Nonlinear PI current control of reluctancesynchronous machines

C.M. Hackl?,†, M.J. Kamper‡, J. Kullick† and J. Mitchell‡

Abstract—This paper discusses nonlinear proportional-integral(PI) current control with anti-windup of reluctance synchronousmachines (RSMs) for which the flux linkage maps are known.The nonlinear controller design is based on the tuning rule“Magnitude Optimum criterion” [1]. Due to the nonlinear fluxlinkage, the current dynamics of RSMs are highly nonlinear and,so, the parameters of the PI controllers and the disturbancecompensation feedforward control must be adjusted online (e.g.,for each sampling instant). The theoretical results are illustratedand validated by simulation and measurement results.

NOTATION

N,R: natural, real numbers. x := (x1, . . . , xn)> ∈ Rn: col-umn vector, n ∈ N. 0n ∈ Rn: zero vector. ‖x‖ :=

√x>x:

Euclidean norm of x. A ∈ Rn×m: real matrix, n,m ∈N, det(A): determinant of A. In ∈ Rn×n: identity ma-trix. C1(I;Y ): space of continuously differentiable functionsmapping I → Y . For modeling of electrical machines, asignal ξ may be represented in the three-phase (a, b, c)-reference frame ξabc :=

(ξa, ξ

b, ξ

c)>

, the stator-fixed s-reference frame ξs :=

(ξα, ξ

β)>

and the arbitrarily rotatingk-reference frame ξk :=

(ξd, ξ

q)>

, which are related by ξk =

Tp(φk)−1ξs = Tp(φk)−1Tcξabc. φk [rad] is the (electrical)

angle of the k-reference frame with respect to the s-referenceframe and Tp(φk) =

[cos(φk) − sin(φk)

sin(φk) cos(φk)

], J =

[0 −1

1 0

]and

Tc = 23

[1 − 1

2− 1

2

0

√3

2−√

32

]are Park, rotation (by π

2 ) and (amplitudecorrect) Clarke transformation matrix, respectively (see [2, 3]).

I. INTRODUCTION

Synchronous machines are widely-spread actuators in indus-try due to their compact design, high efficiency and reliability.In particular, the reluctance synchronous machine (RSM)might be a viable alternative to AC drives [4, 5]: Compared topermanent-magnet synchronous machines, the RSM is simpleto manufacture (e.g., for transversally laminated RSMs, therotor consists of punched and glued iron sheets [6]) and cheap(e.g., rare earth are not necessary [7]). Moreover, due to theiranisotropic flux linkage, saliency-based encoderless controlschemes are applicable [8, 9], which might promote the use ofRSMs in the future. However, in most cases, for the operation

Manuscript uploaded December 31, 2015.∗Corresponding author, e-mail: christoph.hackl@tum.de (All authors, in

alphabetical order, contributed equally to this paper).†C.M. Hackl and J. Kullick are with the research group “Control of

renewable energy systems” (CRES) at the Munich School of Engineering(MSE), Technische Universität München (TUM), Germany.‡M.J. Kamper and J. Mitchell are with the Electric Machines Laboratory

(EMLab) at the University of Stellenbosch, South Africa.

of synchronous machines, an inverter, parameter estimationand adequate control are required. A method for parameter es-timation of RSMs with constant inductances (linear machine)at standstill is presented in [10].

In general, the RSM is highly nonlinear (in particular, inthe flux linkage) and, therefore, RSMs are not easy to controlwith standard control methods (e.g., classical field orientedcontrol [11, Cha. 16]). The (negative) effects of magnetic crosssaturation on the control performance have been investigatedin [12]. The highly nonlinear relation between stator currentsand flux linkages, caused by magnetic saturation and crossmagnetization effects, complicates the system description andcontroller design for RSMs. Moreover, the machines “induc-tances” are severely affected by the rotor dimensions and theresulting cross-magnetization [13]. Often, see e.g. [14–18], theflux linkage is described as a (matrix) product of (possiblynonlinear) inductances and currents, i.e., ψks (iks ) = Lks (iks )iks .However, this kind of modeling has several disadvantages:(a) It cannot reproduce non-zero flux linkages for zero cur-rents, i.e. iks = 02, since ψks (02) = 02, (b) it may lead toa lower control bandwidth or even to instability [19], and (c)its time derivative d

dtψks (iks ) = Lks (iks )iks results in stationary

and transient inductances (which have no physical counterpartand are mathematically questionable [20]).

