- Email: [email protected]
Chaos induced in Brushless DC Motor via current time-delayed feedback
Optik 125 (2014) 6589–6593
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Chaos induced in Brushless DC Motor via current time-delayed feedback Chun-Lai Li a,∗ , Wen Li a , Fu-Dong Li b a b
College of Physics and Electronics, Hunan Institute of Science and Technology, Yueyang 414000, China The Training Center of Hunan Electric Power Corporation, Changsha 410131, China
a r t i c l e
i n f o
Article history: Received 2 November 2013 Accepted 25 June 2014 Keywords: Induce chaos Brushless DC Motor Current time-delayed feedback
a b s t r a c t Chaos in motor system is beneficial in some niche fields such as industrial mixing processes. In this paper, we investigate the issue of inducing chaos in Brushless DC Motor (BLDCM). Based on the differentialgeometry control theory, a nonlinear function of current time-delayed control is derived. Analysis shows that the feedback scheme can induce the BLDCM operating in a stable regime to chaotic state across a large range of domain of control parameters, and if the values of control parameters ˛ and ˇ are set at an appropriate level, the variation of the time delay becomes insensitive to the occurrence of chaos, which is available in the compensation with inaccuracy of time-delay control. © 2014 Elsevier GmbH. All rights reserved.
PACS: 05.45.−a
1. Introduction Chaos is an interesting phenomenon in nonlinear dynamical systems holding the features of apparent randomness and unpredictable behavior [1–5]. The existence of chaos in motor systems was discovered by Kuroe in 1989 [6]. Since then, investigation on chaos of motor drive systems has become a field of active research due to its potential applications in many areas, such as industrial machinery, hybrid vehicle and electrical submersibles thruster drives. Recently, the secure and stable operation of motor drive systems has been researched extensively. Analysis indicates that with systemic parameters falling into a certain area, the motor drive system displays chaotic behavior [1,6–8]. The occurrence of chaos in motors is highly undesirable in most engineering applications for its performance. Thus, an extensive research has been carried out toward the chaos control in several types of motor drive systems [9–11]. However, chaos in motor systems is beneficial in some niche processes, such as fluid mixing and vibratory compactor. In the industrial mixing processes, an approach for improving the mixing is to increase the stirring rate of motors, but it is some times impractical for applications. For example, in the product of
∗ Corresponding author. E-mail address: [email protected] (C.-L. Li). http://dx.doi.org/10.1016/j.ijleo.2014.06.033 0030-4026/© 2014 Elsevier GmbH. All rights reserved.
proteins or many other shear-sensitive macromolecules, a fast stirring can cause high shear rates [12]. In recent years, industrial mixing has been proposed to improve the energy efficiency by using chaotic-motion motor [13]. This mechanical mean not only produces the desired chaotic mixing, but also offers the merits of high controllability and high flexibility. In the process of compact soil and concrete materials, a chaotic compactor can offer better compaction performance than a conventional compactor. There are some works on the chaos inducing in motor systems. Qin et al. [14] investigated the influence of random phase (random noise) on the dynamical behavior of a single-machine-infinite bus power system. It is found that when the phase disturbance is weak, power system is stable. But with the increase of disturbance intensity, power system become unstable and fall into chaos finally. However, random noise is not conducive to control. Wei et al. [15] reported that direct time-delayed current can induce and enhance chaos in permanent magnet synchronous motor, which needs a large control and is seemingly impractical for applications. Zhu studied the issue of chaos inducing of asynchronous motor by using time-delay feedback rotor angular frequency [16]. Analogously, Li studied the chaos induced in permanent magnet DC motor via timedelay feedback rotor speed [17]. Obviously, the methods by Zhu and Li are impractical for applications since the time-delayed rotor speed is not easy to get and measure. In this work, we focus on the issue of inducing chaos in Brushless DC Motor (BLDCM) via nonlinear time-delayed current feedback. Different from the existent works [15–17], the proposed scheme
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C.-L. Li et al. / Optik 125 (2014) 6589–6593
promises the occurrence of chaos according to Li–Yorke criteria [18]. Based on the differential-geometry control theory, an analytical solution of current time-delayed function is derived, which provides a systematic approach for design and completely avoids blind numerical search. By using this approach, we can present a standard procedure for the design of chaos inducing in BLDCM. Our analysis shows that the feedback scheme can induce the BLDCM operating in a stable regime to chaotic state across a large range of domain of control parameters, and if the values of control parameters ˛ and ˇ are set at an appropriate level, the variation of the time delay becomes insensitive to the occurrence of chaos. The characteristic of insensitivity is available in the compensation with inaccuracy of time-delay control.
