Adaptive feedback linearization for position control of a switched reluctance motor: Analysis and simulation

INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING, VOL. 7, 117-136 (1993)

ADAPTIVE FEEDBACK LINEARIZATION FOR POSITION CONTROL OF A SWITCHED RELUCTANCE MOTOR:

ANALYSIS AND SIMULATION

LOTFI BEN AMOR &ole Polytechnique de Montdal, CP 6079, Sumtrsale ‘A’, Montdal (QuCbec) H3C 3A7, Canada

LOUIS-A. DESSAINT AND OUASSIMA AKHRIF i?cole de Technologie Supkrieure. 4750 Ave. Henri-Julien, Montrecll (Qudbec) HZT ZC8, Canada

AND

GUY OLIVIER &ole Polytechnique de Montrecll, CP 6079. Sumusole ‘A’, Montdal (Qudbec) H3C 3A7. Canada

SUMMARY In this paper a non-linear adaptive feedback-linearizing control is designed for a fifth-order model of a three-phase switched reluctance motor (SRM) which includes both electrical and mechanical dynamics. This non-linear adaptive control structure compensates for all the non-linearities between inputs and outputs, allows the use of a linear controller for motion tracking and improves the performance by reducing torque ripple of the SRM. A validated non-linear model of the SRM is used for the system simulation, while the control algorithm

contains an adaptive scheme based on the parametrized model. Simulation results are given to demonstrate the effectiveness of the control method.

KEY WORDS Motor Linearization Adaptive control Torque ripple

1. INTRODUCTION

In recent years, switched reluctance motors (SRMs) have gained increasing popularity because of their simple structure, low cost and the simplicity of the associated unipolar power converter. Moreover, the SRM can produce high torque at low speed. These characteristics make the SRM attractive for direct-drive applications.

SRM models, however, exhibit significant non-linearity and uncertainty. Hence classical linear control schemes cannot provide the high dynamic performances required by variable speed regulation or position tracking. It would thus be interesting to investigate the development of an appropriate design control which can fulfil the required performances for high-precision position control.

This paper was recommended for publication by editor M. J. Grimble

0890-6327/93/020117-20$15 .OO 0 1993 by John Wiley & Sons, Ltd.

Received January 1993

118 L. BEN AMOR ET AL.

Earlier, open-loop strategies were mainly investigated for SRM control. Despite the many attempts that have been made to improve dynamic performances using linear control techniques, ' J high torque ripple was observed owing to the neglected non-linearities.

Torque non-linearities are taken into account in Reference 3 and are handled using Floquet theory. The assumption of constant speed operation is, however, made. In Reference 4 a feedback linearization technique is applied to the full-order SRM model to compensate for the non-linearities. It is assumed that all plant parameters are known and that the motor acceleration is measurable. In Reference 5 feedback linearization techniques are applied to the reduced-order SRM model, where the reduction is achieved using singular perturbation techniques. This method guarantees a high dynamic performance by reducing the torque ripple. The drawback is that it requires off-line knowledge of the torque-position-current characteristics. Moreover, it requires a complex linearizing and decoupling transformation circuit.

The input-output linearization technique6-' is now well established and has been successfully implemented in various practical applications (e.g. the control of flight dynamics' and rigid link robot manipulators lo). Combined with adaptive control techniques, the feedback-linearizing method can be applied to various unknown parameter linearizable systems. ' * - I ' Since this technique is in the early stage of development and since many practical control problems now require adaptive non-linear compensation, it is clearly beneficial to show successful applications of the theory.

To avoid several restrictive considerations previously used, we propose to apply the adaptive non-linear control methodology to the full-order SRM model instead of the reduced-order model. In this way we reduce the effect of modelling errors on the system. We also propose to use on-line estimation of the parameters to avoid the preliminary tests needed by previous applications. With the high-performance microprocessors available nowadays, this can be done without complex hardware circuitry. Using the proposed method, high dynamic performance is achieved (e.g. dramatic torque ripple reduction and accurate position control).

