发表机构
Institute for Advanced Study, Shenzhen University（深圳大学高等研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究结构化自适应LQR中控制器可辨识性，证明其决定对数或平方根局部极小极大遗憾，并构造仅估计可见参数分量的确定性等价策略达到对数上界。
AI 中文摘要
我们研究在结构化自适应线性二次调节中何时能够达到对数遗憾。状态矩阵和输入矩阵仿射地依赖于一个共同的未知参数。我们考虑一个已知标称模型的邻域，该模型是可镇定的，并且通过状态成本相关的输出是可观测的，邻域半径与水平期的四分之一次方成反比。关键条件是控制器可辨识性，它要求最优增益导数在每一个使得闭环动力学在标称最优反馈下保持不变的参数方向上为零。在噪声密度的适当正则性条件下，当标称参数处增益导数非零时，该条件产生对数局部极小极大遗憾；条件不成立则产生平方根局部极小极大遗憾。若标称增益导数为零，则局部极小极大遗憾保持有界。我们证明控制器可辨识性等价于在足够小的不可见扰动下最优增益保持不变。受此不变性启发，我们构造了一个仅估计可见参数分量的确定性等价策略，并在控制器可辨识性下达到对数上界。该策略既不需要水平期也不需要噪声分布，其保证适用于独立同分布、均值为零且协方差有限正定的噪声。
英文摘要
We study when logarithmic regret is attainable in structured adaptive linear-quadratic regulation. The state and input matrices depend affinely on a common unknown parameter. We consider neighborhoods of a known nominal model that is stabilizable and observable through the output associated with the state cost, with radii proportional to the inverse fourth root of the horizon. The key condition is controller identifiability, which requires the optimal gain derivative to vanish in every parameter direction that leaves the closed-loop dynamics unchanged under nominal optimal feedback. Under suitable regularity conditions on the noise density, this condition at the nominal parameter yields logarithmic local minimax regret when the gain derivative there is nonzero. Failure of the condition yields square-root local minimax regret. If the nominal gain derivative vanishes, local minimax regret remains bounded. We prove that controller identifiability is equivalent to the optimal gain remaining unchanged under sufficiently small invisible perturbations of the nominal model. Motivated by this invariance, we construct a certainty-equivalent policy that estimates only the visible parameter component and attains the logarithmic upper bound under controller identifiability. The policy requires neither the horizon nor the noise law, and its guarantee holds for independent, identically distributed noise with zero mean and finite positive definite covariance.
Comments35 pages, 3 figures
