Bifurcations and Chaos in a Permanent-Magnet Synchronous Motor

Transcription

1 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 49, NO., MARCH 00 8 systems having the multiaffine parametric perturbations using uantitative feedback theory. REFERENCES [1] R. M. Steward, A simple graphical method for constructing families of Nyuist diagrams, J. Aeronaut. Sci., vol. 18, pp , [] V. L. Kharitonov, Asymptotic stability of an euilibrium position of a family of systems of linear differential euations, Differ. Eu., vol. 14, no. 11, pp , [] F. N. Bailey, D. Panzer, G. Gu, Two algorithms for freuency domain design of robust control systems, Int. J. Control, vol. 48, no. 5, pp , [4] F. N. Bailey C. H. Hui, A fast algorithm for computing parametric rational functions, IEEE Trans. Automat. Contr., vol. 4, pp , Nov [5] A. Karamancioglu, V. Dzhafarov, C. Ozemir, Freuency response of PID-controlled linear interval systems, Circuits, Syst., Signal Processing, vol. 15, no. 6, pp , [6] C. Hwang J. J. Chen, Computation of the freuency response of interval systems, Circuits, Syst., Signal Processing, vol. 15, no. 11, pp. 1 1, [7] A. C. Bartlett, A. Tesi, A. Vicino, Freuency response of uncertain systems with interval plants, IEEE Trans. Automat. Contr., vol. 8, pp. 99 9, June 199. [8] M. Fu, Computing the freuency response of linear systems with parametric perturbation, Syst. Contr. Lett., vol. 15, pp. 45 5, [9] J. J. Chen C. Hwang, Computing freuency responses of uncertain systems, IEEE Trans. Circuits Syst. I, vol. 45, pp , Mar [10] O. N. Kiselev, L. H. Lan, B. T. Polyak, Freuency responses under parametric uncertainty, Autom. Remote Control, pt., vol. 58, no. 4, pp , [11] J. J. Chen C. Hwang, Robust D-stability analysis of polynomial families with coefficients depending nonlinearly on perturbed parameters, Proc. Inst. Elect. Eng. Part D., vol. 145, no. 1, pp. 7 8, [1] L. Zadeh C. A. Desoer, Linear System Theory. New York: Mc- Graw Hill, 196. [1] S. P. Bhattacharyya, H. Chapellat, L. H. Keel, Robust Control: The Parametric Approach. Englewood Cliffs, NJ: Prentice-Hall PTR, Bifurcations Chaos in a Permanent-Magnet Synchronous Motor Zhong Li, Jin Bae Park, Young Hoon Joo, Bo Zhang, Guanrong Chen Abstract This brief studies dynamic characteristics of a permanent-magnet synchronous motor (PMSM). The mathematical model of the PMSM is first derived, which is fit for carrying out the bifurcation chaos analysis. Then, the steady-state characteristics of the system, when subject to constant input voltage constant external torue, are formulated. Three cases are discussed, for each case, conditions are derived under which the dynamic characteristics of the system are either of steady-state type, limit cycles or chaotic, thus by properly adjusting some system parameters, the system can exhibit limit cycles (LCs) or chaotic behaviors at will. Finally, computer simulations are presented to verify the existence of strange attractors in the PMSM. Index Terms Chaos, Hopf bifurcation, limit cycle (LC), permanent-magnet synchronous motor (PMSM). I. INTRODUCTION Since the 1970s, dynamic characteristics of various motors are widely studied, to deal with starting up, speed control, oscillations of the motors [1] [7]. In the study of dynamic characteristics of motors, there are many problems that remain to be further addressed, such as their low-speed feature, known as the low-freuency oscillations of speed-controlled motors. These problems are closely related to the studies of chaos in nonlinear systems [8]. It is well known that the existing mathematical models of motors are multivariable, nonlinear, strongly coupled, therefore these systems can exhibit complex behaviors [9] [11]. It is now a common belief that understing utilizing the rich dynamics, such as bifurcations chaos, of nonlinear systems have an important impact on the modern technology. A permanent-magnet synchronous motor (PM SM) is a kind of high-efficient high-powered motor. With the development