Cheap and singular controls for linear quadratic regulators
Cheap and singular controls for linear quadratic regulators
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DOI：
10.1109/tac.1987.1104574
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发表时间：
1985-12
期刊：
影响因子：
--
通讯作者：
A. Saberi;P. Sannuti
中科院分区：
文献类型：
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作者：
A. Saberi;P. Sannuti
考虑了一个具有二次代价函数的线性系统，其中一个小参数µ2乘以控制代价。由于控制的廉价性，高增益反馈形式的强控制作用迫使给定系统具有具有高振幅和低振幅相互作用的时间尺度层次的慢速和快速瞬态。通过适当的变量幅度和时间尺度，将这些相互作用归一化，并将所考虑的问题分解为几个最小阶的非奇异子问题，每个子问题只属于一个时间尺度。给出了表征最优性能指标、特征值、轨迹和控制变量为μ¿0的极限行为的完整结果。讨论了非唯一性进入奇异控制的所有不同方法。更重要的是，这里开发的方法允许在设计几个低阶定义良好的子问题方面设计高增益反馈系统。从而为理论的实际应用提供了强大的推动力。
A linear system with quadratic cost function where a small parameter µ2 multiplies the control cost is considered. Due to the cheapness of control, a strong control action in the form of high-gain feedback forces the given system to have slow and fast transients of a hierarchy of time-scales coupled with high and low amplitude interaction. By appropriate amplitude and time-scaling of variables, these interactions are normalized and the considered problem is decomposed into several nonsingular subproblems of minimal order, each pertaining to only one time-scale. Complete results characterizing the limiting behavior of optimal performance index, eigenvalues, trajectory and control variables as µ ¿ 0 are given. All the different ways by which nonuniqueness can pop into singular control are discussed. More importantly, the method developed here allows the design of a high-gain feedback system in terms of the design of several lower order well defined subproblems. Thus it gives a strong impeteus for practical implementation of the theory developed.