position control of permanent magnet stepper motors using conditional servocompensators

8/7/2019 Position control of permanent magnet stepper motors using conditional servocompensators

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Published in IET Control Theory and Applications

Received on 17th June 2008

Revised on 18th October 2008

doi: 10.1049/iet-cta.2008.0243

ISSN 1751-8644

Position control of permanent magnet steppermotors using conditional servocompensatorsS. Seshagiri ECE Department, San Diego State University, San Diego, CA, USA

E-mail: [email protected]

Abstract: The application of the ‘conditional servocompensator’ technique to position control of a permanent

magnet stepper motor is studied. This is a recent approach to the output regulation of minimum-phase non-

linear systems that results in better transient performance over ‘conventional’ servocompensator-based

design. Global regulation results are provided for state-feedback control and semi-global results under output

feedback, with regional results when the control is constrained. The simulation results show that good

tracking performance is achieved, in spite of partial knowledge of the machine parameters.

1 Introduction

Permanent magnet stepper motors (PMSM) have become a popular alternative to the traditionally used brushed DCmotors (BDCM) for many high performance motioncontrol applications for several reasons: better reliability because of the elimination of mechanical brushes, better heat dissipation as there are no rotor windings, higher torque-to-inertia ratio because of a lighter rotor, lower priceand easy interfacing with digital systems [1]. They are now

widely used in numerous motion control applications suchas robotics, printers, process control systems and so on.Some of the drawbacks of PM machines when operated inopen loop are the occurrence of large overshoots and

settling times, especially when the load inertia is high, andthe fact that microstepping is not possible in the open-loopmode of operation. As a result, over the years, many control algorithms that can improve the performance of PMSMs in a closed-loop operation have been examined.

Zribi and Chiasson [2] used the technique of exact feedback linearisation using full state-feedback, with extensions to thepartial state-feedback case in [3, 4], and experimental

validation of the controller in [3]. Adaptive solutions to theproblem, under varying assumptions on the measurable statesand on what parameters in the system are partially or wholly

known, have appeared, for example, in the works of Dawsonand co-workers [1, 5], Khorrami and co-workers [6–9] andothers [10]. A sliding mode controller (SMC) along with

implementation results was reported in Zribi et al. [11]. Inorder to avoid the chattering problem associated with the

‘static’ discontinuous SMC, a ‘dynamic’ or ‘second-order’SMC was proposed, where the discontinuities were relegatedto the derivatives of the control input.

We present a new approach to position control based on a recently proposed technique for the output regulation of minimum-phase non-linear systems [12, 13], whichincorporates a servocompensator as part of a robust SMCdesign. The servocompensation is ‘conditional’ in the sensethat it behaves like a servocompensator only inside theboundary layer. This is shown to improve the transient performance over conventional servocompensator design.

The method applies to MIMO non-linear systems with well-defined relative degree, which are transformable tonormal form uniformly in a compact set of unknownparameters, and minimum phase. The design guaranteesglobal regulation under state-feedback semi-globalregulation under output-feedback and regional regulation

with constrained control, under slightly differing assumptions. The performance of the design is comparableto ideal (discontinuous) SMC, but does not suffer from thedrawback of chattering. In the special case of constant exogenous signals, the conditional servocompensator can besimplified to specially tuned PI/PID controllers with an

anti-windup structure (see [12, Section 6]) for relativedegree one and two systems respectively. Preliminary results were presented in our earlier work [14].

1196 IET Control Theory Appl., 2009, Vol. 3, Iss. 9, pp. 1196–1208

& The Institution of Engineering and Technology 2009 doi: 10.1049/iet-cta.2008.0243

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The rest of this paper is organised as follows. The systemmodel is presented in Section 2, and Section 3 deals withconditional integrator design for the case of (asymptotically) constant exogenous signals, with thedesign done for both the state and output-feedback cases.

An extension to the case of sinusoidal reference signals

using conditional servocompensators is presented in Section4, and conclusions are presented in Section 5. The specificcontribution of this work over [14] are the results of Section 4.1, dealing with internal model perturbations, andthe advantages of the proposed (conditionalservocompensator) approach over a conventional approachin handling such perturbations.

2 System model

A schematic of a PMSM that has a slotted stator with twophases, and a PM rotor is shown in Fig. 1.

The mathematical model of the PMSM is given below [2,3, 11]

di a

dt ¼

1

L(v a ÀRi a þK mv sin( N r f ))

di b

dt ¼

1

L(v b ÀRi b þK mv cos( N r f ))

dv

dt ¼

1

J (K mi b cos( N r f )ÀK mi a sin( N r f )ÀB v À t L)

df

dt ¼ v

9>>>>>>>>>>>=>>>>>>>>>>>;

(1)

where i a is the current in winding A, i b is the current in winding B, f is the angular displacement of the shaft of the motor, v is the angular velocity of the shaft of themotor, v a is the voltage across winding A, v b is the voltageacross winding B, N r is the number of rotor teeth, J is therotor and load inertia, B is the viscous friction coefficient,

L and R are the inductance and resistance, respectively, of the phase windings, K m is the motor torque (back-emf)constant, and t L is the load torque. The model neglects theslight magnetic coupling between the phases, the smallchange in inductance as a function of the rotor position,the detent torque [6], and the variation in inductance

because of magnetic saturation. The DQ transformation [2]from the fixed axes variables (x a , x b ) to the dq axes variables(x d , x q ), defined by

x d

x q

¼def cos( N r f ) sin( N r f )