The paper is organized as follows. In Section 2 a validated mathematical SRM model as well as the electronic commutator are presented. A parametrized model is then developed. In Section 3 the basic theory of the input-output adaptive linearizing technique is reviewed. In Section 4 details of the application of the advanced control technique to the SRM are given. Finally, simulation results are presented in Section 5 .

2. MATHEMATICAL MODEL OF THE SYSTEM

2. I. SRM dynamics

The SRM is a doubly salient, brushless motor with no winding or magnet on its rotor. Currents in stator phases are switched on and off depending on the rotor position as dictated by an electronic commutator (Figure 1). Unlike the DC motor, the SRM exhibits a coupled non-linear structure, since it operates with magnetic saturation in order to maximize the torque/mass ratio. In References 13 and 14 a fundamental treatment of the principle of operation is given. Several models have been proposed. ''

The expression relating voltage and current in each phase j is found using Kirchhoff's laws and is

POSITION CONTROL OF A SWITCHED RELUCTANCE MOTOR 119

u1 Electronic u2

u3 commutator

K = 1,2,

Converter

8 position

Figure 1. Block diagram of system

where Uj , r, Ij and $j are the stator voltage, winding resistance, stator current and flux linkage of phase j respectively. Owing to symmetric location of the poles, the mutual inductance between phases can be neglected, so that the flux linkage of phase j depends on the rotor position 8 and the current Ij only.

First, the flux $j is a non-linear function of the rotor position 8 owing to periodicity of the alignment between stator and rotor poles as the position varies. Secondly, $j is also a non- linear function of the stator current f j owing to magnetic saturation of the iron.

The following expression considers these two non-linearities:

$j(e, 1,) = ~ X P [ - Ijh(@)ll (2)

where

f j ( 8 ) is the Fourier series expansion of the j-phase reluctance and NR is the number of rotor linkages.

The torque produced by phase j is determined by differentiating the coenergy function W,! with respect to 8:

where 4

W,w,Ij) = 1 $ji(e,li) dI j 0

From (4) and ( 5 ) the torque expression is deduced as

The total torque produced by a three-phase SRM can be expressed as 3

j- 1 T ( ~ , I ~ , I ~ , I ~ ) = C Tj(8,I.I (7)

Both mechanical and electrical dynamics can be represented by the expressions

d8 _ - dt -

120 L. BEN AMOR E T A L .

where o and TL are the speed and load torque respectively.

2.2. The electronic commutator

In Reference 5 the electronic commutator is modelled as a system which has the position and the desired torque sign as inputs and the selected phase index K as output. The commutation angles 80 and 00' are internal parameters which are to be optimized for a minimum torque ripple at commutation instants. Since the commutator proposed in Reference 5 cannot operate correctly at negative position, some modifications must be done. In this paper a modified commutator is presented. The output of the modified commutator is expressed as

(1 1) 80 = 80 if sign(Ted) c 0 60 = eo' if sign(Ted) > 0 K = 1 + [ i n t ( x 3NR [Abs(B) + Oo] ) mod 31, where [

If the position is negative, then the following permutation must be done. If K = 3, then phase 2 is selected and vice versa.

2.3. Linear parametrization of the SRM model

An important assumption necessary for application of the control method is that the unknown parameters appear linearly in the non-linear SRM model. Given a non-linear single- input/singlesutput (SISO) system expressed by the model

1 = f (XI + g(x) u, Y = h(x) (12)

with xE IT?" and f, g and h are smooth functions, the parametrization of this non-linear model consists.of expressing the functions f(x) and g(x) as a summation of the product of unknown parameters and measurable functions:

where fi(X) and gl(x) are the measurable functions and pfi and pgj are the unknown parameters. Since the SRM model contains highly non-linear relations between state variables and parameters, the parametrization will result in a high number of parameters and the adaptive control scheme will therefore be very difficult to apply. In order to reduce the number of parameters and make adaptive control possible, the following assumptions must be made.

