~-Ki∫0tω~t1dt1,where ω~=ω-ω∗, with ω∗ the constant desired velocity, ξ=I-I∗, τ∗ represents the desired torque, and symbol ·M stands for the supreme value over the norm. We define the vectorial function H(I∗-I)=[h(I1∗-I1),h(I2∗-I2),h(I3∗-I3)]T, where h(·) stands for the hysteresis nonlinearity depicted in Figure 3. Functions mi(q) are defined in (11) and(16)fζi=αf1-cos⁡ωfζi,∀ζi≤T∗,where ωf is the smallest positive number satisfying(17)1-cos⁡ωfT∗ωfsin⁡ωfT∗=2T∗,which can be solved numerically; αf>0 is chosen as(18)αf=T∗1-cos⁡ωfT∗and T∗ is an arbitrarily small positive constant. Finally, we have (19)β1=1,β2=0,ifζi>T∗,β1=0,β2=1,ifζi≤T∗.There always exist positive scalars Kp, Ki, N, δ, and α, such that the whole state (which includes the velocity error ω~=ω-ω∗) remains bounded and it has an ultimate bound which can be rendered arbitrarily small by using a suitable choice of controller gains. This result stands when starting from any initial condition within a ball whose radius can be arbitrarily enlarged provided that suitable controller gains are chosen.Figure 3
Hysteresis nonlinearity. Slops is assumed to be large but finite.Remark 2.
In Figure4 we depict control scheme in Proposition 1. Note that it is composed of a classical proportional-integral (PI) external loop intended to regulate velocity when motor is submitted to unknown but constant external torque disturbances. An inner electric current loop is driven by both a hysteresis controller and a proportional controller. The former is intended to dominate some terms which do not vanish as the desired equilibrium point is reached. The latter is intended to dominate some cross terms existing between the mechanical and the electrical subsystems. This will become clear in the proof of Proposition 1 which is presented next.
We stress that both hysteresis and proportional controllers are commonly used in practice [11] which shows feasibility of the proposed controller. This is important to stress because this means that insight into how the proposed control scheme can be implemented in practical applications is explained by standard literature on SRM. The main advantage of our approach is that, contrary to standard open-loop operation of SRM, the trigger angles for switching on or switching off the power devices are determined automatically, thus simplifying motor operation at high and low velocities. Finally, the nonlinear terms -k1ωξ and C(q,I)I∗ω are added at the output of the electric current controllers just for stability proof purposes.Figure 4
Control scheme in Proposition1.Remark 3.
In [19] it is shown that function defined in (16) and the choice of parameters described after this expression ensure that by replacing ζi with f(ζi),∀ζi≤T∗, we obtain a function which is continuous up to the first derivative (see Figure 5). On the other hand, note that finding ωf as the numeric solution of the expression after (16) is not to be performed on line. Moreover, since T∗ is a known constant a simple computer program can be written to increase ωf from zero to choose the smallest positive value satisfying such an expression. This can be easily performed graphically since it is not to be done on line.
