SYNCHRONIZAION CRIERION OF CHAOIC PERMANEN MAGNE SYNCHRONOUS MOOR VIA OUPU FEEDBACK AND IS SIMULAION KALIN SU *, CHUNLAI LI College of Physics and Electronics, Hunan Institute of Science and echnology, Yueyang, 414000, China Corresponding author: sukalinvip@126.com (K. Su) Received October 3, 2014 Chaos synchronization of permanent magnet synchronous motor (PMSM) has been central to recent experimental and theoretical investigations. In existing papers, the implementations of synchronization control require the system states for feedback, which are effective but unacceptable in practical application. In this paper, an output feedback controller is proposed for synchronization of PMSM based on the theory of passive control. heoretical analysis shows that the control method makes the synchronization error system between the driving and the response motor systems not only passive but also asymptotically stable. Numerical simulations are provided to verify the effectiveness of the proposed scheme. Key words: permanent magnet synchronous motor; chaos synchronization; output feedback control. PACS Nos.: 05.45.-a; 05.45.Xt; 05.45.Gg. 1. INRODUCION Among various electrical machines, permanent magnet synchronous motor (PMSM) has been studied intensively since it has superior features such as simple structure, low manufacturing cost, more torque per weight and efficiency [1 4]. With the technological development of control and electronic, the PMSM have been widely used in direct-drive robotic applications, electric and hybrid vehicle, especially in industrial applications for low-medium power range. However, the performance of the PMSM is sensitive to external load disturbances and system parameter in the plant. he stability of PMSM is a basic requirement in industrial manufacturing, therefore has received considerable attention. Investigations show that PMSM displays chaotic behavior when motor parameters lie in certain value [2, 3]. Chaos in the PMSM, which decreases the system performance, is highly undesirable in most engineering applications. Many scientists have devoted themselves to find efficient strategies to control chaos in PMSM [5 8]. Rom. Journ. Phys., Vol. 60, Nos. 9 10, P. 1409 1419, Bucharest, 2015
1410 Kalin Su, Chunlai Li 2 More interesting, chaos synchronization of motor, which implies the slave motor is designed to work following the master motor in the same rhyme of the angular phase and (or) the amplitude via an appropriate control scheme, has been attracted to recent theoretical and experimental investigations [9 13]. Although there are large numbers of control programs for synchronization of chaotic systems, investigation about synchronization scheme in chaotic motors is few. For instance, Ge proposed that Brushless DC motors (BLDCM) can be synchronized by backstepping technique [9]. Liu realized chaos synchronization of Brushless DC motors by the variable substitution control strategy [10]. Zhao developed a speed synchronization control strategy for multiple induction motors by employing total sliding mode control method [12]. Verrelli considered the synchronization problem for uncertain permanent magnet synchronous motors based on Fourier approximation theory [13]. All these existing synchronization schemes for chaotic motors require the system states for feedback, which are unrealistic and unacceptable in practical. So, it is significant to find an available method in the real operations for synchronization in motors. In this paper, a feedback controller which only needs to require the knowledge of the system output is proposed for synchronization of permanent magnet synchronous motor (PMSM). he synchronization algorithm is based on the theory of passive control. heoretical analysis shows that the synchronization error system between the driving and the response motor systems is not only passive but also asymptotically stable by the presented controller. Numerical simulations are provided to verify the effectiveness of the proposed design. 2. MODEL OF PMSM he dimensionless mathematical model of PMSM can be described by [2 4] did = id + ωiq + ud dt diq = iq ωid + γω+ uq, dt dω = σ ( iq ω ) L dt where i d and i q denote the stator currents; ω denotes the rotor angular frequency; 2 ud = KLUd / br 2 2 + K / br+ γ, uq = KLUq / br denote the stator voltages with L= Ld = L ; q is the external load torque; γ and σ = bl / JR are the operating L parameters. As considered in the study of bifurcations [2], we take the case u = u = = 0 in our work. d q L (1)
