CN105471345A

CN105471345A - Brushless double-feed-motor chaotic analysis method based on largest Lyapunov exponent

Info

Publication number: CN105471345A
Application number: CN201510823679.9A
Authority: CN
Inventors: 陈集思; 杨俊华; 林卓胜; 陈俊宏
Original assignee: Guangdong University of Technology
Current assignee: Guangdong University of Technology
Priority date: 2015-11-23
Filing date: 2015-11-23
Publication date: 2016-04-06

Abstract

The invention provides a brushless double-feed-motor chaotic analysis method based on the largest Lyapunov exponent. The method is characterized by deriving a non-linear differential equation based on a brushless double-feed-motor d-q shaft mathematic model; taking a reciprocal of a rotor motion time constant as a control parameter of a motor system; according to an actual motor parameter, constructing a chaotic model of a brushless double-feed motor; through calculating the largest Lyapunov exponent of the chaotic model, analyzing a corresponding parameter scope when the motor generates a chaotic phenomenon. Matlab modeling and simulation are used and a parameter condition of the brushless double-feed motor entering into a chaotic motion state is demonstrated.

Description

Chaos analysis method for brushless doubly-fed motor based on maximum Lyapunov exponent

技术领域 technical field

本发明涉及无刷双馈电机(BDFM)混沌现象分析领域，更具体地，涉及一种基于最大李雅普诺夫指数的无刷双馈电机混沌分析方法。 The invention relates to the field of chaotic phenomenon analysis of a brushless doubly-fed motor (BDFM), in particular to a method for analyzing the chaos of a brushless doubly-fed motor based on the maximum Lyapunov exponent.

背景技术 Background technique

无刷双馈电机(brushlessdoubly-fedmachine，BDFM)是一种异步化的交流励磁同步电机，定子上有功率和控制两套绕组，结构上，与绕线式异步电机的定、转子绕组有相似之处，由于其高可靠性和低维护成本，在恶劣的风电场环境下呈现出明显优势。近些年来，关于BDFM的本体研究和控制策略研究方兴未艾，标量控制、矢量控制、直接转矩控制等控制策略获得了较多关注。虽然电机控制策略不断优化改进，但电机在实际运行中仍然存在着不规则运动，如不规则电磁噪声、不规则转矩、转速间歇振荡等问题。随着对电机系统认知的不断提高，发现电机运行中出现的种种不规则运动非常类似于非线性系统的混沌现象。 Brushless doubly-fed machine (BDFM) is an asynchronous AC excitation synchronous motor with two sets of windings for power and control on the stator. In structure, it is similar to the stator and rotor windings of wound asynchronous motors. Due to its high reliability and low maintenance cost, it shows obvious advantages in the harsh wind farm environment. In recent years, the ontology research and control strategy research on BDFM are in the ascendant, and control strategies such as scalar control, vector control and direct torque control have gained more attention. Although the motor control strategy is continuously optimized and improved, there are still irregular motions in the actual operation of the motor, such as irregular electromagnetic noise, irregular torque, and intermittent oscillation of the speed. With the continuous improvement of the cognition of the motor system, it is found that various irregular movements in the operation of the motor are very similar to the chaos phenomenon of the nonlinear system.

继相对论和量子力学之后，混沌学被称为20世纪科学界中又一伟大理论成果，不同于平衡点、周期解、次谐波解和准周期解等行为，混沌是一种随机但有界的稳态行为。Li-Yorke于1975年在“周期三意味着混沌”一文中最早提出混沌概念，文章以数学的严格性分析了任何一维系统中，只要出现规则的周期三，则同一系统中必然会给出其他任意长的规则周期，以及完全混沌的循环。当然，不同文献中，混沌的主要特征表述不尽相同，但一般而言，混沌的主要特征包括：确定系统的内随机性、对初始条件的敏感性、具有正的李雅普诺夫(Lyapunov)指数等。过去几十年，在电机等各个学科领域中开展了广泛的混沌研究，面对由电机本体、功率变换器和控制器等多个复杂系统构成的电机控制系统，混沌分析首先应从电机本体开始，而永磁同步电机混沌运动现象是当前国内外研究的主要热点。 Following the theory of relativity and quantum mechanics, chaos is known as another great theoretical achievement in the scientific community in the 20th century. It is different from behaviors such as equilibrium point, periodic solution, sub-harmonic solution and quasi-periodic solution. Chaos is a random but bounded steady-state behavior. Li-Yorke first proposed the concept of chaos in the article "Period Three Means Chaos" in 1975. The article analyzed any one-dimensional system with mathematical rigor, as long as there is a regular period three, the same system will inevitably give Other arbitrarily long regular periods, and completely chaotic cycles. Of course, the main characteristics of chaos are expressed differently in different literatures, but in general, the main characteristics of chaos include: determining the internal randomness of the system, sensitivity to initial conditions, and having a positive Lyapunov exponent Wait. In the past few decades, extensive chaos research has been carried out in various disciplines such as motors. Facing a motor control system composed of multiple complex systems such as motor body, power converter, and controller, chaos analysis should first start with the motor body. The chaotic motion phenomenon of permanent magnet synchronous motor is the main focus of current research at home and abroad.

目前，对BDFM系统混沌现象研究仍为空白，在某些特定参数条件下，BDFM会否和其他类型电机一样出现混沌现象？本文通过BDFMd-q轴数学模型导出电机系统的非线性微分方程，构建BDFM的七阶非自治系统，从混沌研究角度探讨BFDM中不规则运动发生机理，得到相关结论。 At present, the research on the chaos phenomenon of BDFM system is still blank. Under some specific parameter conditions, will BDFM appear the same chaos phenomenon as other types of motors? In this paper, the nonlinear differential equation of the motor system is derived from the BDFM d-q axis mathematical model, and the seventh-order non-autonomous system of the BDFM is constructed. The mechanism of irregular motion in BFDM is discussed from the perspective of chaos research, and relevant conclusions are obtained.

