Nonlinear adaptive control of permanent magnet step motors

Pergamon ooos109q95)ooo87-9

Auromorica, Vol. 31. No. II, pp. 1595-1604, 1995 Copyright 0 19Y5 Eisev~er Science Lfd

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Nonlinear Adaptive Control of Permanent Magnet Step

Motors*

R. MARINO,? S. PERESADAS and P. TOMEIt

A nonlinear adaptive controller for permanent magnet step motors with unknown parameters is presented that guarantees asymptotic tracking of a reference angle. Simulations show the performance of the adaptive algo-

rithm and a comparison with a classical PID controller.

Key Words-Adaptive control; nonlinear control; parameter estimation; permanent magnet motors: step motors.

Abstract-The problem of accurate positioning of permanent magnet step motors with sinusoidal flux distribution is considered. On the basis of a nonlinear model containing six unknown parameters, a nonlinear adaptive control is designed that guarantees asymptotic tracking of a desired angle reference signal, feeding back the whole state measurements (position, speed and currents). The algorithm may be used for preliminary on-line identification of those parameters that do not vary during operation: persistency of excitation conditions are given that guarantee identification with exponential convergence. When the only unknown parameters are load torque and stator resistance, which vary during operations, persistency of excitation conditions are satisfied when stator currents are not zero: in this case the adaptive control provides on-line exponential parameter estimation and exponential tracking of desired position signals. Simulations show the performance of the adaptive algorithm and a comparison with PID control.

1. INTRODUCTION

Recent advances in nonlinear adaptive control (Kanellakopoulos et al., 1991a, b; Kokotovic, 1991; Marino and Tomei, 1993, 1995) and power electronics, and the availability of low-cost powerful digital signal processors (DSP) now make implementable complex nonlinear control

*Received 9 December 1993; revised 24 February 1994, received in final form 6 June 1995. A preliminary version of this paper was presented at the IFAC Workshop on Motion Control for Intelligent Automation, which was held in Perugia, Italy during October 1992. The Published Proceedings of this IFAC meeting may be ordered from: Elsevier Science Limited, The B&lev&d, Langford Lane, Kidlineton. Oxford OX5 1GB. U.K. This oaoer was recom&enhed for publication in revised form by Associate Editor I. Mareels under the direction of Editor C. C. Hang. Corresponding author Professor Patrizio Tomei. Tel. +39 6 72594451; Fax +39 6 2020519.

t Universita di Roma ‘Tor Vergata’, Dipartimento di Ingegneria Elettronica, Viale della Ricerca Scientifica, 00133 Roma, Italy.

$ Kiev Polytechnical Institute, Department of Electrical Engineering, Prospect Pobedy 37, Kiev 252056, Ukraine.

algorithms in electric drives, designed to meet increasingly more demanding performance re- quirements in applications such as robotics, aerospace, and manufacturing.

There is a clear trend of replacing d.c. motors by a.c., synchronous, permanent magnet and switched reluctance ones, which are more difficult to control but offer several advantages. Since these machines are highly nonlinear and involve parameters that may vary during operation, recently developed nonlinear (Isidori, 1989; Nijmeijer and van der Schaft, 1990) and adaptive (Kokotovic, 1991; Kanellakopoulos et al., 1991; Marino and Tomei, 1993) control techniques have been successfully applied to switched reluctance motors (Illic’-Spong et al., 1987a, b; Taylor, 1991), induction motors (Marino et al., 1993) and permanent magnet stepper motors (Bodson and Chiasson, 1989; Chen and Paden, 1990, 1993; Zribi and Chiasson, 1991; Marino and Tomei, 1992; Bodson et al., 1993). In particular, permanent magnet stepper motors deliver higher peak torque per unit weight and have higher torque-to-inertia ratio than d.c. motors; in addition, they are more reliable and, being brushless machines, require less maintenance. They are especially suitable as direct drives in robotics and aerospace applications, in which motor weight is a crucial factor.

