Exponential stabilization of quasi-one-sided Lipschitz systems with time delay

This paper aims to design Observer-Based (OB) controllers that ensure the exponential stability of a class of nonlinear time-delay systems. The nonlinear part of the system satisfies a weak Quasi-One-Sided Lipschitz (QOSL) condition characterized by the matrices $({\mathcal{L}}_1,{\mathcal{M}}_1,{\mathcal{N}}_1)$, as well as a QOSL condition characterized by the matrices $({\mathcal{L}}_2,{\mathcal{M}}_2,{\mathcal{N}}_2)$. First, we derive a sufficient condition formulated as a Linear Matrix Inequality (LMI) via a Lyapunov–Krasovskii (LK) functional. The main advantage of this design is that the controller and observer gains are computed in a single step. However, its main drawback is that the matrices ${\mathcal{L}}_1$, ${\mathcal{M}}_1$, and ${\mathcal{N}}_1$ are fixed rather than treated as decision variables. To overcome this limitation, we propose an improved design in which the matrices ${\mathcal{L}}_1,{\mathcal{M}}_1$, and ${\mathcal{N}}_1$ are treated as decision-variable matrices with a fixed structure. By using an appropriate decoupling technique, this approach provides greater flexibility in the selection of matrices and reduces conservatism. The efficacy of the developed OB controllers is demonstrated via a suitable numerical example.