(i) The saturation effect is neglected in the SRM model. This leads to a simple relation between the flux, reluctance and current phase of the SRM:

$j(/(e, lj) = LAWj (14)

This assumption will simplify the design of the adaptive controller without degrading the dynamic performance.

(ii) Harmonic analysis of the phase reluctance Lj(0) shows a very low rate of harmonics. This result allows the use of the fundamental term only (Figures 2 and 3).


. . . . , , . , , , , , . . . . , , . . . . . ,

0 0.05 0.1 0.15 0.2 0.25 0.3 ROTOR POSITION (rad)

Figure 2. Phase reluctance (-) and first harmonic (- - -)

HARMONIC ORDER

Figure 3. Spectrum of phase reluctance

(iii) A sixth-order Taylor series approximation is used to express the term (a$j /aI j ) - ’ in a

Using these assumptions, we obtain

parametrized form.

The first assumption implies

This in turn implies

where

a = $SU, P = b/u, j = 1,2,3

1 22 L. BEN AMOR ET AL.

Using (4) , (5) and (14), we obtain

T = 5 ( Y / ~ N R (COSJ I: + C O S ~ I f + COS3 I f ) The load torque can be expressed as

TL = Mgl sin 8 (19)

Using the third assumption, we obtain

( 1 - sin j + 0 2 sin j2 - 0 3 sin j3 + 84 sin j4 - 0 5 sin j5 + 06 sin j 6 ) (20)

The parametrized model can then be expressed as

dw - = [PWIBXl) [Fwl (21)

dt W , d8 d t -=

- d'j= [PF]fix14) [Fj1(14x1) + [ P G ] f i x 7 ) [ G j l ( 7 ~ 1 ) ~ j , j = 1,2,3 dt

where [Pw], [PF] and [PG] are the unknown parameter vectors

PF= [ Pf i 1 1, PG= [ Pgl ; I , Pw=[Pwl] Pa2 Pf 14 Pg7

and (Fw], [Fj] and [Gj] are the measured functions

The unknown parameters pfi , pgi and poi can be expressed as

pfl = ria, Pf2 = Prla, pf3 = P2r/a, pf4 = P r l a Pfs = P ~ ~ I ~ , Pf6 = Psr/a, Pf7 = P6rla, Pfa = B

Pf9 = P 2 , PflO = f i 3 , pf11= P4, P f n = B5 Pf13 = b6, Pf14 = B 7 , Pgl= 1/01. pgz = PI01

pg3 = P2/01, pg4 = P3/01, pgs = P4/% Pg6 = a'/.! POI = aP/ J , pg7 = P 6 / % pw2 = M/ J

The f i j , gij and fwi are given by

f i j = - Ij, f2j = sin jI j , f6j = sin j s I j ,

f3j = -sin j21j, f7j = -sin j61j,

f i j = sin j 3 r j faj= -Nu cos jljw

f i l j = N u cos j sin j601j f i 4 j = -Nu cos j sin j6wIj

f5j = -sin j41j, f9j = NR cos j sin j w l j ,

f i 2 j = - NR cos j sin j4,Ij, fioj = -Nu cos j sin j2wIj,

f i3j = NR cos j sin j s o I j , g i j = 1 , gzj = -sin j , g3j = sin j 2 , g4j = -sin j 3 , gsj = sin j 4

g6j = -sin j ' , g7j =sin j 6 , fw2 = -g l sin 8


3. ADAPTIVE LINEARIZING CONTROL DESIGN

3.1. Basic theory

3. I. 1. Non-adaptive linearizing scheme. In order to simplify the description of the method, we consider a SISO system of the form (12). To achieve the input-output linearization, we must follow three steps.

1. Generate a linear input-output relation by differentiating the output y until the input u

2. Determine the internal dynamics associated with the input-output linearization and study

3. Design a stable controller based on desired error dynamics and determine the non-linear

appears.

its stability.

control law u that will cancel the non-linearities and guarantee tracking.

