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Microstepping and high-performance control of permanent-magnet stepper motors
Energy Conversion and Management 85 (2014) 245–253
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Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman
Microstepping and high-performance control of permanent-magnet stepper motors Sergey Edward Lyshevski ⇑ Department of Electrical and Microelectronic Engineering, Rochester Institute of Technology, Rochester, NY 14623, USA
a r t i c l e
i n f o
Article history: Received 10 March 2014 Accepted 22 May 2014 Available online 14 June 2014 Keywords: Control Energy conversion Electric machine Optimality Stepper motor
a b s t r a c t We examine the problem of control of high-performance drives and servos with permanent-magnet stepper motors. Control of electromechanical systems implies control and optimization of electromechanical transductions and energy conversion. Robust spatio-temporal control algorithms are designed to ensure high efficiency, high-precision microstepping and optimal performance. The system stability, robustness and control design are examined applying an admissibility concept. Nonlinear control guarantees optimal energy conversion in expanded operating envelopes. Our analytic designs are substantiated and verified. A proof-of-concept system is tested and characterized. The high electromagnetic torque and highprecision microstep angular positioning simplify kinematics, enables efficiency, ensures direct-drive capabilities, reduces complexity, etc. For four-phase permanent magnet stepper motors, one may ensure up to 256 microsteps within a 1.8 full step. High efficiency and accurate 2.454 104 rad positioning (25,600 microsteps per revolution) are achieved with high electromagnetic and holding torques. To guarantee high efficiency, optimality and enabled energy conversion capabilities, electromechanical energy conversion and high electromagnetic torque are achieved by applying soft balanced phase voltages. The ripple and friction torques are minimized. The fundamental findings, technology-centric design and experimental results are reported. 2014 Elsevier Ltd. All rights reserved.
1. Introduction Fundamental, applied and practical problems of optimal energy conversion and precision motion control of electromechanical systems are of a great importance. The solution of the aforementioned problems allow one to maximize the efficiency, minimize losses, as well as deploy advanced motion platforms in aerospace, automotive, electronics, energy, manufacturing, power, robotics and other applications. Enabling technologies are under developments. Permanent-magnet stepper motors are widely used in drives and servos due to gearless direct-drive capabilities. Stepper motors are simple, affordable, rugged. Performance and capabilities may be enabled applying recent fundamental and technological advancements ensuring high torque and high power densities. The specified dynamics and high precision microstepping can be achieved developing high electromagnetic torque. Optimal electromagnetic and electromechanical designs, rear-earth permanent magnets and advanced materials enable energy conversion ⇑ Tel./fax: +1 5854754370. E-mail address: [email protected] URL: http://people.rit.edu/seleee/ http://dx.doi.org/10.1016/j.enconman.2014.05.078 0196-8904/ 2014 Elsevier Ltd. All rights reserved.
capabilities. The technology-enhanced stepper motors are controlled by advanced power electronics. Consistent control schemes ensure optimal energy conversion which results in enabled functionality and performance. Coherent device- and system-level solutions imply the use of advanced software, electronics and electromechanical hardware. New trends in motion control of drives and servos foster developments of nonlinear control algorithms consistent with the device electromagnetics and power electronics [1–3]. Sensorless and sensor-centric designs of servos with stepper motors are reported in [4–8] using the quadrature and direct voltages and currents. However, electric machines are controlled changing the phase voltages [1–3,9–12]. Practical concepts in motion control of permanent-magnet stepper motors were developed for full-, half- and quarter stepping. An optimal energy conversion and high-precision microstepping implies the use of advanced-technology motors, electronics and control schemes. Many studies concentrate on modeling [7,8]. The use of the arbitrary reference frame and quadrature-direct quantities serve as a possible inroad to stability analysis and design of feedback linearizing and vector controls [4–8]. While analytic results may be important, practical control schemes use the electromagnetically-consistent
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machine variables [1–3,9–12]. The directly measured phase currents and voltages are used as physically-consistent controllable variables. Due to the algorithmic, software and hardware complexity, it is difficult to implement the spatio-temporal motion control using the mathematically-descriptive quadrature- and direct-components of voltages, fluxes and currents. Optimal energy conversion in stepper motors requires electromagnetically-consistent concepts. The use of adaptive, artificial neural networks, intelligent, sensorless and other control schemes may not ensure adequate energy conversion. Practical solutions require technology-centric fundamental research. These transformative findings result in new knowledge generation and developments which enable engineering and technology. 2. Electromagnetics and electromechanics of permanentmagnet stepper motors By using the laws of electromagnetics and electromechanics, the differential equations for two- and four-phase permanentmagnet stepper motors in the machine variables are derived [1,2]. For two-phase motor, the a and b flux linkages are wa = Lia + wam and wb = Lib + wbm. Assuming a sinusoidal uniform magnetic field of permanent magnets, one has wam = wmcos(RThm) and wbm = wmsin(RThm). We obtain [1,2]
dia 1 ¼ ½ria þ RTwm xm sinðRThm Þ þ ua ; L dt
i did 1 h r r ¼ rid þ LRTiq xm þ urd ; L dt r
i dxm 1 h r ¼ RTwm iq Bm xm T L : J dt
ð3Þ
2 r r r r2 For a positive-definite function V iq ; id ; xm ¼ 12 iq þ id þ x2m , the h dV ðirq ;ird ;xm Þ r2 r r r2 total derivative is ¼ 1L riq þ RTwmJðLJÞ iq xm þ iq urq rid þ dt r
id urd Bm x2m . Therefore, there exists a positive-definite Lyapunov dV ðirq ;ird ;xm Þ r r function V iq ; id ; xm > 0, for which < 0. That is, dV/dt is dt negative-definite for any variations of bounded motor parameters ai(), aimin 6 ai()6aimax, aimin > 0, aimax > 0. Hence, the open-loop permanent-magnet stepper motor is robustly asymptotically stable in the large. The high-fidelity mathematical models and expressions for the electromagnetic torque Te are derived by using a consistent expression for the flux linkages. A nonlinear electromagnetic system, nonuniform and nonstationary magnetic field of permanent magnets and other phenomena result in the equations for wa(hm) and wb(hm) as
wa wb
#
" ¼
L
0
0
L
#"
ia ib
#
3 2X 1 2n1 b cos ðRTh Þ n m 7 6 7 6 n¼1 7; þ wm 6 7 6X 1 5 4 2n1 bn sin ðRThm Þ
ð4Þ
n¼1
dxm 1 ¼ ½RTwm ðia sinðRThm Þ þ ib cosðRThm ÞÞ Bm xm T L ; J dt
ð1Þ
where ia and ib are the a and b phase currents; ua and ub are the applied phase voltages to the a and b windings; xm and hm are the mechanical angular velocity and rotor displacement; TL is the load torque; r and L are the resistance and self-inductance of the stator winding; RT is the number of rotor teeth per stack; wm is the amplitude of the flux linkages established by the permanent magnet; J is the equivalent moment of inertia; Bm is the viscous friction coefficient. The expression for the electromagnetic torque is
T e ¼ RTwm ½ia sinðRThm Þ þ ib cosðRThm Þ:
r
"