Since the states (angular position, angular speed, two-phase currents) are usually available for measurements in permanent magnet step motors (we shall use in the sequel the acronym PMSM), the goal of this paper is the design of an adaptive state feedback control for accurate positioning of PMSM with unknown parameters. We assume sinusoidal flux distribution: in this

1595

R. Marino et al.

case, the dynamics are modelled by nonlinear differential equations that depend on six unknown parameters, namely total inertia J, viscous friction coefficient F, torque load TL, motor torque constant kM, stator winding resistance R and inductance L.

When these parameters are known exactly, the classical control is a nonlinear state feedback algorithm, which is designed after performing the so-called (d, q) transformation for the two-phase currents and voltages. This control may be viewed as a feedback linearizing one (Zribi and Chiasson, 1991) according to a well-established theory (Isidori, 1989; Nijmeijer and van der Schaft, 1990), since the resulting closed loop is linear and controllable in the new coordinates. Experimental results reported in Bodson et al. (1993) show that even in the presence of current sensor noise and not very accurate position measurements (a 2000-line encoder), this control, implemented by commer- cially available technology (20 kHz PWM ampli- fiers, DSP board with less than lo-‘s per instruction), guarantees good tracking performances.

As stated in Bodson et al. (1993) (see also Bodson and Chiasson, 1989), at the present state of the art, identification algorithms are to be performed off-line to determine the parameter values, which are typically not provided by motor manufacturers. To this end, Blauch et al. (1993) proposed and experimentally tested least-squares techniques that require knowledge of currents and speed derivatives. Parameters should be identified off-line for each single motor, raising the difficult commercial problem of control tuning. Moreover two parameters, TL and R, may vary during operation. The contribution given in this paper is to design an adaptive nonlinear state feedback control that ensures asymptotic tracking of position reference signals, in any operating condition, without any knowledge of the six motor parameters. None of the recently developed nonlinear control techniques (Kanellakopoulos et al., 1991a, b; Kokotovic, 1991; Marino and Tomei, 1993) applies to the PMSM model; nevertheless, we show that an adaptive algorithm may be designed, exploiting the physical property that all unknown parameters (except TL) are positive. When, in addition, persistency of excitation conditions are satisfied, the adaptive algorithm provides exponentially convergent parameter estimates while controlling the motor: currents and speed derivatives are not needed. The adaptive algorithm may turn the adaptation off with respect to those parameters that do not vary in a specific operation and have been identified

using persistently exciting inputs. We show how to identify L and R on-line by using the same algorithm, before starting motor operations. In the common situations in which TL and R are the only uncertain parameters, it is shown that, as long as motor currents are different from zero, TL and R are exponentially identified by the control algorithm, which also guarantees that the tracking error exponentially converges to zero.

The key assumption of this paper requires sinusoidal flux distribution, which is achieved by careful mechanical and electromagnetic design of the motor. As is well known, classical controls applied to motors that violate this assumption give rise to torque ripples. A method for torque ripple reduction has been recently proposed in Chen and Paden (1993), where experimental results are also given: current dynamics are neglected, so that electric parameters do not appear. A currently open problem is to design a control algorithm for the true inputs (stator voltages) that reduces torque ripples for motors with non-sinusoidal flux distribution and unknown parameters.

The paper is organized as follows. The dynamical model for PMSM is given in Section 2, in which the modelling simplification due to sinusoidal flux distribution is explained. The classical non-adaptive control scheme is re- viewed in Section 3, while the adaptive control algorithm is given in Section 4 in which the persistency of excitation conditions are dis- cussed. Simulations are reported in Section 5, where the proposed adaptive control is compared with the classical PID control, showing better performance.

2. DYNAMICAL MODEL

In order to clarify the modelling assumptions made in the control design, we first introduce the general model for two-phase PMSM (Kuo and Tal, 1978; Acarnley, 1992). We assume linear magnetic materials, symmetry of the rotor and between the two phases, nonlinear flux density distribution due to air gap geometry only, and negligible magnetic hysteresis and Foucault currents.