where z ∈ Rn is the state variable; y ∈ Rm is the output signal; u(y(t − )) ∈ Rm is the control function to be designed; f (z) and h(z) are smooth vector fields; b(z) is the output function satisfying b(0) = 0. To obtain the output function b(z) and further to determine the control function u(y(t − )), we would introduce an indispensable Lemma [22]. In the first instance, for notation, the Lie derivative of function b(z) with respect to field f (z) can be defined recursively by
Lf b(z) =
Lf2 b(z)
=
2. Description of BLDCM
The dimensionless mathematical model of BLDCM can be described by [19,20]
⎧ did ⎪ = −ıid + ωiq + ud ⎪ ⎪ dt ⎪ ⎪ ⎨
diq = −iq − ωid + ω + uq dt
⎪ ⎪ ⎪ ⎪ ⎪ ⎩ dω = (iq − ω) + i iq − TL d
(1)
dt
f (z)
∂(Lf b(z))
T
∂z ∂(Lf2 b(z))
f (z)
T f (z)
∂z
And the Lie bracket of the two vector fields f (z) and h(z), which can further generate a new smooth vector field, is defined recursively as follows: adf h(z) = adf2 h(z) = adf3 h(z) =
∂h(z) ∂f f (z) − h(z) ∂z ∂z ∂(adf h(z)) ∂z ∂(adf2 h(z)) ∂z
f (z) −
∂f (adf h(z)) ∂z
f (z) −
∂f (adf2 h(z)) ∂z
···
Lemma 1. The controlled system (2) has relative degree n at a point z* and can be described by an nth-order linear differential equation on a neighborhood M of z* if and only if
3. Inducing chaos in BLDCM via current time-delayed feedback 3.1. Preliminaries
We know that a dynamic system with time-delay is of inherently infinite dimension, which leads complicated dynamics obtainable even in a simple system. Wang originated that chaos can be induced in a stable linear system by a simple time-delay feedback control [21]. Then-after Zhou provided a rigorous proof for this method [18]. In accordance with the theory of nonlinear control, if the relative degree of a stable nonlinear system is equal to the order, one can linearize this system exactly. The practicability of the linearization means that the time-delay feedback control for inducing chaos in linear systems can be applied to design for inducing chaos in nonlinear dynamic systems. This is the basic fundamental to obtain an analytical solution of the time-delayed controller if we can hold the linearization transformation for the BLDCM. In this section, we investigate the issue of inducing chaos in BLDCM which operates in a stable regime far away from chaotic behavior via current time-delayed feedback. The significance of our work is to provide a standard design procedure to obtain a current time-delayed feedback controller that ensures the occurrence of chaos in stable BLDCM without exploratory. The controlled nonlinear differential system is considered here, which is the most common and important in practical engineering, and represented as
y = b(z)
Lf3 b(z) =
T
···
where id and iq denote the direct-axis current and quadrature current, respectively; ω denotes the angle speed; ud , uq denote the direct-axis and quadrature stator voltage components, respectively; TL is the load torque; ı, , and are the operating parameters. In our paper, we only take the case ud = uq = TL = 0. The bifurcation diagram for ı ∈ [0.1, 0.8] versus with = 60, = 4.35, = 0.26 is shown in Fig. 1(a). And the typical phase portrait with ı = 0.4 and ı = 0.32 are shown in Fig. 1(b) and (c).
⎧ ⎨ dz = f (z) + h(z)u(y(t − )) dt ⎩
∂b(z) ∂z
(2)
(I) rank h(z) adf h(z)
···
adfn−1 h(z) = n for z ∈ M.
(II) = span{h(z), adf h(z), . . ., adfn−2 h(z)} is involutive on neighborhood M. In this case, the output function y = b(z) of system (2) is a solution of the partial differential equations: ∂b(z) [h(z), adf h(z), . . ., adfn−2 h(z)] = 0 ∂z
3.2. Derivation of current time-delayed feedback controller The controlled BLDCM can be expressed as the form as (2)
⎧ ⎨ dz = f (z) + h(z)u(y(t − )) dt ⎩
(3)
y = b(z)
in which
⎡ T
z = (id , iq , ω) ,
⎢
−ıid + ωiq
f (z) = ⎣ −iq − ωid + ω (iq − ω) + id iq
⎤
⎡ ⎤ 0
⎥ ⎦ , h(z) = ⎣ 0 ⎦ 1
(4)
C.-L. Li et al. / Optik 125 (2014) 6589–6593
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Fig. 1. Bifurcation diagram and phase portrait of system (1): (a) bifurcation diagram; (b) phase portrait with ı = 0.4; and (c) phase portrait with ı = 0.32.
It follows from the definition of Lie derivative and Lie bracket that
⎡
⎤
⎢ ⎥ ∂h ∂f f (z) − h(z) = ⎢ i − ⎥ ⎣d ⎦ ∂z ∂z
adf h(z) =
adf2 h(z)
−iq
=
∂(adf h(z))
⎡ ⎢
∂z
(7)
⎥ ⎥ ⎦
Obviously, the control scheme in our work is practical for applications since this scheme comprises the time-delayed current but excludes the time-delayed rotor speed.