Differentiating y with respect to time until the input appears, we obtain

y ( r ) = Ljh(x) + L,Lj-'h(x)u (22)

where Lfh and Lgh are the Lie derivatives of h with respect to f and g respectively6 and r, the relative degree, is the smallest integer such that L,L)h((x) = 0 for i = 0, ..., r - 2 and L,L;-'h(x) # 0 v x c IR". The linearizing control law is then

which yields

u=y'

If r is equal to the order of the system (r= n), then the system is exactly feedback- linearizable or input-state-linearizable. If r < n, then one must study the stability of the internal dynamics rendered unobservable. We will show that the system we study is exactly feedback-linearizable so that there is no internal dynamics.

For MIMO systems the same method is applied for each output. Consider a p-inputlp- output non-linear system of the form

X = f ( x ) + gl (x)ul+ . * ' + gp(x)up, Y = [ ~ i ... yplT= [hi (x) ... hp(x)IT (25)

where xE R", u c Rp, yE IRp and f ( x ) , g j ( X ) and hj are smooth functions, j = 1, ...,p. Differentiating yj with respect to time until at least one input appears, we obtain

With at least one of the Lpi(LT-'hj) # 0 Vxc I?", the relative degree corresponding to yj is rj. Define the p x p matrix A ( x ) as

Lp, (L;' - 'h i ) . . . LE, (L;' - ' h 1 )

Ls, (LjTp- hp) . . . L,( L;P- hp) A ( x ) =


Then equation (26) can be written as

If A ( x ) is bounded away from singularity, then the linearizing control law is

Lj'hi u = - A ( x ) - ' [ i h ] + A ( x ) - ' v

Lf p

u = [Ul ... U p ] T (30) T v = [ V I ... u q T = [yrl ... yJp] , If Zj-1 rj = n, then exact input state linearization is achieved.

3.1.2. Adaptive linearizing scheme. The principal drawback of the linearizing control scheme is that it is based on knowledge of the non-linear functions f ( x ) and g(x) . If there is any uncertainty in the knowledge of these functions, the cancellation of the non-linear terms is not exact and the resulting input-output equation is not linear. We suggest the use of parameter adaptive control to get asymptotically exact cancellation. The adaptive control method is based on the parametrized model of the system.

SISO with relative degree r = 1. Consider a SISO system of the form (12) with relative degree r = 1. From (13) we define

where p x and a, are the estimates of pfi and pgj respectively at time t . Consequently, the control law u in (23) is replaced by

u=& (- 0 + v )

where

Let us define

If the control law used for tracking is


where a is a design parameter, then we can show that the error ( e = y - y m ) satisfies

b+(re= +'w (33)

(34)

(35)

Consider the candidate Lyapunov function

Y = v ( e , 4) = i e 2 + 4 4 '4 V = -(re2 + 4'4 + 4'eW

To verify the stability condition, i.e. V < 0, we make the term 4'4 + +'eW equal to zero by choosing the following adaptation law which guarantees convergence of the tracking error to zero as f + 00:

4= -eW (36)

f i= - 4 = e w (37)

If we suppose that the real parameters are constant or change very slowly, then

The output and parameter errors e and 4 converge exponentially to zero if W is sufficiently rich, i.e.12

S + b

s 3a l ,a2 ,6>0 al l2 J WWTdt)(r21 (38)

SISO wifh relative degree r > 1. In this case the linearizing control law (23) is replaced by

where LTh and m a r e the estimates of L>h and L,Lj-'h respectively. The parameter vector is augmented with the relative degree as shown in Reference 12 and -

O=y&+a1(y;-'- L j - ' h ) + . - . + a r ( y m - y ) (40)

e r + a I e r - ' + . . ' + a , e = 4 ' W l + 4 ' ~ 2 = 4 ~ ~ (41)

As for the SISO system of relative degree one, we can deduce the adaptive error equation

where 6, 0, e(3), . . . , e(r- ' ) are not measurable. The so-called augmented error scheme is necessitated. l 1 This can be done by filtering the regressor vector W and the scalar product <P, w.