Also note thatI∗ is well-posed since mi(q)τ∗/dLi(q)/dq is always positive or zero, and hence ζi≥0, because mi(q)>0 is used only if dLi(q)/dq and τ∗ have the same sign (see (9) and (11)). Furthermore, according to paragraph after (11), mi(q)→0 faster than dLi(q)/dq→0 [19]. In this respect, notice that Li(q) is a function proposed and introduced by designer trying to model magnetic flux saturation in motor as ψi(q,Ii)=ψsarctan⁡(βLi(q)Ii). Hence, Li(q) is a known function proposed by the designer. Thus, once Li(q) is defined dLi(q)/dq can be computed. On the other hand, mi(q) is proposed as a known polynomial which, hence, can also be proposed such that mi(q)→0 faster than dLi(q)/dq→0.Figure 5
Approximation ofζi with f(ζi) for all ζi≤T∗.Remark 4.
We obtain the closed loop dynamics in the following. This is achieved by simply replacing controller in Proposition1 in the plant dynamics (2)–(4). As stressed in Remark 2, the proposed control scheme is simple. One important factor contributing to this feature is that we are not required to feedback term I˙∗, which includes a large number of computations, but we just dominate such terms. The price to pay for this controller simplicity is that additional algebraic manipulation is required in both the closed loop equations and the stability analysis to show how this term domination is possible.Now let us obtain the closed loop dynamics of system described in Proposition1. Replacing (12) in (2) and adding and subtracting some convenient terms, we have (20)Dq,Iξ˙=HI∗-I-Rξ-αξ-k1ωξ-Cq,Iξω-RI∗-Dq,II˙∗,(21)I˙i∗=I˙ia∗+I˙ib∗+I˙ic∗+I˙id∗,I˙ia∗=β12exp⁡σi-1β2Li2q-1/2×-2β2Li3qdLiqdqexp⁡σi-1ω~+1β2Li2qexp⁡σi∂σi∂qω~+∂σi∂ω~ω~˙+∂σi∂zω~,I˙ib∗=β2dfζidζi-2β2Li3qdLiqdqexp⁡σi-1ω~+1β2Li2qexp⁡σi∂σi∂qω~+∂σi∂ω~ω~˙+∂σi∂zω~,I˙ic∗=β12exp⁡σi-1β2Li2q-1/2-2β2Li3qdLiqdqexp⁡σi-1ω∗+1β2Li2qexp⁡σi∂σi∂qω∗,I˙id∗=β2dfζidζi-2β2Li3qdLiqdqexp⁡σi-1ω∗+1β2Li2qexp⁡σi∂σi∂qω∗,σi=2βLi2qmiqτ∗ψsdLiq/dq,for i=1,2,3. We stress that I˙∗ is continuous according to definition of mi(q) and f(ζi) in Section 3 and in Proposition 1, respectively. On the other hand, adding and subtracting term ∑i=13((ψs/2βLi2q)(dLi(q)/dq))ln⁡(1+β2Li2(q)Ii∗2) in (4) and replacing Ii∗ from (13), we have(22)τ=∑i=13ψs2βLi2qdLiqdqln⁡1+β2Li2qIi2-∑i=13ψs2βLi2qdLiqdqln⁡1+β2Li2qIi∗2+τ∗+β2ϕ,ϕ=∑i=13ψs2βLi2qdLiqdqln⁡1+β2Li2qf2ζi-τ∗,where ϕ≤ϕ¯ for some ϕ¯ which decreases to zero as T∗>0 approaches zero (see [19], Remark 1). Replacing (15), last expression can be written as(23)τ=∑i=13ψs2βLi2qdLiqdqln⁡1+β2Li2qIi2-∑i=13ψs2βLi2qdLiqdqln⁡1+β2Li2qIi∗2-Kpω~-Ki∫0tω~t1dt1+β2ϕ.Hence, from (3) it is obtained that(24)Jω~˙+bω~=Φ-Kpω~-Kiz+β2ϕ,where(25)Φ=∑i=13ψs2βLi2qdLiqdqln⁡1+β2Li2qIi2-∑i=13ψs2βLi2qdLiqdqln⁡1+β2Li2qIi∗2,(26)z=∫0tω~t1dt1+bω∗+τLKi.Thus, the closed loop dynamics is given by (20), (24), and (26). Finally, let us say that using the mean value theorem we can write(27)Φ=∑i=13ψs2βLi2qdLiqdqln⁡1+β2Li2qIi2-ln⁡1+β2Li2qIi∗2,=∑i=13ψs2βLi2qdLiqdq∂∂Iiln⁡1+β2Li2qIi2Ii=I¯iIi-Ii∗,=I¯TCq,I¯ξ, where I¯=[I¯1,I¯2,I¯3]T with I¯i being a point belonging to line joining Ii and Ii∗. Note that there always exists a positive constant KCI¯ such that(28)I¯TCq,I¯≤KCI¯,∀q∈R,I¯∈R3.


