This paper proposes an actuator security index based on the definition of perfect undetectability. This index can help a control system operator to localize the most vulnerable actuators in the networked control system, which can then be secured. Particularly, the security index of an actuator equals the minimum number of sensors and actuators that needs to be compromised, such that a perfectly undetectable attack against that actuator can be conducted. A method for computing the index for small scale networked control systems is derived, and it is shown that the index can potentially be increased by placing additional sensors. The difficulties that appear once the system is of a large scale are then outlined: the problem of calculating the index is NP--hard, the index is vulnerable to system variations, and it is based on the assumption that the attacker knows the entire model of the system. To overcome these difficulties, a robust security index is introduced. The robust index can be calculated in polynomial time, it is unaffected by the system variations, and it can be related to both limited and full model knowledge attackers. Additionally, we analyze two sensor placement problems with the objective to increase the robust indices. We show that both of these problems have submodular structures, so their suboptimal solutions with performance guarantees can be obtained in polynomial time. Finally, the theoretical developments are illustrated through numerical examples.
Download