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Voltage-fed permanent-magnet stepper motor control via position-only feedback P. Krishnamurthy and F. Khorrami Abstract: A robust adaptive nonlinear dynamic controller is designed to achieve practical stabilisation for position tracking error of a voltage-fed permanent-magnet stepper motor. The control design is an output-feedback design that utilises only rotor position measurements. Rotor velocity and stator phase currents are not available for feedback. Furthermore, the only motor parameter that is required to be known is the time constant of the electrical subsystem. Adaptations are utilised so that no other knowledge of motor parameters is required. The proposed controller is a fourth-order dynamic compensator and is robust to load torques, friction, cogging forces and other disturbances satisfying certain bounds. Practical stabilisation of the tracking error is achieved with global boundedness of all closed-loop signals. Furthermore, under the condition that the torque disturbances are locally linearly bounded by a function of rotor position, the designed controller achieves asymptotic stabilisation of the rotor position. These results can also be extended to other classes of motors. 1 Introduction Closed-loop control schemes for permanent-magnet (PM) stepper motors have been extensively investigated [1 – 3]. The majority of results utilise full state feedback. Position control was achieved in [4, 5] using feedback linearisation relying on an exact knowledge of motor dynamics and parameters. To handle certain parametric uncertainties, adaptations were introduced with a feedback linearising control law in [6]. Adaptive state feedback was also addressed in [7]. An adaptive controller robust to parametric and dynamic uncertainties such as nonsinusoidal ﬂux density due to geometric imperfections of the motors was presented in [8]. Partial state feedback designs utilising measurement of various subsets of the state are also of interest. We will consider the case where only the rotor position is measured. Although current sensors are usually part of the inner current loop in the motor ampliﬁers, their outputs are not easily accessible to the user without tapping and possibly modifying the ampliﬁer circuitry. The tachometer signals are usually noisy and it is desirable to eliminate velocity measurement. These considerations have fuelled interest in the output-feedback control problem eliminating rotor velocity and stator current measurements. To obviate rotor velocity measurement, a nonlinear speed observer utilising position and current measurements with linear error dynamics was constructed in [9, 10]. These results were extended to the adaptive case in [11] with the rotor position and stator winding currents being measured. In [12], a current-level output to parametric and dynamic uncertainties such as friction, load torque and cogging torque was designed. feedback controller robust q IEE, 2004 IEE Proceedings online no. 20040622 doi: 10.1049/ip-cta:20040622 Paper ﬁrst received 27th August 2003 and in revised form 1st April 2004 The authors are with the Control/Robotics Research Laboratory (CRRL), Department of Electrical and Computer Engineering, Polytechnic University, Six Metrotech Center, Brooklyn, NY 11201, USA the electromechanical parameters of The output-feedback control problem without current measurements has been investigated in recent years. In [13], under the assumption that lower and upper bounds for all motor parameters are known, a controller utilising the rotor position and velocity was proposed. In [14], a robust adaptive controller was designed utilising the rotor position and velocity for feedback without requiring any knowledge of the motor. This design was extended to the nonsinusoidal ﬂux distribution case in [15]. However, in this more general case, it was necessary to assume knowledge of upper and lower bounds on the time constant of the electrical subsystem. A controller which requires only position measurements was designed in [16] assuming exact knowledge of all motor parameters. In [17], a