iq2 − id2 + ( − )id + + 2
A=
⎡ =
adf2 h(z)
h(z) adf h(z) 0
−iq
4. Numerical analysis of BLDCM with current time-delayed feedback
⎤
(1 − − ı)iq
⎢ 0 i − (1 + − ı)i − − ⎣ d d
⎥ ⎦
iq2 − id2 + ( − )id + + 2
1
After transformation of elementary row, one can determine the rank of the matrix A as rank[A] = 3. So, the first condition of Lemma 1 is satisfied. Following from the defined vector fields and Lie bracket, we obtain the following expressions:
⎡
⎢
⎡ ⎢
0
= rank ⎣ 0
[h(z)
−iq
0
rank h(z) adf h(z) = rank ⎣ 0 rank h(z) adf h(z)
⎤ ⎥
id − ⎦ = 2
(1 − − ı)iq
id −
(1 + − ı)id − −
⎤ ⎥ ⎦=2
iq 2 − id2 + ( − )id + + 2
So, the linear space = span{h(z), adf h(z)} is involutive in M. The controlled system holds the relative degree 3 at the point z*. In the light of Lemma 1, we obtain the following output differential equations: ∂b(z) ∂b(z) h(z) = =0 ∂z ∂ω ∂b(z) ∂b(z) ∂b(z) ∂b(z) adf h(z) = −iq =0 + (id − ) + ∂z ∂id ∂iq ∂ω which lead to the solution b(z) = a1 id2 + a2 id + b1 iq2 . When setting a1 = b1 , a2 = −2b1 , we have b(z) = b1 id2 − 2b1 id + b1 iq2
In this section, we carry out the numerical simulations to confirm the effectiveness of the proposed analytical control function of time-delay feedback for inducing chaos in BLDCM. Through careful bifurcation analysis aligning with the variation of control parameters, we examine the regular pattern of chaos in BLDCM across a parametric domain. The effects about the variation of the controller parameters ˛, ˇ, can provide guidance concerning how to select the control parameters to induce chaos in BLDCM. For the convenience in later of this section, the system’s parameters are fixed as ı = 0.32, = 60, = 4.35, = 0.26, with which the BLDCM operates in a stable regime, and these parameters are not considered for the effect on the system dynamics. 4.1. Dynamics analysis of BLDCM by varying parameter ˛
1 adf h(z)]
−iq
1
(6)
u(y(t − )) = ˛ sin[ˇ(0.01id2 (t − ) − 1.2id (t − ) + 0.01iq2 (t − ))]
⎤
(1 − − ı)iq
= ⎢ (1 + − ı)i − − ⎣ d
b(z) = 0.01id2 − 1.2id + 0.01iq2
To this end, according to the analysis presented in [22], we can choose a small-amplitude time-delay feedback in the form u(y(t − )) = ˛ sin[ˇy(t − )] to induce the BLDCM to be chaotic. This, in turn, implies that we can take the feedback controller
∂f (adf h(z)) f (z) − ∂z
and we get
Considering = 60 and letting b1 = 0.01, one achieve
(5)
Our purpose here is to study how the control gain ˛ affects the system behaviors. In numerical analysis, we let the control gain ˛ varies in the region [–10, 10], and consider three cases of control parameters, Case I: ( = 0.3, ˇ = 5.6), Case II: ( = 0.3, ˇ = 1.6) and Case III: ( = 1.2, ˇ = 5.6) for the comparison of the influences on inducing chaos. The global bifurcation diagram of the controlled BLDCM system (3) for Case I is displayed in Fig. 2(a) and closer views particularly for ˛ ∈ [0.5, 1.1] and ˛ ∈ [2.9, 3.4] are depicted in Fig. 2(b) and (c). Similarly, the global bifurcation diagram and the corresponding partial enlarged view for Case II and Case III are plotted in Figs. 3 and 4, respectively. It is known from Fig. 2 that the line dots near the origin correspond to simple periodic motions, and the cloudy dots far from the origin are associated with chaotic motions. We can note clearly that there are critical values for ˛ = 2.995 and ˛ = 3.331 where a burst of periodic motions arises. Comparing Case II with Case I, the parameter region on occurrence of simple periodic motions near the origin reduces when the value of ˇ decreases, and the number of visible periodic windows increases, as shown in Fig. 3. When increasing the time delay from = 0.3 (Case I) to = 1.2 (Case
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C.-L. Li et al. / Optik 125 (2014) 6589–6593
Fig. 2. (a) Global bifurcation diagram versus ˛ for Case I; (b, c) the partial enlarged view.
Fig. 3. (a) Global bifurcation diagram versus ˛ for Case II and (b) the partial enlarged view.
III), it is found that the effects about the variation of the delay are small, as shown in Fig. 4. We know that there always exists inherent time-delay of mechanical and control operations when processing data in a real-time control, and the characteristic of insensitivity is available in the compensation with inaccuracy of time-delay control. 4.2. Dynamics analysis of BLDCM by varying parameter ˇ We now examine how the controlling parameter ˇ affects the system behaviors through bifurcation analysis. The global bifurcation diagram of the controlled BLDCM system (3) for Case I: (˛ = 1.7, = 0.3), Case II: (˛ = 6.0, = 0.3) and Case III: (˛ = 1.7, = 2.3) versus with ˇ are plotted Fig. 5(a)–(c), respectively. It reveals clearly from Fig. 5(a) that there exists the first periodic response near the origin, and sequential existences of chaos appear intermittently in between the states of periodic response. Note that chaotic motions intermittently occur depending on the selecting of controlling parameter ˇ. Comparing Case II with Case I, the parameter domain on occurrence of periodic motion near the origin reduces when lifting up the value of gain ˛, indicating that a smaller value of ˇ is required to induce chaos in BLDCM. We further note that the parameter domain on occurrence of periodic windows faraway from the origin increases when lifting up the value of gain ˛, which implies less parametric domain of occurrence of chaos holds, as shown in Fig. 5(b).
