394 IEEE TRANSACTIONS ON AUTOXLITIC CONTROL, AUGUST 1969

determination of the appropriate Wiener filter. The analgticd results obtained for t.he general scalar problem indicate that there are rela- tively few instances when t.he Kalman filt.er provides a significant im- provement over the performance of t,he corresponding Wiener filter. This is an indication that the bounds provided should be calculated before a commitment to implement a Kalman filter is made. Since Wiener filters can often be approximated by simpler filters without substantially increasing the mean-square error, t.he resulk of this paper indicate when t,hese simpler filt.ers can be used to replace Kslman filters.

The authors would like to point out that the main t,hrust of this paper is not a criticism of the Kalman filter theory, but rather a criticism of its misuse. We have demonstrated quantitatively that the K h a n fiher will not generally give a substantially bet.ter perform- ance than the simpler time-invariant Wiener filter. However, the Kalman filt,er is applicable to time-varying systems where t.he Wiener theory cannot be used. Indeed, this time-varying sit.uation is essen- tially the proper domain of the Kalman t,heory. A major object,ive of the work reported here was to provide a stimulus for research con- cerned with using the Kalman theory as a basis for evaluating sub- opt,imal filter design in both timevarying and timeinvariant, sit.ua- tions. Some interesting results in this direction have recently been obtained by the second author and mill be reported shortly.

ACKKOWLEDGNENT Helpful discussions with Prof. G. F. Franklin are gratefully

acknowledged.

~ F E F L E X C E S

[ l ] R. E. Kalman “ A new approach to linear filtering and prediction problems. Trans. ASME, J. E a s i Engrg., ser. D, vol. 82, pp. 3545. March 1960.

[a] R. E. Kalmap and R. S. Bucy “New results in linear filtering and predie- tlon theory. Trans. ASME, j . Basic Engrg., ser. D. v 01.83, pp. 95-105,

[3] K. Wiener, The Extrapolation. Interpolation. and Smoothing of Stationary March 1961.

141 J. E. Potter, “hiatrk quadratic solutions,” S I A M J. Applied .Mathematics, Time Sertes. Kern York: Wiley, 1949.

[5] B. D. ‘O;,Anderson:-“An algebraic solut.ion to the spectral factorization vol. 14 pp. 49&501 May 1966.

problem. IEEE Trans. Automatic Conirol, vol. AC-12, pp. 410414.

[8 1 19 1

hugust 1967. H. W. Sorenson, “On the error behavior in linear minimum variance estimation problems,” IEEE Trans. Automatic Control. 1-01. XC12, pp. 557- 562 October 1967.

mith application to state estimation,” Stanford Electronics Laboratories, R. i . Singer, “The design and synthesis of linear multivariable spst.ems

Stanford Univers$y, Stanford, Calif., Tech. Rept.. TR-6302-8. June 1968. R. E. Kalman, Mathematical description of linear dvnamical systems.” S I A M J . Control ser. A vol. 1, no. 2 pp. 152-192, 1963. R. Zurmuhl, katr ieek und. ihre‘ technischen Anwendungen. Berlin:

[ lo] E. Isaacson and E. B. Keller, Analysis of X u m i c a l Methods. N e x York:

[ l l ] F. R. Gantmacher, The Theory of Xatrices, vol. 1. New York: Chelsea,

Springer, 1964.

Wiley, 1966, pp. 10-13.

14.50 [12] R.%ellman, Introdwtion to &fat& Analysis. New Pork: McGraxr~-Hill,

1960. (131 H. L. Van Trees, A. B. Baggeroer, and L. D. Collins. “The application of

state variable and optimal control techniques t o communication systems,” presented a t the 1968 W‘escon Conf., Los -4ngelw, Calii., August 20-23, 1968.

Mathematical Model of a Stepping Motor Operating as a Fine Positioner Around a Given Step

PI. A. DELGADO

Abstract-A h e positioning technique based on the small dis- placement that can be obtained around a stepping position by applying differential currents to the motor windings is discussed. A linear mathematical model of the motor operating in this fashion is derived from the analysis of the motor and coniirmed by a frequency response test of the motor.

