集员估计理论、方法及其应用
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摘要
尽管集员估计理论、方法及其应用的研究已经取得了一定的成果,但是这方面的研究并非完善。本论文在分析现有研究成果的基础上,对集员估计理论、方法及其应用开展了进一步的研究。主要研究工作围绕现有集员估计方法的改进,新的集员估计方法的提出以及集员估计理论、方法在故障诊断领域的应用开展。本论文可分为以下几部分:
第一部分,对椭球集员滤波算法进行了分析和改进。一方面,对线性系统最优椭球集员滤波算法进行了改进,给出了两种数值稳定性比较好的次优算法。两种算法分别采用Cholesky分解和奇异值分解技术使椭球形状矩阵保持正定性。同时,为了避免受病态矩阵求逆的影响,在量测更新过程中两种算法均采用求具有次最小容积椭球的方法。另一方面,提出了一种平方根非最优椭球集员滤波算法,并给出了实现滤波计算的心动阵列结构。平方根非最优椭球集员滤波算法能使椭球形状矩阵保持正定性,因而它也具有较好的数值稳定性。如果状态的维数为n,在心动阵列上的滤波计算的复杂度为O(n)。因此,滤波计算的效率得到显著提高。此外,由于处理单元在不同阶段完成不同的运算功能,因而处理器的利用率也比较高。
第二部分,对全对称多胞形集员估计算法进行了分析和改进。在原始的全对称多胞形与带的交集的外界计算方法的基础上给出了改进外界计算方法。改进的外界计算方法给出了新的包含二者交集的全对称多胞形族,然后将此族中具有最小容积的全对称多胞形作为二者交集的外界。并且,根据全对称多胞形容积计算公式还可以证明改进外界计算方法得到的多胞形不会比原始方法大。在以上工作的基础上,得到了改进的全对称多胞形集员滤波算法和改进的全对称多胞形时变系统集员辨识算法。改进算法的精度比原始算法要高,尤其是当噪声服从重尾分布时,此优势更加明显。
第三部分,系统化的研究了以先进模式分类器为基础的集员辨识算法。一方面,以提高参数可行集的近似描述效果为目的,对原有的基于最小二乘支持向量机的集员辨识算法进行了改进。改进算法采用加权最小二乘支持向量机建立逼近方程误差向量的加权l∞范数与参数向量之间的复杂函数关系的模型,然后根据所建模型和可行的方程误差向量的加权l∞范数导出近似参数可行集。为了评估集员辨识结果的优劣,给出了反映近似边界接近精确边界程度的指标。与原算法相比,改进算法具有更好的近似描述能力。另一方面,将集员辨识视为设计模式分类器对参数空间中的点进行分类的过程,构造了两种新的集员辨识算法。两种算法分别通过训练模式分类器有监督的局部线性嵌入+最近均值分类器和有监督的等距映射+k近邻分类器在参数空间建立决策函数。此决策函数对参数空间中的点按是否属于可行集进行分类。由于模式分类器具有较小的分类错误率,因而由所有被判为可行集内的点组成的集合可以很好的逼近可行集。
第四部分,研究了两类带有未知但有界噪声的非线性系统的故障诊断问题。第一类问题为带有未知但有界噪声的非线性离散时间系统的建模与故障检测问题。针对这一问题,将集员辨识算法与智能模型(如径向基函数神经网络或Takagi-Sugeno模糊模型)相结合给出了非线性系统的建模与故障检测方法。在系统建模方面,此方法可以对系统进行鲁棒建模,并有效的预测实际系统的输出范围;在故障检测方面,此方法可以有效的检测突变故障和缓变故障,并且比阈值通过统计残差序列而设定的故障检测方法更具鲁棒性。第二类问题为在状态空间模型下带有未知但有界噪声的非线性离散时间系统传感器故障的检测与隔离问题。首先,提出一种非线性系统椭球集员滤波算法。此算法具有计算量小的特点,因此采用它估计状态可以实现在线故障检测与隔离。然后,给出了基于集员滤波的故障检测与隔离方法。其中包括一个故障检测方法和两个故障隔离方法。本方法可以有效的对传感器故障进行检测与隔离,并且比基于扩展卡尔曼滤波的方法更具鲁棒性。
Although the research work on set membership estimation theory, methods and their applications has achieved some results, there is a lot of work left to be done. Based on analyzing the existing results, further research work is carried out in this dissertation. The research work here focuses on existing methods improvement, new methods presentation, and applications of set membership estimation theory and methods to fault diagnosis. This dissertation is composed of four parts.
