This paper presents a new technique for the chaoiztion of switched reluctance motor (SRM) drivers. Based on the chaotic modeling of SRM drives, effects of feedback controller gains on the stability of the rotor speed are investigated. In accordance, a control strategy combining piecewise proportional feedback and time-delayed feedback is proposed to produce bound-controllable oscillation around any specific rotor speed. To estimate the continuous influence of a certain control parameter, a Poincaré map sampling at every extreme points is constructed, then the bifurcation diagrams are drawn. Theoretical analysis and numerical simulation for a practical 12/8 motor driver are also given.published_or_final_versio

Chau, KT

Chen, JH

Huang, JF

HKU Scholars Hub

Title Chaoization of switched reluctance motor drivesAuthor(s) Huang, JF; Chen, JH; Chau, KTCitation Icems 2005: Proceedings Of The Eighth International ConferenceOn Electrical Machines And Systems, 2005, v. 1, p. 504-508Issued Date 2005URL http://hdl.handle.net/10722/45890Rights Creative Commons: Attribution 3.0 Hong Kong LicenseChaoization of Switched Reluctance Motor DrivesJ.F. Huang', J.H. Chen , and K.T. Chau2'Department of Electrical Engineering, Tsinghua University, Beijing 100084, P. R. of ChinaDepartment of Electrical and Electronic Engineering, The University ofHong Kong, Hong Kong, P.R. of ChinaAbstract This paper presents a new technique forthe chaoiztion of switched reluctance motor (SRM)drivers. Based on the chaotic modeling of SRMdrives, effects of feedback controller gains on thestability of the rotor speed are investigated. Inaccordance, a control strategy combining piecewiseproportional feedback and time-delayed feedback isproposed to produce bound-controllable oscillationaround any specific rotor speed. To estimate thecontinuous influence of a certain control parameter, aPoincare map sampling at every extreme points isconstructed, then the bifurcation diagrams aredrawn. Theoretical analysis and numericalsimulation for a practical 12/8 motor driver are alsogiven.I. INTRODUCTIONRecently, a number of research activities on chaos inmotor drives have been carried out. Most of them arebased on the identification of chaos, the avoidance ofchaos and the stabilization of chaos in various types ofelectric motor drives [1-6]. Many methods deriving formthe idea of parameter disturbing, namely OGY method,are employed to stabilize an aperiodic or chaotic rotorspeed into a periodic or regular steady state [7-9].However, in some industrial occasions chaotic motionsin motor drives are highly desired to improve theeffectiveness and efficiency, for example, mixing andgrinding. Such tries, so-called anti-control of chaos orchaoization have been barely reported. In this paper,rather than negatively avoiding the occurrence of chaosin motors, a positive idea to utilize the chaotic motion forsome niche applications will be presented.Recent experiences have proved that conventionalcontrollers can also induce chaos in motor drivers. Thispaper firstly presents the analysis of the effect ofparameters of the controller on the stability of dynamicsystem. It can be shown that some chaotic behaviorspotentially offer distinct advantages of the reduction ofsubharmonics, electromagnetic interference (EMI) andacoustic noise. However, this kind of chaotic motionsuffers a limited range of oscillation of the rotor speed.To amplify its chaotic domain, a new control technique,named time-delayed feedback control, has beencombined with the modified proportional feedback. Thisnovel controller has the advantages that only parametersneed to adjust and no inherent parameter of the motor isnecessary to change. It can be easily implemented withdigital controller and has high flexibility. Despite of thebenefits above, time-delayed feedback control bringsome obstacles into the analysis and computation. Tofacilitate the controller design, the switching effectshould be ignored and hence, other methods to estimateits stability must be proposed.In section II, the modeling of chaos in switchedreluctance motor (SRM) drives