Article
Automation & Control Systems
Douglas A. Allan, James Rawlings, Andrew R. Teel
Summary: Incremental input/output-to-state stability (i-IOSS) is a popular characterization of detectability for nonlinear systems, and any system that admits a robustly stable state estimator must be i-IOSS. Storage functions with associated converse theorems are used to characterize other input-to-state-like properties.
SIAM JOURNAL ON CONTROL AND OPTIMIZATION
(2021)
Article
Automation & Control Systems
Zhiyong Chen
Summary: The LaSalle-Yoshizawa theorem is a significant tool for ensuring convergence of a nonlinear time-varying adaptive system. It guarantees boundedness and convergence when the derivative of a Lyapunov function is negative semidefinite. For a nonlinear system with an external input, the input-to-state stability (ISS) Lyapunov theorem reveals boundedness of system solutions when the derivative of an ISS Lyapunov function is negative definite with an input term. This technical communique provides a counter-example to prove that a certain boundedness property is not guaranteed when the derivative of an ISS Lyapunov function is negative semidefinite with an input term.
Article
Engineering, Mechanical
Cong Wu
Summary: In this paper, Lyapunov's first and second instability theorems for Caputo fractional-order systems are presented with proofs and illustrated by examples, based on recent advances in the continuation of solution and Caputo fractional derivative of Lyapunov functions along trajectories.
NONLINEAR DYNAMICS
(2022)
Article
Mathematics
A. R. Humphries, F. M. G. Magpantay
Summary: The study presents Lyapunov stability and asymptotic stability theorems for steady state solutions of general state-dependent delay differential equations using Lyapunov-Razumikhin methods. By replacing DDEs with nonautonomous ODEs and utilizing a contradiction argument along with the Arzela-Ascoli theorem, the asymptotic stability result is established without the need for auxiliary functions. The results are applied to a state-dependent model equation including the Hayes equation as a special case, providing insights into the stability domain and basin of attraction.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Huijuan Li
Summary: This paper estimates the domain of attraction of the origin of two interconnected nonlinear systems by introducing an auxiliary system and constructing Lyapunov functions. A local version of small gain theorem is proposed to facilitate checking of conditions. An estimate for the domain of attraction for the whole system is obtained based on the proposed small gain theorem.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2021)
Article
Physics, Mathematical
Kiho Park, Mark Piraino
Summary: This study demonstrates that typical cocycles over irreducible subshifts of finite type follow several limit laws with respect to unique equilibrium states for Holder potentials, including the central limit theorem and the large deviation principle. Analytic dependence of the top Lyapunov exponent on the underlying equilibrium state is also established. The transfer operator and its spectral properties play crucial roles in establishing these limit laws.
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2022)
Article
Mathematics, Applied
Abdon Atangana, Muhammad Altaf Khan
Summary: We have studied a special class of ordinary differential equations with Caputo fractional global derivative as the differential operators. These equations are generalizations of well-known differential equations with Caputo fractional derivative. We have introduced new inequalities that can be applied in various fields where these equations find applications. Nagumo's principles have been used to establish the existence and uniqueness of solutions for this class of equations with additional conditions. We have also proposed a numerical scheme based on the midpoint principle for solving these equations numerically. The presented illustrative examples show excellent results.
Article
Energy & Fuels
Shaohua Yang, Keng-Weng Lao, Hongxun Hui, Yulin Chen
Summary: In this article, a frequency regulation scheme for power systems is proposed to deal with multiple emergency events, based on coordinate transformation technique and Lyapunov theorem. The effectiveness of the scheme is verified through case studies.
