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REGULAR PAIRINGS FOR NON-QUADRATIC LYAPUNOV FUNCTIONS AND CONTRACTION ANALYSIS

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arXiv 2024年

作者： Proskurnikov, Anton V. Bullo, Francesco Department of Electronics and Telecommunications Politecnico di Torino Turin Italy Center for Control Dynamical Systems and Computation UC Santa Barbara Santa BarbaraCA93101 United States

Recent studies on stability and contractivity have highlighted the importance of semi-inner products, which we refer to as "pairings", associated with general norms. A pairing is a binary operation that relates the derivative of a curve’s norm to the radius-vector of the curve and its tangent. This relationship, known as the curve norm derivative formula, is crucial when using the norm as a Lyapunov function. Another important property of the pairing, used in stability and contraction criteria, is the so-called Lumer inequality, which relates the pairing to the induced logarithmic norm. We prove that the curve norm derivative formula and Lumer’s inequality are, in fact, equivalent to each other and to several simpler properties. We then introduce and characterize regular pairings that satisfy all of these properties. Our results unify several independent theories of pairings (semi-inner products) developed in previous work on functional analysis and control theory. Additionally, we introduce the polyhedral max pairing and develop computational tools for polyhedral norms, advancing contraction theory in non-Euclidean spaces. © 2024, CC BY-NC-ND.

关键词： Lyapunov functions

Comparative Analysis of GOA, WCA and PSO Tuned Fractional Order-PID controller for Cart-Inverted Pendulum

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Comparative Analysis of GOA, WCA and PSO Tuned Fractional Or...

Power and Renewable Energy Conference (IPRECON), IEEE International

作者： Neelam Verma Sudarshan K. Valluru Department of Electrical Engineering Center for Control of Dynamical Systems and Computation Delhi Technological University Delhi India

control of cart based inverted pendulum system is a classical problem of control system and is widely used to verify the effectiveness of control algorithms. In this paper, fractional order PID (FOPID) controller has been used to control the cart position and pendulum angle. Three optimization algorithms namely, particle swarm optimization (PSO), water cycle algorithm (WCA) and grasshopper optimization algorithm (GOA) has been used to tune the parameters of the FOPID controller for underline problem. Results shows that, FOPID controller was able to control both cart position and pendulum angle effectively. Further, it has also been shown that GOA tuned FOPID controller performs better as compared to WCA and PSO tuned FOPID controller. The entire system was simulated using MATLAB2018a.

关键词： Renewable energy sources Software packages control systems Particle swarm optimization Optimization Matlab

Non-euclidean monotone operator theory and applications

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The Journal of Machine Learning Research

The Journal of Machine Learning Research 2024年第1期25卷 14872-14904页

作者： Alexander Davydov Saber Jafarpour Anton V. Proskurnikov Francesco Bullo Center for Control Dynamical Systems and Computation University of California Santa Barbara Santa Barbara CA Department of Electrical Computer and Energy Engineering University of Colorado Boulder Boulder CO Department of Electronics and Telecommunications Politecnico di Torino Turin 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 ℓ1 or ℓ1 norms. This paper provides a natural generalization of monotone operator theory to finitedimensional non-Euclidean spaces. The key tools are weak pairings and logarithmic norms. We show that the resolvent and reected 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.

关键词： non-Euclidean norms monotone operator theory fixed point equations nonexpansive maps

A sufficient condition for 2-contraction of a feedback interconnection

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arXiv 2024年

作者： Ofir, Ron Bullo, Francesco Margaliot, Michael the School of Elec. Eng. Tel Aviv University 69978 Israel the Center for Control Dynamical Systems and Computation UC Santa Barbara Santa BarbaraCA93106 United States

Multistationarity – the existence of multiple equilibrium points – is a common phenomenon in dynamical systems from a variety of fields, including neuroscience, opinion dynamics, systems biology, and power systems. A recently proposed generalization of contraction theory, called k-contraction, is a promising approach for analyzing the asymptotic behaviour of multistationary systems. In particular, all bounded trajectories of a time-invariant 2-contracting system converge to an equilibrium point, but the system may have multiple equilibrium points where more than one is locally stable. An important challenge is to study k-contraction in large-scale interconnected systems. Inspired by a recent small-gain theorem for 2-contraction by Angeli et al., we derive a new sufficient condition for 2-contraction of a feedback interconnection of two nonlinear dynamical systems. Our condition is based on (i) deriving new formulas for the 2-multiplicative [2-additive] compound of block matrices using block Kronecker products [sums], (ii) a hierarchical approach for proving standard contraction, and (iii) a network small-gain theorem for Metzler matrices. We demonstrate our results by deriving a simple sufficient condition for 2-contraction in a network of FitzHugh-Nagumo neurons. © 2024, CC BY.

关键词： dynamical systems

Contractivity of the Method of Successive Approximations for Optimal control

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arXiv 2022年

作者： Smith, Kevin D. Bullo, Francesco Center for Control Dynamical Systems and Computation UC Santa Barbara Santa BarbaraCA93101 United States

Strongly contracting dynamical systems have numerous properties (e.g., incremental ISS), find widespread applications (e.g., in controls and learning), and their study is receiving increasing attention. This work starts with the simple observation that, given a strongly contracting system, its adjoint dynamical system is also strongly contracting, with the same rate, with respect to the dual norm, under time reversal. As main implication of this dual contractivity, we show that the classic Method of Successive Approximations (MSA), an indirect method in optimal control, is a contraction mapping for short optimization intervals or large contraction rates. Consequently, we establish new convergence conditions for the MSA algorithm, which further imply uniqueness of the optimal control and sufficiency of Pontryagin’s minimum principle under additional assumptions. Copyright © 2022, The Authors. All rights reserved.