Àsin( N r f ) cos( N r f )

x a

x b

changes the frame of reference from the fixed phase axes toaxes that are moving with the rotor. The direct current i d

corresponds to the component of the stator magnetic fieldalong the axis of the rotor magnetic field, whereas thequadrature current i q corresponds to the orthogonalcomponent. Defining the states, inputs and outputs as

x 1 ¼ i d , x 2 ¼ i q , x 3 ¼v , x 4 ¼f , u 1 ¼ v d ,

u 2 ¼ v q , y 1 ¼ i d , and y 2 ¼f

it can be verified that (1) can be rewritten in the form of thefollowing state model for the PMSM

_x 1 ¼Àk1x 1 þ k5x 2x 3 þ k6u 1_x 2 ¼Àk1x 2 À k5x 1x 3 À k2x 3 þ k6u 2_x 3 ¼ k3x 2 À k4x 3 À d 0_x 4 ¼ x 3

y 1 ¼ x 1 y 2 ¼ x 4

(2)

where the constants k1– k6 and d 0 are related to N r , J , B , L, R ,K m and t L by

k1 ¼R

L, k2 ¼

K mL

, k3 ¼K m J

, k4 ¼B

J , k5 ¼ N r ,

k6 ¼1

Land d 0 ¼

t L J

3 Conditional integrator design

We consider the case of asymptotically constant exogenoussignals, that is, it is desired that the rotor angular positionand direct-axis current track given references f d(t ) and

I dd (t ) that tends to constant values as t 1 [11]. It canbe verified that the system (2) has full vector relative degreer ¼ {1, 3}, globally in R 4. In ideal sliding mode control, a choice of sliding surface functions would have been

s 1 ¼ e 1, e 1 ¼def

y 1 À I dd ,

s 2 ¼ k21e 2 þ k2

2 _e 2 þ €e 2 , e 2 ¼def

y 2 À f d

(3)

Figure 1 Schematic of a two-phase PMSM

IET Control Theory Appl., 2009, Vol. 3, Iss. 9, pp. 1196 – 1208 1197

doi: 10.1049/iet-cta.2008.0243 & The Institution of Engineering and Technology 2009

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with k21 and k2

2 chosen such that the polynomial

l2þ k2

2lþ k21

is Hurwitz, and the control designed to force the trajectories

to reach the surfaces s i ¼ 0 in finite time and remain on themthereafter. However, as is well known, this method suffersfrom the drawback of chattering, and can exciteunmodelled high-frequency dynamics and degrade systemperformance. Replacing the discontinuous (ideal) control by a continuous approximation in a boundary layer of thesliding surface reduces chattering but at the expense of a finite steady-state error. Asymptotic regulation can berecovered in the continuous sliding mode control (CSMC)by augmenting a conventional integrator

_s i ¼ e i

as part of the sliding surface but (i) requires a redesign of thesliding surface parameters and (ii) degrades transient performance in comparison to ideal SMC, in part becauseof the increase in system order as a result of the integrator,and in part because of the interaction of the integrator withcontrol saturation, which leads to the well known problemof windup. In [12], we presented a ‘conditional integrator’design that introduces integral action conditionally, that is,only inside the boundary layer, thereby recovering asymptotic regulation of ideal SMC, while not degrading its transient performance. The design in [12] was presented

for a general MIMO non-linear system, modelled by

_x ¼ f (x , u ) þXm

i ¼1

g i (x , u )[u i þ d i (x , u , w)]

y i ¼ hi (x , u ), 1 i m

with state x [ R n, input u [ R m, output y [ R m, u a vector of unknown constant parameters belonging to the compact set Q , R p , w(t ) a piecewise continuous exogenous signalbelonging to a compact set W , R q , f (Á) and g i (Á) smooth

vector fields, and piecewise continuous disturbances d i (Á).

In the rest of this section, we directly present the design in[12] for the PMSM model (2).

3.1 State-feedback design

In order to recover the asymptotic regulation of ideal SMCand also retain its transient performance, we modify thesliding surfaces (3) as follows

s 1 ¼ k10s 1 þ e 1

s 2 ¼ k20s 2 þ k2

1e 2 þ k22 _e 2 þ €e 2

(4)

where s 1 and s 2 are the outputs of the conditional

integrators

_s 1 ¼ Àk10s 1 þ m1sat(s 1=m1), s 1(0) [ [Àm1=k1

0, m1=k10]

_s 2 ¼ Àk20s 2 þ m2sat(s 2=m2), s 2(0) [ [Àm2=k2

0, m2=k20]

(5)

k10, k2

0 . 0, the values of k21 and k2

2 are retained from the idealSMC design (3), and m1, m2 are ‘sufficiently small’ positiveconstants representing the widths of the boundary layers for s 1 and s 2, respectively. To see the relation of (5) to integralcontrol, observe that inside the boundary layer {js i j mi },(5) reduces to

_s 1 ¼ e 1

_s 2 ¼ k21e 2 þ k2

2 _e 2 þ €e 2

which implies that e i ¼

0 at equilibrium. Thus, (5) representsa ‘conditional integrator’ that provides integral action only inside the boundary layer.

