Data Analytics and Condition Monitoring Methods for Nonstationary Batch Processes — Current Status and Future
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摘要: 间歇过程作为制造业的重要生产方式之一, 其高效运行是智能制造的优先主题. 为了保障生产过程的高效运行, 面向间歇生产的过程数据解析与状态监控算法在最近三十年间得到大家的广泛关注, 发展速度稳步提升. 但由于间歇过程本身的多重时变大范围非平稳运行复杂特性, 以及对状态监控与故障诊断要求的提高, 现有的理论和方法仍面临着挑战. 本文从分析间歇过程的特性出发, 从数据解析的角度, 总结了近三十年来非平稳间歇过程高性能监控研究的发展. 一方面对间歇过程监控领域几种经典的方法体系进行了总结和梳理, 另一方面揭示了尚存在的问题以及未来可能的研究思路和发展脉络.Abstract: Batch process is an important class of manufacturing processes. Its condition operation has been given high priority for smart manufacturing, which closely depends on the automatic condition monitoring and fault diagnosis. Great efforts have been made in the research on data analytics and high-efficiency monitoring algorithms with significant development for batch processes during the past thirty years. However, due to its complex characteristics and increasing requirements on monitoring and diagnosis precision, there are still many challenging problems in this field. In this paper, starting from the nature of batch process and data analystics, we address the challenges in this field, review the development of monitoring and diagnosis strategies, analyze several classical algorithms, and discuss the future development of batch process high-efficiency monitoring.
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Key words:
- Data analytics /
- intelligent manufacturing /
- batch process /
- nonstationary /
- process monitoring /
- fault diagnosis /
- machine learning
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感应电机交流传动系统广泛应用于各工业领域, 其中新能源汽车等场合普遍采用转子磁链定向的控制方式, 获得精确的转子磁链是实现交流传动系统高性能控制的关键[1].矢量控制策略是当前应用最为广泛的感应电机控制方法, 虽然采用矢量控制的感应电机交流传动系统具有很好的控制性能, 但在设计控制器过程中需要精确的电机参数来实现定、转子控制的解耦.无论是感应电机转子磁链定向矢量控制, 还是其他的非线性控制策略, 都需要转子磁链矢量的幅值和相位[2].转子磁链矢量的检测和获取方法分为直接法和间接法.直接法是在感应电机定子内表面装贴霍尔元件或者在电机槽内埋设探测线圈等直接检测转子磁链, 但由于工艺和技术难度较大, 实际的矢量控制系统中不适用直接法[3].间接法是检测感应电机的定子电压、电流及转速等容易获得的物理量, 利用转子磁链观测模型, 实时计算转子磁链的幅值和相位[4].而由于观测模型不够精确, 控制系统中的延迟问题以及电机参数变化的影响等, 使提高转子磁链观测精度成为提高交流传动系统控制性能的关键问题之一.
为了提高转子磁链观测的精度, 很多专家学者不懈努力, 进行了深入的研究.提出了电压模型法和电流模型法的转子磁链观测器、U-I法磁链观测器、全阶磁链观测器、扩展卡尔曼滤波器、自适应观测器和滑模观测器等方法[5-13].电压模型法模型结构简单, 计算过程是纯积分, 其估算结果受积分初值和输入信号的直流偏移影响很大, 导致结果存在误差, 且电压模型法依赖电机的定子电阻参数, 其受温度等因素影响较大, 也会产生估算误差[5].电流模型法依赖电机的定子电流和转速参数, 同时磁链估算过程需要转子参数, 鲁棒性差[6].电压模型法和电流模型法都是基于开环算法的磁链估算, 观测精度受限.文献[7]提出了一种基于改进U-I法的磁链观测方法, U-I法不需要转速和转子参数, 具有较好的鲁棒性, 但其对定子电阻的摄动较敏感且存在积分漂移, 造成观测误差.文献[8]提出了一种全阶磁链观测器设计方法, 通过观测得到的电机转速等相关参数, 设计合理的估算模型, 得到磁链观测值, 设计过程的极点配置以及受电机参数影响较大, 限制了其观测的精度.文献[9]提出了一种基于扩展Kalman滤波器的转子磁链观测方法, 能够有效减少噪声对磁链观测精度的影响, 但其对参数变化的敏感性及估算过程需要大量的数学计算, 限制了其在实际工程中应用.文献[10]提出了一种基于模型参考自适应系统的自适应磁链观测方法, 改善磁链观测的精度, 但其受系统参数影响较大.文献[12]提出了一种非线性滑模磁链观测方法, 具有较强的抗干扰性, 但其存在的抖振问题无法消除, 极大地限制了其使用范围.
转子磁链幅值和相位的准确估计是构建感应电机交流传动矢量控制系统的关键环节.磁链幅值估计实现系统的磁链控制, 转子位置观测实现矢量控制系统的坐标变换, 从而完成感应电机转矩和励磁控制的解耦[14].中立型系统理论是基于中立型延迟系统的一种理论, 而中立型延迟系统是一种能够精确描述延迟系统的模型, 模型中既包括状态延迟, 也包括状态微分延迟, 使得对延迟系统的描述更加精确[15].本文将中立型系统理论应用到感应电机控制系统中, 解决由于观测模型不够精确、系统控制中的延迟问题以及电机参数变化的影响, 导致转子磁链观测精度不高的问题, 实现系统的高性能控制.
