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Extending dynamic mode decomposition to data from multiple outputs

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arXiv 2021年

作者： Boddupalli, Nibodh Department of Mechanical Engineering Center for Control Dynamical Systems and Computation University of California Santa BarbaraCA United States

System identification based on Koopman operator theory has grown in popularity recently. Spectral properties of the Koopman operator of a system were proven to relate to properties like invariant sets, stability, periodicity, etc. of the underlying system. Estimation of these spectral objects has become widely accessible with the numerous algorithms developed in recent years. We show how one such algorithm – extended dynamic mode decomposition – can be used on data from multiple outputs of a system. These outputs that are functions of state are called observables in the literature and could be known outputs like the state itself or unknown outputs like data from sensors used in systems of biological interest. To this end, we approximate the desired observables and their iterates in time using minimizers of regularized least-squares problems which have analytic solutions with heuristic provisions for expected estimates. © 2021, CC BY-SA.

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A Tale of Two Doses: Model Identification and Optimal Vaccination for COVID-19

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A Tale of Two Doses: Model Identification and Optimal Vaccin...

IEEE Conference on Decision and control

作者： Raphael Chinchilla Guosong Yang Murat K. Erdal Ramon R. Costa João P. Hespanha Center for Control Dynamical Systems and Computation University of California Santa Barbara USA Federal University of Rio de Janeiro Brazil

ISBN: (纸本)9781665436601

We address the model identification and the computation of optimal vaccination policies for the coronavirus disease 2019 (COVID-19). We consider a stochastic Susceptible– Infected–Removed (SIR) model that captures the effect of multiple vaccine treatments, each requiring a different number of doses and providing different levels of protection against the disease. We show that the inclusion of vaccination data enables the estimation of the state of the model and key model parameters that are otherwise not identifiable. This estimates can, in turn, be used to design strategic approaches to vaccination that aim at minimizing the number of deaths and the economic cost of the disease. We illustrate these results with numerical examples.

关键词： COVID-19 Costs computational modeling Conferences Stochastic processes Estimation Data models

Non-Euclidean Contraction Theory for Robust Nonlinear Stability

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arXiv 2021年

作者： Davydov, Alexander Jafarpour, Saber Bullo, Francesco The Department of Mechanical Engineering The Center for Control Dynamical Systems and Computation University of California Santa Barbara93106-5070 United States

We study necessary and sufficient conditions for contraction and incremental stability of dynamical systems with respect to non-Euclidean norms. First, we introduce weak pairings as a framework to study contractivity with respect to arbitrary norms, and characterize their properties. We introduce and study the sign and max pairings for the `1 and `∞ norms, respectively. Using weak pairings, we establish five equivalent characterizations for contraction, including the one-sided Lipschitz condition for the vector field as well as logarithmic norm and Demidovich conditions for the corresponding Jacobian. Third, we extend our contraction framework in two directions: we prove equivalences for contraction of continuous vector fields and we formalize the weaker notion of equilibrium contraction, which ensures exponential convergence to an equilibrium. Finally, as an application, we provide (i) incremental input-to-state stability and finite input-state gain properties for contracting systems, and (ii) a general theorem about the Lipschitz interconnection of contracting systems, whereby the Hurwitzness of a gain matrix implies the contractivity of the interconnected system. Copyright © 2021, The Authors. All rights reserved.

关键词： dynamical systems

ONLINE FEEDBACK OPTIMIZATION AND SINGULAR PERTURBATION VIA CONTRACTION THEORY∗

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arXiv 2023年

作者： Cothren, Liliaokeawawa Bullo, Francesco Dall’Anese, Emiliano Department of Electrical Computer and Energy Engineering University of Colorado Boulder BoulderCO United States Center for Control Dynamical Systems and Computation UC Santa Barbara Santa BarbaraCA93101 United States Department of Electrical and Computer Engineering Boston University BostonMA02215 United States