Since

€e 2 ¼ k3x 2 À k4x 3 À d 0 À f (2)d

is required to construct s 2, the parameters k3, k4 and d 0 willneed to be known. We assume that k1, k2 and k6 areunknown, corresponding to uncertainties in resistance R and inductance L of the phase windings, whereas k5, being the number of rotor teeth, is precisely known. Let

u ¼ [u 1, u 2, u 3] T ¼ [k1, k2, k6] T

denote the vector of unknown parameters. It can be verifiedthat the expressions for _s i take the form

_s 1 ¼ k10(Àk1

0s 1 þ m1sat(s 1=m1))

þ F 1(x , e 1, I (1)dd , u ) þ a 11(x , u )u 1

_s 2 ¼ k20(Àk2

0s 2 þ m2sat(s 2=m2))

þ F 2(x , e 2, _e 2, €e 2, f (3)

d

, u ) þ a 22(x , u )u 2

where

F 1(Á) ¼ Àu 1x 1 þ k5x 2x 3 À I (1)dd

F 2(Á) ¼ k21 _e 2 þk2

2 €e 2 Àf (3)d À k4(k3x 2 À k4x 3 À d 0) À k3u 1x 2

À k3 k5x 1x 3 À k3u 2x 3

and a 11(Á) ¼ u 3, a 22(Á) ¼ k3u 3.

The control u is taken as

u i ¼À F i (x , e , 4 ) À b i (x , e , 4 )sat(s i =mi )

a ii , i ¼ 1, 2 (6)



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where

e T ¼ [e 1, e 2, _e 2, €e 2], 4 T ¼ [ I (1)dd , f

(3)d ], a 11(Á) ¼ u 3,

and a 22(Á) ¼ k3u 3

^

u 3.

0 being a nominal value of u 3. To facilitate thediscussion, we choose the ‘nominal control component’F i (Á) to cancel all known/nominal terms in F i (Á), that is

^F 1(Á) ¼ Àu 1x 1 þ k5x 2x 3 À I (1)dd

F 2(Á) ¼ k21_e 2 þ k2

2€e 2 À f (3)d À k4(k3x 2 À k4x 3 À d 0) À k3u 1x 2

À k3 k5x 1x 3 À k3u 2x 3

where u 1 . 0 and u 2 . 0 are nominal values of u 1 and u 2,respectively. However, note that other choices of ^F i (Á),including F i (Á) ¼ 0, are possible.

To make the choice of b i (Á) precise, suppose that

u i [ [u mi , u M i ], where 0 , u mi , u M

i and u mi and u M i are

known. Let

Di (Á) ¼ F i (Á) À (u 3=u 3) F i (Á)

and r i (x , e , 4 ) be such that

supu 3 Di (Á)

u 3

@ i (x , e , 4 ), i ¼ 1, 2

where the supremum is taken over all x , e [ R 4, u i [[u mi , u M

i ] and 4 [ R 2. The functions b i are then chosen as

b i (Á) ¼ @ i (Á) þ q i , q i . 0

This choice guarantees that

s i _s i , Àq i

whenever js i j . mi , so that the trajectories reach the sets

s i m

i in finite time, and stay there thereafter. A standardSMC argument can then be used to prove that thecontroller (4) – (6) achieves global regulation, provided m1

and m2 are sufficiently small. In particular, we havethe following result. The proof of this theorem for a general MIMO system, stated as [15, Theorem 3.1] and[16, Theorem 2], can be found in either of these tworeferences.

Theorem 1 (Boundedness and error regulation): For any bounded initial state x (0) of the PMSM model (2), thestate x (t ) of the closed-loop system under the state-

feedback control (4)– (6) is bounded for all t ! 0.Moreover, there exists mÃi . 0, such that, for mi [ (0, mÃi ],limt 1 e (t ) ¼ 0.

A simplification of the design results if we make the choice^F i (Á) ¼ 0, b i (Á) ¼ ki , with the constants ki chosen such that

maxu 3F i (Á)

u 3

, ki , i ¼ 1, 2

where the maximisation is taken over all x , e and 4 in somecompact subsets of R 4 and R 2, respectively, andu i [ [u mi , u M

i ]. In this case, the controller is simply

u i ¼Àki

a ii

sat(s i =mi ) ¼

def À M i sat(s i =mi ), i ¼ 1, 2

This design, while having a simple structure, is also naturalif the control is required to be bounded, as is the case inreal applications, and is a design choice that we will also

pursue in the section on output feedback, but for a different reason. For this case, we only have regionalresults, with the region of attraction determined by constraints on the choice of ki . If the gains ki can bechosen sufficiently large, we can achieve semi-global stabilisation. Both these results, along with theglobal result we mentioned earlier, are covered in[16, Theorem 2].

