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d control methods for

servo drives. Following from the structure (PF-type) of model r

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op with changing gain (moment of

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hootless as well as fast speed changing

compensation caused by jump in load. The algorithm even keeps i

ts stability at fast changing, jump-

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ed to provide constant loop gain in

speed control loop with changing parameters exposed to a signif

icant load. The approaching block

diagram of the adaptive control

can be seen as an extended vers

ion of the PF controller, so one of the

adaptation factors (which is the free parameter of the adaptive

control) is given. Both model

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do not need acceleration measuring

(thanks to the first-order m

odel). Simulation and experimental

results demonstrate that the proposed methods are promising too

ls to speed control of electrical

drives

Számel, László

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INVESTIGATION OF ADAPTIVE PDF SPEED CONTROL FOR MOTION CONTROL    Laszlo Szamel Budapest University of Technology and Economics Department of Electric Power Engineering Egry József u. 18, H-1111-Budapest (Hungary) E-mail: szamel.laszlo@vet.bme.hu   Abstract. This paper proposes two model reference adaptive PF (PDF) speed control methods for servo drives. Following from the structure (PF-type) of model reference parameter adaptive control was developed to provide constant loop gain in speed control loop with changing gain (moment of inertia and/or torque factor) it makes it easier to reach overshootless as well as fast speed changing compensation caused by jump in load. The algorithm even keeps its stability at fast changing, jump-like load torque. Model reference signal adaptive control is used to provide constant loop gain in speed control loop with changing parameters exposed to a significant load. The approaching block diagram of the adaptive control can be seen as an extended version of the PF controller, so one of the adaptation factors (which is the free parameter of the adaptive control) is given. Both model reference adaptive controls drawn up can be easily implemented because the adaptation algorithms do not need acceleration measuring (thanks to the first-order model). Simulation and experimental results demonstrate that the proposed methods are promising tools to speed control of electrical drives.    Keywords. Adaptive control, Adjustable speed drives, Motion control, Pseudo-derivative feedback, Servo drives, Switched reluctance drives, Variable speed drives.     1.  INTRODUCTION   Generally the controller of the speed loop in motion control systems is of PI-type. The integrator of the controller eliminates the error caused by the step change in load. The setting of speed controller is difficult as the closed-loop transfer functions are not identical to step changes in the load, as well as the reference signal. It seems to be preferable to use PF-type controller (proportional gain in a separate feedback) instead of the traditional PI one. It means relatively simple solution [2], [3], [4]. Phelan named this structure “Pseudo-derivative feedback” (PDF) control [5].  In motion control systems there is robustness against parameter changes and disturbance rejection of main interest. The model reference adaptive control has the following features:   • It enables the compliance of the system with varying operational conditions possible and ensures the behavior of the controlled system according to the prescribed reference model. • It means such a special type of adaptive systems which results in nonlinear control systems. This is the reason why the analytical analysis is completed by the Lyapunov stability criterion or by hyper-stability principle. • Its design and application are closely related to the using of computer methods. • Simple implementation of the control algorithm.   In this paper the application of two model reference adaptive PF control methods to servo drives with switched reluctance motors are presented.   2.  SPEED CONTROL OF SERVO DRIVES  A model reference adaptive control is used for the speed control. Such an adaptive control has been succesfully elaborated by using a suitable chosen Lyapunov function to compensate the gain of the speed control loop [6], [7].  