robust controller was designed using only position measurements and assuming that upper and lower bounds for all motor parameters are known. We will address the problem of designing a robust adaptive controller for voltage-fed PM stepper motors. Only rotor position measurements are used for feedback. This extends our earlier results in [14] where the rotor position and velocity needed to be measured. Further- more, the only motor parameter that is required in the controller design is the time constant of the electrical the other parameters is subsystem. No knowledge of in [14] does not need this required. The controller parameter to be known. However, in the position-only feedback case, there are no techniques currently available results with this parameter being to obtain global completely unknown. interesting to note that extending the design of [14] to the nonsinusoidal ﬂux distribution case [15] required assuming that upper and lower bounds were available on the electrical time It appears that uncertainty in the electrical constant. time constant is a fundamental obstacle in achieving global output-feedback results for this class of systems. The control design is complicated by the lack of theoretical tools in the nonlinear control literature that can be applied to this system since it is not transformable to any adaptive nonlinear output-feedback canonical form. Therefore, an essentially different design methodology has is It IEE Proc.-Control Theory Appl., Vol. 151, No. 4, July 2004 499

to be explored. Techniques developed here can be utilised to generalise the standard nonlinear output-feedback canonical form along the lines in our work in [18, 19]. These techniques can be extended to the control design for other types of motors. The controller proposed is a fourth-order dynamic compensator and achieves global stability with practical stabilisation of the tracking error. Furthermore, if the torque disturbances can be bounded by a function of the rotor position that is locally linearly bounded around the origin, the designed controller guarantees asymptotic stabilisation. 2 Electromechanical model and problem statement The mechanical dynamics of a PM stepper with two phases may be modelled in the state space as follows [20, 21]: _yy ¼ o _oo ¼ 1 J ðt Do tlÞ ð1Þ where y is the rotor position, o the rotor speed, t the electromagnetic torque, D the viscous damping factor, and tl the sum of all load torques, cogging forces, and other modelling uncertainties. J is the rotor inertia. t is a function of currents and rotor position given by iT @L @y i ð2Þ with the symmetric positive deﬁnite inductance matrix L and current vector i deﬁned as For simplicity, assume R1 ¼ R2 ¼ R: The ﬂux linkages are given by: f¼ ½ f1 f2 ff T ¼ Li ð10Þ Using (2) – (10), the torque and the current dynamics are obtained as (note that the last term involving i2 in the f torque equation represents the disturbance torque due to cogging): t ¼ i1ifLm1Nr sinðNryÞ þ i2ifLm1Nr cosðNryÞ 2Lf4Nr sinð4NryÞi2 f _ii1 ¼ 1 L0 _ii2 ¼ 1 L0 fu1 i1R þ ifoNrLm1 sinðNryÞg fu2 i2R ifoNrLm1 cosðNryÞg It will be convenient to deﬁne the state variables: x2 ¼ o x1 ¼ y x3 ¼ L0i1 x4 ¼ L0i2 ð11Þ ð12Þ ð13Þ ð14Þ The dynamics of the PM stepper motor can be written as: _xx1 ¼ x2 _xx2 ¼ k1½x3 sinðNrx1Þ þ x4 cosðNrx1Þ k2x2 tn _xx3 ¼ gx3 þ k3x2 sinðNrx1Þ þ u1 _xx4 ¼ gx4 k3x2 cosðNrx1Þ þ u2 where ð3Þ k1 ¼ ifNrLm1 JL0 k3 ¼ ifLm1Nr tn ¼ 1 J k2 ¼ D J g ¼ R L0 ðtl þ 2Lf4Nri2 f sinð4Nrx1ÞÞ ð15Þ ð16Þ the t ¼ 1 2 2 4 3 5 L ¼ L11 L12 L1f L12 L22 L2f L1f Lff i ¼ i1 ½ L2f if i2 T ð4Þ where i1 and i2 are the stator phase currents and if is the ﬁctitious constant rotor current provided by the permanent magnet. Ljj; j ¼ 1; 2; is the self-inductance of the jth winding, j ¼ 1; 2: L12 is the mutual inductance between the two windings. Ljf; j ¼ 1; 2; is the mutual inductance between the jth phase and the ﬁctitious rotor winding. Lff is the self-inductance of the ﬁctitious rotor winding. Under the usual assumptions of sinusoidal ﬂux distribution and that the variations of the stator self-inductance with position and the mutual inductance between