Nonlinear dynamic analysis and modeling of fractional permanent magnet synchronous motors

Abstract

This paper is a step forward to generalize the fundamentals of the relationship between fractional-order system modeling and integer order system modeling. First, we establish a fractional-order mathematical modeling of permanent magnet synchronous motor from the view of engineering applications. Obviously, the effect of fractional order is the key factor for extra freedom, more flexibility and novelty. As a pioneering work, we analyze the effect of fractional order on the bifurcation point, and give its basic law, which is a fundamental work to optimize the use of the previous integer-order research results, and a bridge between fractional-order system and integer order system. Furthermore, we discuss the beginning point and ending point of chaos and the size of the attractors with varying fractional-order, respectively. In addition, nonlinear dynamics analyses of the presented fractional order system are also given. Finally, numerical simulations are given which match the analytical discussion.

Keywords

Fractional system mathematical modeling nonlinear dynamics

1. Introduction

Permanent magnet synchronous motor (PMSM) is widely used in high performance applications due to high efficiency, high power density, small size, and simple structure (Chang et al., 2011; Wei and Wang, 2013; Liu and Zhu, 2014; Ananthamoorthy and Baskaran, 2013). However, for those cases where low vibration, low acoustic noise, and high accurateness are highly demanded, magnetically induced vibration is a major concern (Shanmugasundram et al., 2013). Consequently, most of the prior contributions are major in new control methods with a good ability of robustness, and not try to solve its essential problem-basic mathematical model and vibration analysis (Choi and Jung, 2012; Yu et al., 2013; Arumugam et al., 2014). Because of this concern, it is essential to build new models approaching the engineering application and understand the excitation and vibration behaviors.

Fractional calculus dates back to more than four centuries, however, great improvements in the study of fractional calculus were made in the most recent twenty years (Li et al., 2013), because fractional calculus depends on the history of the function, which is more realistic and suitable for modeling, analyzing, and synthesizing and for solving many problems in basic natural sciences. Now, the fractional-order generalization has been applied to nonlinear dynamics, mathematics, physics, mechanics, control theory and engineering, agriculture, electromagnetic (Bhalekar et al., 2012; Chen et al., 2012, 2013, 2014; Jia et al., 2013; Golmankhaneh et al., 2013; Ho et al., 2013; Kayedi-Bardeh et al., 2012; Irandoust-pakchin et al., 2013; Grahovac and Spasic, 2013; Lopes and Machado, 2013; Romanovas et al., 2013; Jalloul et al., 2013; Wu and Baleanu, 2014; Liu et al., 2014; Zhang et al., 2014; Teng et al., 2014; Faieghi et al., 2014a, 2014b; Wu et al., 2014). The challenges in realizing the fractional element prevail through research and practice. Many realizations and patents of the fractional-order element using different technologies have been investigated. Therefore, we try to establish a fractional-order mathematical modeling of PMSM.

Motivated by the above discussions, this paper brings attention to establishing a fractional-order mathematical modeling of PMSM, and discussing nonlinear dynamic behavior. This first point is to build a novel fractional-order mathematical modeling of PMSM from the view of engineering applications. Second, we study the nonlinear dynamics of the above fractional-order modeling of PMSM in detail, especially the effect of fractional order on bifurcation point, which is a prior work. Furthermore, we discuss the beginning point and ending point of chaos and the size of the attractors with varying fractional-order, respectively.

This paper is organized as follows: section 2 build a fractional-order nonlinear mathematical modeling of PMSM, in section 3, nonlinear dynamics of the above PMSM are studied in detail and section 4 concludes this paper.

2. System modelling

Here, we suppose that PMSM have uniform air gap. Also, the other assumptions are:

The stator winding has wye connection, and balanced three-phase sine currents flow in the three-phase symmetric winding.

Stator-flux-orientation has sine distribution, for that we neglect the effect of harmonic distortion and saturation flux.

We also neglect the effect of magnetic hysteretic losses and eddy current loss.

We set the damping of the rotor winding is zero.

