Linear Dynamical Systems Research Articles

Reconstructing rodent brain signals during euthanasia with eigensystem realization algorithm (ERA)

We accurately reconstruct the Local Field Potential time series obtained from anesthetized and awake rats, both before and during CO2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_2$$\\end{document} euthanasia. We apply the Eigensystem Realization Algorithm to identify an underlying linear dynamical system capable of generating the observed data. Time series exhibiting more intricate dynamics typically lead to systems of higher dimensions, offering a means to assess the complexity of the brain throughout various phases of the experiment. Our results indicate that anesthetized brains possess complexity levels similar to awake brains before CO2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$_2$$\\end{document} administration. This resemblance undergoes significant changes following euthanization, as signals from the awake brain display a more resilient complexity profile, implying a state of heightened neuronal activity or a last fight response during the euthanasia process. In contrast, anesthetized brains seem to enter a more subdued state early on. Our data-driven techniques can likely be applied to a broader range of electrophysiological recording modalities.

Analysis of Dynamic Systems Through Artificial Neural Networks

Parameter identification techniques for linear and nonlinear dynamic systems currently show a clear orientation toward black box models, with Artificial Neural Networks occupying a prominent place there. This paper presents a procedure for identifying linear dynamic systems parameters in two stages: in the first, a regressive model is fitted from the excitation and response time records, and in the second, its parameters are identified (matrixes of stiffness and damping) and dynamic characteristics (vibration frequencies and modes) based on the previous model. Artificial Neural Networks of the Adaline type and multilayer Perceptions are used for the first stage. The second stage is fully formulated through matrix algebra, which facilitates its systematic implementation and makes it independent of the complexity or dimension of the studied system. The proposed procedure is intended to operate from experimental records, so special attention is paid to the sensitivity of the results to the data interval and noise in the input signals. For the latter, various noise levels were incorporated into the correct responses obtained under ideal conditions, which respond to Gaussian distribution functions with a null mean and specified standard deviation. The proposed procedure justification, the results with the regressive models, and a study of the sensitivity of the results to the variation in the available data quality are presented.

On-board model predictive control for autonomous lane keeping with fuzzy preview distance: Design and experiment

This paper concerns with the development of computationally efficient lane keeping control method for the autonomous vehicles. To obtain autonomous lane keeping, model predictive control (MPC) scheme was intensively investigated in the previous studies owing to its inherent advantage of dealing with constrained multivariable systems. However, the tradeoff problem between the good tracking performance and the low computational complexity is inevitably raised in developing of MPC-based lane keeping technology. To alleviate the conflict between performance and cost, an on-board MPC with fuzzy preview distance is designed in this study. A linear system dynamics model is used in the MPC design to reduce the computational cost, and a fuzzy logic algorithm is developed to select an appropriate preview distance for enhancing the MPC performance. Further, hardware-in-the-loop test is adopted to explore the effectiveness and efficiency of the proposed control method. In comparison to the proportional-integral controller, the experimental results show that the MPC is more sensitive to the selective value of fixed preview distance. Since the significant impact of preview distance selection on the MPC-based lane keeping performance, the fuzzy logic algorithm is of the essence in terms of selecting the appropriate preview distance for MPC enhancement under different vehicle speed and road curvature. Eventually, experimental results validate that the proposed fuzzy preview distance algorithm can effectively improve the MPC-based lane keeping performance for autonomous vehicles subject to limited computational resource.

Assessment of Reliability of Power Systems Under Adverse Weather Condition Using Markov System Dynamic Method

Abstract We aim to extend the Markov System Dynamic method (MSD) to evaluate the reliability of repairable systems in two-state weather (2SW) models. Increasing the number of components in real practices has limited the use of the Markov method, because of increasing the number of equations and the complexity of the problem. There is no such restriction in the MSD method. Thus, it can be applied to complex systems. Theoretically, we indicate that the 2SW model is a type of linear dynamical system. Therefore, we can use MSD to solve it. Through an example, the validity of the proposed method is confirmed in comparison to the Markov method for the 2SW model. Since the MSD method does not use equations, it can be a preferred alternative for calculating the reliability of 2SW models. Accordingly, we apply the MSD method for assessing the reliability of a four-component system.