the stator windings are negligible, the terms in the inductance matrix are: L11 ¼ L22 ¼ L0 L12 ¼ 0 L1f ¼ Lm0 þ Lm1 cosðNryÞ L2f ¼ Lm0 þ Lm1 sinðNryÞ Lff ¼ Lf0 þ Lf4 cosð4NryÞ ð5Þ ð6Þ ð7Þ ð8Þ where Nr is the integer number of rotor teeth. The electrical subsystem is modelled as: uj ¼ Rjij þ dfj dt j ¼ 1; 2 ð9Þ where uj; fj and Rj are the voltage, ﬂux linkage and resistance in the jth phase, respectively. The control design will be carried out under following assumptions: Assumption 1: g is a known positive constant. k1; k2 and k3 are unknown constants. Assumption 2: tn can be bounded as: jtnj2 kðr0 þ rðy2ÞÞ ¼ kðr0 þ rðx2 1ÞÞ ð17Þ where k and r0 are unknown constants and r is a known class K1 function. Note that a continuous function a : ½0; aÞ ! ½0;1Þ is said to belong to class K if it is strictly increasing and að0Þ ¼ 0: It is said to belong to class K1 if a ¼ 1 and aðrÞ ! 1 as r ! 1: Remark 1: From the deﬁnition of tn in (16), it is seen that tn satisﬁes the bound in (17) if tl does. This implies that position-dependent load torques (which can also depend on velocity in a bounded manner) can be handled, e.g. a load depending on gravity such as N sinðyÞ: The proposed controller design also compensates for friction torques Do with unknown friction coefﬁcient D. Furthermore, any load torque component of form q fðy; oÞ with f being a known function and q an unknown parameter can also be handled 500 IEE Proc.-Control Theory Appl., Vol. 151, No. 4, July 2004

through a straightforward modiﬁcation of the advocated design. We now intend to address the problem of controller design with the control objective being to make y practically track a given thrice-differentiable bounded reference signal yrefðtÞ with bounded derivatives up to third-order while keeping all closed-loop signals bounded given that only y is available for feedback and only parameter g is known, i.e. we seek to design dynamic control laws for u1 and u2 utilising measurement of x1 to make the tracking error ðx1 yrefÞ converge to ½e; e for any arbitrary prespeciﬁed e; while keeping all closed-loop signals bounded. 3 Observer design Since the system, (15), is not transformable to any output feedback canonical form, the standard observer design techniques cannot be used. A reduced-order observer is now proposed which generates estimates of the unmea- sured states, x2; x3 and x4: The observer structure is given by: _^xx^xx2 ¼ a1 ^xx2 ^xx3 sinðNrx1Þ þ ^xx4 cosðNrx1Þ _^xx^xx3 ¼ g^xx3 þ u1 _^xx^xx4 ¼ g^xx4 þ u2 ð18Þ where a1 is a positive design parameter. The observer errors are deﬁned as: x2 þ a2x1 e2 ¼ ^xx2 1 k1 e3 ¼ ^xx3 x3 k3 Nr e4 ¼ ^xx4 x4 k3 Nr cosðNrx1Þ sinðNrx1Þ ð19Þ where a2 ¼ ða1 k2Þ=k1 is a (not necessarily positive) constant. Remark 2: It is seen from the observer error deﬁnitions in (19) that the actual estimates for x2; x3 and x4 are k1ð^xx2 þ a2x1Þ; ^xx3 ðk3=NrÞ cosðNrx1Þ; and ^xx4 ðk3=NrÞ sinðNrx1Þ; respectively. These cannot be implemented since they depend on the unknown parameters. However, since our main objective is controller design, we do not need to actually realise estimates for the unmeasured states. It will be shown that the dynamic signals that are generated from the dynamics in (18) are sufﬁcient to achieve the control objective. estimates ‘virtual’ The observer error dynamics are: _ee2 ¼ a1e2 e3 sinðNrx1Þ þ e4 cosðNrx1Þ tn þ a1a2x1 þ 1 k1 _ee3 ¼ ge3 gk3 cosðNrx1Þ Nr _ee4 ¼ ge4 gk3 sinðNrx1Þ Nr e3 þ k3 Nr 2 þ 6 e2 ga1 Vo ¼ 1 2 Differentiating (21): The stability of this observer can be analysed using the Lyapunov function: ð20Þ 2þe2 4 ð21Þ tn k3 Nr k3 Nr cosðNrx1Þ ð22Þ e2 2 2þ2a1a2 2x2 e2 1 h ½1cosðNrx1Þ2 þ c minðl;1Þ ð23Þ ð24Þ _VVo¼e2 a1e2e3sinðNrx1Þþe4cosðNrx1Þþa1a2x1þ 1 k1 cosðNrx1Þ 4 tn þ12 a1 þ12 a1 e3þ k3 þ 12 ga1 Nr þe4 ge4gk3 Nr e3þ k3 Nr ge3gk3 Nr sinðNrx1Þ 2þe2 ¼a1e2 212 a1 þe2½e3sinðNrx1Þþe4cosðNrx1Þ e3þ k3 þe2 a1a2x1þ 1 k1 Nr sinðNrx1Þ e4 k3 Nr e3þ k3 Nr 2þe2 a1e2 212 a1 e3þ k3 Nr sin2ðNrx1Þþa1 2þ 2k2 e2 a1N2 2þ 2 t2 e2 n a1k2 8 i 2þe2 e3þ k3 þ 6 a1 Nr þsin2ðNrx1Þ a1 2 2 e2 a1 2 £ x2 1þr x2 þ 6k2 a1N2 2þe2 þ 4 a1 þa1 þa1 2þe2 8 8 1 4 3 r 4 3 r 1 4 e3þ k3 Nr þr0 þ a1 8 4 