The $\epsilon$-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 $\epsilon$-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 $\epsilon$-capacity of the gain matrix K. The set of all possible gain matrix that makes the system insensitive to all disturbances is noted $\Phi\cdot$. The characterization of $\Phi\cdot$ is investigated, and we propose an algorithmic approach that allows to determine if a control law is belongs to $\Phi\cdot$ or not. Numerical simulations are given.