Optimal Steering of a Linear Stochastic System to a Final Probability Distribution, Part II
Optimal Steering of a Linear Stochastic System to a Final Probability Distribution, Part II
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DOI：
10.1109/tac.2015.2457791
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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
我们讨论了以最小能量将线性随机系统的状态引导到有限水平上的指定分布的问题，以及以最小功率将状态维持在无限水平上的平稳分布的问题。对于这两个问题，允许控制和高斯噪声通道是不同的，因此，本文的结果超出了以前在概率和控制方面的工作的范围。本文的第一部分讨论了扰动和控制通过同一通道进入的特殊情况。在此，我们用有限水平情形下的动力耦合Riccati方程组和平稳情形的代数条件给出了最优性的充分条件。然后，我们讨论了这两个问题的可行性问题。对于有限水平情形，只要系统是可控的，我们证明了在没有对随机扰动方向性的任何限制的情况下，总是可以在任何指定的有限时间间隔内将状态引导到任意的高斯分布。对于定常无限视界的情况，通过恒定的状态反馈将状态保持在任意的高斯分布并不总是可能的。结果表明，可容许平稳高斯分布的协方差可用一个类Lyapunov方程来刻画，实际上，它们与以合适的平稳有色噪声作为输入所能获得的一类平稳状态协方差重合。最后，我们讨论了如何在数值上计算合适的控制的问题。我们提出了一种求解耦合Riccati方程组的方法，将最优控制表示为两种情况下的(凸)半定规划的解的形式。最后，我们用一个例子来引导惯性粒子分布的状态协方差在有限区间内服从一个可接受的平稳高斯分布，然后通过恒定增益状态反馈控制保持在该平稳分布上。
We address the problem of steering the state of a linear stochastic system to a prescribed distribution over a finite horizon with minimum energy, and the problem to maintain the state at a stationary distribution over an infinite horizon with minimum power. For both problems the control and Gaussian noise channels are allowed to be distinct, thereby, placing the results of this paper outside of the scope of previous work both in probability and in control. The special case where the disturbance and control enter through the same channels has been addressed in the first part of this work that was presented as Part I. Herein, we present sufficient conditions for optimality in terms of a system of dynamically coupled Riccati equations in the finite horizon case and in terms of algebraic conditions for the stationary case. We then address the question of feasibility for both problems. For the finite-horizon case, provided the system is controllable, we prove that without any restriction on the directionality of the stochastic disturbance it is always possible to steer the state to any arbitrary Gaussian distribution over any specified finite time-interval. For the stationary infinite horizon case, it is not always possible to maintain the state at an arbitrary Gaussian distribution through constant state-feedback. It is shown that covariances of admissible stationary Gaussian distributions are characterized by a certain Lyapunov-like equation and, in fact, they coincide with the class of stationary state covariances that can be attained by a suitable stationary colored noise as input. We finally address the question of how to compute suitable controls numerically. We present an alternative to solving the system of coupled Riccati equations, by expressing the optimal controls in the form of solutions to (convex) semi-definite programs for both cases. We conclude with an example to steer the state covariance of the distribution of inertial particles to an admissible stationary Gaussian distribution over a finite interval, to be maintained at that stationary distribution thereafter by constant-gain state-feedback control.