# An Improved FFIP Method Based on Mathematical Logic and SysML
## 1 Introduction
By using this method, a designer can analyze the system function effectively, realize the path of failure propagation, and reduce the influence of human subjectivity through simulation [17]. Simulation and reasoning methods in FFIP are derived from qualitative physics and qualitative reasoning. In other words, the finite state of system behavior is represented, and then the fault propagation path is deduced based on the qualitative relationship between function and behavior [18]. In recent years, FFIP has been constantly expanding and improving. General speaking, FFIP has the following advantages:  
1. It does not require the users to make the hypothesis of causality in advance [19], and it avoids the subjective influence of human. 2. It can identify the functional failures caused by global interaction, and simulate any number of failures. 3. It can help engineers identify the key components of the system and quantify the related risks of specific components. Since FFIP was proposed [17], it has been applied in many fields and developed continuously. In reference [20], a method of automatically generating an event tree based on the FFIP framework was formed, and an automatic tool was developed. In reference [21], the FFIP method is extended to generate training and test data sets for the development of failure detection systems based on a data-driven machine learning method. Reference [22] shows that the FFIP method can effectively evaluate software and hardware. Additionally, in reference [23], human factors are considered, and the impact of human error on system failure is analyzed by using FFIP. In reference [24], external events are considered, the Time-Based Failure Flow Evaluator (TBFFE) is developed based on FFIP method. In reference [25], the Function Failure Propagation Potential Method (FFPPM) is proposed by extending FFIP with graph theory and matrix theory. Of course, the FFIP still has its own shortcomings. First, the FFIP method is suitable for the early design stage of the system, but MBSA pursues a unified model in the whole life cycle of the system, from design to use to inspection. Secondly, the FFL of FFIP is not specific and clear for the description and lacks the standardized description. On top of that, the FFIP method uses graphics to describe the model, missing a standardized and unified modeling language. SysML is a multipurpose standard modeling language, and can support the detailed description, analysis, design, validation, and verification of various complex industrial systems, such as hardware, software, information, process, personnel and facilities and other systems. A SysML diagram contains many elements that can be combined with each other, which can represent the structure, function and behavior of the system in a standardized way [26, 27, 28]. In recent years, researchers have combined some parts of FFIP with SysML. In reference [29], the FFL of FFIP is integrated into the Block Definition Diagram (BDD) to evaluate and compare the input and output streams. In reference [30], StateCharts are used to further expand the FFL. The framework of FFIP is expanded by clearly listing the hierarchical relationship among system state, function state, component state, command signal and sensor signal. In the above literatures, FFIP has been combined with SysML, but they only used SysML to expand FFL, and did not combine with the whole modeling process of FFIP. In this paper, we introduce mathematical logic and SysML to extend the FFIP method from two aspects: the applicable stage of the method and the formation of standardized expression. Firstly, we introduce the concept of failure deviation and extend the FFL. Then, we use mathematical logic to normalize the failure logic. In addition, we combined FFIP with SysML. On one hand, we combine SysML with the whole modeling process of FFIP and propose an integral modeling method under the MBSA framework. It includes building the structure model, function model and behavior model in FFIP with SysML.