dib 1 ¼ ½rib RTwm xm cosðRThm Þ þ ub ; dt L
dhm ¼ xm ; dt
i diq 1 h r r ¼ riq RTwm xm LRTid xm þ urq ; L dt
ð2Þ
The stator resistance r, self-inductance L, amplitude the flux linkages wm, viscous friction coefficient Bm, and equivalent moment of inertia J vary due to varying temperature and loads. These variations are bounded, and, r 2 [rmin rmax], L 2 [Lmin Lmax], wm 2 [wmmin wmmax], Bm 2 [Bmmin Bmmax] and J 2 [Jmin Jmax]. These bounded parameters aimin 6 ai()6aimax are always positive, ai()>0, aimin > 0 and aimax > 0. To perform the stability analysis, we use the arbitrary reference frame which rotates at synchronous angular velocity [1,2]. The Park transformation yields the quadrature- and direct-axis compo r i sinðRThm Þ cosðRThm Þ ia nents of currents qr ¼ and volatcosðRThm Þ sinðRThm Þ ib id r uq sinðRThm Þ cosðRThm Þ ua ges ¼ . Using (1) and (2), one urd cosðRThm Þ sinðRThm Þ ub obtains [1,2]
where bn are the positive coefficients which describe the nonlinear electromagnetics. For example, for the ideal sinusoidal electromagnetic coupling, ensured by the Aerotech, Kollmorgen, Shinano Kenshi and other permanent-magnet synchronous machines, b1 = 1 and "bn = 0 for "n > 1 (n = 2, 3, 4, . . .). For high electromagnetic loadings, inadequate design and gaps (spacing) between magnets, "bn–0 with b1 > b2 > b3. . . The electromagnetic system, electromagnetic loadings and other effects affect Te. From (4), the expression for the electromagnetic torque is
" T e ¼ RTwm ia
1 X ð2n 1Þbn sinðRThm Þ cos2n2 ðRThm Þ n¼1
# 1 X 2n2 ðRThm Þ : þib ð2n 1Þbn cosðRThm Þ sin
ð5Þ
n¼1
Using the expressions for the electromagnetic torque (2) and (5), one finds the balanced current and voltage sets which maximize the electromagnetic torque Te and minimize the torque ripple. From (2), the balanced current and voltage sets are derived for: (i) Full step operation with the angular displacement 2p/RT, p/RT or p/2RT, which depends on the motor and windings designs; (ii) Continuous motor rotation. In particular, we have
ia ¼ iM sinðRThm Þ; ib ¼ iM cosðRThm Þ;
iMmin iM iMmax ;
ua ¼ uM sinðRThm Þ; ub ¼ uM cosðRThm Þ; uMmin uM uMmax ;
ð6Þ
where iM and uM are the rated current and voltage. The balanced current and voltage sets (6) suit the operation of stepper motors in the full operating envelope including high electromagnetic loadings. The derived expressions for Te yield the balanced fed phase currents and applied voltages which ensure the full-, half-, quarter- and microstepping [1,2,12,13]. Assuming 2p/RT displacement for a full step, using (2) and (5), one has
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(
(
ia ¼ iM sgnðsin RThm Þ; 8hm 2 0 ib ¼ iM sgnðcos RThm Þ; 8h m 2 0
ua ¼ uM sgnðsin RThm Þ; 8hm 2 0 ub ¼ uM sgnðcos RThm Þ; 8h m 2 0
(
2p RT ; 2p RT
ia ¼
(
2p RT ; 2p RT
iM ; 8hm 2 0 p iM ; 8hm 2 RT
ua ¼
p
(
RT ; 2p RT
uM ; 8hm 2 0 p uM ; 8hm 2 RT
p
ib ¼
(
RT ; 2p RT
8h m 2 0 p iM ; 8hm 2 2RT
iM ;
ub ¼
p
8hm 2 0 p uM ; 8hm 2 2RT
uM ;
2RT ; 3p 2RT
8hm 2
p
2RT ; 3p 2RT
3p 2RT
8hm 2
2p RT
3p 2RT
;
2p RT
:
For the half- and quarter steps, the phase voltages ua and ub are
8 8 3p p 7p p > > < uM ; 8hm 2 8RT RT < uM ; 8hm 2 0 2RT ;8hm 2 4RT 5p 2p 3p 3p ua ¼ ; ub ¼ uM ; 8hm 2 4RT uM ; 8hm 2 4RT RT 2RT > > : : 0 otherwise 0 otherwise
For two- and four-phase stepper motors, one examines the electromagnetic torque Te. For the bipolar stepper motors with the phase currents (ia, ib) and voltages (ua, ub), or, for unipolar motor with a ; ub ; u b Þ, one obtains the resulting expressions ðia ; ia ; ib ; ib Þ and ðua ; u for full step, 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, 1/128 and 1/256 microstepping. Depending on the stepping mode, the phase currents and voltages depend on the rotor angular displacement hm. Therefore, a ; ub ; u b Þ, are time- and (ia, ib) and (ua, ub), or, ðia ; ia ; ib ; ib Þ and ðua ; u spatially-dependent. For two- and four-phase permanent magnet stepper motors, one obtains the spatio-temporal balanced current and voltage sets to guarantee optimal energy conversion and microstepping repositioning. We have
ia ¼ iM ua ðhm Þ; 8hm 2 HT ib ¼ iM ub ðhm Þ; 8hm 2 HT
;
ua ¼ uM ua ðhm Þ; 8hm 2 HT ub ¼ uM ub ðhm Þ; 8hm 2 HT
ð7Þ
and
8 ia ¼ iM ua ðhm Þ; > > > <i ¼ i u a M a ðhm Þ; > i ¼ i M ub ðhm Þ; b > > : b ðhm Þ; ib ¼ iM u
8hm 2 HT 8hm 2 HT ; 8hm 2 HT 8hm 2 HT
8 ua ¼ uM ua ðhm Þ; > > > u ¼ u M ub ðhm Þ; b > > : b ¼ uM u b ðhm Þ; u
8hm 2 HT 8hm 2 HT ; 8hm 2 HT 8hm 2 HT
ð8Þ
where the period is T 2 [0 2p/RT], T 2 [0 p/RT] or T 2 [0 p/2RT]. If bidirectional loads are applied, the closed-loop systems must be designed to prevent microstepping inadequacy and guarantee tracking. 3. Design of control schemes Permanent magnet stepper motors are open-loop stable. They ensure stable-equilibrium stepping within the open-loop configuration. However, missed steps, microstepping inconsistency, step mismatch and other inadequacies arise at bidirectional loading, high loads, considerable friction, variable inertia, fast repositioning, etc. To overcome the aforementioned drawbacks, closed-loop systems must be designed to ensure tracking of the angular velocity xm, displacement hm, etc. Using the reference vector r 2 R Rb , one examines the system output vector y 2 Y Rb . The tracking error e 2 E Rb is given as
eðtÞ ¼ rðtÞ yðtÞ:
ð9Þ
For drives y = xm with e = ex. For servos y = hm with e = eh. The angular acceleration am = dxm/dt is of importance for servos in addition to angular displacement hm. Hence, y = [hm am]T and
2p RT
8 8 p p p > > < uM ; 8hm 2 2RT RT < uM ; 8hm 2 0 2RT 3p 2p p 3p : and ua ¼ ; ub ¼ uM ; 8hm 2 RT uM ; 8hm 2 2RT RT 2RT > > : : 0 otherwise 0 otherwise
e = [eh ea]T. In high-performance servos, the rate of change of am is specified, and dam/dt can be used as an output. One has y = [hm am dam/dt]T, e = [eh ea eda/dt]T. It was found that the mathematical model is described by nonlinear differential equations (1) with states x = [ia ib xm hm]T, x 2 X Rn and controls u = [ia ib]T or u = [ua ub]T, u 2 U Rm . The mathematical model (1) can be refined using consistent expressions for the flux linkages, nonlinear magnetization, secondary effects, etc. The governing equations of motion are
dx=dt ¼ Fðx; u; rÞ þ Uðx; u; r; z; dÞ; y ¼ Cx;
ð10Þ
where x 2 X Rn , r 2 R Rb and y 2 Y Rb are the system state, reference and output vectors; u 2 U Rm is the control vector with the closed admissible set U ¼ fu 2 Rm ; umin 6 u 6 umax ; umax > 0; umin 6 0g; z 2 Z Rq and d 2 D Rs are the vector of time-varying bounded uncertainties, perturbations and disturbances; F(x, u, r) and U(x, u, r, z, d) are the nonlinear map as defined by consistent electromagnetics and electromechanics; C 2 Rbn is the output matrix. The differential equations (10) provide the model mapping M() which describes the physical system M (). The parameter variations, unmodeled dynamics and other uncertainties are bounded. For time-varying bounded uncertainties z 2 Z and disturbances d 2 D, there exists a norm of U(), and, ||U(x, u, r , z, d)|| 6 q(t, ||x||), where qðÞ : RP0 RP0 ! RP0 is the continuous Lebesgue measurable function. Our goal is to synthesize a control law using a consistent model M() or for a physical system M (), such that the error vector e(t) with E0 ¼ fe0 2 Rb g # E Rb evolves in the specified closed set [15]
Sxe ðdÞ ¼ fe 2 Rb ; x 2 Rn : e0 2 E0 ; x 2 XðX 0 ; U; R; Z; DÞ; x0 2 X 0 ; u 2 U; r 2 R; z 2 Z; d 2 D; y 2 Y; y 2 Y; t 2 ½t0 ; 1ÞjkxðtÞk qx ðt; kx0 kÞ þ qr ðkrkÞ þ qz ðkzkÞ þ qd ðkdkÞ þ d; keðtÞk qe ðt; ke0 kÞ þ qr ðkrkÞ þ qy ðkykÞ þ d; d 0; 8x 2 X;
8e 2 EðE0 ; R; YÞ; 8t 2 ½t 0 ; 1Þg Rb Rn :
ð11Þ
Here, qe ðÞ : RP0 RP0 ! RP0 and qx ðÞ : RP0 RP0 ! RP0 are the continuous strictly increasing functions with respect to t, and, monotonically decreasing with respect to the second argument, limt?/qj(t, ()) = 0; qj ðÞ : RP0 ! RP0 are the continous strictly increasing functions, qj(0) = 0.
