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Jian-Xin Xu Tong Heng Lee Qing-Wei Jia Mao Wang

Wang, Jian-Xin Xu Tong Heng Lee Qing-Wei Jia Mao(1998) 'On adaptive robust backstepping controlschemes suitable for PM synchronous motors', International Journal of Control, 70: 6, 893 920

10.1080/002071798222028

http://dx.doi.org/10.1080/002071798222028

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On adaptive robust backstepping control schemes suitable for PM

synchronous motors

JIAN-XIN XU, TONG HENG LEE, QING-WEI JIA andMAO WANG

This paper presents new adaptive robust control strategies suitable for permanentmagnet (PM) synchronous motors. By using the backstepping approach, both theadaptive and robust controllers are appropriately designed to ensure global stabil-ity. Taking into account the variation ranges of system parameters, the adaptiverobust methods are further developed to achieve better tracking performance. It isshown that the synthesized adaptive robust control schemes developed here canretain advantages of both adaptive and robust control schemes and overcome theirshortcomings. To avoid discontinuous control laws which may cause problems inboth theoretical and practical aspects, a continuous adaptive robust controlmethod associated with a -modication scheme is also proposed to guaranteeboth the uniform boundedness of the system and suitably designated trackingprecision. The paper includes simulation studies demonstrating the performanceof the proposed control schemes.

1. I ntroduction

Various adaptive control algorithms for nonlinear systems with parametricuncertainties have been proposed and developed over the last decade (Taylor et al.1989, Sastry and Isidori 1989, Narendra and Annaswamy 1989, Kokotovic 1991).On the other hand, the problem of designing stabilizing control for nonlinear

systems containing interval uncertainties has also been the central subject of researchon robust control strategies (Gutman 1979, Corless and Leitmann 1981, Barmish etal. 1983, Slotine and Li 1992, Utkin 1992). More recently, by using backsteppingapproaches, adaptive and robust control methods have been extended respectively tocascaded nonlinear systems (Kanellakopoulos et al. 1991, Qu 1993, Jiang and Pomet1995). There are also some reports on the synthesis of adaptive and robust controlschemes, see Ioannou and Sun (1996) for instance. However, little has been done onthe design of robust adaptive backstepping control for cascaded nonlinear systems in

the presence of both parametric and interval uncertainties. The main objective of thispaper is to develop new adaptive robust control schemes incorporating backsteppingdesign approaches for a class of commonly encountered servo-actuators with non-linear dynamicsthe permanent magnet synchronous motors.

As a typical nonlinear control system, the eld of motor control is such a broadresearch area that many nonlinear control techniques can be applied to address thedierent motor control problems. For instance, nonlinear control such as feedbacklinearization, adaptive, robust and sliding mode control schemes have been well

0020-7179/98$12.00 1998 Taylor & Francis Ltd.

INT. J. CONTROL, 1998, VOL. 70, NO. 6, 893920

Received 7 April 1997.Department of Electrical Engineering, National University of Singapore, 10 Kent Ridge

Crescent, Singapore 119260.

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developed for motor control (Liao et al. 1990, Dawson et al. 1994, Fujimoto andKawamura 1995). Adaptive robust control schemes with specied tracking errorbounds were presented in Liao et al. (1990) and Brogliato and Neto (1995). How-ever, those methods are only suitable for one-step design of the systems. In otherwords, their algorithms cannot be extended to the backstepping design which

requires that the input be dierentiable at each step and consists of at least twosteps. Through the use of ingenious ideas like integrator backstepping, it is nowpossible to design controllers for nonlinear electromechanical systems such as elec-tric motors actuating a robotic load (Dawson et al. 1994). Since the permanentmagnet synchronous motors are widely applied to drive fast dynamic loads, werestrict our interest to some of the recent nonlinear control work in this area forposition or velocity tracking control problems.

The dynamics of the permanent magnet synchronous motor can be presented by

a dynamic electrical subsystem and a dynamic mechanical subsystem, which arenonlinear dierential equations. Strictly speaking, most control methods for perma-nent magnet synchronous motors are only locally stable because the d-axis current isassumed to be zero and the design procedure is based on the reduced model. InFujimoto and Kawamura (1995), a robust control scheme based on two-degree-of-freedom control with sliding mode is proposed for PM DC motors, where the delayof electrical response is ignored. In Dawson et al. (1994), both the adaptive back-stepping and robust backstepping approaches are applied to permanent magnet

synchronous motors which are of the parametric-pure feedback form, i.e. the con-troller design is based on the reduced model. It is well known that backsteppingdesign methods essentially require the parametric-pure feedback form. In this paper,we apply backstepping approaches to design adaptive and robust controllers forpermanent magnet synchronous motors in which the nonlinear dynamical equationsare not exactly in the parametric-pure feedback form. Instead of only zeroing d-axiscurrent, the extra d-axis control input voltage is used to deal with the nonlinearcoupling part of the dynamics as well. Hence, the proposed controller ensures the

strictly global stability of the whole electromechanical systems. Taking into accountthe range of system parametric variations, the adaptive and robust approaches arefurther synthesized to form an adaptive robust controller for the permanent magnetsynchronous motor. The robust technique is used to suppress the motor parametricuncertainties which usually are much smaller than the load changes. Since it isdicult to give upper bound of the load variations and an overlarge estimate ofthe upper bound will result in excessive control authority, adaptive technique isadopted to tackle the payload uncertainty.

