We study the state estimation for cyber-physical systems (CPSs) whose communication channels are subject to mixed denial-of-service (DoS) and false data injection (FDI) attacks. Cyber-attacks compromise the security and privacy of sensor and communication information. Unlike systems subject to only one single type of attacks (either DoS or FDI attacks), systems under mixed attacks will make the implementation of the optimal state estimation infeasible. We first obtain the optimal estimator for CPSs under mixed cyber-attacks. The optimal estimator consists of an exponentially growing number of components, and thus its computation effort exponentially grows in time. To efficiently compute the optimal estimate, we propose an approximate estimator by using the generalized pseudo-Bayesian algorithm. We prove that for a stable system, both the optimal estimator and the proposed approximate estimator are secure; and theoretically characterize the boundedness of the distance between the optimal and the approximate estimates. A simulation example is presented to illustrate the effectiveness of the proposed methods in guaranteeing secure state estimation when the privacy of sensor and communication information is at the risk of mixed cyber-attacks.

我们研究其通信通道受到混合拒绝服务（DoS）和错误数据注入（FDI）攻击的网络物理系统（CPS）的状态估计。网络攻击损害了传感器和通信信息的安全性和隐私性。与仅受到一种单一攻击（DoS或FDI攻击）的系统不同，受到混合攻击的系统将使实现最佳状态估计不可行。我们首先获得混合网络攻击下CPS的最佳估计量。最佳估计器由数量成倍增长的组件组成，因此其计算工作随时间呈指数增长。为了有效地计算最优估计，我们通过使用广义伪贝叶斯算法提出了一种近似估计器。我们证明，对于一个稳定的系统，最优估计器和提议的近似估计器都是安全的；并且从理论上描述了最佳估计值和近似估计值之间的距离的有界性。给出了一个仿真示例，以说明当传感器和通信信息的隐私面临混合网络攻击的风险时，所提出方法在保证安全状态估计方面的有效性。