is described withoutignoring the switching effect nor accepting roughassumptions. In section III, based on the modeling ofchaos, the chaotic operation with lower ripples of thetorque and rotor speed is realized. In section IV, withignoring the switching effect, the modeling ofchaoization in SRM drives is derived for a controllablechaotic motion. In section V, to estimate the continuousinfluence of a certain control parameter, the Poincaremap is constructed; then the bifurcation diagrams areproposed and analyzed.II. MODELLING OF CHAOS IN SWITCHED RELUCTANCEMOTOR DRIVESDue to the double salient structure of switchedreluctance motors, nonlinear phenomena, like fringingand saturation, are severe during the time the rotor andstator are overlapped. So the flux linkage is related withboth phase current and rotor position. The most commonmodeling of flux linkage, current and rotor position is asmooth surface interpolated by second-order Lagrangeinterpolation functions using data measuredexperimentally or computed by finite element method.A. State EquationsFig. 1 shows the schematic diagram of a three-phaseSRM which is adopted for exemplification.Fig. 1 Schematic diagram of SRM drives.Since SRM phase windings are conducted in turn,only two adjacent phase windings have currents at thesame instant given the dwell angle Od of every phasewinding equal to the commutation angle O0. In order to504simplify the calculation of the Poincare map, the rotorposition 0 rather than time t is selected as theindependent variable of the system. Hence, the systemequation becomes [11]:|dc (-Bco + Te (0, V-,, V-,0U,)-s)Tl) IfJo )dOdO = (-Ri2(0,6T')+ul(0))6/coldO =(R2 (o - ol I 2 ) + U2 (o - Ss ) Y)1(1)ao=00pi, (0, W-l )d l-' i2(°O( 0,2)dW2(2)where co is the rotor speed, ik(, VJk) and k (0, ik) (k =1,2) are the kth phase current and flux linkage, R is thephase resistance, B is the viscous damping, J is the loadinertia, T, is the load torque.B. Control EquationsFor each conductive phase winding, thecorresponding current control is achieved by applying acurrent hysteresis controller, as Fig. 2.(0ref G>A(0- Al- Bi- ci-A2-B2-C200s1215~5.5010.0:IUs 1150l2NmTABLE I.SIMULATION PARAMETERS=~~ 89s 1525kgm2 0.O5Nm/tV , S50radsgrads'0.02Q0.0 lV rads'When proportional gain ki is 1, 10, 22 V/rads'Irespectively, the corresponding trajectories of torqueversus rotor speed are shown as Fig. 3-5. It indicates thatthe system exhibits fundamental, subharmonic andchaotic operations.Fig. 2 SRM hysteresis controller.The current reference signal v,(0), the current controlsignal vi(0) and the hysteresis band signal are given asfollows:tVv (0) kp (C - 4(0)) + f ki (c - (0))dOVi (0) g=giik (0)3AV gziiwC is the reference speed. The current hysteresiscontroller outputs the signal to turn on or off thoseupper-leg power switches (Al, B1 and Cl), dependingon the phase commutation. When (vi -vi) > Av, theupper-leg switch being the same phase of the turn-onlower-leg switch is off until (vi - vi) < -Av, then it is on.III. CHAOIZATION OF SWITCHED RELUCTANCE MOTORDRIVESChanging system parameters such as kp, ki and gi, theappearances and qualities of solutions can be quitedifferent, varying from fundamental solution tosubharmonic, even chaos. In this paper, parameters froma practical 12/8 SRM and its controller have been used tosimulate the real operation numerically.Fig. 3 Fundamental trajectory of Te versus co.49-86 49-87 49-88 49-89 49.9a (rads'I)Fig. 4 Subharmonic-4 trajectory of Te versus co.161514131211109 49-94 49-945 49-95 49-955 49w(rads'I)Fig. 5 Chaotic trajectory of Te versus w.5051The Waveforms and power spectra PT of thefundamental and chaotic electromagnetic torques Te arerespectively shown in Fig. 6 and 7. Due to the broadfrequency band of the chaotic operation, the later torquehas more uniform distribution in the view of powerspectrum. Comparing Fig. 6(b) and 7(b), they can befound that chaotic motion can reduce the annoying peaksduring commutation, and hence reduce the abrasion andnoise.t (s)(a)o 04o - 030 020 01100 200 300f(Hz)(b)400 5(Fig. 6 Waveform and power spectrum of the fundamental