optimization of speed control system for electrical drives with elastic coupling (deur 1998)

8/7/2019 Optimization of Speed Control System for Electrical Drives with Elastic Coupling (Deur 1998)

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Proceedings of the 1998 IEEEInternational Conference on Control ApplicationsTrieste, Italy 1-4 September 1998 WPOlOptimization of Speed Control System for E lectrical Drives with E lastic C oupling

Joiko Deur, Tomislav KolediC and Nedjeljko PeriCFaculty ofMechanica l Eng. and N av d Architecture,University of Zagreb, I. LuEiCa 5 , 10000 Zagreb, Croat ia

Siemens d.d., Heinzelova 70 a, 10000 Zagreb, CroatiaFacul ty of Electrical Engineering and Computing, University o f Zagreb, Un ska 3, 10000 Zagreb, Croat ia

E-mail: josko .deur@fsb .hr, nedj elj [email protected]

AbstractAn alytxal procedure of speed control systemoptimization for electrical dnves with elastic coupling ispresented. The optimization and comparative analysis ofthe control system are provided with the aid of thedamping optimum for four state controller types ofdifferent order, assuming a wide range of variation of

characteristic process parameters ratios. The behaviour ofthe optimized system with the respect to reference valueand disturbanc e has been examined by comp utersimulation and experimentally.1. Introduction

Controlled electrical drives are usually implementedusing a cascade control structure with a proportional-integral (PI) speed controller tuned according to thesymmetrical optimum [l]. However, it is generallyrecognized that electrical drives with considerabletransmission elasticity utilizing such a speed controlsystem tend to develop undesirable poorly dampedtorsional vibrations [2]. It is therefore important toresearch the possibility of torsional vibrations damping,while retaining the simple and widely used PI speedcontroller but modaing the control system optimizationprocedu re talung into accoun t the transmission elasticity.If well-damped response of an optimized speed controlsystem with a PI controller is impossible to obtain forgiven process parameters, the PI controller must beexpand ed by a ddin g stabilization feedback paths. If only asingle feedback path is introduced for the coupling torque[4] or the differential speed [5], the so-called PLn and PISOcontrollers are obtained. Both of these controller typesincluding the PI controller actually function as a reduced-order state controllers. The full-order state controller withtwo additional feedback paths for the load speed andtorsional angle provides additional capability 141.Introduction of additional feedback paths imposes the needfor additional sensors. However, this problem may beovercomed by using either the Luenberger state estimator[5 ] or a partial estimator of the coupling torque [4].Control system optimization is the next step followingselection of appropriate controller structure. The aim ofthe optimization procedure is to obtain well-dampedsystem response at least possible response time andefficient disturbance rejection (i.e. compensation of loadtorque effect). For this purpose, it is convenient to applythe simple and powerful analytical optimization methodutilizing the dam ping optimum, which has been originallydescribed in [6], modlfied for discrete-time control systemsin 271 and widely used in optimizing controlled electricaldrives [4].0-7803-4104-X/98/$10.0001998 IEEE

In h s contribution an analytical optimizationprocedure is presented and the respective behaviour of thecontrol system analyzed for four Me re n t state controllertypes, assum ing a wide range of variation of characteristicratios of process parameters. It is assumed tha t all the statevariables of the mechanical system are directlymeasurable. However, the optimization results beingreadily applicable to the case of state estimator ap plicationbecause of separation principle. The behaviour of theoptimized control system has been examined by computersimulation and ex yerimentally.2. Structure of control system

The considered digital speed control system (Fig. 1)has a cascade structure in which the speed controller setsthe reference value m lR for the inner torque (current)control loop [11.

UFig. 1. Block diagram of speed control systemProcessThe mechanical system of a large number of electricaldrives with significant transmission elasticity mayconveniently be described by an elastic two-mass system(Fig. 2) [ l , 31. All system quantities are referred to theshaft and n ormalized [4]. Friction and backlash effects areneglected.The tr ansfer h c t i o n of the mech anical system iswhere

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Because of the very small value of the damping ratios 6an d c2 (only a few percentages) [4, 81, additionalconjugate-co mplex pairs of poles and zeros occur in thetransfer function (1) of the elastic two-mass system incontrast to a stiff system. The presence of a weakly-damped pair of poles is manifested by the development ofweakly-damped torsional vibrations in the inadequatelycontrolled electrical drives. On the o ther hand, the weakly-damped pair of zeros exerts a compensating (stabilizing)effect. Since these zeros do not occur in the transferfunction between the motor torque ml and the load speedm,the control system is more convenient to implementwith a feedback path for the motor speed q [ 2 ] .