Comparing Case III with Case I, it is found that the effects about the variation of the delay are small, as shown in Fig. 5(c). 4.3. Dynamics analysis of BLDCM by varying parameter Also we are concerned with how the controlling parameter of time delay affects the system behaviors. Particularly we take an interest in if inducing chaos is widely available in parametric region of . Resembling the above analysis, we consider three cases of control parameters, Case I: (˛ = 1.7, ˇ = 5.6), Case II: (˛ = 8, ˇ = 5.6) and Case III: (˛ = 1.7, ˇ = 9) for the comparison of the influences on inducing chaos. Fig. 6(a) shows the global bifurcation of a small control gain ˛ = 1.7 for ∈ [0, 6]. The first existence of the chaotic state is allocated in the interval of 0 < < 0.2, and sequential existences of chaos appear intermittently in between complicated quasiperiodic motions. Note that chaotic motions intermittently occur depending on the selecting of . In other words, the state of system response is sensitive to the control parameter of time delay when feedback control gain ˛ is set at the small value for inducing chaos. We note that the increased control gain ˛ can extend the parametric domain of chaos. If the value of control gain ˛ is set at an appropriate level, the variation of the time delay becomes insensitive to the occurrence of chaos. As the control gain is lifted up to ˛ = 8, the global bifurcation is plotted in Fig. 6(b). In this case, the existence of chaotic state persists throughout the whole parametric domain.
Fig. 4. (a) Global bifurcation diagram versus ˛ for Case III; (b) the partial enlarged view.
C.-L. Li et al. / Optik 125 (2014) 6589–6593
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Fig. 5. Global bifurcation diagram versus ˇ for (a) Case I; (b) Case II; and (c) Case III.
Fig. 6. Global bifurcation diagram versus for (a) Case I; (b) Case II; and (c) Case III.
However, when increasing the value of controlling parameter ˇ, the number of visible quasi-periodic windows increases. When lifting up the gain from ˇ = 5.6 (Case I) to ˇ = 9 (Case III), the global bifurcation is shown in Fig. 6(c), which implies less parametric domain of chaos holds. 5. Conclusion In this paper, we focus on the issue of inducing chaos in Brushless DC Motor (BLDCM) via nonlinear time-delayed current feedback. Different from the existent works [15–17], the proposed scheme promises the occurrence of chaos according to Li–Yorke criteria. Based on the differential-geometry control theory, an analytical solution of current time-delayed function is derived, which provides a systematic approach for design and completely avoids blind numerical search. By using this approach, we can present a standard procedure for the design of chaos inducing in BLDCM. Through careful bifurcation analysis aligning with the variation of control parameters ˛, ˇ, , we examine the regular pattern of chaos in BLDCM across a parametric domain. The effects about the variation of the controller parameters can provide guidance concerning how to select the control parameters for inducing chaos in BLDCM. Acknowledgments This work was supported by Hunan Provincial Natural Science Foundation of China (Grant No. 10JJ2044) and the Research Foundation of Education Bureau of Hunan Province, China (Grant No. 13C372). References [1] Z. Li, J. Park, B. Zhang, G. Chen, Bifurcations and chaos in a permanent-magnet synchronous motor, IEEE Trans. Circuits Syst. I 49 (2002) 383–387. [2] C. Li, L. Wu, H. Li, Y. Tong, A novel chaotic system and its topological horseshoe, Nonlinear Anal. Model. Control 18 (2013) 66–77.
[3] C. Li, Tracking control and generalized projective synchronization of a class of hyperchaotic system with unknown parameter and disturbance, Commun. Nonlinear Sci. Numer. Simul. 17 (2012) 405–413. [4] C. Li, H. Li, Robust control for a class of chaotic and hyperchaotic systems via linear state feedback, Phys. Scr. 85 (2012) 025007. [5] C. Li, Y. Tong, H. Li, K. Su, Adaptive impulsive synchronization of a class of chaotic and hyperchaotic systems, Phys. Scr. 86 (2012) 055003. [6] Y. Kuroe, S. Hayashi, Analysis of bifurcation in power electronic induction motor drive system, in: Proceedings of the IEEE Power Electronics Specialists Conference, 1989, pp. 923–930. [7] Z. Jing, C. Yu, G. Chen, Complex dynamics in a permanent-magnet synchronous motor model, Chaos Solitons Fractals 22 (2004) 831–848. [8] C. Li, S. Yu, X. Luo, Fractional-order permanent magnet synchronous motor and its adaptive chaotic control, Chin. Phys. B 