## 5. Stability Analysis

In this section we present the proof of Proposition1 employing Lyapunov stability analysis. First, a positive definite and decrescent function is introduced in Section 5.1. Then, time derivative of this function along the closed loop system trajectories is computed in Section 5.2. There, it is also shown that the domain of attraction can be enlarged arbitrarily by suitably selecting the controller gains. Finally, the above findings are exploited in Section 5.3 in order to complete the proof of Proposition 1 by invoking Theorem 4.18, pp. 172, in [21]. Hence, the basic idea behind this procedure is simple. However, as stressed in Remark 4, the algebraic manipulation becomes elaborated because we do not feed back the complex term I˙∗ and we have to show how these noncancelled terms are dominated. Moreover, showing that the domain of attraction can be arbitrarily enlarged introduces additional algebraic steps.


### 5.1. A Positive Definite and Decrescent Function

The scalar function(29)Vω~,z,ξT=12Jω~2+γJzω~+Ki2z2+γ2b+Kpz2+12ξTLqξ,where γ>0 is a constant and L(q)=diag{L1(q),L2(q),L3(q)}, can be bounded as(30)c1x2≤V1ω~,z,ξ≤c2x2,x=ω~,z,ξTT,P=JγJγJKi,(31)c1=12min⁡λmP,Lm1>0,c2=12max⁡λMP+γb+Kp,LM2>0,Lm1=infq∈R⁡λmLq,LM2=supq∈R⁡λMLq.Note that the only condition required for (30) to stand is 0<γ<Ki/J.


### 5.2. Time Derivative ofV(y)

It is possible to verify, after some algebraic manipulations, that the time derivative ofV(x), defined in (29), along the trajectories of the closed loop system (20), (24), and (26), is given as(32)V˙=-b+Kp-γJω~2+ω~+γzI¯TCq,I¯ξ+β2ω~+γzϕ-γKiz2-12ξTdLqdqξω+ξTΔ-Rξ-RI∗-αξ+HI∗-I-k1ωξ-Dq,II˙∗,where Δ is a diagonal positive definite matrix given as(33)Δ=LqD-1q,I=diag⁡1+β2L12qI12βψs,1+β2L22qI22βψs,1+β2L32qI32βψs.Replacing k1 from (14), we can write (32) as(34)V˙≤-b+Kp-γJω~2+ω~+γzI¯TCq,I¯ξ+β2ω~+γzϕ-γKiz2-ξTΔr+αξ+ξTΔ-RI∗+HI∗-I-Dq,II˙∗.Moreover, we can write(35)-ξTΔDq,II˙∗=-ξTΔDq,II˙AB∗-ξTΔDq,II˙CD∗,I˙AB∗=I˙1a∗+I˙1b∗,I˙2a∗+I˙2b∗,I˙3a∗+I˙3b∗T,I˙CD∗=I˙1c∗+I˙1d∗,I˙2c∗+I˙2d∗,I˙3c∗+I˙3d∗T.Hence,(36)V˙≤-b+Kp-γJω~2+ω~+γzKCI¯ξ+β2ω~+γzϕ-γKiz2-1βψsr+αξ2-ξTLqI˙AB∗+ξTΔ-RI∗+HI∗-I-Dq,II˙CD∗.If we assume that(37)N>rIi∗+βψsLiqI˙iCD∗,i=1,2,3,then(38)ξTΔ-RI∗+HI∗-I-Dq,II˙CD∗=∑i=13ξi-rΔiIi∗+ΔihIi∗-Ii-LiqI˙iCD∗≤6maxi⁡ΔiNδ,where Δi is the i-th diagonal entry of diagonal matrix Δ, since, according to Figure 3, we have hIi∗-Ii=Nsign⁡(Ii∗-Ii)=-Nsign⁡(ξi), if ξi>δ; that is, ξih(Ii∗-Ii)=-Nξi, if |ξi|>δ, and