This the first time that the problem of global asymptotic stability analysis, and mixed \(H_{\infty }\) and passive control for a class of control fractional-order nonlinear systems has been studied in this paper. By using the Lyapunov direct method and some properties of fractional calculate, we propose sufficient conditions to ensure the unforced system to be asymptotically stable with mixed \(H_{\infty }\) and passivity performance level. Further, mixed \(H_{\infty }\) and passive control design with an appropriate gain matrix has been derived to achieve the stabilization for fractional-order system with nonlinear perturbations and order \(0 < \alpha < 1\). These conditions are in the form of linear matrix inequalities, which therefore can be efficiently solved by using existing convex algorithms. The effectiveness of our results is illustrated through three numerical examples.

本文首次研究了一类控制分数阶非线性系统的全局渐近稳定性分析以及混合（H _ {\ infty} \）和被动控制的问题。通过使用Lyapunov直接方法和分数计算的某些性质，我们提出了充分的条件，以确保无强迫系统在混合\（H _ {\ infty} \）和无源性能水平下能够渐近稳定。此外，已经推导了具有适当增益矩阵的混合\（H _ {\ infty} \）和被动控制设计，以实现具有非线性扰动和阶\（0 <\ alpha <1 \）的分数阶系统的稳定性。。这些条件采用线性矩阵不等式的形式，因此可以通过使用现有的凸算法来有效解决。通过三个数值示例说明了我们的结果的有效性。