Koopman Spectrum for Cascaded Systems

This paper was co-written by our founder and lead-scientist, Igor Mezic along with Aimdyn scientist and colleague, Ryan Mohr.


This paper considers the evolution of Koopman principal eigenfunctions of cascaded dynamical sys- tems. If each component subsystem is asymptotically stable, the matrix norms of the linear parts of the component subsystems are strictly increasing, and the component subsystems have disjoint spectrums, there exist perturbation functions for the initial conditions of each component subsystem such that the orbits of the cascaded system and the decoupled component subsystems have zero asymptotic relative error. This implies that the evolutions are asymptotically equivalent; cascaded compositions of stable systems are stable. These results hold for both cascaded systems with linear component subsystem dy- namics and linear coupling terms and nonlinear cascades topologically conjugate to the linear case. We further show that the Koopman principal eigenvalues of each component subsystem are also Koopman eigenvalues of the cascaded system. The corresponding Koopman eigenfunctions of the cascaded system are formed by extending the domain of definition of the component systems’ principal eigenfunctions and then composing them with the perturbation function.



AIMdyn Scientists Analyze Data Driven Modal Decompositions

The Dynamic Mode Decomposition (DMD) is a tool of trade in computational data driven analysis of fluid flows. More generally, it is a computational device for Koopman spectral analysis of nonlinear dynamical systems, with a plethora of applications in applied sciences and engineering. Its exceptional performance triggered developments of several modifications that make the DMD an attractive method in data driven framework. This work offers further improvements of the DMD to make it more reliable, and to enhance its functionality. In particular, data driven formula for the residuals allows selection of the Ritz pairs, thus providing more precise spectral information of the underlying Koopman operator, and the well-known technique of refining the Ritz vectors is adapted to data driven scenarios. Further, the DMD is formulated in a more general setting of weighted inner product spaces, and the consequences for numerical computation are discussed in detail. Numerical experiments are used to illustrate the advantages of the proposed method, designated as DDMD_RRR (Refined Rayleigh Ritz Data Driven Modal Decomposition).

Aimdyn Named a Top Program by Air Force Office of Scientific Research

Aimdyn was recently listed in the Air Force Office of Scientific Research’s publication as one of their top research programs from 1951 to present day.

The honor was bestowed upon Aimdyn after a research project led by Aimdyn founder Dr. Igor Mezic directly assisted in the discovery of new methods for passive mitigation of thermoacoustic instabilities in jet engines.

This and other related successes were foundational in forming the UTRC, UCSB. Caltech, Stanford, Princeton, Yale, PlainSight and Aimdyn team that won a 2006 DARPA project where 40 researchers attempted to use new methods or ergodic theory, operator theory, and spectral graph theory to develop methodology and tools for analysis and design of complex dynamical systems robust to uncertainty.

Major a accomplishments of the project were new methods for quantifying uncertainty in dynamical systems that were much faster than previous known methods as well as the discovery of new, efficient unmanned aerial vehicle search methods that had near-linear scaling in system size.

These methods are currently being used in UTRC’s Autonomous and Intelligent Systems program.

AFOSR logo

AIMdyn Team Speaks at 2017 SIAM Conference 

This week AIMdyn’s team of scientists will present their research at the 2017 SIAM Conference on Applications of Dynamical Systems.

The conference will be held from May 21 to May 25 and features presentations from a large number of scientists who are experts in their respective fields.

Featured below we have a brief description of our teams’ material for the conference as well as their speaking times.

Extensions of Koopman operator theory: Stochastic Dynamical Systems and Partial Differential Equations

Speaker: Igor Mezic

Talk time: Tuesday, May 23, 8:30 – 8:55 AM

Symposium: MS79 Koopman Operator Techniques in Dynamical Systems: Theory

Abstract: We extend the theory of Koopman operators to the case of random dynamical systems. We show that the stochastic Koopman operator for random linear systems, defined using expectation over the state space of the underlying random process, admits eigenfunctions related to expectation of eigenvectors of the system. We use this to, via conjugacy, provide a theory for a large class of random dynamical systems. We also describe an extension of the Koopman operator theory for partial differential equations, where we define the notion of eigenfunctionals of the Koopman operator and describe consequences for reduced order modeling of the dynamics.

Koopman Spectrum for Cascaded Dynamical Systems

Speaker: Ryan Mohr

Co-author: Igor Mezic

Talk time: Tuesday, May 23, 9:00 – 9:25 AM

Symposium: MS79 Koopman Operator Techniques in Dynamical Systems: Theory

Abstract: We investigate the Koopman operator spectrum for cascaded dynamical systems – systems constructed by wiring subsystems together such that downstream systems do not influence upstream systems. Under precise conditions on the spectrums of the subsystems and norms of the matrices, we show that the cascaded system is asymptotically equivalent – with zero asymptotic relative error – to the decoupled system started from a perturbed initial condition. We use these results to show that the Koopman principal eigenfunctions of each subsystem can be extended to eigenfunctions for the cascaded system by composing them with the perturbation function. Using a topological conjugacy argument, we show that these results hold for nonlinear cascaded systems as well.

Baroreflex Physiology Using Koopman Mode Analysis

Speaker: Maria Fonoberova

Co-authors: Igor Mezic, Senka Macesic, Nelida Crnjaric-Zic, Zlatko Drmac, Aleksander Andrejcuk

Talk time: Thursday, May 25, 2:45 – 3:10 PM

Symposium: MS156 Applications of Koopman Operator Theory in Dynamical Systems: 

Abstract: We propose new methods for the evaluation of eigenvalues of the Koopman operator family of the non-autonomous dynamic systems. The first step in the development is a new data-driven method for very accurate evaluation of eigenvalues in the hybrid linear non-autonomous case. Then, the approach is extended to continuous linear non-autonomous systems and non-autonomous systems in general. We also propose a relationship between eigenvalues and eigenvalues computed by Arnoldi-like methods on large sets of snapshots. We apply the new approach to baroreflex physiology, i.e. to the resonant breathing which is used in PTSD treatment. For the resonant breathing, we have both data and parameterized mathematical model. The model incorporates a delay and thus is infinite dimensional, hybrid, with a stochastic input. Its asymptotic dynamics is close to quasi-periodic. The applied new methods give an improved insight to the eigenvalues of the related Koopman operator family.


Aimdyn Meets with DARPA and Other Agencies to Discuss Dynamical Data Analysis

The meeting was called, “A Data Driven Operator Theoretic Framework for Space-time Analysis of Process Dynamics,” and funded by DARPA. 

There were participants from Princeton, Courant Institute of the New YorkUniversity, University of Washibgton, and visitors from e.g. the Hughes Research Labs, the United Technologies Research Center and the Institute for Disease Modeling funded by the Gates Foundation. 

The exciting meeting lasted from 8 in the morning to 7 in the evening, and the participants discussed the directions they are pursuing and progres they have made in the operator-theoretic approach to dynamical data analysis. Examples were drawn from human physiology data, fluid dynamics measurements, combustion, climate and other applications. The project continues with high intensity and numerous collaborations.