Manuscripb received October: 24,1968; revised December 27. 1965. The author is with the Umvac Division of the Sperry Rand Corporation,

Blue Bell, Pa.

STEP - TO- STEP POSITIONING

j \ I I I I I I I

I STEP 1 STEP 2 STEP 3 STEP 4

STEPPING MOTOR I

POSIT!ONIKG LOAD , I

I i STEP -TO -STEP POSITIONING I I PLUS FINE POSlTlONlRG I

I I I I

I

I I I I

I

€ 1’: 0 €2 ‘3 €4

6, ; iSTEP POSlTlONl , - I OESIRED POSITION),

Fig. 1. Two posidoning problems. In upper case desired positions are evenly spaced; in IoTer case interspaces are not equal.

INTRODUCTION

There are two basic families of stepping motors: t.he mechanical detent and t.he magnetic detent [l], [2]. The mechanical detent types are primarily rotary solenoid-operated indexing devices. They have a mechanism that transforms dc pulses into discrete predet.ermined shaft angular displacements. At the end of each step, the shaft is secured by a mechanical detent..

The magnetic detent types are recognizable electric motors, usually ac motors specially adapted for dc service. In these, detenting is accomplished magnetically by the int.eraction of t,wo magnetic fields: a permanent one and a dc pulse excited one.

A very interest.ing stepping device of the magnetic detent t,ype is an adaptation of the ac synchronous induct.or motor. This motor has a permanent magnet rotor, and the stepping operation is accam- plihed by esciting hhe st.ator windings m-ith a sequence of square wave de pulses.

The particular device studied in this paper is a synchronous inductor mot.or [3], [4] (SLO-SYN HS50) in which 200 switching operat.ions or steps cause one revolut.ion of t.he rotor. As specified by the manufacturer, t.he motor has an accuracy of =!=3 percent of one step displacement, t.hat is, of 1.8 degrees, and the error is non- cumulative.

The digital driving sequence of t.he motor permits direct inter- face m-ith many types of digital control networks Kith a minimum of hardware [ 5 ] . This, together with ease of use, reliability, high values of driving and detent, (holding) torque, and great accuracy, makes it very attractive for a large number of open-loop positioning control systems. A typical application of t.his kind would be the posit.ioning of a load in discret,e, accurately interspaced, desired positions. However, there are many cases where high positioning accuracy is required along with unevenly spaced positions (Fig. l), when it. becomes necessary to use some form of h e podt.ioning after the load has been stepped close to t,he desired location in an open-loop fashion.

The purpose of this paper is to show a fine posit.ioning concept, and its application t.0 the synchronous inductor motor and to develop a linear mathematical model of the motor operat.ing in this mode.

STEPPISG OPERATION OF THE SYNCHRONOUS IKDUIXOR MOTOR

The charact,eristics of t,he synchronous inductor motor and t,he several excitation techniques for open- and closed-loop stepping have been described in the literature [3]-[7]. Here we are going to consider only a few fundamental det.ails regarding its design and basic st.epping operat.ion.

Fig. 2 is a cross section of the motor. The design has a two-phase st.ator punched in eight 5-tooth poles. There are four field windings for each phase.

SHORT PAPERS 395

Fig. 4. Connection for operation a3 phase-swit.ched stepping motor with single winding per phase.

1 ICY ROTATION ICCX R O T A T l O N l

Fig. 2. Cross section of sr.nchronous inductor motor. Flux path for one of t.he phases is indicated. Fig. 5. Connection for operation as phase-switched stepping

motor with bifilar windings.

I

Fig. 3 . 3 Replacement o f sinusoid by square wave sequence. A-phase -4;

circulation of current in a phase. B-phase B: signs + and - represent tr:o possible opposite directions of

The rot,or has t.wo isolated and identical 50-t.ooth discs mounted on a cylindlical permanent magnet., magnetized axiall>-, creating 0pposit.e polarities in the two discs. These t.wo disc.s are geomet.rically offset by one toot,h width. When operating on ac, t.his SO-tooth rotor design produces the same speed as a conventional synchronous machine with 100 poles. Therefore, {,he expression for the speed of the synchronous inductor motor is

s = - 60 f n

where n is the number of teet,h on the rotor, f is the ac supply fre- quency, and S is the rpm.