By using switching functions, the proposed adaptive robust controller ensuresglobal stability of the whole electromechanical system. However, incorporating dis-continuous functions in the controller may give rise to problems related to the exist-ence and uniqueness of solutions as well as problems in practical implementation. Todeal with these problems, the controller with continuous functions instead of switch-ing functions is further developed and analysed, which guarantees the uniformboundedness of the system tracking error. Moreover, The -modication scheme(Xu et al. 1997) is used to cease parameter adaptation in accordance with the adap-tive robust control law. In this method, the tracking precision can be made arbitra-

rily small by choosing suitable control design parameters.This paper is organized as follows. Section 2 describes the dynamic model of

permanent magnet synchronous motor. Section 3 gives the adaptive backstepping

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design procedure and stability analysis. Section 4 presents the robust backsteppingdesign method and relevant analysis. Section 5 details the synthesis of adaptive androbust control schemes using switching functions. Section 6 presents the adaptiverobust backstepping design by using continuous functions to approximate theswitching functions adopted in 5 as well as the -modication scheme. Simula-

tion-based studies on the proposed methods are given in 7.

2. Model of permanent magnet synchronous motor

A permanent magnet synchronous motor is described by the following subsys-tems: (1) a dynamic mechanical subsystem, which for the purposes of this discussionincludes a single-link robot manipulator and the motor rotor; (2) a dynamic elec-

trical subsystem which includes all of the motors relevant electrical eects.

d

dt= x (1)

dx

dt=

1

J(Ld - Lq)Id + uf[ ]Iq - T sin { } (2)

dId

dt

=1

Ld

(ud + xLqIq - RId) (3)

dIq

dt=

1

Lq(uq - xLdId - RIq - xuf) (4)

where (1) and (2) present the dynamics of mechanical subsystem, and (3) and (4) arethe dynamic electrical subsystem. In these equations, uq and ud are the input controlvoltages, Id and Iq are the motor armature current, R is the stator resistance, Ld andLq are the self-inductances, Jis the inertia angular momentum, and uf is the ux dueto permanent magnet. For the above electromechanical model, we assume that thetrue states (i.e. , x, Id and Iq) are all measurable. This model is obtained by usingcircuits theory principles and a particular dq reference frame. The control objective isto develop a link position tracking controller for the electromechanical dynamics of(1)(4) despite parametric uncertainty. In the paper we assume that all the motorparameters are unknown.

Remark 1: It should be noted that it is not pompous to refer to the above `pendu-

lum model as a single link direct drive robot. In this paper, the single-link robotmanipulator is regarded as a general load for motor. The main purpose of this paperis not to deal with the dierent load for the permanent magnet synchronous motor,but rather to develop a strictly globally stable adaptive robust controller and demon-strate the usefulness of the backstepping techniques for permanent magnet synchro-nous motors. Actually, the proposed method is applicable to any load which can beexpressed by unknown parameters and known function of states, for instance thevelocity tracking control problem in which the load can be expressed as

a + bx+ cx2

, where a, b, and c are unknown parameters.Remark 2: In most existing control schemes for the PM synchronous motors, thecontrollers are designed based on the following reduced model

Control schemes for PM synchronous motors 895

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d

dt= x (5)

dx

dt=

1

JufIq - T sin ( ) (6)

dIq

dt =

1

Lq (uq - RIq - xuf) (7)In this reduced model, component ud is adjusted to regulate current state Id and it issupposed that Id exactly equals zero. Based on this reduced model, the position orvelocity control is only directly related to the command voltage uq. Obviously, basedon the reduced model, the control design will result in only a locally stable controller.

3. Adaptive backstepping control

In this section, the adaptive backstepping technique is used to develop a strictlyglobally stable controller for permanent magnet synchronous motors under theassumption that all the unknown system parameters are constant. The design pro-cedures are presented in detail as follows.

3.1. Design of auxiliary reference current IrefqFor the given desired tracking state d(t) , dene a quantity s1 to be

s1 = ce + e, e = - d, c > 0 (8)where d(t) is at least twice continuously dierentiable. Dierentiating (8), multi-plying by J and substituting the mechanical subsystem dynamics of (2), yields

Js1 = Jce + (Ld - Lq)Id + uf[ ]Iq - Tsin- Jxd (9)Divide (9) by uf and rearrange terms to yield

Js1 = a

>

1 u 1 + Iq + L

IdIq (10)

where a1 and u 1 are dened as

a>

1 = [J, - T], u>

1 = [ce - xd, sin] (11)J

, T

and L

are dened as

J

=J

uf, T

=

T

uf, L

=

Ld - Lquf

Dene an auxiliary reference current Irefq as

Irefq = - a>1 u 1 - k1s1 (12)where k1 is a positive constant gain. The parameter estimate a1 = [a11,a12]

>

isgenerated by the following adaptive law

a1 = G1u 1s1 (13)G1 R

22 is a positive denite matrix.

Remark 3. In the backstepping techniques, embedded control variable is an im-

portant concept. In the parametric-pure feedback form, the embedded control signalis designed as a full feedforward compensator which would provide good trackingperformance. But in our problem, due to the nonlinear term L

IdIq shown in (10), the

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embedded control can not be designed as a full feedforward compensator. Thus thenamed auxiliary reference current Irefq is designed as a partial compensator. Thenonlinear term L

IdIq will be treated later by using adaptive techniques.

3.2. Design of control voltage ud

In the adaptive backstepping method, the control voltage ud is designed not onlyto regulate the current Id, but also to compensate the nonlinear term L

IdIq in (10).

To design ud, s2 is selected as s2 = Id. Dierentiating s2 with respect to time, multi-plying by Ld and substituting the electrical subsystem dynamics (3), we have

Lds2 = a>

2 u 2 + ud (14)

where a2 and u 2 are given by

a>2 = [- R, Lq], u >2 = [Id,xIq] (15)The control voltage ud and the corresponding adaptive law are chosen to be

ud = - a>

2 u 2 - ^Ls1Iq - k2s2 (16)

a2 = C2u 2s2 (17)

^L= gs1s2Iq (18)

where k2 and g are positive constant gains. C2 R22 is a positive denite matrix.

