Expected Cramér–Rao Bound Optimization for RIS-Aided ISAC Systems With Phase-Shift Errors: A Stochastic Optimization Approach

In reconfigurable intelligent surface (RIS)-aided integrated sensing and communication (ISAC) systems, beamforming design based on minimizing the Cramér-Rao bound (CRB) for target direction-of-arrival (DoA) estimation is pivotal for sensing capability enhancement. However, due to practical hardware limitations, phase-shift errors (PSEs) exist at the RIS reflectors and cause performance deterioration. To reduce the adverse impact of PSEs, we develop a novel stochastic optimization (SO)-based expected CRB (ECRB) minimization framework, where the ECRB is defined as the expectation of CRB taken over random PSEs, statistical channel state information (CSI), and historical DoA estimates following known prior distributions. Specifically, we formulate an SO problem to minimize the ECRB, subject to an ergodic achievable sum-rate (EASR) constraint. To solve this non-convex problem, we propose a novel penalty-based block stochastic gradient descent (PBSGD) method. In this method, we first introduce a penalty factor to move the EASR constraint into the objective function. The optimal penalty factor is rigorously proved to be determinable via a bisection search. Then, we design a projected block stochastic gradient descent process to update transmit beamformers and RIS phase shifts alternately with guaranteed convergence. Simulation results demonstrate that our proposed method outperforms several benchmarks, including random phase-shift design, sensing-only beamforming, and state-of-the-art semidefinite relaxation (SDR)-based CRB optimization techniques, while exhibiting enhanced robustness against PSEs.