For the purpose of simulation, we use the following valuesfor the system parameters, obtained from [11]:R ¼ 19:1388V, L ¼ 40 mH, K m ¼ 0:1349 Nm= A, J ¼

4:1295 Â 10kgm2

, B ¼ 0:0013 Nm=rad=s and N r ¼ 50. The load torque is assumed to be t L ¼ 0:2kgm2

=s2. For the purpose of controller design, R and L are assumedto be unknown, with nominal values of R ¼ 20V andL ¼ 35 mH, respectively. Also, R [ [R m, R M ], with R m ¼19V, and R M

¼ 21V, and L [ [Lm, L M ], with Lm¼

30mH, and L M ¼ 40 mH. The current reference is taken

as I dd (t ) ¼ 0 A [17]. The values of the controller parametersare taken as k1

0 ¼ 20, k20 ¼ 100, k2

1 ¼ 7:5 Â 104, k22 ¼ 550,

m1 ¼ 0:1, and m2 ¼ 50. Initial values for all the states aretaken as zero. The functions b i (Á) are taken as

b 1(Á) ¼ [(R M À^

R )jx 1j þ (L M À^

L) N r jx 2x 3j þ q 1]=^

L,

q 1 ¼ 2:1 k10m1L M

b 2(Á) ¼ [(R M À R )k3jx 2j þ (L M

À L)jk21_e 2 þ k2

2€e 2

À k4(k3x 2 À k4x 3 À d 0) À k3 k5x 1x 3j þ q 2]=L,

q 2 ¼ 2:1 k20m2L M

The desired angular position f d(t ) ¼ 0:03142[u (t )þu (t À 0:5)], where u (t ) is the unit step function. Theresults of the simulation are shown in Fig. 2. From the

figure, it is clear that good tracking performance with very little overshoot is achieved, independent of the magnitude of f d(t ).



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In the absence of integral control, it is well known that

decreasing the width m of the boundary layer results inperformance that is ‘close’ to that of ideal SMC. The abovesimulation suggests that this is true with integral control as

well, provided the conditional integrator is used. In particular,one can show that the continuous SMC design withconditional integrators ‘recovers the performance’ of thecorresponding ideal (discontinuous) SMC without integralcontrol

s 1 ¼ e 1

s 2 ¼ k21e 2 þ k2

2 _e 2 þ €e 2

u i ¼À ^F i (x , e , 4 ) À b i (x , e , 4 ) sgn(s i )

a ii , i ¼ 1, 2 (7)

so that we have the following result. As with Theorem 1, thistheorem has been stated as [15, Theorem 3.2] and [16,

Theorem 3], and the proof of the theorem can be found ineither of these references.

Theorem 2 (Performance recovery of ideal SMC): Let x Ã(t ) be the state of the closed-loop system under the idealSMC control (7) and x (t ) be part of the state of the

closed-loop system under the continuous SMC (4)– (6). Then, under the hypotheses of Theorem 1, given any compact subset S of R 4, with x Ã(0) ¼ x (0) [ S , there exists

mÃi . 0, such that, mi [ (0, mÃi ] ) kx Ã(t ) À x (t )k ¼ O (kmk)

8 t ! 0, where m ¼ [m1, m2] T.

As noted in [12, 15, 16], the state-feedback CSMC withconditional integrators can be thought of as the following two-step modification to ideal SMC:

1. Take s i ¼ s Ãi þ ki 0s i , where s Ãi ¼ 0 is the sliding surface

under ideal SMC, and s i is the state of the conditionalintegrator s i ¼ Àki

0s i þ mi sat(s i =mi ), mi , ki 0 . 0 and mi

‘sufficiently small’.

2. Replace sgn(s Ãi ) in ideal SMC with sat(s i =mi ).

Such a design then recovers the performance of idealSMC, as stated in Theorem 2. In light of this observation,it becomes imperative to explain the advantage of theproposed technique over ideal SMC. In our design, as a result of inclusion of integral control, we do not require theboundary layer widths m1 and m2 to be arbitrarily small inorder to achieve asymptotic regulation, but only ‘smallenough’ to stabilise the equilibrium point (and also smallenough for performance recovery of ideal SMC).Consequently, we expect that the method will be lesssensitive to the problem of chattering. As a demonstration

of this statement, consider a sampled-data implementationof the above controller, that is, we assume that the inputsto the controller are sampled and held signals, with a

Figure 2 Tracking error performance under state-feedback integral control



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zero-order hold, and likewise for the controller outputs. Weredo the previous simulation and compare the results against ideal SMC. To simplify the simulation, we exploit theflexibility in the choice of F i (Á)and b i (Á), and simply designthe control as u i ¼ À M i sgn(s i ). For the purposes of simulation, we let M 1 ¼ 50, M 2 ¼ 500, and the sampling

period is assumed to be T ¼ 0:1 ms. The results are shownin Fig. 3 (only error e 2 is plotted, since it corresponds tothe variable f of physical interest), and we see that asymptotic regulation is lost with the ideal SMC with thesampled-data implementation, and there is considerablechattering in the control v q . By contrast, asymptoticregulation is retained with the CSMC with conditionalintegrator, and there is no chattering in the control.