of permanent-magnet materials, its uniue advantage is increasingly evident, it is widely used in motor drive, various servo systems household appliances. In this pursuit, however, research on bifurcation chaotic phenomena of the PMSM is still behind the rapidly evolving trend of nonlinear sciences engineering. In this brief, we further study present some new observations on the strange attractors of the PMSM. First of all, the mathematical model of the PMSM is derived, which is a three-dimensional autonomous euation with only two uadratic terms, this model is fit for carrying on bifurcation chaos analysis. Secondly, the steadystate characteristics of the motor, when subject to constant input voltage external torue, are formulated. A third-order polynomial euation /0$ IEEE Manuscript received November 16, 000; revised July 17, 001 August 0, 001. This work was supported by the Research Project Brain Korea 1. The work of B. Zhang was supported by the National Natural Science Foundation of China under Grant This paper was recommended by Associate Editor T. Saito. Z. Li J. B. Park are with the Dept. of Electric Electronic Engineering, Yonsei University, Seoul , Korea ( zhongli@control.yonsei.ac.kr; jbpark@control.yosei.ac.kr.). Y. H. Joo is with the School of Electronic Information Engineering, Kunsan National University, Kunsan, Chonbuk , Korea ( yhjoo@kunsan.ac.kr). B. Zhang is with the College of Electrical Power, South China University of Technology, Guangzhou, , P.R. China ( epbzhang@scut.edu.cn). G. Chen is with the Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong ( gchen@ee.cityu.edu.hk). Publisher Item Identifier S (0)075-4.

2 84 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 49, NO., MARCH 00 is also derived whose solutions correspond to the steady-state values of the motor angular velocity. Furthermore, based on the Hopf bifurcation condition, we discuss how to determine the parameters of the PMSM in three different cases, with which the system can exhibit such desired dynamic characteristics as limit cycles (LCs) chaotic behaviors. Finally, computer simulations are presented to verify the presence of strange attractors in the PMSM. II. THE SYSTEM MODEL The dynamics of a PMSM can be modeled, based on the d- axis [1], [6], [1], as For the sake of simplicity, in this brief, we only study the dynamic characteristics of the smooth-air-gap PMSM, namely, L d L L in the model. Thus, (4) becomes d ~ i d d~t 0~ i d +~! ~ i +~u d d ~ i d~t 0~ i 0 ~! ~ i d + ~! +~u d~! d~t (~ i 0 ~!) 0 ~ T L By calculating the euilibrium of this system, we obtain (5) di d dt (u d 0 R 1 i d +!L i )L d di dt (u 0 R 1 i 0!L d i d 0! r)l d! dt [np ri + np(l d 0 L )i d i 0 T L 0!]J i d ;i! are the state variables, which represent currents motor angular freuency, respectively, u d u the direct- uadrature-axis stator voltage components, respectively, J the polar moment of inertia, T L the external load torue, the viscous damping coefficient, R 1 the stator winding resistance, L d L the direct uadrature-axis stator inductors, respectively, r the permanentmagnet flux, n p the number of pole-pairs. By applying an affine transformation a time-scaling transformation (1) x ~x; () t ~t () x [i d i!] T ; ~x [ ~ i d ~ i ~!] T d ! b L L d k n p r bk k L R 1, we obtain a system of euations in a dimensionless form 0 r ; kl ~u d 1 R 1 k u d; d ~ i d d~t 0~ i d +~! ~ i +~u d d ~ i d~t 0~ i 0 ~! ~ i d + ~! +~u d~! d~t (~ i 0 ~!) +" ~ i d ~ i 0 ~ T L J T L: 1 J ; ~u 1 R 1k u ; " n pb k (L d 0 L ) J (4) ~ i d ~! + ~ i ~! + ~! + ~! +~u d; (6) ; (7) ~! +(~u d 0 +1)~! + 0 ~u 0: (8) Solving (8) can determine ~!, then substituting it into (6) (7) can determine ~ i d ~ i, thus the euilibria are derived. III. BIFURCATIONS AND CHAOS IN THE PMSM SYSTEM Hopf bifurcation occurs when the corresponding Jacobian matrix has a pair of purely imaginary eigenvalues, with the remaining eigenvalues having nonzero real parts [1] [18]. For the PMSM system, its Hopf