Degree of controllability for linear unstable systems
Degree of controllability for linear unstable systems
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DOI：
10.1177/1077546314545101
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发表时间：
2016-04
影响因子：
2.8
通讯作者：
Haemin Lee;Youngjin Park
中科院分区：
文献类型：
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作者：
Haemin Lee;Youngjin Park
可控度是检验给定系统可控程度的度量，弥补了传统可控性检验方法的不足。可控程度的定义有很多种，最广泛使用的衡量标准是可控性格拉米安。该措施的物理意义是改变状态的最小输入能量。然而，这种方法很难用于不稳定系统，因为格拉米安的稳态值无法定义。因此，本文提出了不稳定系统的可控度。该措施的定义是将状态从任意初始条件改变为任意最终条件的最小控制输入能量。推导结果表明，测度是两项之和，一项用稳定子系统的格拉米亚表示，另一项用具有反稳定子系统镜像极点的子系统的格拉米亚表示。这些格拉米矩阵可以通过求解两个李亚普诺夫方程来获得。一个简单的数值例子显示了所提出的可控程度测量的有用性。
The degree of controllability is a measure to check how controllable a given system is, which makes up for the weak points of traditional methods for checking controllability. There are many definitions of the degree of controllability, and the most widely used measure is represented by controllability Gramian. This measure has physical meaning of minimum input energy to change the states. However, it is hard to use this method for unstable systems because the steady-state value of Gramian cannot be defined. Therefore, the degree of controllability for unstable systems is proposed in this paper. The definition of the measure is minimum control input energy to change the state from an arbitrary initial condition to an arbitrary final condition. The derivation result shows that the measure is the sum of two terms, one represented by Gramian of a stable subsystem and the other represented by Gramian of a subsystem that has mirror image poles of an anti-stable subsystem. These Gramians can be obtained by solving two Lyapunov equations. A simple numerical example shows the usefulness of the proposed measure of degree of controllability.