Remark 4: Usually the control voltage ud is designed to assure Id vanish. In theproposed method, ud is used to handle the nonlinear term IdIq while ensuring that Idtends to zero, which will be shown later. This causes non-zero of Id in the adaptationperiod but the global stability of the closed-loop system is guaranteed from thebeginning.

3.3. Design of control voltage uq

The last step of the proposed design procedure is to choose control voltage uq toensure that the actual current Iq approaches I

refq . Dene s3 as

s3 = Iq - Irefq (19)

Dierentiating (19), multiplying by Lq and substituting (4), we obtain

Lqs3 = uq + a>

4 u 4 (20)where the unknown constant parameter vector a4 R

7and the known regression

vector u 4 R7 are dened (see Appendix A) as

a>

4 = - Ld, - R, - uf, Lquf

J, Lq

L

J, - Lq

T

J, Lq[ ]

u>

4 = [xId, Id, x, (k1 + ca11)Iq, (k1 + ca11)IdIq,

(k1 + ca11) sin, k1 (ce - xd) + s1u>

1G-1

1 u 1 + Ax]Design the control voltage uq with the corresponding adaptive law as


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uq = - a>

4 u 4 - k3s3 - s1 (21)

a4 = G4u 4s3 (22)where k3 is a positive constant gain, C4 R

77is a positive denite matrix.

The proposed adaptive backstepping controller ensures good tracking perform-ance for the closed-loop electromechanical system dynamics as described by thefollowing theorem.

Theorem 1: The proposed adaptive controller ensures global asymptotic positiontracking for the permanent magnet synchronous motor (1)(4), as shown

limt

e = 0 (23)

limte = 0 (24)

Proof: See Appendix B. h

As a summary, the control algorithm is given below.

Step 1. Calculating s1 and vector u 1; then the auxiliary reference current Irefq is

generated together with adaptation ofa1 as follows

Irefq = - a

>

1 u 1 - k1s1 (25)

a1 = G1u 1s1 (26)Step 2. Calculating u 2; then control law for the voltage ud and the adaptive

mechanisms for a2 and L

are given as

ud = - a>

2 u 2 - ^Ls1Iq - k2s2 (27)

a2 = G2u 2s2 (28)

L= gs1s2Iq (29)Step 3. Calculating s3, Ax, and vector u 4; then the control law for voltage uq and the

associated adaptation law are designed as

uq = - a>

4 u 4 - k3s3 - s1 (30)

a4 = G4u 4s3 (31)

Remark 5: In the above design procedure, the position tracking controller is de-veloped by dening s1 = ce + e. It is easy to extend the proposed control pro-cedure to velocity tracking control problems by dening s1 = e. Thecorresponding function vectors u 1, u 2, and u 4 are modied with c = 0.

4. Robust backstepping control

In adaptive control, the unknown parameters are considered to be constant.Unfortunately, some parameters of PM synchronous motors may be time varying

within certain known bounds, for instance the stator resistor R. In this section,taking into account the variations of system parameters, a robust position controllerfor the PM synchronous motor dynamics (1)(4) are developed. The pre-requirement

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for the robust control method is that bounds for all the parametric quantities in theelectromechanical system are available.

In the subsequent derivation procedure, unless otherwise specied, the notationsare dened the same as that for the adaptive backstepping controller. First, theauxiliary reference current I

refq is selected as

Irefq = - a

>

1 u 1 - k1s1 - vI (32)

where a1 is a constant parameter vector which represents a `best-guess for theunknown parameter vector a1 dened in (11). vI is an auxiliary robust controllerused to compensate for the mismatching between a1 and a1:

vI =s1q

2I

is1imqIm + eI(33)

where eI > 0 is a constant design parameter, and qI and qIm are positive scalarfunctions dened as

qI = iu 1i~a1b, qIm = iu 1im ~a1b (34)

where i i denotes the Euclidean norm, and i im is a norm dened to be

iwim = w>

w + d - d , for all w R n (35)with dbeing a positive constant. ~a1b is a bounding constant selected according to

~a1b > i~a1i = ia1 - a1i (36)

The second step is the design of the voltage control ud. In the robust controller,ud is designed to be

ud = - a>

2 u 2 - Ls1Iq - k2s2 - vd (37)

where k2 is a positive constant gain. The auxiliary controller vd is designed as

vd = (~Lb|s1Iq|+ qd) signs2 (38)

where qd is a positive scalar function dened as

qd = iu 2i~a2b (39)

~a2b > i~a2i = ia2 - a2i (40)

Similarly, ~Lb is a bounding constant chosen to be

~Lb > |

~L| = |L

- L

| (41)

The last step in the procedure is to design the voltage control uq. Similar to theadaptive control design procedure, dierentiating (19), multiplying by Lq and sub-stituting (4), we obtain

Lqs

3= u

q+a

>

5u

5(42)

where the unknown constant parameter vector a5 R7

and the known regressionvector u 5 R

7are dened as (see Appendix C)


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a>

5 = - Ld, - R, - uf, Lquf

J, Lq

L

J, - Lq

T

J, Lq[ ]

u>

5 = [xId, Id, x, (k1 + ca11 + H3)Iq, (k1 + ca11 + H3)IdIq,

(k1 + ca11 + H3) sin,k1 (ce - xd) + Ax + H4]

Now design the robust voltage control uq to be

uq = - a>

5 u 5 - k3s3 - s1 - vq (43)

where k3 is a positive constant gain. The auxiliary controller vq is designed as

vq = qq signs3 (44)

where qq is a positive scalar function dened as the following

qq = iu 5i~a5b, ~a5b > i~a5i = ia5 - a5i (45)

The proposed robust voltage controller ensures good position tracking perform-ance for the electromechanical system dynamics (1)(4) as described by the followingtheorem.

