Koopman-operator Observer-based Estimation of Pedestrian Crowd Flows

Benosman, Mouhacine; Mansour, Hassan; Huroyan, Vahan

doi:10.1016/j.ifacol.2017.08.2428

Cited by 7 publications

(5 citation statements)

References 21 publications

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“…In particular, the nonlinear system is nonlinearly observable if the pair (A, C H ) is observable, which can be determined via the rank condition of the corresponding observability matrix. These ideas have been applied to study pedestrian crowd flow [17], extended further to input-output nonlinear systems [168] resulting in bilinear or Lipschitz formulations, and used to compute controllability and reachability [65]. The observability and controllability Gramians, which are used to examine the degree of observability and controllability, can be computed in the lifted observable space given by a dictionary of functions [193].…”

Section: Observability and Controllabilitymentioning

confidence: 99%

“…This work has been extended to input-output systems yielding Lipschitz or bilinear observers, resulting from the representation of the unforced system in eigenfunction coordinates [168], and for constrained state estimation based on a moving horizon analogous to model predictive control [171]. Building on kernel DMD and the Koopman observer form, a Luenberger Koopman observer has been applied to pedestrian crowd flow to estimate the full state from limited measurements [17]. The probabilistic perspective based on the Perron-Frobenius operator allows one to incorporate uncertainty into observer synthesis.…”

Section: Observer Synthesis For State Estimationmentioning

confidence: 99%

“…Building on kernel DMD and the Koopman observer form, a Luenberger Koopman observer has been applied to pedestrian crowd flow to estimate the full state from limited measurements [17].…”

Section: Observer Synthesis For State Estimationmentioning

confidence: 99%

See 2 more Smart Citations

Data-Driven Approximations of Dynamical Systems Operators for Control

Kaiser

Kutz

Brunton

2020

Lecture Notes in Control and Information Sciences

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The Koopman and Perron Frobenius transport operators are fundamentally changing how we approach dynamical systems, providing linear representations for even strongly nonlinear dynamics. Although there is tremendous potential benefit of such a linear representation for estimation and control, transport operators are infinite-dimensional, making them difficult to work with numerically. Obtaining low-dimensional matrix approximations of these operators is paramount for applications, and the dynamic mode decomposition has quickly become a standard numerical algorithm to approximate the Koopman operator. Related methods have seen rapid development, due to a combination of an increasing abundance of data and the extensibility of DMD based on its simple framing in terms of linear algebra. In this chapter, we review key innovations in the data-driven characterization of transport operators for control, providing a high-level and unified perspective. We emphasize important recent developments around sparsity and control, and discuss emerging methods in big data and machine learning.

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Section: Observability and Controllabilitymentioning

confidence: 99%

Section: Observer Synthesis For State Estimationmentioning

confidence: 99%

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Data-Driven Approximations of Dynamical Systems Operators for Control

Kaiser

Kutz

Brunton

2020

Lecture Notes in Control and Information Sciences

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“…The latter has been introduced by the pioneer paper [47], where the aforementioned perturbation of the transport equation is used to derive diffusion models (parabolic) and models with finite speed of propagation (hyperbolic) [48]. The use of entropy estimate is important not only toward an assessment of the physical properties of the solution, but also toward the interpretation of empirical data [49] for systems, where the available data are often not complete [50].…”

Section: Research Perspectivesmentioning

confidence: 99%

On Entropy Dynamics for Active “Living” Particles

Ełaiw

Alghamdi

Bellomo

2017

Entropy

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This paper presents a modeling approach, followed by entropy calculations of the dynamics of large systems of interacting active particles viewed as living-hence, complex-systems. Active particles are partitioned into functional subsystems, while their state is modeled by a discrete scalar variable, while the state of the overall system is defined by a probability distribution function over the state of the particles. The aim of this paper consists of contributing to a further development of the mathematical kinetic theory of active particles.

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“…The underlying nonlinear system is then considered nonlinearly observable if the pair (A, C H ) is observable, which can be determined via the rank condition of the corresponding observability matrix. These ideas have been applied to study pedestrian crowd flow [26], extended further to input-output nonlinear systems resulting in bilinear or Lipschitz formulations [316], and used to compute controllability and reachability [120]. The observability and controllability Gramians can also be computed in the lifted observable space [317].…”

mentioning

confidence: 99%

Modern Koopman Theory for Dynamical Systems

Brunton¹,

Budišić²,

Kaiser³

et al. 2021

Preprint

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The field of dynamical systems is being transformed by the mathematical tools and algorithms emerging from modern computing and data science. First-principles derivations and asymptotic reductions are giving way to data-driven approaches that formulate models in operator theoretic or probabilistic frameworks. Koopman spectral theory has emerged as a dominant perspective over the past decade, in which nonlinear dynamics are represented in terms of an infinitedimensional linear operator acting on the space of all possible measurement functions of the system. This linear representation of nonlinear dynamics has tremendous potential to enable the prediction, estimation, and control of nonlinear systems with standard textbook methods developed for linear systems. However, obtaining finite-dimensional coordinate systems and embeddings in which the dynamics appear approximately linear remains a central open challenge. The success of Koopman analysis is due primarily to three key factors: 1) there exists rigorous theory connecting it to classical geometric approaches for dynamical systems, 2) the approach is formulated in terms of measurements, making it ideal for leveraging big-data and machine learning techniques, and 3) simple, yet powerful numerical algorithms, such as the dynamic mode decomposition (DMD), have been developed and extended to reduce Koopman theory to practice in real-world applications. In this review, we provide an overview of modern Koopman operator theory, describing recent theoretical and algorithmic developments and highlighting these methods with a diverse range of applications. We also discuss key advances and challenges in the rapidly growing field of machine learning that are likely to drive future developments and significantly transform the theoretical landscape of dynamical systems.

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Koopman-operator Observer-based Estimation of Pedestrian Crowd Flows

Cited by 7 publications

References 21 publications

Data-Driven Approximations of Dynamical Systems Operators for Control

Data-Driven Approximations of Dynamical Systems Operators for Control

On Entropy Dynamics for Active “Living” Particles

Modern Koopman Theory for Dynamical Systems

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