Regarding the control of RSMs, two main ideas havebeen subject to extensive research in the past years: DirectTorque Control (DTC), as first proposed by Boldea [21] andLagerquist [22] as Torque Vector Control (TVC) and (classi-cal) vector control as discussed e.g. in [23–25]. While DTC isknown for its robustness and fast dynamics [26], it producesa high current distortion leading to torque ripples [27]. Incontrast, vector control improves the torque response [28] andthe efficiency of the system [7], but good knowledge of thesystem parameters is required for implementation. In [29], acompletely parameter-free adaptive PI controller is proposedwhich guarantees tracking with prescribed transient accuracy.The controller is applied to current control of (reluctance) syn-chronous machines but measurement results are not provided.

Other control approaches (see [28] and [17]) track theinductances online to adjust the current references to achievea higher control accuracy. In [14, 30] a control scheme isproposed, where the PI control parameters are continuouslyadapted to the actual system state. This improves the overallcurrent dynamics. However, the described system model re-quires measurement or estimation of the machine’s stationaryand transient inductances (which are not easy to measure sinceboth are not physical quantities) and the implementation isbased on an approximation of the measured data (as the data

arX

iv:1

512.

0930

1v2

[cs

.SY

] 8

Mar

201

6

−50

5−5

05

−1

0

1

B

ids [A]iqs [A]

ψd s

[Wb]

(a) d-component ψds of the stator flux linkage.

−50

5−5

05−0.5

0

0.5

B

ids [A]iqs [A]

ψq s

[Wb]

(b) q-component ψqs of the stator flux linkage.

Fig. 1: Nonlinear flux linkage of a real RSM (obtained from FEM data).

capacity is limited on the test system). In [15], a predictivetorque controller is proposed which takes into account themagnetic cross saturation of the RSM.

Contribution of this paper: In this paper, it is assumed thatthe nonlinear flux linkage maps are known (e.g. as look-uptables). Such look-up tables can easily be computed fromthe FEM data and, if 2D interpolation is used, their size“can be fairly small” [31]. Hence, the nonlinear relationbetween stator currents and flux linkages is known and canbe visualized as shown in Fig. 1. Moreover, the differentialinductances (see Fig. 2) may be approximated by numericalinterpolation and differentiation of the flux map with respectto the stator currents. The look-up tables for the flux linkagesand the differential inductances will be used for online adjust-ment of (i) the current PI controllers tuned according to the“Magnitude Optimum criterion” [1] and (ii) the disturbancecompensation (feedforward control). The online adjustmentguarantees an almost identical closed-loop system responseof the current dynamics over the whole operation range. Maincontributions of this paper are: (i) generic model of RSMsand the derivation of the current dynamics with differentialinductances (see Sec. II-A; iron losses or hysteresis effectsare not considered), (ii) generic model of two-level voltagesource inverter without imposing the assumption of balancedphase voltages (see Sec. II-B), (iii) theoretical derivation ofthe proposed nonlinear PI controller design with anti-windupaccording to the “Magnitude Optimum criterion” with onlineadjustment of the controller parameters and of the disturbancecompensation (see Sec. III), and (iv) validation of the proposedapproach (modeling, controller design, and implementation)and illustration of the control performance of the nonlinearPI controllers and the nonlinear disturbance compensation bysimulation and measurement results (see Sec. IV).

−50

5

−50

50

0.2

0.4

ids [A]iqs [A]

Ld s

[H]

(a) d-inductance Lds := ∂ψds /∂i

ds .

−50

5

−50

50

0.2

ids [A]iqs [A]

Lq s

[H]

(b) q-inductance Lqs := ∂ψqs /∂i

qs .

−50

5

−50

5

−2

0

2

·10−2

ids [A]iqs [A]

Ldq

s[H

]

(c) cross-coupling inductance Ldqs :=

∂ψds /∂i

qs .

−50

5

−50

5

−2

0

2

·10−2

ids [A]iqs [A]

Lqd

s[H

]

(d) cross-coupling inductanceLqds := ∂ψ

qs /∂i

ds .

Fig. 2: Nonlinear differential inductances of a 1,1 kW RSM (computed fromthe FEM flux linkage data shown in Fig. 1).

II. MODELING AND PROBLEM FORMULATION

RSMs are considered which are subject to unknown distur-bances (e.g. load torques). Friction is neglected. The RSM isactuated by a two-level voltage source inverter (VSI) with aconstant DC-link voltage. Control objectives is stable, fast andaccurate current reference tracking of some bounded reference(e.g., provided by a maximum-torque-per-Ampere algorithm)with (almost) identical closed-loop current dynamics over thewhole operation range of the RSM such that the design ofouter loop controllers remains simple.