Article
Mathematics
Jiamin Xing, Xue Yang, Yong Li
Summary: We introduce the Q(s)-index and Gamma for a symplectic orthogonal group Q(s) and Q(s) invariant subset Gamma of R2n, and prove that ind S2n-1 = n. Using this result, we study multiple rotating periodic orbits of Hamiltonian systems. For an orthogonal matrix Q, a Q-rotating periodic solution z(t) has different forms depending on the structure of Q. Under a non-resonant condition, we prove that the Hamiltonian system admits at least n Q-rotating periodic orbits on each energy surface near the equilibrium, which can be seen as a Lyapunov type theorem on rotating periodic orbits.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Mikhail Anikushin
Summary: The study presents a version of the Frequency Theorem applicable to semilinear parabolic equations, allowing the construction of quadratic Lyapunov functionals and inertial manifolds. The well-known Spectral Gap Condition is shown to be a particular case of a frequency inequality, which can be used to solve problems related to semilinear parabolic equations and develop a new theory combining geometric approaches. The study also explores the optimality of frequency inequalities and their relationship with previous and recent results in the field.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2022)
Article
Computer Science, Interdisciplinary Applications
Dejun Zhu
Summary: This paper investigates the practical stability problem of nonlinear stochastic delayed systems with G-Brownian motion (GSDSs) and proposes new sufficient conditions. By employing stochastic analysis technique, Razumikhin-type theorem, and vector G-Lyapunov function, the qualitative behavior and quantitative properties of the systems can be described. Two examples are presented to verify the feasibility of the theoretical results.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2022)
Article
Mathematics
Anton Gorodetski, Victor Kleptsyn
Summary: The study involves random products of SL(2,R) matrices dependent on a parameter in a non-uniformly hyperbolic regime. It shows that under monotone parameter dependence, the Lyapunov exponents of the random product satisfy specific conditions, and provides a purely geometrical proof of Spectral Anderson Localization for discrete Schrodinger operators on a one-dimensional lattice with random potentials.
ADVANCES IN MATHEMATICS
(2021)
Article
Mathematics
John R. Graef, Kadda Maazouz, Moussa Daif Allah Zaak
Summary: The authors investigate a nonlinear fractional pantograph boundary value problem with a variable order Hadamard fractional derivative and establish the existence and uniqueness results. This type of model is suitable for applications in strongly anomalous media. They also derive a generalized Lyapunov-type inequality for the problem and utilize fractional calculus and Krasnosel'skii's fixed point theorem in their proofs. An example is provided to illustrate their approach.
Article
Automation & Control Systems
Huifang Min, Shengyuan Xu, Baoyong Zhang, Qian Ma, Deming Yuan
Summary: This paper investigates the fixed-time stability theorem and state-feedback controller design for stochastic nonlinear systems. An improved fixed-time Lyapunov theorem is proposed with a more rigorous and reasonable proof procedure. A state-feedback controller is skillfully designed based on the backstepping technique and the addition of a power integrator method. It is proved that the proposed controller can render the closed-loop system fixed-time stable in probability.
IEEE-CAA JOURNAL OF AUTOMATICA SINICA
(2022)
Article
Engineering, Mechanical
Hongyan Zang, Xinxin Zhao, Xinyuan Wei
Summary: This paper explores the generation of pseudorandom numbers using chaotic maps, constructing high-order polynomial chaotic maps based on the Li-Yorke theorem, and designing a pseudorandom number generator with increased chaotic parameters. Testing shows that the generator produces sequences with good randomness, uniformity, complexity, and sensitivity to initial parameters, demonstrating high-quality sequence generation capabilities.
NONLINEAR DYNAMICS
(2022)
Article
Computer Science, Artificial Intelligence
Wenxue Zhang, Erick J. Rodriguez-Seda, Shankar A. Deka, Massinissa Amrouche, Di Zhou, Dusan M. Stipanovic, George Leitmann
JOURNAL OF INTELLIGENT & ROBOTIC SYSTEMS
(2020)
Article
Automation & Control Systems
Milan R. Rapaic, Mirna N. Kapetina, Dusan M. Stipanovic
Summary: This paper proposes a flexible and efficient methodology for constructing norm-bounded optimal receding horizon control laws for a group of agents. The control laws utilize polynomial expansion and appropriate subspace projection to derive closed-form solutions and have been investigated for their properties and their link to a control strategy based on avoidance functions.
ASIAN JOURNAL OF CONTROL
(2022)
Article
Automation & Control Systems
Wenxue Zhang, Dusan M. Stipanovic, Di Zhou
Summary: This paper presents a closed-form cooperative avoidance control design for 3-dimensional rigid-body agents, integrating collision risk with motion and posture information to improve efficiency and reliability of avoidance.
