This article studies two problems related to observability and efficient constrained sensor placement in linear time-invariant discrete-time systems with partial state observations: (i) We impose the condition that both the set of outputs and the state that each output can measure are pre-specified. We establish that for any fixed $k>2$, the problem of placing the minimum number of sensors/outputs required to ensure that the structural observability index is at most $k$, is NP-complete. Conversely, we identify a subclass of systems whose structures are directed trees with self-loops at every state vertex, for which the problem can be solved in linear time. (ii) Assuming that the set of states that each given output can measure is given, we prove that the problem of selecting a pre-assigned number of outputs in order to maximize the number of states of the system that are structurally observable (i.e., to maximize the size of the observable subgraph) is also NP-hard. As an application, we identify suitable conditions on the system structure under which there exists an efficient greedy strategy, which we provide, to obtain a $(1-\frac{1}{e})$-approximate solution. An illustration of the techniques developed for this problem is given on the benchmark IEEE 118-bus power network containing roughly 400 states in its linearized model.

本文研究了与具有部分状态观察的线性时不变离散时间系统中的可观察性和有效约束传感器放置相关的两个问题：（i）我们强加了输出集和每个输出可以测量的状态都预先确定的条件。 -指定的。我们确定对于任何固定$克>2$, 放置最少数量的传感器/输出以确保结构可观察性指数最多的问题 $克$, 是 NP 完全的。相反，我们确定了一个系统的子类，其结构是在每个状态顶点都有自循环的有向树，对于这些系统，问题可以在线性时间内解决。(ii) 假设每个给定输出可以测量的状态集是给定的，我们证明了选择预先分配的输出数量以最大化系统结构上可观察的状态数量的问题（即，最大化可观察子图的大小）也是 NP-hard。作为一个应用程序，我们在系统结构上确定合适的条件，在该条件下，我们提供了一个有效的贪婪策略，以获得一个$(1-\frac{1}{\mathrm{电子}})$- 近似解。在基准 IEEE 118 总线电源网络上给出了针对此问题开发的技术的说明，该网络在其线性化模型中包含大约 400 个状态。