In this paper, we provide a novel contraction-theoretic approach to analyze two-time scale systems, including those commonly encountered in Online Feedback Optimization (OFO). Our framework endows these systems with several robustness properties, enabling a more comprehensive characterization of their behaviors. The primary assumptions are the contractivity of the fast subsystem and the reduced model, along with an explicit upper bound on the time-scale parameter. For two-time scale systems subject to disturbances, we show that the distance between solutions of the nominal system and solutions of its reduced model is uniformly upper bounded by a function of contraction rates, Lipschitz constants, the time-scale parameter, and the variability of the disturbances over time. Applying these general results to the OFO context, we establish new individual tracking error bounds, showing that solutions converge to their time-varying optimizer, provided the plant and steady-state feedback controller exhibit contractivity and the controller gain is suitably bounded. Finally, we explore two special cases: for autonomous nonlinear systems, we derive sharper bounds than those in the general results, and for linear time-invariant systems, we present novel bounds based on induced matrix norms and induced matrix log norms. Copyright © 2023, The Authors. All rights reserved.

关键词： State feedback

Multistability and anomalies in oscillator models of lossy power grids

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arXiv 2022年

作者： Delabays, Robin Jafarpour, Saber Bullo, Francesco Center for Control Dynamical Systems and Computation Uc Santa Barbara Santa BarbaraCA93106-5070 United States School of Electrical and Computer Engineering Georgia Institute of Technology AtlantaGA30332-0250 United States

The analysis of dissipatively coupled oscillators is challenging and highly relevant in power grids. Standard mathematical methods are not applicable, due to the lack of network symmetry induced by dissipative couplings. Here we demonstrate a close correspondence between stable synchronous states in dissipatively coupled oscillators, and the winding partition of their state space, a geometric notion induced by the network topology. Leveraging this winding partition, we accompany this article with an algorithms to compute all synchronous solutions of complex networks of dissipatively coupled oscillators. These geometric and computational tools allow us to identify anomalous behaviors of lossy networked systems. Counterintuitively, we show that loop flows and dissipation can increase the system's transfer capacity, and that dissipation can promote multistability. We apply our geometric framework to compute power flows on the IEEE RTS-96 test system, where we identify two high voltage solutions with distinct loop flows. © 2022, CC BY.

关键词： Complex networks

Modeling and Contractivity of Neural-Synaptic Networks with Hebbian Learning

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arXiv 2022年

作者： Centorrino, Veronica Bullo, Francesco Russo, Giovanni Scuola Superiore Meridionale Naples Italy The Department of Mechanical Engineering The Center for Control Dynamical Systems and Computation University of California Santa Barbara United States The Department of Information and Electric Engineering and Applied Mathematics University of Salerno Italy

This paper is concerned with the modeling and analysis of two of the most commonly used recurrent neural network models (i.e., Hopfield neural network and firing-rate neural network) with dynamic recurrent connections undergoing Hebbian learning rules. To capture the synaptic sparsity of neural circuits we propose a low dimensional formulation. We then characterize certain key dynamical properties. First, we give biologically-inspired forward invariance results. Then, we give sufficient conditions for the non-Euclidean contractivity of the models. Our contraction analysis leads to stability and robustness of time-varying trajectories - for networks with both excitatory and inhibitory synapses governed by both Hebbian and anti-Hebbian rules. For each model, we propose a contractivity test based upon biologically meaningful quantities, e.g., neural and synaptic decay rate, maximum in-degree, and the maximum synaptic strength. Then, we show that the models satisfy Dale's Principle. Finally, we illustrate the effectiveness of our results via a numerical example. Copyright © 2022, The Authors. All rights reserved.