发明内容 Contents of the invention

本发明提供一种基于最大李雅普诺夫指数的无刷双馈电机混沌分析方法，该方法从混沌研究角度探讨BFDM中不规则运动发生机理，得到相关结论。 The invention provides a method for analyzing the chaos of a brushless doubly-fed motor based on the maximum Lyapunov exponent. The method discusses the mechanism of irregular motion in BFDM from the perspective of chaos research and obtains relevant conclusions.

为了达到上述技术效果，本发明的技术方案如下： In order to achieve the above-mentioned technical effect, the technical scheme of the present invention is as follows:

一种基于最大李雅普诺夫指数的无刷双馈电机混沌分析方法，包括以下步骤： A method for analyzing the chaos of a brushless doubly-fed motor based on the maximum Lyapunov exponent, comprising the following steps:

S1：基于无刷双馈电机d-q轴数学模型，导出非线性微分方程； S1: Based on the d-q axis mathematical model of the brushless double-fed motor, derive the nonlinear differential equation;

S2：以转子运动时间常数的倒数为电机系统的控制参量，根据实际电机参数构建无刷双馈电机的混沌模型； S2: Take the reciprocal of the rotor motion time constant as the control parameter of the motor system, and construct the chaos model of the brushless doubly-fed motor according to the actual motor parameters;

S3：通过计算混沌模型的最大Lyapunov指数，分析电机出现混沌现象时所对应的参数范围。 S3: By calculating the maximum Lyapunov exponent of the chaotic model, analyze the parameter range corresponding to the chaotic phenomenon of the motor.

进一步地，所述步骤S1的过程如下： Further, the process of step S1 is as follows:

21)BDFM在转子速d-q轴同步坐标系中各绕组的电压方程为： 21) The voltage equation of each winding of BDFM in the rotor speed d-q axis synchronous coordinate system is:

式中，u_dp、u_qp、u_dc、u_qc、u_dr、u_qr为功率绕组、控制绕组和转子绕组的d、q轴电压分量，i_dp、i_qp、i_dc、i_qc、i_dr、i_qr为d、q轴电流，ψ_dp、ψ_qp、ψ_dc、ψ_qc、ψ_dr、ψ_qr为d、q轴磁链，r_p、r_c、r_r为等效电阻，p_p、p_c为功率绕组和控制绕组的极对数，ω_r为转子转速； In the formula, u _dp , u _qp , u _dc , u _qc , u _dr , u _qr are d and q axis voltage components of power winding, control winding and rotor winding, i _dp , i _qp , i _dc , i _qc , i _dr and i _qr are d and q axis currents, ψ _dp , ψ _qp , ψ _dc , ψ _qc , ψ _dr , ψ _qr are d and q axis flux linkages, r _p , rc , _{r r} _are equivalent resistances, p _p and p _c are the number of pole pairs of power winding and control winding, ω _r is the rotor speed;

22)在两相转子速d-q轴模型中，功率绕组、控制绕组和转子绕组的磁链方程为： 22) In the two-phase rotor speed d-q axis model, the flux linkage equations of power winding, control winding and rotor winding are:

式中，L_sp、L_sc为d-q坐标下定子各绕组自感，M_rp、M_rc为定子绕组与转子绕组间互感； In the formula, L _sp and L _sc are the self-inductance of each winding of the stator under the dq coordinates, M _rp and M _rc are the mutual inductance between the stator winding and the rotor winding;

23)BDFM在d-q坐标下的电磁转矩方程为： 23) The electromagnetic torque equation of BDFM in d-q coordinates is:

T_e＝T_ep+T_ec＝p_pM_rp(i_qpi_dr-i_dpi_qr)+p_cM_rc(i_qci_dr+i_dci_qr)(7) T _e ＝T _ep +T _ec ＝p _p M _rp (i _qp i _dr -i _dp i _qr )+p _c M _rc (i _qc i _dr +i _dc i _qr )(7)

式中，T_e为总电磁转矩，T_ep、T_ec为功率绕组、控制绕组产生的电磁转矩； In the formula, T _e is the total electromagnetic torque, T _ep and T _ec are the electromagnetic torque generated by the power winding and the control winding;

24)BDFM的机械运动方程可表示为： 24) The mechanical motion equation of BDFM can be expressed as:

式中，J、D分别为转子机械惯量和转动阻尼系数，T_L为负载转矩； In the formula, J and D are the rotor mechanical inertia and rotational damping coefficient respectively, T _L is the load torque;

25)将式(4)～(6)代入(1)～(3)可得： 25) Substitute formulas (4)～(6) into (1)～(3) to get:

设T_r为转子运动时间常数，σ₁、σ₂、σ₃为扩散系数： Let T _r be the time constant of rotor movement, and σ ₁ , σ ₂ , σ ₃ be the diffusion coefficients:

令电机系统中的各状态变量[i_dp，i_qp，i_dc，i_qc，ψ_dr，ψ_qr，ω_r]^T＝[x₁，x₂，x₃，x₄，x₅，x₆，x₇]^T，[u_dp，u_qp，u_dc，u_qc，u_dr，u_qr，T_L]^T＝[0，0，0，0，0，0，0]^T。将式(6)、(7)、(9)～(13)整理化简可得BDFM电机系统的一阶微分方程组： Let each state variable in the motor system [i _dp , i _qp , i _dc , i _qc , ψ _dr , ψ _qr , ω _r ] ^T = [x ₁ , x ₂ , x ₃ , x ₄ , x ₅ , x ₆ , x ₇ ] ^T , [u _dp , u _qp , u _dc , u _qc , u _dr , u _qr , T _L ] ^T ＝[0, 0, 0, 0, 0, 0, 0] ^T . The first-order differential equations of the BDFM motor system can be obtained by sorting and simplifying equations (6), (7), (9)-(13):

由式(14)可看出，BDFM电机系统具有多变量、非线性和强耦合的特点。方程组中各项变量的系数用矩阵形式可表示为： It can be seen from formula (14) that the BDFM motor system has the characteristics of multivariable, nonlinear and strong coupling. The coefficients of each variable in the equation system can be expressed in matrix form as:

进一步地，所述步骤S2的过程如下： Further, the process of step S2 is as follows:

31)在式(14)中引入转子转速PD控制项： 31) Introduce the rotor speed PD control item in formula (14):

令上式中ω_ref＝0，可得： Let ω _ref =0 in the above formula, we can get:

32)令：其中T_r ^*为转子运动时间常数的估计值，则有： 32) order: where T _r ^* is the estimated value of the rotor motion time constant, then:

T_r ^*＝kT_r(16) T _r ^* = kT _r (16)

将式(15)、(16)代入式(14)中转子绕组的一阶微分方程组，可构建BDFM的七阶非自治系统： Substituting equations (15) and (16) into the first-order differential equations of the rotor winding in equation (14), the seventh-order non-autonomous system of BDFM can be constructed:

进一步地，所述步骤S3的过程如下： Further, the process of step S3 is as follows:

41)令最大Lyapunov指数为： 41) Let the maximum Lyapunov exponent be:

其中，m为积分次数，τ为积分时间； Among them, m is the number of integration times, τ is the integration time;

42)根据无刷双馈电机实际参数，将各参数代入式(17)，以式(19)为计算原理，在Matlab平台上编写最大Lyapunov指数的求解程序，可得BDFM系统的最大Lyapunov指数谱，从指数谱中得出BDFM系统进入混沌运动状态。 42) According to the actual parameters of the brushless doubly-fed motor, substitute each parameter into formula (17), use formula (19) as the calculation principle, write the solution program of the maximum Lyapunov exponent on the Matlab platform, and obtain the maximum Lyapunov exponent spectrum of the BDFM system , the BDFM system enters the state of chaotic motion from the exponential spectrum.

与现有技术相比，本发明技术方案的有益效果是： Compared with the prior art, the beneficial effects of the technical solution of the present invention are:

本发明基于无刷双馈电机d-q轴数学模型，导出非线性微分方程；以转子运动时间常数的倒数为电机系统的控制参量，根据实际电机参数构建无刷双馈电机的混沌模型；通过计算混沌模型的最大Lyapunov指数，分析电机出现混沌现象时所对应的参数范围。采用Matlab建模仿真，论证了无刷双馈电机进入混沌运动状态时的参数条件。 The present invention derives nonlinear differential equations based on the d-q axis mathematical model of the brushless double-fed motor; takes the reciprocal of the rotor motion time constant as the control parameter of the motor system, and constructs a chaos model of the brushless double-fed motor according to actual motor parameters; by calculating the chaos The maximum Lyapunov exponent of the model is used to analyze the parameter range corresponding to the chaotic phenomenon of the motor. Using Matlab modeling and simulation, the parameter conditions when the brushless doubly-fed motor enters the chaotic motion state are demonstrated.

附图说明 Description of drawings

图1为本发明中BDFM系统最大Lyapunov指数谱； Fig. 1 is the maximum Lyapunov exponent spectrum of BDFM system among the present invention;

图2(a)为BDFM系统功率绕组电流的d-q轴分量混沌动态响应仿真波形图； Fig. 2(a) is the simulation waveform diagram of the chaotic dynamic response of the d-q axis component of the power winding current of the BDFM system;

图2(b)为BDFM系统控制绕组电流的d-q轴分量混沌动态响应仿真波形图； Figure 2(b) is the simulation waveform diagram of the chaotic dynamic response of the d-q axis component of the BDFM system control winding current;

图2(c)为BDFM系统转子绕组磁链的d-q轴分量混沌动态响应仿真波形图； Figure 2(c) is the simulation waveform diagram of the chaotic dynamic response of the d-q axis component of the rotor winding flux linkage of the BDFM system;

图2(d)为BDFM系统转子转速混沌动态响应仿真波形图； Figure 2(d) is the simulation waveform diagram of the chaotic dynamic response of the rotor speed of the BDFM system;

图3(a)为BDFM系统(idp-iqp)吸引子相平面图； Figure 3(a) is the attractor phase plane diagram of the BDFM system (idp-iqp);

图3(b)为BDFM系统(idp-ψqr)吸引子相平面图； Figure 3(b) is the attractor phase plane diagram of the BDFM system (idp-ψqr);

图3(c)为BDFM系统(idp-ω)吸引子相平面图； Figure 3(c) is the attractor phase plane diagram of the BDFM system (idp-ω);

图3(d)为BDFM系统(ψqr-ω)吸引子相平面图； Figure 3(d) is the attractor phase plane diagram of the BDFM system (ψqr-ω);

图3(e)为BDFM系统(iqp-ψqr-ω)吸引子相空间图。 Fig. 3(e) is the attractor phase space diagram of the BDFM system (iqp-ψqr-ω).

具体实施方式 detailed description

附图仅用于示例性说明，不能理解为对本专利的限制； The accompanying drawings are for illustrative purposes only and cannot be construed as limiting the patent;

为了更好说明本实施例，附图某些部件会有省略、放大或缩小，并不代表实际产品的尺寸； In order to better illustrate this embodiment, some parts in the drawings will be omitted, enlarged or reduced, and do not represent the size of the actual product;

对于本领域技术人员来说，附图中某些公知结构及其说明可能省略是可以理解的。 For those skilled in the art, it is understandable that some well-known structures and descriptions thereof may be omitted in the drawings.

下面结合附图和实施例对本发明的技术方案做进一步的说明。 The technical solutions of the present invention will be further described below in conjunction with the accompanying drawings and embodiments.