A. Generic model of reluctance synchronous machine (RSM)

The machine model in the rotating k-reference frame isgiven by (see, e.g., [11, Sec. 16.1])

uks = Rsiks +ωkJψ

ks

(iks)+ d

dtψks

(iks),

ddtωk =

p

Θ

[mm(iks )− ml

], d

dtφk = ωk

(1)

where uks := (uds , uqs )> [V]2 are the applied stator voltages

(see Sec. II-B), Rs [Ω] is the resistance of the stator windings,iks := (ids , i

qs )> [A]2 are the stator currents and ψks :=

(ψds , ψqs )> [Wb]2 are the stator flux linkages (functions of iks ).

The k-reference frame rotates with electrical angular frequencyωk = pωm[rad/s] of the rotor where p [1] is the number ofpole pairs and ωm[rad/s] denotes the mechanical angular fre-quency of the machine (for details see, e.g., [3]). Furthermore,

Θ [kgm3] is the inertia, mm(iks ) := 32p (iks )>Jψks

(iks)

is themachine torque, and ml [Nm] is a load torque.

B. Generic model of the voltage source inverter (VSI)

The control input to the electrical drive system (i.e. RSMand VSI) in Fig. 4 is the reference stator voltage uabcs,ref . The(two-level) VSI generates the stator phase voltages uabcs =(uas , u

bs , u

cs)> which are then applied to the RSM terminals.

1) Switching model of the VSI: The considered two-levelVSI is modeled as follows

ultls (t) :=

(uabs (t)

ubcs (t)ucas (t)

)= udc

[1 −1 00 1 −1−1 0 1

]sabcs (t). (2)

Its output voltages are the stator line-to-line voltages ultls anddepend on the switching vector sabcs and the DC-link voltageudc > 0 [V]. The switching vector sabcs ∈ 000, 001, . . . , 111comprises eight different states and is generated via pulsewidth modulation (PWM) or space vector modulation (SVM).Applying the Clarke transformation to the three-phase statorvoltages uabcs yields (see also [3])

uss :=

(uαs

uβs

)= Tcu

abcs = 2

3

[12

0 − 12

0√32

0

]

︸ ︷︷ ︸=:T

ltlc

ultls , (3)

which shows that the stator voltages uss can directly becomputed via the line-to-line Clarke transformation matrixT ltlc and the line-to-line voltages ultls . Moreover, α- and β-component of the stator voltages uss do not depend on thezero component of the stator voltage u0s :=

√23 (uas +ubs +ucs)

which, in general, is non-zero.

Remark 1. For balanced stator voltages (i.e. u0s = 0), thefollowing holds [3]

uabcs (t) :=

(uas (t)

ubs(t)ucs(t)

)=udc3

[2 −1 −1−1 2 −1−1 −1 2

]sabcs (t). (4)

For the remainder, it is assumed that a regularly sampled,symmetrical PWM1 (for details see [32, Sec. 8.4]) is imple-mented. Due to this PWM, on average, the VSI output voltageis saturated by u := udc

2 > 0 [V] [32, Sec. 8.4.5], i.e.

‖uks ‖ = ‖uss‖ = ‖uabcs ‖ ≤ udc

2 =: u. (5)

For e.g. SVM with over-modulation or PWM with third-harmonics injection, u ≤ 2

3udc increases [33, Sec. 8.4].2) Inverter delay (VSI dynamics): In general, the voltage

generation is subject to a delay Tdelay ∝ 1/fs > 0 [s], i.e.uks (t) = uks (t − Tdelay) ([34], this holds for all referenceframes), which – on average – is inversely proportional tothe switching frequency fs > 0 [Hz]. In the Laplace domain,the delay can be approximated by a first-order lag system

uks (s) = e−sTdelay · uks,ref(s) ≈ 11+sTdelay

uks,ref(s) (6)

since, for sTdelay 1, the following approximation holds

e−sTdelay =(∑∞

i=0(sTdelay)

i

i!

)−1 ≈ 11+sTdelay

.The VSI delay depends on the implementation of the

modulation scheme (e.g. on FPGA, DSP or micro-processor)and varies within the interval Tdelay ∈

[1

2fs, 3

2fs

][34]. It can

further be shown that the VSI delay leads to a coupling be-tween the d- and q-component of the reference stator voltageswhich may be compensated for under certain assumptions [35].

1In this paper, the injection of e.g. third harmonics is not considered [32,Sec. 8.4.6].

For the remainder of this paper, however, this coupling effectwill be neglected (see Assumption 3).