If the alternating excitation is replaced by a de excitation which is witthed in a +step sequence, the motor s6eps in increments of 1.8 degrees. In Fig. 3 t.he two-phase sinusoidal excit.ation is shown re- placed by an overlapping square wave excitation.

In t,he operation of the synchronous inductor motor as a stepping motor, detent,ing is accomplished magnetically by interact,ion of the constant magnetic fields creat.ed by the dc excited two-phase stator windings and the unidirectional flux of the permanent, magnet rotor. Fig. 4 shows the basic connection and the required switching cycle for forward and reverse operat.ion.

Most. types of electronic circuits employed for saitching are simplified by using a single ended power supply. Consequently, bifilar windings are needed for saitching, as shown in Fig. 5 . Thus,

each of the original windings has been center t,apped and the current is switched bet.n-een the t.wo legs.

Linear transfer functions for a single step have been developed [2], [6], [SI. Morreale [i] and Segov [9] have approached the study of the single step dynamics from a nonlinear point of view, and Segov has also extended his study to the derivation of a nonlinear model for the dynamics of a series of steps.

FINE POSITIONING M O D E

Let 11s first define a st,epping position as t.he particular magnetic null obtained experimentally by circulating nominal st,eady state currents through two given windings of the motor A 1 or A Z and B1 or E?. It was found that a very small displacement of t,his magnetic null around the stepping position can he obhined by applying differential current,s to t.he Pame two windings of the motor, t,hat is, by super- imposing +i, a small current increment. on the nominal steady state current in one of t.he windings, and -2 on t.he other winding. Reversing the different.ial currents reverses t.he direction of t,he displacement of the magnetic null. Let. us now define the desired position a5 the magnetic null --here t.he load is t.o be fine posit.ioned by the motor.

ANALOG MODEL

Fig. 6 is a n analog model of t.he synchronous inductor motor. For simplicity the size of t.he st.eps in t.he analog model is 90 degrees as compared to the 1.S degrees step size of the motor.

The variables indicated in Fig. 6 can be d e h e d as

e

I I1

I2

dl 4% dt dpm

instantaneous angular position of rotor with reference to the siepping position (radians) angular distance betaeen desired position of rot.or and the stepping position (radians) nominal steady st.ate current in windings (amperes) current in winding B1 (amperes) current in winding AI (amperes) vector flux creat.4 by current. I1 circulating in B1 (webers) vector flax creat.ed by current IZ circulat,ing in AI (webers) dl + 42 vector unidirectional flux due t.0 rot.or permanent magnetism (webers).

The vector dPm always tends t.0 align itself winth vectm dt, ac- complishing the alignment. in the steady st.ate. When currents are balanced, that is, when I1 = I2 = I , t.he direction of dt coincides wit,h t.he Gdegree line. In these condit.ions, the rotor is locked on the stepping position.

396 IEEE TR~XSACTIONS ON AUTOMATIC CONTROL, AUGUST 1969

IYS'AKTA#EOUS POSITION

STEPPING ?OSITIOH

DESIRED POSlTlOW

M i h.'

Fig. 6. Analog model of motor.

If two differential currents +i and --i are superimposed upon current I in windings I31 and d l , respectively, t.hen

and vector $ 1 is displaced to the magnetic null which is deviated tJd radians from the stepping position. The rotor follows this displace- ment set.t.ling on the new magnetic null. If t.he differential currents are reversed: the direction of displacement of the rotor is reversed with respect to the stepping position.