3 Synchronization criterion of chaotic permanent magnet synchronous motor via output feedback 1411 When applying equilibrium condition, one obtain three equilibria of system (1), as S 1 =(0, 0, 0), S 2,3=( γ 1, ± γ 1, ± γ 1). he previous investigation results show that, with the motor parameters σ and γ lying in certain area, all the three equilibria become unstable, and the PMSM exhibits chaos. he bifurcation diagram for γ [14, 24] versus with σ =5.46 and the typical chaotic attractor with γ = 20, σ = 5.46 are shown in Figure 1. Fig. 1 Bifurcation diagram and chaotic attractor of system (1): (a) bifurcation diagram; (b) chaotic attractor. 3. BASIC HEORY OF PASSIVE CONROL In this section, the basic conception of passivity of nonlinear affine system and passivity-based control method are summarized. Here, the following nonlinear differential equation is considered:
1412 Kalin Su, Chunlai Li 4 x = f( x) + g( xu ), (2) y = h( x) n m m where x R is the state variable; y R is the output; u R is the external input; f ( x ) and g( x ) are smooth vector fields; hx ( ) is a smooth mapping. he conception of passivity can be described as below [14]. Definition 1. If there exists a continuously differentiable positive semidefinite storage function V(x) satisfying u y V ( x) (3) then system (2) is said to be passive. Moreover, if u y V ( x) + y ρ( y) and y ρ ( y ) > 0 (4) for any y 0, we called the output is strictly passive. And the output is strictly passive if u y V ( x) + ( y) (5) for some positive definite function ( y). If system (2) is strictly passive or it s output is strictly passive and zero-state observable, then the origin of system (2) is asymptotically stable with u = 0. Furthermore, if the storage function V(x) is radially unbounded, the origin is globally asymptotically stable. Definition 2. If x = 0 is an asymptotically stable equilibrium of f ( x) and Lgh (0) is nonsingular, then system (2) is a minimum phase system. When system (2) is minimum phase and the distribution spanned by the vector fields g ( x), g ( x ) ( ) 1 2 gm x is involutive, system (2) can be represented as the following generalized form z = f ( z) + q( z, w) w, (6) w = α( z, w) + β( z, w) u where z = ϑ( x), β (, zw) is nonsingular for any (, zw. ) he physical meaning of passive system is that the energy of the nonlinear affine system can t increase unless an external source is supplied. In other words, a passive system cannot store more energy than that supplied externally. he passive system is stable naturally. We can utilize the input-output relationship based on energy-related considerations to analyze stability properties. he problem for passivity-based control is to design a feedback control scheme for achieving the passivity. It s necessary that system (2) is a minimum
5 Synchronization criterion of chaotic permanent magnet synchronous motor via output feedback 1413 phase system when utilizing feedback passivity. herefore, the system (6) cannot be made passive by feedback if its zero-dynamics is unstable. 4. SYNCHRONIZAION OF PMSM VIA OUPU FEEDBACK In order to observe the synchronization behavior of the two identical chaotic motor systems, we assume that the drive system is given as (1) with ud = uq = L = 0. And the response system with controller is expressed by di d = i d + ω iq dt diq = i q ω id + γω + u, (7) dt dω = σ ( i q ω ) dt where u is the control function to be designed. Defining the synchronization error e 1 = i d id, e 2 = i q iq, e 3 = ω ω, then one can obtain the error dynamical system de1 = e1+ i qe3+ ωe2 dt de2 = e2 + γe3 i de3 ωe1+ u. (8) dt de3 = σ ( e2 e3) dt It follows that if controlled system (8) is stabilized, then synchronization error e will converge to a zero equilibrium point, which means that the trajectories of response system with the controller u asymptotically synchronize the trajectories of the driving system. In order to achieve the goal, we introduce a dynamical variable e 3, which is an estimator of e 3. Define z = [ z1, z2, z3] = [ e1, e3, e 3 e3], y = e2, then system (8) can be represented as
