RBF Neural Networks Based Quasi Sliding ModeController and Robust Speed Estimation for PM

Synchronous MotorsL. Ciabattoni∗, M.L. Corradini†, M. Grisostomi∗, G. Ippoliti∗, S. Longhi∗, G. Orlando∗

∗Dipartimento di Ingegneria Informatica, Gestionale e dell’AutomazioneUniversita Politecnica delle Marche, 60131 Ancona, Italy

Email: {gianluca.ippoliti, sauro.longhi, giuseppe.orlando}@univpm.it†Scuola di Scienze e Tecnologie, Universita di Camerino,

via Madonna delle Carceri, 62032 Camerino (MC), Italy, Email: letizia.corradini@unicam.it

Abstract—This paper presents a neural networks based dis-crete time variable structure control and a robust speed estimatordesigned for a Permanent Magnet Synchronous Motor (PMSM).Radial basis function neural networks are used to learn aboutuncertainties affecting the system. A cascade control scheme isproposed which provides accurate speed tracking performance.In this control scheme the speed estimator is a robust digitaldifferentiator that provides the first derivative of the encoderposition measurement. The analysis of the control stability isgiven and the ultimate boundedness of the speed tracking erroris proved. The controller performance has been evaluated bysimulation using the model of a commercial PMSM drive.Simulations show that the proposed solution produces good speedtrajectory tracking performance.

I. INTRODUCTION

Permanent Magnet Synchronous Motors (PMSMs) have animportant role in motion control applications in the low andmedium power range (e.g. robotics and machine tool drives).The desired features of PMSMs are fast dynamical response,high torque to weight ratio, linear dependence of the torque onone component of the current in a suitable reference frame [1],[2]. To achieve fast four quadrant operation, smooth startingand acceleration, the Field Oriented Control (FOC), or vectorcontrol, is used in the design of the PMSM drive [3], [4].Standard linear design methods for FOC consist of a properlytuned cascade configuration of Proportional Integral (PI) speedand current controllers. Therefore accurate information regard-ing the motor parameters and load conditions is necessaryto guarantee good drive performance in terms of precision,bandwidth and disturbance rejection [5].In electric drive systems, electromechanical parameter vari-

ations are a well recognized problem and to overcome thisdrawback different robust control techniques have been fol-lowed, such as, for instance, adaptive and H∞ control, integra-tor backstepping and neural networks based control [6]–[10].In recent years, a widely and successfully applied nonlinear

control approach is the Variable Structure Control (VSC)method [11]–[14]. It is well known that VSC methods providenoticeable robustness and invariance properties to matcheduncertainties [12], and are computational simpler with respect

to other robust control approaches. VSC schemes are typicallybased on high frequency switching of the control signal but,as stated in [12], this well-known implementation drawbackof VSC does not cause difficulties for electric drives since theon-off operation mode is the only admissible one for powerconverters.On the contrary, control signals of the switching type gives

problems to the cascade control structure of FOC. In fact, FOCis realized with two current controllers in inner control loopsand a speed controller in an outer control loop. The speedcontroller provides the reference current for one of the twoinner current control loops; this reference current correspondsto the required motor torque. The cascade-based architecture isrepresented by the block scheme shown in Fig. 1 (the meaningof the signals and blocks shown in Fig. 1 will be explainedthroughout this paper). As stated in [12], for the outer speedcontrol loop VSC techniques can not be applied, since thereference input of the inner current control loop should havebounded time derivatives.To overcome the problem of the cascade control structure

different approaches to VSC have been followed, such as, forinstance, the ‘direct speed control’ of [12] and the ‘second-order sliding-mode technique’ of [11]. Anyway, in our opinionthe continuous time approaches presented in the above papersdoes not allow an appropriate formulation of the problem ina sampled-data systems context, which is doubtless useful inview of the practical discrete-time implementation on the DSPof a real motor drive.A possible solution is presented in this paper, where a

control system based on Discrete-Time VSC (DTVSC) [15]–[18] is designed. Following the approach proposed in [11],the considered solution is a direct generalization of theconventional PI-based cascade control scheme for PMSMvector control, with every PI controller being replaced by theDTVSC. The introduction of DTVSC, in fact, allows to takedirectly into account the issue of control law digitalization andto ensure robustness with respect to disturbances and modeluncertainties.Nevertheless, the discrete time counterpart of sliding mode