THEORETICAL TRASSFER FUNCTION

The variables and constants involved in the equat,ions that describe the operation of the synchronous inductor motor in t.he fine positioning mode are d e h e d as

Laplace t.ransform of t.he differential voltage applied to one winding of the motor (volts) Laplace transform of the back E M F voltage developed in one winding of the motor (volts) 4s) - UdS) Laplace transform of the differential current in one minding of the motor (amperes) Laplace transform of t.he instantaneous angular position of the motor shaft with reference to the stepping posit.ion (radians) Laplace t.ransform of t.he angular distance betaeen t.he desired position of the motor shaft wit.h reference to t,he stepping position (radians)

Laplace transform of the torque (ounce-inches) Laplace transform variable magnetic null displacement const,ant (radians per ampere) Back E M F constant (volts per radian per second) restoring torque constant (ounce-inches per radian) t,otal load moment, of inertia, referred to motor shaft (ounce-inch-seconds2) resistance of one winding (ohms) inductance of one minding (henries) LIR, time constant of winding (seconds).

W S ) - e(s)

" ' " ' 1 , , ~ t Fig. i . Synchronous inductor motor operating in fine positioning mode.

In the folloxing equations, the initial conditions are assumed to be zero. The Laplace transform of t.he voltage in one of the motor windings is

V A S ) = d s ) + V A S ) (3 )

where

ff(s) = ( E + sL)i(s)

ve(s ) = K&(s).

The s m d angular displacement tJd of the magnetic null necessary to go from t.he stepping posit,ion to the desired position is obtained by applying differential currents +i and --i to windings AI and B,, respectively. The relationship between ea and i was experimentally proven to be linear

8 d ( S ) = Ri*i(S) . (6 1

The rotor, when separated by an amount ea from its rest position, manifests B rest.oring torque which tends to bring it.to a rest on the new magnetic null when d8erent.ial currents are applied to the windings. The restoring torque is roughly sinusoidal, but in the region of interest it can be approximat,ed by a straight. line

These equations define the block diagram of Fig. 5. The theoret- ical transfer function is

The following numerical values of the constants involved in this transfer funct.ion were determined experimentally

K, = 4880 ounce-inches per radian Ki = 0.00837 radians per ampere K, = 0.21 volt per radian per second T~ = 0.000909 second J = 0.01326 ounce-inch-seconds2 R = 5.7 ohms.

The constant K , was found by measuring the torque developed when t.he rotor was slightly displaced from the stepping posit.ion. This stepping position was obtained by circulating a dc current of 2 amperes in the two motor windings. Fig. 8 shows the restoring torque curve for one step and Fig. 9 for four f u l l steps.

The constant Ki was determined by measuring t,he angular dis- placement t?d obtained when differentid currents +i and -i were circulated in the motor windings, and the constant. K, was computed by measuring the value of t,he back E M F voltage at Merent. rotor speeds. This back E M F voltage was defined by the crossing point, on the screen of an oscilloscope, of the two volt.age sine waves obtained from the motor when operated as a generat.or. Inserting these values into the transfer function we obtain

e(s) 0.00146 _ - - Y,(s) 2469.10-l'~~ 4- 2717.10-9~2 + 1 2 1 7 . 1 0 3 + 1 . (9)

SHORT PAPERS 397

4 TORQUE -ANGULAR DEVlATlON CHARACTERSTICS

x WL-

#

2

0.00 001 0.02 r - STEP

8, ANGULAR DEVIATION (RADIAN1

Fig. 8. Resoring torque versus angular deviation characteristics for different. values of dc current in rrindings of motor.

TORQUE-ANGULAR DEVIATION CHARACTERISTIC

I v ::{ e C U R R E N T DIFFERENTIAL PER CURRENT WINDING i =CONSTANT = 2 AMPERE = 0

5 80 x

8, ANGULAR DEVlATlON IRAOIANI

Fig. 9. Restoring torque versus angular Characteristics of motor for I = 2 amperes. Four full steps are shown.

-750L-

PHASE PLOT

LOG MAGNITUDE PLOT

,.-40dB/DECADE

io --go : --.

LT c3 W n

- -180 E

4 I n

- -270

I,-60dBIDECAr

100 w LRADIANISECOND)

1000 r0,000

-360 I I I , I , I , , 8 I , . I ,

Fig. 10. Frequency response (Bode plot) of motor operating in s e n 0 mode.