I II - 1

ml, m2q,wzm - coupling torqueAa - torsional angleA 0 = wl-@ - differential speedT M , T M ,c, d( T, = Is

- motor and load torque- motor and load speed

- motor and load mechanical time constant- stiffness and damping coefficient of transmission- base quantity for time)Fig. 2. Elastic two-mass system. Mechanical scheme (a) andblock diagram (b)

With respect to the design of the slower superiinposedspeed control system, the inner torque (current) controlloop of a DC or vector controlled AC drive can beapproximated by the equivalent lag term [I](7)

It is assumed that th e speed is recons tructed by time-differentiation of the measured position. The transferfunction of the measuring term is

where T is sampling time. The corresponding continuous-time transfer function is' (9)e -05Ts 1N z-w (s) 1 - C T SG ( s > = L = - -w ( s ) Ts Q 5Ts+1ni o

The series connection of the sampler and the zero-orderhold has the same transfer function G,(s)=G,,(s). For thepurpose of equivalent continuous-time control systemdesign, all the three process lag terms with their small(parasitic) time cons tants may be approximated by a singlesystem lag term [7]:

with the time constantTTx = T e j + 2 f - = T e i + T I2The dyn amic system behaviour is cha racterized by twodimens ionless coefficients [4]: the ratio of the mom ent ofinertia of the load to motor (the inertia ratio)

and the ratio of the mechanical systein resonancefrequency to the bandwidth of the system lag term (10)(the freque ncies ratio)r j = R,/T;' = T, R,

Controller

dFig. 3. Block diagrams offo ur speed controller types.PI (a) , PLn (b) , PLw(c) and state controller (4

Four differen t types of speed controller re presented byblock diagrams in Fig. 3 are discussed. An integral termensuring steady-state accuracy of the control system withrespect to load torque disturbanc e action is present in mainbranch of all controllers. One or several proportionalterms, depending on the number of measurable mechanicalsystem state variables, are presen t in the fe edback paths.If only the motor speed CO, is measurable, a PIcontroller is applied. After introducing an additionalfeedback path for the coupling torque in [4] or thedifferential speed Aw = CO, - [SI, the PIm and PIAC Ocontrollers are obtained. A state controller is obtained bymeasurement of the motor and load angle al and a2an dreconstruction of the state variables y, y andAa=al-a 2[4].PI, P I m and PIACOare actually reduced-order statecontrollers. The state controller may be re garded a s a full-order state controller since only the parasitic state of thesystem term (l o) remains uncontrolled.

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3. Optimization of control systemThe transfer function of the closed-loop system isderived from the block diagrams of the control system,mechanical system and controllers (Figs. 1-3) and thesimplified transfer function (10) of the system lag term:

Different equations are obtained for the characteristicpolynomial coefficients a,,...,a5 for different controllertypes. By equating the lower coefficients a,, ...,a*, withthe corresponding lower (dom inant) coefficients of thecharacteristic fifth-order polynomial of the dampingoptimum [6,4]:

+D3D:T3s3 + D 2 C 2 s 2+Q+1 (15)the equations for total Y controller parameters (Table 1)and equivalent time constant T, (Table 2) are obtained.The optimal controller parameters are calculated byinserting the optimal value 0.5 for the dominantcharacteristic ratios:

D2=...=Dy+,= 0.5 . (16)The following equations for the nondominantcharacteristic ratios are valid irrespective of controllertype:

Dividing (17) by ( IS ) yields the relation betweenequivalent and system time constant

Since the state controller has Y = 4 parameters, it canset all characteristic ratios (16) at the optimal value 0.5.The control system step response has a well-damped, so-called quasiaperiodic form with an overshoot of 6% andtime to first steady state value of 1.8Te.Th e PI controller has the least number of parametersand can therefore set the optimal value of only twodominant characteristic ratios D 2 an d D3. This w ill sufficeto retain a well-damped response provided that thefollowing two conditions are satisfied:

D4 < 1.4D40pt= 0.7 ; (20)D5-0.6-___A = D , D , > 1 ; (289(29)T , = 3,STI -4TI .

Equation (27) indicates that in the case of soft coupling theresponse time depends on the load resonance frequencyRo2,i.e. tends to increase with increasing inertia ratio YM(Fig. 5a). In the case of stiff coupling the expected result(29) is obtained in accordance with symmetrical optimum[11, which is a special case of the dampin g optimum .The area of favorable reference response damping isobtained in (rM, rEM)plane by the conditions (20) and(21), assuming that the critical value Am%is not exceededby more than 15% (Fig. 4). Weakly-damped (oscillating)response is obtained at low inertia ratios (approx. rM< 1)and high frequencies ratios (approx. rBM> 1) as evidencedby responses shown in Figs. 5a and 6a. At highfrequencies ratios rEbf,significant and almost undampedtorsional vibrations are observed, especially in the loadstep response.