Denoting the rotor angle by 8, we define the electric angle 0, =pt9, which gives the main magnetic field periodicity for a machine with p poles symmetrically placed. Permanent magnets may be modelled by an equivalent rotor winding supplied by a fictitious current source, which is denoted by if. Let (a, 6) denote the stator reference frame, so that u = [u, ub], i = [i, i,]

and 9 = [h &I are stator voltage, current and

Adaptive control of step motors 1597

flux vectors respectively. Denoting the rotor flux by @,f, the motor flux and currents are related by +a [I[ L + uw Li?(~) Lo + L7fcq

+b = Ln(e) L + Lb(e) Lo + Lbde) $f Lo + L,f@) LO + Lbf(e) Lm + Lf@) 1 ia &I x ib [1 r1 ‘L(8) ib , (1)

lf if

in which L, L,, and Lm are constants, L + L,(B) and L + Lb(e) are stator winding self- inductances, L,,,(8) is the stator mutual inductance, the matrix

L + La(e) Ls(e) = [ L,,,(e) L + Lb(e) 1

(2)

is symmetric and positive-definite, Lm + L,-(e) is the self-inductance of the fictitious excitation winding, and Lo + L&e) and L,, + Lb@) are the mutual inductances between stator and rotor. For doubly salient motors, the elements of the matrix L(0) in (1) satisfy the following conditions:

(9 all entries of the matrix L(8) are smooth periodic functions of 8 and contain the dominant fundamental harmonic;

(ii) motor symmetry implies that

L$pe) = Lb(2Pe + $)?

&f(Pe) = Lbf(pe + $);

(iii) L,(8), Lb(e) and L,,,(8) are periodic with frequency 2p;

(iv) the mutual inductances Laf(e) and Lb@) are periodic with frequency p.

From energy balance arguments (White and Woodson, 1959; Fitzgerald er al., 1971; Deltoro, 1985; Miller, 1989; Acarnley, 1992), the motor torque T is equal to the derivative of the electromagnetic energy with respect to 8, namely

= (L& + LA&)& + ($L$z + L&& + iLLi;)

+ tL;if, (3)

in which the derivative with respect to 8 is denoted by a prime. The first term in (3) is the fundamental component of the motor torque, and is due to the interaction between stator currents and permanent magnet excitation. The second term is due to stator salient poles, while

the third term, known as the detent torque, is due to rotor salient poles: they are both equal to zero when there is no saliency in the motor.

The rotor shaft dynamics are given by

Je=T-F&T,, (4)

in which J is the total inertia of the rotor and of the load, F is the viscous friction coefficient, TL is the load torque and T is given by (3). The electromagnetic dynamics are given by

I,& = -R,i + u, (5)

in which the 2 X 2 matrix R, = diag [R, R] is the matrix of stator winding resistances, which are assumed to be equal. Differentiating with respect to time the flux vector

JI = w)i + if [

Lcl + Lf(@

Lo + Lbf(e) 1 obtained from (1) and substituting in (5), we have

di dt = L;‘(e)

(6)

in which w = & The term

is usually called the back electromotive force. Equations (4) and (6) constitute the dynamical model of a PMSM with both rotor and stator salient poles. In this paper we shall only deal with PMSM with no saliency (i.e. L,(8) = 0, L,(8) = 0, Lb(e) = 0 and Ldf3) = 0 in (1) and a sinusoidal flux density distribution (i.e. Laf(8) = L,cospe and Lb@)= L1 sinpe in (1)) which are characterized by the inductance matrix

L(e) =

[

L 0 Lo + L, c0spe 0 L L, + L, sinpe .

Lo + L, c0spe Lo + L, sinpe LfO I

(7)

In this special case the dynamical model (4), (6) becomes

de ;=o,

p(-’ ~~sinpe+ibcosp8)-~w-~,

(8)

1598 R. Marino et al.

dib R . kM

in which w is the rotor speed and kM =pL& is the motor torque constant.

