Åström-Wittenmark自校正调节器在输入增益未知时的强一致性与最优跟踪
Strong Consistency, Optimal Tracking and Sharp Rates for the Åström-Wittenmark Self-Tuning Regulator with Unknown Input Gain
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中文总结 AI 辅助
研究Åström-Wittenmark自校正调节器在输入增益未知时的强一致性与最优跟踪，提出无需参考激励或探测信号的方法，证明参数估计强一致且跟踪误差收敛至噪声方差。
中文摘要 AI 辅助
我们研究了输入增益未知的Åström-Wittenmark自校正调节器的强一致性和最优跟踪问题。现有结果在参考信息满足增长条件下，建立了未修改递归的稳定性、最优跟踪和参数一致性。其他方法通过调整反馈中使用的估计或添加衰减探测信号来获得性能保证。对于一类具有鞅差噪声的最小相位线性系统，我们在无需参考激励、增益调整或额外探测的情况下，建立了普通最小二乘与确定性等价控制的联合保证。对于每个有界参考，平均输入和输出能量几乎必然有界，平均平方跟踪误差几乎必然收敛到噪声方差，并且当至少估计两个参数时，所有参数估计都是强一致的。对于一致性结果，关键在于噪声的辅助最小二乘预测与参考之间累积平方差的对数下界。如果增益误差持续存在，该噪声信息和实际最小二乘递归将对同一加权平方和给出不相容的下界和上界。这一矛盾在稳定性和完全参数一致性之前建立了增益收敛性。
英文摘要
We study strong consistency and optimal tracking of the Astrom-Wittenmark self-tuning regulator with unknown input gain. Existing results establish full parameter consistency for the unmodified recursion under additional growth conditions on reference information. Other approaches obtain performance guarantees by adjusting the estimates used in feedback or adding decaying probing signals. For a class of minimum-phase linear systems with martingale difference noise, we prove that ordinary least squares with certainty-equivalent control achieves stability and optimal tracking almost surely for each bounded reference, without reference excitation, gain adjustment or added probing. When at least two parameters are estimated, all parameter estimates are strongly consistent, with squared estimation error norm $O(\log\log\log n/\log n)$ almost surely. An exact law of the iterated logarithm under zero reference shows that this rate cannot in general be improved. We also derive a logarithm law for cumulative squared tracking residuals that makes their dependence on reference energy explicit. A chi-square limit yields asymptotically exact parameter confidence ellipsoids for each bounded reference. For consistency, we prove a logarithmic lower bound on the cumulative squared difference between an auxiliary least-squares prediction of the noise and the reference. If the gain error persisted, this bound and the actual least-squares recursion would give incompatible bounds on the same weighted squared sum. This contradiction establishes gain consistency before stability.
发表机构
- Institute for Advanced Study, Shenzhen University（深圳大学高等研究院）
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