ξih(Ii∗-Ii)≤Nδ, for all ξi∈R. Also note that, according to the global Lipschitz property of right hand of (24), expressions for I˙i∗ in (20), and the fact that σi,i=1,2,3, depend on τ∗ which is an unbounded function of ω~ and z, there is a positive continuous function η(r1) which grows as r1=ω~2+z2 grows, such that we can write(39)-ξTLqI˙AB∗≤ηr1ξω~+z+ξ+β2ϕ.Thus,(40)V˙≤-y¯TQy¯+6maxi⁡ΔiNδ+β21+γ+ηr1ϕ¯y¯,(41)y¯=ω~,z,ξT.Entries of matrix Q are given as(42)Q11=b+Kp-γJ,Q22=γKi,Q33=r+αβψs-ηr1,Q12=Q21=0,Q13=Q31=-12KCI¯-12ηr1,Q23=Q32=-γ2KCI¯-12ηr1.Matrix Q is positive definite if and only if all of its principal minors are positive; that is,(43)b+Kp-γJ>0,d2=b+Kp-γJγKi>0,(44)det⁡Q=d2Q33+Q13Q21Q32-Q31Q22-Q23Q11Q32-Q12Q31>0.Assume for a moment that λm(Q)>0. Using some constant 0<Θ<1, we can rewrite (40) as(45)V˙≤-λmQx2+6maxi⁡ΔiNδ+β21+γ+ηr1ϕ¯x,≤-1-ΘλmQx2-ΘλmQx2+6maxi⁡ΔiNδ+β21+γ+ηr1ϕ¯x,≤-c31-ΘλmQx2-c41-ΘλmQx2+6maxi⁡ΔiNδ,∀x≥β21+γ+ηr1ϕ¯ΘλmQ,(46)≤-c31-ΘλmQx2,∀x≥μ0=max⁡β21+γ+ηr1ϕ¯ΘλmQ,6maxi⁡ΔiNδc41-ΘλmQ,for some positive numbers c3 and c4 such that c3+c4=1.Expressions in (43) are always satisfied using large enough values for Kp>0 and Ki>0 and a small γ>0. On the other hand, we remark that any singularity in I˙ia∗ and I˙ic∗ as ζi→0,i=1,2,3, is avoided if T∗>0 is chosen. Define μ as the radius of an open ball D={x∈R5∣x<μ} around the origin of the state space where λm(Q)>0. The simplest estimate of the domain of attraction Ωc⊂D is given as [21], pp. 317,(47)Ωc=x∈R5∣Vx<c=minx=μ⁡Vx.Hence, according to (30), we can compute c=minx=μV(x)=c1μ2. A more conservative estimate of the domain of attraction Ωc′, represented by a ball of radius r2, can be found by bounding V(x)≤c2x2<c1μ2, which results in Ωc′={x∈R5∣c2x2<c1μ2}. Thus, this new estimate of the domain of attraction contains the set(48)Ωc′=x∈R5∣x<r2=c1c2μ.Replace η(r1) with η(μ) in (44). From the resulting expression we obtain(49)d2r+αβψs>d2ηr1-Q13ημQ21Q32ημ-Q31ημQ22+Q23ημQ11Q32ημ-Q12Q31ημ.Since Q33 is the only entry of Q depending on α and η(μ) does not depend on α, it is clear that we can enlarge arbitrarily μ by enlarging α. Using this and the fact that c1 and c2 do not depend on α, we conclude that radius r2=(c1/c2)μ can be enlarged arbitrarily by enlarging α. Moreover, once r2 is given, N can be chosen such that (37) is satisfied. This proves that λm{Q}>0 and (46) stand in a domain which can be arbitrarily enlarged by using a large enough α. Now, we are ready to prove Proposition 1.