Theorem 2: The proposed robust controller ensures globally bounded position track-ing for the permanent magnet synchronous motor (1)(4). The tracking trajectory

will enter the setD = {s1,s2,s3 : s21 + s22 + s23 < k

eI}

and remain in it, where k

is a positive constant

k

=max {J

, Ld, Lq}

2min {k1,k2,k3}min{J, Ld, Lq}

. (46)

Proof: See Appendix D. h

Remark 6: Note that the bound of the residual set can be made arbitrarily smallby choosing a smaller eI at the expense of increasing the control authority in vI.

5. Adaptive robust backstepping control using switching function

In 3 and 4, adaptive and robust control methods for permanent magnet syn-chronous motors have been developed. We notice that, in the robust control scheme,the voltage control signal uq tends to be excessively large due to the terms whichcontain the derivative of the auxiliary controller vI used in the auxiliary referencecurrent I

refq , and is inevitable in the robust backstepping control scheme. Besides, the

robust backstepping control scheme becomes very complicated in comparison withthe adaptive control scheme.

In practical applications, the parameters Jand T are constants but may vary in awide range due to the variation of payload. On the other hand, the unknown motorparameters have much less deviations from its rated values (nominal values) incomparison with that of load. Therefore, it would be more appropriate for us todeal with the unknown parameters J and T by using adaptive techniques andtreating the bounded motor parameters by using robust methods. In this way, we

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can greatly simplify the controller design and reduce the control authority becausethe vI term is removed from the auxiliary reference current I

refq .

In the adaptive robust control method, auxiliary reference current Irefq is designedusing the adaptive method given by (12) and (13)

I

ref

q = -^

a

>

1 u 1 - k1s1 (47)a1 = G1u 1s1 (48)

G1 is chosen the same as previous. ud is designed the same as the robust controlvoltage given by (37)(41), namely

ud = -a>

2 u 2 - Ls1Iq - k2s2 - vds (49)

vds

= (~Lb|s

1I

q|+ q

d

) sign s2

(50)

Based on the relation (20)

Lqs3 = uq + a>

4 u 4 (51)

the control voltage uq now is designed as

uq = - a>

4 u 4 - k3s3 - s1 - vqs (52)

where k3 is a positive constant gain. The auxiliary controller vqs is designed as

vqs = qq signs3 (53)

where qq is a positive scalar function dened as follows

qq = iu 4i~a4b, ~a4b > i~a4i = ia4 - a4i (54)

For the adaptive robust controller (47)(54), we have the following theorem.

Theorem 3: The proposed adaptive robust control law (47)(54) ensures globalasymptotic position tracking for the permanent magnet synchronous motor (1)(4),i.e.

limt

e = 0 (55)

limte = 0 (56)

Proof: See Appendix E. h

Remark 7: It is worthwhile pointing out that, owing to the absence of the vIterm in the auxiliary reference current Irefq , the required control uq now is muchsimpler than that of the robust one. In the robust control scheme, uq is a functionof u 5 which includes terms such as xd xd. In the adaptive robust scheme, uq is afunction of u 4 which only contains terms such as xd. Hence much lower controlproles can be expected, especially for the cases where fast system responses arerequired.

Remark 8: Although the above controller guarantees the global stability of thesystem (1)(4), problems may be caused by the incorporation of switching func-tions. On the switching surface there might be some problems related to existence


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and uniqueness of solutions. From the practical point of view, discontinuous func-tions also give rise to implementation problems and may excite high frequencymodes of unmodelled dynamics. Hence our objective in the next section is to re-place the discontinuous switching functions by continuous functions.

6. Continuous adaptive robust backstepping control with -modication scheme

In this section, continuous functions are used in the adaptive robust scheme toreplace the signum functions. A new update scheme, referred to as the -modica-tion scheme, is proposed to cease parameter adaptation in accordance with theadaptive robust control law. The novel property possessed by this new adaptiverobust backstepping control scheme is that, it guarantees the uniform boundedness

of the system and at the same time, is able to allow the tracking precision to bearbitrarily designated through the choice of suitable control design parameters.

First, the auxiliary reference current Irefq is designed as

Irefq = - a

>

1 u 1 - k1s1 (57)

a1 = G1u 1s1 - G1a1 (58)where G1 is chosen the same as previous and is dened as

=0 |s1| e0ks1 (e0 - |s1|) otherwise{ (59)

where ks1 is a positive constant and e0 is a small positive value determined by thedesired tracking error bound. If the tracking error bound is specied by

|e|= |- d| emin,then

e0 = cemin

The robust control voltage ud is designed to be

ud = - a>

2 u 2 - Ls1Iq - k2s2 - vds (60)

where k2 is a positive constant gain. The auxiliary controller vds is designed as

vds =s2s

21I

2q

~L

2b

is1s2Iqi

~

Lb + e1

+s2q

2d

is

2iq

d+ e

2

(61)

where qd and~Lb are dened in (39) and (41) respectively. e1 > 0 and e2 > 0 are

design parameters.Based on the relation (20)

Lq s3 = uq + a>

4 u 4 (62)

the control voltage uq now is designed as

uq = - a>

4 u 4 - k3s3 - s1 - vqs (63)

where k3 is a positive constant gain. The auxiliary controller vqs is designed to be

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vqs =s3q

2q

is3iqq + e3(64)

where e3 > 0 is a design parameter.For the above adaptive robust controller, we have the following theorem.