1414 Kalin Su, Chunlai Li 6 dz1 = z1+ i qz2 + ω y dt dz2 = σ ( y z2) dt. (9) dz3 = σ z3 + yh( y) dt dy = y + γ z2 i d z2 ωz1+ u dt When write system (9) in the normal form (6), we have z = f ( z) + q( z, w) w, (10) w = α( z, w) + β( z, w) u where f ( z) = [ z + i z, σz, σz ], q (, ) [,, ( 0 z y = ω σ h y )] 0 1 q 2 2 3 α ( z, y) = y+ γ z2 i d z2 ωz1, β ( z, y) = 1 Suppose that the bound of i q is B, namely, i B q <. We define a Lyapunov function candidate η 2 1 2 1 2 V0( z) = z1 + z2 + z3 2 2 2 for the zero dynamics z = f ( z), where 2 0 η 4 σ /B. aking the derivative for V ( ) 0 z with respect to time yields V0 ( z) V 0( z) = f0( z) z = ηz ( z + i z ) σz σz 2 2 1 1 q 2 2 3 2 2 2 z1 i q z1z2 z2 z3 2 2 2 z1 Bi z1 z2 z2 z3 2 2 2 z1 2 i z1 z2 z2 z3 2 2 ηiz1 σ iz2 σz3 = η + η σ σ η + η σ σ η + ησ σ σ = ( ) < 0 herefore, the origin of the zero dynamics is stable. hus, system (9) is a minimum phase system. heorem 1. For the controlled error dynamical system (9), if the control scheme is designed as
7 Synchronization criterion of chaotic permanent magnet synchronous motor via output feedback 1415 u = ( σ + γ) e 3 + i d z2 + v, (11) hy ( ) = σ + γ where v is the external input signal, then the controlled system (9) is a passive system and globally asymptotically stabilized at the zero equilibrium point. Namely, the trajectories of response system with the controller u asymptotically synchronize the trajectories of the driving system. Proof. Choose the storage function as 1 2 V( z) = V0 ( z) + y 2 he function V( z ) is positive definite and radially unbounded. aking derivative of V( z ) with respect to time along the trajectory of the controlled error system (9), we can obtain V0( z) V0( z) V ( z) = f0( z) + q0( z, y) y+ yy z z V0 ( z) [ q 0( z, y ) + α( z, y ) + β( z, y ) u ] y z = [ ω z1+ σz2 + h( y) z3 y+ γ z2 i d z2 ωz1+ u] y = [( σ + γ) z + h( y) z y i z + u] y 2 3 d 2 When consider the control scheme, one have. (12) 2 V( z) ( y+ v) y = y + vy Namely, vy V 2 ( z) + ( y), where ( y) = y. So, the output of system (9) with the control scheme (11) is strictly passive. hus, system (8) will be asymptotically stabilized at the zero equilibrium point by using the control scheme (11). Remark. It s known that a dynamic system is more stable with lower energy. On the other hand, we know that a passive system cannot store more energy than that supplied externally. herefore, we have the conclusion from (12) that with the increasing of control parameter v, the stability of the controlled system (8) will reduce, although the synchronization is achieved, and that the initial values don t produce effect on the stability of synchronization. 5. NUMERICAL SIMULAION o demonstrate and verify the validity of the proposed synchronization
1416 Kalin Su, Chunlai Li 8 scheme, some numerical simulations are presented in this section. For comparing conveniently, in all the process of numerical simulation, the ODE45 method in Matlab is adopted to solve the nonlinear systems. Since the initial values don t affect the stability of synchronization, so in all the numerical process, we set the i (0), i (0), ω (0) = (3, 1, 12), initial values of systems (1) and (7) ( d q ) ( i' d(0), i' q(0), ω '(0)) = (1, 3, 3 ). First, the control parameter is set to be v = 0.2. he corresponding simulation results are shown in Fig. 2. Fig. 2 (a) shows the state trajectory; Fig. 2 (b) shows the synchronization error; and Fig. 2 (c) shows the time graphic of the control function u. As we know that the synchronization errors converge asymptotically to zero, and the two motor systems are indeed synchronized.
9 Synchronization criterion of chaotic permanent magnet synchronous motor via output feedback 1417 Fig. 2 Synchronization with v = 0.2: (a) state trajectory; (b) synchronization error; (c) control function u. hen, we select a larger control parameter as v = 10. he corresponding simulation results are shown in Figs. 3 (a), (b) and (c). As we know that the two motor systems are synchronized with a larger control parameter. However, both the synchronization errors and energy consumption of control function increase in comparison with a less parameter v.
1418 Kalin Su, Chunlai Li 10 Fig. 3 Synchronization with v = 10: (a) state trajectory; (b) synchronization error; (c) control function u. 6. CONCLUSION In this paper, a feedback controller which only needs to require the knowledge of the system output is proposed for synchronization of PMSM based on the theory of passive control. heoretical analysis shows that the synchronization error system between the driving and the response motor systems is not only passive but also asymptotically stable by the presented controller. Numerical simulations are provided to verify the effectiveness of the proposed design.
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