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Operator learning offers a robust framework for approximating mappings between infinite-dimensional function spaces. It has also become a powerful tool for solving inverse problems in the computational sciences. This chapter surveys…

In this paper we give a geometrical framework for the design of observers on finite-dimensional Lie groups for systems which possess some specific symmetries. The design and the error (between true and estimated state) equation are explicit…

In this paper, we study connections between the classical model-based approach to nonlinear system theory, where systems are represented by equations, and the nonlinear behavioral approach, where systems are defined as sets of trajectories.…

We introduce a novel, practically relevant variation of the anomaly detection problem in multi-variate time series: intrinsic anomaly detection. It appears in diverse practical scenarios ranging from DevOps to IoT, where we want to…

Inverse problems are concerned with the reconstruction of unknown physical quantities using indirect measurements and are fundamental across diverse fields such as medical imaging, remote sensing, and material sciences. These problems serve…

Alignment of the strapdown inertial navigation system (INS) has strong nonlinearity, even worse when maneuvers, e.g., tumbling techniques, are employed to improve the alignment. There is no general rule to attack the observability of a…

This work provides a framework for nonlinear model-free control of systems with unknown input-output dynamics, but outputs that can be controlled by the inputs. This framework leads to real-time control of the system such that a feasible…

This work proposes a detectability condition for linear time-varying systems based on the exponential dichotomy spectrum. The condition guarantees the existence of an observer, whose gain is determined only by the unstable modes of the…

A novel approach is introduced for deriving exact solutions to nonlinear systems of ordinary differential equations. This method consists of four parts. In the initial part, the examined nonlinear differential equation system is transformed…

A simply structured distributed observer is described for estimating the state of a discrete-time, jointly observable, input-free, linear system whose sensed outputs are distributed across a time-varying network. It is explained how to…

The paper deals with joint state and parameter estimation for nonlinear continuous-time systems. Based on a guaranteed LPV approximation, the set adaptive observers design problem is solved avoiding the exponential complexity obstruction…

We survey work on the paradigm called "computing by observing." Its central feature is that one considers the behavior of an evolving system as the result of a computation. To this end an observer records this behavior. It has turned out…

Time series interpretation aims to provide an explanation of what is observed in terms of its underlying processes. The present work is based on the assumption that the common classification-based approaches to time series interpretation…

The paper introduces a method to solve inverse problems for hyperbolic systems where the leading order terms are non-linear. We apply the method to the coupled Einstein-scalar field equations and study the question whether the structure of…

Inverse problems are central to a wide range of fields, including healthcare, climate science, and agriculture. They involve the estimation of inputs, typically via iterative optimization, to some known forward model so that it produces a…

This paper introduce the notion of output contraction that expands the contraction notion to the time-varying nonlinear systems with output. It pertains to the systems' property that any pair of outputs from the system converge to each…

This paper addresses the issue of inversion in cases where (1) the observation system is modeled by a linear transformation and additive noise, (2) the problem is ill-posed and regularization is introduced in a Bayesian framework by an a…

In this paper, we develop a representation-theoretic formulation of discrete-time linear systems. We show that such systems are naturally viewed as representations of time groups acting on vector spaces, thereby endowing the state space…

An observability problem for linear autonomous distributed systems in the class of linear operations is considered. A criterion of observability with respect to terminal state has been proved. A connection with observability with respect to…

In this work, sample-based observability of linear discrete-time systems is studied. That is, we consider the case where the system output measurements are not available at every time instance. It is shown that some discrete-time systems…