An infinite series representation for fundamental matrix groups with its application in high-order continuous-time linear systems

In this paper, an infinite series representation is developed for fundamental matrix groups arising in high-order continuous-time linear time-invariant systems. With the aid of this representation, a transition property with respect to the time variable is discovered for the fundamental matrix groups. In addition, a further analysis is provided for the state response of high-order continuous-time linear time-invariant systems with the transition property as a tool. By using this new formula of state response, a model reduction method is presented to design control laws for high-order fully actuated systems with input-delays. It can be observed that the property of full actuation could be fully leveraged due to the usage of the proposed transition property.