基于最优权值切换的复合无速度传感器算法
Algorithm for Composite No-Speed Sensor Based on Optimal Weight Switching
摘要:为改善永磁同步电机传统复合无速度传感器算法在算法切换过渡区间内算法权值不能做出实时调整导致观测性能较差的问题,分析了超螺旋滑模控制以及脉振高频注入法的原理,提出了一种基于粒子群优化算法与复合无速度传感器相结合的智能转速观测策略。利用粒子群优化算法优化算法切换过程中的权值,获取在过渡区间内的最优权值,并基于该最优权值获取永磁同步电机的实时转速。在MATLAB/Simulink中仿真结果表明,该控制系统提升了转速观测算法在过渡区间内的观测准确性与实时性,以及算法切换过程中的观测脉动,优化了永磁同步电机控制系统的性能。
Abstract:To improve the issue of poor observation performance caused by the inability to make real-time adjustments to the algorithm weights in the transition zone of the traditional composite sensorless algorithm for permanent magnet synchronous motors, the principles of super-spiral sliding mode control and pulse high-frequency injection method are analyzed. An intelligent speed observation strategy based on particle swarm optimization algorithm combined with the composite sensorless method is proposed. The particle swarm optimization algorithm is used to optimize the weights during the algorithm switching process, obtaining the optimal weights in the transition zone, and real-time speed of the permanent magnet synchronous motor is acquired based on these optimal weights. The simulation results in MATLAB/Simulink show that this control system enhances the accuracy and real-time performance of the speed observation algorithm in the transition zone, as well as the observation fluctuations during the algorithm switching process, optimizing the performance of the permanent magnet synchronous motor control system.
关键词:永磁同步电机;无速度传感器;智能控制;最优控制
Keywords: Permanent magnet synchronous motor; sensorless; intelligent control; optimal control
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永磁同步电机(PMSM)根据转子与永磁体的结构与分布情况可以分为表贴式和内置式两种类型。相对于内置式PMSM,表贴式PMSM的制造成本低,结构简单;转子具有良好的隐极特性,从而使得电机永磁体的磁场分布更接近于标准的正弦分布,电机的性能更易得到发挥;且表贴式PMSM的d-q轴具有相同的电感系数,更易建立数学模型。基于上述特性对表贴式PMSM进行研究与分析并建立数学模型,并设定如下理想条件:
Permanent Magnet Synchronous Motors (PMSM) can be divided into two types: surface-mounted and interior-mounted, based on the structure and distribution of the rotor and permanent magnets. Compared to interior-mounted PMSMs, surface-mounted PMSMs have lower manufacturing costs and simpler structures; the rotor has good saliency characteristics, which allows the magnetic field distribution of the motor's permanent magnets to be closer to a standard sine distribution, making it easier to achieve optimal motor performance. Additionally, the d-q axes of surface-mounted PMSMs have the same inductance coefficients, facilitating the establishment of mathematical models. Based on these characteristics, research and analysis of surface-mounted PMSMs will be conducted, and a mathematical model will be established under the following ideal conditions:
定子绕组Y型接法,三相绕组对称分布,各绕组轴线在空间互差120°。
The stator winding is connected in a Y-type configuration, with the three-phase windings symmetrically distributed, and the axes of each winding are spatially offset by 120°.
转子上的永磁体在定转子气隙内产生的主磁场沿气隙圆周呈正弦分布,转子没有阻尼绕组。
The permanent magnets on the rotor generate a main magnetic field in the air gap between the stator and rotor, which has a sinusoidal distribution along the circumference of the air gap, and the rotor has no damping windings.
忽略定子绕组的的齿槽对气隙磁场分布的影响。
Ignore the effect of the stator winding slots on the air gap magnetic field distribution.
铁芯磁导率无穷大,忽略铁芯的磁滞损耗与涡流损耗。
The magnetic permeability of the iron core is infinite, ignoring the hysteresis loss and eddy current loss of the iron core.
忽略绕组电阻与绕组电感变化的影响。
Ignore the effects of changes in winding resistance and winding inductance.
d-q坐标系下定子电压方程:
Stator voltage equation in the d-q coordinate system:
磁链方程:
Magnetic flux equation:
电磁转矩方程:
Electromagnetic torque equation:
机械运动方程:
Mechanical Motion Equation:
式中:—d轴电压;—q轴电压;—定子绕组电阻;—d轴电流;—q轴电流;—微分算子;—d轴磁链;—q轴磁链;—电机电角速度;—d轴电感;—q轴电感;—永磁体磁链;—电磁转矩;—负载转矩;—磁极对数;—摩擦系数;—电机机械角速度;—电机转子转动惯量。
1 低速区间速度估算算法
1Low-speed range speed estimation algorithm
永磁同步电机无速度传感器算法的核心原理是利用电机运行过程中的反电动势信号来估算电机的实时转速,而反电动势的大小与电机的转速成正相关的关系,也正是这种关系造成了当电机处于零速或低速时,反电动势信号较小,系统的信噪比较小等问题,难以获取较为准确的估算信号,因此需要在系统中额外注入电信号,并根据其引起的响应估算电机的实时转速。
The core principle of the sensorless algorithm for permanent magnet synchronous motors is to estimate the real-time speed of the motor using the back electromotive force (EMF) signal during the motor's operation. The magnitude of the back EMF is positively correlated with the motor's speed. This relationship causes issues such as a smaller back EMF signal when the motor is at zero or low speed, resulting in a lower signal-to-noise ratio, making it difficult to obtain a more accurate estimation signal. Therefore, it is necessary to inject additional electrical signals into the system and estimate the motor's real-time speed based on the responses they generate..