978-1-4244-5226-2/10/$26.00 ©2010 IEEE 2402

control design has received only a limited attention [15]–[18]. Indeed, compared with continuous time sliding-modestrategies, the design problem in discrete-time has receivedmuch less coverage in the literature. This is due to its majordrawback, consisting in the presence of a sector, of widthdepending on the available bound on uncertainties, whererobustness is lost because the sliding mode condition cannotbe exactly imposed any longer. For this reason, only ultimateboundedness of trajectories can be guaranteed, and the largestare the uncertainties affecting the system, the wider is thebound on trajectories which can be guaranteed.As a possible solution to this problem, this paper pro-

poses the design of a discontinuous control law, within thesector, based on an estimation, as accurate as possible, ofuncertainties affecting the system. Owing to their learningcapabilities, Neural Networks (NNs) will be used here toperform this approximation. It is well known in fact, that thelearning ability of neural networks has been widely utilized inelectromechanical systems to make controllers learn nonlinearcharacteristics of plants through experimental data, without aprior knowledge of their parameters and structure. In particularthe fusion of sliding mode control with neural networkshas been investigated by many researchers and a literaturesurvey can be found in [19]. Moreover recently, a considerableamount of research in the field of NNs-based control of electricdrives has been carried out, in order to exploit the propertyof NNs to learn complex nonlinear mappings [20], [21]. Inthis paper Radial Basis Function Networks (RBFNs) havebeen considered. These networks have been widely used fornonlinear system identification [22], [23] because of havethe ability both to approximate complex nonlinear mappingsdirectly from input-output data with a simple topologicalstructure that avoid lengthy calculations [23] and to revealhow learning proceeds in an explicit manner.High performance control of PMSM drives also requires

the knowledge of the shaft speed [11], [24]. The standardbackward-difference method for speed estimation, using sam-pled position measurements provided by a digital incrementalencoder, gives high errors in particular at low speed [25],[26]. To improve the basic differentiation algorithms, sliding-mode approaches have been considered in [27], [28] andan interesting robust real-time digital differentiator based onsecond-order sliding mode is presented in [11]. Moreover, anumber of results have been given considering the use ofnonlinear observers to overcome the problem of ineffectivespeed measurement [29]–[31].In this paper, a quasi sliding mode speed estimator is

presented, aimed at supporting the discussed robust controlarchitecture. The differentiator, coupled with the controller, isshown to guarantee the ultimate boundedness of both the speedestimation error and the tracking error.Summing up, the features of the NN-based DTVSC tech-

nique combined with the robust speed estimator are exploitedin this work to design the cascade-based architecture shownin Fig. 1, where it can be identified the external speed DTVScontrol loop, two internal current DTVS control loops and

the speed estimator which is a robust differentiator based onsliding modes. The task is to make the speed error ω∗r − ωr

to tend to zero as close as possible.The paper is organized as follows. The motor dynamics is

presented in Section II. In Section III some preliminaries aregiven and details on the considered robust speed estimatorand NN-based DTVS controller are discussed. Results onnumerical tests are reported in Section IV. The paper endswith comments on the performance of the proposed controlscheme.