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Using the criteria imposed on the Lyapunov pair [2,14,15] for a positive-definite function V(e, x), one may define the set Sxe Rb Rn where the state x(t) and error e(t) vectors are bounded. For the system, which evolves in XE, using V(e, x), one has
Using the performance measure c 2 RP0 , c > 0, for the physical system M (xm, y, e, u) in the evolution envelope XE(X0, E0, U, R, Z, D), we define the performance goal
Sxe ¼ fe 2 Rb ; x 2 Rn jVðe; xÞ > 0; dVðe; xÞ=dt < 0g Rb Rn :
The desired performance is achieved if the closed-loop response triple (xm, e, u) lies in the set
ð12Þ The stability is guaranteed if XE Sxe. Here, XE is a subset of Sxe, but not equal to Sxe. The system evolves in an operating envelope XE(X0, E0, U, R, Z, D) with x 2 X, e 2 E, r 2 R, u 2 U, z 2 Z and d 2 D. Linear and nonlinear PID, vector, adaptive, model-based, relay, linearizing, sliding mode and other control laws are developed [2–12]. Some control schemes have a limited practicality due to abstract problem formulation and overall inadequateness. We focus on practical control concepts using the first principles of electromagnetics and electromechanics. This ensures overall optimal performance and achievable capabilities by means of optimal energy conversion. Robust performance-seeking control laws should ensure precise tracking, stability, minimum-time dynamics, disturbance attenuation, etc. Design and optimization are formulated as search problems in a high-dimensional space. The performance functional with a positive-definite integrand W(x, e, u)
J ¼ lim
min
Z
T!1x2X;e2E;u2U
T
Wðx; e; uÞdt;
Wðx; e; uÞ > 0;
t0
JðÞ : X E U ! R0
8 ua ¼ uua ðhm Þ; > > > < a ¼ uu a ðhm Þ; u > u ¼ u u > b b ðhm Þ; > : b ¼ uu b ðhm Þ; u
8h m 8h m 8h m 8h m
ð13Þ
2 HT 2 HT 2 HT
;
u¼
satu0M
M p ;Lp X k;l¼1
kpð2k1Þ;ð2l1Þ e proportional
2k1 2l1
ðtÞ þ
NX i ;M i ;Li j;k;l¼1
1. Mathematical conjectural modeling mapping M() obtained by using the first principles of electromagnetics and electromechanics; 2. Physical system M () with the state, output, error, control and other variables and quantities which are directly measured, asserted, examined and used. For physical systems, using the directly measured physical quantities xm and e, we minimize
min
T!1xm 2X;e2E;u2U
Z
8x 2 X;
8e 2 E;
8u 2 U on t 2 ½t0 ; t f Þ:
J S ðcÞ ¼ fðxm ; e; uÞ 2 X m E UjJðxm ; e; uÞ cg:
ð16Þ
ð17Þ
Using the performance specification set JS(c) and measurements set MS = {(xm, e, u) 2 Xm E U}, the control law (15) must be synthesized on a class of implementable, electromagnetic- and electromechanic-consistent algorithms CS. The optimal minimal-complexity performance-seeking bounded control and the corresponding performance measure are found by solving the nonlinear optimization problem [15]
cmin ¼
min ðcÞ
ð18Þ
;
JðÞ:XEU _ Subject to x¼Fðx;u;rÞþ Uðx;u;r;z;dÞ;u¼/ðt;xm ;eÞ Mðxm ;y;e;uÞ;C S 2C;C S \J S ðcÞ\MS
where C is the set of controllers. The analytically designed control law should ensure optimal closed-loop system performance and capabilities as specified by the performance functional (14) and measure c. We synthesize a bounded proportional-integral tracking control law, as well as a control law with the state feedback. For four-phase unipolar stepper motors, we have
Z
Z ... Ni
! ki
j;ð2k1Þ;ð2l1Þ integral
e
2k1 2l1
dt ;
ð19Þ
2 HT
forms a hypersurface. A control law u(t) must ensure a broad spectrum of requirements as given by the performance functional (13). We minimize (13) subject to the system dynamics. The modelcentric and physical designs imply the use of:
J ¼ lim
Jðxm ; e; uÞ 6 c;
T
Wðxm ; e; uÞdt;
Wðxm ; e; uÞ > 0;
uðtÞ ¼ satu0M
M p ;Lp X k;l¼1
NX i ;M i ;Li
þ
j;k;l¼1
2k1
kpð2k1Þ;ð2l1Þ e 2l1 ðtÞ
Z
proportional
Z ... Ni
! ki
2k1
j;ð2k1Þ;ð2l1Þ integral
e 2l1 dt þ Kx m
;
state feedback
where Mp, Lp, Mi, Li and Ni are the positive integers specified by the designer; kp(2k1),(2l1) and kij,(2k1),(2l1) are the proportional and integral feedback coefficients; K is the matrix of feedback coefficients. For a physical system M (), and its model M(), the bounded control (19) is the optimal tracking control law with respect to performance measure c if for (r, x, e, u) 2 R X E U:
t0
JðÞ : X E U ! R0 ;
ð14Þ
subject to M (t, xm, y, e, u). The minimal-complexity performanceseeking bounded tracking control law can be designed and reconfigured in near-real-time during operation of a physical system M () by using the experimental data. The model M() can be used in the preliminary design. For M (), we have
u ¼ /ðt; xm ; eÞ; u 2 U;
8xm 2 X;
8e 2 E;
8t 2 ½t 0 ; 1Þ;
ð15Þ
where /() is the continuous or piecewise-continuous real-analytic one-to-one function which is defined by the hardware bounds.
1. Closed-loop system M () evolves in (14), XE 2 Sxe; 2. Performance goals (16) and (17) are met. In (13) and (14), the performance integrands W() are positivedefinite. The proposed concept compliments the Lyapunov theory. Let the negative-definite dV(e, x)/dt is dV(e, x)/dt = W(xm, e, u) < 0. The positive-definite function V(e, x) is used. The feedback coefficients are derived by solving nonlinear equations or inequalities [14,15]. The positive-definite performance measure c > 0 depends on W(xm, e, u) > 0 and dV(e, x)/dt < 0. The positive-definite V(e, x) > 0 relevant to the performance functional J > 0.
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Example 3.1. Using the quadratic positive-definite integrands with positive-definite diagonal matrices Qx, Qe and G, R1 dV J ¼ minx2X;e2E;u2U 0 ðxT Q x x þ eT Q e e þ uT GuÞdt. Let ¼ dt T T T ðx Q x x þ e Q e e þ u GuÞ. We use u = /(xm, e), u 2 U. For a positive-definite function Vðe; xÞ ¼ xT K x x þ eT K e e, define nega_ From the tive-definite dV ¼ ðx_ T K x x þ xT K x x_ þ e_ T K e e þ eT K e eÞ. dt derived dV(e,x)/dt, we have xT Q x x þ eT Q e e þ /T ðxm ; eÞG/ðxm ; eÞ ¼ _ e = r y = r Cxm. The unknown x_ T K x x þ xT K x x_ þ e_ T K e e þ eT K e e, positive-definite matrices Kx and Ke, as well as the feedback coefficients in (19), can be found [15]. For varying rapidly-changing operating envelopes and scenarios, the control law (19) can be re-designed and reconfigured in near real time. The nonlinear optimization problem is solved by applying the specifications, requirements and capabilities as defined by hardware, performance functional and goals subject to the physical system M (t, xm, y, e, u). The feedback gains can be periodically defined. Control (19) can be reconfigured with the allowed rate depending on the algorithmic solutions and processing hardware. One can solve identification and optimization problems in near-real-time. By applying the identification methods [15], the parameter identification takes 0.03 s. The nonlinear optimization takes 0.02 s. By using the concept reported, the tracking control law (19) can be redesigned and reconfigured
within 0.05 s. The solution of the Riccati equations or Lyapunov inequalities requires 0.05 s. However, one designs the full-state control laws u = /(t, x, e) which can be implemented only by using additional sensors and observers. Due to the sensor and hardware complexity, many state variables are not measured. For nonlinear systems with rapidly-varying nonlinear maps, it is difficult to design nonlinear observers. Hence, the proposed optimal minimal-complexity performance-seeking control laws ensure the following advantages: (1) Technology compliance and practicality of optimal energy conversion in physical electromechanical and energy systems; (2) High performance and enabling capabilities, such as high efficiency, low losses, optimal electromagnetic loading, and fast dynamics; (3) Hardware and software simplicity and scalability; (4) Co-design consistency and overall design specificity; (5) Optimization and adaptive reconfiguration of energy management in near-real-time; (6) Consistent assessment of performance and capabilities; (7) Robustness and optimality using directly-measured variables; (8) Hardware, software, analytic, algorithmic and numeric advantages in design of complex energy management systems; (9) Design tractability and consistency using high-fidelity models and highly-descriptive physical energy/power/electromechanical systems.
Table 1 Permanent-magnet stepper motor parameters. Parameter
r
L
wm
Bm
J
Value
0.4–0.58 ohm
6.9 104 to 7.5 104 H
6.4 103 to 4.9 103 N m/A
9.2 104 N m s/rad
8.1 105 kg m2
Fig. 1. Schematic diagram and hardware of a servo system.
ua ia ub ua
ib
ub
Fig. 2. 1.8 full step motor operation with the assigned repositioning 200 steps per a revolution. The angular acceleration is 6.28 rad/s. The pulse-width modulated phase a ; u b Þ are applied ensuring r = [rh ra]T, rh = 0.0314 rad per step, ra = 6.28 rad/s. voltages ðua ; ub ; u
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The servo system schematics and hardware are reported in Fig. 1. The pointing system consists of microcontroller, power electronics, stepper motor and kinematics. The bidirectional time-varying loading TL(t) is. Using the reference r and the mechanical angular displacements hm (measured by the encoder), the controller develops the pulse-width modulated signals which drive high-frequency MOSFETs. The phase voltages are applied from a dual full-H-bridge topology MOSFET motor driver with the 500 kHz switching frequency, 97% efficiency, 50 V, 3 A rated and 6 A peak. The TI DRV8412 motor driver varies the average phase a ; u b ) which are applied to the motor windings. voltages (ua ; ub ; u The analog motor driver is controlled by a TMS320F28035 32-bit fixed-point microcontroller which guarantees an integrated
4. Experimental validation and substantiation Consider a servo for which high efficiency, high-precision tracking, accurate angular positioning and optimal electromagnetics must be guaranteed. Our goal is to achieve optimal energy conversion as well as up to 0.000245 rad repositioning which corresponds to 25,600 microsteps per revolution. We use a four-phase 8-lead P22NSXA Pacific Scientific permanent-magnet stepper motor with two 50-teeth-each rotor-stack, 1.8 full-step, 2.7 V (unipolar), 4.6 A (unipolar) and 1 N m (rated). A shaft-mount incremental encoder with resolution 30,000 cycles per revolution measures the displacement hm. The angular velocity xm can be estimated by counting the pulses within the sampling rate.
ia ib ua
θm
(b)
(a)
Fig. 3. (a) A pulse-width modulated phase voltage ua train. The varying-width pulses are controlled by a control function u (20) which varies the duty ratio of the output stage MOSFETs. The average phase voltages, applied to the motor windings, are controlled by changing the duty ratio. The soft switching and optimal electromagnetic loading are ensured by applying the sinusoidal, on average, phase voltages; (b) Phase currents and measured hm.
ia ua ub
ib
ua
θm
ub
a ; u b Þ are applied. The phase currents ia and ib in Fig. 4. Half step motor operation with the assigned 400 steps per a revolution repositioning. The PWM phase voltages ðua ; ub ; u the motor windings are reported. The variations of the angular displacement and microstepping hm are reported with the motor loadings ±0.5 N m: The angular displacements hm is measured by the incremental encoder.
ua
ia ub ua
ib
ub
Fig. 5. Quarter step motor operation with the assigned 800 steps per a revolution repositioning.