脉振高频电压注入法控制原理分析
Analysis of the Control Principle of High-Frequency Voltage Injection Method for Pulse Vibration
脉振高频电压注入法是指在同步旋转坐标系的直轴上(也就是d轴)注入高频正弦电压,所以注入信号在静止坐标系中是一个脉振的高频电压信号。注入后,对交轴(也就是q轴)高频电流进行调制解调,最终得到转子位置和速度信息。由于额外注入电信号的频率远高于永磁同步电机的角频率,因此可以将电机等效为阻感性负载,此时d-q坐标系下永磁同步电机的高频电压方程为:
The pulse voltage injection method refers to injecting a high-frequency sine voltage along the direct axis (the d-axis) in a synchronous rotating coordinate system, so the injected signal is a pulsed high-frequency voltage signal in the stationary coordinate system. After injection, the high-frequency current on the quadrature axis(the q-axis) is modulated and demodulated, ultimately obtaining rotor position and speed information. Since the frequency of the additional injected electrical signal is much higher than the angular frequency of the permanent magnet synchronous motor, the motor can be equivalent to a resistive-inductive load. At this time, the high-frequency voltage equation of the permanent magnet synchronous motor in the d-q coordinate system is:
(1) |
式中,、、、为d,q轴高频电压及高频电流分量;、、、为高频注入下d,q轴定子阻感;为微分算子。
In the formula, 、 、 、 are the d and q axis high-frequency voltage and high-frequency current components; 、 、 、 are the d and q axis stator inductances under high-frequency injection; is the differential operator.
当采用频率为的高频注入信号且系统处于稳定状态时,上述方程可改写为:
When the frequency is for the high-frequency injection signal and the system is in a stable state, the above equation can be rewritten as:
(2) |
式中,、为高频注入下d-q轴阻抗。
In the formula, and are the d-q axis impedances under high-frequency injection.
则高频电压和电流在估计转速同步参考坐标系下的关系为:
The relationship between high-frequency voltage and current in the estimated speed synchronous reference coordinate system is:
(3) |
式中,、、、为在估计转速同步参考坐标系下d-q轴高频电压及高频电流分量;为转子实际位置与估计位置的偏差角。
In the formula, 、 、 、 are the high-frequency voltage and high-frequency current components on the d-q axis in the estimated speed synchronous reference frame; is the deviation angle between the actual rotor position and the estimated position.
在d轴(估计转速同步参考坐标系)注入高频电压信号:
Inject a high-frequency voltage signal in the d-axis (estimated speed synchronous reference coordinate system):
(4) |
式中,为注入的高频电压信号模值。
In the formula, is the amplitude of the injected high-frequency voltage signal.
最终可得高频注入下的交直轴电流:
The final obtainable direct and alternating axis currents under high-frequency injection:
(5) |
式中,、为d-q轴半差高频阻抗与平均高频阻抗,定义为:
In the formula, and are the d-q axis half-differential high-frequency impedance and the average high-frequency impedance, defined as:
(6) |
由上述分析过程可以看出,如果d-q轴的阻抗不相等,就是不等于0,那么高频注入下的d-q轴电流都和位置误差角有关,如果转子误差角度为0°时,d轴电流和平均阻抗有关,且d轴电流不为0,但是q轴电流为0。即可以在q轴电流为0的时候,得到确切的位置信息,因此可以对q轴的高频电流信号进行处理,得到永磁同步电机转子的实际位置信息。
From the above analysis processit can be seen that ifd-qaxis impedances are not equal, that is not equal to 0, then thed-qaxis currents are related to the positionerrorangle if the rotor error angle is 0°the d-axis currentis related to the average impedance,andd-axis current is not 0,but the q-axis current is 0.That is, when the q-axis current is 0, accurate position information can be obtained, thereforethe high-frequency currentsignal of the q-axiscan be processed to obtainthe actual position information of thepermanent magnet synchronous motor rotor.
高频信号解调及控制策略
High-Frequency Signal Demodulation and Control Strategy
在稳态下,假设不计PMSM定子线圈d-q轴之间的高频电阻的差异,则q轴电流高频分量为:
Understeadystate, assuming the difference in high-frequency resistance between the d-q axis of the PMSM stator coil is negligible, thenthehigh-frequency component of the q-axis current is:
(7) |
式中,为半高差高频电感。
In the formula, is the half-height difference high-frequency inductor.