3. FEEDBACK LINEARIZING CONTROL

A classical control technique for PMSM modelled by (8) is based on Park’s transformation (Park, 1929; Leonhard, 1985; Zribi and Chiasson, 1991), that is, the transformation of the vectors u and i expressed in the fixed stator frame (a, b) into vectors expressed in a frame (d, q) that rotates along the fictitious excitation vector if directed as the d axis:

xd [I [ cos pe sinpe x, = I[ 1 (9) -% -sinpe cospe xb ’

The dynamics (8) expressed in terms of currents and voltages in rotating (d, q) coordinates become

d8 -= dt w’

dw kM -_=-i F TL

did R ud

-iii- L ---ii,+pwi,+-_

L

&,_ R. dt - - L % -poi,-+o+T.

Denoting the rotor shaft acceleration by

k M. F TL ff=JLq-JW-J, (11)

the dynamics (10) become in the new coordinates (6, 0, (Y, id)

dw -$= a,

$=2(-Ri,-Lp&--krw +u,) (12)

- s (kMiq - Fw - TL),

did R -= --id +poi, +ff_d. dt L L

Defining the nonlinear feedback control (see Zribi and Chiasson, 1991)

Ud = Rid - LpWi, - k&id + Lv,,

uq = Ri, + &mid + kMw

+ g (kMiq - Fw - TL) M

(13)

+g(-k,,f3-k,,o-k,,a+v,), M

the closed loop becomes a linear controllable system

did - = -k& + vd, dt

(14)

X = -k,, 8 - kqzw - kq+ + vq.

In conclusion, the overall state feedback control

47 [I [ cos pe sin@ -’ ud = ub -sinpe c0spe I[ 1 uq

(15) with (&, uq) given by (13), transforms the original system (8) into the linear controllable one (14). Choosing kd > 0, (kql, kq2, kq3) such that the polynomial s3 + kq3s2 + kq2s + kql has all zeros with negative real part (and in the desired position), and the reference signals

didr vd = k&jr + - ,

dt

d28 d38 vq=kq,O.+kqz~+kq++~~

exponential tracking of the desired position O,(t) is achieved. It is a common practice (referred to as ‘field weakening’) to assign a reference for the current id that is related to the reference velocity

W,.

ldr = idrtWrh

to comply with voltage saturation at high speed (Leonhard, 1985). Note that the state feedback control (15) requires knowledge of all six parameters involved in the model (8).


4. ADAPTIVE FEEDBACK LINEARIZATION

The dynamical model (8), or the equivalent one (lo), contains five positive parameters, J, kM, F, R and L, and the unknown parameter TL: their exact values may not be easily determined. In particular, the torque load TL depends on applications and may vary from case to case, and the resistance R varies with temperature, so that their off-line measurements may differ from their on-line true values. The remaining parameters J, kM, F and L may be known with some tolerances within each class of motors. Consequently, the feedback linearizing control (15) does not lead to the linear closed-loop system (14) if the above parameters are not exactly known: in this case asymptotic tracking of e,(t) may not be obtained.

and Tomei (1993), the reference for iq as (k2 > 0 is a control parameter)

iqr = $,[c$~w + $3 + 8, - k,(w - b,) - k,G],

(18)

so that, denoting Tq = iq - iqr and substituting iq = i,, + Tq into (17), we obtain

6=-k$+G,

;=._k &+b; 2 J ’

In this section we develop an adaptive control that guarantees asymptotic tracking of a smooth reference signal e,(t) for a PMSM modelled by (8), or equivalently by (lo), without requiring knowledge of any one of the above six parameters. Denoting by 6 = 0 - 8, the tracking error, from the first equation in (10) we have

B = 6J - 4,. in which Wan, 1 lj 5 3, are known functions. Defining

Following the ‘backstepping’ procedure intro- duced by Kanellakopoulos et al. (1991), define the reference for w as (k, >O is a control parameter)

and taking (lo), (16) and (18) into account, the dynamics of iq and id = id - idr are given by

wr= &- k,& (16)

so that, denoting the speed error by i3 = o - w,,

-: 1 lq = -c$4iq -pwid - c&w + - uq

4%

the first two equations as

in (10) may be rewritten - &[$,w + & + & - k,(o - b,) - k26]

G=-k,e+5,

kM . g=-,~-TO-4-s,+k,(w-8,). J

(17)