3.2 Output feedback

Suppose that the angular velocity v of the motor shaft isunavailable for feedback. It is easy to verify that for thesystem to have a uniform vector relative degree and betransformable to normal form, none of the positiveconstants ki and d 0 need to be exactly known. Accordingly,

we will assume that in the present case, in addition to theresistance R and inductance L, the parameters K m, B , J andthe load torque t L are all unknown, and take the vector u of unknown parameters as

u ¼ [k1, k2, k3, k4, k6, d 0] T

Since v is unavailable for feedback, so is _e 2 ¼ v À _f d .Furthermore, even if v were available for feedback,since €e 2 ¼ k3i q À k4v À d 0 À f 2d , and k3, k4 and d 0 areunknown, €e 2 would be unavailable for feedback. Therefore,

we estimate _e 2 and €e 2 in (4) using the high-gain observer

(HGO) [18]

_z1 ¼ z2 þ a1(e 2 À z1)=e

_z2 ¼ z3 þ a2(e 2 À z1)=e 2

_z3¼ a

3(e

2À z

1)=e 3

(8)

where the positive constants a1, a2 and a3 are chosen suchthat the polynomial l3

þ a1l2þ a2lþ a3 is Hurwitz. We

replace _e 2, €e 2 and also s 2 in (5) by their estimates z2, z3, and

s 2 ¼ k20s 2 þ k2

1e 2 þ k22z2 þ z3 (9)

respectively. Finally, motivated in part by the goal of simplifying the design, and in part by the need to work

with saturated controls (this is required in order to prevent the peaking phenomenon associated with HGOs [18]), andas mentioned in the previous subsection, we make use of

the flexibility in choosing ^F i , and let ^F i ¼ 0, and simply choose b i (Á) as a constant M i , which is equal to themaximum physically allowable value for the controlcomponent ju i j. One can also modify the more generalcontroller (6) for the output-feedback case, without thepreceding simplification (10). This essentially requiressaturating the control (6) outside a compact set of interest,and the details can be found, for example, in [12, Section4.2]. With this choice, the final expression for the controlthen becomes

u 1 ¼ À M 1sat(s 1=m1)

u 2 ¼ À M 2sat(^s 2=m2)(10)

The results of [12, Theorem 1] show that provided m1, m2

and e are ‘sufficiently small’, and M 1, M 2 can be chosen‘sufficiently large’, the controller (10) can achieve semi-global regulation. Moreover, when M 1 and M 2 are limited

Figure 3 Effect of sampled-time implementation on ideal SMC and CSMC with conditional integrator



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in magnitude, there is a trade-off between the controlmagnitude and the region of attraction. A performancerecovery result similar to Theorem 2, which says that theoutput-feedback CSMC controller with integral action (10)recovers the performance of a corresponding state-feedback ideal SMC, can be found in [12, Theorem 2]. Although a

precise statement for the regional case is harder to state without additional terminology (but can be found in [12]), we can state the corresponding results for semi-globalregulation and performance recovery, and do so below.

Theorem 3 (Stability and performance): Consider theclosed-loop system consisting of the PMSM model (2) andthe output-feedback control (4), (5), (8)–(10), and supposethe controller gains M i can be chosen arbitrarily large.Given compact sets M , R 4, and N , R 3 with e (0) [ M and z(0) [ N , there exist mÃi . 0, such that for eachm ¼ [m1, m2] T with mi mÃi , there exists e Ã ¼ e Ã(m) . 0,

such that, for e e Ã, all state variables of the closed-loopsystem are bounded, and limt 1 e (t ) ¼ 0. Furthermore, if x Ã(t ) be part of the state of the closed-loop system under the ideal state-feedback SMC u i ¼ À M i sgn(s Ãi ), and x (t )that with the output-feedback continuous SMC withconditional integrators, with x (0) ¼ x Ã(0), then, for every t . 0, there exists mÃ . 0, and for each m withkmk

1[ (0, mÃ], there exists e Ã ¼ e Ã(m) . 0, such that, for

mi [ (0, mÃ] and e [ (0, e Ã], kx (t ) À x Ã(t )k t 8 t ! 0.

For the purpose of simulation, we let K m, J , B , N r , t L, k1

0, k20, k2

1, k22, m1, m2 and I dd retain their values from

the previous simulation. Also we take R ¼ 19:5V,L ¼ 30mH, and f d ¼ 0:03142 rad . The HGO gainsare taken as a1 ¼ 17, a2 ¼ 80 and a3 ¼ 100, and thesaturation levels for the controls as M 1 ¼ 50, M 2 ¼ 500.

We compare the performance of the output-feedback controller with the partial state-feedback design

u i ¼ À M i sat(s i =mi ), which makes use of measurements of e 1, e 2, _e 2 and €e 2. This could, for instance, be the case whenthe full state x is available for feedback and the parametersk3, k4 and d 0 are known. Fig. 4a shows the results of thesimulation for e ¼ 10À4, and we see that good tracking performance is achieved by the output-feedback controller,

which uses minimal information about the system. Fig. 4b shows the effect of e on the recovery of the state-feedback performance, and it is clear from the figure that the error e 2under output-feedback approaches the error e 2 under state-feedback as e tends to zero.

