Minimal Sensor Placement for Generic State and Unknown Input Observability

TL;DR

The paper tackles minimizing dedicated sensors to achieve GSIO for structured LTI systems with unknown inputs. It develops refined DM-decomposition-based conditions to separate sensor-placement into two steps, proves the problem is NP-hard and hard to approximate within $(1-o(1))\log(n)$, and derives polynomial-time upper and lower bounds tied to the auxiliary system $\hat{A}$ via $H(\hat{A})$. A special polynomial-time solvable case ($q=1$ with self-loops) is presented, along with a practical two-stage heuristic that combines maximum matching and a greedy set-cover approach for the general case. The results provide actionable bounds, an exact polynomial-time solution for a key case, and a scalable approximate algorithm, with implications for safeguarding against zero-dynamics attacks in networked control systems.

Abstract

This paper addresses the problem of selecting the minimum number of dedicated sensors to achieve observability in the presence of unknown inputs, namely, the state and input observability, for linear time-invariant systems. We assume that the only available information is the zero-nonzero structure of system matrices, and approach this problem within a structured system model. We revisit the concept of state and input observability for structured systems, providing refined necessary and sufficient conditions for placing dedicated sensors via the Dulmage-Mendelsohn decomposition. Based on these conditions, we prove that determining the minimum number of dedicated sensors to achieve generic state and input observability is NP-hard, which contrasts sharply with the polynomial-time complexity of the corresponding problem with known inputs. We also demonstrate that this problem is hard to approximate within a factor of $(1-o(1)){\rm{log}}(n)$, where $n$ is the state dimension. Notwithstanding, we propose nontrivial upper and lower bounds that can be computed in polynomial time, which confine the optimal value of this problem to an interval with length being the number of inputs. We further present a special case for which the exact optimal value can be determined in polynomial time. Additionally, we propose a two-stage algorithm to solve this problem approximately. Each stage of the algorithm is either optimal or suboptimal and can be completed in polynomial time.

Figures (6)

• Figure 1: (a) The bipartite digraph of $(A,B,C)$ in Example \ref{['ex.bd']}. (b) $\mathcal{D}(\mathcal{B}^{'}(A,B,C))$ of Example \ref{['ex.bd']} with 4 strongly connected components, where blue lines represent s-edges.
• Figure 2: The bipartite digraph of $\mathcal{D}(\mathcal{B}^{'}(A,B,C_{1}))$ in Example \ref{['example_NP']}, where red lines represent maximum matching and blue lines represent s-edges.
• Figure 3: (a) The digraph of $(A,B)$ in Example \ref{['ex.bounds']}. (b) The digraph of the auxiliary system corresponding to $\hat{A}$ in Example \ref{['ex.bounds']}.
• Figure 4: (a) and (b) represent the same structured system with two different minimum sensor placements to achieve structural observability. Blue vertices belong to $\mathcal{V}_{ess}(U,Y)$. (b) is GSIO but (a) is not.
• Figure 5: The network topology of $(A^{'},e_{i},C_{min})$ and $(\hat{A}^{'},e_{i},C_{min})$ in Subsection \ref{['Exam.1']}. Blue vertices represent $\mathcal{V}_{ess}(\{u\},Y_{min})$.

Theorems & Definitions (25)

• Definition 1
• Lemma 1
• Definition 2
• Lemma 2
• Example 1
• Lemma 3
• Lemma 4
• Theorem 1
• Definition 3
• Lemma 5