Computer Aided Robustness Analysis of Dynamic Systems

The [formula omitted]-compound of a difference–algebraic system

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.

Uncertainty of parameters in control systems was described by interval variables, and some idea of the non-probabilistic reliability method that we proposed in previous papers was used for stability analysis and synthesis of uncertain systems. A new method, called robust reliability method, was presented for quadratic stability analysis and robust stabilization of dynamic systems with interval parameters. In the method, the performance function for robust reliability calculation was established by criterion of quadratic stability, and a solving method in the form of optimization for robust reliability was presented. The proposed stabilization method can take both the required energy supply for control and the control robustness into account. This makes the reasonable strategy of compromising between the control cost and system reliability become possible. By the method, robust reliability measure of the stability degree of uncertain control systems can be provided, and the maximum allowable ranges of the basic parameters can be obtained. The presented method is based on the linear matrix inequality (LMI) approach, and is convenient to implement Three numerical examples were provided to illustrate the validity and efficiency of the presented method.

In this study stochastic analysis of non-linear dynamical systems under α-stable,
\nmultiplicative white noise has been conducted. The analysis has dealt with a special class of α-stable
\nstochastic processes namely sub-Gaussian white noises. In this setting the governing equation either of the
\nprobability density function or of the characteristic function of the dynamical response may be obtained
\nconsidering the dynamical system forced by a Gaussian white noise with an uncertain factor with α/2-
\nstable distribution. This consideration yields the probability density function or the characteristic function
\nof the response by means of a simple integral involving the probability density function of the system
\nunder Gaussian white noise and the probability density function of the α/2-stable random parameter. Some
\nnumerical applications have been reported assessing the reliability of the proposed formulation. Moreover
\na proper way to perform digital simulation of the sub-Gaussian α-stable random process preventing
\ndynamical systems from numerical overflows has been reported and discussed in detail.

Chapter 8 Numerical analysis of dynamical systems

In two recent papers, Brayton and Tong developed some significant results which are the basis of a constructive approach in the stability analysis of dynamical systems. Although their algorithm is powerful, it taxes the capabilities of most modern computers when the dimension of a system is high or even when the dimension is moderate in size. In this paper we remove these difficulties to a certain extent by generalizing the work of Brayton and Tong to interconnected dynamical systems. In our approach, the analysis of complex dynamical systems (which may be of high dimension) is accomplished in terms of the qualitative properties of the free subsystems and the interconnections of such systems.

In the first part of the article is a methodology for implementing a model-driven safety analysis of dynamical technology systems. The safety analysis is performed in the development of control system, mainly aiming at critical processes of system operation. I divided my methodology into five steps. These steps of the methodology was carried out in a hierarchical sequence. The objective of the second part of the article is to demonstrate the principle of the SQMD method concept for performing safety analysis on the example of a dynamical system. By SQMD method will be developed a models of safety-critical processes by which they will be carried out on-line monitoring. The method combining qualitative and quantitative procedures of modelling and taking both methods advantages.

We address the stability analysis of composite hybrid dynamical feedback systems of the type depicted in Fig. 1, consisting of a block (usually the plant) which is described by an operator L and of a finite-dimensional block described by a system of ordinary differential equations (usually the controller). We establish results for the well-posedness, attractivity, asymptotic stability, uniform boundedness, asymptotic stability in the large, and exponential stability in the large for such systems. The hypotheses of these results are phrased in terms of the I/O properties of L and in terms of the Lyapunov stability properties of the subsystem described by the indicated ordinary differential equations. The applicability of our results is demonstrated by means of general specific examples (involving C 0 -semigroups, partial differential equations, or integral equations which determine L ).

The scope of this paper is to apply a model-correction-based strategy for efficient reliability analysis of uncertain dynamical systems based on a low-fidelity (LF) model whose outcomes are corrected in a probabilistic sense to represent the more realistic outcomes of a high-fidelity (HF) model. In the model-correction approach utilized, the LF model is calibrated to the HF model close to the so-called most probable point in standard normal space, which allows a more realistic assessment of the considered complex dynamical system. Since only few expensive limit state function evaluations of the HF model are required, an efficient reliability analysis is enabled. In an application example, the LF model describes an existing single-span railway bridge modelled as simply supported Euler–Bernoulli beam subjected to moving single forces representing the axle loads of a moving train. The HF modelling approach accounts for the bridge–train interaction by modelling the passing train as mass-spring-damper system, however increasing the computational effort of the limit state function evaluations. Failure probabilities evaluated with the model-correction approach are contrasted and discussed with failure probabilities of the sophisticated bridge–train interaction model evaluated with the first-order reliability method (FORM). It is demonstrated that the efficiency of the method depends on the correlation between the LF and the HF model. A comparison of the results of FORM and the model-correction-based approach shows that the latter provides reliable failure probability prediction of the HF model while leading to a significant reduction in computational effort.

