Stabilization of linear systems with unknown dynamics is a canonical problem
in adaptive control. Since the lack of knowledge of system parameters can cause
it to become destabilized, an adaptive stabilization procedure is needed prior
to regulation. Therefore, the adaptive stabilization needs to be completed in
finite time. In order to achieve this goal, asymptotic approaches are not very
helpful. There are only a few existing non-asymptotic results and a full
treatment of the problem is not currently available.
In this work, leveraging the novel method of random linear feedbacks, we
establish high probability guarantees for finite time stabilization. Our
results hold for remarkably general settings because we carefully choose a
minimal set of assumptions. These include stabilizability of the underlying
system and restricting the degree of heaviness of the noise distribution. To
derive our results, we also introduce a number of new concepts and technical
tools to address regularity and instability of the closed-loop matrix.

利用随机线性反馈的新方法，本文针对具有未知动态的线性系统稳定问题，通过随机方法提供了稳定的有限时间概率保证。

有限时间自适应 LQ 系统稳定化

Finite Time Adaptive Stabilization of LQ Systems

We consider adaptive control of the Linear Quadratic Regulator (LQR), where
an unknown linear system is controlled subject to quadratic costs. Leveraging
recent developments in the estimation of linear systems and in robust
controller synthesis, we present the first provably polynomial time algorithm
that provides high probability guarantees of sub-linear regret on this problem.
We further study the interplay between regret minimization and parameter
estimation by proving a lower bound on the expected regret in terms of the
exploration schedule used by any algorithm. Finally, we conduct a numerical
study comparing our robust adaptive algorithm to other methods from the
adaptive LQR literature, and demonstrate the flexibility of our proposed method
by extending it to a demand forecasting problem subject to state constraints.

本文提出了一种自适应控制的方法，可用于处理 Linear Quadratic Regulator 中未知的线性系统和需求预测的问题，算法的时间复杂度为多项式级别，且在控制中有很好的保障。

线性二次型调节器的鲁棒自适应控制遗憾界

Regret Bounds for Robust Adaptive Control of the Linear Quadratic  Regulator

We study the problem of adaptive control of a high dimensional linear
quadratic (LQ) system. Previous work established the asymptotic convergence to
an optimal controller for various adaptive control schemes. More recently, for
the average cost LQ problem, a regret bound of ${O}(\sqrt{T})$ was shown, apart
form logarithmic factors. However, this bound scales exponentially with $p$,
the dimension of the state space. In this work we consider the case where the
matrices describing the dynamic of the LQ system are sparse and their
dimensions are large. We present an adaptive control scheme that achieves a
regret bound of ${O}(p \sqrt{T})$, apart from logarithmic factors. In
particular, our algorithm has an average cost of $(1+\eps)$ times the optimum
cost after $T = \polylog(p) O(1/\eps^2)$. This is in comparison to previous
work on the dense dynamics where the algorithm requires time that scales
exponentially with dimension in order to achieve regret of $\eps$ times the
optimal cost.
We believe that our result has prominent applications in the emerging area of
computational advertising, in particular targeted online advertising and
advertising in social networks.

研究高维线性二次（LQ）系统的自适应控制问题，提出一种实现遗憾界为 O (p√T) 的自适应控制方案，并指出该方法在计算广告领域具有突出的应用价值。

高维线性二次系统的高效强化学习

Efficient Reinforcement Learning for High Dimensional Linear Quadratic  Systems

A technique is introduced for estimating unknown parameters when time series
of only one variable from a multivariate nonlinear dynamical system is given.
The technique employs a combination of two different control methods, a linear
feedback for synchronizing system variables and an adaptive control. The
technique is shown to work even when the unknown parameter appears in the
evolution equations of the variables other than the one for which the time
series is given. The technique not only esablishes that explicit detailed
information about all system variables and parameters is contained in a scalar
time series, but also gives a way to exract it out under suitable conditions.
Illustrations are presented and effect of noise is studied.

本文介绍了一种通过非线性动力系统中单变量时间序列来估计未知参数的技术，该技术采用两种不同的控制方法，线性反馈和自适应控制，可以即使当未知参数出现在除所给的时间序列之外的其他变量的演化方程式中时也能工作。本技术不仅确认了关于所有系统变量和参数的显式详细信息都包含在一个标量时间序列中，而且还提供了在适当条件下提取出这些信息的方法。最后，本文给出了实例并研究了噪声的影响。