1. 感应电机数学模型及中立型系统理论原理
1.1 感应电机数学模型
感应电机的数学模型具有高阶、非线性、强耦合等特征.矢量控制系统建立在感应电机的动态模型上, 在同步旋转$ M$-$T$坐标系中, 当电机转子磁链矢量与$T$轴重合时, 即为按转子磁链定向.通过同步旋转坐标系按转子磁链方向定向, 实现感应电机转矩和磁通的解耦控制.
本文基于感应电机在$ M$-$T$坐标系下的状态方程, 提出一种电机转子磁链观测方法.选择电机的定子电流和转子磁链为状态变量, 以电压变量为输入, 转子磁链为输出, 建立感应电机数学模型[16].
$$ \begin{eqnarray} \left\{ \begin{aligned} &\dot{\Psi}_r = -\lambda_1\Psi_r + \lambda_2i_{sm} \\ &\dot{i}_{sm} = \lambda_3\Psi_r - \lambda_5i_{sm} + \lambda_6i_{st} + \lambda_7u_{sm} \\ &\dot{i}_{st} = -\lambda_4\Psi_r - \lambda_5i_{st} - \lambda_6i_{sm} + \lambda_7u_{st} \end{aligned} \right. \end{eqnarray} $$ (1) $$ \lambda_1 = \frac{1}{T_r}, \lambda_2 = \frac{L_m}{T_r}, \lambda_6=\omega_s, \lambda_7 = \frac{1}{\sigma L_s} $$ $$ \lambda_3 = \frac{L_m}{\sigma L_s L_r T_r}, \lambda_4 = \frac{L_m}{\sigma L_s T_r}, \lambda_5 = \frac{R_s L^{2}_r + R_r L^{2}_m}{\sigma L_s L^{2}_r} $$ 式中, $u_{sm}$和$u_{st}$为定子电压; $i_{sm}$和$i_{st}$为定子电流; $R_s$和$R_r$分别为定子和转子绕组电阻; $L_m$为定子和转子同轴等效绕组互感; $L_s$和$L_r$分别为定子和转子等效两相绕组自感; $\Psi_r$为转子磁链矢量; $n_p$为极对数; $T_L$为负载转矩; $T_e$为电磁转矩; $J$为电机的转动惯量; $\sigma$为电机的漏磁系数, $\sigma=1-L^{2}_m/(L_sL_r)$; $T_r$为转子电磁时间常数, $T_r=L_r/R_r$; $\omega$为转子转动角速度; $\omega_s$为同步角速度, 感应电机矢量控制辅助方程为
$$ \begin{eqnarray} \left\{ \begin{aligned} &\dot{\omega} = \frac{n^{2}_p L_m}{JL_r}i_{st}\Psi_r - \frac{n_p}{J}T_L \\ &T_e = \frac{n_p L_m}{L_r}i_{st}\Psi_r \\ &\omega_s = \omega + \frac{L_m}{T_r\Psi_r}i_{st}(t) \end{aligned} \right. \end{eqnarray} $$ (2) 1.2 中立型系统理论
中立型系统理论是一种针对解决工程实践中延迟问题的理论, 中立型系统模型对系统描述的精确度要优于当前其他的建模理论.标称延迟系统方程为:
$$ \begin{eqnarray*}\dot{{\pmb x}}(t) = f({\pmb x}(t), {\pmb X}(t-d), {\pmb U}(t))\end{eqnarray*} $$ 而中立型延迟系统方程为:
$$ \begin{eqnarray*}\dot{{\pmb x}}(t) - G\dot{{\pmb x}}(t-d) = f({\pmb x}(t), {\pmb X}(t-d), {\pmb U}(t))\end{eqnarray*} $$ 因为系统含有微分差分算子$\dot{D}(t, {\pmb x}_t) = \dot{{\pmb x}}(t) - G\dot{{\pmb x}}(t-d)$, 使得标称延迟系统的许多理论成果无法简单地推广到中立型延迟系统中去.
中立型延迟系统模型方程[15]:
$$ \begin{eqnarray} \left\{ \begin{aligned} &\dot{{\pmb x}}(t) - G\dot{{\pmb x}}(t-d) = A{\pmb x}(t) + C {\pmb x}(t-d) + \\&f(t, {\pmb x}(t), {\pmb X}(t-d)) + B {\pmb u}(t) \\ &{\pmb y}(t) = D {\pmb x}(t) \\ &{\pmb x}(t) = {\pmb \phi}(t), t\in[-d, 0] \end{aligned} \right. \end{eqnarray} $$ (3) 其中, ${\pmb x}(t)\in {\bf R}^{n}$为状态向量; $d>0$为延时时间; $A, C, G$和$B$为维数适当的常数矩阵, 且满足$\parallel G\parallel < 1$; ${\pmb \phi}(t)\in([-d, 0], {\bf R}^{n})$为向量初值函数; $f\in C([0, +\infty], {\bf R}^{n})$为不可观测的非线性不确定扰动, 且满足:
$$ \begin{eqnarray*}\left\{ \begin{aligned} &f(t, 0, 0) = 0 \\ &\parallel f(t, {\pmb x}(t-d))\parallel \leq \alpha\parallel {\pmb x} \parallel + \beta\parallel {\pmb x}(t-d)\parallel \end{aligned} \right.\end{eqnarray*} $$ 其中, $\alpha, \beta$为已知常数.