实施例1 Example 1

步骤S1的过程如下： The process of step S1 is as follows:

步骤S2的过程如下： The process of step S2 is as follows:

T_r ^*＝kT_r(16) T _r ^* = kT _r (16)

步骤S3的过程如下： The process of step S3 is as follows:

41)令最大Lyapunov指数为： 41) Let the maximum Lyapunov exponent be:

42)根据无刷双馈电机实际参数取p_p＝3，p_c＝1，r_p＝1.732Ω，r_c＝1.079Ω，r_r＝0.473Ω，L_p＝714.8mH，L_c＝121.7mH，L_r＝132.6mH，M_rp＝242.1mH，M_rc＝59.8mH，J＝0.01kg·m²，D＝0.1；并令k_p＝4，k_d＝40，T_L＝0，系统初值为[x₁，x₂，x₃，x₄，x₅，x₆，x₇]＝[0.5，0.5，0.5，0.5，0.5，0.5，0.5]，将各参数代入式(17)，以式(19)为计算原理，在Matlab平台上编写最大Lyapunov指数的求解程序，可得BDFM系统的最大Lyapunov指数谱(如图1所示)，从指数谱中得出BDFM系统进入混沌运动状态。 42) According to the actual parameters of the brushless double-fed motor, p _p = 3, p _c = 1, r _p = 1.732Ω, r _c = 1.079Ω, r _r = 0.473Ω, L _p = 714.8mH, L _c = 121.7mH , L _r ＝132.6mH, M _rp ＝242.1mH, M _rc ＝59.8mH, J＝0.01kg·m ² , D＝0.1; and k _p ＝4, k _d ＝40, T _L ＝0, the initial system The value is [x ₁ , x ₂ , x ₃ , x ₄ , x ₅ , x ₆ , x ₇ ]=[0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5], each parameter is substituted into formula (17), Using formula (19) as the calculation principle, write the solution program for the maximum Lyapunov exponent on the Matlab platform, and the maximum Lyapunov exponent spectrum of the BDFM system can be obtained (as shown in Figure 1), and the BDFM system enters the state of chaotic motion from the exponent spectrum .

参数k在[1.6,1.7]、[2.0,2.1]的变化范围中李氏指数由负变正，表明BDFM系统进入混沌运动状态。为了进一步分析和验证BDFM系统的混沌状态，取控制参量k＝2.5，最大Lyapunov指数λ_max＝0.7518，采用四阶五阶的龙格-库塔算法，在Matlab平台上求解其微分方程组(17)，分别得到BDFM功率绕组、控制绕组电流i_dp、i_qp、i_dc、i_qc和转子磁链ψ_dr、ψ_qr的混沌仿真波形，如图2所示。从图中可以发现，各状态变量的波形显示出随机但有界的振荡，同时这些波形是非周期性的，具有明显的混沌特性。 When the parameter k changes from negative to positive in the range of [1.6,1.7], [2.0,2.1], it indicates that the BDFM system enters a state of chaotic motion. In order to further analyze and verify the chaotic state of the BDFM system, the control parameter k = 2.5, the maximum Lyapunov exponent λ _max = 0.7518, and the fourth-order and fifth-order Runge-Kutta algorithm are used to solve the differential equations on the Matlab platform (17 ), the chaotic simulation waveforms of BDFM power winding, control winding current i _dp , i _qp , i _dc , i _qc and rotor flux linkage ψ _dr , ψ _qr are respectively obtained, as shown in Figure 2. It can be found from the figure that the waveforms of each state variable show random but bounded oscillations, and these waveforms are non-periodic and have obvious chaotic characteristics.

在Matlab平台上求解其微分方程组(17)，分别得到BDFM功率绕组、控制绕组电流i_dp、i_qp、i_dc、i_qc和转子磁链ψ_dr、ψ_qr的混沌仿真波形，如图2(a)-(d)所示。从图中可以发现，各状态变量的波形显示出随机但有界的振荡，同时这些波形是非周期性的，具有明显的混沌特性。 The differential equations (17) are solved on the Matlab platform, and the chaotic simulation waveforms of BDFM power winding, control winding current i _dp , i _qp , i _dc , i _qc and rotor flux linkage ψ _dr , ψ _qr are respectively obtained, as shown in Figure 2 (a)-(d) shown. It can be found from the figure that the waveforms of each state variable show random but bounded oscillations, and these waveforms are non-periodic and have obvious chaotic characteristics.

在状态空间中定性研究电机系统时，不仅可以刻划定、转子电流和磁链等每一状态量的具体轨道，而且需要刻划一切可能轨道的集合，弄清轨道的类型和分布，以整体把握电机动态系统的运动规律和特性。为此，在i_dp-i_qp、i_dp-ψ_qr、i_dp-ω、ψ_qr-ω二维平面空间和i_qp-ψ_qr-ω三维空间中观察各状态量的轨道集合，如图3(a)-(e)所示。由图中可以看出，相邻轨道不断分离、靠近，经不断拉伸和折叠后形成混沌吸引子。拉伸特性导致了轨道运动的长期不可预测性，而折叠特性使得轨道运动有界，因此，相图中随机且有界的轨道集合表明在该参数条件下，BDFM系统进入了混沌运动状态。 When qualitatively studying the motor system in the state space, it is not only possible to delineate the specific orbit of each state quantity such as rotor current and flux linkage, but also to describe the set of all possible orbits, to clarify the type and distribution of the orbit, and to use the overall Grasp the motion laws and characteristics of the motor dynamic system. To this end, observe the orbital set of each state quantity in i _dp -i _qp , i _dp -ψ _qr , i _dp -ω, ψ _qr -ω two-dimensional plane space and i _qp -ψ _qr -ω three-dimensional space, as shown in 3(a)-(e). It can be seen from the figure that the adjacent orbits are continuously separated and approached, and after being continuously stretched and folded, a chaotic attractor is formed. The stretching property leads to the long-term unpredictability of the orbital motion, while the folding property makes the orbital motion bounded. Therefore, the random and bounded orbital collection in the phase diagram indicates that under this parameter condition, the BDFM system enters a state of chaotic motion.

相同或相似的标号对应相同或相似的部件； The same or similar reference numerals correspond to the same or similar components;

附图中描述位置关系的用于仅用于示例性说明，不能理解为对本专利的限制； The positional relationship described in the drawings is only for illustrative purposes and cannot be construed as a limitation to this patent;

显然，本发明的上述实施例仅仅是为清楚地说明本发明所作的举例，而并非是对本发明的实施方式的限定。对于所属领域的普通技术人员来说，在上述说明的基础上还可以做出其它不同形式的变化或变动。这里无需也无法对所有的实施方式予以穷举。凡在本发明的精神和原则之内所作的任何修改、等同替换和改进等，均应包含在本发明权利要求的保护范围之内。 Apparently, the above-mentioned embodiments of the present invention are only examples for clearly illustrating the present invention, rather than limiting the implementation of the present invention. For those of ordinary skill in the art, other changes or changes in different forms can be made on the basis of the above description. It is not necessary and impossible to exhaustively list all the implementation manners here. All modifications, equivalent replacements and improvements made within the spirit and principles of the present invention shall be included within the protection scope of the claims of the present invention.

