# Robustness Estimation of Large Deviations in Linear Discrete-time Systems with Control Signal Delay

### Nina Vunder, Natalia Dudarenko

#### Abstract

The article deals with robustness estimation of large deviations in free motion of linear discrete-time systems to parameter variations of the state matrix. A tracking discrete-time system with the modal control law is considered in the paper. The modal control law is designed taking into account the value of delay and the deviation. It is assumed that parameters of the system are linearly dependent on the uncertainties. The problem is solved with the state space approach and the sensitivity theory methods. An upper bound estimation of trajectory deviations for discrete-time systems is obtained. The estimation contains the condition number of the eigenvectors matrix of the system state matrix. Therefore, sensitivity functions of singular values of the eigenvectors matrix are used to calculate the robustness estimation of the deviations. Based on the obtained equations, an algorithm for the robustness estimation of large deviations in linear discrete-time systems with parametric uncertainties is proposed. Two cases of control signal delay are considered in the paper. The first case relates to predictable delay of control signal, and the second one relates to unpredictable delay of control signal. The results are supported with an examples.

Download