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首页> 外文期刊>IEEE Transactions on Automatic Control >Optimal Steering of a Linear Stochastic System to a Final Probability Distribution, Part II
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Optimal Steering of a Linear Stochastic System to a Final Probability Distribution, Part II

机译：线性随机系统到最终概率分布的最佳控制，第二部分

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摘要

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 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 - articles to an admissible stationary Gaussian distribution over a finite interval, to be maintained at that stationary distribution thereafter by constant-gain state-feedback control.
机译：我们解决了将线性随机系统的状态转向具有最小能量的有限范围内的指定分布的问题，以及将状态保持在具有最小功率的无限范围内的稳态分布上的问题。对于这两个问题，允许将控制通道和高斯噪声通道区分开，从而使本文的结果在概率和控制方面都超出了先前工作的范围。在第一部分介绍的本工作的第一部分中，已经解决了扰动和控制通过相同的通道进入的特殊情况。在此，我们为动态耦合Riccati方程组中的系统提供了最优的充分条件。有限水平情况和平稳情况的代数条件。然后，我们针对两个问题解决的问题。对于有限水平的情况，只要系统是可控制的，我们证明在对随机干扰的方向性没有任何限制的情况下，始终可以在任何指定的有限时间间隔内将状态转向任意高斯分布。对于平稳的无限地平线情况，通过恒定的状态反馈并不总是能够将状态保持在任意的高斯分布。结果表明，可容许的平稳高斯分布的协方差由某个类Lyapunov方程表征，实际上，它们与可以由适当的平稳有色噪声作为输入获得的平稳状态协方差的类别重合。我们最终解决了如何通过数值计算合适的控件的问题。通过以两种情况的（凸）半定程序的解形式表示最优控制，我们提出了一种解决耦合Riccati方程组的方法。我们以一个示例结束，以将惯性物品的状态协方差引导到有限间隔内的允许的平稳高斯分布，然后通过恒定增益状态反馈控制将其保持在该平稳分布。

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