solution:(a) Waveform of torque; (b) Power spectrum.1.1211t (s)(a)100 200 300f(Hz)IV. CHAOIZATION COMBINING TIME-DELAYED ANDPIECEWISE PROPORTIONAL FEEDBACKAs shown above, proportional gains restrain thevariation range of the rotor speed and prohibit the motorto oscillate in a large area though it can in fact improvethe frequency of oscillation. To obtain the requiredoscillation in a desired range, other control techniquesmust be used. Time-delayed feedback can transform anordinary differential equation into an infinite-dimensional differential equation, and therefore has theadvantage of introducing infinite kinds of dynamicalcharacteristics into the original systems [10].We add a sinusoidal function of the state variable intothe reference signal:VC (0) = k, (oi - 0)) + f k (ic - (0))dO {AA~J4)+ pSin(J(c6c -oi(0 - r))In the meantime, the proportional gain kp has alsobeen modified as following. When the rotor speed liesright in the desired range, no proportional feedback isadded to the reference signal, which means that thesystem is overwhelmed by time-delayed feedbackcontrol. Once the rotor speed leaves the desired domain,the linear feedback will pull it back.I Ac +wc 0)kp (no) = 0Awc + cC 0)CV c I< A cc0c 'C Awc0)t > Awc(5)Setting a desired range for the rotor speed, such as[48, 52] rads-', different performances of the rotor speedcan be regulated by changing the magnitude andfrequency of the sinusoidal function and the delay timeas well. When the frequency a is set with 1 and 3respectively, the system operates in the periodic andchaotic motions as show in Fig. 8 and 9. It can be foundthat the oscillation boundaries of both periodic andchaotic rotor speed, namely [49, 53] rads-', are near tothe desired rang. It indicates that the system has acontrollable chaotic motion. As shown in Fig. 8, it isinteresting that the periodic solution is actually aheteroclinic trajectory which has two unstable saddlepoints locating respectively at 0)=47.5 rads-' and=52.5 rads'1.It should be noted that the commutation angle hasbeen chosen as the unit for rotor position because muchwider oscillation of rotor speed in spatial and temporalconcept are what are now focused on.54ct3-504846400 500(b)Fig. 7 Waveform and power spectrum of the chaotic solution:(a) Waveform of torque; (b) Power spectrum.40 80 120 160 200Fig. 8 Waveform of periodic solution.506u - U -1)-* eug: ,J ,> L > ,;Ji K ,A&dw,-Au-dl U. ., -,- JL I5450~48F460 40 80 120 160 200Fig. 9 Waveform of chaotic solution.V. SIMPLIFIED DESIGN MODELLINGWhen the rotor speed oscillating in a much widerrange in spatial and temporal concept is the main objectto be considered, the minor variations caused by PWMswitching operation can often be ignored withoutaffecting the result much. Another reason to give up theoriginal analysis modeling is its high order and heavycomputation time. Assume the phase current follows thereference signal quick enough; the state equations can besimplified as a second-order differential equation:dOI = co{ dt (6)dc-= (-Bco + Te (0, i)-T,) I JdtAssume the phase current stays constant in a samplingperiod, the discretized control equation can be write as:ni, (n) = gi [kp ()c - w(n)) + E ki (wt - o(m))+ ,uSin(u(w6c - o(n - N))]In order to analyze the influence of a systemparameter, the bifurcation diagrams should be gainedwith other parameters fixed and changing the chosenparameter continuously. Construction of Poincare map isa prerequisite to draw bifurcation diagrams. Here, a newmethod based on extremum points to construct Poincaremap is proposed. Select the Poincare section as::= {(O,c): 6(t) = O} (8)Hence, the Poincare map P: R2-+R2 is defined as:xn+1 = P(Xn) (9)The correspondence between the continuouswaveform of the rotor speed and its discrete mappingpoints derived from Poincare map is illustrated withexemplification of the chaotic motion as shown in Fig.10. If the mapping has two fixed points, which means amaximum point and a minimum point, it indicates thatthe motor operates in periodic solution; if the motorspeed is like the waveform in Fig. 10, then when thesystem is stable a set of dense points will be got insteadof two points.Keep other parameters as listed in the table 1, andvary the ,u and kp, the corresponding bifurcationdiagrams can be achieved as shown in Fig. 11 and Fig.12. Increasing ,u, the rotor speed transfers from periodicsolution into chaos as shown in Fig. 11. Contrary to theFig. 