S.E. Lyshevski / Energy Conversion and Management 85 (2014) 245–253
251
ua ub
ia
ua ub
ib
(a)
ua ub ua
ia
ib
ub
(b)
ua ub ua
ia
ib
ub
(c)
ua ub ua
ia
ib
ub
(d)
ua ub ua
ia
ib
ub
(e) Fig. 6. (a) 1/8 stepping: 1600 steps per a revolution repositioning; (b) 1/16 stepping: 3200 steps per revolution; (c) 1/32 stepping: 6400 steps per a revolution repositioning; (d) 1/64 stepping: 12800 steps per a revolution; (e) 1/128 stepping: 25,600 steps per a revolution repositioning.
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S.E. Lyshevski / Energy Conversion and Management 85 (2014) 245–253
modular processing, control, interfacing, peripheral, programming and other capabilities. The motor parameters are found in the operating temperature region [20 120]C are given in Table 1. These parameters are bounded as aimin 6 ai()6aimax, and ai() 2 [aimin aimax]. The high-precision angular positioning and fast repositioning are required. The references are the angular displacement hm and angular acceleration am = dxm/dt. Thus, y = [hm am]T. The angular displacement hm is directly measured, while the angular accelera hm and reference tion am is estimated. For the output y ¼ am r eh r h ¼ h m . A r ¼ h , the tracking error vector is e ¼ ra ea ra am nonlinear control law is designed to guarantee full-step repositioning as well as microstepping. Using (19), we have
Z Z 1 u ¼ satu0M kp1 e5h þ kp2 eh þ kp3 ea þ ki1 eh dt þ ki2 ea dt ; 8 ua ¼ uua ðhm Þ; 8hm 2 HT > > > ub ¼ uub ðhm Þ; 8hm 2 HT > > : b ¼ uu b ðhm Þ; 8hm 2 HT u
ð20Þ
The proportional and integral feedback gains are found by solving the nonlinear optimization problem [15]. For the functional R1 2 2 J ¼ minx2X;e2E;u2U 0 ia þ i2a þ ib þ i2b þ 1 108 e2h þ 1 104 e2a þ u2a þ 2b Þdt, we specify the performance measure c 6 1 within 2a þ u2b þ u u 2p [rad] angular displacement during full-step and microspetting repositioning. Using the MATLAB environment, the nonlinear optimization problem (18) is solved. We have kp1 = 2.9 102, kp2 = 4.7 104, kp3 = 0.13, ki1 = 3.4 102 and ki2 = 8.1 103. The experimental results for the full-step repositionings are reported in Fig. 2. For each step r = [rh ra]T, rh = 0.0314 rad per step and ra = 2p = 6.26 rad/s. The efficiency is 67% at the rated load a and u b are applied as ±1 N m. The phase voltages ua, ub, u 17.5 kHz pulse-width-modulated pulses from the output stage with two H-bridge topology motor drivers. These pulse-width moda ; u b Þ, as well as the representative ulated phase voltages ðua ; ub ; u phase currents (ia, ib), are illustrated in Fig. 2. Implementing a balanced voltage set with a soft sinusoidal switching, we maximize the electromagnetic toque Te, minimize the torque and current ripples, maximize the efficiency, reduce losses and heat, optimize electromagnetic loading, ensure the soft power transistor switching, reduce noise and minimize vibration. The average sinusoidal phase voltages are applied to the phase windings to ensure a soft switching. A single pulse-width modulated ua train is illustrated in Fig. 3a. The control law has the angular acceleration feedback. The acceleration am is estimated by measuring hm. The reference full-step repositioning hm is reported in Fig. 3b. Performing prefiltering of hm by using the third-order notch filter, one finds am = d2hm/dt2. The closed-loop system output y = [yh ya]T tracks the reference vector r = [rh ra]T, rh = 0.0314 rad per step, ra = 6.26 rad/s. The tracking error e = [eh ea]T is bounded and converges to zero. Despite the rated bidirectional loading with TL = ± 1 N m, the specified accurate positioning hm and angular acceleration are guaranteed. The half and quarter step motor repositioning are studied. We apply the balanced phase voltages. The voltages and currents are illustrated in Figs. 4 and 5. For the half stepping, the specified displacements are achieved as illustrated in Fig. 4. The angular displacement hm is directly measured by the high-precision incremental encoder. A consistent motor operation, high-accuracy stepping and specified displacements are guaranteed despite the applied varying bidirectional load torque TL, TL 2 [0.25 0.75] N m. a ; u b Þ, defined Using the spatio-temporal phase voltages ðua ; ub ; u by the control law (20), the 1/8- and 1/16 microstepping
a ; u b Þ and curoperations are achieved. The phase voltages ðua ; ub ; u rents (ia, ib) are illustrated in Figs. 6a and b. We guarantee highaccuracy microstepping, including 1/32-, 1/64- and 1/128 stepping modes. The phase voltages and currents are illustrated in Fig. 6c–e. A consistent motor operation, high-accuracy microstepping and specified displacements are guaranteed with the designed control law despite the bidirectional load torque TL and disturbances. The proposed minimal-complexity control with spatio-temporal phase-voltage switching (20) guarantees high efficiency, optimal electromagnetics and high-precision microstepping in the expanded operating envelope XE with bidirectional loads. The motor dynamics is accomplished with minimum losses, vibration and noise due to a balanced voltage sets and soft switching. The disturbances are attenuated, and, high acceleration is achieved. High electromagnetic and holding torques are ensured with minimal torque and current ripples. Despite the rated load torque TL, there are no missed steps. The tracking errors eh and ea converge to zero. At TL = ± 1 N m, in the full- and half-step operation, the motor efficiency is 67% and 62%. The experimental results provide evidence of high motor efficiency with g1,1/2,1/4,1/8,1/16-step 2 [51 67]%, soft electromagnetic loading, fast and high-accuracy microstepping, low vibration and noise, etc. 5. Conclusions Electromagnetics and electromechanics of permanent-magnet stepper motors are examined to develop consistent control schemes which guarantee optimal and controlled energy conversion. Using the device physics, control schemes are derived to ensure high efficiency, optimal electromagnetics, stability, fast high-precision microstepping, disturbance rejection, optimal dynamics, fast repositioning, etc. Consistent electromagnetic, toque production and energy conversion analyses are performed. Optimal energy conversion is ensured. We consistently examined electromagnetics, torque production and motion dynamics. A nonlinear spatio-temporal motion control problem is solved. By applying the balanced displacement-dependent voltage sets, the electromagnetic torque was maximized while minimizing the ripple torque. Control schemes are verified and experimentally validated. The closed-loop system is robust to parameter variations, perturbations and loads. The experimental results substantiate the effectiveness and practicality of the reported design. This paper reports transformative findings in the design of optimal electromechanical systems. Our results ensured fundamental advancements, guaranteed engineering practicality and enabled technology transfer. Acknowledgements The author sincerely acknowledges the valuable comments and suggestions made by the reviewers. These cohesive reviews were very helpful and assisted the author. References [1] Krause P, Wasynczuk O. Electromechanical motion devices. New York: McGraw-Hill Book Company; 1989. [2] Lyshevski SE. Electromechanical systems and devices. Boca Raton, FL: CRC Press; 2008. [3] Wale JD, Pollock C. Hybrid stepping motors and drives. Power Electron J 2001;15(1):5–12. [4] Demirtas M, Karaoglan AD. Optimization of PI parameters for DSP-based permanent magnet brushless motor drive using response surface methodology. Energy Convers Manage 2012;56:104–11. [5] Hasanien HM. FPGA implementation of adaptive ANN controller for speed regulation of permanent magnet stepper motor drives. Energy Convers Manage 2011;52(2):1252–7. [6] Marino R, Peresada S, Tomei P. Nonlinear adaptive control of permanent magnet step motors. Automatica 1995;31:1595–604.
S.E. Lyshevski / Energy Conversion and Management 85 (2014) 245–253 [7] Speagle R, Dawson, DM. Adaptive tracking control of a permanent magnet stepper motor driving a mechanical load. In: Proc Southeastern Conference; 1993. [8] Zribi M, Chiasson J. Position control of a PM stepper motor by exact linearization. IEEE Trans Autom Control 1991;36(5):620–5. [9] Bodson M, Chiasson J, Novotnak RT, Rekowski RB. High performance nonlinear feedback control of a permanent magnet stepper motor. IEEE Trans Control Syst Technol 1993;1(1):5–13. [10] Kim W, Yang C, Chung CC. Design and implementation of simple field-oriented control for permanent magnet stepper motors without qd transformation. IEEE Trans Magn 2011;47(10):4231–4.