JOURNAL OF THE FRANKLIN INSTITUTE-ENGINEERING AND APPLIED MATHEMATICS
(2021)
Article
Automation & Control Systems
Shankar A. Deka, Dusan M. Stipanovic, Claire J. Tomlin
Summary: This article presents both theory and experiments on real-time attacks, with a focus on adversarial inputs for recurrent neural networks. Drawing inspiration from dynamical systems theory, it approaches this as a control problem and provides illustrative examples to support the theoretical discussions.
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY
(2022)
Article
Automation & Control Systems
Theodoros Mamalis, Dusan Stipanovic, Petros Voulgaris
Summary: This letter proposes a stochastic learning rate scheme by introducing multiplicative stochasticity to the learning rate of stochastic optimization algorithms. Theoretical and empirical results show significant optimization performance gains compared to deterministic learning rate versions.
IEEE CONTROL SYSTEMS LETTERS
(2022)
Article
Computer Science, Artificial Intelligence
Massi Amrouche, Dusan M. Stipanovic
Summary: This article provides a mathematical formulation for describing and designing activation functions in deep neural networks. The methodology accurately characterizes the desired activation functions and addresses the issue of gradient vanishing or exploding during training. The problem of finding desired activation functions is converted into an infinite-dimensional optimization problem, which is then solved by solving a partial differential equation. Bounds that guarantee the optimality of the designed activation function are also provided. Relevant examples with state-of-the-art activation functions are included to illustrate the methodology.
IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS
(2022)
Proceedings Paper
Automation & Control Systems
Thiago Marinho, Massi Amrouche, Dusan Stipanovic, Venanzio Cichella, Naira Hovakimyan
Summary: Biological evidence shows animals can avoid imminent collisions using looming stimuli, while robots require relative distance measurement for collision avoidance. This study proposes a control framework for a unicycle-like vehicle in a 2D plane, achieving collision avoidance with theoretical guarantees of minimum separation.
2021 AMERICAN CONTROL CONFERENCE (ACC)
(2021)
Article
Operations Research & Management Science
Dusan M. Stipanovic, Mirna N. Kapetina, Milan R. Rapaic, Boris Murmann
Summary: In this paper, a specific discrete-time nonlinear and time-invariant system is analyzed with a focus on stability properties, as it models gated recurrent unit neural networks commonly used in machine learning applications. By treating the system as a convex combination of discrete-time systems, the analysis focuses on stability, linearization, and multiple equilibria of the system. Results specific to gated recurrent unit neural network models are derived, with a connection between local stability analysis and learning emphasized.
JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
(2021)
Proceedings Paper
Automation & Control Systems
Shankar A. Deka, Dusan M. Stipanovic, Claire J. Tomlin
2020 59TH IEEE CONFERENCE ON DECISION AND CONTROL (CDC)
(2020)
Proceedings Paper
Automation & Control Systems
Wenxue Zhang, Dusan M. Stipanovic, Di Zhou
2020 AMERICAN CONTROL CONFERENCE (ACC)
(2020)
Proceedings Paper
Engineering, Industrial
Aleksandra Lekic-Vervoort, Milovan Majstorovic, Leposava Ristic, Dusan Stipanovic
2020 IEEE 29TH INTERNATIONAL SYMPOSIUM ON INDUSTRIAL ELECTRONICS (ISIE)
(2020)
Proceedings Paper
Automation & Control Systems
Yichuan Li, Nikolaos M. Freris, Petros Voulgaris, Dusan Stipanovic
2020 AMERICAN CONTROL CONFERENCE (ACC)
(2020)
Proceedings Paper
Automation & Control Systems
Massinissa Amrouche, Thiago Marinho, Dusan Stipanovic
2020 EUROPEAN CONTROL CONFERENCE (ECC 2020)
(2020)
Article
Automation & Control Systems
Erick J. Rodriguez-Seda, Dusan M. Stipanovic
IEEE CONTROL SYSTEMS LETTERS
(2020)
Article
Automation & Control Systems
Zhengyuan Zhou, Jonathan R. Shewchuk, Dusan Stipanovic, Haomiao Huang, Claire J. Tomlin
SIAM JOURNAL ON CONTROL AND OPTIMIZATION
(2020)