where 3 1 c ¼ minðl; 1Þ max 2a1a2 2 þ 8k2 ; a1 with l 1 being a positive design freedom. the observer error states ðe2; e3 þ ðk3=NrÞ; e4Þ are rendered input-to-state- practically-stable [22] with respect to x1: From (24), seen that 2k a1k2 ð25Þ is it 1 4 Controller design In this Section, the control inputs, u1 and u2; are designed using robust adaptive observer backstepping technique [23] applied to the subsystem: _xx1 ¼ k1ð^xx2 þ a2x1 e2Þ _^xx^xx2 ¼ a1 ^xx2 ^xx3 sinðNrx1Þ þ ^xx4 cosðNrx1Þ _^xx^xx3 ¼ g^xx3 þ u1 _^xx^xx4 ¼ g^xx4 þ u2 ð26Þ The backstepping design proceeds by considering lower- dimensional subsystems and designing virtual control inputs (or equivalently, state transformations). The virtual control input in the ﬁrst step is ^xx2: In the second step, both ^xx3 and ^xx4 appear in the derivative of ^xx2 and are used as virtual control inputs. The virtual control law designed in the second step can be interpreted as a virtual control law for (scaled) torque IEE Proc.-Control Theory Appl., Vol. 151, No. 4, July 2004 501

1 þ y2 z2 ref where _VV2 0 1z1n1 z2n2 þðb1 þ b2 ^bb b4Þb2 þð _^bb^bbþ s ^bbÞb4 þ z2fb3 ^xx3 sinðNrx1Þþ ^xx4 cosðNrx1Þgþ 2lz2 þ ln2 ¼ 0 1ðz1Þþ 2e2 1z1n1 z2n2 þðb1 þ b2 ^bb b4Þb2 þð _^bb^bbþ s ^bbÞb4 2 þ d2 1 þ z2 cosðNrx1Þðb3 cosðNrx1Þþ ^xx4Þþ 2e2 þ z2 sinðNrx1Þðb3 sinðNrx1Þ ^xx3Þþ 2lz2 2 þ d2 1 þ ln2 1ðz1Þ ¼ 0 1z1n1 z2n2 þðb1 þ b2 ^bb b4Þb2 þð _^bb^bbþ s ^bbÞb4 z2 sinðNrx1Þz3 þ z2 cosðNrx1Þz4 þ 2lz2 þ 2e2 2 þ d2 1 þ ln2 1ðz1Þ ð35Þ i 1Þ2z3 1 1Þ2 z2 2 1ð1 þ 0 i 1Þ2 b3ð ^bb; z1; z2; ^xx2; _yyrefÞ ¼ 0 1 2 ref 1 2l þ k2 1 4 1a2 þ k2 2l b2 ¼ k1 þ k2 ref þ _yy2 d2 ¼ 2y2 h 1 þ 02 b2ð ^bb; z1; z2Þ ¼ ð0 1Þz2 ^bb2ð1 þ 0 þ b1 þ b2 1 þ b2 _yyref þ 0 1z1 a1 ^xx2 b1 h þ z2½ ^bb þ ð1 þ 0 1Þz1z2 ^bb2ð1 þ 0 1 þ b2 b1 þ b2 s ^bbð1 þ 0 1Þz1 þ n2ðz2Þ 1Þz1 b4ðz1; z2Þ ¼ z2ð1 þ 0 1 1 z3 ¼ ^xx3 b3 sinðNrx1Þ z4 ¼ ^xx4 þ b3 cosðNrx1Þ where the control estimate along with a commutation scheme. In the third inputs u1 and u2 appear and can be step, designed. Step 1: For the ﬁrst step, a Lyapunov function: 1 z2 1 V1 ¼ 1 2k1 ð27Þ is used where z1 ¼ x1 yref and 1 is a class K1 function to be chosen later in the design process. Differentiating (27) (note that in the following, f 0ðsÞ denotes @fðsÞ=@sÞ : z1ð^xx2 þ a2x1 e2Þ 1 k1 1 z2 1 z1 ð28Þ Straightforward algebra yields the inequalities: _yyref0 1 þ a2 02 2 1 z2 z2 1 4 1 þ e2 z2 1 þ _yy2 z2 2 1 z2 1 1 z2 1 a20 0 1 z2 1 _yyref0 _VV1 ¼ 0 z1x1 aa20 1 z2 1 02 z1e2 1 1 z2 1 4 02 z1 1 1 z2 1 4k2 1 z1 ^xx2 þ b1 0 b1 ¼ max aa2; 1 z2 1 1 z2 1 1 ð29Þ k1 where aa2 ¼ ða1 þ k2Þ=k1 and ¼ ½a1=ð16cÞ: Thus, (28) reduces to: _VV1 0 2 þ d1 ð30Þ 1 þ e2 z2 1 z2 1 1 z2 1 ref a2 2 4 ref z1z2 z1n1ðz1Þ þ 0 þ ðb1 ^bbÞ 0 1 z2 1 ref þ _yy2 d1 ¼ y2 þ 1 þ 1 ð31Þ 4k2 4 1 1ÞÞz1 þ n1ðz1Þ where ^bb is a Deﬁne z2 ¼ ^xx2 þ ^bbð1 þ 0 1ðz2 dynamic in counteracting the effect of the unknown b1 and n1 is a function which satisﬁes sn1ðsÞ 08s 2 R: Therefore, (30) may be rewritten as: signal which aids _VV1 0 1 z2 1 1 z2 1 2 þ d1 1 þ e2 z2 n ð32Þ Step 2: A new Lyapunov function is deﬁned as V2 ¼ V1 þ ð1=2Þz2 _VV2 0 þ z2 þ ^xx4 cosðNrx1Þ a1 ^xx2 þ b1ð ^bb; z1Þ ½k1ð^xx2 þ a2x1 e2Þ _yyref þ _^bb^bb 1 þ 0 1 z2 1 þ ðb1 ^bbÞ 0 2 which satisﬁes: z1n1ðz1Þ þ 0 ^xx3 sinðNrx1Þ o z1z2 þ e2 2 þ d1 þ 02 ð33Þ 1 z2 1 1 z2 1 1 z2 1 1 z2 1 þ 02 1 z2 1 z2 1 z1 þ 02 where, for notational convenience, we have introduced: b1ð ^bb; z1Þ ¼ ^bb 1 þ 0 1 z2 1 þ 2z2 100 1 z2 1 þ n0 1ðz1Þ ð34Þ We shall also drop the arguments of b1; 1; etc. for ease and clarity of notation. Remark 3: b1 is non-negative if ^bb; 0 1 are non- negative. The ﬁnal choice of the design freedoms will ensure that b1 0: 1 and n0 1; 00 Rearranging terms and overbounding, (33) reduces to (for notational convenience, we drop arguments of functions whenever no confusion will result): 502 ð36Þ s is a positive design parameter, and n2 ¼ c2z2 with c2 being a positive design parameter. Step 3: This is the last step of backstepping. In this step, the control inputs, u1 and u2; will be designed. For this step, a new Lyapunov function is deﬁned as: 3 þ 1 z2 2 ð37Þ z2 4 Before differentiating this Lyapunov function, it will be useful to ﬁnd _bb3: V3 ¼ V2 þ 1 2 k1ð^xx2 þ a2x1 e2Þ n @b3 @z2 þ b1 _bb3 ¼ b5 þ @b3 @z1 where b5 ¼ @b3 @ ^bb þ^xx4 cosðNrx1Þ þ _^bb^bbð1 þ 0 _yyref þ @b3 @z2 _^bb^bb @b3 @z1 a1 ^xx2 ^xx3 sinðNrx1Þ o 1Þz1 b1 þ @b3 @ _yyref fa1 ^xx2 ^xx3 sinðNrx1Þ þ ^xx4 cosðNrx1Þg _yyref þ @b3 @ ^xx2 ð38Þ €yyref ð39Þ The partial derivatives of b3 in (38) and (39) are evaluated with b1 and b3 interpreted as symbolic functions of the indicated arguments, i.e. algebraic relationships between the arguments such as the one between z2 and ^xx2 are not utilised IEE Proc.-Control Theory Appl., Vol. 151, No. 4, July 2004