Fig. 1. Block scheme of the proposed cascade controller (FOC)

II. MOTOR DYNAMICS

In the (d, q) reference frame, synchronously rotating withthe motor rotor, the electrical equations of motion of apermanent-magnet synchronous motor can be written as [12],[32]:

diddt

= −R

Lid + ωeiq +

1

Lud (1)

diqdt

= −R

Liq − ωeid −

1

Lλ0ωe +

1

Luq (2)

where id and iq are the d− axis and q− axis stator currents,respectively; ud and uq are the d− axis and q − axis statorvoltages, respectively; R is the winding resistance and L =Ld = Lq is the winding inductance on axis d and q; λ0 is theflux linkage of the permanent magnet and ωe is the electricalangular speed of the motor rotor.The electrical torque τe and the mechanical power P of the

motor are given by:

τe = Ktiq (3)P = τeωr (4)

in which Kt = 32λ0Nr is the torque constant with Nr the

number of pole pairs and ωr is the mechanical angular speedof the motor rotor. The developed torque of the motor isproportional to the iq current because of the assumption thatthere is no reluctance torque in the considered PMSM.

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The mechanical motion equation of the motor is describedby:

Jdωr

dt+ Bωr = τe − τ� (5)

dθr

dt= ωr (6)

where J is the mechanical inertia of the motor and load, B isthe coefficient of viscous friction, τ� is the load torque and θr

denotes the mechanical angular position of the motor rotor.For the electrical angular position/speed and the mechanical

angular position/speed, these relations hold: θe = Nrθr, ωe =Nrωr.

III. CONTROL DESIGNThe field-oriented control scheme of Fig. 1 is based on the

measure of two phase currents (ia and ib) and of the motorrotor position (θr) given by encoder measurements. The speedand current controllers are DTVSCs and the speed estimatorblock is a robust differentiator, based on sliding modes, aimedat estimating angular velocity from angular position data givenby an encoder.

A. PreliminariesA RBFN with input pattern χ ∈ R

m and a scalar outputy ∈ R implements a mapping h : R

m → R according to

y = h(χ) = λ0 +

L∑i=1

λiφ (‖χ− ci‖) (7)

where φ(·) is a given function from R+ to R, ‖ · ‖ denotes

the Euclidean norm, λi ∈ R, i = 0, 1, · · ·, L are the weightsor parameters, ci ∈ R

m, i = 1, 2, · · · , L, are the radial basisfunctions centers (called also units or neurons) and L is thenumber of centers [23].In this paper the RBFN is used for the estimation of the

uncertainties affecting the electric motor drive. The uncertaintydynamics can be taken into account through the network inputpattern χ, that must be composed of a proper set of systeminput and output samples acquired in a finite set of past timeinstants [33] as specified in (29).Theoretical investigation and practical results show that the

choice of the non-linearity φ(·), a function of the distancedi between the current input χ and the centre ci, does notsignificantly influence the performance of the RBFN [23].Therefore, the following Gaussian function is considered:

φ(di) = exp(−d2

i /β2i

), i = 1, 2, · · ·, L (8)

where di = ‖χ− ci‖ and the real constant βi is a scaling or“width” parameter [23].

B. Robust differentiationConsider a measurable signal x1(t) described by the fol-

lowing differential equations:{x1(t) = x2(t)x2(t) = w(t)

(9)

where w(t) is an unknown external variable correspondingto the second derivative of the signal x1(t). Our aim is toachieve an estimate of the first derivative x2(t) of x1(t) atgiven sampling time instants kTc, k = 1, 2, . . . ,. Integratingthe second equation of (9) one gets:

x2(k + 1) = x2(k) +

∫ (k+1)Tc

kTc

w(τ)dτ = x2(k) + v(k)

and integrating the first equation of (9):

x1(k + 1) = x1(k) + Tcx2(k)

+

∫ (k+1)Tc

kTc

∫ t

kTc

w(τ)dτdt = x1(k) + Tcx2(k) + w(k).

Therefore it holds:{x1(k + 1) = x1(k) + Tcx2(k) + w(k)x2(k + 1) = x2(k) + v(k)

(10)

Assumption 3.1: It is assumed that a positive constant Wis known such that:

sup

∫ t

kTc

w(τ) = W � Tc maxτ>0

|w(τ)| ∀t ∈ [kTc (k + 1)Tc].