C. Control objective: Current reference tracking

Control objective is stator current reference tracking of somegiven, possibly discontinuous but bounded current referenceiks,ref such that asymptotic tracking is achieved for constantreferences. Therefore, nonlinear proportional-integral (PI) con-trollers and a nonlinear disturbance compensation (feedfor-ward control) will be implemented. In view of the satura-tion (5) of the VSI (saturated output voltage), the PI controllersshould incorporate an anti-windup strategy. Moreover, due tothe nonlinear system dynamics, the controller parameters andthe disturbance compensation terms will be adjusted online toachieve (almost) identical closed-loop current dynamics overthe whole operation range of the RSM. Controller tuning isdone according to the tuning rule of the “Magnitude Optimumcriterion”. The achievement of the control objective shall beassured under the following assumptions:

Assumption 1 (Properties of the flux linkages). The fluxlinkages ψks [Wb]

2 are continuously differential functions ofthe stator currents iks [A]

2 only2, i.e. ψks (·) ∈ C1(R2;R2)

Assumption 2 (System knowledge and measured signals).Resistance Rs and flux linkages ψks (iks ) as depicted in Fig. 1are known (e.g. from FEM or measurements). Moreover, statorcurrents iabcs (t), mechanical angle φm(t) and mechanicalangular speed ωm(t) are available for feedback.

Assumption 3. The VSI has the following properties: (i) Itsdynamics are sufficiently fast (i.e., Tdelay 1 s is smallcompared to the stator dynamics of the RSM), such that (a) theapproximation as first-order lag system as in (6) is reasonableand (b) the reference voltage coupling is negligible; (ii) ItsDC-link voltage is constant, i.e. udc(t) = udc > 0 V, and (iii)Its (average) output voltage is limited, i.e. (5) holds true.

III. CURRENT CONTROLLER DESIGN

Controller design is discussed in two steps (i) currentdynamics decoupling via disturbance compensation feedfor-ward control and (ii) nonlinear PI controller design withanti-windup according to the “Magnitude Optimum criterion”(controller parameters are updated at each sampling instant).The derivation is shown in the time-continuous case, whereasimplementation is done in the discrete-time case (see Sec. IV).

A. Current dynamics

Since model-based current controllers will be designed,the current dynamics of the RSM are required. Due to thenonlinear current dynamics, the model will be derived in statespace3. In view of Assumption 1, a differential inductance ma-trix Lks [H]

2×2 can be introduced to represent the derivativesof the flux linkage with respect to the stator current iks , i.e.

2The flux linkages do not depend on the electrical/mechanical rotor angle.3An analysis in the frequency domain (transfer functions) is not admissible.

Definition 1. The differential inductance matrix is given by

Lks (iks ):=

∂ψ

ds (i

ks )

∂ids

∂ψds (i

ks )

∂iqs

∂ψqs (i

ks )

∂ids

∂ψqs (i

ks )

∂iqs

∣∣∣∣∣∣iks

:=

[Lds (iks ) Ldqs (iks )

Lqds (iks ) Lqs (iks )

](7)

and has the following properties for all admis-sible stator currents: (p1) It is positive definite(see [36]), i.e. Lds (iks ) > 0, Lqs (iks ) > 0 and

det(Lks (iks )

):= Lds (iks )Lqs (iks )−M(iks )2 > 0 and (p2)

It is symmetric (see [6, 37]), i.e. Lks (iks ) = Lks (iks )> and

M(iks ) := Ldqs (iks ) = Lqds (iks ) where M(iks ) [H] is themutual (cross-coupling) inductance of the RSM.

Note that, in view of Assumption 2, the differential induc-tance matrix can be computed numerically from the FEM data(as shown in Fig. 1). Due to numerical differentiation, thecross-coupling inductances might slightly differ (i.e. Ldqs 6=Lqds , see Fig. 2). Invoking Assumption 1 and Definition 1 andcomputing the time derivative of the flux linkages (by applyingthe chain rule) leads to the following expression

ddtψ

ks (iks ) = ∂ψ

ks (i

ks )

∂iks

ddti

ks

(7)= Lks (iks ) d

dtiks . (8)

Inserting (8) into (1) and solving for ddti

ks leads to the

nonlinear current dynamics as follows

ddti

ks = Lks (iks )−1 ·

[uks −Rsi

ks − ωkJψ

ks (iks )

], (9)

where the inverse of the inductance matrix4 is given by

Lks (iks )−1 =1

det(Lks (iks ))

[Lqs (i

ks ) −M(i

ks )

−M(iks ) L

ds (i

ks )

]. (10)

Remark 2. In view of properties (p1) and (p2) the inductancematrix Lks (iks ) nonlinearly depends on the stator currents iks ,is non-singular (hence invertible) and has real and positiveeigenvalues for all currents iks [38, Proposition 5.5.20].

The current dynamics (9) are nonlinear and coupled dueto the inverse of the inductance matrix (10) and due to thenonlinear term ωkJψ

ks (iks ) of the back electro-motive (EMF).

