带任意加性和乘性噪声的多目标最优控制
Multi-Objective Optimal Control with Arbitrary Additive and Multiplicative Noise
摘要：
本文研究了具有任意概率分布和给定二阶矩的加性和乘性噪声动力系统的多目标最优控制问题。目标由状态和控制器中的二次约束给出，其中二次型可以是不确定的，因此不一定是凸的。我们表明，该问题可以转化为一个半定的计划，因此凸。该最优化问题是关于用作状态和控制器的协方差阵的某个变量进行优化的。我们证明了仿射控制器是最优的，并且依赖于最优的协方差阵。此外，我们证明了最优控制器是线性的，如果所有的二次型是凸的控制变量。有限和无限地平线的情况下，解决方案。给出了具有加性和乘性噪声的动力系统均方可镇定的一个充要条件。该条件是一个类似李雅普诺夫条件，其解再次由状态和控制变量的协方差阵给出。最后用一个例子说明了结果。
英文摘要：
In this paper, we consider the problem of multi-objective optimal control of a dynamical system with additive and multiplicative noises with given second moments and arbitrary probability distributions. The objectives are given by quadratic constraints in the state and controller, where the quadratic forms maybe indefinite and thus not necessarily convex. We show that the problem can be transformed to a semidefinite program and hence convex. The optimization problem is to be optimized with respect to a certain variable serving as the covariance matrix of the state and the controller. We show that affine controllers are optimal and depend on the optimal covariance matrix. Furthermore, we show that optimal controllers are linear if all the quadratic forms are convex in the control variable. The solutions are presented for both the finite and infinite horizon cases. We give a necessary and sufficient condition for mean square stabilizability of the dynamical system with additive and multiplicative noises. The condition is a Lyapunov-like condition whose solution is again given by the covariance matrix of the state and the control variable. The results are illustrated with an example.
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