We reparameterize by introducing new parameters defined as

Let JJ~ and 4i= 4i - $i, 1 sis3, denote parameter estimates and parameter errors respectively. Since & in (17) is an unknown parameter, the backstepping procedure given by Kanellakopoulos et al. (1991) to design adaptive controls does not apply. However, since +1 is a positive parameter, we may define, as in Marino

+ & + $ + (k, + k,)& - k, k,6 1 (20)

+ &(biq - 42@ - 44(kl + k2 - $2), 1 did,

ld= -4&+poiq +-uUd--. 46 dt

Note that we now have in (19) and (20) seven unknown parameters 4i, 1 I i 5 7, which depend on the six physical given ones. In (20), when w, is less than the rated speed, the reference idr is set to zero, while when w, is greater than the rated speed, id, = w(&), so that

didr dw . . -=- 8 dt de, r’

with w a function chosen to achieve ‘field weakening’ (see Leonhard, 1985). The feedback


controls (ud, u,) are defined as (k3 >O, k4 > 0 are current control gains and y is a positive real)

uq = 4, {

$A& +pwi, + $5w

+ $,[&w + $3 + 6, - k,(w - 6,) - k26]

~*W+~~+~+(k,+kZ)ijr-k,kZlj 1 (21) ,. n .

- 4*(47&j - $2~ - &)(kl + k2 - 8,)

didr && - pwi, - k$d + -

dt ,

which, when substituted into (20), together with (19), give

6=-k,6+6,

- F (C$.& + puid + 4,” 6

+ t$ ,[$2w + ~$3 + 8, - k,(w - 6,) - k,ij]

- k& - y-‘&G I

ii - k& - y-1$+j (22)

+ [w32 w33 w34 w35 W36/46 w371

_- z” ($& -pOi, - k4Fd + 2) 6

’ -k4ti + [w44 w46/46]

[ 44 4 6 1 ,

which may be rewritten in matrix form as

[rl-[1:*: _,,:k, i3 _!J]

0 000

%,I41 w2 w23 0 +

0 w32 w33 w34

0 0 0 w,

r $1 1 0 0

0 0

w35 w36/46

0 w46146

P (23)

or, in compact notation (x’ = [6 6 Tq $lT),

x” =A.f + WD-‘4, (24)

in which W is called the regressor matrix (and is known), while D = diag [&, 1, 1, 1, 1, 46, l] is a positive-definite diagonal matrix.

Consider the function

v = f=pf + ,$=A-ID-‘4, (25)

in which 4 = [$, . . . $7]T and P and A are positive-definite diagonal matrices:

P = diag [I, 1, Y, rl,

A = diag [A,, h2, h3, h4, A5, h6, A71.

The time derivative of (25) is

ri = X-=(A*P + PA)X + 2r=PWD-‘c$

+ 24=A-‘D-‘~5.

Since)he ptrameters +i are constant, it follows that 4 = -4, so that, defining the adaptation algorithm

f = AWTPf, (26)

and substituting it in the expression for li, we obtain

with

ri = x’=(A=P + PA)f, (27)

Assuming that the desired reference O,(t) and its time derivatives up to order three are bounded, it follows from (25) and (27) that (provided that k,k2 > $) 8, 9, Tq, Fd and $i, 1 Si 57, are bounded for any t 2 2. Hence 8, o, id and iq are bounded, and 6, &, i, and id are also bounded. Since, from (25) and (27), 6, 6, $ and ;sd are L2


signals, by Barbalat’s lemma (see Narendra and Annaswamy, 1989) it follows that

m lim [I W) = 0

(28) l+m Qt) .

G(t) If there exists a positive constant T such that the 7 X 7 matrix

I

r+T

W’(z)W(z) dr (29) I

is positive-definite for any t 2 0 (persistency of excitation condition) then P = 0, 4 = 0 is a globally exponentially stable equilibrium point for the linear time-varying system (see Narendra and Annaswamy, 1989)

x” = A4 + WD-‘&,

c$ = -AWTPZ (30)

In conclusion, the adaptation algorithm (26) is given explicitly by

$1 = A,&+&0 - 4, - e, + kl(ClJ - 4,) + k*O],

& = AJ-3~ - w&(k, + k2 - &)yrq],

& = h3[-2 -(k, + kz - &)~$,y~~],

$4 = -h4y(iqrq + i,g,

& = -hgy6J$,

J6 = -A6 y’q ; (

1 ,. f$& +p6& + &co

+ &$,o + & + a, - kl(w - 6,) - k,ij]

d30 &+$3+-I

dt3 (31)

+ (k, + k2)& - klk26 1 - &(47iq - dk - &)(k, + kz - 8,)

4, = h7dq + A7y(kl + k2 - &)$liq~q.