bifurcation chaotic behavior are discussed under the following three cases. A. ~u d ~u ~ T L 0 In this case, system (5) is identical to the Lorenz euation. This case can be thought of as that, after an operating period of the system, the external inputs are set to zero. Applying the euilibrium conditions to (6) (8), it is determined that the origin is an euilibrium state that other two nontrivial euilibria exist, if > 1, which are defined by ~ i e d ~ i e ~! e p p 0 1 : (9) A little linear analysis shows that the origin is stable if 0 < < 1 loses stability in a pitchfork bifurcation at 1, creating the two nontrivial euilibria which are initially stable. To determine the stability of these nontrivial euilibria we look at the Jacobian matrix 01 ~! ~ i 0~! 01 0 ~ i d which has eigenvalues given by the roots of D() +(+) +[1+ + ( ~ i d 0 ) +~! ] + (1 + ~ i ~! + ~ i d 0 +~! ) +(+) +( + ) +(0 1) 0 (10) when evaluated at the nontrivial euilibria. Note that since the two nontrivial euilibria are symmetric, their stability must be the same. For the bifurcation of the two nontrivial euilibria, i.e., values of the parameters for which either 0or jw is the solution of the eigenvalue euation, setting 0we find 1, giving the pitchfork bifurcation

3 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 49, NO., MARCH which we already know about. Setting jw euating real imaginary parts of the euation we find 1) 10: If 10, (8) has three real roots, two of them are eual the remainder is negative i.e., 0w +( + )w 0; 0( + )w +( 0 1) 0 ~! 1 0 4; ~! ~! : Substituting them into (6) (7), respectively, we obtain the two corresponding euilibria as follows: ( 0 1) w ( + ): + Rearranging terms a little, this implies that there is a Hopf bifurcation at ( +4) h 0 provided w > 0 at this value of, i.e., provided ( +1) w > 0 0 (11) which will always be satisfied provided >. So, the corresponding eigenvalues are given by 1 0( +) ; 6j ( +1) ( 0 ) : (1) Therefore, h corresponds to a Hopf bifurcation point of the system, for close enough values 6 h, euilibria are surrounded by a limit cycle, for > h all three euilibria become unstable. B. ~u ~ T L 0But ~u d 60 In this case, if 0 u d > 1, the two euilibria are defined by ~ i e d ~ i e ~! e p ~u d 6 p ~u d (1) It follows the same procedure as in Section A that we have ~u dh (14) the eigenvalues corresponding to the two nontrivial euilibria are 1 0( +) ; 6j (( + 1))( 0 ). Therefore, ~u d ~u dh corresponds to a Hopf bifurcation point of the system, for close enough values ~u d 6 ~u dh, euilibria are surrounded by a limit cycle, for 0~u d > 0~u dh all euilibria become unstable. C. ~u d ; ~u T ~ L are in General Applying the euilibrium conditions to (6) (8), we can evaluate the euilibria, denoted by (x; y; z). Then, if yz (1 + ) +( + )(x 0 ) +z holds, which follows the same procedure as in Subsection A, the system exhibits a limit cycle, while when T ~ L takes some deterministic values, by adjusting ~u d or ~u properly, the system exhibits chaos. If the euilibria are restricted in the real domain, 1 0 must hold, 1() +(p) ;p(~u d 0 +1)0 ( ~ T L ) ; ( ~ T L ) 7 0 [( ~ T L )(~u d 0 +1)] +( ~ T L 0 ~u ). Here, we consider two subcases. (4) 0 ~ T L (4) +~u d; 0(4) + + +~u d ; p ~u d +~ 0 a ~u d + ~ a ~ a ~ + ;0(4) ; 7 a 0 1 a(0 +1)+a 0 ~u : Without loss of generality, ~u ~ T L can take deterministic values, so we can adjust ~u d properly, such that 10. Thus, the expression for 1 is 1 + p ~ 0 a ~u d 1 7 ~u d ~ a ~u d + ~u d +~ ~ 6 a ~ ~u d ~ 7 ~ 0: 4 Then, ~u d can be obtained by numerical evaluation. ) 1 < 0: In this case, if ~u ~ T L take deterministic values, ~u d can be properly adjusted, such that 1 < 0. Here, we assume that ~u ~ T L are both given. From (8), we obtain three roots ~! ~! ~! p + p + p + p + p + p! 0! 0! 0! 0 p! 0 (01+ j). By substituting them into (6) (7), respectively, the euilibria can be obtained. If ~u d is evaluated such that yz (1 + ) +( + )(x 0 ) +z holds, then we can find the Hopf bifurcation point.