4 Conditional servocompensatordesign

The integral control designs of the previous section was done with the goal of point-to-point motion of the PMSM.In many positioning applications, the desired trajectory

for the position is a sinusoid [1]. Specifically, supposethat the desired trajectory asymptotically converges tof d (t ) ¼ r 0 sin(v 0t ), which is generated by the neutrally stable exosystem

_w ¼0 v 0

Àv 0 0

w ¼

def S 0w, w T(0) ¼ [0, r 0], r (t ) ¼ w1

More general reference trajectories can be considered, as long as they satisfy the conditions of [13, Assumption 3]. Weshow that the conditional servocompensator design of [13],

which is a natural extension of the conditional integrator

design of [12], can be applied to this case. Before weproceed further on the control design, we explain the key points briefly. First, by studying the dynamics of the systemon the zero-error manifold, a linear internal model isidentified, which generates the trajectories of the exosystem,along with a number of higher-order harmonics generated

Figure 4 Tracking error performance under the output-feedback integral control a State/output feedback integral controlb Performance recovery of sfb as 1 tends to zero



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by the non-linearities of the system. As before, the idea is toreplace ideal discontinuous SMC with a continuous version,but to augment a servocompensator as part of the sliding surface, so as to recover the asymptotic error convergence of ideal SMC. As with the conditional integrator design, thegoal is to introduce the servocompensator conditionally,

only inside the boundary layer. The reason to do so is asbefore, namely that a conventional servocompensator design, while guaranteeing steady-state error convergence,does so at the expense of transient performance, while theconditional servocompensator design retains the transient performance of ideal SMC.

To continue with the design, our first goal is to identify a suitable linear internal model that generates the steady-state

values of the control inputs v d and v q . As before, thedesired reference for the current i d is a constant I dd . It canbe verified that with steady-state values x 1ss(t ) ¼ I dd and

x 4ss(t ) ¼ r 0 sin(v 0t ), respectively, for x 1 and x 4, the steady-state values of x 2 and x 3 are given by

x 2ss ¼ [d 0 þ k4r 0v 0 cos(v 0t ) À r 0v 20 sin(v 0t )]=k3

and x 3ss ¼ r 0v 0 cos(v 0t )

respectively, and the steady-state values of the control inputsv d and v q are given by

u 1ss ¼ g 1 þ g 2 cos(v 0t ) þ g 3 sin(2v 0t ) þ g 4 cos(2v 0t )

u 2ss ¼ g 5 þ g 6 sin(v 0t ) þ g 7 cos(v 0t )

for some constants g 1 to g 7. The steady-state values of thecontrol u i ss satisfy identities of the form

Lq s x ¼ c 0x þ c 1Ls x þ Á Á Á þ c q À1Lq À1

s x (11)

where Ls x ¼ (@x =@w)S 0w, and the (characteristic)polynomial

lq À c q À1l

q À1À Á Á Á À c 0

has distinct roots on the imaginary axis. In particular, u 1ss

does so with q ¼ 5, c 0 ¼ 0, c 1 ¼ À4v 4

0, c 2 ¼ 0, c 3 ¼ À5v 2

0and c 4 ¼ 0, whereas u 2ss does so with q ¼ 3, c 0 ¼ 0,c 1 ¼ Àv 20 and c 2 ¼ 0. An explanation of the identity (11)can be found in the paragraph following [13, Assumption5]. Roughly speaking, it guarantees the existence of a linear internal model that generates the trajectories of theexosystem, along with a number of higher-order deformations that result from the plant non-linearities. Let

S 1 ¼

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

0 À4v 40 0 À5v 20 0

26666664

37777775

, S 2 ¼

0 1 0

0 0 1

0 Àv 20 0

2

64

3

75

be the internal model matrices corresponding to u 1ss and u 2ss,and

J 1 ¼ [ 0 0 0 0 1 ] T, J 2 ¼ [ 0 0 1 ] T

It can be verified that the steady-state control inputs u i ss ¼

x i are generated by the internal models (@t i =@w)S 0w ¼ S i t i ,x i ¼ Gi t i , where t ¼ [x i Ls x i Á Á ÁLq À2

s x i Lq À1s x i ]

T, and Gi ¼

[10 Á Á Á 0]1Âq . We take s 1 and s 2 as outputs of theconditional servocompensators

_s i ¼ (S i À J i K i 0)s i þmi J i sat(s i =mi ) (12)

where

s 1 ¼ K 10 s 1 þ e 1

s 2 ¼ K 20 s 2 þ k21e 2 þ k2

2_e 2 þ €e 2(13)

The matrices K i 0 are chosen such that S i À J i K i 0 are Hurwitz (which is always possible since the pair (S i , J i ) is controllable),the scalars k2

1 and k22 are chosen such that the polynomial

x 2 þ k22x þ k2

1 is Hurwitz, s 1 ¼ s 1, s 2 ¼ K 20s 2 þ k21e 2þ

k22z2 þ z3, where z2 and z3 are estimates of _e 2 and €e 2,

respectively, provided by the high-gain observer (8).

As with the conditional integrator design, (12) represents a perturbation of the exponentially stable system_s ¼ (S À JK 0)s , with the norm of the perturbationbounded by the small parameter m, and thereforeks i k ¼ O(mi ), provided s i (0) ¼ O(mi ). With statefeedback, (i.e. when s ¼ s ) inside the boundary layer js j m, (12) reduces to _s ¼ S s þ Je a , where the‘augmented error’ e a is alinear combination of the tracking error and its derivatives, and it is clear that it generates thesteady-state trajectories when e a ¼ 0, that is, providesservo-action inside the boundary layer.

To continue with the design, the control then is taken as in

(10), that is

u i ¼ À M i sat(s i =mi ) (14)

As before, we can choose the nominal component of thecontrol to be non-zero, and possibly error dependent, andalso choose the ‘switching’ component as state/error dependent. We do not pursue that design here for the sakeof simplicity. This completes the design of the controller.

