Given a linear dynamical system affected by noise, we study the problem of optimally placing sensors (at design time) subject to a sensor placement budget constraint in order to minimize the trace of the steady-state error covariance of the corresponding Kalman filter. While this problem is NP-hard in general, we consider the underlying graph associated with the system dynamics matrix, and focus on the case when there is a single input at one of the nodes in the graph. We provide an optimal strategy (computed in polynomial time) to place the sensors over the network. Next, we consider the problem of attacking (i.e., removing) the placed sensors under a sensor attack budget constraint in order to maximize the trace of the steady-state error covariance of the resulting Kalman filter. Using the insights obtained for the sensor placement problem, we provide an optimal strategy (computed in polynomial time) to attack the placed sensors. Finally, we consider the scenario where a system designer places the sensors under a sensor placement budget constraint, and an adversary then attacks the placed sensors subject to a sensor attack budget constraint. The resilient sensor placement problem is to find a sensor placement strategy to minimize the trace of the steady-state error covariance of the Kalman filter corresponding to the sensors that survive the attack. We show that this problem is NP-hard, and provide a pseudopolynomial-time algorithm to solve it.

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网络系统中用于卡尔曼滤波的弹性传感器放置：复杂性和算法

给定一个受噪声影响的线性动力学系统，我们研究在传感器放置预算约束下优化放置传感器（在设计时）的问题，以最大程度地减少相应卡尔曼滤波器的稳态误差协方差的踪迹。尽管此问题通常是NP难题，但我们考虑与系统动力学矩阵关联的基础图，并关注图中节点之一上有单个输入的情况。我们提供了一种最佳策略（以多项式计算），可以将传感器放置在网络上。接下来，我们考虑在传感器攻击预算约束下攻击（即移除）放置的传感器的问题，以便最大程度地跟踪所得卡尔曼滤波器的稳态误差协方差的轨迹。利用针对传感器放置问题获得的见解，我们提供了一种优化策略（以多项式时间计算）来攻击放置的传感器。最后，我们考虑系统设计人员将传感器放置在传感器放置预算约束下，然后对手受到传感器攻击预算约束的攻击来攻击放置的传感器的情况。有弹性的传感器放置问题是找到一种传感器放置策略，以最小化对应于在攻击中幸存的传感器的卡尔曼滤波器的稳态误差协方差的轨迹。我们证明这个问题是NP难的，并提供了一个伪多项式时间算法来解决。然后，对手受到传感器攻击预算约束，攻击放置的传感器。有弹性的传感器放置问题是找到一种传感器放置策略，以最小化对应于在攻击中幸存的传感器的卡尔曼滤波器的稳态误差协方差的轨迹。我们证明这个问题是NP难的，并提供了一个伪多项式时间算法来解决。然后，对手会受到传感器攻击预算的约束，攻击放置的传感器。有弹性的传感器放置问题是找到一种传感器放置策略，以最小化对应于在攻击中幸存的传感器的卡尔曼滤波器的稳态误差协方差的轨迹。我们证明这个问题是NP难的，并提供了一个伪多项式时间算法来解决。