MOTOR FREQUENCY RESPOKSE The frequency response of the motor (Fig. 10) shows a resonant

frequency at 6iO radians per second with a damping ratio E = 0.1. It also shows a break frequency from -40 to -60 decibels per decade at. approximately 1000 radians per second due to t.he motor windings, and a steady state gain of 0.00146. The exyenmental transfer function was synthesized from this frequency response test as

The theoretical t,ransfer funct.ion of the synchronous inductor motor, operat.ing in fine positioning mode, developed in this paper agrees very accurately with the experimental transfer function obtained from a frequency response test of the motor. Therefore, it can be concluded that the theoretical t.ransfer function is an accurate representat.ion of the dynamics of the motor in t,he fine positioning mode.

The next step in the study of the synchronous inductor motor and its digital posit.ioning applicat,ions is the development of a nonlinear mathemat,ical model for the stepping opemtion using computer simulation. The int.emelationship between this stepping model and the fine positioning model offered here could also be considered in order to study the coarsefine cha.ngeover.

ACKNOWLEDGNEST The author gratefully acknowledges the assistance of several of

his colleagues at. Univac, and, in particular, that, of T. J. B. Han- nom and R. Tickell.

REFERENCES [I] J. Proctor, “Stepping motors moPe in.” Prod. Engrg., 1.01. 34, pp. 74-78. Feb-

121 8. J.- Bailey, “Incremental servos: pt. b-operation and analysis,” Control

I31 N. L. Morgan, “Versatile inductor motor for industrial control problems,”

ruarv 1963.

Engrg., rol. 7, pp, 97-102, December 1960.

Plant Etwro.. vol. 16. DD. 143-146. dnnn 18fi2. I41 +. E. Sno;v’don,,and*E. El. kladsin; “Characteristics of a synchronous

Inductor motor. AIBE Trans. (Applications and Industry), 001. 81, pp. 1-5. March lM2. [51 T..R:,..~-~.- ~

redriksen, “Nen dere1opment.s and applications of the closer-loop

161 R. €3. Kieburtz. “Step motor-t,he next advance in cont.ro1 systems,” I E E B

[7] A. 0. Morreale, Theory and operation of stepsen-o motors,” El%. Design

steppmg motor,” Preprints, 1966 JACC (Seattle, Wash.). pp. 7582.

Trans. AutorndiqdControl. 1701. AC-9, pp. 98-104, January 1964.

News. Julv 1963. [8] J. P. O’Dinobue, “Transfer function for a stepper motor,” Conirol Engrg.,

pp. 103-104. NFvember 1961. 191 B. 0. Segov, Dinamiks sistem czifrovoho prohramnoho upravilinia z kro-

motor).” Automatiunanni Electroprimd Ukr. Acad. SCI.. no. 4, pp. 38-61, kovim dvihunom (Dynamics of digital routine contro1,systems with a step

1959.

Finite Differences to Implement the Solution for Optimal Control of Distributed Parameter Systems

GEORGE L. K(UsIC, JR., NEMBER, IEEE

Abstract-Finite difference methods are used to implement the solution to optimal control of distributed parameter systems. The control is assumed to be intrinsic to the partial differential equation (PDE) as well as continuous, such that the calculus of variations is used to obtain the control law. Several important principles are developed to formulate the difference approximations to the partial differential equations which describe the system and the control law. An iterative method of solution is employed on these two equations. The convergence of the iteration is assured by stability considerations of the finite difference expressions.

I. PROBLEM fk4TEMENT

Consider the distributed parameter system described by the nonconstant coefficient, linear partial differential equation (PDE) in two independent var iabh

CONCLUSION where ai’ are nonconstant coefficient operators such that f i l are A synchronous inductor mot.or operating as a pha5e-saitched cont,inuous on the domain of the problem R: { 0 5 t 5 T, 0 X 11,

stepper has a wide variet.y of open-loop digital positioning applica- men the requires fine positioning about a given Manuscript received January 31, 1968; revised July 1,1968, and Hovember2g,

The author is with the Department oi Electrical Engineering, University of the principle of displacement, of the rot,or by application of diffe- 1968. rential currents to the st.at,or windings can be employed snccessfllly. Pittsburgh, Pittsburgh, pa.