For the further derivation, it is convenient to introduce theauxiliary inductances (which also depend nonlinearly on iks )

Lds(iks ) := det(L

ks (i

ks ))

Lqs (i

ks )

> 0 ∧ Lqs(iks ) := det(Lks (i

ks ))

Lds (i

ks )

> 0 (11)

and the “disturbance voltages”

uks,dist(ωk,uks , i

ks ) :=

(uds,dist(ωk,u

ks , i

ks )

uqs,dist(ωk,u

ks , i

ks )

)

:=

ωkψ

qs (i

ks )− M(i

ks )

Lqs (i

ks )

(uqs −Rsi

qs − ωkψ

ds (i

ks ))

−ωkψds (i

ks )− M(i

ks )

Lds (i

ks )

(uds −Rsi

ds + ωkψ

qs (i

ks ))

. (12)

The disturbance voltages uks,dist comprise all coupling termsand depend on the applied stator voltages uks , the statorcurrents iks and the angular velocity ωk, respectively. Insertingthe newly introduced quantities above into (9) and solving

4I.e., Lks (iks )L

ks (i

ks )−1

= Lks (i

ks )−1

Lks (i

ks ) = I2 for all iks ∈ R2.

for ddt i

ds and d

dt iqs , allows to rewrite the current dynamics

component-wise in the more compact form

ddt i

ds = 1

Lds(i

ks )

(uds −Rsi

ds + uds,dist(ωk,u

ks , i

ks )),

ddt i

qs = 1

Lqs(i

ks )

(uqs −Rsi

qs + uqs,dist(ωk,u

ks , i

ks )).

(13)

B. Controller structure (see Fig. 3)

The proposed controller structure (following the idea ine.g. [11, Sec. 7.1.1], [20] or [3] with the same notation ashere) consists of two parts, i.e.

uks,ref = uks,pi︸︷︷︸PI controller output

+ uks,comp︸ ︷︷ ︸disturbance compensation

. (14)

Hence, the stator voltage reference uks,ref = (uds,ref , uqs,ref)

>

– the control input to the VSI/electrical drive system –is the sum of the disturbance compensation uks,comp =

(uds,comp, uqs,comp)> and the output uks,pi = (uds,pi, u

qs,pi)

> ofthe current PI controllers. Important to note that both controllerparts will require online tracking of the flux linkages ψks (iks )(see Fig. 1) and of the differential inductances Lds (iks ), Lqs (iks )and M(iks ) = Lqds (iks ) = Ldqs (iks ) (see Fig. 2). Both parts willbe discussed in more detail in the following.

C. Disturbance compensation (feedforward control)

To obtain decoupled system dynamics in (13) for controllerdesign, the “disturbance” uks,dist(t) as in (12) is compensatedfor by introducing the (ideal) feedforward compensation

uks,comp(t) :=

(uds,comp(t)uqs,comp(t)

)= −uks,dist(t+ Tdelay)

c s uks,comp(s) ≈ c0(1+sTdelay)

1+sT0︸ ︷︷ ︸=:Fcomp(s)

uks,dist(s), (15)

where 0 < c0 ≤ 1 and 0 < T0 Tdelay. In viewof (12), (15) requires knowledge of the resistance Rs, theinductances Lds (iks ), Lqs (iks ) and M(iks ) (available as look-uptable as shown in Fig. 2), the stator voltage uks (which can beapproximated by the stator voltage reference uks,ref ), the statorcurrent iks (measured), the stator flux linkage ψks (available aslook-up table as shown in Fig. 1) and the electrical angularvelocity ωk = pωm (ωm measured). Actual flux linkagesψks (iks ) and inductances Lds (iks ), Lqs (iks ) and M(iks ) are trackedonline based on the actual current measurements.

Note that, in view of Assumptions 2 and 3, for c0 = 1and small T0 Tdelay, the compensator Fcomp(s) in (15)approximates the inverse of the VSI dynamics (6) and allowsto compensate for the disturbance uks,dist (almost) ideally bythe feedforward term uks,comp (see [3]). So, nonlinear butdecoupled currents dynamics are obtained as follows

ddt i

ds = 1

Lds(i

ks )

(uds −Rsi

ds

),

ddt i

qs = 1

Lqs(i

ks )

(uqs −Rsi

qs

).

(16)

Remark 3 (Linear RSM). For constant auxiliary inductances,i.e. Lds(i

ks ) = Lds > 0 and Lqs(i

ks ) = Lqs > 0, the current

uks,ref

−

uksu

ks,pi i

ksi

ks,ref

uks,distu

ks,comp

Kkp (i

ks ) K

ki

1 Tdelay R−1s T

ks (i

ks )

Fig. 3: Block diagram of the nonlinear current closed-loop system.

dynamics (16) become linear decoupled first-order lag systemswith the transfer functions

ids (s)

uds (s)

=1Rs

1+sTds

∧ iqs (s)

uqs (s)

=1Rs

1+sTqs, (17)

which share the same system gain 1/Rs but have different timeconstants T ds := Lds/Rs and T qs := Lqs/Rs.