由上述公式可以看出,不仅和位置角的误差有关,还受的影响,是一个时变量。为了提取q轴上高频电流中的估计位置的误差,需要按照如下策略进行解调制。首先需要经过带通滤波器,因为高频估计电流都和有关,所以需要通过带通滤波器BPF得到高频电流中与半差高频电感相关的高频电流。然后通过乘法器,对q轴的高频电流分量进行解调制,相当于经过带通滤波器的信号再乘以以得到的平方,这个平方可以化为,对于高频信号而言,如果经过低通滤波器,则。所以最后可以把经过滤波,解调,再滤波之后的电流信号表示为:
From the above formula, it can be seen that is not only related to the error of the position angle , but is also influenced by , making it a time-variable. To extract the estimated position error from the high-frequency current on the q-axis, demodulation needs to be carried out according to the following strategy. First, a band-pass filter is required, as the high-frequency estimated current is related to , so a band-pass filter (BPF) is needed to obtain the high-frequency current related to the semi-difference high-frequency inductance. Then, through a multiplier, the high-frequency current component on the q-axis is demodulated, which is equivalent to taking the signal that has passed through the band-pass filter and multiplying it by to obtain the square of , which can be expressed as . For high-frequency signals, if passed through a low-pass filter, then . Therefore, the current signal after filtering, demodulating, and then filtering can be represented as:
(8) |
若转子位置信息估计误差足够小,可以线性化估算器的输入为:
If the estimation error of the rotor position information is sufficiently small, the input of the estimator can be linearized as:
(9) |
式中,为误差增益,取决于高频注入电压及PMSM的参数。
In the formula, isthe error gain, which depends on the high-frequency injection voltage and the parameters of the PMSM.
对高频载波信号的解调过程如图1所示。
The demodulationprocess of high-frequency carriersignals is shown inFigure1.
图1 高频载波信号的解调过程
Figure1 The demodulation process of high-frequency carrier signals
Figure 1 Demodulation of high frequency carrier signal
那么最终可以得到跟转子位置误差息息相关的电流信息,可利用基于PI调节器的转子位置估算对其进行进一步处理就可以得到位置信息。基于PI调节器的转子位置估算器原理框图如图2所示。
Thus, we can ultimately obtain information related to the rotor position error current information,which can be further processed usingPIcontroller-based rotor position estimationto obtain position information. The principle block diagram of the PI controller-based rotor position estimator is shown inFigure2.
此方法相当于一个锁相环原理,当系统工作在稳定状态时,相位差不会随着时间变化而变化,此时处于锁定状态,既可以实现对PMSM转子位置的跟踪,又能够降低因滤波环节引起相位的滞后效应。
This method is equivalent to a phase-locked loop principle. When the system operates in a stable state, the phase difference does not change over time, and at this point, it is in a locked state, which can achieve tracking of the PMSM rotor position while also reducing the phase lag effect caused by the filtering process.
图2 转子位置估算器
Figure2 Rotor Position Estimator
Figure 2 Rotor Position Estimator
利用PI调节器构成的PLL系统控制框图如图3所示。
Using PIadjustersto formthe control block diagram of the PLL systemas shown inFigure3.
图3 PLL系统控制框图
Figure3 PLL System Control Block Diagram
Figure 3 Control block diagram of PLL system
其中,LPF滤波器为一阶低通滤波器,其期望带宽为,其传递函数形式为:
Among them, the LPF filter is a first-order low-pass filter, with a desired bandwidth of , and its transfer function is:
(10) |
PI控制器的传递函数形式为:
The transfer function form of the PI controller is:
(11) |
图示控制框图的闭环传递函数为:
The closed-loop transfer function of the illustrated control block diagram is:
(12) |
将闭环传递函数的三个极点均配置为,则PI控制器的比例系数与积分系数、可以整定为:
The three poles of the closed-loop transfer functionare set to , then the proportional and integral coefficients of the PI controller , can be tuned to:
(13) |
2 高速区间速度估算算法
2 High-speed interval speed estimation algorithm
永磁同步电机处于高速区间时,电机旋转带来的反电动势较大,信噪比也较大,可以较为准确的获取电机的反电动势信号来实现对转速的实时估算。在众多算法中基于滑模控制原理的无速度传感器算法表现出了相对优良的性能。在该算法的基础上如何抑制滑模控制中滑模面切换带来的抖振问题是提高速度估算准确性与稳定性的关键。
When the permanent magnet synchronous motor is in the high-speed range, the back electromotive force generated by the motor's rotation is relatively large, and the signal-to-noise ratio is also high, allowing for a more accurate acquisition of the back electromotive force signal to achieve real-time estimation of the speed. Among various algorithms, the sensorless algorithm based on sliding mode control principles has demonstrated relatively excellent performance. The key to improving the accuracy and stability of speed estimation lies in how to suppress the chattering problem caused by the switching of the sliding surface in sliding mode control based on this algorithm.