Model reference parameter adaptive control  The adaptive control of servo-drives with a cascade arrangement is most effective when it is applied in the inner loop containing the effect of variable parameters directly, i.e. the inertia ( mJ ) and/or torque factor ( mk ). The speed control implemented by PI controller is of cascade arrangement in fact as it contains an inner, proportional feedback loop (PF controller, [4], Fig.1.). A one-storage proportional element can describe this inner loop neglecting the time constant of closed current control loop. By this our adaptation algorithm is simplest.   Relation between accelerating current determining dynamic torque and angular velocity can be given by the following transfer function:   sAsY iii t =− )()(,ω  , (1) wheremmi JkA = . Arrangement of control circuit can be seen in the following figure:               il     ωr             ωri     i   ω sAi  siK  pK  pK   Fig. 1. Block scheme of parameter adaptive PF speed control  where  ω  is the speed,  rω  is the speed reference signal,  i  is the current of the motor,  li  is the current equivalent to the load-torque.  Section determined by the transfer function )()(, sY tii−ω  is fed back by a proportional member of gain PK . The task is to change the gain PK  in such a way that the product Pi KA  should remain constant despite the changing of parameter iA .  Transfer factor of the inner closed loop is given by the reciprocal ( PK/1 ) of feedback member that is not constant because of the change in torque factor. Consequently loop gain of the outer speed control loop would change as well. In order to get a one-storage element with unity transfer factor we have to insert a member with gain PK  between the integrator of the PF controller and the reference signal of the inner loop. First-order reference model with time constant mT  gets sum of the input signals of above member ( riω ) and the signal lmω  compensating the load effect for the model. So dynamics of reference model can be described by the following differential equation:   lmrimmmT ωωωω +=+& . (2)  Dividing (2) with mT  and applying designation mm Tq /1= , we get the following equation.   )( lmrimmmm qq ωωωω +=+&  (3)  The differential equation of the first-order controlled section is as follows:   liriipip iAAKAK −=+ ωωω  )( )(& . (4)  Factor pK  can be described as the sum of p0K  determined by mean iA  and pK∆  accomplished by the adaptation algorithm. So:  qqAKKAK ipp0ip ∆+=∆+=  )( , (5)  where P0K , and q  are constant. In this case we assume that the change of iA  is slow from the viewpoint of adaptation and therefore the effect of this change can be neglected. Substituting (5) into (4) we get:   liri iAqqqq −∆+=∆++ ωωω  )( )(& . (6)  By using (3) and (6) and substituting expression of model error ωωε −= m  the dynamic equation will be:   lilmmrim iAqxxq ++−+−= ωωωεε   & , (7)  where mqqqx −∆+= )( . Dynamic of model error should by asymptotically stable to follow the system with model proposed. For determination of q∆  the following Lyapunov function should be composed:   ( )22  21 xV βε += , (8)  where β  is a positive number.  When choosing the Lyapunov function both purposes, i.e. the termination of the model error ( ωωε −= m ) and loop gain deviation have been taken account. The time derivative of the Lyapunov function is:   xxV &&&     βεε += . (9)  Substituting (7) into (9) the following equation is valid:   .   )(     2xxiAqxxqVlilmmrim&&βωεωεωεε+++−+−= (10)  If   0      =+− xxxx ri &βωεωε , (11)  that is   βωωε /)( −= rix&  (12)  and   0)( <+ lilmm iAq ωε , (13)  then   2εmqV −<& . (14)  The above equation is a negative definite function that shows the asymptotic stability of the error dynamic (7). By using (5), (7) and (12) the following adaptation algorithm is true:  )(  ωωεγ −=∆ ripK& , (15)  where γ  may be an arbitrary positive number. The inequality (13) shows how we have to change the signal lmω  representing the load of model. If   millm TAi max    then   ,0 −<> ωε , (16)  respectively if   millm TAi max    then   ,0 −>< ωε .  Model reference signal adaptive control The controlled loop has been approximated by an integral element. Time constant of the closed current control loop has been neglected. The control consists of a P-element with the gain pK . Input of P-element contains not only the control error signal but an adaptation signal as well (g). Applying the signal adaptation control, a P type controller with pK  gain can ensure zero speed error as the adaptation signal can produce a current reference signal to compensate the loading current at zero speed error.     g      il   ωr        i    ω     pK  sAi   Fig. 2. Initial block scheme of signal adaptive speed control  Regarding the block diagram of control loop following differential equation is valid for the closed loop:   ( ) .  