1.3 中立型系统理论在感应电机中的应用
电机控制系统中, 由于电力电子器件的存在, 以及控制器响应时间等, 控制信号会产生延迟现象, 出现开关动作、信号响应等和系统模态不同步, 这种现象称为异步切换[17].当开关频率较高时, 异步切换现象对系统性能的影响可以忽略.而在大功率低开关频率条件下, 异步现象会造成电机的电流畸变、发热等问题.中立型系统在建模时不需要考虑异步切换现象, 故本文将中立型系统理论应用到感应电机系统建模中, 建立感应电机中立型延迟系统数学模型.
$$ \begin{equation} \dot{{\pmb x}}(t) = \begin{bmatrix} A &B \end{bmatrix}\begin{bmatrix} {\pmb x}(t) \\{\pmb u}(t) \end{bmatrix} \end{equation} $$ (4) 令式(3)中系数矩阵$A = C$.其中
$$ \begin{align*} &\dot{{\pmb x}}(t) = \begin{bmatrix} \dot{\Psi}_r(t) \\\dot{i}_{sm}(t) \\\dot{i}_{st}(t) \end{bmatrix}, \quad {\pmb x}(t) = \begin{bmatrix} \Psi_r(t) \\ i_{sm}(t) \\i_{st}(t) \end{bmatrix}\\&{\pmb u}(t) = \begin{bmatrix} u_{sm}(t) \\u_{st}(t) \end{bmatrix}, \quad A = \begin{bmatrix} -\lambda_1 & \lambda_2 & 0 \\ \lambda_3 & -\lambda_5 & \lambda_6 \\ -\lambda_4 & -\lambda_6 & -\lambda_5 \end{bmatrix} \\&B = \begin{bmatrix} 0 & 0 \\\lambda_7 & 0 \\0 & \lambda_7 \end{bmatrix}\end{align*} $$ 在电机控制系统中, 由于异步切换现象的存在, 状态滞后对系统的影响具有不确定性.但在一定范围内, 影响程度取决于对应的系数矩阵元素的大小, 本文定义$\mu(\chi)$为影响因子, 表示系统中状态滞后对系统影响程度的度量的物理量, 其中$\chi$为对应的系数矩阵.这里:
$$ \begin{equation*}\mu(\chi) = \frac{1}{\parallel \chi\parallel_\infty}\end{equation*} $$ 考虑状态滞后对电机控制系统的影响, 得到感应电机标称延迟系统方程(5), 其中$\tau$为总延时时间.
$$ \begin{equation} \dot{{\pmb x}}(t) = \begin{bmatrix} (1-\mu)A & \mu C & B \end{bmatrix}\begin{bmatrix} {\pmb x}(t) \\{\pmb x}(t-\tau) \\{\pmb u}(t) \end{bmatrix} \end{equation} $$ (5) 式(5)由莱布尼兹公式得:
$$ \begin{align} &\left[ {\begin{array}{*{20}{c}} 1&-\mu C\int \end{array}} \right]\begin{bmatrix} \dot{{\pmb x}}(t) \\\dot{{\pmb x}}(t-d) \end{bmatrix} =\nonumber\\& \qquad \left[ {\begin{array}{*{20}{c}} (1-\mu)A & \mu C & B \end{array}} \right]\left[ \begin{array}{c} {\pmb x}(t) \\{\pmb x}(t-d) \\{\pmb u}(t) \end{array} \right] \end{align} $$ (6) 式中, $d$为系统状态延时时间, $d=\tau/2$.
表达式(6)中含有积分项, 这里用$N$个小直角梯形的面积之和极限逼近积分项的值, $N\in Z$, 又由数值均值定理, 得到表达式(7).
$$ \begin{align} &\int^{-\frac{\tau}{2}}_0\dot{{\pmb x}}(t-d+\theta){\rm d}\theta\approx\nonumber\\& \qquad\frac{d}{N}\sum\limits_{i=1}^N\dot{{\pmb x}}\left(t-d-\frac{(2i-1)\cdot\tau}{4N}\right) \end{align} $$ (7) 对于任意的$d$, 存在$N\geq M, M\in Z$, 使不等式$\mu\frac{d}{N}\leq\mu(\chi)$成立.
由上式得到感应电机中立型转子磁链模型方程(8).
$$ \begin{align} &\left[ {\begin{array}{*{20}{c}} 1 & -\mu C \cdot \frac{d}{N}\end{array}} \right]\begin{bmatrix} \dot{{\pmb x}}(t) \\\sum\limits_{i=1}^N\dot{{\pmb x}}(t-d-\varepsilon) \end{bmatrix} =\nonumber\\& \qquad \left[ {\begin{array}{*{20}{c}} (1-\mu)A & \mu C & B \end{array}} \right]\left[ \begin{array}{c} {\pmb x}(t) \\{\pmb x}(t-d) \\{\pmb u}(t) \end{array} \right] \end{align} $$ (8) 其中, $\varepsilon=\frac{(2i-1)d}{2N}$.
