We study a fault detection problem in the context of Markovian Jump Linear Systems (MJLS). The major contribution of this paper is the design of a mode-dependent mixed $H_{\infty }/H_{-}$ Fault Detection Filter (FDF) under the MJLS formulation, which is obtained by solving a Linear and Bilinear Matrices Inequalities (LMI / BMI) optimization problem. As a secondary contribution, we present a simulation of the theoretical results using a Control Moment Gyroscope unit, which is extensively used as an actuator for attitude control of spacecraft and satellites. The attitude control of satellites is a class of problem that is within the framework of MJLS, since it allows us to model the loss of communication between the various components of the network system. Therefore, designing Fault Detection Filters under this framework provides reliable solutions. From the simulations results, it is possible to observe that the proposed mixed $H_{\infty }/H_{-}$ fault detection solution outperformed the approach using only the $H_{\infty }$ norm, showing that it could be a valuable alternative for the fault detection problem.

我们研究了马尔可夫跳跃线性系统（MJLS）上下文中的故障检测问题。本文的主要贡献是依赖于模式的混合设计$H_{\infty }/H_{-}$通过解决线性和双线性矩阵不等式（LMI / BMI）优化问题而获得的MJLS公式下的故障检测滤波器（FDF）。作为第二贡献，我们介绍了使用控制力矩陀螺仪单元对理论结果的模拟，该单元被广泛用作航天器和卫星姿态控制的执行器。卫星的姿态控制是MJLS框架内的一类问题，因为它允许我们对网络系统各个组件之间的通信损耗进行建模。因此，在此框架下设计故障检测过滤器可提供可靠的解决方案。从仿真结果可以看出，提出的混合$H_{\infty }/H_{-}$ 故障检测解决方案仅使用 $H_{\infty }$ 规范，表明它可能是故障检测问题的宝贵替代方案。