in evaluating the partial derivatives. Differentiating (37): _VV3 0 1z1n1 z2n2 þ ðb1 þ b2 ^bb b4Þb2 þ ð _^bb^bb þ s ^bbÞb4 1ðz1Þ k1 þ b1 ð^xx2 þ a2x1 e2Þ 1 þ ln2 z2 sinðNrx1Þz3 þ z2 cosðNrx1Þz4 þ 2lz2 þ z3 g^xx3 þ u1 b5 þ @b3 @b3 @z1 @z2 sinðNrx1Þ þ z4 g^xx4 þ u2 þ b5 þ @b3 @z1 cosðNrx1Þ þ b1 b3Nr sinðNrx1Þk1ð^xx2 þ a2x1 e2Þ b3Nr cosðNrx1Þk1ð^xx2 þ a2x1 e2Þ k1ð^xx2 þ a2x1 e2Þ @b3 @z2 þ 2e2 2 þ d2 ð40Þ Deﬁning b6 ¼ @b3 @z1 b7 ¼ @b3 @z1 þ b1 þ b1 @b3 @z2 @b3 @z2 1a2 þ k2 2l h þ k2 þ k2 b3 ¼ k2 1 1 1 4 2l c2 6 1þ ^bb2 1þ 02 b8 ¼ z2 3b2 2 1 i d3 ¼ 4y2 þ z2 h ref þ _yy2 7 1þ ^bb2 1þ 02 4b2 ref 1 i ð43Þ Note that bi; i¼ 1; . . . ; 8 depend only on the states that are available and do not involve any unknown parameters. The control inputs, u1 and u2; are now speciﬁed as: u1 ¼ z2 sinðNrx1Þþ gb3 sinðNrx1Þþ b5 sinðNrx1Þ n3ðz3Þ ð ^bbþ b4Þz3b2 1 u2 ¼ z2 cosðNrx1Þ gb3 cosðNrx1Þ b5 cosðNrx1Þ h 6 1þ ^bb2 1þ 02 h 7 1þ ^bb2 1þ 02 i i 1 n4ðz4Þ ð ^bbþ b4Þz4b2 ð44Þ where n3 and n4 are functions such that sniðsÞ cis28s2R; i¼ 3; 4 with c3 and c4 being positive design freedoms. Using (44), (42) simpliﬁes to: _VV3 0 1z1v1 1 z2v2 z3v3 z4v4 g z2 2 b4 þ ðb1 þ b2 þ b3 ^bb b4Þðb2 þ b8Þ þ _^bb^bb þ s ^bb þ 6lz2 1ðz1Þ þ 4e2 1 þ 3lv2 2 þ d3 3 þ z2 4 ð45Þ _^bb^bb ¼ s ^bb þ b2 þ b8 ð46Þ 5 Stability analysis Stability of the closed-loop system is analysed using the composite Lyapunov function: whose derivative satisﬁes _VV 0 4 Vo ð47Þ 3 þ z2 1z1n1 1 2 V ¼ V3 þ 1 2 ð ^bb b1 b2 b3Þ2 þ 16 a1 z2n2 z3n3 z4n4 g z2 þ ðb1 þ b2 þ b3 ^bb b4Þðb2 þ b8Þ þ ð _^bb^bb þ s ^bbÞb4 þ 6lz2 2 þ d3 þ ð ^bb b1 b2 b3Þ _^bb^bb 1ðz1Þ þ 4e2 þ lx2 8e2 e3 þ k3 1 þ r0 þ lr x2 Nr 4e2 3 þ z2 z2n2 z3n3 z4n4 g z2 2þe2 e3 þ k3 Nr 1 þ 3ln2 2 32 a2 1 1z1n1 1 2 ð ^bb b1 b2 b3Þ2 32 a2 1 þ d4 1 þ 3ln2 ref þ r 4y2 1ðz1Þ þ lr 4z2 ðb1 þ b2 þ b3Þ2 ð48Þ ð49Þ ref þ r0 þ s 2þe2 þ _yy2 ref 4 1 4 2 4 1 0 s þ 8lz2 2 where d4 ¼ 6y2 2 Before proceeding, we need the following lemma. Lemma 1: Given any class K1 function, FðsÞ; possible to ﬁnd a class K1 function, s ! 0þ; and a constant, s 0; to satisfy: is ~FFðsÞ ¼ O½s as it sinðNrx1Þ þ b3Nr cosðNrx1Þ cosðNrx1Þ b3Nr sinðNrx1Þ ð41Þ The dynamics for ^bb are designed as: and overbounding, (40) reduces to: 1 1 þ z4 i h 6 1þ ^bb2 1þ 02 3b2 z2 _VV3 0 z2n2 þðb1 þ b2 ^bb b4Þb2 þð _^bb^bbþ s ^bbÞb4 1z1n1 1 2 2 þ d3 1 þ 3ln2 1ðz1Þþ 4e2 þ 6lz2 i h þ z3 z2 sinðNrx1Þ g^xx3 þ u1 b5 sinðNrx1Þ 6 1þ ^bb2 1þ 02 þ ^bbz3b2 i h z2 cosðNrx1Þ g^xx4 þ u2 þ b5 cosðNrx1Þ 7 1þ ^bb2 1þ 02 þ ^bbz4b2 i h þðb3 ^bbÞ 7 1þ ^bb2 1þ 02 4b2 n 0 z2n2 þðb1 þ b2 þ b3 ^bb b4Þðb2 þ b8Þ 1z1n1 1 h 2 þ z3 6 1þ ^bb2 1þ 02 þð ^bbþ b4Þz3b2 h þ z4 z2 cosðNrx1Þ g^xx4 þ u2 7 1þ ^bb2 1þ 02 þb5 cosðNrx1Þþð ^bbþ b4Þz4b2 þ 6lz2 z2 sinðNrx1Þ g^xx3 þ u1 b5 sinðNrx1Þ 2 þ d3 þð _^bb^bbþ s ^bbÞb4 1ðz1Þþ 4e2 1 þ 3ln2 io io þz2 n 1 1 1 1 ð42Þ where IEE Proc.-Control Theory Appl., Vol. 151, No. 4, July 2004 503

FðsÞ ~FFðsÞ þ s 8s > 0 Proof: s ¼ sups1 FðsÞ and: ~FFðsÞ ¼ Fð1Þs FðsÞ for s 1 for s > 1 ð50Þ ð51Þ satisfy (50) A Applying lemma 1, we ﬁnd a class K1 function, ~rrðsÞ; and a positive constant, cr; such that ~rrðsÞ ¼ O½s as s ! 0þ and ð52Þ rðsÞ ~rrðsÞ þ cr 1 2 Thus, (48) reduces to: _VV 0 s þ 8lz2 1ðz1Þ þ l ~rr 4z2 1z1n1 1 2 ð ^bb b1 b2 b3Þ2 32 a2 1 1 þ 3ln2 þ d5 where d5 ¼ 6y2 ref þ r 4y2 ref þ r0 þ s z1n1ðz1Þ c11 z2 2l 8z2 1 þ 3n2 with a constant c1 > 0 and with 0 being non-negative for all z1 2 R: 0 1 z2 1 þ _yy2 ref 2 1 Finally, 1 and n1 are chosen to ensure that: ðb1 þ b2 þ b3Þ2 þ cr ð54Þ 1ðz1Þ þ ~rr 4z2 1ðz2 1ðz2 1Þ; 00 1 1Þ; and n0 ð55Þ 1ðz1Þ z2n2 z3n3 z4n4 g z2 3 þ z2 e3 þ k3 Nr 4 4e2 2þe2 2 4 ð53Þ 1 1 1 z2 1 m z2 1 c10 1ðz2 ð56Þ 1Þ; 00 (55) reduces to: ¼ 2l 8 þ 3c2 Remark 4: In this remark, we provide a proof that functions 1 2 K1 and n1 can be chosen such that (55) is satisﬁed with c1 being a positive constant, and with 1Þ and n0 1ðz1Þ being non-negative for all 1ðz2 0 z1 2 R: Choosing, for instance, n1ðz1Þ ¼ c1z1 with c1 > 0; and deﬁning: 1 þ ~rr 4z2 z2 1 m z2 z2 ð57Þ Note that mðsÞ is continuous and mðsÞ ¼ O½s as s ! 