Simultaneous design of $H_{2}$ controller and observer in the presence of disturbance for linear systems

In this work, we design a simultaneous H
₂
controller and a state observer for linear systems in the presence of disturbance using linear matrix inequality (LMI) technique. A linear state feedback is proposed to stabilize the states of the control system. In addition, the problem of stabilizing the system states and optimizing the observer design by computing H
₂
is combined with the linear matrix inequality method. After obtaining state controller and observer values using LMI and applying them to the system, it is observed that stability and performance of the closed loop system against disturbance are guaranteed. The simulation results of two numerical examples are presented to show the effectiveness and feasibility of the presented method in this article.