2.1 滑模无速度传感器算法
2.1 Slipform without Speed Sensor Algorithm
PMSM在两相静止坐标系下的系统方程为:
The system equations of the PMSM in a two-phase stationary coordinate system are:
(14) |
式中,,可以看作α-β坐标系下感应电动势。
In the formula, , can be regarded as the induced electromotive force in the α-β coordinate system.
为了获得扩展反电动势的估计值,SMO可设计为:
In order to obtain an estimate of the extended back electromotive force,SMOcan be designed as:
(15) | |
(16) |
式中,、为电流观测值;为滑模增益。
In the formula, and are the current observation values; k is the sliding mode gain.
可得电流误差状态方程为:
The current error state equation can be obtained as:
(17) |
式中,;为电流观测误差。
In the formula, ; isthe currentobservationerror.
采用滑模观测器对电流进行估计,其滑模面函数定义为:
The sliding mode observer is used to estimate the current, and its sliding mode surface function is defined as:
(18) |
当满足以下条件时,滑模观测器进入滑动模态。
The sliding mode observer enters the sliding mode when the following conditions are met.
(19) |
当滑模增益满足该不等式时,则
When the sliding mode gain satisfies this inequality, then
(20) | |
(21) |
可以观察到,估算得出的反电动势中含有高频切换信号。并且,采用基于反正切函数的估算方法时,会将该高频切换信号直接引入反正切函数的除法运算中,结果导致高频抖振现象的产生
It can be observed that the estimated back electromotive force contains high-frequency switching signals. Moreover, when using an estimation method based on the arctangent function, this high-frequency switching signal is directly introduced into the division operation of the arctangent function, resulting in the occurrence of high-frequency oscillation phenomena.
2.2 改进型滑模无速度传感器算法
2.2 Improved Sliding Mode Algorithm without Speed Sensor
对于传统的一阶滑模观测器,估算的反电动势因符号函数的影响,存在抖振现象,从而导致估算的动子速度存在跟踪误差,虽然通过低通滤波能减小抖振,但会造成相位延迟。为此,引入超螺旋算法,来解决传统SMO造成的抖振现象。超螺旋算法为:
Fortraditional first-ordersliding modeobservers,the estimatedback electromotive force due tothe sign functionhaschatteringphenomena,which leads to the estimatedrotorspeedhavingtracking errors,althoughlow-passfiltering can reducechattering, itwillcause phase delay. To address this,the super spiral algorithm is introducedto solvethe chattering phenomena caused bytraditional SMO.The super spiralalgorithm is:
(22) |
式中,为状态变量,为估计值与实际值之间的误差;为滑模增益。
In the , is the state variable , the estimated value and the actual value are the errors ; is the sliding mode gain .
将超螺旋算法中的状态变量分别用PMSM估算的电流信号、替换掉,得到基于超螺旋算法的PMSM滑模观测器:
Replace thestate variables in thesuper spiral algorithm with the current signals estimated by PMSM and obtain the PMSM sliding mode observer based on the super spiral algorithm:
(23) |
电流的误差方程为:
The error equation of the currentis:
(24) |
式中,;为电流观测误差。
In the formula, ; isthe currentobservationerror.
当ST-SMO稳定时,估计值等于实际值,即PMSM电流估计误差和变化率近似为零()。此时,可得到超螺旋滑模观测器估算的永磁同步电机反电动势:
When ST-SMO is stable,theestimatedvalue equals the actual value,that isPMSM currentestimationerror andrate of changeare approximatelyzero ( ).At this time, one canobtainthe superhelical sliding modeobserverestimatedpermanent magnet synchronous motorback electromotive force:
(25) |
式中,为边界函数;为常数,用来减小模糊控制输入的误差微分造成的影响。
In the formula, is the boundary function; is a constant used to reduce the impact of the error differential caused by fuzzy control input.
为解决滑模控制中的高频抖振问题,超螺旋控制算法采用串联高阶滑模以确保输出的连续性,其特点在于只需要滑模量信息,而不必要求一阶导数的信息。
To solve the high-frequency chattering problem in sliding mode control, the super-spiral control algorithm uses a series of higher-order sliding modes to ensure the continuity of the output, characterized by the requirement of only sliding mode quantity information, without needing information on the first derivative.
3 过渡区间切换算法
3 Transition Zone Switching Algorithm
3.1 滞环切换法
3.1 Hysteresis Switching Method
滞环切换法原理简单,实现起来十分方便,它在预先设置好的切换转速点处直接完成高低转速算法间的切换。该方法在切换点及其附近转速区间处,由于两种算法的误差叠加,对PMSM转子位置的估计会产生较大的偏差,从而使整个控制系统出现失稳、抖振等情况,滞环切换的工作原理如图4所示。
The hysteresis switching method has a simple principle and is very convenient to implement. It directly completes the switching between high and low speed algorithms at the pre-set switching speed points. Near the switching points and within the speed range, due to the error accumulation of the two algorithms, the estimation of the PMSM rotor position can have significant deviations, leading to instability and oscillation in the entire control system. The working principle of hysteresis switching is shown in Figure 4.