### 5.3. Proof of Proposition1

Taking into consideration (30) and (46), we can invoke Theorem 4.18, pp. 172, in [21] to conclude that, given an arbitrary initial state x(t0)∈Ωc′, we can always find controller gains such that the closed loop system state x satisfies(50)xt≤β0xt0,t-t0,∀t0≤t≤t0+T,(51)xt≤α1-1α2μ0,∀t≥t0+T,where β0(·,·) is a KL function and T≥0 depends on x(t0) and μ0. We stress that once r2 is fixed, Ii∗, i=1,2,3, is bounded; that is, Ii=ξi+Ii∗, maxi⁡{Δi} and η(r1) are also bounded. Recall that N in (37) is already fixed. Then, we can choose a small enough δ in order to render 6maxi⁡{Δi}Nδ/c4(1-Θ)λm{Q} arbitrarily small. On the other hand, according to (16) and paragraph before (24), ϕ¯ decreases to zero as T∗>0 approaches zero; hence, β2(1+γ+η(r1))ϕ¯/Θλm{Q} can be rendered arbitrarily small by choosing a small enough T∗. Thus, μ0 can always be rendered arbitrarily small. Since α1-1(α2(·)) is a K∞ function, then the ultimate bound in (51) tends to zero as T∗>0 and δ>0 approach zero.Note that, as was explained in paragraph after (46), conditions (43) and (44) can be satisfied by finite controller gains only if T∗>0, that is, to keep (46) valid. This means that the closed loop system has an ultimate bound which cannot be reduced to zero but can be rendered arbitrarily small by a suitable choice of controller gains. Recall that this result stands semiglobally. This completes the proof of Proposition 1. Finally, we emphasize that the conditions to guarantee Proposition 1 are summarized by 0<γ<Ki/J, (37), (43), (44), and some small constants T∗>0, δ>0.Remark 5.
Note that use of the desired electric current(52)I∗=I1∗,I2∗,I3∗T,Ii∗=ζi,ifdLiqdq≠00,ifdLiqdq=0,(53)ζi=exp⁡2βLi2qmiqτ∗/ψsdLiq/dq-1β2Li2q,for i=1,2,3, renders the corresponding time derivative unbounded when τ∗=0 and τ˙∗≠0, a common condition when a change in the sense of rotation is commanded. This can be seen from the product of the last term between brackets and factor at the left in the following expression:(54)I˙i∗=12exp⁡σi-1β2Li2q-1/2×-2β2LiqdLiqdqexp⁡σi-1ω+1β2Li2qexp⁡σi∂σi∂qω+∂σi∂τ∗τ˙∗,σi=2βLi2qmiqτ∗ψsdLiq/dq,for i=1,2,3, if Li(q)≠0. This problem was pointed out by [19] where the authors have introduced function f(·), defined in (16), to replace · when τ∗≤T∗, for some T∗>0. This prevents such a singularity from appearing. This function replacement is instrumental for result in Proposition 1 in the present paper and it explains definition of I∗ in (13). Finally, we remark that this class of singularity has been pointed out since the work by [8].Remark 6.
An important difference of our approach with respect to that of other authors [10, 12–15] is that, aside from term C(q,I)I∗ω, we do not require to complete by feedback of an error equation for the electrical subsystem of motor. This allows us to avoid necessity of feedback of complex functions of the state such as I˙∗ which we only have to dominate. Aside from an important simplification of the control law this is also instrumental to successfully present stability proof of Proposition 1.Remark 7.
Note that functionV(ω~,z,ξT), introduced in (29), contains the quadratic term (1/2)ξTL(q)ξ in order to cope with the motor electric dynamics. This is a clever choice since L(q) does not represent the motor inductance matrix but simplifies the stability proof. As a matter of fact, if the quadratic function (1/2)ξTD(q,I)ξ is employed, because matrix D(q,I) does represent the motor inductance matrix, when computing the time derivative of this function, the fact that D(q,I) depends on I complicates the stability analysis.