Let

1 M ( s ) = S' + a1sr-l + ..* + a r

Then

e = ~ ( s ) 4 'W

Define the augmented error

el = e + [P'M(s) w - M(s)P' W I P=++P

(43)


The augmented error can be replaced by

el = e + [ + T ~ ( s ) ~ - ~ ( s ) + T ~ ~ (4)

el = +Tt (47)

t = M ( s ) w (48)

Replacing (43) in (46), we deduce

where

Equation (47) is a standard form in adaptive control and we can then apply the gradient algorithm to find the adaptive law. l6

The gradient adaptation

or in the normalized form

with PO > 0.

law can be expressed in the standard form

B = -ci,=PoelE

. Poelt p = -+= 1 + tTE

Using either of these forms leads to a stable adaptation control. To show this, let V be a Lyapunov candidate function

v = ; +T+ > 0

V = -Pee: < 0 (51)

Using the standard form, we obtain

Using the normalized form, we obtain

For MIMO systems we use the same procedure for each output. The feedback control law can be made adaptive by replacing the control law (29) by

(53)

u = [u, ... UPIT, o = [Ol ... ElpIT

4. APPLICATION TO POSITION CONTROL OF THE SRM

4.1. Control objectives

to the unselected ones. In the rest of this paper, the index K will refer to the selected phase, and K - 1 and K + 1

At each rotor position and desired torque sign the control objectives are defined as follows. (i) The control UK should be the only one responsible for producing the desired motion.

(ii) The effect of the unselected phase currents must be decoupled from torque production. (iii) The currents ZK- 1 and ZK+ I should be forced to decay quickly to zero.


By using the adaptive linearization technique, these objectives must be achieved. The system formed by the switched reluctance motor (SRM) and the electronic commutator

can be presented as a three-input/three-output system of fifth order. The inputs of the system are UK as a motion control input and U K - ~ and UK+I as current-stabilizing inputs for the unselected phases.

According to the control objectives defined before, the position 8 and the unselected phase currents IK-I and I K + ~ must be taken as outputs. Then the state space model of the parametrized system will be

0 d8 dt -=

dw -= [ P o ] T I F w ] dt

(54)

( 5 5 )

&x) has a similar form with [PG] replaced by [El. The system can be represented as

i = F ( x ) + G ( x ) u , Y = h ( x )

4.2. Linearization

Each output must be differentiated with respect to time until at least one input appears.

(1) For i = 1, hl ( X I = e, Vh1= [l 0 0 0 01

Let

JQ = hi ( x ) = 8


Then

y, = w

j 4 = [Pw]"Fw]

where

I T 3 7 sin j ~ f w -g / cos (elw c - - NR F/" cos jIf J = l 2

Thus

yf" = [PPl]'Wl+ [PPZ]'WZ

The relative degree corresponding to y1 is therefore rl = 3.

The relative degree is rz = 1 .


(3) For i = 3, ~ ~ ( X ) = I K + I , Vhz= [0 0 0 0 1 )

Let ~3 = h3(x) = IK+ 1

3

i = 1 h = L F ~ ~ ( X ) + (LGfi3)Ui = [PFITIFK+lI + [PFI'[GK+dUK+i (72)

The relative degree is r3 = 1 and we can verify that C ri = n; rl + r2 + r3 = 5. Hence there is no internal dynamics and we can achieve a full linearization of the system. Let

(73)

a = [PW]T[FO] (74)

(75)

Z = [YI Y P yf2' yz y31T= [d w a IK-1 1x+11'

with

0 = [OK OK- 1 OK+ 11' = [Yf3) yjl) '

I LEhi [PPll' Wl LG, (L$hi LG2(L$hi LG, (Lshi B(X) = [ :2j = [ [PFI '[PK-II] , = [ Lo,hz L G ~ L G , ~

[pF]'[FK+ 11 LGI L ~ h 3 Lo,h

(76)

(77) It can be shown that the new state z and the new input vector u satisfy a linear time-invariant

Since A(x) , B(x) , u, y:, yz and y3 contain real parameters, the linearizing control law will