0þ: Hence, a smooth positive nondecreasing function ~mm can s 2 ½0;1Þ: A R be chosen such that smooth positive nondecreasing function ~mmc can be chosen such that ~mmcðsÞ ~mm0ðsÞ for s 2 ½0;1Þ: Choosing 1ðsÞ ¼ 0 ~mmcðwÞdw; the inequalities c11ðsÞ mðsÞ ðs=c1Þ½ ~mmð0Þ þ 1ðsÞs ¼ c11ðsÞ þ s2 ~mmcðsÞ c11ðsÞ are satisﬁed and c10 1ðsÞ for s 2 ½0;1Þ: With this choice of 1; the function 0 is positive for all s 2 ½0;1Þ: Moreover, since 0 1 is a 1ðsÞ is positive for monotonically increasing function, 00 positive s. Furthermore, note that with this choice of 0 1Þz1n1ðz1Þ ¼ c10 1ðz2 1Þz2 and n1; the term 0 1 1 is a class K1 function of z2 1: s ~mmðsÞ mðsÞ 1ðz2 for 1 s Remark 5: The parameter estimate ^bb is initialised as a positive real number. From remark 3 and remark 4, b1 is seen to be positive. Also, b2 and b8 are positive which implies that ^bb and hence b1 remain positive. With 1 and n1 chosen to satisfy (55), (53) reduces to: _VV 1 4e2 2 s z2n2 z3n3 z4n4 g z2 e3 þ k3 Nr 0 1z1n1 1 2 ð ^bb b1 b2 b3Þ2 32 a2 1 3 þ z2 2þe2 2 4 2 4 þ d5 ð58Þ Theorem 1: The proposed dynamic controller guarantees the global boundedness of all closed-loop signals and achieves practical stabilisation of the tracking error. 4 2 4 2 Proof: Equation (58) can be written as: _VV UðwÞ þ d5 ð59Þ where w¼ðz1; z2; z3; z4; ^bb b1 b2 b3; e2; e3 þ k3=Nr; e4Þ and UðwÞ¼ 1 þ 4e2 2 2þe2 3 þ z2 z2n2 þ z3n3 þ z4n4 þ g z2 e3 þ k3 Nr 0 1z1n1 þ 1 2 þ s ð ^bb b1 b2 b3Þ2 þ 32 a2 1 ð60Þ is a smooth positive-deﬁnite and radially unbounded function of w: Note that V is also a smooth positive-deﬁnite and radially unbounded function of w: From (47) and (55), it follows that UðwÞ PV where P¼ minðk1c1; c2; 2c3 þ 2g; 2c4 þ 2g;s;a1=2;g=3Þ: Applying the comparison lemma [24]: ð61Þ From (54) and (61), using the boundedness of yrefðtÞ and its derivative, it follows that V and hence w are bounded along the trajectories of the solutions of the system. By routine signal-chasing, it is seen that the boundedness of w implies the boundedness of all closed-loop signals. Furthermore, solutions tend to the compact set in which: VðtÞ Vð0Þe d5ðsÞds Pt þ PðtsÞ Z e 0 t ð62Þ where dd5 ¼ lim supt!1 d5 is a positive constant since d5 is bounded. In the set (62), the tracking error z1 satisﬁes the inequality: V P1 dd5 1 2 1 z2 1 P1k1 dd5 ð63Þ The practical tracking property can be inferred from (63). Let an arbitrary e > 0 be given. 1 will be designed to regulate the tracking error z1 to ½e; e: First, c1 >0 and a smooth class K1 function GðsÞ are chosen satisfy 1ðz1Þþ ~rrð4z2 1Þ: The discus- c1G0ðsÞs c1GðsÞ 2l½8z2 1 þ 3n2 sion in remark 4 guarantees that such a choice is possible. Choose ~cc1 ¼ max 1;2P1k1 ;n1ðsÞ¼ c1s dd5=G 2 and 1ðsÞ ¼ ~cc1GðsÞ: With this choice of c1 and 1; (55) is satisﬁed which ensures, as proved by the Lyapunov analysis presented above, that the solutions tend asymptotically to a compact set in which (62) and (63) are satisﬁed. From (63): to e s ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ G1 2P1k1 dd5 ~cc1 jz1j e ð64Þ A Theorem 2: If yref 0; r0 ¼ 0 and rðsÞ ¼ O½s as s ! 0þ; the proposed dynamic controller with s ¼ 0 achieves asymptotic stabilisation of the rotor position to zero. 504 IEE Proc.-Control Theory Appl., Vol. 151, No. 4, July 2004