Claims

1. a kind of brushless doubly-fed machine chaos analysis method based on maximum Lyapunov exponent, it is characterized in that, comprises the following steps:

S1: Based on the d-q axis mathematical model of the brushless double-fed motor, derive the nonlinear differential equation;

S2: Take the reciprocal of the rotor motion time constant as the control parameter of the motor system, and construct the chaos model of the brushless doubly-fed motor according to the actual motor parameters;

S3: By calculating the maximum Lyapunov exponent of the chaotic model, analyze the parameter range corresponding to the chaotic phenomenon of the motor.

2. the brushless doubly-fed machine chaos analysis method based on maximum Lyapunov exponent according to claim 1, is characterized in that, the process of described step S1 is as follows:

21) The voltage equation of each winding of BDFM in the rotor speed d-q axis synchronous coordinate system is:

\{\begin{matrix} {u u}_{d d p p} = = {r r}_{p p} {i i}_{d d p p} + + \frac{{dψ dψ}_{d d p p}}{d d t t} - - {p p}_{p p} {ω ω}_{r r} {ψ ψ}_{q q p p} \\ {u u}_{q q p p} = = {r r}_{p p} {i i}_{q q p p} + + \frac{{dψ dψ}_{q q p p}}{d d t t} + + {p p}_{p p} {ω ω}_{r r} {ψ ψ}_{d d p p} \end{matrix} - - - - - - ((11))

\{\begin{matrix} {u u}_{d d c c} = = {r r}_{c c} {i i}_{d d c c} + + \frac{{dψ dψ}_{d d c c}}{d d t t} - - {p p}_{c c} {ω ω}_{r r} {ψ ψ}_{q q c c} \\ {u u}_{q q c c} = = {r r}_{c c} {i i}_{q q c c} + + \frac{{dψ dψ}_{q q c c}}{d d t t} + + {p p}_{c c} {ω ω}_{r r} {ψ ψ}_{d d c c} \end{matrix} - - - - - - ((22))

\{\begin{matrix} {u u}_{d d r r} = = {r r}_{r r} {i i}_{d d r r} + + \frac{{dψ dψ}_{d d r r}}{d d t t} \\ {u u}_{q q r r} = = {r r}_{r r} {i i}_{q q r r} + + \frac{{dψ dψ}_{q q r r}}{d d t t} \end{matrix} - - - - - - ((33))

In the formula, u _dp , u _qp , u _dc , u _qc , u _dr , u _qr are d and q axis voltage components of power winding, control winding and rotor winding, i _dp , i _qp , i _dc , i _qc , i _dr and i _qr are d and q axis currents, ψ _dp , ψ _qp , ψ _dc , ψ _qc , ψ _dr , ψ _qr are d and q axis flux linkages, r _p , rc , _{r r} _are equivalent resistances, p _p and p _c are the number of pole pairs of power winding and control winding, ω _r is the rotor speed;

22) In the two-phase rotor speed d-q axis model, the flux linkage equations of power winding, control winding and rotor winding are:

{{\begin{matrix} {ψ ψ}_{d d p p} = = {L L}_{s the s p p} {i i}_{d d p p} + + {M m}_{r r p p} {i i}_{d d r r} \\ {ψ ψ}_{q q p p} = = {L L}_{s the s p p} {i i}_{q q p p} + + {M m}_{r r p p} {i i}_{q q r r} \end{matrix} - - - - - - ((44))

\{\begin{matrix} {ψ ψ}_{d d c c} = = {L L}_{s the s c c} {i i}_{d d c c} + + {M m}_{r r c c} {i i}_{d d r r} \\ {ψ ψ}_{q q c c} = = {L L}_{s the s c c} {i i}_{q q c c} - - {M m}_{r r c c} {i i}_{q q r r} \end{matrix} - - - - - - ((55))

\{\begin{matrix} {ψ ψ}_{d d r r} = = {L L}_{r r} {i i}_{d d r r} + + {M m}_{r r p p} {i i}_{d d p p} + + {M m}_{r r c c} {i i}_{d d c c} \\ {ψ ψ}_{q q r r} = = {L L}_{r r} {i i}_{q q r r} + + {M m}_{r r p p} {i i}_{q q p p} - - {M m}_{r r c c} {i i}_{q q c c} \end{matrix} - - - - - - ((66))

In the formula, L _sp and L _sc are the self-inductance of each winding of the stator under the dq coordinates, M _rp and M _rc are the mutual inductance between the stator winding and the rotor winding;

23) The electromagnetic torque equation of BDFM in d-q coordinates is:

T _e ＝T _ep +T _ec ＝p _p M _rp (i _qp i _dr -i _dp i _qr )+p _c M _rc (i _qc i _dr +i _dc i _qr )(7)

In the formula, T _e is the total electromagnetic torque, T _ep and T _ec are the electromagnetic torque generated by the power winding and the control winding;

24) The mechanical motion equation of BDFM can be expressed as:

\frac{{dω dω}_{r r}}{d d t t} = = \frac{11}{J J} (({T T}_{e e} - - {T T}_{L L} - - {Dω Dω}_{r r})) - - - - - - ((88))

In the formula, J and D are the rotor mechanical inertia and rotational damping coefficient respectively, T _L is the load torque;

25) Substitute formulas (4)～(6) into (1)～(3) to get:

\{\begin{matrix} {u u}_{d d p p} = = {r r}_{p p} {i i}_{d d p p} + + {L L}_{s the s p p} \frac{{di di}_{d d p p}}{d d t t} - - {p p}_{p p} {ω ω}_{r r} {L L}_{s the s p p} {i i}_{q q p p} \\ + + {M m}_{r r p p} \frac{{di di}_{d d r r}}{d d t t} - - {p p}_{p p} {ω ω}_{r r} {M m}_{r r p p} {i i}_{q q r r} \\ {u u}_{q q p p} = = {p p}_{p p} {ω ω}_{r r} {L L}_{s the s p p} {i i}_{d d p p} + + {r r}_{p p} {i i}_{q q p p} + + {L L}_{s the s p p} \frac{{di di}_{q q p p}}{d d t t} \\ + + {p p}_{p p} {ω ω}_{r r} {M m}_{r r p p} {i i}_{d d r r} + + {M m}_{r r p p} \frac{{di di}_{q q r r}}{d d t t} \end{matrix} - - - - - - ((99))

\{\begin{matrix} {u u}_{d d c c} = = {r r}_{c c} {i i}_{d d c c} + + {L L}_{s the s c c} \frac{{di di}_{d d c c}}{d d t t} - - {p p}_{c c} {ω ω}_{r r} {L L}_{s the s c c} {i i}_{q q c c} \\ + + {M m}_{r r c c} \frac{{di di}_{d d r r}}{d d t t} + + {p p}_{c c} {ω ω}_{r r} {M m}_{r r c c} {i i}_{q q r r} \\ {u u}_{q q c c} = = {p p}_{c c} {ω ω}_{r r} {L L}_{s the s c c} {i i}_{d d c c} + + {r r}_{c c} {i i}_{q q c c} + + {L L}_{s the s c c} \frac{{di di}_{q q c c}}{d d t t} \\ + + {p p}_{c c} {ω ω}_{r r} {M m}_{r r c c} {i i}_{d d r r} - - {M m}_{r r c c} \frac{{di di}_{q q r r}}{d d t t} \end{matrix} - - - - - - ((1010))

\{\begin{matrix} {u u}_{d d r r} = = {M m}_{r r p p} \frac{{di di}_{d d p p}}{d d t t} + + {M m}_{r r c c} \frac{{di di}_{d d c c}}{d d t t} + + {r r}_{r r} {i i}_{d d r r} \\ + + {L L}_{r r} \frac{{di di}_{d d r r}}{d d t t} \\ {u u}_{q q r r} = = {M m}_{r r p p} \frac{{di di}_{q q p p}}{d d t t} - - {M m}_{r r c c} \frac{{di di}_{q q c c}}{d d t t} + + {r r}_{r r} {i i}_{q q r r} \\ + + {L L}_{r r} \frac{{di di}_{q q r r}}{d d t t} \end{matrix} - - - - - - ((1010))

Let T _r be the time constant of rotor movement, and σ ₁ , σ ₂ , σ ₃ be the diffusion coefficients:

{T T}_{r r} = = \frac{{L L}_{r r}}{{r r}_{r r}} - - - - - - ((1212))

\{\begin{matrix} {σ σ}_{11} = = 11 - - \frac{{M m}_{r r p p}^{22}}{{L L}_{s the s p p} {L L}_{r r}} \\ {σ σ}_{22} = = 11 - - \frac{{M m}_{r r c c}^{22}}{{L L}_{s the s c c} {L L}_{r r}} \\ {σ σ}_{33} = = 11 - - \frac{{M m}_{r r p p} {M m}_{r r c c}}{{L L}_{r r}} \end{matrix} - - - - - - ((1313))

Let each state variable in the motor system [i _dp , i _qp , i _dc , i _qc , ψ _dr , ψ _qr , ω _r ] ^T = [x ₁ , x ₂ , x ₃ , x ₄ , x ₅ , x ₆ , x ₇ ] ^T , [u _dp , u _qp , u _dc , u _qc , u _dr , u _qr , T _L ] ^T ＝[0, 0, 0, 0, 0, 0, 0] ^T . The first-order differential equations of the BDFM motor system can be obtained by sorting and simplifying equations (6), (7), (9)-(13):

\{\begin{matrix} 00 = = {η η}_{11} {\overset{\cdot &Center Dot;}{x x}}_{11} + + {η η}_{22} {x x}_{11} - - {η η}_{33} {x x}_{22} {x x}_{77} + + {η η}_{44} {x x}_{33} \\ - - {η η}_{55} {x x}_{44} {x x}_{77} - - {η η}_{66} {x x}_{55} + + {η η}_{77} {x x}_{66} {x x}_{77} \\ 00 = = {η η}_{11} {\overset{\cdot \cdot}{x x}}_{22} + + {η η}_{33} {x x}_{11} {x x}_{77} + + {η η}_{22} {x x}_{22} - - {η η}_{55} {x x}_{33} {x x}_{77} \\ - - {η η}_{44} {x x}_{44} + + {η η}_{88} {x x}_{55} {x x}_{77} - - {η η}_{66} {x x}_{66} \\ 00 = = {γ γ}_{11} {\overset{\cdot &Center Dot;}{x x}}_{33} + + {γ γ}_{22} {x x}_{11} - - {γ γ}_{33} {x x}_{22} {x x}_{77} + + {γ γ}_{44} {x x}_{33} \\ - - {γ γ}_{55} {x x}_{44} {x x}_{77} - - {γ γ}_{66} {x x}_{55} + + {γ γ}_{77} {x x}_{66} {x x}_{77} \\ 00 = = {γ γ}_{11} {\overset{\cdot &Center Dot;}{x x}}_{44} - - {γ γ}_{33} {x x}_{11} {x x}_{77} - - {γ γ}_{22} {x x}_{22} + + {γ γ}_{55} {x x}_{33} {x x}_{77} \\ + + {γ γ}_{44} {x x}_{44} - - {γ γ}_{88} {x x}_{55} {x x}_{77} + + {γ γ}_{66} {x x}_{66} \\ 00 = = {ϵ ϵ}_{11} {\overset{\cdot \cdot}{x x}}_{55} + + {ϵ ϵ}_{22} {x x}_{55} - - {ϵ ϵ}_{33} {x x}_{11} - - {ϵ ϵ}_{44} {x x}_{33} \\ 00 = = {ϵ ϵ}_{11} {\overset{\cdot &Center Dot;}{x x}}_{66} + + {ϵ ϵ}_{22} {x x}_{66} - - {ϵ ϵ}_{33} {x x}_{22} - - {ϵ ϵ}_{44} {x x}_{44} \\ 00 = = {ϵ ϵ}_{11} {\overset{\cdot &Center Dot;}{x x}}_{77} + + {ϵ ϵ}_{55} {x x}_{77} + + {ϵ ϵ}_{66} (({x x}_{22} {x x}_{33} + + {x x}_{11} {x x}_{44})) \\ - - {ϵ ϵ}_{77} (({x x}_{55} {x x}_{22} - - {x x}_{66} {x x}_{11})) - - {ϵ ϵ}_{88} (({x x}_{55} {x x}_{44} + + {x x}_{66} {x x}_{33})) \end{matrix} - - - - - - ((1414))