Then

o = B(x ) + A(X)U

relation by expressing the new linear system in Brunowsky canonical form.6

be expressed in terms of the estimated parameters as u = f(x) + &x)O

where

f y x ) = -A-'(x)B(x), B(x) = A-l(x)

4.3. Linear controller design and adaptation law

4.3. I . Position-tracking adaptive controller. The control law for position tracking is

OK = dld - a1t - 012) - a3e (79)

where e = 8 - dd, h = w - Od, t = ii - Old, a1, at and a 3 are design parameters and

d = Z 1 ( x ) = [ K y [ F O ] , & = y1

Let


This leads to the adaptive error equation

.e+ alt + a2i + age = rpT w Consider the third-order filter represented by the transfer function

1 M ( s ) = s3 + a1s2 + a2s + a3

e = M ( s ) 4 T W

We express the error e as

Using the notion of augmented error, we have

el = e + [ [ lFIT~(s) W - ~ ( s ) [ p P 7 ' ~ l

It was shown previously that under some assumptions we can express el as

el = # T f , where [ = M ( s ) W (86)

€ can easily be calculated by simple integration of the state space corresponding to M ( s ) with

Equation (86) is a standard form in adaptive control. We then can use the gradient algorithm input W.

as presented previously to obtain the adaptation law

To achieve a zero steady state position error, the following linear control law is used:

OK = dld - a1 - a26 - aje - X 1' e dt to

4.3.2. Current-stabilizing adaptive controller. The parameter vectors corresponding to the unselected phase currents IK-I and IK+I are [PF] and [PG] deduced from (57) and (58). The adaptation law for position tracking is based on the parameter vectors [PPI] and [PP2] which contain pwl, pwl [PF] and pwl [ P G ] . Then we can deduce [ P I and [ E l as

The control law for current stabilization is

uK-l= -CIK- ,

v K + 1 = -cIK+I

The linearizing control laws for the unselected phases UK-I and U K + ~ are expressed as where C is a positive design parameter.

[ - [ E I T F K + 1 + UK+ 11 1

" " + ' = ~ (93)


Figure 4. Block diagram of adaptive linearizing control scheme

whereas the linearizing control law for the selected phase VK is expressed as

(94) x [GK-I COSK-IIK-iUK-i + GK+I C O S K + I I K + I U K + ~ I I

The block diagram of the adaptive linearizing control scheme is shown in Figure 4.

5 . SIMULATION RESULTS

In order to evaluate the performance and feasibility of the advanced control technique presented in this paper, a single-link SRM directdrive manipulator is considered. The dynamics of the system is modelled by (8)-(10) and the control law used in the simulation is defined in (92)-(94). The real parameters used for the system simulation are listed in Table I. The desired position trajectory shown in Figure 5 is taken to be 1 -57 rad rotation of the link with variable payload during a 1 -6 s time interval.

Table I. Motor and load real parameters

Parameter NR r ILS i7 b J M I Value 25 0.30 0.25 Wb 0-024A-' 0*019A-' 1 kgm' 4 kg 0 - 5 m


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

TIME (s) Figure 5. Desired position

0 0.2 0.4 0.6 0.8 I 1.2 1.4 1.6 1.8 2

TIME (s) Figure 6. Position error

The maximum transient tracking error is 8 mrad (0-005Vo) and, as shown in Figure 6, a zero steady state error is obtained owing to the controller integrator.

In Figure 7 the developed torque in open-loop operation is shown. Without the adaptive non-linear control, the torque ripple is very high and can reach more than 200% of the average torque. This torque ripple causes high overvoltage in the stator windings. Using the linearizing control technique, the torque ripple is dramatically reduced as shown in Figure 8.

