Advanced Estimation Algorithms for Nonlinear Systems: Design Methods and Applications

A dynamical system models how a system evolves over time, governed by specific laws or equations. Ensuring the stability of such systems, under a given control action, is vital for predicting their long-term behavior. Stability analysis methods, such as Lyapunov’s theory and Input-to-State Stability (ISS), provide essential tools to assess whether a system will remain stable. When direct measurements of all system states are not possible, estimation techniques, such as observers and estimators, play a crucial role in reconstructing internal states from available measurements, enabling effective feedback control. Then, developing advanced estimation schemes and observer design methods is the main motivation of this thesis. Towards this end, three major contributions are proposed in this thesis, as detailed below: \begin{enumerate} \item Analysis of Robust Moving Horizon Estimation (MHE) Schemes: Since the incremental exponential input/output-to-state stability (i-EIOSS) property is required to synthesize the parameters of the cost function in the MHE context, then, first, two numerical design procedures are proposed to ensure that a nonlinear system achieves i-EIOSS property and to compute the associated i-EIOSS parameters. Then, the robust stability of moving horizon estimation (MHE) is proven. Novel design conditions are established and advanced prediction techniques are introduced. \item Contributions to LMI-based and high-gain-based Observers: This chapter is split into two parts: the first part deals with observer design for nonlinear systems via Linear Matrix Inequalities (LMIs). The main goal consists of showing that for some families of nonlinear systems, the LMI-based observer design techniques always provide exponential convergent observer. The second part deals with high-gain observer methodology. A novel design method is proposed for systems with arbitrary nonlinear structures contrary to the standard results on high-gain observer methodology developed for triangular nonlinearities. \item Contributions to observer design for nonlinear systems with delayed measurements: The main idea behind consists of using a dynamic extension technique to transform a system with delayed nonlinear outputs into a system with linear outputs and a delay-dependent integral term in the dynamic process. Based on this transformation, novel and less conservative synthesis conditions are established, ensuring the exponential convergence of the observer despite the large values of the delay in the output measurements.