Finite‐time estimation of second‐order linear time‐invariant systems in the presence of delayed measurement

In this article, we design a nonlinear observer for second‐order linear time‐invariant (LTI) systems in the observable form with measurement delay. The design guarantees the convergence of the error state to the origin within a finite‐time that depends on the initial conditions. To accomplish this, we reformulate the original system into a cascade ODE–PDE system where the PDE part is a transport equation that models the effect of the delay on the output. We construct the nonlinear gains in a way that ensures the error system to be finite‐time stable (FTS). To prove this, we use an invertible backstepping transformation to convert the error system into a target system which is shown to be finite‐time stable using Lyapunov‐based analysis and homogeneity tools. We use the inverse transformation to transfer this property to the error system.