The $ε$-capacity of a gain matrix and tolerable disturbances: Discrete-time perturbed linear systems

Discrete-time linear systems with perturbed initial state are considered. A disturbance that infects the initial state is said to be ε-tolerable if the corresponding output signal is relatively insensitive to their effects. In this paper, we will define a new set that characterize each gain matrix K and the associated feedback control law ui = Kxi , this set will be called the ε-capacity of the gain matrix K. The set of all possible gain matrix that makes the system insensitive to all disturbances is noted Φ. The characterization of Φ is investigated, and we propose an algorithmic approach that allows to determine if a control law is belongs to Φ or not. Numerical simulations are given.