D. Nonlinear PI controller design with anti-windup

It is well known that PI(D) controllers in the presence ofinput saturation may exhibit integral windup (in particular forlarge initial errors) leading to large overshoots and/or oscilla-tions in the closed-loop system response (see, e.g., [39, 40]).Therefore, a simple but effective anti-windup strategy (similarto ‘conditional integration’, see e.g. [40]) is implementedwhich stops integration of the integral control action if thecontrol input (here uks or uks,ref ) exceeds the admissible range.For this, the “anti-windup decision function”

fu(uks,ref) :=

0, ‖uks,ref‖ ≥ u

( (5)= udc

2

),

1, ‖uks,ref‖ < u(18)

is combined with the nonlinear PI controller as followsddtξ

ki = fu

(uks,ref

)Kk

i

(iks,ref − iks

),

uks,pi = ξki +Kkp (iks )

(iks,ref − iks

).

(19)

where ξki = (ξdi , ξqi )> are the integrator outputs of the PI

controllers. The controller gains (merged in the gain matrices)

Kkp (iks ) :=

[kdp(i

ks ) 0

0 kqp(i

ks )

]∧ Kk

i :=

[kdi 00 k

qi

](20)

are tuned according to the “Magnitude Optimum criterion”(see [1] or [11, p. 81,82]) as follows

d-gains: kdp(iks ) = Lds(i

ks )

2Tdelay> 0 [Ω]

kdi = Rs

2Tdelay> 0 [V/(As)]

q-gains: kqp(iks ) = Lqs(i

ks )

2Tdelay> 0 [Ω]

kqi = Rs

2Tdelay> 0 [V/(As)] .

(21)

Note that the integrator gains kdi and kqi are constant.

Remark 4. The basic idea of the Magnitude Optimum isto compensate for the large time “constants” T ds (iks ) :=Lds(i

ks )/Rs and T qs (iks ) := Lqs(i

ks )/Rs of the system dynam-

ics (16) [1]. Since, Lds(iks ) and Lqs(i

ks ) are varying with

the stator currents, an online adjustment of the controllerparameters is required.

Remark 5. The use of the discontinuous anti-windup functionas in (18) may lead to chattering [39]. Chattering wasnot observed in the simulations and experiments, howeverif chattering occurs, the use of a Lipschitz continuous anti-windup function, as proposed in [29, 41], might be beneficial.

Remark 6. If, for the current measurement, a first-order low-pass filter Ff(s) = 1

1+s Tfwith filter time constant Tf 1 s

is implemented to filter out noise, the small time constantsTdelay and Tf can be combined to one small time constantT ′delay := Tdelay + Tf and, in (21), T ′delay should be usedinstead of Tdelay [11, Sec. 7.1.1.5].

IV. SIMULATIVE AND EXPERIMENTAL VALIDATION

In this section, the proposed nonlinear PI controller (19)with anti-wind up (18) and online parameter adjustment (21)and disturbance compensation (15) with online tracking of theflux linkage and the differential inductances are implementedin Matlab/Simulink and at a laboratory setup. Key data ofimplementation, simulation and hardware is collected in Tab. I.Simulation and measurement results illustrate the effectivenessof the proposed controller structure with online parameteradjustment. For the same current reference trajectory, thecontrol performance is illustrated for two scenarios over thewhole operation regime of the RSM: (i) Operation at idlespeed: simulation results and measurement results(see Fig. 6), and (ii) Operation with and without disturbancecompensation (feedforward control): and mea-surement results (see Fig. 7). To ease comparability of theresults, simulation and measurement results are both depictedin Figures 6 where the top large subfigure shows the overallduration of the experiment including the quantities (from topto bottom): direct currents ids & ids,ref , quadrature currents iqs& iqs,ref , norm of the control input ‖uks,ref‖ (& actuator limitu = udc

2 ), and machine angular velocity ωm. The three smallersubfigures at the bottom show different details (zoomed inexcerpts) of the overall experiment including the quantities(from top to bottom): direct current ids & ids,ref and directreference voltage uds,ref (& actuator limit u = udc

2 ), quadraturecurrents iqs & iqs,ref and quadrature reference voltage uqs,ref (&actuator limit u = udc

2 ), and machine angular velocity ωm.