In the first part, set membership state estimation algorithms by ellipsoids are analyzed and improved. On one hand, the optimal set membership state estimation algorithm by ellipsoids for linear systems is improved, and two numerically stable suboptimal algorithms are proposed. In order to keep the shape-defining matrixes of the ellipsoids positive definite, cholesky decomposition and singular value decomposition are used in their computation. Besides, a subminimal-volume ellipsoid in the observation update is computed to circumvent inverse of ill-conditioned matrix. On the other hand, a square root non-optimal set membership state estimation algorithm by ellipsoids is given, and the parallel computation of state estimation using systolic arrays is schemed. The square root non-optimal algorithm can keep the shape-defining matrixes of the ellipsoids positive definite, so it is also numerically stable. If the state dimension is n, the parallel computation of state estimation using systolic arrays is of O(n) complexity. So the efficiency of computation is improved. Since the processing elements are enabled different functions at different stages, high processor utilization is achieved.
In the second part, set membership estimation algorithms by zonotopes are analyzed and improved. Improved outer bound computation methods of intersection of a zonotope and a strip are pointed out based on the original ones. They give new families of zontopes containing the intersection and select the minimal-volume one as the outer bound. And, it can be proved by zonotope volume formulas that the improved outer bound computation methods can give smaller zonotopes than the original ones. Based on the above work, an improved set membership state estimation algorithm by zonotopes and an improved set membership identification algorithm by zonotopes for time-varying parameterized systems are obtained. The improved algorithms are more accurate than the original ones especially in the presence of uniformly heavy-tailed noise.
In the third part, set membership identification algorithms based on advanced pattern classifiers are studied systematically. On one hand, the original set membership identification algorithm by least squares support vector machines is improved for better characterization of the feasible parameter sets. In the improved algorithm, a weighted least squares support vector regression is solved to build a model which approximates the complex functional relationship between the weighted l∞norms of the equation-error vectors and the given parameter vectors, and then the approximate feasible parameter set is obtained according to this model and the feasible weighted l∞norms of the equation-error vectors. Besides, an index reflecting the closeness between the approximate boundary and the true boundary is also given to evaluate the results of the proposed algorithm. Comparing with the original algorithm, the improved algorithm has better performance. On the other hand, two new set membership identification algorithms are given by viewing set membership identification as construction a pattern classifier to decide which class a point belongs to. In the two algorithms, supervised locally linear embedding and supervised isomap are combined with nearest mean classifier and k-nearest neighbor classifier respectively to build decision functions in the parameter space. The points in the parameter space can be divided into two classes according to whether they are in the feasible parameter set or not by decision functions. Since the classification errors are small, the set of all the points that are decided to be in the feasible parameter set can well approximate the feasible parameter set.
In the fourth part, two fault diagnosis problems for nonlinear systems with unknown but bounded noises are studied. The first problem is about modeling and fault detection for nonlinear discrete-time dynamical systems with unknown but bounded noises. For this problem, a modeling and fault detection method is proposed using set membership identification algorithms and intelligent models such as radial basis function neural networks and Takagi-Sugeno fuzzy models. From system modeling aspect, this method can model nonlinear systems robustly and predict the bounds of the actual system outputs easily. While from fault detection aspect this method can detect abrupt and incipient faults effectively and is more robust than the methods where the thresholds are determined by statistical properties of residual errors. The second problem is about sensor fault detection and isolation of nonlinear discrete-time dynamical systems based on state space models with unknown but bounded noises. First, a set membership state estimation algorithm by ellipsoids for nonlinear systems is proposed. Since the algorithm has light computational load, on-line fault detection and isolation can be realized when it is applied to state estimation. Then, one fault detection method and two fault isolation methods based on set membership state estimation are given. The methods can detect and isolate sensor faults effectively and are more robust than the methods based on extended Kalman filtering.