## 5.1. A Positive Definite and Decrescent Function

The scalar function(29)Vω~,z,ξT=12Jω~2+γJzω~+Ki2z2+γ2b+Kpz2+12ξTLqξ,where γ>0 is a constant and L(q)=diag{L1(q),L2(q),L3(q)}, can be bounded as(30)c1x2≤V1ω~,z,ξ≤c2x2,x=ω~,z,ξTT,P=JγJγJKi,(31)c1=12min⁡λmP,Lm1>0,c2=12max⁡λMP+γb+Kp,LM2>0,Lm1=infq∈R⁡λmLq,LM2=supq∈R⁡λMLq.Note that the only condition required for (30) to stand is 0<γ<Ki/J.


## 5.2. Time Derivative ofV(y)

It is possible to verify, after some algebraic manipulations, that the time derivative ofV(x), defined in (29), along the trajectories of the closed loop system (20), (24), and (26), is given as(32)V˙=-b+Kp-γJω~2+ω~+γzI¯TCq,I¯ξ+β2ω~+γzϕ-γKiz2-12ξTdLqdqξω+ξTΔ-Rξ-RI∗-αξ+HI∗-I-k1ωξ-Dq,II˙∗,where Δ is a diagonal positive definite matrix given as(33)Δ=LqD-1q,I=diag⁡1+β2L12qI12βψs,1+β2L22qI22βψs,1+β2L32qI32βψs.Replacing k1 from (14), we can write (32) as(34)V˙≤-b+Kp-γJω~2+ω~+γzI¯TCq,I¯ξ+β2ω~+γzϕ-γKiz2-ξTΔr+αξ+ξTΔ-RI∗+HI∗-I-Dq,II˙∗.Moreover, we can write(35)-ξTΔDq,II˙∗=-ξTΔDq,II˙AB∗-ξTΔDq,II˙CD∗,I˙AB∗=I˙1a∗+I˙1b∗,I˙2a∗+I˙2b∗,I˙3a∗+I˙3b∗T,I˙CD∗=I˙1c∗+I˙1d∗,I˙2c∗+I˙2d∗,I˙3c∗+I˙3d∗T.Hence,(36)V˙≤-b+Kp-γJω~2+ω~+γzKCI¯ξ+β2ω~+γzϕ-γKiz2-1βψsr+αξ2-ξTLqI˙AB∗+ξTΔ-RI∗+HI∗-I-Dq,II˙CD∗.If we assume that(37)N>rIi∗+βψsLiqI˙iCD∗,i=1,2,3,then(38)ξTΔ-RI∗+HI∗-I-Dq,II˙CD∗=∑i=13ξi-rΔiIi∗+ΔihIi∗-Ii-LiqI˙iCD∗≤6maxi⁡ΔiNδ,where Δi is the i-th diagonal entry of diagonal matrix Δ, since, according to Figure 3, we have hIi∗-Ii=Nsign⁡(Ii∗-Ii)=-Nsign⁡(ξi), if ξi>δ; that is, ξih(Ii∗-Ii)=-Nξi, if |ξi|>δ, and ξih(Ii∗-Ii)≤Nδ, for all ξi∈R. Also note that, according to the global Lipschitz property of right hand of (24), expressions for I˙i∗ in (20), and the fact that σi,i=1,2,3, depend on τ∗ which is an unbounded function of ω~ and z, there is a positive continuous function η(r1) which grows as r1=ω~2+z2 grows, such that we can write(39)-ξTLqI˙AB∗≤ηr1ξω~+z+ξ+β2ϕ.Thus,(40)V˙≤-y¯TQy¯+6maxi⁡ΔiNδ+β21+γ+ηr1ϕ¯y¯,(41)y¯=ω~,z,ξT.Entries of matrix Q are given as(42)Q11=b+Kp-γJ,Q22=γKi,Q33=r+αβψs-ηr1,Q12=Q21=0,Q13=Q31=-12KCI¯-12ηr1,Q23=Q32=-γ2KCI¯-12ηr1.Matrix Q is positive definite if and only if all of its principal minors are positive; that is,(43)b+Kp-γJ>0,d2=b+Kp-γJγKi>0,(44)det⁡Q=d2Q33+Q13Q21Q32-Q31Q22-Q23Q11Q32-Q12Q31>0.Assume for a moment that λm(Q)>0. Using some constant 0<Θ<1, we can rewrite (40) as(45)V˙≤-λmQx2+6maxi⁡ΔiNδ+β21+γ+ηr1ϕ¯x,≤-1-ΘλmQx2-ΘλmQx2+6maxi⁡ΔiNδ+β21+γ+ηr1ϕ¯x,≤-c31-ΘλmQx2-c41-ΘλmQx2+6maxi⁡ΔiNδ,∀x≥β21+γ+ηr1ϕ¯ΘλmQ,(46)≤-c31-ΘλmQx2,∀x≥μ0=max⁡β21+γ+ηr1ϕ¯ΘλmQ,6maxi⁡ΔiNδc41-ΘλmQ,for some positive numbers c3 and c4 such that c3+c4=1.Expressions in (43) are always satisfied using large enough values for Kp>0 and Ki>0 and a small γ>0. On the other hand, we remark that any singularity in I˙ia∗ and I˙ic∗ as ζi→0,i=1,2,3, is avoided if T∗>0 is chosen. Define μ as the radius of an open ball D={x∈R5∣x<μ} around the origin of the state space where λm(Q)>0. The simplest estimate of the domain of attraction Ωc⊂D is given as [21], pp. 317,(47)Ωc=x∈R5∣Vx<c=minx=μ⁡Vx.Hence, according to (30), we can compute c=minx=μV(x)=c1μ2. A more conservative estimate of the domain of attraction Ωc′, represented by a ball of radius r2, can be found by bounding V(x)≤c2x2<c1μ2, which results in Ωc′={x∈R5∣c2x2<c1μ2}. Thus, this new estimate of the domain of attraction contains the set(48)Ωc′=x∈R5∣x<r2=c1c2μ.Replace η(r1) with η(μ) in (44). From the resulting expression we obtain(49)d2r+αβψs>d2ηr1-Q13ημQ21Q32ημ-Q31ημQ22+Q23ημQ11Q32ημ-Q12Q31ημ.Since Q33 is the only entry of Q depending on α and η(μ) does not depend on α, it is clear that we can enlarge arbitrarily μ by enlarging α. Using this and the fact that c1 and c2 do not depend on α, we conclude that radius r2=(c1/c2)μ can be enlarged arbitrarily by enlarging α. Moreover, once r2 is given, N can be chosen such that (37) is satisfied. This proves that λm{Q}>0 and (46) stand in a domain which can be arbitrarily enlarged by using a large enough α. Now, we are ready to prove Proposition 1.