Theorem 4: By choosing the control gain k1 (e+ c1) /e20 with e= e1 + e2 + e3and c1 > 0, the proposed continuous adaptive robust control law (57)(64) ensuresthat the system trajectory as well as the parameter estimation error will enter the set

D = {s1,s2,s3, ~a1 : |s1| e0;s21 + s22 + s23 + ~a>

1~a1 < k

(ks1e0a

>

1 a1 + e)} (65)

in a nite time, where k

is dened to be

k

= max {J, Ld, Lq,(C- 11 )}2 min {k0,k1,k2,k3}min{J

, Ld, Lq,

(C- 11 )}

(A) and

(A) indicate the maximum and minimum eigenvalues of the matrix Arespectively. k0 > 0 is a constant to be explained later.

Proof: The following positive denite function V is selected

V = 12

Js

21 +

1

2Lds

22 +

1

2Lqs

23 +

1

2

~

a

>

1

G

- 11

~

a1 (66)

where

~a1 = a1 - a1Taking the derivative of V along the trajectory of the overall electromechanicaldynamics (1)(4) with the adaptive robust control law (57)(64), we have

V = Js1 s1 + Lds2 s2 + Lqs3s3 +~a>

1 ~a1

= - k1s21 + s1 (Iq - Irefq ) + Ls1IdIq + s2 (a>2 u 2 - a>2 u 2 - Ls1Iq - k2s2 - vds)

+ s3 (a>

4 u 4 - a>

4 u 4 - k3s3 - s1 - vqs) + ~a>

1 a1

= - k1s21 + s1 (L

- L)IdIq - k2s22 + s2 (a2 - a2)

>

u 2 - s2vds

- k3s23 + s3 (a4 - a4)>

u 4 - s3vqs + ~a>

1 a1

- k1s2

1 - k2s2

2 - k3s2

3 +

~

Lb|s1IdIq|+ qd|s2|+ qq|s3| - s2vds - s3vqs - ~a

>

1

^

a1

- k1s21 - k2s22 - k3s23 + e1 + e2 + e3 + ~a>

1 a1

- k1s21 - k2s22 - k3s23 + e+ ~a>

1 (a1 - ~a1)

- k1s21 - k2s22 - k3s23 + e- ~a>

1~a1 + ~a

>

1 a1

- k1s21 - k2s22 - k3s23 - 12~a>

1~a1 +

12 a

>

1 a1 + e (67)

where

e= e1 + e2 + e3


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From (59), if |s1| e0, we have

V - k1s21 - k2s22 - k3s23 + e (68)If we choose k1 to be

k1 e+ c1e20

where c1 is an arbitrary small positive constant, then we can easily get

V - c1 "|s1| > e0 (68)Notice that V is a continuous function, so obviously there exists a constant e0 suchthat:

0 < e0 < e0

V < 0 "|s1| > e0 (69)From (67), we can see that V < 0 if:

k1s21 > 12a

>

1 a1 + e

Note that a1 and e are constants and is a function ofs1 in terms of the denition

(59). Hence e0 can be easily determined by solving the equationk1s

21 =

12a

>

1 a1 + e

Substituting in terms of (59) and replacing s1 by e0, we get:

k1e0

2= 12 ks1 (e0 - e0)a

>

1 a1 + e (70)

The positive solution of equation (70) is the desired e0 :

e0 = - k

s1a>

1 a1 +(ks1a

>

1 a1)2

+ 4k1 (ks1a>

1 a1e0 + 2e) 2k1As a consequence, when |s1| > e

0, the system will enter |s1|> e0 in a nite time. On

the other hand, when |s1| e0, it is obvious that

ks1 (e0 - e0) ks1e0 (71)

From (59), (67) and (71) we obtain

V - k V + 12 a>1 a1 + e

- k V + ks1e0a>

1 a1 + e "|s1| e0 (72)

where

k =2 min {k0,k1,k2,k3}

max {J, Ld, Lq,(G- 11 )}

k0 =12 ks1 (e0 - e0) > 0 (73)

By integrating (72) we can establish that

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V(t) e-kt

V(t = 0) +1

k(ks1e0a

>

1 a1 + e) (73)

which implies that s1, s2, s3 and ~a1 converge exponentially to the residual set

D

= {s1,s2,s3, ~a1 : s21 + s22 + s23 + ~a>

1~a1 < k

(ks1e0a

>

1 a1 + e)} (74)

where

k

=max {J

, Ld, Lq,(G- 11 )}

2 min {k0,k1,k2,k3}min{J, Ld, Lq,

(G- 11 )}

Finally, from (68), (69) and (74) we can conclude that s1,s2,s3, and ~a1 convergeexponentially to the residual set

D = {s1,s2,s3, ~a1 : |s1| e0;s21 + s22 + s23 + ~a>

1~a1 < k

(ks1e0a

>

1 a1 + e)} h

Remark 9: Analogous to the selection of the gain k1, we can choose gains k2 andk3 such that the tracking error bounds of the quantities s2 and s3 can be speciedthrough the design.

Remark 10: In Liao et al. (1990) and Brogliato and Neto (1995), the synthesis ofthe adaptive and robust control schemes with specied tracking error bounds werepresented. However, those methods are valid only for one-step design of thesystems. In other words, their algorithms cannot be extended to the backsteppingdesign which requires that the input be dierentiable at each step and consists ofat least two steps. It should be noted that, the control design for the PM synchro-nous motor is essentially two steps: the ctitious auxiliary reference current inputand the actual voltage inputs ud and uq; thus the method proposed here is applic-able.