Usually, we use d-q rotor coordinate system, which will synchronous rotate with rotor. Therefore, we can easily get the space vector of PMSM as shown in Figure 1. Thus, we can get the voltage equation, ferromagnetic chain equation and torque equation as follows

u_{d} = \frac{d ψ_{d}}{d t} - ω ψ_{q} + {Ri}_{d}

(1)

u_{q} = \frac{d ψ_{q}}{d t} + ω ψ_{d} + {Ri}_{q}

(2)

ψ_{d} = L_{d} i_{d} + ψ_{f}

(3)

ψ_{q} = L_{q} i_{q}

(4)

T_{e} = \frac{3}{2} p (ψ_{d} i_{q} - ψ_{q} i_{d}) = \frac{3}{2} p [ψ_{f} i_{q} + (L_{d} - L_{q}) i_{d} i]

(5)

In addition, mechanical motion equation is

J \frac{d ω_{m}}{d t} = T_{e} - T_{L} - b ω_{m}

(6)

All of the parameters are described in detail in Table 1.

Table 1.

The PMSM parameters.

Parameter	Nomenclature	Unit
L_d	Stator inductance in the q axis	H
L_q	Stator inductance in the d axis	H
J	Moment of inertia	$kg \cdot m^{2}$
b	Viscous damping coefficient	$N / (rad / s)$
R	Stator phase resistance	Ω
u_d	d-axis stator voltage	$V$
u_q	q-axis stator voltage	$V$
T_L	Load torque	$N \cdot m$
$ψ_{d}$	d-axis flux	Weber
$ψ_{q}$	q-axis flux	Weber
i_d	d-axis stator currents	A
i_q	q-axis stator currents	A
$ψ_{f}$	Rotor permanent magnet flux	Weber
T_e	Electromagnetic torque	$N \cdot m$
$ω_{m}$	Rotor angular frequency	rad/s
p	Rotor pole pairs
ω = p $ω_{m}$	Electrical angular frequency	rad/s

From equation (1) to equation (6), we can get the equation of state as

[\frac{{di}_{d}}{d t} \frac{{di}_{q}}{d t} \frac{d ω_{m}}{d t}] = [- R / L_{d} p ω_{m} 0 - p ω_{m} - R / L_{q} - p ψ_{f} / L_{q} 0 p ψ_{f} / J b / J] [i_{d} i_{q} ω_{m}] + [u_{d} / L_{d} u_{q} / L_{q} - T_{L} / J]

(7)

We set $L_{d} = L_{q} = L$ , and get

{\frac{{di}_{d}}{d t} = \frac{1}{L} (- {Ri}_{d} + Lp ω_{m} + u_{d}) \frac{{di}_{q}}{d t} = \frac{1}{L} [- {Ri}_{q} - p ω_{m} ({Li}_{d} + ψ_{f}) + u_{q}] \frac{d ω_{m}}{d t} = \frac{1}{J} (p ψ_{f} i_{q} + b ω_{m} - T_{L})

(8)

Using affine invariance and time scaling property, one gets

{x = φ y + ξ t = τ \cdot \overset{\land}{t}

(9)

where

x = (x_{1} x_{2} x_{3}) = (i_{q} i_{d} ω_{m}), y = (y_{1} y_{2} y_{3}), ξ = (ξ_{1} ξ_{2} ξ_{3}), φ = (σ_{1} 000 σ_{2} 000 σ_{3}), τ = \frac{L}{R}, σ_{1} = \frac{b}{p^{2} τ ψ_{f}}, σ_{2} = σ_{1}, σ_{3} = \frac{1}{p τ}, ξ_{1} = 0, ξ_{2} = - \frac{ρ Lb σ_{3} + p ψ_{f}^{2}}{pL ψ_{f}}, ξ_{3} = 0

By simplifying equation (9), we get

{\frac{{dy}_{1}}{d \overset{\land}{t}} = - y_{1} - y_{2} y_{3} + ρ y_{3} + {\overset{\land}{u}}_{q} \frac{{dy}_{2}}{d \overset{\land}{t}} = - y_{2} + y_{1} y_{3} + {\overset{\land}{u}}_{d} \frac{{dy}_{3}}{d \overset{\land}{t}} = σ (y_{1} - y_{3}) - {\overset{\land}{T}}_{L}