21 (2012) 100506. [9] H. Ren, D. Liu, Nonlinear feedback control of chaos in permanent magnet synchronous motor, IEEE Trans. Circuits Syst. II 53 (2006) 45–50. [10] D. Wei, X. Luo, Passivity-based adaptive control of chaotic oscillations in power system, Chaos Solitons Fractals 31 (2007) 665–671. [11] A. Zaher, A nonlinear controller design for permanent magnet motors using a synchronization-based technique inspired from the Lorenz system, Chaos 18 (2008) 013111. [12] D.J. Lamberto, F.J. Muzzio, P.D. Swanson, A.L. Tonkovich, Using time-dependent rpm to enhance mixing in stirred vessels, Chem. Eng. Sci. 51 (1996) 733–741. [13] M.M. Alvarez-Hernández, T. Shinbrot, J. Zalc, F.J. Muzzio, Practical chaotic mixing, Chem. Eng. Sci. 57 (2002) 3749–3753. [14] Y. Qin, X. Luo, D. Wei, Random-phase-induced chaos in power systems, Chin. Phys. B 19 (2010) 050511. [15] D. Wei, B. Zhang, D. Qiu, X. Luo, Effects of current time-delayed feedback on the dynamics of a permanent-magnet synchronous motor, IEEE Trans. Circuits Syst. II 57 (2010) 456–460. [16] H. Zhu, J. Chen, Z. Wang, Anti-control of chaos in induction motor drives by time delay feedback, Proc. CSEE 24 (2004) 156–159. [17] J. Li, J. Chen, K. Chau, Anticontrol of chaos in permanent magnet DC motor, Proc. CSEE 26 (2006) 77–81. [18] T. Zhou, G. Chen, Q. Yang, A simple time-delay feedback anticontrol method made rigorous, Chaos 14 (2004) 662–668. [19] N. Hemati, Strange attractors in brushless DC motor, IEEE Trans. Circuits Syst. I 41 (1994) 40–45. [20] N. Hemati, M. Leu, A complete model characterization of brushless DC motors, IEEE Trans. Ind. Appl. 28 (1992) 172–180. [21] X. Wang, G. Chen, X. Yu, Anticontrol of chaos in continuous-time systems via time-delay feedback, Chaos 10 (2000) 771–779. [22] A. Isidori, Nonlinear Control Systems, 3rd ed., Springer-Verlag, Berlin, 1995.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Chaos induced in Brushless DC Motor via current time-delayed feedback Chun-Lai Li a,∗ , Wen Li a , Fu-Dong Li b a b
College of Physics and Electronics, Hunan Institute of Science and Technology, Yueyang 414000, China The Training Center of Hunan Electric Power Corporation, Changsha 410131, China
a r t i c l e
i n f o
Article history: Received 2 November 2013 Accepted 25 June 2014 Keywords: Induce chaos Brushless DC Motor Current time-delayed feedback
a b s t r a c t Chaos in motor system is beneficial in some niche fields such as industrial mixing processes. In this paper, we investigate the issue of inducing chaos in Brushless DC Motor (BLDCM). Based on the differentialgeometry control theory, a nonlinear function of current time-delayed control is derived. Analysis shows that the feedback scheme can induce the BLDCM operating in a stable regime to chaotic state across a large range of domain of control parameters, and if the values of control parameters ˛ and ˇ are set at an appropriate level, the variation of the time delay becomes insensitive to the occurrence of chaos, which is available in the compensation with inaccuracy of time-delay control. © 2014 Elsevier GmbH. All rights reserved.
PACS: 05.45.−a
1. Introduction Chaos is an interesting phenomenon in nonlinear dynamical systems holding the features of apparent randomness and unpredictable behavior [1–5]. The existence of chaos in motor systems was discovered by Kuroe in 1989 [6]. Since then, investigation on chaos of motor drive systems has become a field of active research due to its potential applications in many areas, such as industrial machinery, hybrid vehicle and electrical submersibles thruster drives. Recently, the secure and stable operation of motor drive systems has been researched extensively. Analysis indicates that with systemic parameters falling into a certain area, the motor drive system displays chaotic behavior [1,6–8]. The occurrence of chaos in motors is highly undesirable in most engineering applications for its performance. Thus, an extensive research has been carried out toward the chaos control in several types of motor drive systems [9–11]. However, chaos in motor systems is beneficial in some niche processes, such as fluid mixing and vibratory compactor. In the industrial mixing processes, an approach for improving the mixing is to increase the stirring rate of motors, but it is some times impractical for applications. For example, in the product of
∗ Corresponding author. E-mail address: [email protected] (C.-L. Li). http://dx.doi.org/10.1016/j.ijleo.2014.06.033 0030-4026/© 2014 Elsevier GmbH. All rights reserved.