7. Simulation studiesThe permanent magnet synchronous motor with following parameter is used to

demonstrate the control performance

Ld = 25.0 10- 3 H, Lq = 30.0 10- 3 H, J= 1.625 10- 3 Kg m2,

uf = 0.90 N m/A, R = 5.0 V, T = 2.2816 Kg A m/N s2

a>

1 = [0.0018,- 2.535]The following control parameters

G1 = 1.2I22, c = 5.0

k1 = 8.0, k2 = 2.0, k3 = 2.0are used for all the proposed control schemes. The estimatesa1,a2 and a4 are chosen15% larger than their real values, while ~a2b, ~a4b and ~a5b are chosen based on theassumption that each of the electromechanical parameters in (2) and (5) would be o

20% from the estimates a2,a4 and a5. The desired trajectory isd =

p2

(1 - e- 0.1t3

) sinp5

t( )


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The initial states are (0) = 0.1 rad, x(0) = 0.1rad /s, Id(0) = 0.05A, Iq (0) = 0.01A.The initial values of all the parameter estimates are set zero.

7.1. Adaptive backstepping control

The control law (25)(31) is found to yield a `good position tracking perform-ance with the following control parameters

C2 = 0.05I22, g= 0.05, C4 = 0.05I77The control inputs ud and uq are given in gure 1. The tracking error is shown in

gure 2. Figure 3(a) shows the corresponding armature current Iq compared withauxiliary reference current I

refq while Id is shown in gure 3(b). It is easy to see that

the current Iq asymptotically converges to Irefq . During the adaptation period, ud is

used to counteract the eect of the nonlinear term IdIq in the mechanical subsystem.

7.2. Robust backstepping control

Taking into account the variations of system parameters, the robust controlmethod is used with the following control parameters

eI = 0.1, ~a1b = 0.5, ~a2b = 1.0~Lb = 0.002,

~a5b = 3.0

The simulation results are shown in gures 46. It is obvious that robust controlmethod is found to yield the better position tracking performance compared with the

906 J.-X. Xu et al.

Figure 1. (a) Evolution of input voltage uq; (b) Evolution of input voltage ud.

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Figure 2. (a) Position tracking error e; (b) Velocity tracking error e.

Figure 3. (a) Evolution of current Iq (solid-line) and Irefq (dot-line); (b) Evolution of

current Id.

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908 J.-X. Xu et al.

Figure 4. (a) Evolution of input voltage uq; (b) Evolution of input voltage ud.


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adaptive control method, as indicated in gure 5. However, the control voltage uq,shown in gure 4(a), is much higher than that of adaptive control method due to theterms which contain the derivative of the auxiliary controller vI.

7.3. Adaptive robust backstepping control method using switching function

The additional control parameter

~a4b = 3.0

is selected in this control scheme. The parameter estimate a1 is generated by theadaptation law (48). Simulation results are depicted in gures 79.

We can see that the control scheme achieves good tracking performance (gure

8), while the control voltage uq is greatly reduced compared with the robust back-stepping control signal in gure 4(a). Due to the incorporation of switching func-tions, high frequency chattering phenomenon can still be observed in the controlproles.

7.4. Continuous adaptive robust backstepping control method with -modicationscheme

The following control parameters are chosen in the adaptive robust controlscheme proposed in 6:

e1 = e2 = e3 = 2.0, ks1 = 8


Figure 6. (a): Evolution of current Iq; (b) Evolution of current Id.

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910 J.-X. Xu et al.

Figure 7. (a) Evolution of input voltage ud; (b) Evolution of input voltage uq.


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The parameter estimate a1 is generated by the adaptation law (48). The simulationresults are shown in gures 1013.

We can see that the tracking performance (gure 11) is similar to that of 7.3, butthe chattering phenomenon has been reduced signicantly, hence the control signals

are much smoother (as shown in gures 10 and 12). The parameter estimate a1 isshown in gure 13.

Remark 11: The control `chattering is highly dependent on the choice of the de-sign parameters e1, e2 and e3. The greater the three parameters, the smaller the`chattering, but the greater the size of the residual set D (65). Notice that if theparameters are set to zero, the control scheme in 6 is exactly the same as that in 5.

Remark 12: Comparing the four control schemes developed in this paper, it isclear that the continuous adaptive robust control scheme is the best in the senseof making `balance in terms of the tracking accuracy, control algorithm complex-ity, control signal authority and smoothness, and design degrees of freedom.

8. C on clu sio ns

In this paper, four control schemes are developed for a permanent magnet syn-chronous motor which is not exactly in the parametric-pure feedback form. The

adaptive backstepping control is rst proposed to achieve strictly globally asympto-tic convergence of the system tracking error. Taking into account parameter vari-ations in electromechanical system, the robust backstepping control scheme is


Figure 9. (a) Evolution of current Id; (b) Evolution of current Iq.

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912 J.-X. Xu et al.

Figure 10. (a) Evolution of input voltage ud; (b) Evolution of input voltage uq .


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Figure 12. (a) Evolution of current Id; (b) Evolution of current Iq.

Figure 13. (a) Evolution of parameter a11; (b) Evolution of parameter a12.

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proposed which achieves better tracking performance at the expense of larger controlauthorities. To obtain better tracking control performance and yet retain low controlproles, the adaptive and robust backstepping control scheme using switching func-tions are synthesized. In the adaptive robust controller, the robust part is used tosuppress the variations of system parameters with upper bounds while the unpre-

dictable load is estimated by the adaptive part. In order to smooth the control inputsignals, we further develop the continuous adaptive robust backstepping controlmethod associated with the -modication scheme, which achieves signicant reduc-tion of the control chattering and at the same time guarantees the uniform bound-edness of the system tracking error with specied accuracy. Simulation studiesconrm the validity of the four proposed control schemes.