lirpipi iAgKAKA −+=+ ωωω&  (17)  The feature of the closed speed control loop has been taken into consideration by a parallel control model to be expressed by a first order proportional element. The differential equation of the first order system is:   rmmmm qq ωωω =+& , (18)  where index m refers to the model and mq is the reciprocal of model time constant.  Using (17) and (18) and introducing the expression ωωε −= m for model error, the dynamic equation for the model error is as follows:   ( )( ) ( )gKiAKAqq plirpimm −+−−=+ ωωεε  & . (19)  The adaptation signal g(t) can be written in the following form:   ( ) )(   (t))( 21 tggtg r +−= ωω . (20)  Substituting (20) for (19):   ( ) 21 bbq rm +−+−= ωωεε& , (21)  where   ( ))(1 11 tgKAqb pim +−= ,  ( ))(22 tgKiAb pli −= .  Let us compose the following Lyapunov function to produce the signal )(1 tg  and )(2 tg :   ( )2222112 2121 bbV ββε ++= , (22) where 1β  and 2β  are positive constants. Time-derivation of the Lyapunov function is:   222111 bbbbV &&&& ββεε ++= . (23)  Substituting (21) for (23):   ( ).         222111212bbbbbbqV rm&&&ββεεωωε+++−+−= (24)  If   ( ) 11 / βεωω −−= rb& , (25)  and   22 / βε−=b& ,  then   2εmqV −=& . (26)  and it ensures asymptotical stability of the model error. On the basis of (21), (25) and by assuming that variation of iA  can be neglected compared to the speed of adaptation the following adaptation algorithm is valid:   ( )  )( 11 ωωεγ −= rtg& ,   )( 22 εγ=tg& , (27)  where 1γ  and 2γ are positive constants, the free parameters of adaptation. Taking relations (20), (27) into consideration the following equation comes true:   ( ) ( ) ∫∫ +−−= dtdttg rr      )( 21 εγωωεωωγ . (28)  A block diagram of the control circuit introducing adaptation signal g(t) furthermore g1(t) = const. can be seen in Fig. 3. Taking the structure of control: it contains two parts. In the first part the reference signal is led through a first order system and a PI controller with variable gain and integration time. The second one is a differentiating filter which takes effect only on changing of reference signal. The gain and differentiation time are also changing. The adaptation gain factor 2γ  gives the reciprocal of integrating time constant of controller type PI, assuming g1(t) = 0. To fulfill the constant integrating time constant it is preferable to substitute 2γ  by ( ))(1 12 tg+γ . In such a way the neglect of time constant of current control loop can be compensated.                     il  ωr    ε     i   ω       sAi  sp2Kγ  ( )1p g1K +  ( )1pm g1KT +s  mT11s+   Fig. 3. Block scheme of the signal adaptive speed control with adaptation signal  The approaching block diagram of the adaptive control can be seen as extended version of the PI controller, so 2γ , one of the adaptation factors (which is the free parameter of the adaptive control) is given. Contraction of model-filtered reference signal and the PI controller can be transformed into a so-called PF controller when integration time of PI controller equals to the time constant of the model (Fig.4.).                        il ωr       i   ω sAi  ( )s1p2 g1Kγ +  ( )1p g1K +  ( )m1pmT1g1KTss++  Fig. 4. Block scheme of signal adaptive PF speed control  (extended version)  The basic structure of the signal adaptive speed controller is also PF type which on the one hand provides overshootless with its structure. Moreover, it ensures fast compensation of speed variation caused by a jump in motor load.  The transfer-function between ω and rω is as follows:   )1(111111122222gKAssTssTsTpimmmr++++++=γγγωω . (29) If the adaptation signal g(t) furthermore g1(t) = const., b1=0 (at the end of the adaptation), the following equation is valid:  ( )Pi KAg 211γ=+ . (30)  Substituting (30) into (29):   222222111111γγγωωssTssTsTmmmr +++++= . (31)  Choosing 21γ=mT , the transfer function is as follows:   mr sT+=11ωω . (32)  So the system follows the model without time delay.   3.  DRIVE SYSTEM  Block scheme of the examined drive system with switched reluctance motor (SRM) [1] is shown in Fig. 5.  Fig. 5. Block scheme of drive system The supply unit consists of three main blocks, namely the RECTIFIER, the FILTER and the INVERTER. The inverter is a pulsed width modulated (PWM) one, marked by QP in the figure and it contains a one-one switching transistor per phase and a brake chopper, not shown in the figure. The common point of phase windings is supplied by the PWM inverter. It is of autonomous operation and has an inner current control loop. The other ends of phase windings are connected to the phase switching transistors   4.  