proteins or many other shear-sensitive macromolecules, a fast stirring can cause high shear rates [12]. In recent years, industrial mixing has been proposed to improve the energy efficiency by using chaotic-motion motor [13]. This mechanical mean not only produces the desired chaotic mixing, but also offers the merits of high controllability and high flexibility. In the process of compact soil and concrete materials, a chaotic compactor can offer better compaction performance than a conventional compactor. There are some works on the chaos inducing in motor systems. Qin et al. [14] investigated the influence of random phase (random noise) on the dynamical behavior of a single-machine-infinite bus power system. It is found that when the phase disturbance is weak, power system is stable. But with the increase of disturbance intensity, power system become unstable and fall into chaos finally. However, random noise is not conducive to control. Wei et al. [15] reported that direct time-delayed current can induce and enhance chaos in permanent magnet synchronous motor, which needs a large control and is seemingly impractical for applications. Zhu studied the issue of chaos inducing of asynchronous motor by using time-delay feedback rotor angular frequency [16]. Analogously, Li studied the chaos induced in permanent magnet DC motor via timedelay feedback rotor speed [17]. Obviously, the methods by Zhu and Li are impractical for applications since the time-delayed rotor speed is not easy to get and measure. In this work, we focus on the issue of inducing chaos in Brushless DC Motor (BLDCM) via nonlinear time-delayed current feedback. Different from the existent works [15–17], the proposed scheme
6590
C.-L. Li et al. / Optik 125 (2014) 6589–6593
promises the occurrence of chaos according to Li–Yorke criteria [18]. Based on the differential-geometry control theory, an analytical solution of current time-delayed function is derived, which provides a systematic approach for design and completely avoids blind numerical search. By using this approach, we can present a standard procedure for the design of chaos inducing in BLDCM. Our analysis shows that the feedback scheme can induce the BLDCM operating in a stable regime to chaotic state across a large range of domain of control parameters, and if the values of control parameters ˛ and ˇ are set at an appropriate level, the variation of the time delay becomes insensitive to the occurrence of chaos. The characteristic of insensitivity is available in the compensation with inaccuracy of time-delay control.
where z ∈ Rn is the state variable; y ∈ Rm is the output signal; u(y(t − )) ∈ Rm is the control function to be designed; f (z) and h(z) are smooth vector fields; b(z) is the output function satisfying b(0) = 0. To obtain the output function b(z) and further to determine the control function u(y(t − )), we would introduce an indispensable Lemma [22]. In the first instance, for notation, the Lie derivative of function b(z) with respect to field f (z) can be defined recursively by
Lf b(z) =
Lf2 b(z)
=
2. Description of BLDCM
The dimensionless mathematical model of BLDCM can be described by [19,20]
⎧ did ⎪ = −ıid + ωiq + ud ⎪ ⎪ dt ⎪ ⎪ ⎨
diq = −iq − ωid + ω + uq dt
⎪ ⎪ ⎪ ⎪ ⎪ ⎩ dω = (iq − ω) + i iq − TL d
(1)
dt
f (z)
∂(Lf b(z))
T
∂z ∂(Lf2 b(z))
f (z)
T f (z)
∂z
And the Lie bracket of the two vector fields f (z) and h(z), which can further generate a new smooth vector field, is defined recursively as follows: adf h(z) = adf2 h(z) = adf3 h(z) =
∂h(z) ∂f f (z) − h(z) ∂z ∂z ∂(adf h(z)) ∂z ∂(adf2 h(z)) ∂z
f (z) −
∂f (adf h(z)) ∂z
f (z) −
∂f (adf2 h(z)) ∂z
···
Lemma 1. The controlled system (2) has relative degree n at a point z* and can be described by an nth-order linear differential equation on a neighborhood M of z* if and only if
3. Inducing chaos in BLDCM via current time-delayed feedback 3.1. Preliminaries
We know that a dynamic system with time-delay is of inherently infinite dimension, which leads complicated dynamics obtainable even in a simple system. Wang originated that chaos can be induced in a stable linear system by a simple time-delay feedback control [21]. Then-after Zhou provided a rigorous proof for this method [18]. In accordance with the theory of nonlinear control, if the relative degree of a stable nonlinear system is equal to the order, one can linearize this system exactly. The practicability of the linearization means that the time-delay feedback control for inducing chaos in linear systems can be applied to design for inducing chaos in nonlinear dynamic systems. This is the basic fundamental to obtain an analytical solution of the time-delayed controller if we can hold the linearization transformation for the BLDCM. In this section, we investigate the issue of inducing chaos in BLDCM which operates in a stable regime far away from chaotic behavior via current time-delayed feedback. The significance of our work is to provide a standard design procedure to obtain a current time-delayed feedback controller that ensures the occurrence of chaos in stable BLDCM without exploratory. The controlled nonlinear differential system is considered here, which is the most common and important in practical engineering, and represented as
y = b(z)
Lf3 b(z) =
T
···
where id and iq denote the direct-axis current and quadrature current, respectively; ω denotes the angle speed; ud , uq denote the direct-axis and quadrature stator voltage components, respectively; TL is the load torque; ı, , and are the operating parameters. In our paper, we only take the case ud = uq = TL = 0. The bifurcation diagram for ı ∈ [0.1, 0.8] versus with = 60, = 4.35, = 0.26 is shown in Fig. 1(a). And the typical phase portrait with ı = 0.4 and ı = 0.32 are shown in Fig. 1(b) and (c).
⎧ ⎨ dz = f (z) + h(z)u(y(t − )) dt ⎩
∂b(z) ∂z
(2)
(I) rank h(z) adf h(z)
···
adfn−1 h(z) = n for z ∈ M.