Appendix A: Denitions of a4 and }4

Dierentiating (19), yields


From (12), we have

Irefq = - (s1C1u 1)>

u 1 - a>

1 u 1 - k1s1

= - s1u>

1 C1u 1 - a>

1 u 1 -k1

J

(a>

1 u 1 + Iq + L

IdIq) (76)

For a>

1 u 1, it followsa>

1 u 1 = a11[c( x- xd) - xd]+ a12xcos

=ca11

J(Ld - Lq)IdIq + ufIq - Tsin[ ]- a11 (cxd + xd) + a12xcos

= a>

3 u 3 + Ax (77)

where a3, u 3 and Ax are dened by

a>

3 =Ld - Lq

J,uf

J,- T

J[ ] (78)u>

3 = ca11[IdIq,Iq, sin] (79)

Ax = - a11 (cxd + xd) + a12xcos (80)

Multiplying (75) by Lq and substituting (4), we obtain

Lq s3 = uq - xLdId - RId - xuf

+ Lq s1u>

1 G-1

1 u 1 + a>

3 u 3 + Ax +k1

J

(a>

1 u 1 + Iq + L

IdIq)[ ]= uq + a

>

4 u 4 (81)


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a>

4 = - Ld, - R, - uf, Lquf

J, Lq

L

J, - Lq

T

J, Lq[ ]

u>

4 = [xId, Id, x, (k1 + ca11)Iq, (k1 + ca11)IdIq, (k1 + ca11) sin ,k1 (ce - xd) + s1u

>

1 C- 11 u 1 + Ax]

Appendix B : P roof of T heor em 1

First, dene the following positive denite functions V1 and V2

V1 =12 J

s21 +

12

(a1 - a1)>

C- 11 (a1 - a1) (82)

V2 =

1

2 Lds

2

2 +

1

2 (a2 -^

a2)

>

C

- 12 (a2 -

^

a2) (83)It follows that

V1 = Js1 s1 - (a1 - a1)>

G- 11 a1

= s1 (a>

1 u 1 + Iq + L

IdIq) - (a1 - a1)>

G- 11 a1

= s1 (a>

1 u 1 + Irefq + Iq - Irefq + L

IdIq) - (a1 - a1)

>

G- 11 a1

= s1 (a1 - a1)>u 1 - k1s1 + Iq - Irefq + LIdIq[ ] - (a1 - a1)>C- 11 a1

= - k1s21 + s1 (Iq - Irefq ) + Ls1IdIq (84)

and

V2 = Lds2s2 - (a2 - a2)>

G- 12 a2

= s2[(a2 -^

a2)

>

u

2 -^

Ls1s2Iq - k2s2]- (a2 -^

a2)

>

C- 12

^

a2

= - k2s22 - ^Ls1s2Iq (85)

To complete the analysis of the adaptive backstepping controller, the followingpositive denite function V3 is selected

V3 = V1 + V2 + 12 Lqs23 +

12 (a4 - a4)

>

C- 14 (a4 - a4) + 12g-

1 (L

- ^L

) 2 (86)

Taking the derivative of V3 along the trajectory of the overall electromechanical

dynamics (1)(4) and substituting V1, V2 yieldsV3 = V1 + V2 + Lqs3 s3 - (a4 - a4)

>

C- 14 a4 - g- 1 ( L-

^L) ^L

= - k1s21 + s1 (Iq - Irefq ) + Ls1IdIq - k2s22 - ^L

s1s2Iq

+ s3 (- a>

4 u 4 - k3s3 - s1 + a>

4 u 4) - (a4 - a4)>

G- 14 a4 - g- 1 (L-^

L

) ^L

= - k1s21 - k2s22 - k3s23 + s1 (Iq - Irefq ) + s1s2Iq (L- L) - s3s1 - g- 1 (L- L) L

= - k1s21 - k2s22 - k3s23 (87)


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This fact implies that s1, Id (s2), s3, and the parameter estimates are bounded andsquare integrable. Consequently, it is easy to verify that s1, s2 and s3 are bounded.These facts, together with Barbalats Lemma, allow us to conclude that

limt

s1 = 0

limt

s2 = 0

limt

s3 = 0

From the denition of s1, it shows that the control task has been achieved by thiscontroller and the control system is global stable, i.e.

limt

e = 0

limte = 0

Appendix C: D enitions of a5 an d }5

Dierentiating (19), yields


From (32), we have

Irefq = - a>

1 u 1 - k1 s1 - vI (89)For vI, it follows that

vI = s1q

2I + 2s1 u

>

1 u 1~a1b

is1imqIm + eI-

s1q2I

is1imqIm + eI( )2 is1im

qIm + i

s1imqIm( )

= s1q2I

is1imqIm + eI+

2s1 u>

1 u 1~a1b

is1imqIm + eI

-s1q

2Iis1im u

>

1 u 1~a1b

u >1 u 1 + d( ) is1imqIm + eI( )2

-s

21q

2Is1qIm

s21 + d( ) is1imqIm + eI( )

2

= H1 s1 + H2 u>1 u 1 (90)

where H1 and H2 are given by

H1 =q

2I

is1imqIm + eI-

s21q

2IqIm

s21 + d( ) is1imqIm + eI( )

2(91)

H2 =

2s1 ~a1b

is1imqIm + eI -

s1q2Iis1im ~a1b

u >1 u 1 + d( ) is1imqIm + eI( ) 2(92

)

In terms of denitions of s1 in (8) and u 1 in (11), we have

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vI = H1[ce + x- xd]+ H2[(c2

e - cxd) (x- xd) + xsincos+ xd xd - c xde]= [H1 + H2 (c2e - cxd)]x+ H1 (ce - xd)