In the special case in which all parameters are known (or identified) except TL and R, the adaptive control law (21), (31) becomes

{

i? kM U,=L -ii,+pOid+--w

L L

&. = A,[ -JG -; (k, + k2 - $)yFq],

R = -A4 yL(iqTq +” i&),

(32)

and the system (22) becomes

6=-k$+i3,

k pL ;=-k,~+>;q_--, J J

kM 5, = -k3iq _ y-l _ &j J

i? fd= -k4rd-id-.

L

(33)

The persistency of excitation condition (29) becomes in this case

2

k, + k2 - 5

) 1 $ iq M

dr>O VtrO,

i: + is

and is satisfied if there exists a positive real T such that

I

f+T

[i:(2) + i:(z)] dr > 0 Vt 2 0 ,

i.e. in all physical situations: this guarantees that (x’, c$~, c$~) = 0 is a globally exponentially stable equilibrium point for the closed-loop system.

If in (21) and (31) we set 4,(O) = 0, i = 1,2, 3,5,7, and O,(t) = e(O) Vt 10, then $i(t) = 0, i = 1, 2, 3, 5, 7, e(t) = 0(O), W(t) = 0, i4(t) = 0 Vt I 0, for the closed-loop system (10) with initial conditions (e(O), w(O), id(O), i,(O)) = 0, for every i&(t): in fact, there is no electromechanical energy conversion. In this case the closed-loop dynamics are

$4 = -A,&,

and the controls are

u,(t) = 0,

8& - k& + 2). (35)

Both c$4(t) and &(t) tend asymptotically to zero provided that &(t) is, for instance, a sinusoidal function of time. This implies that both R and L are asymptotically estimated: hence these parameters can be preliminarly estimated on-line. Since L does not vary during operations (while R


does), we set in (31)

c&(t) = 0 Vt 2 0,

h(O) = R/L,

so that the resulting algorithm is of order six.

5. SIMULATION RESULTS

In this section we report simulations in which the proposed adaptive control (21), (31) and on-line estimator (34) are applied to a PMSM with sinusoidal flux distribution, whose data are listed in Table 1. The performance obtained is compared with those achievable by a classical PID control.

In Fig. 1 the typical position reference trajectory O,(t) with trapezoidal speed reference q(t) is given, which is to be tracked by the motor angle, while a constant load torque of 2 N m (which is unknown to the controller) is applied at t = 0.15 s.

The adaptive control algorithm (21), (31), (36) is compared with the following PID control:

ud = -PLO& - k,(i, - idr)

- ks I

’ [id(z) - idr(T)] dr

0

uq = kMw - k4(iq - iqr) (37)

- k5 I

’ [i,(z) - i&)1 dr 0

with

idr = 0

tqr= -- k” M

{ k,( 8 - or) + k,l’ [e(z) - @r(r)] dr a

+ b(m - 4 I and (all values are in SI units)

k, = 80000, kz = 65kl, k3 = 500,

k,=$, k5=E T’

T = 0.0005.

T is the equivalent time constant of the current loops: in fact, with the above choice of k4 and kg, the current loop transfer function is

iq(s) id(S) l -=-

with break

i&) - i&(S) - 1 + Ts ’

frequency l/T = 2000 rad SC’. The

Table 1. Motor data

Number of phases Number of pole pairs Torque constant Stator resistance Stator self-inductance Total rotor-load inertia

2 p=6 k,=2NmA-’ R=352 L=O.O06H J = 0.01 kg m*

speed loop transfer function (obtained with k, = k2 = 0) is

w(s) 1 -= w,(s) (T/k&’ + (l/k& + 1 ’

in which k3 is chosen so that l/k, = 4T. The position loop transfer function with k2 = 0 (neglecting the current loop dynamics) is

e(s) (kg/Q + 1 --= f%(s) (l/k+* + (k,/k,)s + 1 ’

where the gain k, is chosen so that the natural resonant frequency is -\/r;T = 285 rad SC’ and the damping ratio is equal to 0.9.