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[11] Krishamurthy P, Khorrami F. Robust adaptive voltage-fed permanent magnet step motor control without current measurements. IEEE Trans Control Syst Technol 2003;11(3):415–25. [12] Lyshevski SE. Motion control of electromechanical servo-devices with permanent-magnet stepper motors. Mechatronics 1997;7(6):521–36. [13] Kim W, Shin D, Chung CC. Microstepping with nonlinear torque modulation for permanent magnet stepper motors. IEEE Trans Control Syst Technol 2013;21(5):1971–9. [14] Khalil HK. Nonlinear systems. NJ: Prentice Hall; 1996. [15] Lyshevski SE. Control systems theory with engineering applications. Boston, MA: Birkhauser; 2001.
Contents lists available at ScienceDirect
Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman
Microstepping and high-performance control of permanent-magnet stepper motors Sergey Edward Lyshevski ⇑ Department of Electrical and Microelectronic Engineering, Rochester Institute of Technology, Rochester, NY 14623, USA
a r t i c l e
i n f o
Article history: Received 10 March 2014 Accepted 22 May 2014 Available online 14 June 2014 Keywords: Control Energy conversion Electric machine Optimality Stepper motor
a b s t r a c t We examine the problem of control of high-performance drives and servos with permanent-magnet stepper motors. Control of electromechanical systems implies control and optimization of electromechanical transductions and energy conversion. Robust spatio-temporal control algorithms are designed to ensure high efficiency, high-precision microstepping and optimal performance. The system stability, robustness and control design are examined applying an admissibility concept. Nonlinear control guarantees optimal energy conversion in expanded operating envelopes. Our analytic designs are substantiated and verified. A proof-of-concept system is tested and characterized. The high electromagnetic torque and highprecision microstep angular positioning simplify kinematics, enables efficiency, ensures direct-drive capabilities, reduces complexity, etc. For four-phase permanent magnet stepper motors, one may ensure up to 256 microsteps within a 1.8 full step. High efficiency and accurate 2.454 104 rad positioning (25,600 microsteps per revolution) are achieved with high electromagnetic and holding torques. To guarantee high efficiency, optimality and enabled energy conversion capabilities, electromechanical energy conversion and high electromagnetic torque are achieved by applying soft balanced phase voltages. The ripple and friction torques are minimized. The fundamental findings, technology-centric design and experimental results are reported. 2014 Elsevier Ltd. All rights reserved.
1. Introduction Fundamental, applied and practical problems of optimal energy conversion and precision motion control of electromechanical systems are of a great importance. The solution of the aforementioned problems allow one to maximize the efficiency, minimize losses, as well as deploy advanced motion platforms in aerospace, automotive, electronics, energy, manufacturing, power, robotics and other applications. Enabling technologies are under developments. Permanent-magnet stepper motors are widely used in drives and servos due to gearless direct-drive capabilities. Stepper motors are simple, affordable, rugged. Performance and capabilities may be enabled applying recent fundamental and technological advancements ensuring high torque and high power densities. The specified dynamics and high precision microstepping can be achieved developing high electromagnetic torque. Optimal electromagnetic and electromechanical designs, rear-earth permanent magnets and advanced materials enable energy conversion ⇑ Tel./fax: +1 5854754370. E-mail address: [email protected] URL: http://people.rit.edu/seleee/ http://dx.doi.org/10.1016/j.enconman.2014.05.078 0196-8904/ 2014 Elsevier Ltd. All rights reserved.
capabilities. The technology-enhanced stepper motors are controlled by advanced power electronics. Consistent control schemes ensure optimal energy conversion which results in enabled functionality and performance. Coherent device- and system-level solutions imply the use of advanced software, electronics and electromechanical hardware. New trends in motion control of drives and servos foster developments of nonlinear control algorithms consistent with the device electromagnetics and power electronics [1–3]. Sensorless and sensor-centric designs of servos with stepper motors are reported in [4–8] using the quadrature and direct voltages and currents. However, electric machines are controlled changing the phase voltages [1–3,9–12]. Practical concepts in motion control of permanent-magnet stepper motors were developed for full-, half- and quarter stepping. An optimal energy conversion and high-precision microstepping implies the use of advanced-technology motors, electronics and control schemes. Many studies concentrate on modeling [7,8]. The use of the arbitrary reference frame and quadrature-direct quantities serve as a possible inroad to stability analysis and design of feedback linearizing and vector controls [4–8]. While analytic results may be important, practical control schemes use the electromagnetically-consistent
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S.E. Lyshevski / Energy Conversion and Management 85 (2014) 245–253
machine variables [1–3,9–12]. The directly measured phase currents and voltages are used as physically-consistent controllable variables. Due to the algorithmic, software and hardware complexity, it is difficult to implement the spatio-temporal motion control using the mathematically-descriptive quadrature- and direct-components of voltages, fluxes and currents. Optimal energy conversion in stepper motors requires electromagnetically-consistent concepts. The use of adaptive, artificial neural networks, intelligent, sensorless and other control schemes may not ensure adequate energy conversion. Practical solutions require technology-centric fundamental research. These transformative findings result in new knowledge generation and developments which enable engineering and technology. 2. Electromagnetics and electromechanics of permanentmagnet stepper motors By using the laws of electromagnetics and electromechanics, the differential equations for two- and four-phase permanentmagnet stepper motors in the machine variables are derived [1,2]. For two-phase motor, the a and b flux linkages are wa = Lia + wam and wb = Lib + wbm. Assuming a sinusoidal uniform magnetic field of permanent magnets, one has wam = wmcos(RThm) and wbm = wmsin(RThm). We obtain [1,2]
dia 1 ¼ ½ria þ RTwm xm sinðRThm Þ þ ua ; L dt
i did 1 h r r ¼ rid þ LRTiq xm þ urd ; L dt r
i dxm 1 h r ¼ RTwm iq Bm xm T L : J dt
ð3Þ
2 r r r r2 For a positive-definite function V iq ; id ; xm ¼ 12 iq þ id þ x2m , the h dV ðirq ;ird ;xm Þ r2 r r r2 total derivative is ¼ 1L riq þ RTwmJðLJÞ iq xm þ iq urq rid þ dt r
id urd Bm x2m . Therefore, there exists a positive-definite Lyapunov dV ðirq ;ird ;xm Þ r r function V iq ; id ; xm > 0, for which < 0. That is, dV/dt is dt negative-definite for any variations of bounded motor parameters ai(), aimin 6 ai()6aimax, aimin > 0, aimax > 0. Hence, the open-loop permanent-magnet stepper motor is robustly asymptotically stable in the large. The high-fidelity mathematical models and expressions for the electromagnetic torque Te are derived by using a consistent expression for the flux linkages. A nonlinear electromagnetic system, nonuniform and nonstationary magnetic field of permanent magnets and other phenomena result in the equations for wa(hm) and wb(hm) as
wa wb
#
" ¼
L
0
0
L
#"
ia ib
#
3 2X 1 2n1 b cos ðRTh Þ n m 7 6 7 6 n¼1 7; þ wm 6 7 6X 1 5 4 2n1 bn sin ðRThm Þ
ð4Þ
n¼1
dxm 1 ¼ ½RTwm ðia sinðRThm Þ þ ib cosðRThm ÞÞ Bm xm T L ; J dt
ð1Þ
where ia and ib are the a and b phase currents; ua and ub are the applied phase voltages to the a and b windings; xm and hm are the mechanical angular velocity and rotor displacement; TL is the load torque; r and L are the resistance and self-inductance of the stator winding; RT is the number of rotor teeth per stack; wm is the amplitude of the flux linkages established by the permanent magnet; J is the equivalent moment of inertia; Bm is the viscous friction coefficient. The expression for the electromagnetic torque is
