Linear Dynamical Systems Research Articles

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.

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 Po´lya-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.

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.

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.

Controllability of the time-varying fractional dynamical systems with a single delay in control

In this article, we explored the controllability of fractional dynamical systems with a single delay in the control function with the Caputo fractional derivative. It is the first work in which the author studies the controllability of a time-varying fractional dynamical system with a delay in the control function. We develop the necessary and sufficient criteria for the solution representation of controllability of time-varying fractional linear dynamical systems by utilizing the Grammian matrix. We use Schauder’s fixed point theorem to establish sufficient conditions for the controllability of time-varying nonlinear fractional dynamical systems. With the help of successive approximation techniques, numerical examples validate the theoretical results.

Learning the Causal Structure of Networked Dynamical Systems under Latent Nodes and Structured Noise

This paper considers learning the hidden causal network of a linear networked dynamical system (NDS) from the time series data at some of its nodes -- partial observability. The dynamics of the NDS are driven by colored noise that generates spurious associations across pairs of nodes, rendering the problem much harder. To address the challenge of noise correlation and partial observability, we assign to each pair of nodes a feature vector computed from the time series data of observed nodes. The feature embedding is engineered to yield structural consistency: there exists an affine hyperplane that consistently partitions the set of features, separating the feature vectors corresponding to connected pairs of nodes from those corresponding to disconnected pairs. The causal inference problem is thus addressed via clustering the designed features. We demonstrate with simple baseline supervised methods the competitive performance of the proposed causal inference mechanism under broad connectivity regimes and noise correlation levels, including a real world network. Further, we devise novel technical guarantees of structural consistency for linear NDS under the considered regime.

Model order reduction, a novel method using krylov sub-spaces and genetic algorithm

Model Order Reduction (MOR) of complex and large systems in Electrical engineering, continuous to be an attractive field for Engineers and Scientists over the last few decades, this complexity of models makes the control designs and simulation using Computer Aided Design (CAD) more and more difficult and consuming a lot of time. There for, accurate, robust and fast algorithms for simulation are needed. The goal of MOR is to replace the original system by an appropriate reduced system which preserves the main properties of the original one such that stability and passivity. Several analytical MOR techniques have been proposed in the literature over the past few decades, to approximate high order linear dynamic systems like Krylov sub-space techniques and SVD (Singular Value Decomposition) techniques. However, most of these techniques lead to computationally demanding, time consuming, iterative procedures that usually result in non-robustly stable models with poor frequency response resemblance to the original high order model in some frequency ranges. Recently a set of new techniques based on Artificial Intelligence (AI) were proposed in [1] for MOR. This article considers the problem of model order reduction of Linear Time In varying (LTI) systems. It is described by first and second order ordinary differential equations model. A tow steps method for model order reduction of LTI systems is proposed here, which combined features of an analytic technique (Krylov approach) and an AI technique (Genetic Algorithm). In the first step, the size of the original model is reduced to an intermediate order, using an analytical technique based on Krylov sub-spaces. In the final step of the reduction process, an AI approach based on Genetic Algorithm (GA) is applied to obtain an optimized nominal model.

Nonlinear optical dynamics in Heisenberg space: Directional curves and recursive inquiry

This paper delves into the exploration of directional recursion operators within the realm of regular space curves modeled by Heisenberg systems. The central objective is to introduce a myriad of recursive flows, encompassing ferromagnetic and antiferromagnetic solutions, alongside a family of general normalization operators in the normal and binormal directions. The study employs the extended compatible and inextensible flow model of curves to examine the evolution models, providing a comprehensive understanding of their dynamics. A significant aspect of the investigation involves elucidating the evolution model in terms of anholonomy shapes and their density. The directional recursive operator, a focus of this study, demonstrates distinct results compared to traditional approaches. The reliability and applicability of the obtained results extend to the examination of various linear and nonlinear continuous dynamical systems.

Discover an accurate approximation of dynamical system without prior information and customized design

Modeling dynamical systems is fraught with challenges when data can be collected but thorough analysis of the mechanism is absent. We design a method to discover unknown dynamical systems from data. The method discovers an accurate approximation of the model without the prior information and the customized design for each problem. The identification steps are straightforward as bringing in the data and then obtaining the model. The method begins with the simple idea that the equations of motion of many practical problems are Riemann integrable functions. For this reason, the Fourier series can decompose the equations of motion. In order to improve the accuracy, we design an extension that helps us to approximate unknown functions by the Fourier series with a high rate of convergence. The idea converts the difficulty of modeling the dynamical system into finding its Fourier series approximation. Convenient procedures enable the modeling of different problems. Numerical examples show that the new method discovers linear and nonlinear dynamical systems in the same steps and without the prior information.

