Controllability and observability of fractional linear systems with multiple different orders

In this paper, we study the controllability and observability for fractional linear systems with multiple different orders. The controllability and observability Gramian matrix of the system are obtained respectively. Based on non-singularity of the Gramian matrices, the necessary and sufficient conditions that the systems is controllable or observable are obtained. Some existing results on controllability and observability are extended to the case of multi-order fractional linear systems. The results obtained will be important in the research field of fractional dynamical systems.