## 5.3. Proof of Proposition1

Taking into consideration (30) and (46), we can invoke Theorem 4.18, pp. 172, in [21] to conclude that, given an arbitrary initial state x(t0)∈Ωc′, we can always find controller gains such that the closed loop system state x satisfies(50)xt≤β0xt0,t-t0,∀t0≤t≤t0+T,(51)xt≤α1-1α2μ0,∀t≥t0+T,where β0(·,·) is a KL function and T≥0 depends on x(t0) and μ0. We stress that once r2 is fixed, Ii∗, i=1,2,3, is bounded; that is, Ii=ξi+Ii∗, maxi⁡{Δi} and η(r1) are also bounded. Recall that N in (37) is already fixed. Then, we can choose a small enough δ in order to render 6maxi⁡{Δi}Nδ/c4(1-Θ)λm{Q} arbitrarily small. On the other hand, according to (16) and paragraph before (24), ϕ¯ decreases to zero as T∗>0 approaches zero; hence, β2(1+γ+η(r1))ϕ¯/Θλm{Q} can be rendered arbitrarily small by choosing a small enough T∗. Thus, μ0 can always be rendered arbitrarily small. Since α1-1(α2(·)) is a K∞ function, then the ultimate bound in (51) tends to zero as T∗>0 and δ>0 approach zero.Note that, as was explained in paragraph after (46), conditions (43) and (44) can be satisfied by finite controller gains only if T∗>0, that is, to keep (46) valid. This means that the closed loop system has an ultimate bound which cannot be reduced to zero but can be rendered arbitrarily small by a suitable choice of controller gains. Recall that this result stands semiglobally. This completes the proof of Proposition 1. Finally, we emphasize that the conditions to guarantee Proposition 1 are summarized by 0<γ<Ki/J, (37), (43), (44), and some small constants T∗>0, δ>0.Remark 5.
Note that use of the desired electric current(52)I∗=I1∗,I2∗,I3∗T,Ii∗=ζi,ifdLiqdq≠00,ifdLiqdq=0,(53)ζi=exp⁡2βLi2qmiqτ∗/ψsdLiq/dq-1β2Li2q,for i=1,2,3, renders the corresponding time derivative unbounded when τ∗=0 and τ˙∗≠0, a common condition when a change in the sense of rotation is commanded. This can be seen from the product of the last term between brackets and factor at the left in the following expression:(54)I˙i∗=12exp⁡σi-1β2Li2q-1/2×-2β2LiqdLiqdqexp⁡σi-1ω+1β2Li2qexp⁡σi∂σi∂qω+∂σi∂τ∗τ˙∗,σi=2βLi2qmiqτ∗ψsdLiq/dq,for i=1,2,3, if Li(q)≠0. This problem was pointed out by [19] where the authors have introduced function f(·), defined in (16), to replace · when τ∗≤T∗, for some T∗>0. This prevents such a singularity from appearing. This function replacement is instrumental for result in Proposition 1 in the present paper and it explains definition of I∗ in (13). Finally, we remark that this class of singularity has been pointed out since the work by [8].Remark 6.
An important difference of our approach with respect to that of other authors [10, 12–15] is that, aside from term C(q,I)I∗ω, we do not require to complete by feedback of an error equation for the electrical subsystem of motor. This allows us to avoid necessity of feedback of complex functions of the state such as I˙∗ which we only have to dominate. Aside from an important simplification of the control law this is also instrumental to successfully present stability proof of Proposition 1.Remark 7.
Note that functionV(ω~,z,ξT), introduced in (29), contains the quadratic term (1/2)ξTL(q)ξ in order to cope with the motor electric dynamics. This is a clever choice since L(q) does not represent the motor inductance matrix but simplifies the stability proof. As a matter of fact, if the quadratic function (1/2)ξTD(q,I)ξ is employed, because matrix D(q,I) does represent the motor inductance matrix, when computing the time derivative of this function, the fact that D(q,I) depends on I complicates the stability analysis.