The performance of the PID control (37) with the reference trajectory given in Fig. 1 are reported in Fig. 2, in which the position errors and the corresponding voltages required are given: solid lines refer to the case in which the parameters R = RN, L=LN, J=JN, kM=kMN and F = FN assume their nominal values listed in Table 1, while dashed lines correspond to the case in which R = 0.5RN, L = 1.5LN, J = l.SJ,, kM = 0.5kMN and F = 0.5F,. A degradation of performance when parameters differ from their nominal value is noticed. To compare the two control algorithms, the adaptive algorithm (21), (31), (36) has been tested for the same set of perturbed parameters corresponding to the dashed line performance of the PID control in Fig. 2. The initial conditions for parameter estimates are I&(O) = JNIkMN, &(O) = FNIJN, &(O) = 0, &,@I) = RN/L, &(O) = kMN/L and n C&(O) = kMN/JN with L being previously identified using the algorithm (34), (35). The tuning parameters of the adaptive control are y = 100, A, = 0.001, hz = 0.15, A3 = 1000, A“ = 1000, AS = 200, A,=500, k, = k2=200 and k3= k4=2000. The adaptive control is tuned so that the current loops and the position loop (whose natural, resonant frequency is in this case w, = m = 200rads-‘, while it is 285 rads-’ with the PID control) have the same dynamic performance obtained by the PID control. Comparison of Fig. 2 (dashed lines) and Fig. 3 shows that the adaptive control gives better performance within the same voltage limits. Note that the performance obtained is also better than the one given by the PID control with nominal parameters (Fig. 2, solid lines).

When the only unknown parameters are TL and R, the adaptive control may be tuned to obtain even better performance. Simulation results of a PMSM with L = LN, J = JN,

kM = km, F = FN and R = 1.2RN, controlled by the adaptive algorithm (32) (with R(0) = RN,


Time (s) Tim (s)

Fig. 1. Reference signals.

Position error I

Time. (s)

Fig. 2. PID control.

Time (a)

pL(0)=O, y = 100, AD= loo0, hz,=300, kl = k2 = 200 and k3 = k, = 2000) are reported in Fig. 4. While the voltage modulus profile is very similar to the one required by the PID control (Fig. 2, solid line) under the same conditions, once resistance and load torque estimates have converged to their true values (see Fig. 4), the position error obtained by the adaptive control is

two orders of magnitude smaller than the one obtained by a PID control.

6. CONCLUSIONS

The main contribution of this paper is to present an adaptive position tracking algorithm for permanent magnet step motors with sinuso-

a04 Position error

om- ’

-aO2-

"040 ai 02 . 03 . 04 9 05 . ‘0 0.1 0.2 0.3 a4 0.5 Time (s) Time. (s)

Fig. 3. Adaptive control.

ai a2 0.3 0.4 a5 ai 0.2 0.3 a4 0.5

Time @I Time (I)

Fig. 4. Adaptive control with only R and TL being unknown.

AUTO 31-11-F

R. Marino et al.

idal flux distribution and unknown parameters, on the basis of state measurements (position, speed and stator currents). Persistency of excitation conditions are given that guarantee, in addition, exponential convergence for the parameter estimates. In the special case in which the only unknown parameters are the constant load torque TL and the stator resistance R, persistency of excitation conditions are satisfied, provided that the stator currents are different from zero, and guarantee exponential convergence to zero both of position tracking error and parameter estimate errors.

Acknowledgements-This work was partly supported by MURST and CNR.

REFERENCES Acarnley, P. P. (1992). Stepping Motors: a Guide to Modern

Theory and Pructice. Peter Peregrinus, London. Blauch, A., M. Bodson and J. Chiasson (1993). High-speed

parameter estimation of stepper motors. IEEE Trans. Control System Technology, CST-1,270-279.

