Steady state programming of controlled nonlinear systems via deep dynamic mode decomposition

Hasnain, Aqib; Boddupalli, Nibodh; Balakrishnan, Shara; Yeung, Enoch

doi:10.23919/acc45564.2020.9147218

Cited by 5 publications

(3 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…They have been used for observer design within the Koopman canonical transform (Surana 2016, Surana & Banaszuk 2016 and within the Koopman reduced-order nonlinear identification and control framework (Kaiser et al 2021), which both typically yield a global bilinear representation of the underlying system. Subsequent research has focused on directly identifying Koopman eigenfunctions , Pan et al 2021) and approximate invariant subspaces (Haseli & Cortés 2023).…”

Section: Extensions and Connection With Koopman Operatorsmentioning

confidence: 99%

Residual dynamic mode decomposition: robust and verified Koopmanism

2023

View full text Add to dashboard Cite

Dynamic mode decomposition (DMD) describes complex dynamic processes through a hierarchy of simpler coherent features. DMD is regularly used to understand the fundamental characteristics of turbulence and is closely related to Koopman operators. However, verifying the decomposition, equivalently the computed spectral features of Koopman operators, remains a significant challenge due to the infinite-dimensional nature of Koopman operators. Challenges include spurious (unphysical) modes and dealing with continuous spectra, which both occur regularly in turbulent flows. Residual dynamic mode decomposition (ResDMD), introduced by Colbrook & Townsend (Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems. 2021. arXiv:2111.14889), overcomes such challenges through the data-driven computation of residuals associated with the full infinite-dimensional Koopman operator. ResDMD computes spectra and pseudospectra of general Koopman operators with error control and computes smoothed approximations of spectral measures (including continuous spectra) with explicit high-order convergence theorems. ResDMD thus provides robust and verified Koopmanism. We implement ResDMD and demonstrate its application in various fluid dynamic situations at varying Reynolds numbers from both numerical and experimental data. Examples include vortex shedding behind a cylinder, hot-wire data acquired in a turbulent boundary layer, particle image velocimetry data focusing on a wall-jet flow and laser-induced plasma acoustic pressure signals. We present some advantages of ResDMD: the ability to resolve nonlinear and transient modes verifiably; the verification of learnt dictionaries; the verification of Koopman mode decompositions; and spectral calculations with reduced broadening effects. We also discuss how a new ordering of modes via residuals enables greater accuracy than the traditional modulus ordering (e.g. when forecasting) with a smaller dictionary. This result paves the way for more significant dynamic compression of large datasets without sacrificing accuracy.

show abstract

Section: Extensions and Connection With Koopman Operatorsmentioning

confidence: 99%

Residual dynamic mode decomposition: robust and verified Koopmanism

2023

View full text Add to dashboard Cite

show abstract

“…Ref. [25] showed that Koopman operators could be used to formulate steady-state control of large-scale biological networks as a data-driven convex program, while Ref. [26] showed that cell growth could be modeled and predicted using Koopman operators that acted on temporal embeddings.…”

Section: Introductionmentioning

confidence: 99%

Stability Analysis of Parameter Varying Genetic Toggle Switches Using Koopman Operators

Harrison

Yeung²

2021

Mathematics

Self Cite

View full text Add to dashboard Cite

The genetic toggle switch is a well known model in synthetic biology that represents the dynamic interactions between two genes that repress each other. The mathematical models for the genetic toggle switch that currently exist have been useful in describing circuit dynamics in rapidly dividing cells, assuming fixed or time-invariant kinetic rates. There is a growing interest in being able to model and extend synthetic biological function for growth conditions such as stationary phase or during nutrient starvation. As cells transition from one growth phase to another, kinetic rates become time-varying parameters. In this paper, we propose a novel class of parameter varying nonlinear models that can be used to describe the dynamics of genetic circuits, including the toggle switch, as they transition from different phases of growth. We show that there exists unique solutions for this class of systems, as well as for a class of systems that incorporates the microbial phenomena of quorum sensing. Further, we show that the domain of these systems, which is the positive orthant, is positively invariant. We also showcase a theoretical control strategy for these systems that would grant asymptotic monostability of a desired fixed point. We then take the general form of these systems and analyze their stability properties through the framework of time-varying Koopman operator theory. A necessary condition for asymptotic stability is also provided as well as a sufficient condition for instability. A Koopman control strategy for the system is also proposed, as well as an analogous discrete time-varying Koopman framework for applications with regularly sampled measurements.

show abstract

“…Finding an infinite dimensional (linear) operator is computationally infeasible, hence many approaches are devised to best approximate the Koopman operator in finite dimensional space [3,33,34,12,35,36,37,38,39,40,41]. Most popular methods include dynamic mode decomposition (DMD) [3,33]; extended dynamic mode decomposition (E-DMD) [12,42]; kernel dynamic mode decomposition (K-DMD) [34], naturally structured dynamic mode decomposition (NS-DMD) [43], Hankel-DMD [17], deep dynamic mode decomposition (deep-DMD) [6,42,44,45,46]. All these methods are data-driven and accuracy of some such approximations are discussed in [47].…”

Section: Introductionmentioning

confidence: 99%

Data-Driven Operator Theoretic Methods for Global Phase Space Learning

Nandanoori

Sinha

Yeung

2020

2020 American Control Conference (ACC)

Self Cite

View full text Add to dashboard Cite

This paper uses data-driven operator theoretic approaches to explore the global phase space of a dynamical system.We defined conditions for discovering new invariant subspaces in the state space of a dynamical system starting from an invariant subspace based on the spectral properties of the Koopman operator. When the system evolution is known locally in several invariant subspaces in the state space of a dynamical system, a phase space stitching result is derived that yields the global Koopman operator. Additionally, in the case of equivariant systems, a phase space stitching result is developed to identify the global Koopman operator using the symmetry properties between the invariant subspaces of the dynamical system and time-series data from any one of the invariant subspaces. Finally, these results are extended to topologically conjugate dynamical systems; in particular, the relation between the Koopman tuple of topologically conjugate systems is established. The proposed results are demonstrated on several second-order nonlinear dynamical systems including a bistable toggle switch. Our method elucidates a strategy for designing discovery experiments: experiment execution can be done in many steps, and models from different invariant subspaces can be combined to approximate the global Koopman operator.

show abstract

Steady state programming of controlled nonlinear systems via deep dynamic mode decomposition

Cited by 5 publications

References 37 publications

Residual dynamic mode decomposition: robust and verified Koopmanism

Residual dynamic mode decomposition: robust and verified Koopmanism

Stability Analysis of Parameter Varying Genetic Toggle Switches Using Koopman Operators

Data-Driven Operator Theoretic Methods for Global Phase Space Learning

Contact Info

Product

Resources

About