4 86 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 49, NO., MARCH 00 Fig. 1. A limit cycle generated at 14:1. Fig.. Limit cycle generated when ~u ~u +: Fig. 4. Chaotic attractor generated when ~u ~u 0. Fig.. Chaotic attractor generated at 0. Furthermore, for the close enough values, euilibria are surrounded by a limit cycle, for a little larger values all euilibria become unstable. IV. SIMULATION RESULTS In this section, we provide some simulation results to verify the existence of chaotic attractors in a PMSM. Note that for Hopf bifurcation to occur at the two nontrivial euilibria, one must have >for the first two cases, i.e., ~u d ~u T ~ L 0 ~u T ~ L 0but ~u d 60. To illustrate some of the viewpoints given above, computer simulation results corresponding to a PMSM are presented. As mentioned above, for every PMSM there exist appropriate values of ~u d ; ~u,, under which the system exhibits limit cycles or chaotic behaviors. In fact, for the case of ~u d ~u T ~ L 0 for every given pair of dimensionless parameters in (5), there correspond infinitely many combinations of motor parameters. For the other two cases, namely, ~u T ~ L 0but ~u d 60 ~u d ; ~u, T ~ L are arbitrary in general, the results are similar. Here, for illustration purpose, we consider a PMSM with the following specifications: L d L L 14:5 mh, R 1 0:9; r 0:01 Nm/A, n p 1;J 4: Kgm, 0:016 N/rad/s. For the case of ~u d ~u T ~ L 0, in practical operation, the PMSM system is actually stable under the above parameter setting. For the purpose of illustrating the proposed methods, we arbitrarily set as 5:46, thus we can determine h 14:9 in terms of the proposed method, which corresponds to a Hopf bifurcation point of the system. Using the trial--error method, we can find while 14:1, which is close to the Hopf bifurcation point, the euilibria are surrounded by a limit cycle as shown in Fig. 1 with the initial value ( ~ i d ; ~ i ; ~!) (0:01; 0:01; 0:01). Furthermore, when 0> h, the system can exhibit chaotic behavior as shown in Fig.. These phenomena can be Fig. 5. The steady state generated at ~u 00:9014. explained as that after a period of operation, the external inputs of the motor are set to zero. The system actually exhibits different dynamic behaviors corresponding to different choices of parameters such as 5:46 together with 14:1 0, respectively. For the case of ~u T ~ L 0 but ~u d 6 0, Figs. 4 show the simulation results with a limit cycle chaos, respectively, which correspond to different values of ~u d, namely, ~u d ~u dh +:4 67 5, ~u d ~u dh 0. For the case of ~u d ; ~u, T ~ L being arbitrary in general, without loss of generality, let ~u T ~ L be fixed, so we can adjust ~u d such that 1 0. We choose the given parameters as above, set T L 1:N1m ~u 1. By letting 10, we obtain, via numerical evaluation, ~u d1 00:9014 ~u d; 00: j0:0769 (ignored). Fig. 5 depicts the simulation result, which shows that the system is stable. For the case of 1 < 0, in order for yz ( + )(x 0 ) + z + to be held, we must have ~! 0:41 6 p 04:755~u d 0 5:9119. For~! 0:41 + p 04:755~ud 0 5:9119, some algebraic operations give ~u d 05:01. Then we can find a nearby value ~u d 016:67, at which the system