(10)

Here, we set

{\overset{\land}{u}}_{d} = 0

{\overset{\land}{u}}_{q} = 0

{\overset{\land}{T}}_{L} = 0

. So, one gets

{\frac{dx}{d t} = - x + yz \frac{dy}{d t} = - y - xz + az \frac{dz}{d t} = b (y - z)

(11)

Since the fractional-order derivative does not rely on the point, but also relay the historical process, the stator currents for PMSM cannot, obviously, changed suddenly, whose value is also dependent on the historical process from an electrical engineers' point of view. Therefore, modeling of fractional-order PMSM is

{\frac{d^{q 1} x}{d t^{q 1}} = - x + yz \frac{d^{q 2} y}{d t^{q 2}} = - y - xz + az \frac{d^{q 3} z}{d t^{q 3}} = b (y - z)

(12)

Taking physical significance into consideration, q₁ should be equal to q₂, for that they are the components of the same current in d-q coordinate system. Obviously, q₃ should be equal to 1 according to the definition of first-order derivative. Therefore, we will not only present the bifurcation diagram of variable x versus parameter b, but also study the effect of fractional-order, q₁ = q₂, in the next section.

3. Nonlinear dynamic analysis

In the following section we will study nonlinear dynamics of the above fractional-order PMSM in detail. Here, the numerical simulation results were carried out by using algorithm Adams-Bashforth-Moulton with a fixed step size of 0.01.

From Figures 2 and 3, the system goes directly through Hopf bifurcation to chaos. Moreover, the time domain waveforms and the diagram of phase trajectory agree with each other. When q₁ = q₂ = 1.00, there are period three between chaos from Figure 4. With the increasing of fractional order q₁ = q₂, a wide variety of bifurcation phenomena occurs. For example, from Figure 8, there is paroxysmal flashes of multi-periods near period two; from Figure 9, the system goes through period two to chaos omission of period one; from Figures 12 and 13, special bifurcation occurs.

On the other hand, for the shape of the attractors, it is similar to Chen's attractor at first with q₁ = q₂ = 1.14. However, from q₁ = q₂ = 1.14 to q₁ = q₂ = 1.35, the attractors are not similar to Lorenz’s attractors or Chen's attractors.

Furthermore, Bifurcation diagrams of x versus fractional orders q₁ = q₂ are also presented, from Figure 14. Interestingly, with the decreasing of fractional order q₁ = q₂, the system routs period-doubling bifurcation to chaos.

From Figures 8 and 9, we also can observe a typical transient phenomena, which are that the maximum amplitude abruptly become larger near b = 3.0.

In the following contents, we will study nonlinear dynamical analysis in detail.

From Figure 2, the system with q₁ = q₂ = 0.95, q₃ = 1 goes into chaotic state at b = 9.2, and there is not any period window. Interestingly, the amplitude of x abruptly increases near b = 9.5, which can be observed obviously comparing Figure 2(b1) and (c₁). From the viewpoint of an electrical engineer, the current appears ripple current at b = 9.0 while it becomes chaotic vibration at b = 9.6 as shown in Figure 2(b1) and (c1), respectively.

From Figure 3, the system with q₁ = q₂ = 0.98, q₃ = 1.00 become chaos from b = 2.75. For b = 2.5, the system goes through quasi-periodicity to a stable point, as shown in Figure 3(a). For b = 3.0, the chaotic attractor of the system is similar to the Lorentz’s attractor. From the viewpoint of an electrical engineer, the current stays a current value at b = 2.5 while it becomes chaotic vibration with high value at b = 3.0 as shown in Figures 3(b1) and (c1), respectively.

From Figure 4, the system with q₁ = q₂ = 1.00, q₃ = 1.00 abruptly goes into chaos without any transient process at b = 1.85, although there are rich period windows from b = 1.85 to b = 1.85. Moreover, the attractor similar to Lorentz system as shown in Figure 4(c2) is bigger than that is shown in Figure 3(c2).