关键词： Hopfield neural networks

Non-Euclidean Monotone Operator Theory and Applications

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arXiv 2023年

作者： Davydov, Alexander Jafarpour, Saber Proskurnikov, Anton V. Bullo, Francesco Center for Control Dynamical Systems and Computation University of California Santa Barbara Santa BarbaraCA93106-5070 United States Department of Electrical Computer and Energy Engineering University of Colorado BoulderCO80309-0020 United States Department of Electronics and Telecommunications Politecnico di Torino Turin10129 Italy

While monotone operator theory is often studied on Hilbert spaces, many interesting problems in machine learning and optimization arise naturally in finite-dimensional vector spaces endowed with non-Euclidean norms, such as diagonally-weighted 1 or ∞ norms. This paper provides a natural generalization of monotone operator theory to finite-dimensional non-Euclidean spaces. The key tools are weak pairings and logarithmic norms. We show that the resolvent and reflected resolvent operators of non-Euclidean monotone mappings exhibit similar properties to their counterparts in Hilbert spaces. Furthermore, classical iterative methods and splitting methods for finding zeros of monotone operators are shown to converge in the non-Euclidean case. We apply our theory to equilibrium computation and Lipschitz constant estimation of recurrent neural networks, obtaining novel iterations and tighter upper bounds via forward-backward splitting. © 2023, CC BY.

关键词： Recurrent neural networks

Topological entropy of switched nonlinear and interconnected systems

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arXiv 2023年

作者： Yang, Guosong Liberzon, Daniel Hespanha, João P. The Department of Electrical and Computer Engineering Rutgers University PiscatawayNJ08854 United States The Coordinated Science Laboratory University of Illinois Urbana-Champaign UrbanaIL61801 United States The Center for Control Dynamical Systems and Computation University of California Santa Barbara Santa BarbaraCA93106 United States

A general upper bound for topological entropy of switched nonlinear systems is constructed, using an asymptotic average of upper limits of the matrix measures of Jacobian matrices of strongly persistent individual modes, weighted by their active rates. A general lower bound is constructed as well, using a similar weighted average of lower limits of the traces of these Jacobian matrices. In a case of interconnected structure, the general upper bound is readily applied to derive upper bounds for entropy that depend only on "network-level" information. In a case of block-diagonal structure, less conservative upper and lower bounds for entropy are constructed. In each case, upper bounds for entropy that require less information about the switching signal are also derived. The upper bounds for entropy and their relations are illustrated by numerical examples of a switched Lotka–Volterra ecosystem *** Codes 37B40, 93C30 (Primary) 93C57 (Secondary) © 2023, CC BY-NC-ND.

关键词： Jacobian matrices

Learning Robust Data-based LQG controllers from Noisy Data

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arXiv 2023年

作者： Liu, Wenjie Sun, Jian Wang, Gang Bullo, Francesco Chen, Jie The National Key Lab of Autonomous Intelligent Unmanned Systems School of Automation Beijing Institute of Technology Beijing100081 China The Beijing Institute of Technology Chongqing Innovation Center Chongqing401120 China The Mechanical Engineering Department The Center of Control Dynamical Systems and Computation UC Santa Barbara CA93106-5070 United States The Department of Control Science and Engineering Tongji University Shanghai201804 China The National Key Lab of Autonomous Intelligent Unmanned Systems Beijing Institute of Technology Beijing100081 China

This paper addresses the joint state estimation and control problems for unknown linear time-invariant systems subject to both process and measurement noise. The aim is to redesign the linear quadratic Gaussian (LQG) controller based solely on data. The LQG controller comprises a linear quadratic regulator (LQR) and a steady-state Kalman observer;while the data-based LQR design problem has been previously studied, constructing the Kalman gain and the LQG controller from noisy data presents a novel challenge. In this work, a data-based formulation for computing the steady-state Kalman gain is proposed based on semi-definite programming (SDP) using some noise-free input-state-output data. Additionally, a data-based LQG controller is developed, which is shown to be equivalent to the model-based LQG controller. For cases where offline data are corrupted by noise, a robust data-based observer gain is constructed by tackling a relaxed SDP. The proposed controllers are proven to achieve robust global exponential stability (RGES) for state estimation and input-to-state practical stability (ISpS) under standard conditions. Finally, numerical tests are conducted to validate the proposed controllers' correctness and effectiveness. © 2023, CC BY.

关键词： Linear time-invariant system