Estimation Methods for Infinite-Dimensional Systems Applied to the Hemodynamic Response in the Brain
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AbstractInfinite-Dimensional Systems (IDSs) which have been made possible by recent advances in mathematical and computational tools can be used to model complex real phenomena. However, due to physical, economic, or stringent non-invasive constraints on real systems, the underlying characteristics for mathematical models in general (and IDSs in particular) are often missing or subject to uncertainty. Therefore, developing efficient estimation techniques to extract missing pieces of information from available measurements is essential. The human brain is an example of IDSs with severe constraints on information collection from controlled experiments and invasive sensors. Investigating the intriguing modeling potential of the brain is, in fact, the main motivation for this work. Here, we will characterize the hemodynamic behavior of the brain using functional magnetic resonance imaging data. In this regard, we propose efficient estimation methods for two classes of IDSs, namely Partial Differential Equations (PDEs) and Fractional Differential Equations (FDEs). This work is divided into two parts. The first part addresses the joint estimation problem of the state, parameters, and input for a coupled second-order hyperbolic PDE and an infinite-dimensional ordinary differential equation using sampled-in-space measurements. Two estimation techniques are proposed: a Kalman-based algorithm that relies on a reduced finite-dimensional model of the IDS, and an infinite-dimensional adaptive estimator whose convergence proof is based on the Lyapunov approach. We study and discuss the identifiability of the unknown variables for both cases. The second part contributes to the development of estimation methods for FDEs where major challenges arise in estimating fractional differentiation orders and non-smooth pointwise inputs. First, we propose a fractional high-order sliding mode observer to jointly estimate the pseudo-state and input of commensurate FDEs. Second, we propose a modulating function-based algorithm for the joint estimation of the parameters and fractional differentiation orders of non-commensurate FDEs. Sufficient conditions ensuring the local convergence of the proposed algorithm are provided. Subsequently, we extend the latter technique to estimate smooth and non-smooth pointwise inputs. The performance of the proposed estimation techniques is illustrated on a neurovascular-hemodynamic response model. However, the formulations are efficiently generic to be applied to a wide set of additional applications.