It can be seen from formula (14) that the BDFM motor system has the characteristics of multivariable, nonlinear and strong coupling. The coefficients of each variable in the equation system can be expressed in matrix form as:

\begin{matrix} η η = = [\begin{matrix} {η η}_{11} & {η η}_{22} \\ {η η}_{33} & {η η}_{44} \\ {η η}_{55} & {η η}_{66} \\ {η η}_{77} & {η η}_{88} \end{matrix}] \\ = = [\begin{matrix} \frac{{L L}_{s the s p p} {L L}_{s the s c c} {σ σ}_{11} {σ σ}_{22}}{{σ σ}_{33}} - - {σ σ}_{33} & \frac{{σ σ}_{33}}{{T T}_{r r}} + + \frac{{r r}_{p p} {L L}_{s the s c c} {σ σ}_{22}}{{σ σ}_{33}} + + \frac{{M m}_{r r p p}^{22} {L L}_{s the s c c} {σ σ}_{22}}{{L L}_{r r} {σ σ}_{33} {T T}_{r r}} \\ \frac{{p p}_{p p} {L L}_{s the s p p} {L L}_{s the s c c} {σ σ}_{11} {σ σ}_{22}}{{σ σ}_{33}} + + {p p}_{c c} {σ σ}_{33} & {r r}_{c c} + + \frac{{M m}_{r r c c}}{{L L}_{r r} {T T}_{r r}} + + \frac{{L L}_{s the s c c} {σ σ}_{22}}{{T T}_{r r}} \\ (({p p}_{p p} + + {p p}_{c c})) {L L}_{s the s c c} {σ σ}_{22} & \frac{{M m}_{r r c c}}{{L L}_{r r} {T T}_{r r}} + + \frac{{M m}_{r r p p} {L L}_{s the s c c} {σ σ}_{22}}{{L L}_{r r} {σ σ}_{33} {T T}_{r r}} \\ \frac{{p p}_{c c} {M m}_{r r c c}}{{L L}_{r r}} - - \frac{{p p}_{p p} {M m}_{r r p p} {L L}_{s the s c c} {σ σ}_{22}}{{L L}_{r r} {σ σ}_{33}} & \frac{{p p}_{c c} {M m}_{r r c c}}{{L L}_{r r}} + + \frac{{p p}_{p p} {M m}_{r r p p} {L L}_{s the s c c} {σ σ}_{22}}{{L L}_{r r} {σ σ}_{33}} \end{matrix}] \end{matrix}

\begin{matrix} γ γ = = [\begin{matrix} {γ γ}_{11} & {γ γ}_{22} \\ {γ γ}_{33} & {γ γ}_{44} \\ {γ γ}_{55} & {γ γ}_{66} \\ {γ γ}_{77} & {γ γ}_{88} \end{matrix}] \\ = = [\begin{matrix} \frac{{L L}_{s the s p p} {L L}_{s the s c c} {σ σ}_{11} {σ σ}_{22}}{{σ σ}_{33}} - - {σ σ}_{33} & {r r}_{p p} + + \frac{{M m}_{r r p p}^{22}}{{L L}_{r r} {T T}_{r r}} + + \frac{{L L}_{s the s p p} {σ σ}_{11}}{{T T}_{r r}} \\ (({p p}_{p p} + + {p p}_{c c})) {L L}_{s the s p p} {σ σ}_{11} & \frac{{σ σ}_{33}}{{T T}_{r r}} + + \frac{{r r}_{c c} {L L}_{s the s p p} {σ σ}_{11}}{{σ σ}_{33}} + + \frac{{M m}_{r r c c}^{22} {L L}_{s the s p p} {σ σ}_{11}}{{L L}_{r r} {σ σ}_{33} {T T}_{r r}} \\ \frac{{p p}_{c c} {L L}_{s the s p p} {L L}_{s the s c c} {σ σ}_{11} {σ σ}_{22}}{{σ σ}_{33}} + + {p p}_{p p} {σ σ}_{33} & \frac{{M m}_{r r p p}}{{L L}_{r r} {T T}_{r r}} + + \frac{{M m}_{r r c c} {L L}_{s the s p p} {σ σ}_{11}}{{L L}_{r r} {σ σ}_{33} {T T}_{r r}} \\ \frac{{p p}_{c c} {M m}_{r r c c} {L L}_{s the s p p} {σ σ}_{11}}{{L L}_{r r} {σ σ}_{33}} - - \frac{{p p}_{p p} {M m}_{r r p p}}{{L L}_{r r}} & \frac{{p p}_{c c} {M m}_{r r c c} {L L}_{s the s p p} {σ σ}_{11}}{{L L}_{r r} {σ σ}_{33}} + + \frac{{p p}_{p p} {M m}_{r r p p}}{{L L}_{r r}} \end{matrix}] \end{matrix}