T e ¼ RTwm ½ia sinðRThm Þ þ ib cosðRThm Þ:
r
"
dib 1 ¼ ½rib RTwm xm cosðRThm Þ þ ub ; dt L
dhm ¼ xm ; dt
i diq 1 h r r ¼ riq RTwm xm LRTid xm þ urq ; L dt
ð2Þ
The stator resistance r, self-inductance L, amplitude the flux linkages wm, viscous friction coefficient Bm, and equivalent moment of inertia J vary due to varying temperature and loads. These variations are bounded, and, r 2 [rmin rmax], L 2 [Lmin Lmax], wm 2 [wmmin wmmax], Bm 2 [Bmmin Bmmax] and J 2 [Jmin Jmax]. These bounded parameters aimin 6 ai()6aimax are always positive, ai()>0, aimin > 0 and aimax > 0. To perform the stability analysis, we use the arbitrary reference frame which rotates at synchronous angular velocity [1,2]. The Park transformation yields the quadrature- and direct-axis compo r i sinðRThm Þ cosðRThm Þ ia nents of currents qr ¼ and volatcosðRThm Þ sinðRThm Þ ib id r uq sinðRThm Þ cosðRThm Þ ua ges ¼ . Using (1) and (2), one urd cosðRThm Þ sinðRThm Þ ub obtains [1,2]
where bn are the positive coefficients which describe the nonlinear electromagnetics. For example, for the ideal sinusoidal electromagnetic coupling, ensured by the Aerotech, Kollmorgen, Shinano Kenshi and other permanent-magnet synchronous machines, b1 = 1 and "bn = 0 for "n > 1 (n = 2, 3, 4, . . .). For high electromagnetic loadings, inadequate design and gaps (spacing) between magnets, "bn–0 with b1 > b2 > b3. . . The electromagnetic system, electromagnetic loadings and other effects affect Te. From (4), the expression for the electromagnetic torque is
" T e ¼ RTwm ia
1 X ð2n 1Þbn sinðRThm Þ cos2n2 ðRThm Þ n¼1
# 1 X 2n2 ðRThm Þ : þib ð2n 1Þbn cosðRThm Þ sin
ð5Þ
n¼1
Using the expressions for the electromagnetic torque (2) and (5), one finds the balanced current and voltage sets which maximize the electromagnetic torque Te and minimize the torque ripple. From (2), the balanced current and voltage sets are derived for: (i) Full step operation with the angular displacement 2p/RT, p/RT or p/2RT, which depends on the motor and windings designs; (ii) Continuous motor rotation. In particular, we have
ia ¼ iM sinðRThm Þ; ib ¼ iM cosðRThm Þ;
iMmin iM iMmax ;
ua ¼ uM sinðRThm Þ; ub ¼ uM cosðRThm Þ; uMmin uM uMmax ;
ð6Þ
where iM and uM are the rated current and voltage. The balanced current and voltage sets (6) suit the operation of stepper motors in the full operating envelope including high electromagnetic loadings. The derived expressions for Te yield the balanced fed phase currents and applied voltages which ensure the full-, half-, quarter- and microstepping [1,2,12,13]. Assuming 2p/RT displacement for a full step, using (2) and (5), one has
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S.E. Lyshevski / Energy Conversion and Management 85 (2014) 245–253
(
(
ia ¼ iM sgnðsin RThm Þ; 8hm 2 0 ib ¼ iM sgnðcos RThm Þ; 8h m 2 0
ua ¼ uM sgnðsin RThm Þ; 8hm 2 0 ub ¼ uM sgnðcos RThm Þ; 8h m 2 0
(
2p RT ; 2p RT
ia ¼
(
2p RT ; 2p RT
iM ; 8hm 2 0 p iM ; 8hm 2 RT
ua ¼
p
(
RT ; 2p RT
uM ; 8hm 2 0 p uM ; 8hm 2 RT
p
ib ¼
(
RT ; 2p RT
8h m 2 0 p iM ; 8hm 2 2RT
iM ;
ub ¼
p
8hm 2 0 p uM ; 8hm 2 2RT
uM ;
2RT ; 3p 2RT
8hm 2
p
2RT ; 3p 2RT
3p 2RT
8hm 2
2p RT
3p 2RT
;
2p RT
:
For the half- and quarter steps, the phase voltages ua and ub are
8 8 3p p 7p p > > < uM ; 8hm 2 8RT RT < uM ; 8hm 2 0 2RT ;8hm 2 4RT 5p 2p 3p 3p ua ¼ ; ub ¼ uM ; 8hm 2 4RT uM ; 8hm 2 4RT RT 2RT > > : : 0 otherwise 0 otherwise
For two- and four-phase stepper motors, one examines the electromagnetic torque Te. For the bipolar stepper motors with the phase currents (ia, ib) and voltages (ua, ub), or, for unipolar motor with a ; ub ; u b Þ, one obtains the resulting expressions ðia ; ia ; ib ; ib Þ and ðua ; u for full step, 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, 1/128 and 1/256 microstepping. Depending on the stepping mode, the phase currents and voltages depend on the rotor angular displacement hm. Therefore, a ; ub ; u b Þ, are time- and (ia, ib) and (ua, ub), or, ðia ; ia ; ib ; ib Þ and ðua ; u spatially-dependent. For two- and four-phase permanent magnet stepper motors, one obtains the spatio-temporal balanced current and voltage sets to guarantee optimal energy conversion and microstepping repositioning. We have
ia ¼ iM ua ðhm Þ; 8hm 2 HT ib ¼ iM ub ðhm Þ; 8hm 2 HT
;
ua ¼ uM ua ðhm Þ; 8hm 2 HT ub ¼ uM ub ðhm Þ; 8hm 2 HT
ð7Þ
and
8 ia ¼ iM ua ðhm Þ; > > > <i ¼ i u a M a ðhm Þ; > i ¼ i M ub ðhm Þ; b > > : b ðhm Þ; ib ¼ iM u
8hm 2 HT 8hm 2 HT ; 8hm 2 HT 8hm 2 HT
8 ua ¼ uM ua ðhm Þ; > > > u ¼ u M ub ðhm Þ; b > > : b ¼ uM u b ðhm Þ; u
8hm 2 HT 8hm 2 HT ; 8hm 2 HT 8hm 2 HT
ð8Þ
where the period is T 2 [0 2p/RT], T 2 [0 p/RT] or T 2 [0 p/2RT]. If bidirectional loads are applied, the closed-loop systems must be designed to prevent microstepping inadequacy and guarantee tracking. 3. Design of control schemes Permanent magnet stepper motors are open-loop stable. They ensure stable-equilibrium stepping within the open-loop configuration. However, missed steps, microstepping inconsistency, step mismatch and other inadequacies arise at bidirectional loading, high loads, considerable friction, variable inertia, fast repositioning, etc. To overcome the aforementioned drawbacks, closed-loop systems must be designed to ensure tracking of the angular velocity xm, displacement hm, etc. Using the reference vector r 2 R Rb , one examines the system output vector y 2 Y Rb . The tracking error e 2 E Rb is given as
eðtÞ ¼ rðtÞ yðtÞ:
ð9Þ
For drives y = xm with e = ex. For servos y = hm with e = eh. The angular acceleration am = dxm/dt is of importance for servos in addition to angular displacement hm. Hence, y = [hm am]T and
2p RT
8 8 p p p > > < uM ; 8hm 2 2RT RT < uM ; 8hm 2 0 2RT 3p 2p p 3p : and ua ¼ ; ub ¼ uM ; 8hm 2 RT uM ; 8hm 2 2RT RT 2RT > > : : 0 otherwise 0 otherwise
e = [eh ea]T. In high-performance servos, the rate of change of am is specified, and dam/dt can be used as an output. One has y = [hm am dam/dt]T, e = [eh ea eda/dt]T. It was found that the mathematical model is described by nonlinear differential equations (1) with states x = [ia ib xm hm]T, x 2 X Rn and controls u = [ia ib]T or u = [ua ub]T, u 2 U Rm . The mathematical model (1) can be refined using consistent expressions for the flux linkages, nonlinear magnetization, secondary effects, etc. The governing equations of motion are
dx=dt ¼ Fðx; u; rÞ þ Uðx; u; r; z; dÞ; y ¼ Cx;
ð10Þ
where x 2 X Rn , r 2 R Rb and y 2 Y Rb are the system state, reference and output vectors; u 2 U Rm is the control vector with the closed admissible set U ¼ fu 2 Rm ; umin 6 u 6 umax ; umax > 0; umin 6 0g; z 2 Z Rq and d 2 D Rs are the vector of time-varying bounded uncertainties, perturbations and disturbances; F(x, u, r) and U(x, u, r, z, d) are the nonlinear map as defined by consistent electromagnetics and electromechanics; C 2 Rbn is the output matrix. The differential equations (10) provide the model mapping M() which describes the physical system M (). The parameter variations, unmodeled dynamics and other uncertainties are bounded. For time-varying bounded uncertainties z 2 Z and disturbances d 2 D, there exists a norm of U(), and, ||U(x, u, r , z, d)|| 6 q(t, ||x||), where qðÞ : RP0 RP0 ! RP0 is the continuous Lebesgue measurable function. Our goal is to synthesize a control law using a consistent model M() or for a physical system M (), such that the error vector e(t) with E0 ¼ fe0 2 Rb g # E Rb evolves in the specified closed set [15]
Sxe ðdÞ ¼ fe 2 Rb ; x 2 Rn : e0 2 E0 ; x 2 XðX 0 ; U; R; Z; DÞ; x0 2 X 0 ; u 2 U; r 2 R; z 2 Z; d 2 D; y 2 Y; y 2 Y; t 2 ½t0 ; 1ÞjkxðtÞk qx ðt; kx0 kÞ þ qr ðkrkÞ þ qz ðkzkÞ þ qd ðkdkÞ þ d; keðtÞk qe ðt; ke0 kÞ þ qr ðkrkÞ þ qy ðkykÞ þ d; d 0; 8x 2 X;
8e 2 EðE0 ; R; YÞ; 8t 2 ½t 0 ; 1Þg Rb Rn :
ð11Þ
Here, qe ðÞ : RP0 RP0 ! RP0 and qx ðÞ : RP0 RP0 ! RP0 are the continuous strictly increasing functions with respect to t, and, monotonically decreasing with respect to the second argument, limt?/qj(t, ()) = 0; qj ðÞ : RP0 ! RP0 are the continous strictly increasing functions, qj(0) = 0.