In this simulation all parameters are assumed to be unknown. The adaptive scheme is used to compensate for all uncertainties. This allows an acceptable motor acceleration estimation as shown in Figure 9. The error between real and estimated acceleration is due to the

POSITION (d)

Figure 7. Developed torque in open loop


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 POSITION (rad)

Figure 8. h e l O ~ e d torque in closed loop (-) and load torque (- - -)

assumptions in the model parametrization. However, this error does not significantly affect the control performance, since it is compensated by a judicious choice of the parameters al, as, a3 and X in the expression of the control UK defined in (88).

At 1 s time a 100% variation in the payload mass is simulated as shown in Figure 10. This variation does not significantly affect the tracking performance and only 0.004 rad error is observed and a zero steady state error is achieved as shown in Figure6. Moreover, the robustness against inertia and stator resistance variation is also verified.

In Figure ll(a) the control voltage obtained in the non-adaptive control scheme is shown. In this control scheme we suppose that all parameters are known; the phase voltage is, as

TIME (s) Figure 9. Real acceleration (-) and estimated acceleration (- - -)

60 1 1 I I 1 I I I 1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 (9

Figure 10. Effect of payload mass variation on developed torque (-) and load torque (- - -)


expected, a smooth function. In the adaptive control case, however, high-frequency switching is present in the control voltage as shown in Figure ll(b). This is due to the fact that the control voltage is calculated to compensate not only for the non-linearities and the uncertainties in the parameters but also for the modelling uncertainties caused by the linear parametrization assumptions. Owing to the inductive nature of the motor, the high switching frequency is reflected as a hysterisis component in the current wave-forms as shown in Figure 12(b).

We can also see from the shape of the current wave-forms in Figure 12(b) how the compensation for the torque non-linearities shown in Figure 7 can be achieved.

Figures ll(b) and 12(b) show that the adaptive control scheme rejects disturbances due to uncertainties without necessarily requiring a higher control effort.

The starting current as seen in Figure 12(b) is relatively high because of the high desired starting acceleration. However, it can be significantly reduced by choosing a smooth desired trajectory defined by (ed, Wd, a d ) .

Finally, useful information obtained from current and voltage wave-forms can be used for converter design.

The particularity of the control objectives defined in Section 4.1 is the operation mode and the structure of the electronic commutator required for SRM operation. This leads to a control strategy with a varying structure where only one of the three possible output vectors can be considered at each commutation interval. The outputs of the system are chosen depending on the electronic commutator output K defined in (1 1):

[d I3 I2IT if K = 1 y = [e I K - l IK+*]*= [e I , Z3]' if K = 2 t [e 12 I,]' if ~ = 3

"0 0.2 0.4 0.6 0.8 I 1.2 1.4 1.6 1.8 2

(4 TIME (s)

"0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 @) TIME 6)

Figure 1 I . One-phase control voltage: (a) non-adaptive case; (b) adaptive case


- 1 I

----- -______ ---_ -. 3 30-

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 (a) TIME (s)

50 I 1 340

30 1: 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

@I TIME (s) Figure 12. Phase currents II (-), 12 (- - -) and I3 (. - * * * a ) : (a) non-adaptive case; (b) adaptive case

It is therefore clear that at each commutation interval the linearizing feedback control is

The following three possible new co-ordinate vectors z and state feedback vectors u have based on one of three possible coordinate transformations and state feedbacks.

been defined depending on the value of K [e 01 z3 z2lT

z = [e O1 L ~ + ~ ] ~ = [e 01 I# [e 01 i2

[ul u3 u21T if K = 1 and u = [UK w-1 u ~ + 1 ] [UZ u1 uglT if K = 2 TI [u3 u2 ul]' if K = 3

[ In the application at hand it is important to show that during tracking there is a transition

from one feedback control structure to another at each commutation. Figure 10 shows some torque spikes at commutation instants 1-08 and 1-49 s. These

torque spikes are caused by the transition in the control structure. The effect of such transition can be reduced by an optimal choice of the commutation angle. Unlike the phase currents, the position-tracking error shown in Figure 6 is not affected by these transitions, since the position is chosen to be a permanent output independently of the electronic commutator output.