Examining reachability criteria for fractional dynamical systems with mixed delays in control utilizing [formula omitted]-Hilfer pseudo-fractional derivative

The main objective of the present article is to employ the ψ-Hilfer pseudo-fractional derivative (HPFD) to examine the reachability criteria for fractional dynamical systems with mixed delays in control of order ϑ∈(0,1) and type ϱ∈[0,1]. we derived the sufficient and necessary conditions for the reachability criterion of fractional linear dynamical systems by utilizing the positiveness of Grammian matrices, which are defined by the Mittag-Leffler functions. The sufficient conditions for the reachability criteria of fractional nonlinear dynamical systems are obtained by using Banach’s fixed point theorem. To help grasp the theoretical results, only a limited number of numerical examples are provided.

Extensive numerical simulations on competitive growth between the Edwards-Wilkinson and Kardar-Parisi-Zhang universality classes

Abstract In this work, extensive numerical simulations and scaling analysis are performed to investigate competitive growth between the linear and nonlinear stochastic dynamic growth systems, which belong to the Edwards-Wilkinson (EW) and Kardar-Parisi-Zhang (KPZ) universality classes, respectively. The linear growth systems include the EW equation and the random deposition with surface relaxation (RDSR) model, the nonlinear growth systems involve the KPZ equation and typical discrete models including ballistic deposition (BD), Etching, and restricted solid-on-solid (RSOS). The scaling exponents are obtained in both the (1+1)- and (2+1)-dimensional competitive growth with the nonlinear growth probability $p$ and the linear proportion $1-p$. Our results show that, when $p$ changes from 0 to 1, there exist non-trivial crossover effects from EW to KPZ universality classes based on different competitive growth rules. Furthermore, the growth rate and the porosity are also estimated within various linear and nonlinear growth of cooperation and competition.

On the smoothness of principal angles between subspaces and their application to angular values of dynamical systems

In this work we provide detailed estimates of maximal principal angles between subspaces and we analyse their smoothness for smoothly varying subspaces. This leads to a new definition of angular values for linear dynamical systems in continuous time. We derive some of their properties complementary to the theory of angular values developed in Beyn, et al. [Angular values of nonautonomous and random linear dynamical systems: Part I – fundamentals, SIAM J. Appl. Dyn. Syst. 21(2) (2022), pp. 1245–1286; Angular values of nonautonomous linear dynamical systems: Part II – reduction theory and algorithm, SIAM J. Appl. Dyn. Syst. 22(1) (2023), pp. 162–198] for discrete time systems. The estimates are further employed to establish upper semicontinuity of angular values for some parametric model examples of discrete and continuous type.

Projection of diabetes morbidity and mortality till 2045 in Indonesia based on risk factors and NCD prevention and control programs

Diabetes Mellitus is one of the biggest health problems in Indonesia but the research on the disease’s projection is still limited. This study aimed to make a projection model of prevalence and mortality of diabetes in Indonesia based on risk factors and NCD programs. The study was a quantitative non-experimental study through multiple linear regression models and system dynamics. The baseline projection was created by 2018 data and projections until 2045 involved the dynamization of risk factors and programs, population, and case fatality rate. The model was created from 205 districts data. This study used secondary data from Basic Health Research, BPJS Kesehatan, NCD programs, and Ministry of Health. The prevalence of diabetes in Indonesia is estimated to increase from 9.19% in 2020 (18.69 million cases) to 16.09% in 2045 (40.7 million cases). The prevalence will be lower to 15.68% (39.6 million) if interventions of programs were carried out, and to 9.22% (23.2 million) if the programs were added with prevention of risk factors. The projected number of deaths due to diabetes increases from 433,752 in 2020 to 944,468 in 2045. Deaths due to stroke among diabetes increases from 52,397 to 114,092 in the same period. Deaths from IHD among diabetes increase from 35,351 to 76,974, and deaths from chronic kidney disease among diabetes increase from 29,061 to 63,279. Diabetes prevalence and mortality in Indonesia rise significantly in Indonesia and can be reduced by intervention of several programs and risk factors. This study findings could be source of planning and evaluation of Diabetes prevention and control program at national and provincial level in the future related to risk factors control and program development.

Insights on equidistributed real spectral values in second-order delay systems: Perspectives in partial pole placement

In this work, we show that the coexistence of the maximal number of real spectral values of generic single-delay retarded second-order differential equations guarantees the realness of the rightmost spectral value. From a control theory standpoint, this entails that a delayed proportional-derivative (PD) controller can stabilize a delayed second-order differential equation. By assigning the maximum number of negative roots to the corresponding characteristic function (a quasipolynomial), we establish the conditions for asymptotic stability. If the assigned real spectral values are uniformly distributed, we specify a necessary and sufficient condition for the rightmost root to be negative, thus guaranteeing the exponential decay rate of the system’s solutions. We illustrate the proposed design methodology in the delayed PD control of the damped harmonic oscillator. It is worth mentioning that this work represents a natural continuation of Amrane et al. (2018) and Bedouhene et al. (2020), addressing the problem of coexisting real spectral values for linear dynamical systems including delays in their models.