I direct my research towards exploring new mathematical methods and tools for the characterization, control, and monitoring of complex systems with a particular focus on application in engineering and bioengineering. I contributed in several areas, both in fundamental topics and applications. For instance, the fundamental contributions consist of developing and analyzing fractional models, designing asymptotic and non-asymptotic estimation methods for different classes of systems, and developing a quantum signal analysis tool for the efficient processing of pulse-shaped signals. From an application point of view, I developed new mathematical models and control and estimation methods for the optimization of water desalination and wastewater systems. I also developed new mathematical models and analysis tools for sensing and monitoring biomedical signals such as cardiovascular and neuro signals
Core axes:
My main research activities are in the development of estimation methods and algorithms for different classes of systems including, nonlinear Ordinary Differential Equations (ODEs), Differential Algebraic Equations (DAEs), Partial Differential Equations (PDEs) and Fractional Differential Equations (FDEs). The choice of these equations has been influenced by their ability to describe a wide range of real-world phenomena where the estimation problems are important but challenging. I have been interested in two types of estimation approaches; asymptotic estimation based on observers and non-asymptotic estimation based on modulating functions and more recently on observers with prescribed- time convergence properties. In addition, I have been interested in developing signal analysis tools based on the spectrum of the Schrodinger operator.
1- Observer-based estimation:
Observers are well-known in control theory for state estimation in finite dimensional dynam- ical systems. Given the distinctive feature and the main advantage of operating recursively, over the past few years, there has been a continuous effort to design observer-like algorithms for the solution of estimation problems for different classes of systems. We developed observers for state, parameters and input estimation for nonlinear ODE, PDE, coupled ODE/PDE, DAE and fractional PDE.
2- Non-asymptotic estimation methods:
With my team, we have been interested in non-asymptotic estimation for different classes of systems, with a particular interest in modulating functions-based methods given its advantages of providing nonasymptotic estimation, simplifying the problem into an algebraic system, and its robustness against noise. We studied the mathematical properties of this method and extended the concept to different classes of systems including partial differential equations and fractional differential equations.
3- Adaptive signal analysis based on the squared eigenfunctions of a Schrodinger operator:
I work on developing a new adaptive signal/image reconstruction, analysis, and denoising method where the signal is decomposed into signal-dependent functions. These functions are the L2-normalized squared eigenfunctions associated with the discrete spectrum of a Schrodinger operator, the potential of which is considered to be the signal/image. These signal-dependent functions provide a good approximation of the signal and exhibit interesting localization properties. They supply new parameters that can be used to extract relevant features of signal variations. They also constitute an efficient analysis tool as the information about the signal is continuously reflected in these localized functions. We named this method SCSA for Semi-Classical Signal Analysis. Several publications, software, and patents resulted from this project.
Application axes:
Neurophysiological Signals’ Assessment and Monitoring
Monitoring Athletes’ Stress
Prediction and Personalized Treatment of Vascular Diseases