第一部分,对椭球集员滤波算法进行了分析和改进。一方面,对线性系统最优椭球集员滤波算法进行了改进,给出了两种数值稳定性比较好的次优算法。两种算法分别采用Cholesky分解和奇异值分解技术使椭球形状矩阵保持正定性。同时,为了避免受病态矩阵求逆的影响,在量测更新过程中两种算法均采用求具有次最小容积椭球的方法。另一方面,提出了一种平方根非最优椭球集员滤波算法,并给出了实现滤波计算的心动阵列结构。平方根非最优椭球集员滤波算法能使椭球形状矩阵保持正定性,因而它也具有较好的数值稳定性。如果状态的维数为n,在心动阵列上的滤波计算的复杂度为O(n)。因此,滤波计算的效率得到显著提高。此外,由于处理单元在不同阶段完成不同的运算功能,因而处理器的利用率也比较高。
第二部分,对全对称多胞形集员估计算法进行了分析和改进。在原始的全对称多胞形与带的交集的外界计算方法的基础上给出了改进外界计算方法。改进的外界计算方法给出了新的包含二者交集的全对称多胞形族,然后将此族中具有最小容积的全对称多胞形作为二者交集的外界。并且,根据全对称多胞形容积计算公式还可以证明改进外界计算方法得到的多胞形不会比原始方法大。在以上工作的基础上,得到了改进的全对称多胞形集员滤波算法和改进的全对称多胞形时变系统集员辨识算法。改进算法的精度比原始算法要高,尤其是当噪声服从重尾分布时,此优势更加明显。
第三部分,系统化的研究了以先进模式分类器为基础的集员辨识算法。一方面,以提高参数可行集的近似描述效果为目的,对原有的基于最小二乘支持向量机的集员辨识算法进行了改进。改进算法采用加权最小二乘支持向量机建立逼近方程误差向量的加权l∞范数与参数向量之间的复杂函数关系的模型,然后根据所建模型和可行的方程误差向量的加权l∞范数导出近似参数可行集。为了评估集员辨识结果的优劣,给出了反映近似边界接近精确边界程度的指标。与原算法相比,改进算法具有更好的近似描述能力。另一方面,将集员辨识视为设计模式分类器对参数空间中的点进行分类的过程,构造了两种新的集员辨识算法。两种算法分别通过训练模式分类器有监督的局部线性嵌入+最近均值分类器和有监督的等距映射+k近邻分类器在参数空间建立决策函数。此决策函数对参数空间中的点按是否属于可行集进行分类。由于模式分类器具有较小的分类错误率,因而由所有被判为可行集内的点组成的集合可以很好的逼近可行集。
第四部分,研究了两类带有未知但有界噪声的非线性系统的故障诊断问题。第一类问题为带有未知但有界噪声的非线性离散时间系统的建模与故障检测问题。针对这一问题,将集员辨识算法与智能模型(如径向基函数神经网络或Takagi-Sugeno模糊模型)相结合给出了非线性系统的建模与故障检测方法。在系统建模方面,此方法可以对系统进行鲁棒建模,并有效的预测实际系统的输出范围;在故障检测方面,此方法可以有效的检测突变故障和缓变故障,并且比阈值通过统计残差序列而设定的故障检测方法更具鲁棒性。第二类问题为在状态空间模型下带有未知但有界噪声的非线性离散时间系统传感器故障的检测与隔离问题。首先,提出一种非线性系统椭球集员滤波算法。此算法具有计算量小的特点,因此采用它估计状态可以实现在线故障检测与隔离。然后,给出了基于集员滤波的故障检测与隔离方法。其中包括一个故障检测方法和两个故障隔离方法。本方法可以有效的对传感器故障进行检测与隔离,并且比基于扩展卡尔曼滤波的方法更具鲁棒性。
Although the research work on set membership estimation theory, methods and their applications has achieved some results, there is a lot of work left to be done. Based on analyzing the existing results, further research work is carried out in this dissertation. The research work here focuses on existing methods improvement, new methods presentation, and applications of set membership estimation theory and methods to fault diagnosis. This dissertation is composed of four parts.