关键词： dynamical systems

INPUT-DRIVEN DYNAMICS FOR ROBUST MEMORY RETRIEVAL IN HOPFIELD NETWORKS

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arXiv 2024年

作者： Betteti, Simone Baggio, Giacomo Bullo, Francesco Zampieri, Sandro Department of Information Engineering Università degli Studi di Padova Padova35131 Italy Center for Control Dynamical Systems and Computation University of California at Santa Barbara Santa BarbaraCA93106 United States

The Hopfield model provides a mathematically idealized yet insightful framework for understanding the mechanisms of memory storage and retrieval in the human brain. This model has inspired four decades of extensive research on learning and retrieval dynamics, capacity estimates, and sequential transitions among memories. Notably, the role and impact of external inputs has been largely underexplored, from their effects on neural dynamics to how they facilitate effective memory retrieval. To bridge this gap, we propose a novel dynamical system framework in which the external input directly influences the neural synapses and shapes the energy landscape of the Hopfield model. This plasticity-based mechanism provides a clear energetic interpretation of the memory retrieval process and proves effective at correctly classifying highly mixed inputs. Furthermore, we integrate this model within the framework of modern Hopfield architectures, using this connection to elucidate how current and past information are combined during the retrieval process. Finally, we embed both the classic and the new model in an environment disrupted by noise and compare their robustness during memory *** Codes 37N25 (Primary) 37C75, 34D45 (Secondary) © 2024, CC BY.

关键词： Associative storage

Firing Rate Models as Associative Memory: Excitatory-Inhibitory Balance for Robust Retrieval

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arXiv 2024年

Firing rate models are dynamical systems widely used in applied and theoretical neuroscience to describe local cortical dynamics in neuronal populations. By providing a macroscopic perspective of neuronal activity, these models are essential for investigating oscillatory phenomena, chaotic behavior, and associative memory processes. Despite their widespread use, the application of firing rate models to associative memory networks has received limited mathematical exploration, and most existing studies are focused on specific models. Conversely, well-established associative memory designs, such as Hopfield networks, lack key biologically-relevant features intrinsic to firing rate models, including positivity and interpretable synaptic matrices that reflect excitatory and inhibitory interactions. To address this gap, we propose a general framework that ensures the emergence of re-scaled memory patterns as stable equilibria in the firing rate dynamics. Furthermore, we analyze the conditions under which the memories are locally and globally asymptotically stable, providing insights into constructing biologically-plausible and robust systems for associative memory *** Codes 37N25 (Primary) 34D45, 34D23 (Secondary) © 2024, CC BY.

关键词： Associative storage

On Weakly Contracting Dynamics for Convex Optimization

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arXiv 2024年

作者： Centorrino, Veronica Davydov, Alexander Gokhale, Anand Russo, Giovanni Bullo, Francesco Scuola Superiore Meridionale Italy The Center for Control Dynamical Systems and Computation UC Santa Barbara Santa BarbaraCA93106 United States The Department of Information and Electric Engineering and Applied Mathematics University of Salerno Italy

We analyze the convergence behavior of globally weakly and locally strongly contracting dynamics. Such dynamics naturally arise in the context of convex optimization problems with a unique minimizer. We show that convergence to the equilibrium is linear-exponential, in the sense that the distance between each solution and the equilibrium is upper bounded by a function that first decreases linearly and then exponentially. As we show, the linear-exponential dependency arises naturally in certain dynamics with saturations. Additionally, we provide a sufficient condition for local input-to-state stability. Finally, we illustrate our results on, and propose a conjecture for, continuous-time dynamical systems solving linear programs. Copyright © 2024, The Authors. All rights reserved.

关键词： Dynamics

On the Equivalence of Multi-Agent 2D Coverage control and Leader-Follower Consensus Network

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On the Equivalence of Multi-Agent 2D Coverage Control and Le...

American control Conference (ACC)

作者： Xiaotian Xu Alexander Davydov Yancy Diaz-Mercado Department of Mechanical Engineering University of Maryland College Park MD USA Department of Mechanical Engineering and the Center for Control Dynamical Systems and Computation University of California Santa Barbara CA USA

Coverage control algorithms seek to spatially distribute agents in a domain of coverage, e.g., to minimize proximity to all points. Leader-follower consensus network algorithms use local coordination rules to influence the behavior of a multi-agent system (MAS) as a whole through explicit control of a subset of agents (called leaders) and neighbor interactions. In this paper, the equivalence of these two classes of distributed algorithms for swarm robotics, that were once considered inherently different, is established. We present a swarm robotics application, where the real agents (i.e., the robots) in the domain of coverage are followers; and virtual agents (i.e., the leaders) are introduced based on the domain of coverage. The dynamics of followers are shown to be in the form of a weighted, state-dependent consensus protocol and the dynamics of the leaders (dependent on the evolution of the domain) are provided. Formulating a standard coverage algorithm (i.e., Lloyd’s algorithm) over 2D polygonal domains as a leader-follower consensus protocol makes the structure of the ensemble-level dynamics for the MAS explicit with respect to neighbor interaction. The resultant weighted graph Laplacian may contribute to the future investigation on the performance guarantees of a MAS tracking a time-varying domain. The equivalence of the two classes of algorithms is validated in simulation.

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