Analysis of linear dynamic systems of low rank

Response of multiple coupled dynamic systems

This work delves into the theory of dynamic systems, focusing on the analysis of entropy in both classical and topological contexts. Beginning with an exposition of fundamental concepts in dynamical systems theory, particular attention is given to topological dynamical systems (TDS). The discussion progresses to explore discrete topological entropy and its significance within dynamical systems, culminating in the introduction of topological entropy pressure as a nuanced form of this concept. The study then investigates various applications of topological entropy within dynamic systems, emphasizing its utility in understanding chaotic systems and its role in ergodic theory. A novel theory, termed Topological Ergodic Entropy Theory (TEET), is presented, offering a fresh perspective on the analysis of ergodic dynamical systems. Furthermore, the work introduces the Ergodic Theory of Turbulent Flow (ETTF), which probes the interplay between topological entropy and the ergodic properties of dynamic systems governed by the Navier-Stokes equations. Through these explorations, the findings contribute significantly to the comprehension of the intricate nature of dynamical systems and their diverse applications across mathematics and physics. By scrutinizing topological entropy and its implications in dynamical systems, this research offers novel insights into the chaotic and stochastic behaviors exhibited by these systems. Additionally, the introduction of pioneering theories like ETTF opens up new avenues for understanding and modeling turbulent phenomena, thereby enriching our understanding of complex dynamical processes.

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.

In this work, a the software project AnT is presented. This project is about a simulation and analysis tool for dynamical systems, whereby it was aimed right from the start, that not only a broad spectrum of dynamical systems is supported, but many investigation methods as well. This thesis is structured as follows: In chapter one (Introduction) a motivation and an overview about the main topics and the involved scientific disciplines is given and among others, the notions of simulation and analysis and their prerequisites are defined and described. The modeling and especially the mathematical modeling are considered in more detail as well as the tasks of interactive and non-interactive simulation. Chapter two (The AnT project) provides an overview about the software - its aims, requirements and features. Some historical and technical remarks are given together with some basic principles and facts about the functionality and a short guidance how to use this software and its features. In the third chapter (Supported classes of dynamical systems), the notion of a dynamical system is defined and described in this context and the most important classes of dynamical systems supported by the AnT software package including their main characteristic properties are listed. For each addresses class of dynamical systems at least one typical and illustrative representative is presented. In chapter four (Scanning dynamical systems), the notion of scanning is described and its importance in the field of nonlinear dynamics is emphasized. The different types of scan possibilities and procedures provided by the software are explained as well as the meaning of a scan item and a scan item sequence. Directly connected with the scanning of a dynamical system is the investigation or analysis of the dynamic behavior, because in most of the cases one is not only interested in the investigation under some fixed conditions but under varying conditions. For the analysis of dynamical systems, several methods are developed in the meanwhile. The methods, which are already implemented in the AnT software and hence available, are presented in chapter five (Supported investigation methods). The investigation of dynamical systems is the most important task in the field of nonlinear dynamics, because it represents the basis for prediction and forecasting of the dynamics as well as improvement and enhancement of the used or derived mathematical models. Therefore, a lot of examples are given in this chapter where the supported investigation methods are applied and illustrated. In chapter six (Simulating and investigating dynamical systems), a more detailed description of the software architecture and applied concepts is given. Especially the most important concepts of transitions and machines designed for the simulation and the investigation are explained and illustrated. The necessity of distributed computing is addressed to in chapter seven (Distributed computing), where the capability of the software to run in distributed mode to solve time consuming investigation tasks in parallel on several workstations or nodes of a cluster is described. The used client/server architecture is presented together with the developed network protocol. Many diagrams and figures in this work are based on the result of time consuming computations and could not have been prepared without the important distributed computing feature of the AnT software package. In chapters eight (AnT-gui: the graphical user interface) and nine (Visualization of dynamical systems), the graphical user interface, which guides a user through the initialization phase and the visualization capabilities of the software were presented, before in chapter ten (Numerical aspects of simulation) some remarks about numerics and scientific computing are elucidated. In chapter eleven (Examples), further illustrative examples of the functionality of the software are given including several scans not only with respect to system parameters but also with respect to initial values and even method parameters. The work closes with chapter twelve (Conclusion), where the main advantages of the AnT software package and the future extensions are presented, together with a short overview about related software projects.

A unified study of the applications of Volterra functional series to nonlinear-system analysis is presented with special emphasis on frequency-domain results which either have not been published before, or where rigour had been lacking. In particular, an in-depth analysis of the harmonic and intermodulation frequency components due to each Volterra kernel of a given order will be presented and explicit formulas which complement those already published will be given. The effects of manipulating various kernels, which need not be symmetric, on the overall response, as well as on the composite kernels will be investigated. These results are then applied to the analysis of dynamical systems described by a system of nonlinear state equations and a read-out map