Analytical results for stability and performance of thecontroller (12)– (14), similar to the ones in the previous

section, can be found in [13, Theorems 1 and 2], and arestated in Theorem 3 below for the special semi-globalregulation case.



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Theorem 4 (Stability and performance): Consider theclosed-loop system consisting of the PMSM model (2) andthe output-feedback control (12)– (14), and suppose thecontroller gains M i can be chosen arbitrarily large. Givencompact sets M , R 4, and N , R 3 with e (0) [ M andz(0) [ N , there exist mÃi . 0, such that for eachm ¼ [m1, m2] T with mi mÃi , there exists e Ã ¼ e Ã(m) . 0,such that, for e e Ã, all state variables of the closed-loopsystem are bounded, and limt 1 e (t ) ¼ 0. Furthermore, if x Ã(t ) be part of the state of the closed-loop system under theideal state-feedback SMC u i ¼ À M i sgn(s Ãi ), and x (t ) that

with the output-feedback continuous SMC with conditionalservocompensators (12), with x (0) ¼ x Ã(0), and s i (0) ¼ 0.

Then, for every t . 0, there exists mÃ . 0, and for each m

with kmk1[ (0, mÃ], there exists e Ã ¼ e

Ã(m) . 0, such that,for mi [ (0, mÃ] and e [ (0, e Ã], kx (t ) À x Ã(t )k t 8 t ! 0.

In order to demonstrate the efficacy of the design by

simulation, we repeat the simulation in Section 3.2, with allthe numerical values retained, except for the reference f d(t ),

which is now chosen as f d(t ) ¼ r 0 cos(v 0t ). This is a sinusoidal signal that is simply phase-shifted fromf d (t ) ¼ r 0 sin(v 0t ) by p =2, and chosen this way to have a non-zero initial error e 2(0). Two sets of values of (r 0, v 0) areused in the simulation, (r 0, v 0) ¼ (p =2, 2) and(r 0, v 0) ¼ (p =10, 5). The matrices K i 0 in the conditionalservocompensator (12) are chosen to place the eigenvalues of S 1 À J 1K 10 and S 2 À J 2K 20 at {À1, À2+ j , À3+ j } and{À1, À2+ j }, respectively. We compare the performanceagainst a continuous SMC design that does not include a servocompensator, that is, ^

s 1¼

e 1, ^

s 2¼

k

2

1e 2þ

k

2

2z2þ

z3,

where z2 and z3 are as defined above, andu i ¼ À M i sat(s i =mi ). The results of the simulation are shownin Fig. 5, and we see that good tracking performance isachieved by the output-feedback servocompensator design,

which uses minimal information about the system. Asexpected, the transient performance of the continuous SMC

without servocompensator is close to the one with a servocompensator (indistinguishable in the figure), but thesteady-state error is non-zero without servocompensation,

whereas it tends to zerowith the conditional servocompensator.

In our simulations so far, we have assumed that the load

torque t L is constant. This assumption can be relaxed: (i) for the constant exogenous signals case, since we only requirethat t L be asymptotically constant, and f (t ) and v (t )approach a constant and zero, respectively, we can allow t L ¼ f t (f (t ), v (t )), where f t (Á) is a sufficiently smoothfunction of its arguments, and (ii) for the sinusoidalexogenous signals case, on account of the need for theidentity of the form (11) that the steady-state controlmust satisfy, it can be verified that f t (Á) will have to be a polynomial function of its inputs, and that its form must beknown, that is

f t (f , v )¼ X

i [ I , j [ J aij f

i

v

j

where I , J , Z !0 known, and aij possibly unknown.

4.1 Internal model perturbations

In the sinusoidal exogenous signals case, when thepolynomial condition is violated, for example, when theload torque is of the form t L ¼ N sin(f ) say, then, asshown in [13, Section 5], polynomial approximations may

be used toachieve practical regulation of the error.Successively higher-order approximations will lead to better approximations that make the steady-state error smaller (see[13, Theorem 3] for a precise statement about the effect of ‘size’ of internal model perturbations on the steady-stateerror), but will require successively higher-order servocompensator designs. Ideal SMC does not requiresuch approximations, but suffers from chattering. Replacing the discontinuous control with a continuous approximationthat does not include a servocompensator will result insteady-state errors that are O(m), so that we need to makem smaller in order to decrease the steady-state error, but a too small m will lead to chattering again. The error can bemade smaller by incorporating a servocompensator that isbased on polynomial approximations. However, when a conventional servocompensator driven by the tracking error of the form

_s i ¼ S i s i þ J i e i

is used, then, as previously mentioned, although successively higher-order approximations will result in smaller steady-state errors, this will be at the expense of degradedtransient performance, which becomes progressively worseas the order of the approximation increases. The transient response of the conditional servocompensator design, onthe other hand, is practically independent of theservocompensator order, and, as shown by Theorem 4, issimply close to an ideal SMC design that does not includea servocompensator.