图4 滞环切换
Figure4 Hysteresis Switching
Figure 4 Hysteresis switching method
滞环切换算法包含两个相反的过程,第一个过程是电机的转速上升过程,当电机转速低于ωH时,无速度传感器采用脉振高频注入算法获取转子的转速及位置信息,当电机转速高于ωH时,无速度传感器将切换成模糊超螺旋滑模算法,第二个过程是电机转速下降过程,当电机转速由较高转速下降到ωL之前,电机一直由模糊超螺旋滑模算法获取转子的位置与转速信息,当电机转速低于ωL之后,无速度传感器算法将切换成脉振高频注入算法。
The hysteresis switching algorithm includes two opposing processes. The first process is the motor speed increase process. When the motor speed is belowωH, the sensorless system uses a pulse high-frequency injection algorithm to obtain the rotor's speed and position information. When the motor speed is aboveωH, the sensorless system will switch to a fuzzy super spiral sliding mode algorithm. The second process is the motor speed decrease process. When the motor speed decreases from a higher speed toωL before, the motor continuously uses the fuzzy super spiral sliding mode algorithm to obtain the rotor's position and speed information. When the motor speed drops belowωL, the sensorless algorithm will switch to the pulse high-frequency injection algorithm.
3.2 加权切换法
3.2 Weighted Switching Method
为了避免滞环切换法造成的系统失稳与抖振问题,加权切换法设置了转速切换的过渡区间,并在过渡区间内设置权重函数,当电机的转速值位于过渡区间时,系统不单独由某一种算法获取转子转速及位置信息,而是由两种算法的输出值与权重函数共同决定,从而避免了滞环切换算法中类似于开关式的切换过程,减少了切换过程中的冲击,使电机实现由低转速到高转速的稳定平滑的切换过程。加权切换法的原理图如图5所示。
To avoid system instability and oscillation issues caused by the hysteresis switching method, the weighted switching method sets a transition interval for speed switching and establishes a weight function within this interval. When the motor's speed value is within the transition interval, the system does not solely rely on one algorithm to obtain rotor speed and position information; instead, it is determined by the output values of both algorithms and the weight function. This avoids the switch-like transition process found in hysteresis switching algorithms, reduces the impact during the switching process, and allows for a stable and smooth transition from low to high speed in the motor. The schematic diagram of the weighted switching method is shown in Figure 5.
图5 加权切换
Figure 5Weighted Switching
Figure 5 Weighted switching method
设置为低转速算法的权值,为高转速算法的权值并且两者和为1,则权值函数表述为:
Set as the weight for the low-speed algorithm, as the weight for the high-speed algorithm, and the sum of both is 1, then the weight function is expressed as:
(26) |
当电机的转子角速度位于低速区间时,脉振高频注入法的权重系数为1,此时模糊超螺旋滑模算法不参与无速度传感器的运算,电机转子速度与位置信息仅由脉振高频注入法提供。当电机的转子角速度位于高速区间时,模糊超螺旋滑模算法的权重系数为1,此时脉振高频注入法不参与无速度传感器的运算,电机转子速度与位置信息仅由模糊超螺旋滑模算法提供。当电机转子角速度位于切换区间时,电机转子速度与位置信息由两种算法在考虑其权重系数后综合提供,每种算法的权重系数值由权重函数根据转子转速值提供。
When the rotor angular velocity of the motor is in the low-speed range, the weight coefficient of the pulse vibration high-frequency injection method is 1, at this time the fuzzy super spiral sliding mode algorithm does not participate in the operation of the sensorless speed, and the rotor speed and position information of the motor are only provided by the pulse vibration high-frequency injection method. When the rotor angular velocity of the motor is in the high-speed range, the weight coefficient of the fuzzy super spiral sliding mode algorithm is 1, at this time the pulse vibration high-frequency injection method does not participate in the operation of the sensorless speed, and the rotor speed and position information of the motor are only provided by the fuzzy super spiral sliding mode algorithm. When the rotor angular velocity of the motor is in the switching range, the rotor speed and position information of the motor are provided by both algorithms after considering their weight coefficients, and the weight coefficient value of each algorithm is provided by the weight function based on the rotor speed value.