SIMULATION RESULTS  In Fig. 6 and Fig. 7 two of many executed simulations are shown. Fig. 6 shows the run-up with model reference parameter adaptive control (15), (16), while Fig. 7 with model reference signal adaptive control (28). RECTIFIERMainsFILTER INVERTERReluctanceMotorSwitchedRotorPositionSensorsignalTorque limitSpeed signalAnglePositionDecoderCurrentControllerControllerSpeedControlsignalssignalCurrent signal+_AngleControllerSpeed referenceCurrent referenceQP0 100 200 300 400 500 600 700 8000510152025303540Speed [rad/s]Time [ms]  0 100 200 300 400 500 600 700 800-20246810121416Torque [Nm]Time [ms]  0 100 200 300 400 500 600 700 800024681012141618Current [A]Time [ms]  0 100 200 300 400 500 600 700 800024681012Phase current [A]Time [ms]  Fig. 6. Simulation results with model reference parameter  adaptive speed control 0 100 200 300 400 500 600 700 800051015202530354045Speed [rad/s]Time [ms]  0 100 200 300 400 500 600 700 800-20246810121416Torque [Nm]Time [ms]  0 100 200 300 400 500 600 700 800024681012141618Current [A]Time [ms]  0 100 200 300 400 500 600 700 800024681012Phase current [A]Time [ms]  Fig. 7. Simulation results with model reference signal  adaptive speed control  4.  RESULTS  The tests were completed by the described drive system. The test results have supported our theoretical investigations. The oscillograms in the following figures illustrate some typical starting curves and wave forms. The loading machine was a DC motor. Its inertia is about a triple of that of SRM. Fig. 8 and Fig. 9 show the speed and current curves in the course of starting. The upper curve is the speed (1500 rpm), the lower one is the current flowing in the common point of stator windings (10 A/div). It is related to the no-load operation mode. The experiences show that the model reference parameter adaptive and signal adaptive control suggested in this paper works without overshooting.  Fig. 8. Oscillogram of speed and current with model reference parameter adaptive speed control   Fig. 9. Oscillogram of speed and current with model reference signal adaptive speed control    5.  CONCLUSIONS  To provide constant loop gain in speed control loop with changing parameters (moment of inertia and/or torque factor), parameter and signal adaptive model reference adaptive control was developed.  Following from the structure (PF-type) of model reference parameter adaptive control was developed to provide constant loop gain in speed control loop with changing gain (moment of inertia and/or torque factor) it makes it easier to reach overshootless as well as fast speed changing compensation caused by jump in load. The algorithm even keeps its stability at fast changing, jump-like load torque. Model reference signal adaptive control is used to provide constant loop gain in speed control loop with changing parameters (moment of inertia and/or torque factor) exposed to a significant load. The approaching block diagram of the adaptive control can be seen as an extended version of the PF controller, so one of the adaptation factors (which is the free parameter of the adaptive control) is given.  The adaptive controls suggested in this paper work without overshooting. Though these methods require a longer calculation period it is less sensitive to the variations of parameters. Both model reference adaptive controls drawn up can be easily implemented, because the adaptation algorithms do not need acceleration measuring (thanks to the first-order model). Simulation and experimental results demonstrate that the proposed methods are promising tools to speed control of electrical drives.   6.  REFERENCES  [1] Bose, B.K. - Miller, T.G.E.: Microcomputer Control of Switched Reluctance Motor. IEEE/IAS Annual Meeting. 1985. pp. 542-547.  [2] Diep, N.V. - Szamel, L.: Up-to-date Control Strategy in the Regulators of Robot Drives. PEMC'90, Budapest. 1990. pp. 811-815. [3] Ohm, D.Y.: A PDFF Controller for Tracking and Regulation in Motion Control, Proceedings of 18th PCIM Conference, Intelligent Motion, Philadelphia, October 21-26 1990. pp.26-36. [4] Perdikaris, G.A. - VanPatten, K.W.: Computer Schemes for Modeling, Tuning and Control of DC Motor Drive Systems, PCI Proc., Mar. 1982. pp. 83-96. [5] Phelan, R.M.: Automatic Control Systems, Cornell University Press, New York, 1977. [6] Szamel, Laszlo: Model Reference Adaptive Control of SRM Drives, EPE-PEMC 2002, 10th Int. Power Electronics and Motion Control Conf., Cavtat & Dubrovnik (Croatia), 9-11 September 2002. pp. 466, CD-ROM: Paper No. T11-007 [7] Szamel, Laszlo: Investigation of Model Reference Parameter Adaptive SRM Drives, EPE-PEMC 2004, 11th Int. Power Electronics and Motion Control Conf., Riga (Latvia), 2-4 September 2004. Full paper A95117 on CD_ROM 

Investigation of adaptive PDF speed control for motion control

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