11, increasing kp, the rotor speed transfers fromcn03Schaos into periodic solution as shown in Fig. 12. Itindicates that the dynamic characteristic depends on thebalance of the sinusoidal function of time-delayedfeedback and the proportional feedback gain. It can alsobe found that the boundaries of the chaotic rotor speedfor the different parameter ,u and k, always keep in thesame range. Hence, the chaotic range is controllable.Furthermore, since no periodic window exists in chaoticparameter areas, this chaotic behavior is robust.51504948475654525046464454524 503 4846l 02 0:4 06 0I8 1 1 2t (s)Fig. 10 Illustration of Poincare mapt(s)0 5 1 1 5 2 2 5Fig. 11 Bifurcation of rotor speed versus 1i....,,~ .;~ _0 0 5 1 1 5 2pkpFig. 12 Bifurcation of rotor speed versus kp.VI. CONCLUSIONIn this paper, the effects of parameter variations in aconventional controller on the stability of switchedreluctance motor drives have been firstly investigated. Itcan be seen that some chaotic behaviors potentially offerdistinct advantages of reduction of subharmonics, EMIand acoustic noise. Secondly a feedback controltechnique combining with time-delayed feedback andpiecewise proportional feedback is proposed to achievedesired bound-controllable and robust chaotic motion. Indespite of its benefits, delayed feedback control increasethe computation tremendously. In order to reduce thecomputation time, a simplified chaotic design modeling507has been derived with the assumption of the current loopfollowing the rotor speed loop quick enough. Meantime,a method to construct Poincare map based on extremasampling has been presented. The bifurcation diagramsshow that when a certain parameter of the controllervaries continuously, the dynamic system exhibits asudden bifurcation into chaos.ACKNOWLEDGMENTThis work was supported and funded by NationalNatural Science Foundation of China (50177015).REFERENCES[1] N. Hemati, "Strange attractors in brushless dc motors,"IEEE Transactions on Circuits and Systems - I, Vol. 41,1994, pp. 40-45.[2] T. Asakura, et al., "Chaos detection in velocity controlof induction motor and its control by using neuralnetwork," Proc. of IEEE ICSP, 2000, pp. 1633-1638.[3] Chen, J.H., et al., "Chaos in voltage-mode controlled dcdrive system," International Journal of Electronics,1999,Vol. 86, No. 7, p. 857-874.[4] Zhong Li, et al., "Bifurcations and chaos in apermanent-magnet synchronous motor," IEEETransactions on Circuits and Systems -I, Vol. 49, 2002,pp.383-387.[5] Y. Gao, K.T. Chau, "Design of permanent magnets toavoid chaos in PM synchronous machines," IEEETransactions on Magnetics, Vol. 39, No. 5, 2003,pp.2995-2997.[6] Reiss, J., et al., "A detailed analysis of the nonlineardynamics of the electric step motor IndustrialTechnology," 2002. IEEE ICIT '02. 2002 IEEEInternational Conference on, Volume: 2 , 11-14 Dec.2002 Pages: 1078 - 1083 vol.2.[7] Ito, Shunji, et al., "Abrasive machining under wetcondition and constant pressure using chaotic rotation,"Seimitsu Kogaku Kaishi Journal of the Japan SocietyforPrecision Engineering, Vol. 64, No. 5, 1998, pp748-752.[8] J. Guckenheimer, et al., Nonlinear oscillations,Dynamical Systems, and Birfucations of Vector Fields,(New York: Springer-Verlag), 1983.[9] Parker T.S, Chua L 0, Practical Numerical Algorithmsfor Chaotic Systems, (New York: Springer-Verlag), 1989.[10] G. Chen, et al., From Chaos to Order: Methodologies,Perspectives and Applications, Singapore: WorldScientific, 1998.[11] K.T. Chau, J.H. Chen, "Modeling, analysis, andexperimentation of chaos in a switched reluctance drivesystem" , IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, May 2003,Vol. 50, No. 5, pp712-716.508

Chaoization of switched reluctance motor drives

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