+ H2[xsincos+ xd xd

- (c2e - cxd) xd - ce xd]

=H3

J{[(Ld - Lq)Id + uf]Iq - Tsin }+ H4 (93)

where H3 and H4 are given by

H3 = H1 + H2 (c2

e - c

xd) (94)

H4 = H1 (ce - xd) + H2[xsin cos+ xd xd - (c2

e - cxd) xd - ce xd] (95)

Similar to the derivation of (77), we get

a>

1 u 1 = a11[c( x- xd) - xd]+ a12xcos

=ca11

J

(Ld - Lq)IdIq + ufIq - Tsin[ ]- a11 (cxd + xd) + a12xcos

= a>

3 u 3 + Ax (96)

where

u>

3 = ca11[IdIq,Iq,sin ] (97)

Ax = - a11 (cxd + xd) + a12xcos (98)

Using (93) and (96), Irefq is given by

Irefq = - a>

1 u 1 - k1 s1 - vI

= - a>3 u 3 - Ax -k1

J

(a>

1 u 1 + Iq + L

IdIq)

-

H3

J {[(Ld

-Lq)Id + uf

]Iq

-Tsin

}-H4 (99)

Multiplying (88) by Lq and substituting (4) and (99), we obtain

Lqs3 = uq - xLdId - RId - xuf + Lq a>

3 u 3 + Ax +k1

J

(a>

1 u 1 + Iq + L

IdIq)(+

H3

J{[(Ld - Lq)Id + uf]Iq - Tsin}+ H4 )

= uq + a>

5 u 5 (100)



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a>

5 = [- Ld, - R, - uf, Lquf

J, Lq

L

J, - Lq

T

J, Lq]

u>

5 = [xId, Id, x, (k1 + ca11 + H3)Iq, (k1 + ca11 + H3)IdIq, (k1 + ca11 + H3) sin,

k1(ce -

xd

)+ Ax + H4]

Appendix D: P roof of T heorem 2

The following Lyapunov function candidate V is selected

V = 12 Js

21 +

12 Lds

22 +

12 Lqs

23 (101)

Taking the derivative of V along the trajectory of the overall electromechanicaldynamics (1)(4), substituting (10), (14), (100) and the robust controller (32), (37)and (43) we have

V = Js1s1 + Lds2 s2 + Lqs3s3

= s1 (a>

1 u 1 + Irefq + Iq - Irefq + L

IdIq) + s2 (a

>

2 u 2 - a>

2 u 2 - Ls1Iq - k2s2 - vd)

+ s3 (a>

5 u 5 - a>

5 u 5 - k3s3 - s1 - vq)

= - k1s21 + s1 (a1 - a1)>

u 1 + s1 (Iq - Irefq ) + s1 L

IdIq - s1vI

- k2s22 + s2 (a2 - a2)>

u 2 - Ls1s2Iq - s2vd

- k3s23 + s3 (a5 - a5)>

u 5 - s1s3 - s3vq

= - k1s21 + s1 (a1 - a1)>

u 1 + s1 ( L

- L)IdIq - s1vI

- k2s22 + s2 (a2 - a2)>

u 2 - s2vd

- k3s23 + s3 (a5 - a5)>

u 5 - s3vq

- k1s21 + s1~a>

1 u 1 -s

21q

2I

is1imqIm + eI+ s1

~L

IdIq - ~Lb|s1IdIq|

- k2s22 + s2 ~a>

2 u 2 - qd|s2| - k3s23 + s3 ~a>

5 u 5 - qq|s3| (102)

In terms of the choices of qI, qd and qq, we can obtain the following

V - k1s21 - k2s22 - k3s23 + eI (103)which is upper bounded by the expression

V - kV + eI (104)where k is dened to be

k = 2min {k1, k2,k3}max {J, Ld, Lq}. (105)

By integrating (104), we can establish that

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V(t) e-kt

V(t = 0) +eI

k(1 - e- kt) (106)

which implies that s1,s2 and s3 converge exponentially to the residual setD = {s1,s2,s3 : s21 + s22 + s23 < k

eI}

where

k

=1

k

1

min {J, Ld, Lq}

=max {J

, Ld, Lq}

2min {k1,k2,k3}min {J, Ld, Lq}

. h

Appendix E : P roof of T heor em 3

The following positive denite function W is selected as

W= 12 Js

21 +

12 Lds

22 +

12 Lqs

23 +

12

(a1 - a1)>

C- 11 (a1 - a1) (107)

Taking the derivative of W along the trajectory of the overall electromechanicaldynamics (1)(4) with adaptive robust control law (47)(54), we have

W= Js1 s1 + Lds2 s2 + Lqs3 s3

= - k1s21 + s1 (Iq - Irefq ) + Ls1IdIq + s2 (a

>

2 u 2 - a>

2 u 2 - Ls1Iq - k2s2 - vds)

+ s3 (a>

4 u 4 - a>

4 u 4 - k3s3 - s1 - vqs)

= - k1s21 + s1 (L

- L

)IdIq - k2s22 + s2 (a2 - a2)>

u 2 - s2vds

- k3s23 + s3 (a4 - a4)>

u 4 - s3vqs

= - k1s21 - k2s22 - k3s23 + s1 ~LIdIq - ~Lb|s1IdIq|+ s2 ~a>

2 u 2

- qd|s2|+ s3 ~a>

5 u 5 - qq|s3| (108)

Utilizing (34), (39)(41), and (45), we can obtain

W - k1s21 - k2s22 - k3s23 (109)Similar to Theorem 1, the following results are concluded as

limts1 = 0

limt

s2 = 0

limt

s3 = 0

From the denition of s1, it shows that

limte = 0

limte = 0 h


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920 Control schemes for PM synchronous motors

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