3.3 最优权值切换法
3.3 Optimal Weight Switching Method
(1)粒子群算法的基本原理
(1) Basic Principles of Particle Swarm Algorithm
粒子群算法的灵感来源于对鸟群觅食行为的探究,鸟群可以通过集体信息的共享来找到最优的目的地。在粒子群算法中,每个个体(也就是粒子)在移动过程中都会积累自己的记忆和经验,当它们移动时,会根据自己的经验和记忆来调整方向。并且,群体中的粒子是同时移动的,每个粒子又会将自己的经验与记忆结合其他粒子的经验与记忆来寻找最合适的解决方案,使自己处于最优解中。这种特性使得粒子群算法不仅受到单个粒子演化的影响,还具有群体间的学习和记忆性,从而帮助粒子实现最佳调整。粒子群算法的优点在于收敛速度快、参数少、算法简单易于实现,相较于遗传算法,在解决高维优化问题时能够更快收敛于最优解。因此广泛应用于解决理论研究与工程实践中所面临的参数优化问题。其计算流程如图6所示。
The inspiration for the particle swarm algorithm comes from the exploration of the foraging behavior of bird flocks, which can find the optimal destination through the sharing of collective information. In the particle swarm algorithm, each individual (or particle) accumulates its own memories and experiences during movement, and as they move, they adjust their direction based on their own experiences and memories. Additionally, the particles in the swarm move simultaneously, and each particle combines its own experiences and memories with those of other particles to find the most suitable solution, positioning itself within the optimal solution. This characteristic allows the particle swarm algorithm to be influenced not only by the evolution of individual particles but also by the learning and memory among the group, thereby helping particles achieve optimal adjustments. The advantages of the particle swarm algorithm include fast convergence speed, fewer parameters, and a simple and easy-to-implement algorithm, which can converge to the optimal solution more quickly than genetic algorithms when solving high-dimensional optimization problems. Therefore, it is widely used to address parameter optimization issues faced in theoretical research and engineering practice. Its computational process is shown in Figure 6.
图6 粒子群算法流程图
Figure 6Flowchart of Particle Swarm Algorithm
Figure 6 Algorithmic flow of PSO
假设参与寻优的粒子个数为N,寻优空间的维数为D,则每一个粒子的空间位置向量可以表述为:
Assuming the number of particles participating in the optimization isN, and the dimensionality of the optimization space isD, then the spatial position vector of each particle can be expressed as:
(27) |
每一个粒子的运动速度可以表述为:
The motion speed of each particle can be expressed as:
(28) |
每个粒子的空间位置和运动速度的更新规则为:
The update rules for the spatial position and motion speed of each particle are:
(29) | |
(30) |
式中,代表第个粒子处于第次迭代计算时其运动速度在寻优空间中第维的分量;代表第个粒子处于第次迭代计算时其位置在寻优空间中第维的分量;代表第个粒子处于第次迭代计算时第维的历史最佳位置,即在第次迭代后,第个粒子(个体)搜索到的最优解;代表群体在第次迭代中第维的历史最优位置,即在第次迭代后,整个粒子群体中的最优解;、分别代表粒子的个体学习因子和群体学习因子,作用是调节和的相对重要性;,代表随机产生的一个位于0与1之间的随机数。
(2)变权重系数的优化函数建立
(2) Establishment of the optimization function for variable weight coefficients
本文设置粒子群优化算法的粒子个数,迭代次数,维数,惯性系数,加速度常数、均为0.8。标准化粒子群优化算法的具体实现步骤如下。
The number of particles in the particle swarm optimization algorithm is set , the number of iterations , the dimensions , the inertia coefficient , and the acceleration constants , are all 0.8. The specific implementation steps of the standardized particle swarm optimization algorithm are as follows.
首先随机生成各个粒子的初始寻优速度以及初始寻优位置,并规定粒子位置范围和寻优速度范围所满足的约束条件:
First, randomly generate the initial optimization velocity and initial optimization position of each particle, and specify the constraints that the particle position range and optimization velocity range must satisfy:
(30) |
根据仿真结果分析可得出高频脉振注入法的最佳工作区间为ω<400 r/min,而当ω>700 r/min,模糊超螺旋滑模算法对转速的估计精度较高,因此设置400~700 r/min作为转速过渡区间,并将该过渡区间作为粒子群优化算法的位置范围,将其带入上述约束条件,就可以得到电机在该算法中的寻优位置范围约束和寻优速度范围约束,当每次位置更新或者速度更新时,如果更新后的速度或位置超过了各自约束范围,那么就取这个范围的最大值或最小值为本次更新的寻优速度或者寻优位置更新值:
According to the analysis of the simulation results, the optimal working range for the high-frequency pulse injection method isω<400 r/min,and whenω>700 r/min,the fuzzy super-spiral sliding mode algorithm has a high estimation accuracy for the rotational speed, so set 400~700 r/min as the transitional speed range, and take this transitional rangeas the position range for the particle swarm optimization algorithm, substituting it into the aboveconstraints, we can obtain theoptimizationposition rangeconstraintsandoptimizationspeed rangeconstraints. When eachpositionupdate orspeedupdate occurs, if the updatedspeed or positionexceedstheir respective constraint ranges, then take themaximum or minimum valueof this range as theoptimizationspeed oroptimizationposition update value:
(31) |
在过渡区内使整体的估算误差达到最小,其本质就是寻找两种算法各自的最优权重系数,为此需要建立粒子的适应度函数。建立权重系数的目标函数为:
To minimize the overall estimation error in the transition zone, the essence is to find the optimal weight coefficients for each of the two algorithms, for which it is necessary to establish the fitness function of the particles. The objective function for establishing the weight coefficients is:
(32) | |
(33) |
式中,为低转速算法的权值;为高转速算法的权值;为电机转子角速度的真实值;为电机转子位置的真实值;由脉振高频注入法获取的转子角速度的估计值;为由脉振高频注入法获取的转子位置的估计值;由模糊超螺旋算法获取的转子角速度的估计值;为由模糊超螺旋算法获取的转子位置的估计值。
In the formula, is the weight of the low-speed algorithm; is the weight of the high-speed algorithm; is the true value of the motor rotor angular velocity; is the true value of the motor rotor position; is the estimated value of the rotor angular velocity obtained by the pulse vibration high-frequency injection method; is the estimated value of the rotor position obtained by the pulse vibration high-frequency injection method; is the estimated value of the rotor angular velocity obtained by the fuzzy super-spiral algorithm; is the estimated value of the rotor position obtained by the fuzzy super-spiral algorithm.
