线性连续系统具有时间特性的控制能力
Control Ability with Time Attributy for Linear Continous-time Systems
摘要：
本文定义了线性连续时间（LCT）系统具有时间属性的控制能力，并通过能控域的体积计算进行了分析。首先，提出并证明了LCT系统的开环控制能力、控制策略空间（\textit{i.e.}，控制问题的输入变量的解空间）和某些闭环性能之间的关系定理。这一定理说明了对实际工程问题进行控制能力优化的必要性。其次，讨论了有限时间能控域的低计算复杂度的递归体积计算算法。最后，推导了具有$n$不同和重复实特征值的系统无穷时间能控域的两种解析体积计算，并通过解构体积计算方程，构造了3类形状因子.这些解析的体积和形状因子可以精确地描述可控区域的大小和形状。由于LCT系统的时间属性控制能力与单位输入变量的可控域直接相关，基于这些体积和形状因子的解析表达式，可以方便地计算和优化LCT系统的时间属性控制能力.
英文摘要：
In this paper, the control ability with time attributy for the linear continuous-time (LCT) systems are defined and analyzed by the volume computing for the controllability region. Firstly, a relation theorem about the open-loop control ability, the control strategy space (\textit{i.e.}, the solution space of the input variable for control problems), and the some closed-loop performance for the LCT systems is purposed and proven. This theorem shows us the necessity to optimize the control ability for the practical engineering problems. Secondly, recurssive volume-computing algorithms with the low computing complexities for the finite-time controllability region are discussed. Finally, two analytical volume computations of the infinite-time controllability region for the systems with $n$ different and repeated real eigenvalues are deduced, and then by deconstructing the volume computing equations, 3 classes of the shape factors are constructed. These analytical volume and shape factors can describe accurately the size and shape of the controllability region. Because the time-attribute control ability for LCT systems is directly related to the controllability region with the unit input variables, based on these analytical expressions on the volume and shape factors, the time-attribute control ability can be computed and optimized conveniently.
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