2. 中立型磁链观测器设计
考虑标准型中立型系统[18]:
$$ \begin{equation} \left\{\begin{aligned} &\dot{{\pmb x}}(t)-G\dot{{\pmb x}}(t-d)=A{\pmb x}(t)+C{\pmb x}(t-d)\\ &{\pmb y}(t)=D{\pmb x}(t) \end{aligned} \right. \end{equation} $$ (9) 式中
$$ \begin{align*} A =\, & (1-\mu)\times\\ &\left[{\begin{array}{*{20}{c}} -\frac{1}{T_r}&\frac{L_m}{T_r}&0 \\ \frac{L_m}{\sigma L_sL_rT_r}&-\frac{R_sL^{2}_r+R_rL^{2}_m}{\sigma L_sL^{2}_r}&\omega_s\\ -\frac{L_m}{\sigma L_sL_r}\omega&-\omega_s&-\frac{R_sL^{2}_r+R_rL^{2}_m}{\sigma L_sL^{2}_r}\end{array}} \right]\\ C =\, & \mu\left[ {\begin{array}{*{20}{c}} -\frac{1}{T_r}&\frac{L_m}{T_r}&0 \\ \frac{L_m}{\sigma L_sL_rT_r}&-\frac{R_sL^{2}_r+R_rL^{2}_m}{\sigma L_sL^{2}_r}&\omega_s\\ -\frac{L_m}{\sigma L_sL_r}\omega&-\omega_s&-\frac{R_sL^{2}_r+R_rL^{2}_m}{\sigma L_sL^{2}_r}\end{array}} \right]\end{align*} $$ $$ \begin{align*} G =\, & \mu\frac{d}{N}\times\\ &\left[ {\begin{array}{*{20}{c}} -\frac{1}{T_r}&\frac{L_m}{T_r}&0 \\ \frac{L_m}{\sigma L_sL_rT_r}&-\frac{R_sL^{2}_r+R_rL^{2}_m}{\sigma L_sL^{2}_r}&\omega_s\\ -\frac{L_m}{\sigma L_sL_r}\omega&-\omega_s&-\frac{R_sL^{2}_r+R_rL^{2}_m}{\sigma L_sL^{2}_r}\end{array}} \right]\\ D =\, & \begin{bmatrix} 1&0&0 \end{bmatrix}\end{align*} $$ 设计如下状态观测器:
$$ \begin{align} &\hat{\dot{{\pmb x}}}(t)-G\hat{\dot{{\pmb x}}}(t-d) = A\hat{{\pmb x}}(t)+C\hat{{\pmb x}}(t-d)+\nonumber\\& \qquad L({\pmb y}(t)-D\hat{{\pmb x}}(t)) \end{align} $$ (10) 使得误差动态系统方程(11)渐近稳定.
$$ \begin{equation} \dot{{\pmb e}}(t)-G\dot{{\pmb e}}(t-d) = H {\pmb e}(t) + C {\pmb e}(t-d) \end{equation} $$ (11) 式中, $H=A-LD, {\pmb e}(t)={\pmb x}(t)-\hat{{\pmb x}}(t)$为误差向量, $\hat{{\pmb x}}(t)\in R^{n}$是观测状态, $L$为$n\times q$阶观测器增益矩阵.
设误差动态系统方程的初值条件为
$$ \left\{ \begin{aligned} &{\pmb e}(t_r)={\pmb \varphi}(t_r), -d\leq t_r\leq0 \\ &{\pmb \varphi}(t_r)\in C([-d, 0], R^{n}) \end{aligned} \right. $$ 若存在正定阵$P, Q, R$和矩阵$K$, 使得线性矩阵不等式(12)成立, 则误差动态系统方程(11)渐近稳定.
$$ \begin{equation}\begin{aligned} \begin{bmatrix} 2PA-2PLD+Q+R & D^{\rm T}L^{\rm T}PG-A^{\rm T}PG & PC\\ G^{\rm T}PLD-G^{\rm T}PA & -Q & G^{\rm T}PC \\ C^{\rm T}P & C^{\rm T}PG & -R\end{bmatrix}<0 \end{aligned}\end{equation} $$ (12) 令$K=PL$, 则线性矩阵不等式写为
$$ \begin{equation} \begin{aligned} \begin{bmatrix} 2PA-2KD+Q+R & D^{\rm T}K^{\rm T}G-A^{\rm T}PG & PC\\ G^{\rm T}KD-G^{\rm T}PA & -Q & G^{\rm T}PC \\ C^{\rm T}P & C^{\rm T}PG & -R\end{bmatrix} <0 \end{aligned} \end{equation} $$ (13) 即若存在正定阵$P, Q, R$, 满足线性矩阵不等式(13), 则存在状态观测器(10), 解得观测器增益矩阵$L=P^{-1}K$.
这里以感应电机转子磁链为观测目标, 由感应电机中立型系统方程(8), 设计中立型转子磁链观测器模型, 如图 1.其中系数$A_{11}, $ $A_{12}, $ $C_{11}, $ $C_{12}, $ $G_{11}, $ $G_{12}$为系统方程系数矩阵$A, C, G$的对应元素.
图 1 M-T坐标系下中立型转子磁链观测模型Fig. 1 Observation model of neutral rotor flux linkage in M-T coordinate system3. 稳定性分析
已知状态观测器方程(10), 假设观测器增益系数$L$已知, 证明误差动态系统方程(11)是渐近稳定的.
为此本文提出了一种基于线性矩阵不等式(Linear matrix inequality, LMI)的Lyapunov泛函[19-22], 用来证明方程(11)的稳定性.
$$ \begin{align} V(t, {\pmb e})=\, &[{\pmb e}(t)-G{\pmb e}(t-d)]^{\rm T}P[{\pmb e}(t)-G{\pmb e}(t-d)]+\nonumber\\&\int^{t}_{t-d}{\pmb e}^{\rm T}(\rho)Q{\pmb e}(\rho){\rm d}\rho+\int^{t}_{t-d}{\pmb e}^{\rm T}(\rho)R{\pmb e}(\rho){\rm d}\rho \end{align} $$ (14) 其中, $P, Q, R$为正定对称矩阵.