迭代更新每个粒子的个体最优值以及粒子群的全局最优值。粒子在迭代的过程中运动方向会受到自身以往经验以及与其相近的粒子的群体经验的影响。每个粒子会把当前个体最优值与以往个体最优值进行比较,如果当前个体最优值优于以往最优值,则用当前个体最优值取代以往的最优值,反之,则舍弃本次的最优值。同时,粒子还会把个体最优值和群体最优值进行比较,如果本次得到的个体最优值优于群体最优值,则将个体最优值作为新的群体最优值。粒子群优化算法通常有两种结束条件,迭代次数达到预先设定的最大迭代次数或者输出结果满足可接受的满意解。
Iteratively update each particle's individual best value and the swarm's global best value. During the iteration process, a particle's movement direction is influenced by its past experiences and the collective experiences of nearby particles. Each particle compares its current individual best value with its previous individual best value; if the current individual best value is better than the previous one, it replaces the old best value; otherwise, it discards the current best value. At the same time, the particle also compares its individual best value with the global best value; if the current individual best value is better than the global best value, it becomes the new global best value. The particle swarm optimization algorithm typically has two termination conditions: the number of iterations reaches a pre-set maximum number of iterations, or the output meets an acceptable satisfactory solution.
4 仿真建模及结果分析
4 Simulation Modeling and Result Analysis
在第三章所构建的基于变结构模糊神经网络转速控制器的磁场定向控制系统的基础上,利用无速度传感器获取调速系统转速反馈信息,在Matlab/Simulink中搭建基于最优权值切换法的无速度传感器模型,系统整体仿真模型如图7所示。
Based on the field-oriented control system of the variable structure fuzzy neural network speed controller constructed in Chapter 3, speed feedback information for the speed control system is obtained without a speed sensor. In Matlab/Simulink, a sensorless model based on the optimal weight switching method is built, and the overall simulation model is shown in Figure 7.
图7 系统整体仿真模型
Figure 7Overall Simulation Model of the System
Figure 7 Simulation model of system
包括高速区间估测算法、低速区间估测算法以及过渡区间切换算法在内的各个子模块仿真模型如图7所示。
The simulation models of various sub-modules, including the high-speed interval estimation algorithm, low-speed interval estimation algorithm, and transition interval switching algorithm, are shown in Figure 7.
图8 各子模块仿真模型
Figure 8SubmoduleSimulation Model
Figure 8 Simulation model of subsystem
粒子群算法优化后的最优权重系数函数如图4-11所示。
The optimal weight coefficient function after optimization by the particle swarm algorithm is shown in Figure4-11.
图9 最优权重函数
Figure 9Optimal Weight Function
Figure 9 Optimal weight function
为同时验证所设计算法在高速区间、过渡区间、低速区间的观测准确性以及与滞环算法,加权切换法相比的优越性同时搭建两种传统切换算法的仿真模型子模块,其中滞环算法由Simulink中自带滞环模块实现,加权切换法由查表法实现,仿真实验设置给定工况为:设置负载恒为10 N·m,给定目标转速为3000 r/min,过渡区间为下限为400 r/min上限为700 r/min,电机参数与第三章仿真实验所用电机参数保持一致。
To simultaneously verify the observation accuracy of the designed algorithm in the high-speed range, transition range, and low-speed range, as well as its superiority compared to the hysteresis algorithm and weighted switching method, two traditional switching algorithm simulation model submodules are built. The hysteresis algorithm is implemented using the built-in hysteresis module in Simulink, while the weighted switching method is implemented using a lookup table method. The simulation experiment is set with the given working conditions: the load is set to a constant of 10 N·m, the target speed is set to 3000 r/min, the transition range has a lower limit of 400 r/min and an upper limit of700 r/min, and the motor parameters are consistent with those used in the simulation experiments in Chapter 3.