(II) = span{h(z), adf h(z), . . ., adfn−2 h(z)} is involutive on neighborhood M. In this case, the output function y = b(z) of system (2) is a solution of the partial differential equations: ∂b(z) [h(z), adf h(z), . . ., adfn−2 h(z)] = 0 ∂z
3.2. Derivation of current time-delayed feedback controller The controlled BLDCM can be expressed as the form as (2)
⎧ ⎨ dz = f (z) + h(z)u(y(t − )) dt ⎩
(3)
y = b(z)
in which
⎡ T
z = (id , iq , ω) ,
⎢
−ıid + ωiq
f (z) = ⎣ −iq − ωid + ω (iq − ω) + id iq
⎤
⎡ ⎤ 0
⎥ ⎦ , h(z) = ⎣ 0 ⎦ 1
(4)
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Fig. 1. Bifurcation diagram and phase portrait of system (1): (a) bifurcation diagram; (b) phase portrait with ı = 0.4; and (c) phase portrait with ı = 0.32.
It follows from the definition of Lie derivative and Lie bracket that
⎡
⎤
⎢ ⎥ ∂h ∂f f (z) − h(z) = ⎢ i − ⎥ ⎣d ⎦ ∂z ∂z
adf h(z) =
adf2 h(z)
−iq
=
∂(adf h(z))
⎡ ⎢
∂z
(7)
⎥ ⎥ ⎦
Obviously, the control scheme in our work is practical for applications since this scheme comprises the time-delayed current but excludes the time-delayed rotor speed.
iq2 − id2 + ( − )id + + 2
A=
⎡ =
adf2 h(z)
h(z) adf h(z) 0
−iq
4. Numerical analysis of BLDCM with current time-delayed feedback
⎤
(1 − − ı)iq
⎢ 0 i − (1 + − ı)i − − ⎣ d d
⎥ ⎦
iq2 − id2 + ( − )id + + 2
1
After transformation of elementary row, one can determine the rank of the matrix A as rank[A] = 3. So, the first condition of Lemma 1 is satisfied. Following from the defined vector fields and Lie bracket, we obtain the following expressions:
⎡
⎢
⎡ ⎢
0
= rank ⎣ 0
[h(z)
−iq
0
rank h(z) adf h(z) = rank ⎣ 0 rank h(z) adf h(z)
⎤ ⎥
id − ⎦ = 2
(1 − − ı)iq
id −
(1 + − ı)id − −
⎤ ⎥ ⎦=2
iq 2 − id2 + ( − )id + + 2
So, the linear space = span{h(z), adf h(z)} is involutive in M. The controlled system holds the relative degree 3 at the point z*. In the light of Lemma 1, we obtain the following output differential equations: ∂b(z) ∂b(z) h(z) = =0 ∂z ∂ω ∂b(z) ∂b(z) ∂b(z) ∂b(z) adf h(z) = −iq =0 + (id − ) + ∂z ∂id ∂iq ∂ω which lead to the solution b(z) = a1 id2 + a2 id + b1 iq2 . When setting a1 = b1 , a2 = −2b1 , we have b(z) = b1 id2 − 2b1 id + b1 iq2
In this section, we carry out the numerical simulations to confirm the effectiveness of the proposed analytical control function of time-delay feedback for inducing chaos in BLDCM. Through careful bifurcation analysis aligning with the variation of control parameters, we examine the regular pattern of chaos in BLDCM across a parametric domain. The effects about the variation of the controller parameters ˛, ˇ, can provide guidance concerning how to select the control parameters to induce chaos in BLDCM. For the convenience in later of this section, the system’s parameters are fixed as ı = 0.32, = 60, = 4.35, = 0.26, with which the BLDCM operates in a stable regime, and these parameters are not considered for the effect on the system dynamics. 4.1. Dynamics analysis of BLDCM by varying parameter ˛
1 adf h(z)]
−iq
1
(6)
u(y(t − )) = ˛ sin[ˇ(0.01id2 (t − ) − 1.2id (t − ) + 0.01iq2 (t − ))]
⎤
(1 − − ı)iq
= ⎢ (1 + − ı)i − − ⎣ d
b(z) = 0.01id2 − 1.2id + 0.01iq2
To this end, according to the analysis presented in [22], we can choose a small-amplitude time-delay feedback in the form u(y(t − )) = ˛ sin[ˇy(t − )] to induce the BLDCM to be chaotic. This, in turn, implies that we can take the feedback controller
∂f (adf h(z)) f (z) − ∂z
and we get
Considering = 60 and letting b1 = 0.01, one achieve
(5)
Our purpose here is to study how the control gain ˛ affects the system behaviors. In numerical analysis, we let the control gain ˛ varies in the region [–10, 10], and consider three cases of control parameters, Case I: ( = 0.3, ˇ = 5.6), Case II: ( = 0.3, ˇ = 1.6) and Case III: ( = 1.2, ˇ = 5.6) for the comparison of the influences on inducing chaos. The global bifurcation diagram of the controlled BLDCM system (3) for Case I is displayed in Fig. 2(a) and closer views particularly for ˛ ∈ [0.5, 1.1] and ˛ ∈ [2.9, 3.4] are depicted in Fig. 2(b) and (c). Similarly, the global bifurcation diagram and the corresponding partial enlarged view for Case II and Case III are plotted in Figs. 3 and 4, respectively. It is known from Fig. 2 that the line dots near the origin correspond to simple periodic motions, and the cloudy dots far from the origin are associated with chaotic motions. We can note clearly that there are critical values for ˛ = 2.995 and ˛ = 3.331 where a burst of periodic motions arises. Comparing Case II with Case I, the parameter region on occurrence of simple periodic motions near the origin reduces when the value of ˇ decreases, and the number of visible periodic windows increases, as shown in Fig. 3. When increasing the time delay from = 0.3 (Case I) to = 1.2 (Case
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Fig. 2. (a) Global bifurcation diagram versus ˛ for Case I; (b, c) the partial enlarged view.
