未知增长指数下自适应镇定的基本极限
Fundamental Limits of Adaptive Stabilization with an Unknown Growth Exponent
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中文总结 AI 辅助
本文研究增长指数未知时自适应控制的可镇定临界指数，确定标称指数低于$3\sqrt{3}/2$时可镇定，有限指数集的临界指数为4，揭示增长速率不确定性对可镇定范围的影响。
中文摘要 AI 辅助
自适应控制中的一个基本问题是：具有未知参数的离散时间非线性装置在保持可镇定性的同时，其增长速度能达到多快。当非线性增长指数已知、仅标量系数未知时，现有理论确定4是可镇定性的精确临界指数。本文研究增长指数也未知时，该临界指数的变化情况：对于任意正扰动界，存在一个反馈律可镇定标称参数对邻域内的所有装置，当且仅当标称指数低于$3\sqrt{3}/2$；当指数等于该值时，所有指数不超过该值的紧参数集仍保持可镇定。若指数属于已知有限集，则每个非退化紧系数区间的临界指数仍为4，因此有限指数集与任意短指数区间可具有不同的临界指数，即增长速率的不确定性改变了反馈可镇定的增长范围，而非仅改变不确定性集的大小。在已知正的、有上界且远离零的状态相关乘子下，临界指数仍为$3\sqrt{3}/2$；对于系数和指数均为未知参数函数的系统，当指数不超过该值时，紧参数族可镇定；在这些函数的雅可比矩阵秩为2且指数至少为该值的内部参数点处，不存在紧邻域可镇定。
英文摘要
A basic question in adaptive control is how rapidly a discrete-time nonlinear plant with unknown parameters may grow while remaining stabilizable. When the nonlinear growth exponent is known and only a scalar coefficient is unknown, existing theory identifies 4 as the exact critical exponent for stabilizability. We determine how this critical exponent changes when the growth exponent is also unknown. For any positive disturbance bound, one feedback law stabilizes all plants in some neighborhood of a nominal parameter pair if and only if the nominal exponent is below $3\sqrt{3}/2$. At equality, every compact parameter set whose exponents do not exceed this value remains stabilizable. If the exponent belongs to a known finite set, the critical exponent remains 4 for every nondegenerate compact coefficient interval. Hence finite exponent sets and arbitrarily short exponent intervals can have different critical exponents. Uncertainty in the growth rate therefore changes the range of growth that feedback can stabilize, not merely the size of the uncertainty set. The critical exponent remains $3\sqrt{3}/2$ under a known positive state-dependent multiplier bounded above and away from zero. For a system whose coefficient and exponent are known functions of an unknown parameter, a compact parameter family is stabilizable when its exponents do not exceed this value. At an interior parameter point where the Jacobian of these functions has rank two and the exponent is at least this value, no compact neighborhood is stabilizable.
发表机构
- Institute for Advanced Study, Shenzhen University（深圳大学高等研究院）
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