Algebraic Observer Design for Nonlinear General Conformable Time‐Delay Systems with a Common Conformable Operator

ABSTRACT

This paper addresses algebraic observer design for nonlinear time‐delay systems within the framework of the General Conformable Derivative (GCD). Throughout the paper, a common GCD operator—a single pair formed by the parameter and the weighting function —is applied to every component of the state vector; systems mixing distinct GCD operators across state components are beyond the scope of this work and are discussed as a direction for future research. Building on a Luenberger‐type architecture, we construct a Lyapunov–Krasovskii functional tailored to the conformable time‐scale and derive two key lemmas: (i) the exact GCD derivative of the quadratic term, and (ii) a delay‐functional inequality whose exponential weighting is matched to the conformable time‐scale, so that the functional contributes a pure decay term. These ingredients yield a main theorem: an algebraic convex Linear Matrix Inequality (LMI) that guarantees exponential convergence of the estimation error in the conformable time‐scale and provides the observer gain in closed form from the LMI solution. The result covers Lipschitz nonlinearities and fixed delays, recovers the classical (delay‐free) detectability picture as a special case, and clarifies the relationship between structural properties and feasibility‐based certificates. Numerical examples—including a delayed Duffing‐type oscillator with a comparative study against delay‐free designs—illustrate the theory and confirm that the estimation error decays at least at the certified exponential rate in conformable time.