Fig. 3. (a) Global bifurcation diagram versus ˛ for Case II and (b) the partial enlarged view.
III), it is found that the effects about the variation of the delay are small, as shown in Fig. 4. We know that there always exists inherent time-delay of mechanical and control operations when processing data in a real-time control, and the characteristic of insensitivity is available in the compensation with inaccuracy of time-delay control. 4.2. Dynamics analysis of BLDCM by varying parameter ˇ We now examine how the controlling parameter ˇ affects the system behaviors through bifurcation analysis. The global bifurcation diagram of the controlled BLDCM system (3) for Case I: (˛ = 1.7, = 0.3), Case II: (˛ = 6.0, = 0.3) and Case III: (˛ = 1.7, = 2.3) versus with ˇ are plotted Fig. 5(a)–(c), respectively. It reveals clearly from Fig. 5(a) that there exists the first periodic response near the origin, and sequential existences of chaos appear intermittently in between the states of periodic response. Note that chaotic motions intermittently occur depending on the selecting of controlling parameter ˇ. Comparing Case II with Case I, the parameter domain on occurrence of periodic motion near the origin reduces when lifting up the value of gain ˛, indicating that a smaller value of ˇ is required to induce chaos in BLDCM. We further note that the parameter domain on occurrence of periodic windows faraway from the origin increases when lifting up the value of gain ˛, which implies less parametric domain of occurrence of chaos holds, as shown in Fig. 5(b).
Comparing Case III with Case I, it is found that the effects about the variation of the delay are small, as shown in Fig. 5(c). 4.3. Dynamics analysis of BLDCM by varying parameter Also we are concerned with how the controlling parameter of time delay affects the system behaviors. Particularly we take an interest in if inducing chaos is widely available in parametric region of . Resembling the above analysis, we consider three cases of control parameters, Case I: (˛ = 1.7, ˇ = 5.6), Case II: (˛ = 8, ˇ = 5.6) and Case III: (˛ = 1.7, ˇ = 9) for the comparison of the influences on inducing chaos. Fig. 6(a) shows the global bifurcation of a small control gain ˛ = 1.7 for ∈ [0, 6]. The first existence of the chaotic state is allocated in the interval of 0 < < 0.2, and sequential existences of chaos appear intermittently in between complicated quasiperiodic motions. Note that chaotic motions intermittently occur depending on the selecting of . In other words, the state of system response is sensitive to the control parameter of time delay when feedback control gain ˛ is set at the small value for inducing chaos. We note that the increased control gain ˛ can extend the parametric domain of chaos. If the value of control gain ˛ is set at an appropriate level, the variation of the time delay becomes insensitive to the occurrence of chaos. As the control gain is lifted up to ˛ = 8, the global bifurcation is plotted in Fig. 6(b). In this case, the existence of chaotic state persists throughout the whole parametric domain.
Fig. 4. (a) Global bifurcation diagram versus ˛ for Case III; (b) the partial enlarged view.
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Fig. 5. Global bifurcation diagram versus ˇ for (a) Case I; (b) Case II; and (c) Case III.
Fig. 6. Global bifurcation diagram versus for (a) Case I; (b) Case II; and (c) Case III.
However, when increasing the value of controlling parameter ˇ, the number of visible quasi-periodic windows increases. When lifting up the gain from ˇ = 5.6 (Case I) to ˇ = 9 (Case III), the global bifurcation is shown in Fig. 6(c), which implies less parametric domain of chaos holds. 5. Conclusion In this paper, we focus on the issue of inducing chaos in Brushless DC Motor (BLDCM) via nonlinear time-delayed current feedback. Different from the existent works [15–17], the proposed scheme promises the occurrence of chaos according to Li–Yorke criteria. Based on the differential-geometry control theory, an analytical solution of current time-delayed function is derived, which provides a systematic approach for design and completely avoids blind numerical search. By using this approach, we can present a standard procedure for the design of chaos inducing in BLDCM. Through careful bifurcation analysis aligning with the variation of control parameters ˛, ˇ, , we examine the regular pattern of chaos in BLDCM across a parametric domain. The effects about the variation of the controller parameters can provide guidance concerning how to select the control parameters for inducing chaos in BLDCM. Acknowledgments This work was supported by Hunan Provincial Natural Science Foundation of China (Grant No. 10JJ2044) and the Research Foundation of Education Bureau of Hunan Province, China (Grant No. 13C372). References [1] Z. Li, J. Park, B. Zhang, G. Chen, Bifurcations and chaos in a permanent-magnet synchronous motor, IEEE Trans. Circuits Syst. I 49 (2002) 383–387. [2] C. Li, L. Wu, H. Li, Y. Tong, A novel chaotic system and its topological horseshoe, Nonlinear Anal. Model. Control 18 (2013) 66–77.
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