Detection-Control Games under Hidden Modes: Resilience-Induced Blindness Phenomenon

Technical Summary: Detection–Control Games under Hidden Modes

Problem Formulation
This paper addresses the challenge of resilient control for cyber-physical systems (CPS) operating under hidden operational modes, which may represent nominal conditions, degraded states, faults, or attacks. The core problem challenges the traditional "separation principle" in control design, where a detector is optimized for inference accuracy and a controller is optimized for regulation performance independently. The authors argue that in hidden-mode systems, the detector and controller are intrinsically coupled: the controller shapes the statistical structure of the measurements used for detection, while the detector's belief updates dictate the controller's feedback gains. This coupling creates a noncooperative game between two decision-makers with divergent objectives: the detector seeks informative belief updates, while the controller seeks to minimize regulation costs.

Methodology
The authors formulate the interaction between the detector and controller as a dynamic noncooperative game.

• System Model: The study focuses on a two-mode linear Gaussian system where the mode $\theta_k \in \{0, 1\}$ is unobserved. The system dynamics and measurement functions are mode-dependent.
• Detector Design: The detector maintains a posterior belief $\pi_k$ over the modes. It utilizes a "tempered Bayesian" update rule, where a parameter $\beta$ controls the sensitivity to new measurement evidence. The detector minimizes a cost function balancing belief stability (penalizing abrupt changes) and likelihood fit.
• Controller Design: The controller employs a belief-dependent Linear Quadratic Regulator (LQR). It computes a control input $u_k$ based on a state estimate and the current belief $\pi_k$, effectively treating the system as a convex combination of the two modes.
• Game-Theoretic Analysis: The interaction is analyzed via the Nash equilibrium concept. The authors demonstrate that while the detector and controller may be individually optimal given the other's policy, the joint equilibrium can be globally inefficient.
• Theoretical Characterization: For the linear Gaussian case, the authors derive a quantitative bound linking the controller's "resilience margin" (a measure of how well the controller suppresses state deviations) to the log-likelihood evidence generated by the system.

Key Contributions

1. Game-Theoretic Formulation: The paper establishes detection and control as coupled decision problems, clarifying why separated designs can lead to locally rational but globally inefficient outcomes (Nash equilibrium) due to objective misalignment.
2. Resilience-Induced Blindness Phenomenon: The primary theoretical contribution is the characterization of a phenomenon where a controller with a large resilience margin (highly effective at suppressing state deviations in a fixed mode) inadvertently suppresses the mode-dependent variability in measurements. This suppression reduces the statistical separation between modes, slowing the accumulation of log-likelihood evidence and causing the detector's belief to adapt slowly.
3. Quantitative Bound: The authors derive a specific bound (Theorem 1) showing that a larger resilience margin $m_\theta$ leads to smaller contraction factors in state trajectories. This reduces the innovation difference between modes, thereby shrinking the upper bound on the log-likelihood ratio update speed.
4. Mitigation Strategy: The paper proposes tuning the detector's objective (specifically increasing the sensitivity parameter $\beta$) to partially counteract the blindness effect, allowing the belief to adapt more rapidly despite the controller's suppression of information.

Results
Numerical simulations on a two-mode linear Gaussian system with unstable dynamics illustrate the phenomenon:

• Performance Reversal: In a fixed-mode scenario, increasing the control penalty (reducing controller aggressiveness) degrades regulation performance as expected. However, under mode switching, the trend reverses: controllers with smaller penalties (more aggressive, higher resilience margins) result in higher total state costs. This occurs because aggressive controllers suppress the very information needed to detect the mode switch, leading to prolonged belief mismatch and suboptimal control gains.
• Mitigation: Increasing the detector sensitivity parameter $\beta$ (e.g., from 1 to 6) accelerates belief adaptation. This reduces the duration of gain mismatch and partially restores the expected performance trend, though it does not fully eliminate the phenomenon because the controller continues to shape the available information.

Significance and Claims
The paper claims that the "resilience-induced blindness" phenomenon is an intrinsic closed-loop effect that arises when detection and control are designed independently. It highlights that a controller optimized for robust regulation in a fixed mode can degrade the overall system's ability to detect and respond to hidden mode changes. The authors conclude that achieving optimal resilient performance requires a co-design approach that aligns regulation performance with the need for informative measurements and rapid belief adaptation, rather than relying on the separation principle. The work provides a theoretical foundation for understanding why high-performance controllers can paradoxically lead to poor detection and degraded switched-mode performance.