Low conservative stability criteria for discrete-time Lur’e systems with sector and slope constrained nonlinearities

Lur’e system is an important nonlinear system in engineering, which is composed of a feedback connection of a linear dynamic system and a nonlinear system that satisfies the sector constraint. In this paper, for nominal and uncertain discrete Lur’e systems with time-varying delays, the absolute and robustly absolute stability are studied in detail with the nonlinearity satisfying the sector and slope constraints. Firstly, an augmented Lyapunov–Krasovskii functional (LKF) is designed, in which some augmented vectors are selected to increase the coupling information between the delay intervals and other system state variables. At the same time, some effective integral terms of nonlinear state variables are extended in the LKF according to the characteristics of sector constraint and slope constraint. Secondly, a general lemma of summation inequality for free-weighted matrices is given, which adds some coupling information between vectors and extends the search range for solving linear matrix inequalities (LMIs). Thirdly, by synthesizing the general summation inequality and the newly constructed LKF, different absolute stability criteria are derived respectively, and the absolute stability criteria are extended to robustly absolute stability cases with uncertain parameters. The stability criteria reduce the conservatism of some previously proposed results. Finally, some common numerical examples and simulations are used to verify the results of this paper.

New Approach and Integration for The Introduction of Dynamic Systems : Methodology, Models, And Relations

Introduction: This article is the result of the research “Analysis of the dynamic systems component of the engineering programs in control and automation and technology in industrial electronics,” developed at the Francisco José de Caldas District University in the years 2022-2023. Problem: The document addresses the lack of a clear and comprehensive structure in the analysis of linear dynamic systems, as well as the absence of an innovative approach to addressing these topics in traditional courses, which hinders their understanding and application in different disciplines. Objective: The objective is to present a novel approach to analyzing linear dynamic systems, providing a complete framework of concepts, relationships, and important tools in this analysis, as well as a literature review based on over seventeen years of experience in courses and studies of this kind. Methodology: A journey of concepts is proposed from signals and systems, through modeling with methods such as black box, white box, gray box, to linear analysis based on examples with calculation of differential equations, state representation, transfer function, and block diagrams. Results: The document aggregates and articulates all the concepts of dynamic systems, along with the relationships and tools used, offering a more practical and intuitive approach to understanding the material. Conclusion: The document provides a comprehensive and articulated view of key concepts in the analysis of linear dynamic systems, highlighting an innovative approach that facilitates their understanding. Its main contribution is to aggregate and articulate these concepts, along with the tools and relationships used, to offer a more practical and clear approach for their study. Originality: The originality lies in proposing a novel and structured approach to analyzing linear dynamic systems, addressing the lack of clarity and completeness in traditional approaches. Limitations: Although the document proposes a novel approach, it does not delve into specific aspects of some topics covered, which could limit the detailed understanding of certain concepts. Deepening: The document could delve into the practical application of the proposed concepts and tools in real cases of dynamic systems, as well as into the comparison with traditional approaches to highlight the differences and advantages of the proposed new approach.

Analytical modelling and design of linear controlled dynamic systems

A generalized canonical mathematical model of multidimensional controlled mining transportation complexes and other technical systems in form by A.M. Letov is considered. A principle of symmetry and an algebraic concept are in the basis of the developed analytical methods of design. A principle of symmetry is realized on a set of indexes of roots of a characteristic equation of the system and on a set of indexes of phase coordinates of the mathematical model. The problem of quality of dynamic processes in time is reduced to an algebraic problem of distribution of roots of a corresponding characteristic equation in a complex plane. An analogy in a procedure of transformation of a characteristic determinant into a polynomial and a structure of elementary symmetrical polynomials of roots is established. A new formulation of an analytical representation of change of phase coordinates in time in a form of ordered determinants with respect to indexes of roots and indexes of phase coordinates is obtained based on residue theorem. An illustration of the developed analytical method of design is performed on a special case of a well-known controlled technical system of the fourth order.

Controllability of time-varying fractional dynamical systems with distributed delays in control

Abstract The objective of this study is to assess controllability results in the realm of Caputo fractional derivatives, specifically focusing on time-varying linear and nonlinear fractional dynamical systems with distributed control delays. In the setting of a linear system, using the Grammian matrix's positive definiteness makes it possible to ascertain both the necessary and sufficient conditions. It is possible to determine, for nonlinear systems, the sufficient conditions for the presence of a solution by making use of the Schauder fixed point theorem. In addition to that, the paper discusses the controllability of a nonlinear integrodifferential system for a particular case. A few examples and the graphs that correspond to them are shown here to make it easier to conceptualize the conclusions of the theoretical discussion.