令${\pmb\eta}_1={\pmb e}(t), {\pmb\eta}_2={\pmb\eta}_3={\pmb e}(t-d)$, 则表达式为
$$ \begin{align} V(t, {\pmb e})=\, &({\pmb\eta}_1-G{\pmb\eta}_2)^{\rm T}P({\pmb\eta}_1-G{\pmb\eta}_2)+\nonumber\\&\int^{t}_{t-d}{\pmb e}^{\rm T}(\rho)Q{\pmb e}(\rho){\rm d}\rho+\int^{t}_{t-d}{\pmb e}^{\rm T}(\rho)R{\pmb e}(\rho){\rm d}\rho \end{align} $$ (15) 对式(15)求导, 得
$$ \begin{align} \dot{V}(t, {\pmb e})=\, &2({\pmb\eta}_1-G{\pmb\eta}_2)^{\rm T}P(\dot{{\pmb\eta}_1}-G\dot{{\pmb\eta}_2})+\nonumber\\&{\pmb\eta}_1^{\rm T}Q{\pmb\eta}_1-{\pmb\eta}_2^{\rm T}Q{\pmb\eta}_2+{\pmb\eta}_1^{\rm T}R{\pmb\eta}_1-{\pmb\eta}_3^{\rm T}R{\pmb\eta}_3 \end{align} $$ (16) 由中立型系统方程式(9), 式(16)可以表达为
$$ \begin{align} \dot{V}(t, {\pmb e})=\, &2({\pmb\eta}_1-G{\pmb\eta}_2)^{\rm T}P(H{\pmb\eta}_1-C{\pmb\eta}_2)+\nonumber\\&{\pmb\eta}_1^{\rm T}(Q+R){\pmb\eta}_1-{\pmb\eta}_2^{\rm T}Q{\pmb\eta}_2-{\pmb\eta}_3^{\rm T}R{\pmb\eta}_3 \end{align} $$ (17) $$ \begin{align} \dot{V}(t, {\pmb e})=\, &2{\pmb\eta}_1^{\rm T}PH{\pmb\eta}_1+2{\pmb\eta}_1^{\rm T}PC{\pmb\eta}_3-2{\pmb\eta}_2^{\rm T}G^{\rm T}PH{\pmb\eta}_1-\nonumber\\&2{\pmb\eta}_2^{\rm T}G^{\rm T}PC{\pmb\eta}_3+{\pmb\eta}_1^{\rm T}(Q+R){\pmb\eta}_1-\nonumber\\&{\pmb\eta}_2^{\rm T}Q{\pmb\eta}_2-{\pmb\eta}_3^{\rm T}R{\pmb\eta}_3 =\nonumber\\&{\pmb\eta}_1^{\rm T}(PH+H^{\rm T}P+Q+R){\pmb\eta}_1+\nonumber\\&2{\pmb\eta}_1^{\rm T}PC{\pmb\eta}_3-2{\pmb\eta}_2^{\rm T}G^{\rm T}PH{\pmb\eta}_1-\nonumber\\&2{\pmb\eta}_2^{\rm T}G^{\rm T}PC{\pmb\eta}_3-{\pmb\eta}_2^{\rm T}Q{\pmb\eta}_2-{\pmb\eta}_3^{\rm T}R{\pmb\eta}_3 \end{align} $$ (18) 将式(18)等价为线性矩阵不等式的形式为式(19).其中${\pmb\eta}(t)=\begin{bmatrix}{\pmb e}(t) & {\pmb e}(t-d) & {\pmb e}(t-d)\end{bmatrix}$, $H=A-LD$.令$K=PL$则式(19)表达为式(20).
若线性矩阵不等式(13)有解, 即存在正定阵$P, Q, R$和矩阵$K$, 得$\dot{V}(t, {\pmb e}) < 0$由Razumikhin型定理[20]得误差动态系统方程(11)渐近稳定.
$$ \begin{align} \dot{V}(t, {\pmb e})={\pmb\eta}^{\rm T}(t)\times \begin{bmatrix} PH+H^{\rm T}P+Q+R & -H^{\rm T}PG & PC\\ -G^{\rm T}PH & -Q & -G^{\rm T}PC \\C^{\rm T}P & -C^{\rm T}PG & -R \end{bmatrix}{\pmb\eta}(t) \end{align} $$ (19) $$ \begin{align} \dot{V}(t, {\pmb e})={\pmb\eta}^{\rm T}(t)\times \begin{bmatrix} 2PA-2KD+Q+R & D^{\rm T}K^{\rm T}G-A^{\rm T}PG & PC\\ G^{\rm T}KD-G^{\rm T}PA & -Q & G^{\rm T}PC \\C^{\rm T}P & C^{\rm T}PG & -R \end{bmatrix}{\pmb\eta}(t) \end{align} $$ (20) 另外, 这里假设式(1)中的感应电机定子和转子绕组互感和绕组电阻等电机参数在一定范围内发生变化时, 记为$L^{\ast}_s, L^{\ast}_r, R^{\ast}_s, R^{\ast}_r$等, 分别对应变化后的方程(8)系数矩阵$A^{\ast}, B^{\ast}, C^{\ast}, D^{\ast}$, 把变化后的系数矩阵带入线性矩阵不等式(13), 求解不等式有解, 即存在正定阵$P^{\ast}, Q^{\ast}, R^{\ast}$和矩阵$K^{\ast}$, 观测器增益矩阵$L^{\ast}=P^{\ast-1}K^{\ast}$, 得误差动态系统方程(11)依然渐近稳定.所以通过分析说明了在一定范围内, 所设计的中立型磁链观测器具有较好的鲁棒性.