Consider the following Unknown Input Observer (UIO):{z(k + 1) = Fz(k)x2(k) = z(k) + Hx1(k) + v(k − 1)

(11)

where z(k) is the observer state, v(k) is an auxiliary input,H = T−1

c , and F is a design constant, with |F | < 1. Thefollowing result holds.Theorem 3.1: Given the system (10) under Assumption 3.1

and the UIO (11), the auxiliary input v(k) = veq(k) + vn(k),with:

veq(k) = −Fz(k)−Hx1(k) (12)

andvn(k) = θ (ε− 2W ) (13)

where θ and ε are design parameters, with |θ| < 1, ε > 2W ,guarantees the attainment of the following condition:

|x2(k)− x2(k)| < ε (14)

Proof: Consider the following recursive quasi-slidingsurface:

s(k+1) = λs(k)+(x2(k+1)−x2(k+1)); 0 < λ < 1. (15)

Since:

|s(k + 1)− λs(k)| = |x2(k + 1)− x2(k + 1)| (16)

the choice of λ in surface (15) ensures that the achievementof a quasi-sliding motion on it guarantees the boundednessof the observation error. A quasi sliding mode on (15) canbe obtained using the auxiliary input v(k) = veq(k) + vn(k).In fact, substituting (12) and (13) in (15), and imposing thecondition |s(k +1)−λs(k)| < ε to ensure the achievement ofthe quasi sliding motion within a sector of width ε, one gets:

|s(k + 1)− λs(k)| < |v(k)−1

Tc

w(k)− vn(k)| < ε (17)

Condition (17) is guaranteed by (13), and the statementfollows.

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C. Discrete-Time Variable Structure ControlThe discretization of the model equations with a sampling

time Tc according to standard techniques gives:

ωe(k + 1) = Aωωe(k) + Bω(Ktiq(k)− τ�) (18)id(k + 1) = Aiid(k) + Biud(k) + f1(ωe, iq, k) (19)iq(k + 1) = Aiiq(k) + Biuq(k)− f2(ωe, id, k) (20)

with

Aω = e−B

JTc , Bω =

1

J

∫ Tc

0

e−B

Jτdτ

Ai = e−R

LTc , Bi =

1

L

∫ Tc

0

e−R

Lτdτ

f1(ωe, iq, k) =

∫ (k+1)Tc

kTc

ωe(τ)iq(τ)dτ � ωe(k)iq(k)Tc;

f2(ωe, id, k) =

∫ (k+1)Tc

kTc

ωe(τ)(id(τ) +λ0

L)dτ

� ωe(k)(id(k) +λ0

L)Tc.

To account for possible model uncertainties, it is assumed thatmodel parameters may differ from their nominal values forsome unknown but bounded quantities:

Aω = Aω + ΔAω ; Bω = Bω + ΔBω;

|ΔAω | ≤ ρAω; |ΔBω| ≤ ρBω

Ai = Ai + ΔAi; Bi = Bi + ΔBi;

|ΔAi| ≤ ρAi; |ΔBi| ≤ ρBi. (21)

Define the following discrete-time sliding surfaces:

sω(k) = (ωe(k)− ω∗e(k)) + λω(ωe(k − 1)− ω∗e(k − 1)) = 0(22)

siq(k) = (iq(k)− i∗q(k)) + λq(iq(k − 1)− i∗q(k − 1)) = 0(23)

sid = id(k) + λdid(k − 1) = 0 (24)

where λω , λq, λd ∈ (−1, 1), ωe(k) is the estimate of ωe(k)provided by the UIO (11), ω∗e(k) is the given reference valuefor the angular velocity, and i∗q(k) will be defined in thefollowing. Note that the achievement of a sliding motion onthe surface sω(k) = 0 guarantees that the estimated speedωe(k) tends to the reference variable. This fact, coupled withthe result of Theorem 3.1 when the variable x2 is the electricalspeed, ensures that the speed tracking error is ultimatelybounded.As well known, a quasi sliding motion on the surface

sω(k) = 0 can be achieved imposing the following discretetime sliding mode existence condition [16], [17]:

sω(k)Δsω(k + 1) < −1

2[Δsω(k + 1)]2 (25)

being Δsω(k + 1) = sω(k + 1) − sω(k). It can be easilyverified that condition (25) is ensured by the control law

i∗q(k) = ieqq (k) + inq (k), where the equivalent control is given

by:

ieqq (k) =

1

BωKt

(ω∗e(k+1)− Aωωe(k)−λω(ωe(k)−ω∗e(k))).