From Figure 5, there is a wide period window, which can be also called periodic zone, from b = 1.29 to b = 2.72. In this regions, the system goes through inversed period doubling bifurcation to period two from chaos. Furthermore, period four and period three occur at b = 2.915 and b = 4.17 as shown in Figure 5(e) and (f), respectively. In other words, the current appears periodic vibration.

The bifurcation changes considerably, although the fractional order change little, from Figure 6. The system with q₁ = q₂ = 1.05, q₃ = 1 goes through inverse period doubling bifurcation to period one from chaos at b = 0.77. Interestingly, the chaotic attractor of the system is not the same as before at b = 2.5 as shown in Figure 6(c). From b = 3.2 to b = 3.4, there is period two window. And then, the system goes through period doubling bifurcation to chaos. From an engineer's viewpoint, the current alternate periodic and chaotic vibration.

From Figure 7, the initial chaotic region has now narrowed. As the increasing of fractional order, there is not period windows. Comparing Figure 7(c) and (d), the magnitude of x obviously changes. From Figure 7(d), the divergence of the attractor occurs, which shows that the shape of the attractor is changing.

From Figure 8, the prior chaotic region disappears for the system with q₁ = q₂ = 1.14, q₃ = 1 the system goes through period doubling bifurcation to chaos. Interestingly, there are flashes of the branch of multi-cycle on the branch of period two, which is extremely rare. For b = 2.556 and b = 2.57, the flashes of period three occurs as shown in Figure 8(c) and (d). Comparing Figure 8(e) and (f), the magnitude of x abruptly becomes bigger, which is a typical transient dynamical phenomena. From an engineer's viewpoint, the current suddenly becomes bigger. Now, the shape of the attractor is similar to Chen’s system as shown in Figure 8(f2).

From Figure 9, the period one disappears for the system with q₁ = q₂ = 1.15, q₃ = 1. So, the system goes though period doubling bifurcation to chaos from period two. Interestingly, the flashes of multi-cycle occurs near period two, which means that the current alternates between different states.

From Figure 10, the bifurcation changes significantly for the system with q₁ = q₂ = 1.20, q₃ = 1, which means that the system goes into chaos directly. Moreover, the shape of the chaotic attractor becomes surprising, which is not similar to typical attractors. From Figure 10(b2), the attractor on y-z plane is centrosymmetric. While, the attractor is axisymmetric from Figure 10(b3) and (b4).

From Figure 11, the system with q₁ = q₂ = 1.25, q₃ = 1 goes into chaos directly, and there is obviously period window. Moreover, the shape of the attractor also changes as shown in Figure 11(b2). From Figure 11(b2), the attractor on y-z plane is centrosymmetric. While, the attractor is axisymmetric from Figure 11(b3) and (b4).

Form Figure 12, the bifurcation phenomena changes momentously for the system with q₁ = q₂ = 1.30, q₃ = 1. More specially, the system goes through turbulence to period one from a quick chaos. Moreover, the shape of the attractor also changes as shown in Figure 12(b). From Figure 12(b2), the attractor on y-z plane is centrosymmetric. While, the attractor is axisymmetric from Figure 12(b3) and (b4).

From Figure 13, the presented system with q₁ = q₂ = 1.35, q₃ = 1 goes from chaos through quasi-periodicity to period one in a strange way. From Figure 13(c1), the current abruptly tends to infinity from period vibration at b = 48.2. It, however, holds periodic vibration at b = 50.1.

For both of b = 2.5 and b = 2.7, the general trend of the bifurcation are the same, which is that the system abruptly becomes chaos, and then goes through inverse period doubling bifurcation to period two. The same process will be occurs again during the next region. The difference is that the flashes of the multi-cycle bifurcation occurs for b = 2.5. Moreover, the system ends with period two for b = 2.5, while it ends with period four for b = 2.7.