A. Implementation in Matlab/Simulink

The implementation in Matlab/Simulink is shown in Fig. 4.Machine model (1) and VSI model (3) with ultls as in (2) andconstant DC-link voltage are implemented in the s = (α, β)-reference frame. The implemented regularly sampled, sym-metrical PWM (for details see [32, Sec. 8.4.6]) applies therequired pulse pattern (switching vector sabcs ) according tothe desired reference stator voltage uks,ref . The implementa-tion of the nonlinear PI controller (19) and the disturbancecompensation (15) is in the k = (d, q)-reference frame.The overall implementation is depicted as block diagram inFig. 4. The feedforward control (15) is implemented with staticcompensator, i.e., Fcomp(s) = 1, and the use of uks,ref insteadof the stator voltages uks (which are not measured in real worldand hence are not available for feedback). For the simulation,

description symbols & values [unit]

RSM (9) pmech = 1,1 kW, Rs = 4,7 Ω, p = 2,ψks as in Fig. 1, ‖iks ‖ ≤ 5 A,nnominal = 1 500 rpm, Irms

nominal ≈ 2 A,U rmsnominal = 400 V (@50 Hz)

Mechanics Θ = 8,1 · 10−3 kg m2 (friction neglected)

VSI udc = 600 V, fs = 5 kHz, Tdelay = 32fs

RT-system Ts = 1/fs (sampling period),measured signals were not filtered!

reference see 1st & 2nd subplots in Fig. 6

disturbance Fcomp(s) = 1 (VSI dynamics neglected),comp. (15) In (12), instead of uks , uks,ref is used

anti-windup u = udc/2 = 300 V (due to symmetrical,PI (19), (18) regularly sampled PWM), ξk,0i = (0, 0)>

Table I: System, implementation and controller data.

rotor

stator

iks,ref

iks i

ss i

abcs = (i

as , i

bs , i

cs )

⊤

ωm

φk =∫pωm dt

uks,ref

uks,comp

uks,pi u

abcs

sabcs

uss,ref u

abcs,ref

udc

αβ

αβ

αβ

αβ

abc

abc

dq

dq

PWM VSI

Clarke

transformation

Park

transformation

−

controller implementation

RSM

Nonlinear PI controller

with anti-windup

hardware

Fig. 4: Block diagram of the implemented simulation model.

an inverse interpolation algorithm is used to extract the actualstator currents iss = Tp(φk)iks from the available flux maps.

B. Implementation at the laboratory setup

1) Laboratory setup: The laboratory setup is depicted inFig. 5. It consists of the RSM (on the left) under test(with flux linkages as shown in Fig. 1) and an inductionmachine (on the right; not controlled for the experiments).Both machines are driven by voltage source inverters of thesame power rating. The implementation is done in C on aPentium-based real-time system. To illustrate and validate thecontrol performance as close as possible with respect to theproposed theoretical controller design presented in Sec. III, nofilters were implemented for current measurement. Hence, themeasured signals in Figures 6 and 7 exhibit noticable noise.However, the control performance is still acceptable, fast, andaccurate over the whole operation range of the RSM.

2) Discrete-time implementation at the real-time system:For the discrete implementation, the nonlinear PI con-troller (19) with anti-windup and the disturbance compensa-tion (15) are discretized by the (explicit) Euler method (i.e.,ddtx(t) ≈ x[k+1]−x[k]

Tsand x(t) ≈ x[k] where x[k] := x(kTs)

and Ts = 1fs 1 s) and implemented in C.

C. Discussion of the results

1) Discussion of Scenario (i) — Idle speed (see Fig. 6):The simulation results are shown in cyan (see inFig. 6) whereas the measurements results are shown in blue

Fig. 5: Reluctance synchronous machine (RSM) and induction machine (IM).

(see in Fig. 6). The control performance of the proposedcontrol strategy for current reference tracking is fast andalmost perfectly accurate. Solely, for higher currents (highertorque), the machine is rotating faster which leads to higherand ramp-like changes of the machine speed and hence tohigher and ramp-like changes of the back EMF. In turn, thecontroller structure generates a higher and ramp-like statorvoltage reference to compensate for the disturbance (12). Thenoise in the reference voltages uds,ref and uqs,ref are due tothe numerical (inverse) interpolation of the inductances andflux linkage values which was based on a look-up table withonly 25x25 supporting points. Nevertheless, on average, thesimulation results very much coincide with the measurementresults. The remaining deviations are due to iron losses andhysteresis effects in the machine which were not consideredin the machine model.

The measured control performance of the proposed controlstrategy for current reference tracking is very similar to thesimulation results. It is fast and almost perfectly accurate.Again, due to higher and more rapid changing torques forhigher currents, the machine speed exhibits faster and ramp-like changes which yields higher and ramp-like changes of theback EMF. The controller must counteract by applying higherand ramp-like stator voltage references.

