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Optimal control of dynamical systems and structures under stochastic uncertainty Stochastic optimal feedback control

机译：随机不确定性下动力系统和结构的最优控制随机最优反馈控制

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摘要

Consider a dynamic mechanical control systems or structure under stochastic uncertainty, as e.g. the active control of a mechanical structure under stochastic applied dynamic loadings. Optimal controls, being most insensitive with respect to random parameter variations, are determined by rinding stochastic optimal controls, i.e., controls minimizing the expected total costs composed of the costs arising along the trajectory, the costs for the control (correction), and possible terminal costs. The problem is modeled in the framework of optimal control under stochastic uncertainty, where the process differential equation depends on certain random parameters having a given probability distribution. Since by computing stochastic optimal controls, random parameter variations are incorporated into the optimal control design, most insensitive or robust controls are obtained. Based on the stochastic Hamiltonian of the optimal control problem under stochastic uncertainty, the class of "H-minimal controls" is determined first by solving a finite-dimensional stochastic program for the minimization of the expected Hamiltonian with respect to the input u(t) at time t. Having a H-minimal control, a two-point boundary value problem with random parameters is formulated for the computation of optimal state-and costate trajectories. Inserting then these trajectories into the H-minimal control, stochastic optimal controls are found, or at least stationary controls satisfying the necessary optimality conditions for a stochastic optimal control. Numerical solutions of the two-point boundary value problem are obtained by (ⅰ) Discretization of the underlying probability distribution of the random parameters, and (ⅱ) Taylor expansion of the expected total costs and the expected Hamiltonian with respect to the random parameter vector at its expectation. The method is illustrated by the stochastic optimal regulation of a robot.
机译：考虑具有随机不确定性的动态机械控制系统或结构，例如在随机施加的动态载荷下对机械结构的主动控制。对随机参数变化最不敏感的最优控制是通过浸入随机最优控制来确定的，即，将期望总成本减至最小的控制包括沿轨迹产生的成本，控制成本（校正）和可能的终端成本费用。在随机不确定性下的最优控制框架中对问题进行建模，其中过程微分方程取决于具有给定概率分布的某些随机参数。由于通过计算随机最优控制，可以将随机参数变化纳入最优控制设计中，因此可以获得大多数不敏感或鲁棒的控制。基于随机不确定性下最优控制问题的随机哈密顿量，首先通过求解有限维随机程序来确定“ H-最小控制”类别，以使预期的哈密顿量相对于输入u（t）最小化在时间t。具有H最小控制，具有随机参数的两点边值问题被公式化为最优状态和代价轨迹的计算。然后将这些轨迹插入到H最小控制中，找到随机最优控制，或者至少找到满足随机最优控制必要的最优性条件的平稳控件。通过（ⅰ）随机参数的潜在概率分布的离散化，以及（ⅱ）相对于随机参数向量的预期总成本和预期哈密顿量的泰勒展开，可以获得两点边值问题的数值解。它的期望。通过机器人的随机最优调节来说明该方法。

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