对抗控制：基础、可扩展性与非线性性
Antagonistic Control: Foundations, Scalability and Nonlinearity
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中文总结 AI 辅助
本文研究对抗控制的最坏情况输入问题，针对线性时不变、正系统及非线性多项式系统分别给出对应可扩展方法，还揭示了安全检测攻击输入的结构局限性。
中文摘要 AI 辅助
本文研究受约束控制输入的最坏-case影响：一个输入旨在最大化某些输出的平均代价，该代价以$L_2$或$L_1$范数衡量，同时在其他输出方面保持有界。该问题模板涵盖了$H_\infty$范数和输出到输出增益等经典指标，出现在对抗控制、安全评估和鲁棒控制领域。对于线性时不变系统，当存在单个约束时，我们提供精确的半定规划（SDP）；当存在多个约束时，提供计算上界的SDP。我们根据系统零点和相对阶推导最坏情况代价无界的充分条件，同时给出构造性闭环修改方案以消除该无界性。从安全角度看，无界值揭示了检测某些攻击输入的结构局限性。对于正系统，我们提供复杂度随状态维度线性增长的可扩展公式：针对二次代价的可扩展SDP，以及针对线性代价的精确线性规划。这些结果通过和平方程序扩展到非线性多项式系统。我们用数值示例说明这些结果。
英文摘要
This paper studies the worst-case impact of constrained control inputs: an input seeks to maximize the average cost of some outputs, measured in the $L_2$ or $L_1$ norm, while remaining bounded in terms of other outputs. This problem template subsumes classical metrics such as the $H_\infty$ norm and the output-to-output gain, and arises in adversarial control, security assessment, and robust control. For linear time-invariant systems, we provide an exact semi-definite program (SDP) when there is a single constraint, and SDPs computing upper bounds when there are multiple constraints. We derive sufficient conditions, in terms of system zeros and relative degrees, under which the worst-case cost is unbounded, together with a constructive closed-loop modification that removes the unboundedness. From a security standpoint, unbounded values reveal structural limitations in detecting certain attack inputs. For positive systems, we provide scalable formulations whose complexity grows linearly in the state dimension: a scalable SDP for quadratic costs, and an exact linear program for linear costs. The results extend to nonlinear polynomial systems via a sum-of-squares program. We illustrate the results with numerical examples.
