Lur´e Lyapunov Function and Absolute Stability Criterion for Lur´e Singularly Perturbed Systems

This technical note investigates the absolute stability problem for Lur´e singularly perturbed systems with multiple nonlinearities. The objective is to determine if the system is absolutely stable for any ε ∈ (0,ε₀), where ε denotes the perturbation parameter and ε₀ is a pre-defined positive scalar. First, an ε-dependent Lur´e Lyapunov function is constructed that facilitates the stability analysis of the singularly perturbed system. Then, a stability criterion expressed in terms of ε-independent linear matrix inequalities (LMIs) is derived. Based on the stability criterion, an algorithm is proposed to compute the stability bound that is shown to be less conservative than those computed using other existing methods. Finally, examples are given to show the feasibility and effectiveness of the obtained method.