Most important, in simulations and measurements, theclosed-loop current dynamics show an (almost) identicaltransient behavior: Over the complete operation range with‖iks ‖ ≤ 5 A, rise and settling times are similar for direct andquadrature currents, respectively. Hence, outer control loops(e.g. for speed and/or position control) can be designed basedon simple dynamical approximations of the underlying closed-loop current dynamics.

2) Discussion of Scenario (ii)—The positive effect of usingthe disturbance compensation (see Fig. 7): The measurementresults with disturbance compensation (15) are shownin blue, whereas the measurement results without dis-turbance compensation (i.e., uks,comp(t) = (0, 0)>) are shownin green. The results clearly illustrate the advantageous effectof the disturbance compensation on the control and trackingperformance. With disturbance compensation, direct andquadrature currents are decoupled and track the reference stepsaccurately, whereas without disturbance compensationthe tracking accuracy is deteriorated (in particular the trackingaccuracy of the quadrature current).

V. CONCLUSION AND FUTURE WORK

In this paper, nonlinear PI controllers with anti-windup andonline parameter adjustment have been proposed and imple-mented for current control of nonlinear reluctance synchronousmachines (RSMs) where the flux linkage maps are known and

available to the control engineer (e.g. in the form of look-uptables). The online adjustment of the PI controller parametersrequires the online tracking of the differential inductances,which are approximated by numerical differentiation of theflux linkage maps with respect to the stator currents. Inaddition, a nonlinear disturbance compensation (feedforwardcontrol) with online tracking of the flux linkages and thedifferential inductances has been proposed and implementedto compensate for the nonlinear cross-coupling between thedirect and quadrature current dynamics. The nonlinear PIcontroller design has been derived in state space and hasbeen tuned according to Magnitude Optimum. The proposedcontroller structure has been implemented in Matlab/Simulinkand at a laboratory test bench. Simulation and measurementresults fit closely, and illustrate and validate the expectedcontrol performance of the proposed nonlinear control strategy.The closed-loop current dynamics exhibit (almost) identicalrise and settling times over the whole operation range of themachine.

Future work will focus on (i) the consideration of iron losses(e.g. due to hysteresis and eddy currents), (ii) a robustnessanalysis (e.g. when the flux linkages are only roughly known),(iii) an analytical or numerical method for maximum-torque-per-Ampere reference current generation, and (iv) the combi-nation with sensorless control methods.

VI. ACKNOWLEDGMENT

The authors are deeply indebted to Dr.-Ing. Peter Lands-mann (Technische Universität München) for very fruitfuldiscussions and the interpolation algorithms for the imple-mentation in Matlab/Simulink and C. This study and themutual research visits in Germany and South Africa weremade possible by the financial support granted by the “Freundeder Technischen Universität München e.V”.

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id s[A

]

0

2

4ids,ref

iq s[A

]

−4

−2

0

2

4

iq

s,ref

‖uk s,ref‖[V

]

0

150

300 ±udc2

ωm[rad/s]

time t [s]0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

−100

−50

0

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100

(a) Overall control performance (from top to bottom): currents ids and iqs , norm reference voltage ‖uks,ref‖, and angular velocity ωm.

id s[A

]

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2

4ids,ref

ud s,ref[V

]

−300

−150

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±udc2

iq s[A

]

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4iq

s,ref

uq s,ref[V

]

−300

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300±

udc2

ωm[rad/s]

time t [s]0.1 0.15 0.2

−100

−50

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(b) Zoom 1.

id s[A

]

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4

ids,ref

ud s,ref[V

]

−300

−150

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±udc2

iq s[A

]

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s,ref

uq s,ref[V

]

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udc2

ωm[rad/s]

time t [s]2.5 2.55 2.6

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id s[A

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]

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s,ref

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]

−300

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300±

udc2

ωm[rad/s]

time t [s]4.9 4.95 5

−100

−50

0

50

100

(d) Zoom 3.

Fig. 6: Simulation results and Measurement results at idle speed (unfiltered!) for the RSM with flux linkage as depicted in Fig. 1: Controlperformance of the proposed nonlinear PI current controller (19) with online parameter adjustment (21).

id s[A

]

0

2

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time t [s]

iq s[A

]

1.22 1.23 1.24 1.25 1.26 1.27

0

2

4iq

s,ref

Fig. 7: Measurement results for the RSM with flux linkage as depicted inFig. 1: Tracking performance (i) with disturbance compensation,i.e. u

ks,comp = −uks,dist = −(uds,dist, uqs,dist)

> (as in (15)) and

(ii) without disturbance compensation, i.e. uks,comp = 02.

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