6. CONCLUSIONS

In this paper an adaptive linearizing control scheme has been presented. The algorithm is based on the parametrized model of the non-linear system. The application to the position control


)f the SRM shows that high performances such as full linearization, a dramatic reduction in orque ripple and a high rejection of disturbances due to variations in payload mass, stator ind inertia are achieved during adaptation. Th eventual implementation of such an advanced :ontrol technique is possible, since motor acceleration detection and (I priori knowledge of >arameters are not required.

REFERENCES

1. Bosc, B. K., T. J. E. Miller, P. M. Szczesny, and W. H. Bicknell, ‘Microcomputer control of switched reluctance motor’, Proc. IEEE IAS Ann. Meet., Toronto, October 1985, IEEE, New York, 1985. pp. 542-547.

2. Curran, R., and G. Mayer, ‘The architecture of the adeptone directdrive robot’, Roc. Am. Control C o d . , Boston, MA, June 1985, pp. 716-721.

3. Ilic-Spong. M., T. J. E. Miller, S. R. MacMinn, and J. S. Thorp, ‘Instantaneous torque control of electric motor drives’, Proc. IEEE Power Electronics Specialists C o d . , Toulouse, 1985, IEEE, New York, 1986, pp. 42-48; also IEEE Trans. Power Electron, P E 2 (I) , January (1987).

4. Ilic-Spong, M., R. Marino, S. M. Paessada, and D. G. Taylor, ‘Feedback linearizing control of switched reluctance motors’, IEEE Trans. Automatic Control, AC-32, 371-379 (1987).

5. Taylor, D. G., M. J. Woolley, and M. Ilk, ‘Design and implementation of a switched reluctance motor’, Proc 17th Symp. on Incremental Motion Control Systems and Devices, Champaign, IL, June 1988. pp. 173-184.

6. Isidori, A., Non Linear Control Systems, 2nd edn, Springer, New York, 1989. 7. Singh, S. N., and W. J. Rugh, ‘Decoupling in a class of nonlinear systems by state variable feedback’, Trans.

8. Freund, E., ‘The structure of dccouplcd nonlinear systems’, Int. J. Control, 21, 651-654. 9. Meyer, G., and L. Cicolani. ‘Application of nonlinear system inverses to automatic flight control design-system

concepts and flight evaluations’, in Kent, P. (ed.). AGARDo-Graph 251, Theory and Applications of Optimal Control in Aerospace Systems, 1980.

10. Asare. H. G., and D. G. Wilson, ‘Design of computed torque model; reference adaptive control for space-based robotic manipulators’, ASME WAM, 1986, pp. 195-204.

11. Sastry, S., and A. Isidori. ‘Adaptive control of linearizable systems’, IEEE Trans. Automatic Control, AC-34,

12. Sastry, S.. and M. Bodson, Adaptive Control Stability Convergence and Robustness, Prentice-Hall. Englmood

13. Majmudar. H., Electromechanical Energy Converter. Allyn and Bacon, Boston, MA, 1%5. 14. Miller, T. J. E., Brushless Permanent Magnet and Reluctanm Motor Driver, Clarendon, Oxford, 1989. IS. Torrey, D. A., and J. H. Lang, ‘Modelling a non linear variable reluctance motor drive’, Proc. IEE. 137, pt.

16. Slotine, J. J. E.. Applied Nonlinear Control, Prentice Hall, Englmrood Cliffs, NJ, 1991. 17. Kanellakapoulos, I., P. V. Kokotovic, and A. S. Morse. ‘Systematic design of adaptive controllers for feedback

18. Akhrif, O., ‘Nonlinear adaptive control with application to flexible structures’, Docroral Thesis, University of

ASME, J. Dyn. Syst. Meas. Control, 94, 323-324 (1972).

1123-1131 (1989).

Cliffs, NJ, 1989.

B, (1990), pp. 314-326.

liiearizable systems’, IEEE Trans. Automatic Control, AC-36. 1123-1131 (1989).

Maryland, 1989.

Documents

Adaptive feedback linearization for position control of a switched reluctance motor: Analysis and simulation