In the first part, set membership state estimation algorithms by ellipsoids are analyzed and improved. On one hand, the optimal set membership state estimation algorithm by ellipsoids for linear systems is improved, and two numerically stable suboptimal algorithms are proposed. In order to keep the shape-defining matrixes of the ellipsoids positive definite, cholesky decomposition and singular value decomposition are used in their computation. Besides, a subminimal-volume ellipsoid in the observation update is computed to circumvent inverse of ill-conditioned matrix. On the other hand, a square root non-optimal set membership state estimation algorithm by ellipsoids is given, and the parallel computation of state estimation using systolic arrays is schemed. The square root non-optimal algorithm can keep the shape-defining matrixes of the ellipsoids positive definite, so it is also numerically stable. If the state dimension is n, the parallel computation of state estimation using systolic arrays is of O(n) complexity. So the efficiency of computation is improved. Since the processing elements are enabled different functions at different stages, high processor utilization is achieved.
In the second part, set membership estimation algorithms by zonotopes are analyzed and improved. Improved outer bound computation methods of intersection of a zonotope and a strip are pointed out based on the original ones. They give new families of zontopes containing the intersection and select the minimal-volume one as the outer bound. And, it can be proved by zonotope volume formulas that the improved outer bound computation methods can give smaller zonotopes than the original ones. Based on the above work, an improved set membership state estimation algorithm by zonotopes and an improved set membership identification algorithm by zonotopes for time-varying parameterized systems are obtained. The improved algorithms are more accurate than the original ones especially in the presence of uniformly heavy-tailed noise.
In the third part, set membership identification algorithms based on advanced pattern classifiers are studied systematically. On one hand, the original set membership identification algorithm by least squares support vector machines is improved for better characterization of the feasible parameter sets. In the improved algorithm, a weighted least squares support vector regression is solved to build a model which approximates the complex functional relationship between the weighted l∞norms of the equation-error vectors and the given parameter vectors, and then the approximate feasible parameter set is obtained according to this model and the feasible weighted l∞norms of the equation-error vectors. Besides, an index reflecting the closeness between the approximate boundary and the true boundary is also given to evaluate the results of the proposed algorithm. Comparing with the original algorithm, the improved algorithm has better performance. On the other hand, two new set membership identification algorithms are given by viewing set membership identification as construction a pattern classifier to decide which class a point belongs to. In the two algorithms, supervised locally linear embedding and supervised isomap are combined with nearest mean classifier and k-nearest neighbor classifier respectively to build decision functions in the parameter space. The points in the parameter space can be divided into two classes according to whether they are in the feasible parameter set or not by decision functions. Since the classification errors are small, the set of all the points that are decided to be in the feasible parameter set can well approximate the feasible parameter set.
In the fourth part, two fault diagnosis problems for nonlinear systems with unknown but bounded noises are studied. The first problem is about modeling and fault detection for nonlinear discrete-time dynamical systems with unknown but bounded noises. For this problem, a modeling and fault detection method is proposed using set membership identification algorithms and intelligent models such as radial basis function neural networks and Takagi-Sugeno fuzzy models. From system modeling aspect, this method can model nonlinear systems robustly and predict the bounds of the actual system outputs easily. While from fault detection aspect this method can detect abrupt and incipient faults effectively and is more robust than the methods where the thresholds are determined by statistical properties of residual errors. The second problem is about sensor fault detection and isolation of nonlinear discrete-time dynamical systems based on state space models with unknown but bounded noises. First, a set membership state estimation algorithm by ellipsoids for nonlinear systems is proposed. Since the algorithm has light computational load, on-line fault detection and isolation can be realized when it is applied to state estimation. Then, one fault detection method and two fault isolation methods based on set membership state estimation are given. The methods can detect and isolate sensor faults effectively and are more robust than the methods based on extended Kalman filtering.
引文
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