Bodson, M. and J. Chiasson (1989). Application of nonlinear control methods to the positioning of a permanent magnet steuoer motor. In Proc. 28th IEEE Co& on Decision and Coktroi, Tampa, FL, pp. 531-532. ’

Bodson, M., J. Chiasson, R. Novotnak and R. Rekowski (1993). High performance nonlinear feedback control of a permanent magnet stepper motor. IEEE Trans. Control System Technology, CST-1,5-14.

Chen, D. and B. Paden (1990). Nonlinear adaptive torque-ripple cancellation for step motors. In Proc. -29th IEEE Co& on Decision and Control. Honolulu. HI. DD. 3319-3324:

. . .

Chen, D. and B. Paden (1993). Adaptive linearization of hybrid step motors. IEEE Trans. Autom. Control, AC-XI, 874-887.

Deltoro, V. (1985). Electric Machines and Power Systems. Prentice-Hall, Englewood Cliffs, NJ.

Fitzeerald. A.. C. Kineslv and A. Kusko (1971). Electrical k&chin&y. McGrawrHil, New York. ’ ’

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

Ilic’Spong, M. T., T. J. E. Miller, S. R. Macminn and J. S. Thorp (1987b). Instantaneous torque control of electric motor drives. IEEE Trans. Power Electronics, PE-2, 55-61.

Isidori, A. (1989). hronlinear Control Systems. Springer- Verlag, Berlin.

Kanellakopoulos, I., P. V. Kokotovic and R. Marino (1991a). An extended direct scheme for robust adaptive nonlinear control. Automatica, 27,247-255.

Kanellakopoulos, I., P. V. Kokotovic and A. S. Morse (1991b). Systematic design of adaptive controllers for feedback linearizable systems. IEEE Trans. Autom. Control, AC-36, 1241-1253.

Kokotovic, P. V. (Ed.) (1991). Foundations of Adaptiue Control. Springer-Verlag, Berlin.

Kuo, B. and J. Tal (1978). Incremental Motion Control. SRL, Champaign, IL.

Leonhard, W. (1985). Control of Electrical Drives. Springer-Verlag, Berlin.

Marino, R. and P. Tomei (1992). Adaptive control of stepper motors via nonlinear extended matching. In Proc. ZFAC Workshop on Motion Control for Zntelligent Automation, Perugia, Italy, pp. 135-139.

Marino, R. and P. Tomei (1993). Global adaptive output-feedback control of nonlinear systems. Part I: linear parameterization. IEEE Trans. Autom. Control, AC-J% 17-32.

Marino, R. and P. Tomei (1995). Nonlinear Control Design-Geometric, Adaptive and Robust. Prentice-Hall, Hemel Hempstead, U.K.

Marino, R., S. Peresada and P. Valigi (1993). Adaptive input-output linearizing control of induction motors. IEEE Trans, Autom. Control, AC-XI, 208-221.

Miller, T. J. (1989). Brushless Permanent-Magnet and Reluctance Motor Drives. Clarendon Press, Oxford.

Narendra, K. S. and A. M. Annaswamy (1989). Stable Adaptive Systems. Prentice-Hall, EngIewood Cliffs, N.J.

Nijmeijer, H. and A. J. van der Schaft (1990). Nonlinear Dynamical Control Systems. Springer-Verlag, Berlin.

Park, R. H. (1929). Two-reaction theory of synchronous machines-generalized method of analysis-part I. AZEE Trans., 48,716727.

Taylor, D. G. (1991). Adaptive control design for a class of doubly-salient motors. In Proc. 30th IEEE Conf on Decision and Control, Brighton, U.K. pp. 2903-2908.

White, D. C. and H. H. Woodson (1959). Electromechanicut Energy Conversion. J. Wiley, New York.

Zribi, M. and J. Chiasson (1991). Position control of a PM stepper motor by exact linearization. IEEE Trans. Autom. Control, AC-36,620-625.

Nonlinear adaptive control of permanent magnet step motors

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