Controllability Criteria on Discrete-Time Impulsive Hybrid Systems With Input Delay

Due to the importance of the hybrid systems, the controllability for linear discrete-time impulsive hybrid systems with input delay (DIHSID) is investigated by resorting to the geometric and algebraic analytical methods in this article. First, the null reachability and controllability geometric conditions are studied. Specifically, the null reachable set and controllable set for every impulsive and switching sequence are obtained by the properties of the invariant subspace. Besides, a new subspace sequence is constructed to analyze the null reachability and controllability of DIHSID. Second, the complete controllability of DIHSID is investigated by algebraic methods. In the form of Gramian matrices, several sufficient complete controllability conditions for DIHSID are established without assuming the nonsingularity of all impulsive matrices. Furthermore, a less conservative complete controllability criterion that is necessary and sufficient is developed by introducing a row matrix of some Gramian matrices. Finally, two illustrating examples show the effectiveness of the developed controllability theory.

Automated summary

This article investigates the controllability of linear discrete-time impulsive hybrid systems with input delay (DIHSID) by using geometric and algebraic analytical methods. The geometric conditions for null reachability and controllability are studied, and the null reachable set and controllable set for every impulsive and switching sequence are obtained. A new subspace sequence is constructed to analyze the controllability of DIHSID. The complete controllability of DIHSID is investigated using algebraic methods, and sufficient conditions for complete controllability are established without assuming nonsingularity of all impulsive matrices. Furthermore, a less conservative criterion for complete controllability is developed by introducing a row matrix of some Gramian matrices. Two examples are provided to demonstrate the effectiveness of the developed controllability theory.