4. 仿真及实验研究
针对感应电机中立型转子磁链观测模型, 利用Matlab/Simulink搭建仿真模型, 借助Simulink/S-函数对中立型转子磁链观测器增益矩阵进行求解, 并从实际应用的角度出发, 设计实验方案, 借助DSP电力电子与电气传动实验平台, 对中立型转子磁链观测器进行实验验证.仿真和实验使用的电机参数一样, 电机参数见表 1.
表 1 电机参数Table 1 Motor parameters参数 数值 额定功率$P_N$ /kW 4 额定电压$U_N$/V 380 额定频率$f_N$/Hz 50 额定电流$I_N$/A 8.8 定子电阻$R_s/{\rm \Omega}$ 1.405 定子电感$L_s$/mH 178 转子电阻$R_r/{\rm \Omega}$ 1.395 转子电感$L_r/$mH 178 定转子互感$L_m/$mH 172.2 极对数$n_p$ 2 额定转速$n/{\rm r}\cdot {\rm min}_{-1}$ 1 440 将表 1电机参数带入感应电机中立型延迟系统方程, 得到方程各项系数矩阵, 利用MATLAB中的LMI工具箱, 求解线性矩阵不等式(13), 解得正定矩阵$P, Q, R$以及矩阵$K$, 即可求出观测器增益矩阵$L$, 如下:
$$ \begin{align*}&P=\begin{bmatrix}19.0204&0.1979&0.0079\\ 0.1979&0.1770&0.0275\\ 0.0079&0.02575&0.0073\end{bmatrix}\\& Q=\begin{bmatrix}39.2067&0.7058&0.9499\\ 0.7058&19.4849&1.1232\\ 0.9499&1.1232&8.2866\end{bmatrix}\\& R=\begin{bmatrix}18.8541&-0.0330&-0.0008\\ -0.0330&20.6600&0.8838\\ -0.0008&0.8838&8.2839\end{bmatrix}\\& K=\begin{bmatrix}-6.1763\\ -8.8302\\ 13.7074\end{bmatrix}, L=\begin{bmatrix}0.0063\\ -0.8371\\ 5.0218\end{bmatrix}\times10^{3}\end{align*} $$ 因为存在正定矩阵$P, Q, R$, 已知$C=\begin{bmatrix}1 & 0 & 0\end{bmatrix}$, 因此, 矩阵$L$中的元素$L_1$即是图 1中的观测器增益.
上述所求正定矩阵及$L$是实时变化的, 上面是电机达到稳定后的一组解, 证明了感应电机中立型延迟系统模型的合理性.
本文利用S-函数, 模拟微处理器运行, 建立基于中立型转子磁链观测器的三相感应电机矢量控制系统仿真模型, 如图 2.仿真时间设为2.0 s.仿真和实验中电流控制器均采用的是PI控制器.
图 2 基于中立型转子磁链观测器的感应电机矢量控制系统仿真模型Fig. 2 Simulation model of vector control system of induction motor based on the observer of neutral rotor flux linkage仿真中转子磁链的真实值由电机模型给出.给定电机转速为500 r/m, 给定转子磁链为0.96 Wb, 电流采样控制频率为4 kHz, 开关频率为500 Hz, 即开关周期为0.002 s, 系统总延时时间[2]为$\tau=1.5T_s=3$ ms.图 3是基于中立型转子磁链观测器的电机转速响应图, 从图 3中可以看出, 电机转速在0.15 s即达到给定值, 响应速度快, 且曲线平滑, 稳定性好; 图 4是转子磁链幅值响应图, 从图 4中可知, 转子磁链在小于0.03 s的时间内即达到稳定值, 响应迅速, 且稳定性好.
图 3 中立型磁链观测方法转速响应图Fig. 3 Speed response diagram of flux linkage observation of the neutral method
图 4 中立型磁链观测方法转子磁链幅值响应图Fig. 4 Magnitude response diagram of rotor flux linkage of flux linkage observation of the neutral method分别采用中立型转子磁链观测器、电压模型法观测器和二阶滑模方法观测器观测转子磁链, 并进行稳态观测对比.电机稳态运行时的转子磁链观测及观测误差波形如图 5和图 6所示.
图 5 感应电机稳态运行时磁链观测波形Fig. 5 Observation waveform of flux linkage of induction motor in steady state operation
图 6 中立型观测方法和二阶滑模观测方法观测磁链对比图Fig. 6 Observational flux linkage contrast diagram of neutral observation method and second order sliding-mode method从图 5中可知, 电机稳态运行时采用中立型转子磁链观测器的磁链观测曲线平滑, 波动小, 观测误差峰-峰值为0.02 Wb, 采用电压模型法的磁链观测曲线波动较大, 观测误差峰-峰值为0.1 Wb, 采用二阶滑模方法的磁链观测误差峰-峰值为0.06 Wb.从图 6中可以看出, 中立型观测方法和二阶滑模观测方法观测磁链的观测值都始终收敛于单位圆内, 满足预先给定值, 而采用中立型观测方法观测磁链的曲线密集程度明显优于采用二阶滑模方法的磁链观测, 从对应波形的放大图中看更加显著.所以本文提出的基于中立型系统理论的转子磁链观测方法的观测精度优于电压模型法和二阶滑模方法的观测精度.
考虑电机参数变化对磁链观测精度的影响, 对设计的中立型转子磁链观测器进行鲁棒性研究.图 7是感应电机转子电阻突变为真实值的1.5倍的磁链观测波形.图 8是感应电机转子电阻突变为真实值的0.5倍的磁链观测波形.