5 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 49, NO., MARCH Fig. 6. Limit cycle generated at ~u 016:67. Fig. 7. Chaotic attractor generated at ~u 00. exhibits a limit cycle, as shown in Fig. 6, further adjusting ~u d to ~u d 00, the system exhibits chaos as shown in Fig. 7. [5] J. de la Ree N. Boules, Torue production in permanent magnet synchronous motor, in Rec. Conf IEEE-IAS Annual Meeting, 1987, pp [6] P. Pillay R. Krishnan, Modeling, analysis simulation of a high performance vector controlled, permanent magnet synchronous motor drive, in Rec. Conf IEEE-IAS Annu. Meeting, 1987, pp [7], Application characteristics of permanent magnet synchronous brushless DC motors for servo drivers, in Rec. Conf IEEE-IAS Annu. Meeting, 1987, pp [8] J. Guckenheimer P. Holmes, Nonlinear Oscillations, Dynamical Systems, Bifurcation of Vector Fields. New York: Springer-Verlag, [9] J. R. Wood, Chaos: A real phenomenon in power electronics, in Rec. IEEE Applied Power Electronics Conf., Mar. 1989, pp [10] J. H. B. Dean D. C. Hamill, Instability, subharmonics chaos in power electronic system, IEEE Trans. Power Electron., vol. 5, pp , July [11] N. Hemati, Stange attractors in brushless DC motors, IEEE Trans. Circuits Syst. I, vol. 41, pp , Jan [1] J. Chen, The Mathematical Model of AC Motors Speed Control Systems. Beijing, China: National Defense Industry Press, 1991, pp in Chinese. [1] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems Chaos. New York: Springer-Verlag, [14] E. Hopf, Bifurcation of a periodic solution from a stationary solution of system of differential euations, Ber. Math. Phys. Klasses Sachs. Akad. Wiss., vol. 94, pp., 194. [15] S. N. Chow J. K. Hale, Methods of Bifurcation Theory. New York: Springer-Verlag, 198. [16] T. S. Parker L. O. Chua, Chaos: A tutorial for engineers, Proc. IEEE, vol. 75, pp , Aug [17] L. O. Chua L. T. Huynh, Bifurcation analysis of Chua s circuit, in Proc. 5th Midwest Symp. on Circuits Systems, 199, pp [18] T. Ueta, H. Kawakami, T. Yoshinaga, Y. Katsuta, A computation of bifurcation parameter values for limit cycles, in Proc IEEE Int. Symp. on Circuits Systems, Hong Kong, June 9 1, 1997, pp V. CONCLUSION The dynamic characteristics of a PMSM have been studied in detail. It was demonstrated that by adjusting system parameters properly, the system could exhibit different dynamic characteristics, such as steady states, limit cycles, chaos. A numerical method has been proposed to adjust these parameters for intended dynamics. Finally, computer simulations were presented to illustrate the presence of different dynamic characteristics in a PMSM. More theoretical studies on the chaotic nature of the PMSM are undertaken based on the mathematical chaos theory [8], [1], [16], along with the control utilization of chaos in a PMSM, which include studying the advantages disadvantages of chaos in a PMSM, further studying control or anticontrol of chaos on purpose in it. ACKNOWLEDGMENT The authors wish to thank the reviewers for their constructive comments based on which the briefr has been improved. REFERENCES [1] D. P. M. Cahill B. Adkins, The permanent magnet synchronous motor, Proc. Inst. Elec. Eng. A, vol. 109, no. 48, pp , 196. [] R. Krishnan A. J. Beutler, Performance design of an axial field permanent magnet synchronous motor servo drive, in Proc. IEEE Ind. Appl. Soc. Annu. Meeting, 1985, pp [] V. B. Honsinger, Permanent magnet machines: Asynchronous operation, IEEE Trans. Power App. Syst., vol. PAS-99, no. 4, pp , July [4] T. J. E. Miller, Transient performance of permanent magnet machines, in Proc. IEEE Ind. Appl. Soc. Annu. Meeting, 1981, pp