Department of Electrical Engineering, Science And Research Branch, Islamic Azad University, Tehran, Iran.
ISA Trans. 2012 Jan;51(1):65-73. doi: 10.1016/j.isatra.2011.07.010. Epub 2011 Sep 14.
This paper presents the stabilization problem of a linear time invariant fractional order (LTI-FO) switched system with order 1<q<2 by a single Lyapunov function whose derivative is negative and bounded by a quadratic function within the activation regions of each subsystem. The switching law is extracted based on the variable structure control with a sliding sector. First, a sufficient condition for the stability of an LTI-FO switched system with order 1<q<2 based on the convex analysis and linear matrix inequality (LMI) is presented and proved. Then a single Lyapunov function, whose derivative is negative, is constructed based on the extremum seeking method. A sliding sector is designed for each subsystem of the LTI-FO switched system so that each state in the state space is inside at least one sliding sector with its corresponding subsystem, where the Lyapunov function found by the extremum seeking control is decreasing. Finally, a switching control law is designed to switch the LTI-FO switched system among subsystems to ensure the decrease of the Lyapunov function in the state space. Simulation results are given to show the effectiveness of the proposed VS controller.
本文提出了一种通过单李雅普诺夫函数来解决线性时不变分数阶（LTI-FO）切换系统的稳定性问题，该函数的导数在每个子系统的激活区域内是负的且受二次函数的限制，其中 1<q<2。切换律是基于滑动扇区的变结构控制提取的。首先，基于凸分析和线性矩阵不等式（LMI），给出并证明了具有阶数 1<q<2 的 LTI-FO 切换系统的稳定性的充分条件。然后，基于极值搜索方法构建了一个导数为负的单李雅普诺夫函数。为 LTI-FO 切换系统的每个子系统设计了一个滑动扇区，使得状态空间中的每个状态都至少在一个滑动扇区内，并且对应于相应的子系统，其中极值搜索控制找到的李雅普诺夫函数在减小。最后，设计了切换控制律，以在子系统之间切换 LTI-FO 切换系统，以确保状态空间中李雅普诺夫函数的减小。给出了仿真结果，以验证所提出的 VS 控制器的有效性。