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S.E. Lyshevski / Energy Conversion and Management 85 (2014) 245–253
Using the criteria imposed on the Lyapunov pair [2,14,15] for a positive-definite function V(e, x), one may define the set Sxe Rb Rn where the state x(t) and error e(t) vectors are bounded. For the system, which evolves in XE, using V(e, x), one has
Using the performance measure c 2 RP0 , c > 0, for the physical system M (xm, y, e, u) in the evolution envelope XE(X0, E0, U, R, Z, D), we define the performance goal
Sxe ¼ fe 2 Rb ; x 2 Rn jVðe; xÞ > 0; dVðe; xÞ=dt < 0g Rb Rn :
The desired performance is achieved if the closed-loop response triple (xm, e, u) lies in the set
ð12Þ The stability is guaranteed if XE Sxe. Here, XE is a subset of Sxe, but not equal to Sxe. The system evolves in an operating envelope XE(X0, E0, U, R, Z, D) with x 2 X, e 2 E, r 2 R, u 2 U, z 2 Z and d 2 D. Linear and nonlinear PID, vector, adaptive, model-based, relay, linearizing, sliding mode and other control laws are developed [2–12]. Some control schemes have a limited practicality due to abstract problem formulation and overall inadequateness. We focus on practical control concepts using the first principles of electromagnetics and electromechanics. This ensures overall optimal performance and achievable capabilities by means of optimal energy conversion. Robust performance-seeking control laws should ensure precise tracking, stability, minimum-time dynamics, disturbance attenuation, etc. Design and optimization are formulated as search problems in a high-dimensional space. The performance functional with a positive-definite integrand W(x, e, u)
J ¼ lim
min
Z
T!1x2X;e2E;u2U
T
Wðx; e; uÞdt;
Wðx; e; uÞ > 0;
t0
JðÞ : X E U ! R0
8 ua ¼ uua ðhm Þ; > > > < a ¼ uu a ðhm Þ; u > u ¼ u u > b b ðhm Þ; > : b ¼ uu b ðhm Þ; u
8h m 8h m 8h m 8h m
ð13Þ
2 HT 2 HT 2 HT
;
u¼
satu0M
M p ;Lp X k;l¼1
kpð2k1Þ;ð2l1Þ e proportional
2k1 2l1
ðtÞ þ
NX i ;M i ;Li j;k;l¼1
1. Mathematical conjectural modeling mapping M() obtained by using the first principles of electromagnetics and electromechanics; 2. Physical system M () with the state, output, error, control and other variables and quantities which are directly measured, asserted, examined and used. For physical systems, using the directly measured physical quantities xm and e, we minimize
min
T!1xm 2X;e2E;u2U
Z
8x 2 X;
8e 2 E;
8u 2 U on t 2 ½t0 ; t f Þ:
J S ðcÞ ¼ fðxm ; e; uÞ 2 X m E UjJðxm ; e; uÞ cg:
ð16Þ
ð17Þ
Using the performance specification set JS(c) and measurements set MS = {(xm, e, u) 2 Xm E U}, the control law (15) must be synthesized on a class of implementable, electromagnetic- and electromechanic-consistent algorithms CS. The optimal minimal-complexity performance-seeking bounded control and the corresponding performance measure are found by solving the nonlinear optimization problem [15]
cmin ¼
min ðcÞ
ð18Þ
;
JðÞ:XEU _ Subject to x¼Fðx;u;rÞþ Uðx;u;r;z;dÞ;u¼/ðt;xm ;eÞ Mðxm ;y;e;uÞ;C S 2C;C S \J S ðcÞ\MS
where C is the set of controllers. The analytically designed control law should ensure optimal closed-loop system performance and capabilities as specified by the performance functional (14) and measure c. We synthesize a bounded proportional-integral tracking control law, as well as a control law with the state feedback. For four-phase unipolar stepper motors, we have
Z
Z ... Ni
! ki
j;ð2k1Þ;ð2l1Þ integral
e
2k1 2l1
dt ;
ð19Þ
2 HT
forms a hypersurface. A control law u(t) must ensure a broad spectrum of requirements as given by the performance functional (13). We minimize (13) subject to the system dynamics. The modelcentric and physical designs imply the use of:
J ¼ lim
Jðxm ; e; uÞ 6 c;
T
Wðxm ; e; uÞdt;
Wðxm ; e; uÞ > 0;
uðtÞ ¼ satu0M
M p ;Lp X k;l¼1
NX i ;M i ;Li
þ
j;k;l¼1
2k1
kpð2k1Þ;ð2l1Þ e 2l1 ðtÞ
Z
proportional
Z ... Ni
! ki
2k1
j;ð2k1Þ;ð2l1Þ integral
e 2l1 dt þ Kx m
;
state feedback
where Mp, Lp, Mi, Li and Ni are the positive integers specified by the designer; kp(2k1),(2l1) and kij,(2k1),(2l1) are the proportional and integral feedback coefficients; K is the matrix of feedback coefficients. For a physical system M (), and its model M(), the bounded control (19) is the optimal tracking control law with respect to performance measure c if for (r, x, e, u) 2 R X E U:
t0
JðÞ : X E U ! R0 ;
ð14Þ
subject to M (t, xm, y, e, u). The minimal-complexity performanceseeking bounded tracking control law can be designed and reconfigured in near-real-time during operation of a physical system M () by using the experimental data. The model M() can be used in the preliminary design. For M (), we have
u ¼ /ðt; xm ; eÞ; u 2 U;
8xm 2 X;
8e 2 E;
8t 2 ½t 0 ; 1Þ;
ð15Þ
where /() is the continuous or piecewise-continuous real-analytic one-to-one function which is defined by the hardware bounds.
1. Closed-loop system M () evolves in (14), XE 2 Sxe; 2. Performance goals (16) and (17) are met. In (13) and (14), the performance integrands W() are positivedefinite. The proposed concept compliments the Lyapunov theory. Let the negative-definite dV(e, x)/dt is dV(e, x)/dt = W(xm, e, u) < 0. The positive-definite function V(e, x) is used. The feedback coefficients are derived by solving nonlinear equations or inequalities [14,15]. The positive-definite performance measure c > 0 depends on W(xm, e, u) > 0 and dV(e, x)/dt < 0. The positive-definite V(e, x) > 0 relevant to the performance functional J > 0.
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Example 3.1. Using the quadratic positive-definite integrands with positive-definite diagonal matrices Qx, Qe and G, R1 dV J ¼ minx2X;e2E;u2U 0 ðxT Q x x þ eT Q e e þ uT GuÞdt. Let ¼ dt T T T ðx Q x x þ e Q e e þ u GuÞ. We use u = /(xm, e), u 2 U. For a positive-definite function Vðe; xÞ ¼ xT K x x þ eT K e e, define nega_ From the tive-definite dV ¼ ðx_ T K x x þ xT K x x_ þ e_ T K e e þ eT K e eÞ. dt derived dV(e,x)/dt, we have xT Q x x þ eT Q e e þ /T ðxm ; eÞG/ðxm ; eÞ ¼ _ e = r y = r Cxm. The unknown x_ T K x x þ xT K x x_ þ e_ T K e e þ eT K e e, positive-definite matrices Kx and Ke, as well as the feedback coefficients in (19), can be found [15]. For varying rapidly-changing operating envelopes and scenarios, the control law (19) can be re-designed and reconfigured in near real time. The nonlinear optimization problem is solved by applying the specifications, requirements and capabilities as defined by hardware, performance functional and goals subject to the physical system M (t, xm, y, e, u). The feedback gains can be periodically defined. Control (19) can be reconfigured with the allowed rate depending on the algorithmic solutions and processing hardware. One can solve identification and optimization problems in near-real-time. By applying the identification methods [15], the parameter identification takes 0.03 s. The nonlinear optimization takes 0.02 s. By using the concept reported, the tracking control law (19) can be redesigned and reconfigured
within 0.05 s. The solution of the Riccati equations or Lyapunov inequalities requires 0.05 s. However, one designs the full-state control laws u = /(t, x, e) which can be implemented only by using additional sensors and observers. Due to the sensor and hardware complexity, many state variables are not measured. For nonlinear systems with rapidly-varying nonlinear maps, it is difficult to design nonlinear observers. Hence, the proposed optimal minimal-complexity performance-seeking control laws ensure the following advantages: (1) Technology compliance and practicality of optimal energy conversion in physical electromechanical and energy systems; (2) High performance and enabling capabilities, such as high efficiency, low losses, optimal electromagnetic loading, and fast dynamics; (3) Hardware and software simplicity and scalability; (4) Co-design consistency and overall design specificity; (5) Optimization and adaptive reconfiguration of energy management in near-real-time; (6) Consistent assessment of performance and capabilities; (7) Robustness and optimality using directly-measured variables; (8) Hardware, software, analytic, algorithmic and numeric advantages in design of complex energy management systems; (9) Design tractability and consistency using high-fidelity models and highly-descriptive physical energy/power/electromechanical systems.