Proof: Under the hypothesis of the theorem, r0 ¼ cr ¼ s ¼ d5 ¼ 0: Hence, reduces to z1 ¼ x1 ¼ 0: A the compact set in (62) The control design procedure can be summarised as 1ðs2Þ; 00 follows: 1. Find ~rr to satisfy (52) with ~rrðsÞ ¼ O½s as s ! 0þ: 2. Pick a positive constant l and functions 1 2 K1 and n1 1ðsÞ being non- to satisfy (55) with 0 negative for all s 2 R: 3. Pick a positive constant c2 and deﬁne function n2ðsÞ ¼ c2s: Pick positive constants c3 and c4 and deﬁne functions n3 and n4 such that sniðsÞ cis2 8s 2 R; i ¼ 3; 4: 4. Pick positive numbers a1 and s: 5. Implement the observer dynamics: 1ðs2Þ and n0 _^xx^xx2 ¼ a1 ^xx2 ^xx3 sinðNrx1Þ þ ^xx4 cosðNrx1Þ _^xx^xx3 ¼ g^xx3 þ u1 _^xx^xx4 ¼ g^xx4 þ u2 ð65Þ 6. Compute the control inputs as u1 ¼ z2 sinðNrx1Þ þ gb3 sinðNrx1Þ þ b5 sinðNrx1Þ n3ðz3Þ ð ^bb þ b4Þz3b2 1 u2 ¼ z2 cosðNrx1Þ gb3 cosðNrx1Þ b5 cosðNrx1Þ h 6 1 þ ^bb2 1 þ 02 h 7 1 þ ^bb2 1 þ 02 1 i i ð ^bb þ b4Þz4b2 n4ðz4Þ ð66Þ i 2 z2 2 i 1ðz1Þ ^bb2 1 þ 0 1Þ2z3 1 þ b2 1 1 1 ^bb2ð1 þ 0 1Þ2 1 1 z1 þ n1ðz1Þ where z1 ¼ x1 yref z2 ¼ ^xx2 þ ^bb 1 þ 0 1 z2 1 z3 ¼ ^xx3 b3 sinðNrx1Þ z4 ¼ ^xx4 þ b3 cosðNrx1Þ h þ n0 b1 ¼ ^bb 1 þ 0 100 þ 2z2 1 z2 1 z2 1 1 b2 ¼ 0 1 þ 02 1 þ b2 1 þ b1 þ b2 z2 h _yyref þ 0 1ð1 þ 0 b3 ¼ 0 1z1 a1 ^xx2 b1 þ z2½ ^bb þ ð1 þ 0 1Þz1z2 b1 þ b2 s ^bbð1 þ 0 1Þz1 _^bb^bb @b3 @z1 n b4 ¼ z2ð1 þ 0 b5 ¼ @b3 @ ^bb þ^xx4 cosðNrx1Þ þ _^bb^bbð1 þ 0 þ @b3 @ ^xx2 b6 ¼ @b3 @z1 þ b1 b7 ¼ @b3 @z1 b8 ¼ z2 6 þ z2 4b2 3b2 7 1Þz1 þ n2ðz2Þ n _yyref þ @b3 @z2 þ b1 h 1 _yyref 1Þz1 b1 o þ @b3 @ _yyref a1 ^xx2 ^xx3 sinðNrx1Þ þ ^xx4 cosðNrx1Þ sinðNrx1Þ þ b3Nr cosðNrx1Þ @b3 @z2 @b3 @z2 cosðNrx1Þ b3Nr sinðNrx1Þ 1 þ ^bb2 1 þ 02 i a1 ^xx2 ^xx3 sinðNrx1Þ o €yyref 7. Implement the parameter update law: _^bb^bb ¼ s ^bb þ b2 þ b8 ð67Þ ð68Þ ^bb serves as a dynamic estimate for b ¼ Remark 6: b1 þ b2 þ b3: Alternatively, separate parameter estimators could be built for b1; b2 and b3 (or indeed for any of the components of these terms). The approach followed here of only estimating ðb1 þ b2 þ b3Þ has the advantage of resulting in a lower-dimensional controller. Thus, we only estimate the size of total uncertainties and not the individual uncertain parameters themselves. Attempting to estimate all the uncertain parameters would result in a controller of higher order. However, in some cases, it might be desirable to separately estimate some of the parameters in an attempt to improve performance at the expense of an increased dynamic order of the controller. Such an extension would be straightforward with the controller developed here. Instead of overbounding all terms involving uncertain parameters during the Lyapunov analysis, the relevant terms could be retained and parameter estimators built using the approach described here. In particular, further performance improve- ment can be obtained (as illustrated using simulations in Section 6) by utilising additional adaptations to compensate for the friction torque and to provide an adaptive feedforward term. In this case, steps 1-5 of the control design summarised above remain unchanged and steps 6 and 7 are modiﬁed to steps 6’ and 7’ provided below. The stability of the resulting closed-loop system can be proved using the Lyapunov function: V ¼ 1 2k1 þ 1 2g2 ð ^bb b1 b2 b3Þ2 4 þ 1 z2 2 2þ 16 a1 2 þ 1 z2 2 ð^aa2 a2Þ2 þ 1 2gz 3 þ 1 z2 2 ^zz 1 k1 þ 1 2 1 z2 1 Vo 6’. Choose positive constant g2 and gz: Choose non-negative constants s2 and sz: Compute the control inputs as: u1 ¼ z2 sinðNrx1Þ þ gb3 sinðNrx1Þ þ b5 sinðNrx1Þ 2 þ ^zz2 i i u2 ¼ z2 cosðNrx1Þ gb3 cosðNrx1Þ b5 cosðNrx1Þ h 6 1 þ ^bb2 1 þ 02 h 7 1 þ ^bb2 1 þ 02 ð ^bb þ b4Þz4b2 ð ^bb þ b4Þz3b2 n3ðz3Þ þ ^aa2 1 1 þ ^aa2 2 þ ^zz2 n4ðz4Þ ð69Þ z1 þ n1ðz1Þ þ ^aa2x1 ^zz_yyref þ n0 1ðz1Þ þ ^aa2 i ^zz2 z2 2 i ^zz2 1Þ2 þ b2 2 þ b2 1 ^aa2 1ð1 þ 0 1Þ2z3 1 1 1Þ2 þ b2 2 þ b2 1 ^aa2 1 1 z2 1 100 1 z2 1 where z1 ¼ x1 yref z2 ¼ ^xx2 þ ^bb 1 þ 0 1 z2 1 z3 ¼ ^xx3 b3 sinðNrx1Þ z4 ¼ ^xx4 þ b3 cosðNrx1Þ b1 ¼ ^bb 1 þ 0 þ 2z2 h b2 ¼ 0 1 þ 02 z2 1 þ b1 þ b2 1 þ b2 1z1 a1 ^xx2 b1 h þ z2½ ^bb þ ð1 þ 0 b1 þ b2 1 þ b2 s ^bbð1 þ 0 1Þz1 þ n2ðz2Þ þ ^aa2 þ sz ^zz_yyref þ g20 1 þ gz0 1z1 ^bb2ð1 þ 0 _yyref þ 0 1Þz1z2 ^bb2ð1 þ 0 b3 ¼ 0 1z1x2 1 1 1 _yyref ^zz€yyref s2 ^aa2x1 _yy2 ref IEE Proc.-Control Theory Appl., Vol. 151, No. 4, July 2004 505