(26)When replacing (26) in sω(k + 1), one obtains:

sω(k + 1) = Bωinq (k) + n(k) (27)

where n(k) summarizes the uncertain terms which cannot betaken into account by the equivalent control (26). As usual,the discontinuous control inq is such that the sliding condition(25) can be imposed exactly only outside a given sector.Inside such sector the sliding condition can be imposed onlyapproximately as:

inq (k) =

⎧⎪⎨⎪⎩

θω

|sω(k)| − ρω

BωKt

if |sω(k)| > ρω

−n(k)

BωKt

if |sω(k)| ≤ ρω

(28)

with |θω| ≤ 1, and with

ρω = (|Bω|+ ρBω)ρτ + ρAωωmaxe + KtρBωimax

q

ρτ being the constant bound of the unknown load which canaffect the motor, i.e. |τ�| ≤ ρτ . Note that ωmax

e and imaxq

are the largest speed achievable by the motor and the largestcurrent which can be supplied, respectively, according to itsconstructive limits.The estimation n(k) of n(k) is performed by a neural

network h : R6 → R of the form (7) with the output

y(k) := n(k) ∈ R and the input pattern χ ∈ R6 defined

as:

χ(k) := [ωe(k) ωe(k−1)ω∗e(k)ω∗e(k−1) inq (k−1)n(k−1)].(29)

The desired output of the network y(k) := n(k) ∈ R, from(27), has the form:

n(k) = sω(k + 1)− Bωinq (k). (30)

The algorithm used for net training minimizes the NN func-tional approximation error e(k) := n(k) − n(k) along theconsidered reference trajectories.The control law i∗q(k) is fed as reference value, which is

the required motor torque, to one of the two inner currentcontrol loops (see Fig. 1). The tracking of such reference isensured by the imposition of a quasi sliding motion of thesurface siq(k) = 0. Following the same lines as before, it canbe easily verified that the sliding condition on siq(k) = 0 isensured by the control law uq(k) = ueq

q (k) + unq (k), where:

ueqq (k) =

1

Bi

[i∗q(k)− Aiiq(k)− λq(iq(k)− i∗q(k))

]. (31)

In this case, to reduce the computational burden required bya second neural network, inside the sector the sliding condition

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is imposed approximately by the approach known as TimeDelay Control [34], [35], obtaining:

unq (k) =

⎧⎪⎪⎨⎪⎪⎩

θq

|siq(k)| − ρq

Bi

if |siq(k)| > ρq

−siq(k)− Biu

nq (k − 1)

Bi

if |siq(k)| ≤ ρq

(32)where |θq| ≤ 1, ρq = ρAii

maxq +ρBiu

maxq +ρ+ωmax

e (imaxd +

λ0

L)Tc, ρ being the bound of Δi∗q(k) = |i∗q(k + 1)− i∗q(k)|.Finally, the achievement of a quasi sliding motion on

sid(k) = 0 guarantees the vanishing of the variable id(k),and is ensured by the control law:

ueqd (k) = −

(Ai + λd)id(k)

Bi

(33)

und (k) =

⎧⎪⎨⎪⎩

θd

|sid(k)| − ρd

Bi

if |sid(k)| > ρd

−sid(k)− Biu

nd(k − 1)

Bi

if |sid(k)| ≤ ρd

(34)where |θd| ≤ 1 and ρd = ρAii

maxd +ρBiu

maxd +ωmax

e imaxq Tc.

For the same reason as above, also in this case the TimeDelay Control approach is used inside the sector.