Аналіз точності самоналаштування динамічної моделі газотурбінного двигуна

One of the most promising directions in the development of aircraft engines is related to the introduction of adaptive automatic control systems (ACS). The defining element of these systems are dynamic mathematical models of engines capable of self-adjustment based on engine operating parameters measured in flight. A number of leading researchers have developed a concept of using such models called STORM (Self-Tuned On-board Real-time Model). However, in the corresponding works, clearly insufficient attention is paid to solving the problem of checking the sufficiency of the information used to ensure the necessary accuracy of the models. This check must be performed a priori (to predict the composition of engine operating modes, and the volume of registered information), as well as posteriori. The subject of this study is the process of forming dynamic mathematical models (MM) of gas turbine engines using real data for the subsequent use of these models to solve problems related to the control and diagnostics of on-board systems. The goal of this study is to determine the dependence of estimation errors of dynamic parameters of mathematical models on influencing factors. Tasks considered in the work: forming the structure of a mathematical model, dividing the identification process into stages according to the structure of the model (estimating the parameters that determine the static and dynamic properties of the object), forming the least-square functional for the assessment task, determining the errors in estimating dynamic coefficients, analyzing influencing factors, and determining dependencies between factors and errors. For this purpose, the methods of the theory of air-jet engines, the theory of linear dynamic systems, and statistical evaluation are used. The following results were obtained: a mathematical model of a turboshaft engine with a reciprocating gas generator was formed, and ratios were obtained that allow determining the errors in estimating the time constant of a reciprocating engine or gas generator. Scientific and practical innovation: for the first time, a ratio was obtained that determines the errors in estimating the time constant based on the specified values of the measurement errors, the intensity of the jump-like change in fuel consumption, and the frequency and duration of observation. These relations are presented in dimensionless coordinates, which makes them universal and able to be applied to any single-stroke turbojet engine or single-stroke gas generator during a priori or a posteriori analysis of results, as well as planning experiments and debugging on-board self-tuning algorithms of models.

New dynamical system for circular satellites relative motion

In this work, a new perturbed dynamical system to describe the satellites relative motion is derived considering that the gravitational attractive force is generated by the continued fraction potential. In this context, the parameter of continuation fraction creates an additional force which is the source of perturbation for motion. Aiming to find a simple and precise mathematical model to characterize this motion while maintaining high accuracy results, a new circular linear perturbed system is also derived under the effect of this parameter. Furthermore, the perturbed linear dynamical system of satellites relative motion is studied, along with finding the conditions of stable and periodic solutions. Finally, these solutions are calculated and their graphical simulation is introduced. We emphasize that the new proposed system could be applied in space rendezvous missions aiming at getting precise descriptions of satellites relative motion.

Bayesian Inference of Recurrent Switching Linear Dynamical Systems with Higher-Order Dependence

Many complicated dynamical events may be broken down into simpler pieces and efficiently described by a system that shifts among a variety of conditionally dynamical modes. Building on switching linear dynamical systems, we develop a new model that extends the switching linear dynamical systems for better discovering these dynamical modes. In the proposed model, the linear dynamics of latent variables can be described by a higher-order vector autoregressive process, which makes it feasible to evaluate the higher-order dependency relationships in the dynamics. In addition, the transition of switching states is determined by a stick-breaking logistic regression, overcoming the limitation of a restricted geometric state duration and recovering the symmetric dependency between the switching states and the latent variables from asymmetric relationships. Furthermore, logistic regression evidence potentials can appear as conditionally Gaussian potentials by utilizing the Pólya-gamma augmentation strategy. Filtering and smoothing algorithms and Bayesian inference for parameter learning in the proposed model are presented. The utility and versatility of the proposed model are demonstrated on synthetic data and public functional magnetic resonance imaging data. Our model improves the current methods for learning the switching linear dynamical modes, which will facilitate the identification and assessment of the dynamics of complex systems.

Weak multi-sensitive compactness and mean sensitivity for linear operators

Let ( X , T ) be a linear dynamical system defined on a Banach space X and L T : K X ∗ → K X ∗ be defined by L T S = TS , where K X ∗ = span { 〈 ⋅ , y ∗ 〉 x : y ∗ ∈ X ∗ , x ∈ X } ¯ . This paper first obtains the equivalence of the (syndetically) weak multi-sensitive compactness between ( X , T ) and ( K X ∗ , L T ) . Then, it is shown that (1) there exists a non-mean-sensitive linear dynamical system ( X , T ) satisfying that ( K X ∗ , L T ) is mean sensitive; (2) There exists a non-topologically-transitive and weakly multi-sensitive compact system, which is not syndetically weakly multi-sensitive compact. Besides, it is proved that every weakly mixing linear dynamical system defined on a separable Banach space is weakly multi-sensitive compact.

Existence and Exponential Decay for a Contact Problem Between Two Dissipative Beams

We deal with the Signorini contact problem between two Timoshenko beams. In this work we use the theory of semigroups to show the existence of solutions that decay uniformly to zero. This method is new and more effective than the widely used energy method. This is because in particular we obtain uniform decay of the solutions to zero for any boundary condition. A second important point is that we can take advantage of stabilization results of others linear dynamic systems with different dissipative mechanisms and apply them through our method for Contact Problems (see Sect. 4). Finally, thanks to Lipschitzian perturbations we can generalize the Signorini problem to more general semi linear problems in a simple way (see Sect. 4.3).

