We study controllability and observability concepts of tempered fractional linear systems in the Caputo sense. First, we formulate a solution for the class of tempered systems under investigation by means of the Laplace transform method. Then, we derive necessary and sufficient conditions for the controllability, as well as for the observability, in terms of the Gramian controllability matrix and the Gramian observability matrix, respectively. Moreover, we establish the Kalman criteria that allows one to check easily the controllability and the observability for tempered fractional systems. Applications to the fractional Chua’s circuit and Chua–Hartley’s oscillator models are provided to illustrate the theoretical results developed in this manuscript.

中文翻译：

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回火分数阶差分系统的可控性和可观测性


我们研究了 Caputo 意义上的回火分数线性系统的可控性和可观察性概念。首先，我们通过拉普拉斯变换方法为所研究的回火系统类制定一个解决方案。然后，我们分别根据 Gramian 可控性矩阵和 Gramian 可观察性矩阵推导出可控性和可观察性的必要和充分条件。此外，我们建立了卡尔曼准则，使人们可以轻松检查回火分数系统的可控性和可观察性。本文提供了分数阶 Chua 电路和 Chua-Hartley 振荡器模型的应用，以说明本手稿中开发的理论结果。