Secure and Robust Beamforming Design for STAR-RIS-aided MU-MIMO ISAC Systems

1. Problem Statement

The paper addresses the challenge of securing Integrated Sensing and Communication (ISAC) systems in the presence of Imperfect Channel State Information (ICSI). Specifically, it focuses on a Multi-User Multiple-Input Multiple-Output (MU-MIMO) downlink scenario assisted by a Simultaneous Transmitting and Reflecting Reconfigurable Intelligent Surface (STAR-RIS).

Key Challenges:

• Dual Functionality: The Base Station (BS) must simultaneously transmit confidential data to multiple intended users (IUs) and perform target sensing, while preventing information leakage to malicious eavesdroppers (Eves).
• Full-Space Coverage: Unlike conventional Reflecting-only RIS (R-RIS) which covers only half-space, STAR-RIS covers both transmission and reflection spaces, introducing complex coupling between transmission and reflection coefficients.
• Channel Uncertainty: In practical scenarios, perfect CSI is unattainable. The system must remain robust against bounded channel estimation errors for communication links, eavesdropper links, and sensing targets.
• Non-Convex Optimization: The joint design of active beamforming (at the BS) and passive beamforming (STAR-RIS phase shifts/amplitudes) under secrecy, Quality of Service (QoS), and sensing constraints results in a highly non-convex, coupled optimization problem with semi-infinite constraints due to CSI uncertainty.

2. Methodology

The authors propose a worst-case robust beamforming design framework to maximize the sum secrecy rate. The solution involves transforming the intractable problem into a solvable form using advanced mathematical tools and an iterative algorithm.

Key Technical Steps:

1. System Modeling:

   • The system employs an Energy Splitting (ES) mode for STAR-RIS, where each element simultaneously splits energy between transmission and reflection ($|\beta_{tn}|^2 + |\beta_{rn}|^2 = 1$).
   • A Norm-Bounded CSI Error Model is adopted, where actual channels are modeled as the estimated channel plus a bounded error term.

2. Problem Formulation (P1):

   • Objective: Maximize the sum secrecy rate of IUs.
   • Constraints:
     • Minimum QoS (rate) for all IUs.
     • Maximum tolerable eavesdropping rate (leakage) for Eves.
     • Minimum Signal-to-Interference-plus-Noise Ratio (SINR) for target sensing echoes.
     • Total transmit power budget.
     • Practical amplitude and phase shift constraints for STAR-RIS elements.

3. Robustification (Handling Uncertainty):

   • The semi-infinite constraints (arising from the need to satisfy conditions for all possible channel errors) are converted into deterministic Linear Matrix Inequalities (LMIs).
   • The S-procedure is utilized to handle the quadratic forms of the channel uncertainty.
   • Schur's Complement and Schur's Lemma are applied to decouple variables and reformulate constraints.

4. Algorithm Design (Alternating Optimization - AO):

   • Since the joint problem is non-convex, an Alternating Optimization (AO) framework is developed to iteratively optimize variables while fixing others.
   • Sub-problem 1 (Active Beamforming): With STAR-RIS coefficients fixed, the active beamforming vectors at the BS are optimized. This is solved as a Semidefinite Programming (SDP) problem.
   • Sub-problem 2 (Passive Beamforming): With BS beamforming fixed, the STAR-RIS phase shifts and amplitudes are optimized.
     • The unit-modulus constraint ($|\theta_{tn}|^2 + |\theta_{rn}|^2 = 1$) is non-convex.
     • Penalty Convex-Concave Programming (CCP) is employed. The non-convex equality is linearized using a first-order Taylor expansion, and a penalty term is added to the objective function to enforce feasibility.
   • Line Search: A line search mechanism is integrated to optimize the parameter controlling the eavesdropper's worst-case rate, ensuring the solution converges to a high-quality local optimum.

3. Key Contributions

1. Novel Framework: First work to propose a robust, secure MU-MIMO ISAC framework using STAR-RIS under Imperfect CSI, addressing the full-space coverage capability of STAR-RIS in a security-critical context.
2. Robust Optimization: Development of a tractable solution that converts semi-infinite robust constraints into finite LMIs using the S-procedure, ensuring system reliability even with bounded channel errors.
3. Efficient Algorithm: Design of an AO algorithm combining SDR (Semidefinite Relaxation) for active beamforming and Penalty CCP for passive beamforming, effectively handling the non-convexity and coupling of variables.
4. Comprehensive Constraints: The formulation simultaneously guarantees user QoS, sensing accuracy (echo SINR), and physical layer security (secrecy rate) under realistic hardware and propagation constraints.

4. Numerical Results

Extensive simulations validate the proposed framework against benchmarks (Random Phase STAR-RIS and Conventional R-RIS).

• Convergence: The proposed algorithm converges monotonically to a stable secrecy rate within a practical number of iterations.
• Performance Gain: The proposed STAR-RIS scheme significantly outperforms conventional R-RIS and random phase schemes. The gap widens as the number of RIS elements ($N$) and BS antennas ($L$) increase.
• Robustness: Under increasing CSI estimation errors ($\rho$), the proposed robust design maintains superior secrecy rates compared to non-robust designs, demonstrating its effectiveness in uncertain environments.
• Trade-offs:
  • QoS vs. Secrecy: As the minimum required user rate ($R_{min}$) increases, the achievable secrecy rate decreases, but the proposed scheme maintains a higher baseline than benchmarks.
  • Sensing vs. Secrecy: Stricter sensing SINR requirements reduce the secrecy rate (due to resource allocation trade-offs). However, increasing the number of BS antennas helps mitigate this loss.
  • Distance: The system performance degrades with distance due to path loss, but the STAR-RIS architecture provides significant gains over R-RIS, especially at shorter distances where signal manipulation is more effective.

5. Significance

This paper makes a significant contribution to the field of 6G wireless networks by:

• Bridging the Gap: It effectively integrates ISAC, Physical Layer Security (PLS), and STAR-RIS, three critical pillars of future networks.
• Practicality: By addressing Imperfect CSI, the proposed solution moves beyond theoretical idealizations, offering a design that is viable for real-world deployment where channel estimation is never perfect.
• Full-Space Advantage: It demonstrates that STAR-RIS is not just a coverage extension tool but a critical enabler for secure and robust communication in complex, obstructed environments where traditional MIMO or R-RIS fail.
• Scalability: The proposed algorithm scales well with the number of users and RIS elements, making it suitable for dense urban and industrial IoT scenarios.

In conclusion, the paper establishes that a robust, joint active-passive beamforming design for STAR-RIS-aided ISAC systems can achieve full-space coverage, high spectral efficiency, and strong security guarantees even in the presence of channel uncertainties and malicious eavesdroppers.