IV. NUMERICAL RESULTSThe proposed DTVS controller and the robust differentiator

have been tested by intensive simulations using the model ofthe Technosoft MBE.300.E500 PMSM, as a preliminary stepbefore experimental validation on the Technosoft MCK28335-Pro DSP motion control kit [36]. The kit is available atthe Advanced Robotics Laboratory of the Dipartimento diIngegneria Informatica, Gestionale e dell’Automazione of theUniversita Politecnica delle Marche.

A. Structure and Validation of Implemented Neural NetworkTraining and testing phases of considered NN have been

performed offline with data acquired on a set of plannedtrajectories chosen with different shapes. In particular a RBFNis designed to estimate the uncertainty n(k) in the speedcontrol loop (see eq. (30)). These data sets are used to train thenet offline by the OLS-based algorithm [23], [37]. Data havebeen also normalized in order to have the same range. Thecomplexity of RBFN (i.e. the number of centers), has beenchosen to match the approximation capability of net to a lownet complexity which is necessary for the implementation ofthe control scheme on a DSP. The number of centers has beenchosen to obtain a good trade off among the time complexityof learning, computation efforts of the resulting NN controlscheme and the accuracy of predictions for obtaining satis-factory control performance. Therefore, hidden layer of thisnet is chosen with a number of 25 centers. Choices operatedon net structure, have been confirmed by cross-validation tests,which have been used to check the performance generalizationof the network in order to avoid the overfit of the data usedfor training the net. Different trajectories not used duringthe training phase have been considered for cross-validation

tests. Trained NN has shown good capability to generalize itspredictions in all considered validation trajectories different tothe trajectories of the training phase.

B. Control PerformanceA sample of the performed speed-tracking simulations is

shown in Fig. 2. In this figure the performance produced by theproposed NN-based DTVS-FOC with the robust differentiatoris illustrated for the motor following a square-wave velocityprofile. A comparison with the performance of a DTVSC-based FOC without the NN, i.e. the Time Delay Controlapproach is used also for the speed control loop, has beenalso made.In particular it is apparent from inspection of Fig. 2 that the

NN-based DTVS-FOC (Fig. 2(a)) produces a lower chatteringduring the constant speed part of the reference trajectorywith respect to the DTVSC-based FOC without the NN (Fig.2(b)). In all the simulations, the sampling frequency has been

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−60

−40

−20

0

20

40

60

Time [s]

Spe

ed [r

ad/s

]

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−60

−40

−20

0

20

40

60

Time [s]

Spe

ed [r

ad/s

]

(b)

Fig. 2. Square-wave velocity profile. Actual (blue continuous line) andreference (red dashed line) velocities: (a) NN-based DTVS-FOC; (b) DTVSC-based FOC without NN

selected as 1 kHz for the velocity control loop and 10 kHzfor the current control loops according to the real devicespecifications.The criterion IAE, i.e. the integral of the absolute value of

the speed tracking error, is used to summarize the above sim-ulation result and also the performance obtained consideringa reference trajectory given by a trapezoidal velocity profile(see Table I).

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TABLE IPERFORMANCE COMPARISON.

Velocity profile NN-based DTVS-FOC DTVSC-based FOCwithout NN

Square-wave 1.778 2.059

Trapezoidal 0.359 0.491

V. CONCLUDING REMARKS

In this paper a NN-based discrete-time VSC is proposedfor the speed control of a PMSM. The approximation ca-pability of NN is used to learn about system uncertainties,and ultimate boundedness of the tracking error is guaranteed.The considered nonlinear control scheme is based on thecascade implementation of discrete-time VS controllers andcan be considered a generalization of the classical PI-basedfield oriented control, where every PI-based controller beingreplaced by the DTVSC. The proposed solution makes also useof a robust differentiator to estimate the shaft speed unavailableto direct measurement. The differentiator, coupled with thecontroller, is shown to guarantee the ultimate boundedness ofboth the speed estimation error and the tracking error. Thepresented solution has been validated by simulations on themodel of a commercial PMSM drive. Simulations show goodspeed trajectory tracking performance. Currently, the developof efficient code on the DSP for the experimental testing ofthe considered control scheme is being carried out.

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