基于多项式混沌的概率鲁棒故障检测及其统计性能极限分析

The industrial applications evaluate fault detection performance with false alarm rate (FAR) and fault detection rate (FDR). In contrast, the conventional robust fault detection approaches rely on robustness and sensitivity criteria which are indirectly related to the above statistical indices, hence the resulting design would not always guarantee satisfactory FAR and FDR. In the probabilistic framework, the optimal design and statistical performance limit analysis with respect to FAR and FDR are not fully solved due to the following challenges. Firstly, it requires to compute the evolving probabilistic distribution of non-Gaussian system behavior, and its approximate computation often relies on randomized sampling which suffers from high online computational complexity. Secondly, it is challenging to deal with the non-Gaussian distribution of the system behavior in the optimal design and performance evaluation. To address these problems for non-Gaussian systems with probabilistic parametric uncertainties, a probabilistic robust fault detection framework is investigated: 1) for residual generation, propose a computationally efficient approach to quantify the evolving probabilistic distributions of non-Gaussian residuals, by exploiting polynomial chaos theory without resorting to online randomized sampling; 2) for residual evaluation, formulate a chance constrained optimization problem by exploiting the polynomial chaos expansions of residuals, such that the detection statistics and thresholds are designed to guarantee a predefined FAR and maximize the FDR; 3) for performance evaluation, analyze the limits of FAR and FDR which are independent of the implemented detection algorithm; 4) demonstrate the effectiveness of the above proposed methods using a lithium battery simulator.

实际工业应用采用误报率和检测率评价故障检测性能；而传统鲁棒故障检测方法使用的鲁棒性、敏感性指标只是上述统计性能的间接反映，因此不一定实现满意的误报率和检测率。随机框架下直接针对误报率和检测率的优化设计和性能极限分析目前并未完满解决，其难点在于：需要计算非高斯系统行为的概率分布演化，其近似计算方法多使用随机采样，导致在线计算复杂度较高；系统行为概率分布的非高斯性也给优化设计和性能评估带来困难。考虑到上述现状，本项目针对带有随机模型参数的非高斯系统，研究概率鲁棒故障检测方法：1）在残差生成中，利用多项式混沌理论提出非高斯残差概率分布演化的高效率计算方法，避免了在线随机采样；2）在残差评价中，利用残差的多项式混沌展开式，将满足误报率并最大化检测率的设计目标描述为机会约束优化问题，设计指示故障的检测量和检测阈值；3）分析独立于具体检测算法的误报率和检测率极限，用于性能评估；4）锂电池仿真系统验证。

在随机不确定因素的影响下，故障检测系统设计如何保证误报率并最大化检测率的问题并未在现有鲁棒故障检测理论中得到完满解决。为此，本项目针对带有随机不确定参数的非高斯系统，利用多项式混沌理论和分布鲁棒优化技术，研究概率鲁棒故障检测方法和统计性能极限分析，取得了以下成果：1）针对一类带有随机不确定参数的线性变参数系统，构造了依赖于不确定参数的多项式等价矩阵，提出了保证误报率水平的故障检测方法；2）针对带有随机不确定参数的动态系统，提出了基于多项式混沌理论的概率鲁棒观测器-控制器优化设计方法，能够处理范数有界、多面体等传统不确定描述无法直接表达的非线性不确定性结构；3）针对非高斯噪声的不确定概率分布，引入概率分布的模糊集合进行描述，提出了基于分布鲁棒优化的等价空间故障检测方法，在满足最坏情况下最大误报率的条件下优化最坏情况下的最小检测率；4）为实现锂电池仿真模型和实验数据上的算法验证，针对参数依赖于荷电状态的锂电池非线性等效电路提出了方向遗忘最小二乘辨识和基于高斯过程回归的线性变参数模型辨识，并在此基础上成功验证了锂电池荷电状态的滚动时域估计算法和概率鲁棒故障检测算法。本项目已正式发表7篇SCI期刊论文（包含1篇Automatica长文）和4篇EI会议论文，已录用待发表2篇SCI期刊论文（包含1篇IEEE Trans. on Automatic Control），已申请发明专利2项（其中1项获得授权）。综上所述，本项目不仅在概率鲁棒故障检测研究上取得了创新性理论成果，而且在锂电池仿真系统和实验数据上验证并得到满意效果，因此具有理论意义和应用价值。

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