主动故障诊断技术对于保证系统的可靠性和安全性方面具有重要意义。实际系统在运行时受到各种不确定性因素的干扰，如何处理这些不确定性因素以提高诊断结果的鲁棒性是主动故障诊断技术的关键问题。目前的不确定性因素建模方式通常只考虑了其随机性或有界性，并分别建模为高斯分布或者某种有界集合的形式。考虑多种建模方式的混合不确定性更接近实际情况，但基于此的主动故障诊断方法较少。因此，本文主要研究混合不确定性因素下的主动故障诊断输入设计方法，具体内容如下： （1）有界-高斯混合不确定性无法给出显式的概率密度函数，难以度量输出分布之间的分离程度。通过引入置信因子，将高斯分布退化为带有置信率的椭球集合，根据zonotope集合与椭球集合的闵可夫斯基和定义输出置信域。最后通过设计输入序列使不同模态的输出置信域相互分离实现主动故障诊断，并证明基于该置信域分离的主动故障诊断方法的准确率大于等于给定的置信率下界。 （2）现有的集分离条件无法应对具有光滑表面的椭球体集合，且相应的混合整数二次规划问题的计算复杂度高。本文定义的置信域为zonotope集合与椭球体集合的闵可夫斯基和的形式，基于置信域分离的方法需要同时处理两种集合的分离条件。首先利用二阶锥的定义，将椭球体转化为二阶锥形式，进而给出带有线性互补约束和二阶锥互补约束的置信域分离条件。最后利用双线性项的凸包络以及重整线性化技术，获得原问题的线性松弛形式，并给出基于分支定界的全局最优解法。同时考虑到前者的计算复杂度问题，通过Fischer-Burmeister光滑函数将互补约束光滑近似为一般的非线性约束，给出一种基于光滑近似的局部最优快速解法。 （3）确定性框架下的主动故障诊断方法只针对一种集合类型的不确定性因素，没有统一的基于集分离的主动故障诊断方法，因此当前方法无法处理多种类型集合混合不确定性。将多面体集合表示成约束zonotope的形式，利用集合缩放因子，给出其相应的广义二阶锥表示方法，并以此定义统一集合表示形式。该形式可以表示任意的凸多面体、椭球体以及两者的闵可夫斯基和。本文同时证明了该统一集合表示形式在线性变换和闵可夫斯基和运算下保持闭合。最后根据统一集合表示形式给出简洁的基于集分离的主动故障诊断输入设计框架。
Active fault diagnosis (AFD) is of great significance in ensuring the reliability and safety of systems. Real systems are disturbed by various uncertainties during operation. How to handle these uncertainties to improve the robustness of diagnosis results is a critical issue in AFD technology. The current modeling methods for uncertainties usually only consider their randomness or boundedness, and model them in the form of Gaussian distributions or some bounded sets, respectively. The hybrid uncertainties, which contain multiple modeling forms, are closer to the real situation, but the AFD methods for hybrid uncertainties are relatively limited. Therefore, this thesis mainly studies the input design methods of AFD under hybrid uncertainties, with the specific contents as follows: (1) Hybrid bounded and Gaussian uncertainties cannot provide an explicit probability density function, and it is difficult to measure the separation of output distributions. By introducing a confidence coefficient, the Gaussian distribution is reduced to an ellipsoidal set with a confidence level, and the confidence domain of output is formulated as the Minkowski sum of a zonotopic set and an ellipsoidal set. The AFD task is achieved by designing an input sequence to separate the output confidence domains of different modes. It is proved that the correct rate of the AFD based on the separation of output confidence domains is no less than the given confidence coefficient. (2) The existing set separation conditions cannot deal with ellipsoidal sets which contain smooth surfaces, and the corresponding mixed integer quadratic program (MIQP) has high computational complexity. The confidence domain defined in this thesis is formulated as the Minkowski sum of the zonotopic set and the ellipsoidal set. The method based on the separation of confidence domains needs to handle the separation conditions of these two kinds of sets simultaneously. First, based on the definition of the second-order cone (SOC), the ellipsoid is converted into the form of SOC. Furthermore, the separation conditions of confidence domains are transformed into a group of linear complementarity and SOC complementarity constraints. Finally, using the convex envelope of bilinear terms and reformulation-linearization technique, the linear relaxation form of the original problem is obtained, and then a branch-and-bound method is provided for the global optimal solution. Meanwhile, considering the computational complexity of the former, the complementarity constraints are smoothed as ordinary nonlinear constraints by the Fischer-Burmeister smoothing function. Then an efficient method based on smoothing approximation is provided for local optimum. (3) AFD method under the deterministic framework can only handle the uncertainties formed as one type of set, and there is no unified AFD method based on the set separation. Therefore, the current method cannot handle hybrid uncertainties formed as multiple types of sets. First, the polytopic set is represented in the form of the constrained zonotope, which is then transformed into the generalized SOC representation based on a scaling factor. Accordingly, a unified set representation is further defined, which can represent any convex polytope, ellipsoid, and the Minkowski sum of both. It is also proved that the unified set representation remains closed under linear transformation and Minkowski sum operations. Finally, utilizing the unified set representation, a concise input design framework for the AFD problem based on the separation of output sets is provided.