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Table 1. Equationsfo r optinial parameters of different con troller types

I I h w controllcr Statc controller 1

Table 2.Equahonsf o r equivalent time constant T, of closed-loop systems utilizing different controlle r types

PIm controllerInserting (22 ) into the equation from Table 2yieldsD4D:D@i!2t2T,c3 - D 3 D 3 2 i 2 c 2+ 1= 0 .

Analysis of the cubic equation (30) provides the followingphysically ac ceptable solution(30 )

(31)P + X3r, = 2p cos(-) cpwhere ( D 2= D 3 = 0 . 5 ) :

43 0 , T,!2:p = (33 )

The solution is real if the following condition is satisfied

The complex so lution obtained for D4=0.5at higher valuesof the ratios rM an d rEbf may be overcom ed by delaying thesystem with decreasing characteristic ratio D 4 according to

D4= min(0.5,D,,,,) . ( 35 )In the case of soft coupling (rEM> a),D 4=D4- is valid which yields p = 0 and finally

Inserting (36) an d (22) into (19) yields after rearranging( 3 7 )

Equation ( 3 7 ) indicates that with increasing frequenciesratio rEM,especially at low inertia ratios rM , the well-damped response condition (21), and even the necessalysystem stability condition ( A


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The impaired control system stability resulting fromviolation of condition (21) may be reversed by delaying thesystem with increasing equivalent time constant accordingto (19):(38)T, -- 4TXr ,=

( ~ 5 ~ 4) m a QD , Amax (55where the estimated value of the nondominant ratio D5 iscomputed using the PI controller optimization procedure(with rM= 1 substituted for rM >1). According to (21), Amaxis bounded by 0.376 and 0.645, so that the following istrue

6.2T, < T , < 10.6T, . (39)Since the equivalent time constant T,no longer satisfiesequation (30), the integral time constant of the controller(unlike the gains KCo1an d Km ) is computed as follows(C=r2=0)

instead according to equation from Table 1.As shown in Fig. 4, the area of favorable referenceresponse damping is significantly extended in comp arisonwith the system utilizing the PI controller, resulting inwell-damped reference response even at higher frequenciesratios r E M 2 1 (Fig. 6b). Torsional v ibrations still occur atthe load step response but they have a significantly smalleramplitude and are more adequately damped than those ofthe PI controller-based system. The benefits of introducinga feedback path for the coupling torque are also reflectedin well-damped system response at low inertia ratios rbf(Fig. 5b), with the exception of a relatively narrow rangeof frequencies ratios around rE M = 0.5 (Fig. 4).

10 -+ PIm controller

0 1 110

Fig.4. Area of well-damped reference response of optimizedcontrol system (whole ( rM,rBM)plane for P I A0 an d state controller)

PIA^ controllertransformed into the biquadratic equationOn inserting (22), the equation from Table 2 is

D 4 D ~ D ~ Q @ ~ 2 T , 4- D2Qi2T,2+ 1= 0 . (41)with the following physically acceptable solution( 0 2 =D3=0.5):

(42)The solution is real if the following condition is satisfied

r lD4 < D J m a X= -.!.-l+r , (43)At higher inertia ratios r M , the value of the characteristicratio D4should be reduced according to (35) from 0.5 toDma.The equivalent time constant changes with variationsof the inertia ratio r M as follows:

4 for O T >---,no e - n o

Comparison of (44) and (27) indicates that at lower inertiaratios rM> 1 it provides faster response, asevidenced by the re sponse s shown in Fig. 5a-c.Since the equivalent time constant described by (42) isindependent of the system time constant T x, the systemstability is impaired at higher frequencies ratios rEM.Aswith the PIm controller, the problem may be solved byincreasing the equivale nt time constan t T, a ccordin g to(38). Althou gh the time constant T, no long er satisfiesequation (40), all the equations for controller parametersgiven in Table 1are still valid.As demonstrated by the responses shown in Figs. 5cand 6c, the optimized controlled system satisfies the well-damped response conditions (20) and (21) over the entireplane (rM,rEM)in Fig. 4. Inferior disturbance rejection athigher frequencies ratios 1) is its only disadvantagein comparison with the PIm controller.State controllerInserting (22) into the equation from Table 2 yields

T , =D5D4D3D2 D5D4 (45)

With optimal values of nondominant characteristic ratiosD4=Ds= 0 . 5 ,equation (45) gives T, = 16Tz.Since the equivalent time constant T,is independent ofthe resonance frequency of the mechanical system Ro, thecontrol system response time in the p resence of extremelysoft coupling (approx. rEM< 0.25) is less than that ofsystem containing PI, PIm or PIAOIcontrollers (compareequations (27) and (44)).On the other hand, in the case of median or stiffcoupling, the response time would be longer anddisturbance rejection considerably worse than thatprovided by a lower order controller (compare T, = 16Tzwith (39) and (29)). This problem may be solved byincreasing the nondominant ratio D5 to the value D5pIwhich is obtained by PI controller optimization. Since atlow inertia ratios rM the PI controller does not performproperly, at rM