根据仿真结果分析,无论是高速区间还是低速区间三种切换算法都可以实现对转速的精确估算,这也验证了脉振高频电压注入法在低速区间的适用性以及模糊超螺旋算法在高速区间对滑模观测法抖振问题的抑制作用,在这两个区间中三种切换算法表现相同的原因是在非过渡区间内三种切换算法在本质上都是仅有一种相同的观测算法参与计算,所以对转速的估算结果自然也是相同的。
According to the analysis of the simulation results, both in the high-speed range and the low-speed range, the three switching algorithms can achieve accurate estimation of the rotational speed. This also verifies the applicability of the pulse high-frequency voltage injection method in the low-speed range and the suppression effect of the fuzzy super-spiral algorithm on the chattering problem of the sliding mode observer in the high-speed range. The reason that the three switching algorithms perform the same in these two ranges is that, in the non-transitional range, all three switching algorithms essentially involve only one identical observation algorithm in the calculation, so the estimation results for the rotational speed are naturally the same.
电机转速估算结果如图10。
The estimated motor speed is shown in Figure 10.
图10 转速估算结果
Figure 10Speed Estimation Results
Figure 10 Estimation of rotational speed
过渡区间内电机转速估算结果局部放大图如图11所示
The local magnified view of the estimated motor speed in the transition interval is shown in Figure 11
图11 转速估算结果局部放大图
Figure 11Local magnification of the rotational speed estimation results
Figure 11 Local magnification of speed estimation
电机转速估算误差情况如图12所示。
The estimation error of the motor speed is shown in Figure 12.
图12 转速误差
Figure 12Speed Error
Figure 12 Rotational speed error
在过渡区间内电机转速估算误差的局部放大图如图13所示。
In the transition intervalthe estimated error of motor speedis shown in the enlarged view in Figure 13.
图13 转速误差局部放大图
Figure 13Local magnification of speed error
Figure 13 Local magnification of rotational speed error
在过渡区间内三种算法则表现出了明显的性能差异,其中滞环切换法表现最差,出现了两次误差峰值,误差分别达到了220 r/min和240 r/min,对应的误差率分别为46%和42%。加权切换法由于采取了更为平滑的切换策略其性能表现也相对较好,与滞环法相比明显抑制了误差峰值的出现,仅出现了一次误差值为70 r/min,误差率为14%左右的观测波动。而基于粒子群优化后的最优权值切换法在过渡区间内一直保持着准确稳定的转速观测效果,没有明显的转速及误差波动。该仿真实验验证了算法的有效性和优越性。
During the transition interval, the three algorithms showed significant performance differences, with the hysteresis switching method performing the worst, resulting in two error peaks with errors reaching 220 r/min and 240 r/min, corresponding to error rates of 46% and 42%. The weighted switching method, due to its smoother switching strategy, also performed relatively well, significantly suppressing the occurrence of error peaks compared to the hysteresis method, with only one error value of 70 r/min and an error rate of about 14%. The optimal weight switching method based on particle swarm optimization maintained accurate and stable speed observation throughout the transition interval, with no significant speed or error fluctuations. This simulation experiment verified the effectiveness and superiority of the algorithm.
5 小结
5 Summary
本章首先根据不同无速度传感器算法的适用转速区间,分析推导了低速区间的高频脉振注入法以及高速区间的滑模估算法原理,并利用模糊逻辑和超螺旋算法对滑模抖振问题进行抑制。而后对粒子群优化算法进行了简要阐述,列写了粒子群优化算法应用于复合型无速度传感器算法在过渡区间进行算法切换时的权重系数的目标函数,并获取了最优权值函数,将该最优权值函数作为过渡区间内两种算法所占权重的的参考。通过在Matlab/Simulink搭建仿真模型,对比传统滞环切换法、加权切换法以及所提的基于粒子群优化的最优权值切换算法的表现,验证了最优权值切换的有效性和优越性。
This chapter first analyzes and derives the principles of the high-frequency pulse injection method in the low-speed range and the sliding mode estimation method in the high-speed range based on the applicable speed ranges of different speed sensorless algorithms. It also suppresses the sliding mode chattering problem using fuzzy logic and hyper-spiral algorithms. Then, a brief explanation of the particle swarm optimization algorithm is provided, listing the objective function of the weight coefficients for the application of the particle swarm optimization algorithm in switching algorithms in the transitional range of the composite speed sensorless algorithm, and obtaining the optimal weight function, which serves as a reference for the weights of the two algorithms in the transitional range. By building a simulation model in Matlab/Simulink and comparing the performance of the traditional hysteresis switching method, the weighted switching method, and the proposed optimal weight switching algorithm based on particle swarm optimization, the effectiveness and superiority of the optimal weight switching are verified.