Pareto optimal filter design with hybrid $H_2/H_{\mathrm{\infty }}$ optimization

In this article, we study the Pareto optimal $H_2$ / $H_{\mathrm{\infty }}$ filter design problem for a generalization of discrete-time stochastic systems. By constructing the estimation equation of the given systems with the estimated signal, a filter error estimation system is obtained. The aim is to obtain a gain matrix $K^{\star }$ that optimizes both performance indicators we set. To deal with this problem, two different upper bounds for two performance indicators are given respectively. The optimal problem therefore is transformed into a Pareto optimal problem with linear matrix inequalities ( $LMIs$) which can be addressed through the $LMI$ toolbox in $MATLAB$.