图 7 $R_r$值变为真实值的1.5倍磁链观测图Fig. 7 Observation diagram of flux linkage with $R_r$ value changed to 1.5-fold real value
图 8 $R_r$值变为真实值的0.5倍磁链观测图Fig. 8 Observation diagram of flux linkage with $R_r$ value changed to 0.5-fold real value由图 7和图 8可知, 当感应电机转子电阻发生突变后, 电机稳态运行时的磁链观测误差在小范围内波动, 有微小增大, 变化率均在5 %以内; 由图 9可知, 当感应电机转子电感发生突变后, 电机稳态运行时的磁链观测误差变化率低于8 %; 此外经验证把上述求得的正定矩阵$P, Q, R$以及矩阵$K$和$L$带入线性矩阵不等式(13), 把转子电阻变化后记为$R_r{'}$, 其作为变量, $R_r{'}\in[0.5R_r, 1.5R_r]$, 带入式(13), 线性矩阵不等式仍然成立, 说明了转子电阻变化对中立型转子磁链观测方法的影响很小, 证明了中立型磁链观测器具有较强的鲁棒性.
图 9 $L_r$值变为真实值的0.5倍磁链观测图Fig. 9 Observation diagram of flux linkage with $L_r$ value changed to 0.5-fold real value综合图 7~图 9以及对应的分析, 可以得到当电机参数发生突变时, 电机稳态运行时对应的磁链观测误差的变化范围见表 2.
表 2 磁链观测误差Table 2 Observation error of flux linkage参数变化量 $-0.5R_r$ $+0.5R_r$ $-0.5L_r$ $+0.5L_r$ $\Delta\Psi_{r\alpha}$变化量$10^{-3}$/kW + 0.9 + 1.0 +1.5 +1.6 $\Delta\Psi_{r\alpha}$变化率 $\leq 5 %$ $\leq 8 %$ 为了研究系统的动态性能, 初始转速给定值为500 r/m, 空载情况下, 在0.5 s时, 调节转速从500 r/m升高到1 000 r/m, 图 10是基于中立型观测方法的电机调速波形.电机以给定转速稳态运行时, 在0.5 s加35 N$\cdot $ m的负载, 图 11是负载阶跃的动态响应波形.
图 10 基于中立型观测方法的电机调速波形Fig. 10 Motor speed regulation waveform based on neutral observation method
图 11 基于中立型观测方法的负载阶跃的响应波形Fig. 11 Response waveform of load step based on neutral observation method从图 10 (a)中可以看出, 电机转速由500 r/m升高到1 000 r/m所用时间约为0.2 s, 说明系统具有较好的转速动态响应性能; 由图 10 (b)和(c)可知, 磁链观测响应速度快, 观测幅值不变, 观测误差很快收敛到稳态误差值0.02 Wb; 电机电流响应迅速, 说明系统具有很好的动态响应性能.
从图 11中可以看出, 当系统外部负载发生变化时, 转子磁链观测幅值在负载突变时有短时微小波动, 很快稳定到给定值; 电机转矩电流响应迅速, 曲线波动较小; 电机三相电流曲线平滑, 响应迅速; 电机实际转矩迅速稳定在给定值35 N$\cdot $ m, 表明系统对外部负载变化具有良好的抗干扰能力, 鲁棒性强, 即采用本文提出的中立型转子磁链观测器的系统具有很好的动态特性.
为了研究应用中立型磁链观测器的感应电机在低速条件下的性能, 这里给出电源频率为5 Hz时的仿真.给定电机转速为150 r/m, 空载条件下, 在0.5 s时让电机停止, 即使得转速降为0 r/m, 图 12是基于中立型观测方法的电机低速条件下的仿真波形.
图 12 基于中立型观测方法的感应电机低速运行波形Fig. 12 Low speed operation waveform of induction motor based on neutral observation method从图 12 (a)可以看出, 在低频状态下电机转速很快稳定到给定值150 r/m, 0.5 s电机制动, 转速很快降为0, 曲线平滑, 说明了低频下系统具有较好的转速动态响应性能; 由图 12 (b)可知, 低频条件下, 电机转速达到稳定值时, 磁链观测误差峰值约为0.02 Wb, 当电机低速制动时, 磁链观测误差开始减小, 且磁链观测误差收敛至0 Wb, 证明了中立型磁链观测方法在低速条件下的有效性.
为了研究延迟问题对磁链观测精度的影响, 这里考虑在仿真中加入延迟时间.因为上述仿真中采用电压模型法观测磁链时并未考虑延迟时间, 而中立型观测方法中已考虑延迟时间.当采用电压模型法的电机仿真中加入延时时间$d$时, 得到如图 13的磁链观测波形.由图 13可知, 当采用电压模型法的仿真中考虑延时时间$d$时, 转子磁链观测波动较大, 观测幅值波动高于$\pm0.24$ Wb, 而磁链观测误差由0.1 Wb增大到大于0.15 Wb, 误差波动率大于50 %, 观测精度显著降低, 电机的稳态及动态性能将受到影响.
为验证中立型转子磁链观测器的可行性, 本文使用"电力电子与电气传动综合实验台"进行实验验证.实验平台及其结构原理图如图 14所示.实验平台由电机、负载、主回路、PC机、TMS320F2812 (DSP)控制板和保护电路等部分组成, 可以选择按转子磁链定向、空间矢量以及脉宽调制等模式实验, 本文选择按转子磁链定向模式, 具有信息采集功能, 具有绘制电机三相电流、转速和磁链的实时波形的能力, DSP控制器具有编程下载执行能力, 电机使用的是三相鼠笼式感应电机, 电机参数、观测器参数及实验给定参数与仿真时相同.实验结果如图 15和图 16所示.