Table 1 Permanent-magnet stepper motor parameters. Parameter
r
L
wm
Bm
J
Value
0.4–0.58 ohm
6.9 104 to 7.5 104 H
6.4 103 to 4.9 103 N m/A
9.2 104 N m s/rad
8.1 105 kg m2
Fig. 1. Schematic diagram and hardware of a servo system.
ua ia ub ua
ib
ub
Fig. 2. 1.8 full step motor operation with the assigned repositioning 200 steps per a revolution. The angular acceleration is 6.28 rad/s. The pulse-width modulated phase a ; u b Þ are applied ensuring r = [rh ra]T, rh = 0.0314 rad per step, ra = 6.28 rad/s. voltages ðua ; ub ; u
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The servo system schematics and hardware are reported in Fig. 1. The pointing system consists of microcontroller, power electronics, stepper motor and kinematics. The bidirectional time-varying loading TL(t) is. Using the reference r and the mechanical angular displacements hm (measured by the encoder), the controller develops the pulse-width modulated signals which drive high-frequency MOSFETs. The phase voltages are applied from a dual full-H-bridge topology MOSFET motor driver with the 500 kHz switching frequency, 97% efficiency, 50 V, 3 A rated and 6 A peak. The TI DRV8412 motor driver varies the average phase a ; u b ) which are applied to the motor windings. voltages (ua ; ub ; u The analog motor driver is controlled by a TMS320F28035 32-bit fixed-point microcontroller which guarantees an integrated
4. Experimental validation and substantiation Consider a servo for which high efficiency, high-precision tracking, accurate angular positioning and optimal electromagnetics must be guaranteed. Our goal is to achieve optimal energy conversion as well as up to 0.000245 rad repositioning which corresponds to 25,600 microsteps per revolution. We use a four-phase 8-lead P22NSXA Pacific Scientific permanent-magnet stepper motor with two 50-teeth-each rotor-stack, 1.8 full-step, 2.7 V (unipolar), 4.6 A (unipolar) and 1 N m (rated). A shaft-mount incremental encoder with resolution 30,000 cycles per revolution measures the displacement hm. The angular velocity xm can be estimated by counting the pulses within the sampling rate.
ia ib ua
θm
(b)
(a)
Fig. 3. (a) A pulse-width modulated phase voltage ua train. The varying-width pulses are controlled by a control function u (20) which varies the duty ratio of the output stage MOSFETs. The average phase voltages, applied to the motor windings, are controlled by changing the duty ratio. The soft switching and optimal electromagnetic loading are ensured by applying the sinusoidal, on average, phase voltages; (b) Phase currents and measured hm.
ia ua ub
ib
ua
θm
ub
a ; u b Þ are applied. The phase currents ia and ib in Fig. 4. Half step motor operation with the assigned 400 steps per a revolution repositioning. The PWM phase voltages ðua ; ub ; u the motor windings are reported. The variations of the angular displacement and microstepping hm are reported with the motor loadings ±0.5 N m: The angular displacements hm is measured by the incremental encoder.
ua
ia ub ua
ib
ub
Fig. 5. Quarter step motor operation with the assigned 800 steps per a revolution repositioning.
S.E. Lyshevski / Energy Conversion and Management 85 (2014) 245–253
251
ua ub
ia
ua ub
ib
(a)
ua ub ua
ia
ib
ub
(b)
ua ub ua
ia
ib
ub
(c)
ua ub ua
ia
ib
ub
(d)
ua ub ua
ia
ib
ub
(e) Fig. 6. (a) 1/8 stepping: 1600 steps per a revolution repositioning; (b) 1/16 stepping: 3200 steps per revolution; (c) 1/32 stepping: 6400 steps per a revolution repositioning; (d) 1/64 stepping: 12800 steps per a revolution; (e) 1/128 stepping: 25,600 steps per a revolution repositioning.
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modular processing, control, interfacing, peripheral, programming and other capabilities. The motor parameters are found in the operating temperature region [20 120]C are given in Table 1. These parameters are bounded as aimin 6 ai()6aimax, and ai() 2 [aimin aimax]. The high-precision angular positioning and fast repositioning are required. The references are the angular displacement hm and angular acceleration am = dxm/dt. Thus, y = [hm am]T. The angular displacement hm is directly measured, while the angular accelera hm and reference tion am is estimated. For the output y ¼ am r eh r h ¼ h m . A r ¼ h , the tracking error vector is e ¼ ra ea ra am nonlinear control law is designed to guarantee full-step repositioning as well as microstepping. Using (19), we have
Z Z 1 u ¼ satu0M kp1 e5h þ kp2 eh þ kp3 ea þ ki1 eh dt þ ki2 ea dt ; 8 ua ¼ uua ðhm Þ; 8hm 2 HT > > > ub ¼ uub ðhm Þ; 8hm 2 HT > > : b ¼ uu b ðhm Þ; 8hm 2 HT u
ð20Þ
The proportional and integral feedback gains are found by solving the nonlinear optimization problem [15]. For the functional R1 2 2 J ¼ minx2X;e2E;u2U 0 ia þ i2a þ ib þ i2b þ 1 108 e2h þ 1 104 e2a þ u2a þ 2b Þdt, we specify the performance measure c 6 1 within 2a þ u2b þ u u 2p [rad] angular displacement during full-step and microspetting repositioning. Using the MATLAB environment, the nonlinear optimization problem (18) is solved. We have kp1 = 2.9 102, kp2 = 4.7 104, kp3 = 0.13, ki1 = 3.4 102 and ki2 = 8.1 103. The experimental results for the full-step repositionings are reported in Fig. 2. For each step r = [rh ra]T, rh = 0.0314 rad per step and ra = 2p = 6.26 rad/s. The efficiency is 67% at the rated load a and u b are applied as ±1 N m. The phase voltages ua, ub, u 17.5 kHz pulse-width-modulated pulses from the output stage with two H-bridge topology motor drivers. These pulse-width moda ; u b Þ, as well as the representative ulated phase voltages ðua ; ub ; u phase currents (ia, ib), are illustrated in Fig. 2. Implementing a balanced voltage set with a soft sinusoidal switching, we maximize the electromagnetic toque Te, minimize the torque and current ripples, maximize the efficiency, reduce losses and heat, optimize electromagnetic loading, ensure the soft power transistor switching, reduce noise and minimize vibration. The average sinusoidal phase voltages are applied to the phase windings to ensure a soft switching. A single pulse-width modulated ua train is illustrated in Fig. 3a. The control law has the angular acceleration feedback. The acceleration am is estimated by measuring hm. The reference full-step repositioning hm is reported in Fig. 3b. Performing prefiltering of hm by using the third-order notch filter, one finds am = d2hm/dt2. The closed-loop system output y = [yh ya]T tracks the reference vector r = [rh ra]T, rh = 0.0314 rad per step, ra = 6.26 rad/s. The tracking error e = [eh ea]T is bounded and converges to zero. Despite the rated bidirectional loading with TL = ± 1 N m, the specified accurate positioning hm and angular acceleration are guaranteed. The half and quarter step motor repositioning are studied. We apply the balanced phase voltages. The voltages and currents are illustrated in Figs. 4 and 5. For the half stepping, the specified displacements are achieved as illustrated in Fig. 4. The angular displacement hm is directly measured by the high-precision incremental encoder. A consistent motor operation, high-accuracy stepping and specified displacements are guaranteed despite the applied varying bidirectional load torque TL, TL 2 [0.25 0.75] N m. a ; u b Þ, defined Using the spatio-temporal phase voltages ðua ; ub ; u by the control law (20), the 1/8- and 1/16 microstepping
a ; u b Þ and curoperations are achieved. The phase voltages ðua ; ub ; u rents (ia, ib) are illustrated in Figs. 6a and b. We guarantee highaccuracy microstepping, including 1/32-, 1/64- and 1/128 stepping modes. The phase voltages and currents are illustrated in Fig. 6c–e. A consistent motor operation, high-accuracy microstepping and specified displacements are guaranteed with the designed control law despite the bidirectional load torque TL and disturbances. The proposed minimal-complexity control with spatio-temporal phase-voltage switching (20) guarantees high efficiency, optimal electromagnetics and high-precision microstepping in the expanded operating envelope XE with bidirectional loads. The motor dynamics is accomplished with minimum losses, vibration and noise due to a balanced voltage sets and soft switching. The disturbances are attenuated, and, high acceleration is achieved. High electromagnetic and holding torques are ensured with minimal torque and current ripples. Despite the rated load torque TL, there are no missed steps. The tracking errors eh and ea converge to zero. At TL = ± 1 N m, in the full- and half-step operation, the motor efficiency is 67% and 62%. The experimental results provide evidence of high motor efficiency with g1,1/2,1/4,1/8,1/16-step 2 [51 67]%, soft electromagnetic loading, fast and high-accuracy microstepping, low vibration and noise, etc. 5. Conclusions Electromagnetics and electromechanics of permanent-magnet stepper motors are examined to develop consistent control schemes which guarantee optimal and controlled energy conversion. Using the device physics, control schemes are derived to ensure high efficiency, optimal electromagnetics, stability, fast high-precision microstepping, disturbance rejection, optimal dynamics, fast repositioning, etc. Consistent electromagnetic, toque production and energy conversion analyses are performed. Optimal energy conversion is ensured. We consistently examined electromagnetics, torque production and motion dynamics. A nonlinear spatio-temporal motion control problem is solved. By applying the balanced displacement-dependent voltage sets, the electromagnetic torque was maximized while minimizing the ripple torque. Control schemes are verified and experimentally validated. The closed-loop system is robust to parameter variations, perturbations and loads. The experimental results substantiate the effectiveness and practicality of the reported design. This paper reports transformative findings in the design of optimal electromechanical systems. Our results ensured fundamental advancements, guaranteed engineering practicality and enabled technology transfer. Acknowledgements The author sincerely acknowledges the valuable comments and suggestions made by the reviewers. These cohesive reviews were very helpful and assisted the author. References [1] Krause P, Wasynczuk O. Electromechanical motion devices. New York: McGraw-Hill Book Company; 1989. [2] Lyshevski SE. Electromechanical systems and devices. Boca Raton, FL: CRC Press; 2008. [3] Wale JD, Pollock C. Hybrid stepping motors and drives. Power Electron J 2001;15(1):5–12. [4] Demirtas M, Karaoglan AD. Optimization of PI parameters for DSP-based permanent magnet brushless motor drive using response surface methodology. Energy Convers Manage 2012;56:104–11. [5] Hasanien HM. FPGA implementation of adaptive ANN controller for speed regulation of permanent magnet stepper motor drives. Energy Convers Manage 2011;52(2):1252–7. [6] Marino R, Peresada S, Tomei P. Nonlinear adaptive control of permanent magnet step motors. Automatica 1995;31:1595–604.
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