Unified optimization of \hbox {H}_{\infty } index and upper stability bound for singularly perturbed systems

In this paper, unified optimization problem for the upper stability bound \(\varepsilon ^{*}\) and the \(\hbox {H}_{\infty }\) performance index \(\gamma \) based on state feedback is considered for singularly perturbed systems. First, a sufficient condition for the existence of state feedback controller is presented in terms of linear matrix inequalities such that the resulting closed-loop system is asymptotically stable if \(0<\varepsilon <\varepsilon ^{*}\) and also guarantees \(\hbox {H}_{\infty }\) performance index. Furthermore, a new algorithm to optimize these two indices simultaneously is proposed based on Nash game theory which transfers multi-objective problem into a single objective problem as well we determines the objective weights. Then an optimal state feedback controller can be derived. Finally, some numerical examples are provided to demonstrate the effectiveness and correctness of the proposed results.