由图 15 (a)可知, 低速条件下, 电机从启动加速到给定值500 r/min, 上升时间约为0.1 s, 这表明系统具有较好的速度动态响应性能; 图 15 (b)说明在转速阶跃过程中电流变化稳定, 说明了中立型转子磁链观测方法的有效性; 由图 15 (c)可知电机转速从500 r/min调节至650 r/min稳定转速所用时间约为0.04 s, 表明电机具有较好的低速动态响应性能.图 16 (a)是高速条件下的实验调速波形图, 电机转速从750 r/min调节至1500 r/min, 所用时间约为0.14 s, 表明电机具有较好的高速动态响应性能; 由图 16 (b)中可以看出, 高速条件下, 转子磁链观测波形近似单位圆, 观测值收敛于圆内, 表明中立型磁链观测器对转子磁链观测的准确性.实验结果和仿真结果的曲线趋势一致, 证明了中立型转子磁链观测方法的切实可行性.
5. 结论
本文提出了一种基于中立型系统理论的感应电机磁链观测方法, 将中立型延迟系统引入到感应电机磁链观测模型中, 运用线性矩阵不等式理论证明了中立型转子磁链观测器的稳定性.通过仿真分析和实验验证, 得出所提方法有效提高了磁链观测精度, 削弱了电机参数变化对磁链观测精度的影响, 解决系统控制延时对磁链观测的影响问题, 增强了系统观测的鲁棒性, 且该观测方法具有参数自整定, 时效性好, 使用范围广的优点, 证明了所提方法和设计转子磁链观测器的可行性.
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图 3 将三维数据展开成二维数据的6种方式
Fig. 3 Unfold the three dimensions data into two dimensions using six different manners
表 1 时段划分方法总结对比
Table 1 The comparison of different phase partition methods
时段划分方法 划分依据 优点 缺点 过程机理法[45, 48, 72] 利用实际间歇工业过程运行机理的变化来划分过程运行时段, 要求一定的专家经验和过程知识. 如果间歇生产过程相对简单或者工程师对此比较熟悉, 则可以比较容易地获取过程机理知识实现时段划分. 工业生产过程往往机理复杂, 很难在短时间内获取相关的知识和经验, 从而极大地限制和约束了其顺利实施施和推广应用. 特征分析方法[73—75] 时段的切换对应引起相应测量变量的变化. 对某些过程变量或从中提取的特征变量进行分析, 借助其沿时间轴上的变化判断时段信息. 指示变量方法是其中一种典型代表. 当时段发生切换或者变化, 过程特性变化, 相应的某些过程变量或是特征变量亦发生显著变化, 可用于指示不同时段. 算法较为简单. 并不是每个工业过程中都存在并能找到这样的“指示”变量. k-means[62—66] 通过相似度度量, 分析不同时间点上的潜在相关特性的相似与不同, 如果时间片具有相似特性则被归到同一类中, 具有显著差异则被分到不同类中. 该方法能够自动划分不同的多个时段, 不需借助任何过程机理和知识. 分类的结果决定于过程相关性在时间方向上的变化规律. 没有考虑间歇过程时段运行的时序性, 因此划分结果中会出现时间上不连续的具有相似过程相关性的时间片被分在同一个聚类中. 时段划分结果可读性有所欠缺, 需要针对划分结果进行进一步的后续处理. 此外, 该划分方法根据距离定义衡量过程相关特性的相似度, 聚类的结果受到相似性衡量指标的影响, 而该指标并不能与过程监测的目的直接相关. MPPCA[74—75] 一种优化策略, 通过对不同时间点进行不断尝试, 分析在该点的划分所得到的局部模型是否能够改善原有模型对数据的重构精度, 以此来确定该点的划分是否合适. 无需过程先验知识条件, 自动划分的各个时段时间连续, 解释性较强. 易陷入局部最优, 导致时段划分结果不能更好的反映过程特性变化. SSPP[76—77] 自动地按照间歇生产过程运行时间顺序捕捉潜在过程特性的发展变化, 通过评估时段划分对监测统计量的影响确定合适的时段划分点. 无需过程先验知识条件, 深入考虑了间歇过程潜在特性的时变性和实际过程运行的时序性以及时段划分结果对于之后监测性能的影响. 对过程时段特性变化的实时捕捉具有一定的时间延迟. 
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表 2 多向分析方法与子时段分析方法对比
Table 2 The comparison of multi-way methods and phase partition methods
方法 优点 缺点 多向分析法 分析方法相对简单, 直接针对展开的二维数据矩阵进行分析, 可借用传统的连续过程方法. 针对整个过程只需要建立一个模型. 无法有效分析过程特性时间上的变化规律. 子时段分析方法 1)可以更细致地揭示过程运行的潜在特征, 更好地体现过程运行的局部特征, 促进对复杂工业过程的了解;
2)在每个子时段可以很容易建立统计分析模型, 结构简单, 模型实用;
3)基于子时段可以很容易建立过程监测模型并实现在线应用而无需预估未知数据;
4)可以提高在线故障检测的精度和灵敏度, 并有利于后续准确的故障隔离和诊断;
5)可以深入分析质量指标和每个时段的具体关系, 找出影响质量的关键时段和预测变量等关键性因素, 有利于产品质量的进一步改进.需要进行时段划分, 分析过程特性在同一个操作周次内的变化.
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