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5. I,. .....0 2 ~1 rM = ...........0.3

0. 2

0.1

0.0

021T::iI,,,....,, .. ,. .0.1 t / s i0 0

0.0 0.2 0 4 0. 6 0.8 1 0 1 2 1 4 1.6 1 8 2.0 C

Fig 5. Responses o f optimized control system with respectto reference step (AWR= 0 .2 at t = 0 )and load step( A m 2 = 0. 1 at t Is) f or d ~ e r e n tcontroller types and

inertia ratios rM (rm = 0.3)

Amax(D,) ID, , for D5 2 0.6(48)D4= { O S , forD,


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PI controller State controllero.04h0.02 ' ,

0.063

t i s /-0.02 . . . . . . - . . . . . . . . . . , - . . . . . , I . m . . . . . . . . . - . . . . . . . . .0 . 0 0 ' 1 012 013 014 0,'s 0.'6bl0.06 1

. . . . . . . , . - . . . . ,. . - , . - - . . . . . . . , . . . _ . _0 0 0 1 0 2 0 3 0 4 0 5 0 6 cl

0 00 . .f I s /4 @ 2 - , , , . . . . . . - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0.0 o.'i 0.L 013 o.'4 0,'s 2 6 a20067

rM = 0.61 0 .@ 4 \0 .02

0 06

vEM = 0.34 0 0 20 000 02

0 0 0 1 0 2 0 3 0 4 0 5 0 6

Fig. 7. Experimental responses of optimized speed control systems with PI (al. bl, cl) and state (u2, b2, c2) controllers with respectto reference step ( A m R = -0.05 at t = 0 ) and load step ( b r , = 0.2 at t = 0.3 s )for die ren t inertia (rd and frequency (rEd ratios

5. ConclusionThe PI speed controller of electrical drives with elasticcoupling designed according to the damping optimumprovides well-damped reference step responses of thecontrol system only if the following two approximateconditions are satisfied: load moment of inertia exceedsmotor moment of inertia (inertia ratio greater than l) , andresonance frequency of the mechanical system is lowerthan the bandwidth of the system term (frequencies ratioless than I) .Significant improvement is obtained by expanding thePI controller with a feedback path for the coupling torque.Such a controller provides oscillating reference response ofthe control system only at frequencies ratio between 0.35and 0.65 at inertia ratio less than 1 . Well-dampedreference response independently of process parameterratios is obtained with a PI controller expanded with afeedback path for the differential speed. Inferiordisturbance rejection at frequencies ratio greater than 1 isits main disadvantage.The state controller combines the favorable propertiesof low order controllers, while providing faster responsesand efficient disturbance rejection in drives with extremelysoft coupling.At relatively high resonance frequency of themechanical system (frequencies ratio greater than 1)disturbance response oscillations are impossible to avoid

independently of the controller applied. Nevertheless,higher-order controllers generally tend to reduce theamplitude of oscillations and improve damping.

6. AcknowledgmentThe authors would like to thank to Ministry of Scienceand Technology and Ministry of Defense of Republic ofCroatia for financial support of this project.

7. ReferencesW. Leonhard, "Control of Electrical Drives",Springer Verlug, Berlin, 1985.D. Schroder, "Requirements in Motion ControlApplications", IFAC W orkshop ' M o t ion Con t rol fo rInel l igent Autom at ion ",Perugia, Italy, 1 992.R. Schonfeld, "Digitale Regelung elektrischerAntriebe", VEB Verlug Technik , Berlin, 1987.U . SchSer, "Entwicklung von nichtlinearenDrehzahl- und Lageregelungen zur Kompensationvon Columb-Reibung und Lose bei einein elektrischangetriebenen, elastischen Zweimassensystem",Disser tat ion, TU Miinchen, 1992.G. Weihrich, "Drehzahlregelung von Gleichstrom-antrieben unter Venvendung eines Zustands- undStorgrossen-Beobachters", Regelungstechnik , H. 11,P. Naslin, "Essentials of Optimal Control", I l G eBooks L td , London, 1968.H.-P. Trondle, "Anwenderorientierte Auslegung vonAbtastreglern nach der Methode derDoppelverhaltnisse", Regelungstechnik , M. 12. S.J. Deur, A. BoiiC, N. PeriC, "Laboratory model ofcontrolled electrical drive with elastic coupling,friction and backlash", accep t ed f o r 1Oth Int. ConJElectr ical Drives and Power Electronics , Dubrovnik,Croatia, 1998.

S. 349-380,H. 12 , S. 392-397, 1978.

384-391, 1978.

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optimization of speed control system for electrical drives with elastic coupling (deur 1998)

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