_^aa^aa2 þ @b3 @ ^zz b4 ¼ z2ð1þ 0 1Þz1 n _^bb^bbþ @b3 b5 ¼ @b3 @ ^aa2 @ ^bb a1 ^xx2 ^xx3 sinðNrx1Þþ ^xx4 cosðNrx1Þ þ @b3 @z2 1Þz1 þ _^aa^aa2x1 _^zz^zz_yyref ^zz€yyref b1 _^zz^zz @b3 @z1 þ _^bb^bbð1þ 0 _yyref n o €yyref þ @b3 @ €yyref þ @b3 @ _yyref þ ^xx4 cosðNrx1Þ b6 ¼ @b3 þ @b3 @z1 @x1 b7 ¼ @b3 þ @b3 @z1 @x1 b8 ¼ z2 6 þ z2 4b2 3b2 7 þ b1 h þ b1 yð3Þ ref þ @b3 @ ^xx2 @b3 @z2 @b3 @z2 sinðNrx1Þþ b3Nr cosðNrx1Þ cosðNrx1Þ b3Nr sinðNrx1Þ i _yyref þ ^aa2 _yyref a1 ^xx2 ^xx3 sinðNrx1Þ 1þ ^bb2 1þ 02 1 þ ^aa2 2 þ ^zz2 7’. Implement the parameter update laws: _^bb^bb ¼ s ^bb þ b2 þ b8 _^aa^aa2 ¼ s2 ^aa2 þ g20 _^zz^zz ¼ sz ^zz gz0 1z1 1z1x1 _yyref ð70Þ ð71Þ Remark 7: While the condition (55) imposes a constraint on the form of 1 and n1; these functions can be scaled by multiplication with arbitrary constants by choosing l appropriately. This freedom can be exploited to avoid unwanted high gains in the controller. Furthermore, l can be made a function of c1 such that the choice of 1 to satisfy (57) is independent of c1: This simpliﬁes the tuning of the parameter c1: Remark 8: The above stability analysis and theorems provide global results under the assumption that the electrical time constant is exactly known. If this parameter is only approximately known, semiglobal stability is achieved by the designed observer=controller (using the available estimate for g) in the following sense. The designed observer=controller with g replaced every- where by an estimate ^gg ensures stability and practical tracking if j^gg gj=g < g" with g" being a positive constant depending on the initial condition, i.e. V(0). A mismatch between the actual and estimated values of the electrical time constant results in the introduction of additional terms ðg ^ggÞ^xx3 and ðg ^ggÞ^xx4 in the observer error dynamics. Thus: _VVo a1 2 1 e2 a1 2þe2 e3 þ k3 Nr þ r0 þ c minðl; 1Þ x2 1 þ r x2 3 þ ^xx2 ð^gg gÞ2 ^xx2 þ 36 g2a1 2 4 1 4 and 506 _VV P2V þ dd5a þ d6 ð^gg gÞ2 g2 3 þ ^xx2 ^xx2 4 where P2 ¼ minðk1c1; c2; 2c3 þ 2g; 2c4 þ 2g; s; a1=2; g=6Þ; dd5a ¼ supt2½0;1Þ d5ðtÞ and d6 ¼ 576"=a2 1: 2 ð dd5a þ dd6Þ 4Þ: An upper bound for M can be and M ¼ supV ddð^xx2 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ p ﬃﬃﬃﬃﬃ p 1 ð2k1 ddÞ 1 Let dd6 be a positive constant. Deﬁne dd ¼ P1 q obtained through the relations below: zz1 ¼ zz2 ¼ zz3 ¼ zz4 ¼ 3 þ ^xx2 ﬃﬃﬃﬃﬃ 2 dd o ^bb^bb ¼ b1 þ b2 þ b3 þ 0 1 ¼ sup 1ðs2Þ 1 ¼ sup 0 00 jsjzz1 jsjzz1 n0 1 ¼ sup 1ðsÞ n0 jsjzz1 zz1 þ nn1 ^xx^xx2 ¼ zz2 þ ^bb^bb 1 þ 0 bb3 ¼ 0 ^xx^xx2 þ bb1 sup t2½0;1Þ 1 h 1zz1 þ a1 ^bb^bb þ 1 þ 0 þ zz2 þ s ^bb^bb 1 þ 0 3 þ 2zz2 4 þ 2bb2 1 3 1 zz1zz2 zz1 þ c2zz2 M 2zz2 jn1ðsÞj 2 dd 00 1ðs2Þ nn1 ¼ sup jsjzz1 bb1 ¼ ^bb^bb 1 þ 0 1 þ 2zz2 100 i h 1 1 þ 0 _yyrefðtÞ þ 0 zz3 1 1 þ 0 1 þ bb2 bb1 þ bb2 ^bb^bb2 1 1 2 1 1 1 þ n0 i 2 ð74Þ If Vð0Þ dd and s ﬃﬃﬃﬃﬃﬃﬃﬃﬃ dd6 d6M j^gg gj=g < g" ¼ then the set fwjV ddg is forward-invariant under the closed-loop dynamics. Hence, for all time t, we have _VVðtÞ P2VðtÞ þ dd5a þ dd6 and the above claims follow easily. 6 Simulation results The efﬁcacy of the proposed controller is demonstrated using simulation for a PM stepper motor with the following parameters: Nr ¼ 50 J ¼ 0:0733 kg m2 L0 ¼ 0:7 mH Lm1 ¼ 5 mH Lf4 ¼ 1:766 mH R ¼ 1 O if ¼ 1 A D ¼ 0:002 kg m2=s tl ¼ 1:7201 sinðx1Þ The desired trajectory for the rotor position is a smooth-start sinusoid, ð1 e0:2t2Þ sinð4tÞ: Initial conditions for all states including the parameter estimates are zero. A bounding function for the torque uncertainties is assumed to be 1Þ ¼ x2 available as rðx2 1: The design parameters in the controller are chosen as: a1 ¼ 400 s ¼ 2000 c1 ¼ 10 c2 ¼ 800 c3 ¼ c4 ¼ 1500 l ¼ 10 ¼ 18z2 niðsÞ ¼ cis 1 i 4 ð75Þ 5 5 1 z2 1 1 ð72Þ ð73Þ The simulation results are shown in Fig. 1. It is seen that the rotor position tracks the desired trajectory with a peak error of 0:089 and a root-mean-square (RMS) error of 0:056: The choice of the form of 1 and n1 is dictated by the structure of the bounding function for the torque uncertain- ties. The constants in 1; n1 and n2 were picked using an approximate linearisation of the system around the origin assuming fast electrical and observer error dynamics to place the poles of the linearised mechanical subsystem at a natural frequency of around 42 Hz with a 0.75 damping ratio. c3; c4 and a1 are chosen large enough to ensure fast IEE Proc.-Control Theory Appl., Vol. 151, No. 4, July 2004

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