Joint learning of linear time-invariant dynamical systems

Linear time-invariant systems are very popular models in system theory and applications. A fundamental problem in system identification that remains rather unaddressed in extant literature is to leverage commonalities amongst related systems to estimate their transition matrices more accurately. To address this problem, we investigate methods for jointly estimating the transition matrices of multiple systems. It is assumed that the transition matrices are unknown linear functions of some unknown shared basis matrices. We establish finite-time estimation error rates that fully reflect the roles of trajectory lengths, dimension, and number of systems under consideration. The presented results are fairly general and show the significant gains that can be achieved by pooling data across systems, in comparison to learning each system individually. Further, they are shown to be robust against moderate model misspecifications. To obtain the results, we develop novel techniques that are of independent interest and are applicable to similar problems. They include tightly bounding estimation errors in terms of the eigen-structures of transition matrices, establishing sharp high probability bounds for singular values of dependent random matrices, and capturing effects of misspecified transition matrices as the systems evolve over time.

Analyzing Observability of Fractional Dynamical Systems Employing ψ-Caputo Fractional Derivative

In this present article, we inquire into the observability of linear and nonlinear fractional dynamical systems in terms of ψ-Caputo fractional derivative. The observability Grammian matrix, which is positive definite and expressed by the Mittag-Leffler functions, is used to obtain the necessary and sufficient conditions of observability of linear fractional dynamical systems, and Banach’s fixed point theorem is used to get the sufficient conditions for the observability of nonlinear fractional systems. Three numerical examples are given to demonstrate the applicability of theoretical results for linear and nonlinear cases.

On the Transformation of a Stationary Fuzzy Random Process by a Linear Dynamic System

In this paper, stationary random processes with fuzzy states are studied. The properties of their numerical characteristics—fuzzy expectations, expectations, and covariance functions—are established. The spectral representation of the covariance function, the generalized Wiener–Khinchin theorem, is proved. The main attention is paid to the problem of transforming a stationary fuzzy random process (signal) by a linear dynamic system. Explicitform relationships are obtained for the fuzzy expectations (and expectations) of input and output stationary fuzzy random processes. An algorithm is developed and justified to calculate the covariance function of a stationary fuzzy random process at the output of a linear dynamic system from the covariance function of a stationary input fuzzy random process. The results rest on the properties of fuzzy random variables and numerical random processes. Triangular fuzzy random processes are considered as examples.

On the Transformation of a Stationary Fuzzy Random Process by a Linear Dynamic System

On the Transformation of a Stationary Fuzzy Random Process by a Linear Dynamic System

On the Transformation of a Stationary Fuzzy Random Process by a Linear Dynamic System

Вданной работе изучены стационарные случайные процессы с нечеткими состояниями. Установлены свойства их числовых характеристик – нечетких ожиданий, ожиданий и ковариационных функций. Обосновано спектральное представление ковариационной функции – обобщенная теорема Винера–Хинчина. Основное внимание уделено задаче о преобразовании стационарного нечетко случайного процесса (сигнала) линейной динамической системой. Получены формулы, связывающие нечеткие ожидания (и ожидания) входных и выходных стационарных нечетко случайных процессов. Разработан и обоснован алгоритм вычисления ковариационной функции стационарного нечетко случайного процесса на выходе линейной динамической системы по ковариационной функции стационарного входного нечетко случайного процесса. Полученные результаты опираются на свойства нечетко случайных величин и числовых случайных процессов. Вкачестве примеров рассмотрены треугольные нечетко случайные процессы.

Learning of Linear Dynamical Systems as a Noncommutative Polynomial Optimization Problem

Learning of Linear Dynamical Systems as a Noncommutative Polynomial Optimization Problem

Distributed Time-Varying Optimization-An Output Regulation Approach.

This article deals with distributed algorithm design for time-varying optimization problems, which include unconstrained time-varying optimization and a special constrained problem commonly known as a resource allocation problem. The time-varying nature exists in the individual cost functions and the demand functions, and they are then captured by neutrally stable linear dynamic systems known as exosystems. To address the time-varying nature, new distributed algorithm structures are developed and two algorithms are designed for distributed time-varying optimization (DTVO) and distributed time-varying optimal resource allocation (DTVORA) to ensure that there exist time-varying solutions and the solution states will converge to the time-varying solutions. The driving terms for tracking the variation in the solutions are designed using exosystem dynamics. Rigorous convergence analyses are carried out using Lyapunov theory, and the examples are included to demonstrate the potential applications of the two proposed algorithms.