We will discuss how the changes of fractional order affect the bifurcation. From Figures 2(a) to 13(a), obvious shifts can be observed, which is that the bifurcation point moves to the left as the fractional-orders of the system increase. In other words, the value of bifurcation parameter b decreases where bifurcation occurs with the increasing of fractional orders. More specially, when the system order satisfies q₁ = q₂ = 0.95, q₃ = 1, the bifurcation point is b = 9.30 (Figure 2(a)), while q₁ = q₂ = 0.98, q₃ = 1, the bifurcation point dramatically shifts to b = 2.75 (Figure 3(a)). When q₁ = q₂ = q₃ = 1, bifurcation happens at b = 1.85 (Figure 4(a)); When q₁ = q₂ = 1.02, q₃ = 1, bifurcation happens at b = 1.3 (Figure 5(a)); When q₁ = q₂ = 1.05, q₃ = 1, bifurcation happens at b = 0.77 (Figure 6(a)); When q₁ = q₂ = 1.10, q₃ = 1, bifurcation happens at b = 0.04, bur soon get rid of chaos and enter period 1 orbit (Figure 7(a)). These sorts of shifts can be observed for a relatively long time before the Hopf bifurcation disappears.

This interesting phenomena can be explained by the stability theory of fractional-order nonlinear system. As we all know that if and only if $| \arg (λ_{i}) | > q π / 2$ is satisfied for all eigenvalues $λ_{i}$ of matrix A, where matrix A is the matrix of state equation for a certain fractional-order nonlinear system, q is the fractional order of the given system. For the above fractional-order PMSM model, the state equations remain unchanged, which means that all eigenvalues $λ_{i}$ stays the same. On the other hand, at the right side of the inequality, the changes of fractional order q are sure to affect the stability region. More specially, the increasing of fractional order q will scale down the stability region. However, the routes to chaos are different for different fractional-orders. More specially, the system go through Hopf bifurcation to chaos for fractional order q₁ = q₂ from 0.95 to 1.10 and from 1.20 to 1.35 with fractional order q₃ = 1, while for fractional order q₁ = q₂ from = 1.14 to 1.15, the system go through period-doubling bifurcation to chaos. Obviously, there is not an exacting mathematical theory to describe the laws and phenomena in the instability region. Therefore, we will study these issues in the following contents, respectively.

Furthermore, we discuss the beginning point and ending point of chaos with varying fractional-order. From Figure 15(a), the beginning point and ending point of main chaotic region both decrease first and then increase with the increasing of fractional-order q₁ = q₂, whose minimum is at about q₁ = q₂ = 1. From Figure 15(b), the beginning point and ending point of the secondary chaotic region decrease until the chaotic region disappear increasing the fractional-order q₁ = q₂.

From Figure 16, we also find another interesting phenomena, which is that the size of the attractor becomes bigger with the increasing of fractional-order q₁ = q₂.

4. Conclusion and discussions

This paper demonstrates the generalized concepts from the very narrow integer order scope system to fractional-order mathematical modeling. More specially, we established a fractional-order mathematical modeling of PMSM from the view of engineering applications. Moreover, we studied the nonlinear dynamics of the above fractional-order nonlinear system including bifurcation diagrams, waveforms in time domain and phase orbits. As a pioneering work, we analyzed the effect of fractional order on the bifurcation point, and gave its basic law, which is a fundamental work to optimize the use of the previous integer-order research results, and a bridge between fractional-order system and integer order system. Moreover, we discuss the beginning point and ending point of chaos and the size of the attractors with varying fractional-order, respectively. Finally, numerical simulations are given which match the analytical discussion.

In the future, we will try to propose new nonlinear dynamical indicators to analyze the effect of the fractional-order on the stability of nonlinear systems, and describe dynamic characters from a different viewpoint. Moreover, we will test the presented fractional-order mathematical modeling by conducting scientific experiments.

Footnotes

Funding

This work was supported by the Scientific Research Foundation of the National Natural Science Foundation (grant numbers 51109180 and 51279167), the National Science and Technology Supporting Plan from the Ministry of Science and Technology of China (grant numbers 2011BAD29B08 and 2012BAD10B02), the Fundamental Research Funds for the Central Universities (grant number 201304030577) and Scientific Research Funds of Northwest A&F University (grant number 2013BSJJ095).

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