Optimal Steering of a Linear Stochastic System to a Final Probability Distribution, Part I
Optimal Steering of a Linear Stochastic System to a Final Probability Distribution, Part I
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DOI：
10.1109/tac.2015.2457784
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发表时间：
2016-05-01
影响因子：
6.8
通讯作者：
Pavon, Michele
中科院分区：
文献类型：
--
作者：
Chen, Yongxin;Georgiou, Tryphon T.;Pavon, Michele
我们考虑将具有完整状态观察的线性动力系统从状态空间中的初始高斯分布转向具有最小能量控制的最终分布的问题。系统通过控制通道随机驱动；这种系统的一个例子是经历随机“白噪声”强迫的惯性粒子。我们证明目标概率分布总是可以在有限时间内实现。最优控制以状态反馈形式给出，并通过求解一对通过其边界值非线性耦合的微分李亚普诺夫方程来显式计算。鉴于其有吸引力的算法性质，这一结果似乎具有多种潜在应用，例如质量控制、工业过程控制以及纳米机械系统和分子冷却的主动控制。控制端点边缘之间的扩散过程的问题由来已久（薛定谔桥），目前控制线性随机系统的情况构成了这样一个薛定谔桥，用于可能的简并扩散。我们的结果为一般高斯-马尔可夫过程提供了第一个可实现的最优控制形式。提供了用于引导惯性粒子和“冷却”随机振荡器的说明性示例。最终结果直接确立了薛定谔桥的属性，作为线性随机系统当前背景下给定边缘之间最可能的随机演化。这项工作的第二部分（即第二部分）解决了随机激励通过可能与用于控制的通道不同的通道进入的一般情况。
We consider the problem of steering a linear dynamical system with complete state observation from an initial Gaussian distribution in state-space to a final one with minimum energy control. The system is stochastically driven through the control channels; an example for such a system is that of an inertial particle experiencing random "white noise" forcing. We show that a target probability distribution can always be achieved in finite time. The optimal control is given in state-feedback form and is computed explicitly by solving a pair of differential Lyapunov equations that are nonlinearly coupled through their boundary values. This result, given its attractive algorithmic nature, appears to have several potential applications such as to quality control, control of industrial processes, as well as to active control of nanomechanical systems and molecular cooling. The problem to steer a diffusion process between end-point marginals has a long history (Schrodinger bridges) and the present case of steering a linear stochastic system constitutes such a Schrodinger bridge for possibly degenerate diffusions. Our results provide the first implementable form of the optimal control for a general Gauss-Markov process. Illustrative examples are provided for steering inertial particles and for "cooling" a stochastic oscillator. A final result establishes directly the property of Schrodinger bridges as the most likely random evolution between given marginals to the present context of linear stochastic systems. A second part to this work, that is to appear as part II, addresses the general situation where the stochastic excitation enters through channels that may differ from those used to control.