In this paper, we study an intelligent reflecting surface (IRS)-aided communication system with single-antenna transmitter and receiver, under imperfect channel state information (CSI). More specifically, we deal with the robust selection of binary (on/off) states of the IRS elements in order to maximize the worst-case energy efficiency (EE), given a bounded CSI uncertainty, while satisfying a minimum signal-to-noise ratio (SNR). In addition, we consider not only continuous but also discrete IRS phase shifts. First, we derive closed-form expressions of the worst-case SNRs, and then formulate the robust (discrete) optimization problems for each case. In the case of continuous phase shifts, we design a dynamic programming (DP) algorithm that is theoretically guaranteed to achieve the global maximum with polynomial complexity $O(L\,{\log L})$, where $L$ is the number of IRS elements. In the case of discrete phase shifts, we develop a convex-relaxation-based method (CRBM) to obtain a feasible (sub-optimal) solution in polynomial time $O(L^{3.5})$, with a posteriori performance guarantee. Furthermore, numerical simulations provide useful insights and confirm the theoretical results. In particular, the proposed algorithms are several orders of magnitude faster than the exhaustive search when $L$ is large, thus being highly scalable and suitable for practical applications. Moreover, both algorithms outperform a baseline scheme, namely, the activation of all IRS elements.

翻译：本文针对单天线的智能反射面(IRS)辅助通信系统，在非理想信道状态信息(CSI)条件下开展研究。具体而言，我们处理IRS单元二进制（开/关）状态的鲁棒选择问题，目的是在给定有界CSI不确定性和满足最小信噪比(SNR)约束的前提下，最大化最坏情况下的能量效率(EE)。此外，我们不仅考虑了连续型IRS相移，还考虑了离散型IRS相移。首先，我们推导了最坏情况信噪比的闭式表达式，然后针对每种情况构建了鲁棒（离散）优化问题。在连续相移情况下，我们设计了一种动态规划(DP)算法，该算法理论上能以多项式复杂度$O(L\,{\log L})$（其中$L$为IRS单元数量）达到全局最优解。在离散相移情况下，我们开发了基于凸松弛的方法(CRBM)，可在多项式时间$O(L^{3.5})$内获得可行（次优）解，并附带后验性能保证。此外，数值仿真提供了有价值的见解并验证了理论结果。特别地，当$L$较大时，所提算法比穷举搜索快数个数量级，因此具有高度可扩展性，适合实际应用。同时，两种算法均优于基准方案（即激活全部IRS单元）。