Max-Min Fair Sensor Scheduling: Game-theoretic Perspective and Algorithmic Solution

TL;DR

The paper tackles fair allocation of limited communication bandwidth among $n$ sensors monitoring independent $n$-process LTI systems to minimize the worst-case remote estimation error. It casts the max-min fair resource allocation as a two-player zero-sum game with $g(\mathbf{w},\mathbf{r})=\sum_i w_i J_i(r_i)$, proves the existence of a pure-strategy Nash equilibrium and a unique game value, and develops a cost-based iterative method that converges to a unique fair allocation under mild assumptions. The framework is specialized to fair sensor scheduling by leveraging single-sensor results, yielding a two-loop algorithm with convergence guarantees and a distributed variant via dual decomposition. Numerical experiments demonstrate convergence and the emergence of equalized estimation performance among unstable processes, validating the approach for practical bandwidth-sharing in remote state estimation. Overall, the work provides a principled, game-theoretic and algorithmic methodology for achieving max-min fairness in sensor scheduling with provable convergence and scalability features.

Abstract

We consider the design of a fair sensor schedule for a number of sensors monitoring different linear time-invariant processes. The largest average remote estimation error among all processes is to be minimized. We first consider a general setup for the max-min fair allocation problem. By reformulating the problem as its equivalent form, we transform the fair resource allocation problem into a zero-sum game between a "judge" and a resource allocator. We propose an equilibrium seeking procedure and show that there exists a unique Nash equilibrium in pure strategy for this game. We then apply the result to the sensor scheduling problem and show that the max-min fair sensor scheduling policy can be achieved.

Figures (3)

• Figure 1: The optimal average estimation error of the single sensor scheduling under constrained communication is piecewise linear, convex and nonincreasing w.r.t. communication rate.
• Figure 2: The bandwidth allocation of each processes.
• Figure 4: The error of $\bm{r}(k)$ in each iteration.

Theorems & Definitions (18)

• Remark 1
• Remark 2
• Theorem 1
• Example 1
• Corollary 1
• Remark 3
• Theorem 2
• Theorem 3
• Remark 4
• Proposition 1