Fundamentals of Event-Continuous System Simulation Theory (200,00 руб.)

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UDC 004.94(075.8) S 55 Reviewers: Professor V. V. Aksenov, D.Sc. (Phys. & Math.), Professor A. A. Voevoda, D.Sc. (Tech.) This publication was conducted within InMotion project (Innovative teaching and learning strategies in open modelling and simulation environment for student-centered engineering education (573751-EPP-12016-1-DE-EPPKA2-CBHE-JP). This project has been funded with support from the European Commission. This publication reflects the views only of the authors, and the Commission cannot be held responsible for any use which may be made of the information contained therein. S 55 Shornikov Yu. V. Fundamentals of Event-Continuous System Simulation Theory : Textbook / Yu. V. Shornikov, D. N. Dostovalov. – Novosibirsk : NSTU Publisher, 2018. – 175 p. ISBN 978-5-7782-3773-5 Effective computer analysis of event-continuous and hybrid systems is addressed. A multipurpose software architecture employing control of the integration step size with regard to the error, stability, and unilateral events is proposed. The problem of synchronization of continuous and discrete processes is dealt with. All new theoretical concepts are tested on heterogeneous applications to biological systems, large electric power systems, mechanical engineering and chemical kinetics problems. ISBN 978-5-7782-3773-5 © © Shornikov Yu.V., Dostovalov D.N., UDC 004.94(075.8) 2018 Novosibirsk State Technical University, 2018

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Contents Preface 1 Event-Continuous Systems 7 9 1.1 Discrete-Continuous Models . . . . . . . . . . . . . . . . . . . . . 9 1.2 Continuous Models . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2.1 Solution Dependence on the Initial Conditions . . . . . . . 17 1.2.2 Lyapunov Stability . . . . . . . . . . . . . . . . . . . . . . 17 1.2.3 Caratheodory’s Conditions . . . . . . . . . . . . . . . . . . 18 1.3 Discrete Models and Zeno Behavior . . . . . . . . . . . . . . . . . 21 1.3.1 Zeno Phenomenon . . . . . . . . . . . . . . . . . . . . . . 22 1.3.2 Harel Statecharts . . . . . . . . . . . . . . . . . . . . . . . 25 1.4 Modes and Events . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.5 Local and Global Behavior . . . . . . . . . . . . . . . . . . . . . . 28 1.6 Discontinuity Classification . . . . . . . . . . . . . . . . . . . . . . 29 1.6.1 Change of Initial Conditions . . . . . . . . . . . . . . . . . 29 1.6.2 Change of the Values of Right-Hand Side Parameters . . . 30 1.6.3 Changing the Right-Hand Side Form without Changing the Set of Continuous State Variables . . . . . . . . . . . . . . 32 1.6.4 Changing a Hybrid System Mode Right-Hand Side along with Changing the Set of Continuous State Variables . . . 33 2 Mathematical Foundations of HS Mode Numerical Analysis 37 2.1 Choosing a Numerical Scheme . . . . . . . . . . . . . . . . . . . . 37 2.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.4 Runge-Kutta Methods . . . . . . . . . . . . . . . . . . . . . . . . 42 2.5 Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.6 Accuracy (Error) Control . . . . . . . . . . . . . . . . . . . . . . . 44 2.7 Stability Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.8 Step Size Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.8.1 Step Size Control with Respect to the Error . . . . . . . . 47 2.8.2 Step Size Control with Respect to the Stability . . . . . . 47

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2.8.3 Step Size Control with Respect to the Error and Stability . 48 2.9 Method of Order Two . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.10 Adams’ Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3 Correct Detection of Discrete Events 53 3.1 Hybrid System’s Singular Regions . . . . . . . . . . . . . . . . . . 53 3.2 Problem of Correct Discrete Event Detection . . . . . . . . . . . . 54 3.3 Linearization and the Relaxation Method in Event Localization . 56 3.3.1 Event Function Linearization . . . . . . . . . . . . . . . . 57 3.3.2 Relaxation Method in Event Detection . . . . . . . . . . . 58 3.4 Ensuring Asymptotic Approaching the Event Surface for Explicit Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.4.1 Detection Algorithm with a One-Step Method of Order Two 60 3.4.2 Adams’ Method in Event Detection . . . . . . . . . . . . . 62 3.4.3 L–Stable Method in Event Detection . . . . . . . . . . . . 66 3.5 Hybrid Systems with Nontrivial Event Functions . . . . . . . . . . 71 4 Software 75 4.1 Architecture of the Modeling and Simulation Environment . . . . 75 4.2 Visual Computer Models . . . . . . . . . . . . . . . . . . . . . . . 78 4.2.1 User-Defined (Macro) Blocks . . . . . . . . . . . . . . . . . 79 4.2.2 Data Import . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.3 Textual Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.3.1 Specification of Discrete Behavior . . . . . . . . . . . . . . 84 4.3.2 Specification of Continuous Behavior . . . . . . . . . . . . 88 4.3.3 Macros in Textual Description . . . . . . . . . . . . . . . . 90 4.4 Block-Textual Models . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.5 Computer Model Analysis . . . . . . . . . . . . . . . . . . . . . . 96 4.5.1 Textual Model Analysis . . . . . . . . . . . . . . . . . . . 96 4.5.2 Visual Computer Model Analysis . . . . . . . . . . . . . . 99 4.6 Graphical Interpretation of Simulation Results . . . . . . . . . . . 102 5 Software Unification 105 5.1 Topicality and Problem Statements . . . . . . . . . . . . . . . . . 105 5.1.1 Chemical Kinetics . . . . . . . . . . . . . . . . . . . . . . . 105 5.1.2 Models with Distributed Properties . . . . . . . . . . . . . 106 5.2 Construction of Chemical Kinetics Differential Equations . . . . . 107 5.2.1 Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.2.2 Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.3 Supported Types of Partial Differential Equations . . . . . . . . . 111 5.3.1 Textual Language LISMA_PDE . . . . . . . . . . . . . . . 112

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5.3.2 Modeling and Simulation of an HS with Distributed Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 6 Modeling and Simulation Examples 117 6.1 Model of Two Tanks with Sluggish Valves . . . . . . . . . . . . . 117 6.2 Interactive Simulation . . . . . . . . . . . . . . . . . . . . . . . . 120 6.3 Production–Distribution System Model . . . . . . . . . . . . . . . 122 6.4 Transient Heat Conduction Model . . . . . . . . . . . . . . . . . . 129 6.5 Ring Modulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 6.6 Biosystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 6.6.1 Modeling and Simulation of Diffusion . . . . . . . . . . . . 136 6.6.2 Computer Modeling and Simulation of the Biliary System 138 A Visual Modeling Languages of the ISMA Environments B Shortened Version of the LISMA_PDE Grammar C List of Handled Semantic Errors 153 161 165 169 171 6.7 Power Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . 144 Bibliography D Symbolic Computer Model of the Production-Distribution System

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