ϵ ϵ = = [\begin{matrix} {ϵ ϵ}_{11} & {ϵ ϵ}_{22} \\ {ϵ ϵ}_{33} & {ϵ ϵ}_{44} \\ {ϵ ϵ}_{55} & {ϵ ϵ}_{66} \\ {ϵ ϵ}_{77} & {ϵ ϵ}_{88} \end{matrix}] = = [\begin{matrix} 11 & \frac{11}{{T T}_{r r}} \\ \frac{{M m}_{r r p p}}{{T T}_{r r}} & \frac{{M m}_{r r c c}}{{T T}_{r r}} \\ \frac{D D.}{J J} & \frac{(({p p}_{p p} + + {p p}_{c c})) {σ σ}_{33}}{J J} \\ \frac{{p p}_{p p} {M m}_{r r p p}}{{L L}_{r r}} & \frac{{p p}_{c c} {M m}_{r r c c}}{{L L}_{r r}} \end{matrix}]

3. the brushless doubly-fed machine chaos analysis method based on maximum Lyapunov exponent according to claim 2, is characterized in that, the process of described step S2 is as follows:

31) Introduce the rotor speed PD control item in formula (14):

{i i}_{q q p p} = = {k k}_{p p} ((ω ω - - {ω ω}_{r r e e f f})) + + {k k}_{d d} \frac{d d ((ω ω - - {ω ω}_{r r e e f f}))}{d d t t}

Let ω _ref =0 in the above formula, we can get:

{x x}_{22} = = {k k}_{p p} {x x}_{77} + + {k k}_{d d} {\overset{\cdot &Center Dot;}{x x}}_{77} - - - - - - ((1515));;

32) order: where T _r ^* is the estimated value of the rotor motion time constant, then:

{T T}_{r r}^{* *} = = {kT kT}_{r r} - - - - - - ((1616))

Substituting equations (15) and (16) into the first-order differential equations of the rotor winding in equation (14), the seventh-order non-autonomous system of BDFM can be constructed:

\{\begin{matrix} {\overset{\cdot &Center Dot;}{x x}}_{11} = = \frac{11}{{η η}_{11}} ((- - {η η}_{22} {x x}_{11} + + {η η}_{33} {x x}_{22} {x x}_{77} - - {η η}_{44} {x x}_{33} + + \\ {η η}_{55} {x x}_{44} {x x}_{77} + + {η η}_{66} {x x}_{55} - - {η η}_{77} {x x}_{66} {x x}_{77})) \\ {\overset{\cdot &Center Dot;}{x x}}_{22} = = \frac{11}{{η η}_{11}} ((- - {η η}_{33} {x x}_{11} {x x}_{77} - - {η η}_{22} {x x}_{22} + + {η η}_{55} {x x}_{33} {x x}_{77} + + \\ {η η}_{44} {x x}_{44} - - {η η}_{88} {x x}_{55} {x x}_{77} + + {η η}_{66} {x x}_{66})) \\ {\overset{\cdot \cdot}{x x}}_{33} = = \frac{11}{{γ γ}_{11}} ((- - {γ γ}_{22} {x x}_{11} + + {γ γ}_{33} {x x}_{22} {x x}_{77} - - {γ γ}_{44} {x x}_{33} + + \\ {γ γ}_{55} {x x}_{44} {x x}_{77} + + {γ γ}_{66} {x x}_{55} - - {γ γ}_{77} {x x}_{66} {x x}_{77})) \\ {\overset{\cdot &Center Dot;}{x x}}_{44} = = \frac{11}{{γ γ}_{11}} (({γ γ}_{33} {x x}_{11} {x x}_{77} + + {γ γ}_{22} {x x}_{22} - - {γ γ}_{55} {x x}_{33} {x x}_{77} - - \\ {γ γ}_{44} {x x}_{44} + + {γ γ}_{88} {x x}_{55} {x x}_{77} - - {γ γ}_{66} {x x}_{66})) \\ {\overset{\cdot &Center Dot;}{x x}}_{55} = = \frac{k k}{{ϵ ϵ}_{11}} ((- - {ϵ ϵ}_{22} {x x}_{55} + + {ϵ ϵ}_{33} {x x}_{11} + + {ϵ ϵ}_{44} {x x}_{33})) \\ {\overset{\cdot &Center Dot;}{x x}}_{66} = = \frac{k k}{{ϵ ϵ}_{11}} [[- - {ϵ ϵ}_{22} {x x}_{66} + + {ϵ ϵ}_{33} (({k k}_{p p} {x x}_{77} + + {k k}_{d d} {\overset{\cdot &Center Dot;}{x x}}_{77})) \\ + + {ϵ ϵ}_{44} {x x}_{44}]] \\ {\overset{\cdot &Center Dot;}{x x}}_{77} = = \frac{11}{{ϵ ϵ}_{11}} [[- - {ϵ ϵ}_{55} {x x}_{77} - - {ϵ ϵ}_{66} (({x x}_{22} {x x}_{33} + + {x x}_{11} {x x}_{44})) + + \\ {ϵ ϵ}_{77} (({x x}_{55} {x x}_{22} - - {x x}_{66} {x x}_{11})) + + {ϵ ϵ}_{88} (({x x}_{55} {x x}_{44} + + {x x}_{66} {x x}_{33}))]] \end{matrix} - - - - - - ((1717)) . .

4. according to the brushless doubly-fed machine chaos analysis method based on maximum Lyapunov exponent according to claim 3, it is characterized in that, the process of described step S3 is as follows:

41) Let the maximum Lyapunov exponent be:

{λ λ}_{m m a a x x} = = \underset{m m &RightArrow; &Right Arrow; ∞ ∞}{lim lim} \frac{11}{m m τ τ} {Σ Σ}_{i i = = 11}^{m m} l l o o g g ((\frac{{d d}_{i i}}{{d d}_{00}})) - - - - - - ((1818))

Among them, m is the number of integration times, τ is the integration time;

42) According to the actual parameters of the brushless doubly-fed motor, substitute each parameter into formula (17), use formula (19) as the calculation principle, write the solution program of the maximum Lyapunov exponent on the Matlab platform, and obtain the maximum Lyapunov exponent spectrum of the BDFM system , the BDFM system enters the state of chaotic motion from the exponential spectrum.