In order to illustrate the above observations, we repeat theprevious simulation, but with the following changes: weassume that the load torque is t L ¼ N sin(f ), where (as in[8]) N ¼ 1.7201 kg m2/s2, and the desired referencetrajectory is, as before, f d (t ) ¼ r 0 cos(v 0t ) with r 0 ¼ 1,

v 0 ¼ 2. It is easy to verify that the steady-state values of the control inputs v d and v q are given by

u 1ss ¼ g 8 þ g 9 sin(2v 0t ) þ g 10 cos(2v 0t ) þ g 11 cos(v 0t )

Â sin( sin(v 0t ))

u 2ss ¼ g 12 sin(v 0t ) þ g 13 cos(v 0t ) þ g 14 sin( sin(v 0t ))

þ g 15 cos(v 0t )cos(sin(v 0t ))

for some constants g 8 to g 15, and it can be verified that thesedo not satisfy identities of the form (11). Consequently, we

replace sin(sin(v 0t )) and cos( sin(v 0t )) (that appear in theterms with constant coefficients g 11, g 14 and g 15) withpolynomial approximations. In particular, viewing sin(v 0t )



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as the argument q of sin(q ) and cos(q ), we employ thefollowing two approximations:

1. Approximation 1: sin(q ) ’ q , cos(q ) ’ 1

2. Approximation 2: sin(q ) ’ q À q 3=3, cos(q ) ’ 1Àq 2=2

It is then straightforward to verify that that the internalmodel matrices S i corresponding to these twoapproximations then are:

Approximation 1

S 1 ¼0 1 00 0 10 À4v 20 0

24 35, S 2 ¼ 0 1Àv 20 0

Approximation 2

S 1 ¼

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

0 À64v 40 0 À20v 20 0

26666664

37777775,

S 2 ¼

0 1 0 0

0 0 1 0

0 0 0 1

À9v 40 0 À10v 20 0

26664

37775

For the first (lower-order) approximation, the matrices K i 0 inthe conditional servocompensator design are chosen to place

Figure 5 Sinusoidal reference tracking using output-feedback servocompensators



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the eigenvalues of S 1 À J 1K 10 and S 2 À J 2K 20 at {À1, À2+ j }and {À1, À2}, respectively. For the second (higher-order)approximation, these are chosen to place the eigenvaluesof S 1 À J 1K 10 and S 2 À J 2K 20 at {À1, À2+ j , À5+ j }and {À2+ j , À5+ j }, respectively. The conventionalservocompensator design requires a complete redesign

of the sliding surface parameters (the details of which canbe found in [19]), and we only mention that in order tomake a meaningful comparison, the controller parametersin the simulation results that we present were chosen sothat the eigenvalues from the conditional design areretained.

The results of the simulation are shown in Fig. 6, with variables q 1 and q 2 indicated being the orders of matricesS 1 and S 2, respectively. The set q 1 ¼ 3, q 2 ¼ 2 correspondsto the lower-order polynomial approximation, whereasq 1 ¼ 5, q 2 ¼ 4 to the higher-order one. Several inferences

can be made from the figure. The subplots in the first row show that the steady-state errors decrease as the polynomialapproximation order increases (i.e. the approximation

becomes better), both for the conditional and for theconventional servocompensator design. Furthermore, thesteady-state errors for the same approximation order arecomparable in both these designs. From the subplots onthe second row, we see, as expected, that the transient performance of the conditional servocompensator designs is

close to that of the continuous SMC without servocompensator (indistinguishable in the figure) andhence to that of an ideal SMC, regardless of the order of the servocompensator (which are shown with q 2 ¼ 2 andq 2 ¼ 4). In other words, increase in servocompensator order does not degrade the transient performance of thedesign. The steady-state error is non-zero without servocompensation, whereas it deceases (significantly) withthe servocompensator (shown for q 2 ¼ 2). The subplot onthe last row clearly demonstrates the advantage of theconditional design over a conventional one. In particular,

while the steady-state performance of the two designs are

comparable (as seen from the subplots in the first row),increasing the order of the servocompensator in order todecrease the steady-state error has the drawback of

Figure 6 Effect of increasing approximation orders on transient and steady-state performance



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degrading the transient response in the conventionaldesign, which clearly gets progressively worse as theservocompensator order increases. The conditionalservocompensator design’s transient performance though, as

we have just observed, is practically independent of theservocompensator order.

5 Conclusions

In this paper, we presented a new approach for positioncontrol of a permanent magnet stepper motor. Our problem formulation allows for the system to containconstant unknown parameters u , and time-varying matcheddisturbances d (t ) that could also possibly depend on u andthe state x . In the case of constant exogenous signals, welooked at both a state-feedback design for global regulation,and an output-feedback design for regional/semi-globalregulation, based on conditional integrators and sliding

mode control. An extension of the design to the case of sinusoidal references using conditional servocompensators

was provided, with both state- and output-feedback designsbeing possible. Analytical results for stability andperformance of the proposed method were provided, withthe performance of ideal SMC as the benchmark.Simulation results show that good tracking performance isachieved in all cases, in spite of partial knowledge of themachine parameters. Advantages of the design over idealSMC as regards the issue of chattering and over conventional servocompensator design as regards transient performance were also demonstrated by simulation.

Lastly, although we specifically considered PMSMs inthis paper, our results can be extended to other types of motors, such as permanent magnet synchronous and DCmotors.

6 Acknowledgments

The author would like to thank Prof. H. K. Khalil for his valuable guidance and his contributions to a preliminary version of this manuscript. This work was supported inpart by a grant from the San Diego State University Foundation. A preliminary version of this paper was

presented at the 2005 American Control Conference [14].

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