Modulated-Integral Observer Design with a Virtual Output Derivative Injection Term

We propose an observer design with a modulated integral correction gain. The observer relies on a virtual output derivative term. The observer correction term shifts the virtual output derivative through the modulating function's properties to induce a constant gain term and an integral modulated gain that only depends on the measurement output. The proposed observer is endowed with an exponentially stable observer and relies on the bounded modulating function condition, which arises from the Lyapunov-based stability and the generalization of the Halanay inequalities. Finally, we provide an illustrative example to demonstrate the observer's performance. A comparison with the standard high-gain observer is provided to demonstrate the superiority of the proposed observer in reducing the peaking phenomenon and rejecting the noise.