## 6. Simulation Results

In this section we present a simulation example intended to give some insight into performance achievable with controller in Proposition1. For this, we use the SRM whose numerical parameters are given by [10, 14] where it is assumed that Li(q)=l0+l1cos⁡Nrq-(i-1)(2π/3): Nr=8, R=5 [Ω], l0=0.03 [H], l1=0.02 [H], J=0.001 [kg-m2], and b=0.02 [Nm/(rad/s)]. Additionally, the flux model parameters were chosen as ψs=0.5 [Wb], β=1.8, which were experimentally identified by [8]. Sharing functions mi(q) were designed similarly as described by [10, 19]; that is, they are composed of a constant unitary value, a constant zero value, a raising polynomial pr(x), and a falling polynomial pf(x)=1-pr(x). As the rising polynomial we used prx=35x4-84x5+70x6-20x7, x=h/qmi (see [10, 19] for definitions of h and qmi and further details). This ensures that all of the requirements in Section 3 are satisfied. The controller gains were set to Kp=0.6, Ki=20, k1=5, α=10, N=30, δ=0.02, T∗=0.1, αf=0.1632, and ωf=27.85. All of the initial conditions were set to zero.The desired velocity was designed as follows. A ramp takes the desired value from0 [rad/s] at t=0 [s] to 50 [rad/s] at t=0.15 [s]. It remains constant at 50 [rad/s] from t=0.15 [s] to t=0.4 [s]. Then, a ramp takes it from 50 [rad/s] at t=0.4 [s] to -50 [rad/s] at t=0.7 [s]. Finally, the desired velocity remains constant at -50 [rad/s] for t≥0.7 [s]. The load torque was selected to be constant at τL=-4 [Nm] from t=1 [s] to t=1.4 [s] and τL=0 elsewhere. This allows to study performance when step changes in the load torque are applied. We stress that the above specified load torque is unknown for the controller and, despite this, its effect is successfully rejected by the integral part of the controller, as observed in Figure 6. We consider that this feature represents an important improvement with respect to results by [13, 14] where any information is not given on the load torque that they have considered. Moreover, a zero load torque due to friction is considered by [14]. On the other hand, although a nonzero constant (for all time) load torque is reported in simulations presented by [12], performance during the transient response cannot be evaluated from such a test.Figure 6
Simulation results when using controller in Proposition1.
(a)
Motor velocity and desired velocity
(b)
Electric current in phase 1
(c)
Electric current in phase 2
(d)
Electric current in phase 3The simulation results obtained when using controller in Proposition1 are presented in Figures 6 and 7. We can see that velocity ω and the desired velocity ω∗ remain almost identical for most of the time. Important differences between these variables only appear when step changes in the load torque are applied. However, such differences disappear very fast which proves the good disturbance rejection properties of controller in Proposition 1. We stress that a zero steady state velocity error is achieved when the desired velocity is constant although Proposition 1 only ensures an ultimate bound in velocity error. Reason for this is that the presence of friction and a load torque ensures that ζi>T∗>0 in steady state.Figure 7
Simulation results when using controller in Proposition1 (cont.).
(a)
Applied voltage in phase 1
(b)
Applied voltage in phase 2
(c)
Applied voltage in phase 3In Figures6 and 7, we also present the applied voltages and electric currents through all the three motor phases. Note that these variables have important changes when step changes in the load torque appear and when the desired velocity is described by ramps. We stress that the desired torque τ∗ becomes zero at some point of time which is close to time when velocity is zero. This is because the only load torque applied before t=1 [s] is that due to friction, and this load torque smoothly passes through zero when velocity changes direction. We realize that, despite this, any voltage (or electric current) does not become large when velocity passes though zero. This proves that singularity referred to in Remark 5 is not present in controller in Proposition 1.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 ofdLi(q)/dq changes for those phases, which would generate torque in the opposite direction if electric currents through such phases were not forced to be zero very fast. SRM are provided with electronic drivers which are able to provide such negative voltage values as described by [22]. These voltage spikes are also present in [14] although the voltage spikes in Figure 7 of the present paper are larger. This is because we are using a large load torque τL=-4 [Nm] (load torque in [14] is not specified).As a second test, in Figures8, 9, and 10 we present some simulation results when two-step desired velocities are commanded. The first one is applied at t=0 and changes desired velocity from 0 to 25 [rad/s] and the second one changes desired velocity from 25 [rad/s] to 0 at t=0.9 [s]. As expected, large electric current peaks appear which are required to accelerate and deaccelerate motor in order to follow the commanded velocity changes. Note that these current spikes are always positive and only appear in certain motor phases, that is, those phases which, according to sharing functions, are able to produce suitable torque according to the motor position angle. Notice that voltage may be negative. This is due to the fact that electric current must be rendered zero very fast in phases which, according to sharing functions, must not contribute to generated torque. If electric current is not zero in such phases then those phases would contribute with a torque in the opposite direction more than required. A positive electric current is rendered zero very fast by applying a large negative voltage.Figure 8
Additional simulation results when using controller in Proposition1. ω is in rad/s and τ is in Nm.
(a)
Motor velocity and generated torque
(b)
Motor velocity and generated torque (zoom)Figure 9
Additional simulation results when using controller in Proposition1 (cont.).
(a)
Electric current in phase 1
(b)
Electric current in phase 2
(c)
Electric current in phase 3Figure 10
Additional simulation results when using controller in Proposition1 (cont.).
(a)
Voltage applied at phase 1
(b)
Voltage applied at phase 2
(c)
Voltage applied at phase 3


## 7. Conclusions

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.



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*Source: 3280468-2018-01-02.xml*