Thermodynamic Modeling of Large-Scale Interconnected Systems

2011 ◽  
Author(s):  
Wassim M. Haddad ◽  
Sergey G. Nersesov
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Thermodynamic Modeling ◽  
Energy Flow ◽  
Large Scale ◽  
Type Inequality ◽  
Dynamical System Theory ◽  
Flow Models ◽  
Energy Equipartition ◽  
Definition Of

This chapter describes the thermodynamic modeling of large-scale interconnected dynamical systems. Using compartmental dynamical system theory, it develops energy flow models possessing energy conservation and energy equipartition principles for large-scale dynamical systems. It then gives a deterministic definition of entropy for a large-scale dynamical system that is consistent with the classical definition of entropy and shows that it satisfies a Clausius-type inequality leading to the law of nonconservation of entropy. It also introduces the notion of ectropy as a measure of the tendency of a dynamical system to do useful work and grow more organized. It demonstrates how conservation of energy in an isolated thermodynamic large-scale system leads to nonconservation of ectropy and entropy. Finally, the chapter uses the system ectropy as a Lyapunov function candidate to show that the large-scale thermodynamic energy flow model has convergent trajectories to Lyapunov stable equilibria determined by the system initial subsystem energies.

Thermodynamic Modeling for Discrete-Time Large-Scale Dynamical Systems

2011 ◽  
Author(s):  
Wassim M. Haddad ◽  
Sergey G. Nersesov
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Thermodynamic Modeling ◽  
Discrete Time ◽  
Large Scale ◽  
Type Inequality ◽  
Second Law Of Thermodynamics ◽  
Second Law ◽  
Thermodynamic Models ◽  
Definition Of

This chapter describes the thermodynamic modeling of discrete-time large-scale dynamical systems. In particular, it develops nonlinear discrete-time compartmental models that are consistent with thermodynamic principles. Since thermodynamic models are concerned with energy flow among subsystems, the chapter constructs a nonlinear compartmental dynamical system model characterized by conservation of energy and the first law of thermodynamics. It then provides a deterministic definition of entropy for a large-scale dynamical system that is consistent with the classical thermodynamic definition of entropy and shows that it satisfies a Clausius-type inequality leading to the law of entropy nonconservation. The chapter also considers nonconservation of entropy and the second law of thermodynamics, nonconservation of ectropy, semistability of discrete-time thermodynamic models, entropy increase and the second law of thermodynamics, and thermodynamic models with linear energy exchange.

Control Vector Lyapunov Functions for Large-Scale Impulsive Systems

2011 ◽  
Author(s):  
Wassim M. Haddad ◽  
Sergey G. Nersesov
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lyapunov Function ◽  
Lyapunov Functions ◽  
Large Scale ◽  
Lyapunov Theory ◽  
System Stability ◽  
Dynamical System Theory ◽  
Control Vector ◽  
Vector Lyapunov Functions

This chapter extends the notion of control vector Lyapunov functions to impulsive dynamical systems. Vector Lyapunov theory has been developed to weaken the hypothesis of standard Lyapunov theory to enlarge the class of Lyapunov functions that can be used for analyzing system stability. In particular, the use of vector Lyapunov functions in dynamical system theory offers a very flexible framework since each component of the vector Lyapunov function can satisfy less rigid requirements as compared to a single scalar Lyapunov function. Using control vector Lyapunov functions, the chapter develops a universal hybrid decentralized feedback stabilizer for a decentralized affine in the control nonlinear impulsive dynamical system that possesses guaranteed gain and sector margins in each decentralized input channel. These results are used to develop hybrid decentralized controllers for large-scale impulsive dynamical systems with robustness guarantees against full modeling and input uncertainty.

Large-Scale Impulsive Dynamical Systems

2011 ◽  
Author(s):  
Wassim M. Haddad ◽  
Sergey G. Nersesov
Keyword(s):  
Dynamical Systems ◽  
Lyapunov Functions ◽  
Large Scale ◽  
Nonlinear Dynamical Systems ◽  
Supply Rate ◽  
Nonlinear Dynamical ◽  
Dissipation Inequality ◽  
Vector Lyapunov Functions ◽  
Definition Of ◽  
Stability Results

This chapter develops vector dissipativity notions for large-scale nonlinear impulsive dynamical systems. In particular, it introduces a generalized definition of dissipativity for large-scale nonlinear impulsive dynamical systems in terms of a hybrid vector dissipation inequality involving a vector hybrid supply rate, a vector storage function, and an essentially nonnegative, semistable dissipation matrix. The chapter also defines generalized notions of a vector available storage and a vector required supply and shows that they are element-by-element ordered, nonnegative, and finite. Extended Kalman-Yakubovich-Popov conditions, in terms of the local impulsive subsystem dynamics and the interconnection constraints, are developed for characterizing vector dissipativeness via vector storage functions for large-scale impulsive dynamical systems. Finally, using the concepts of vector dissipativity and vector storage functions as candidate vector Lyapunov functions, the chapter presents feedback interconnection stability results of large-scale impulsive nonlinear dynamical systems.

Large-Scale Discrete-Time Interconnected Dynamical Systems

2011 ◽  
Author(s):  
Wassim M. Haddad ◽  
Sergey G. Nersesov
Keyword(s):  
Dynamical Systems ◽  
Discrete Time ◽  
Large Scale ◽  
Discrete Energy ◽  
Flow Balance ◽  
Subsystem Level ◽  
Coupling Strengths ◽  
Definition Of ◽  
Energy Dissipated ◽  
Discrete Time Dynamical Systems

This chapter develops vector dissipativity notions for large-scale nonlinear discrete-time dynamical systems. In particular, it introduces a generalized definition of dissipativity for large-scale nonlinear discrete-time dynamical systems in terms of a vector dissipation inequality involving a vector supply rate, a vector storage function, and a nonnegative, semistable dissipation matrix. On the subsystem level, the proposed approach provides a discrete energy flow balance in terms of the stored subsystem energy, the supplied subsystem energy, the subsystem energy gained from all other subsystems independent of the subsystem coupling strengths, and the subsystem energy dissipated. The chapter also develops extended Kalman–Yakubovich–Popov conditions, in terms of the local subsystem dynamics and the interconnection constraints, for characterizing vector dissipativeness via vector storage functions for large-scale discrete-time dynamical systems.

scholarly journals What is the dynamical hypothesis?

1998 ◽  
Vol 21 (5) ◽  
pp. 633-634 ◽  
Author(s):  
Nick Chater ◽  
Ulrike Hahn
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Turing Machines ◽  
Definition Of ◽  
Dynamical Hypothesis

Van Gelder's specification of the dynamical hypothesis does not improve on previous notions. All three key attributes of dynamical systems apply to Turing machines and are hence too general. However, when a more restricted definition of a dynamical system is adopted, it becomes clear that the dynamical hypothesis is too underspecified to constitute an interesting cognitive claim.

scholarly journals Ergodic Axiom: The Ontological Mistakes in Economics

2015 ◽  
Vol 5 (1) ◽  
pp. 47-65
Author(s):  
Ladislav Andrášik
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Large Population ◽  
Mathematical Formalism ◽  
Dynamical System Theory ◽  
Real Economy ◽  
Knowledge Based ◽  
Network Process ◽  
Ergodic Dynamical System ◽  
Living Organisms

Abstract There are several ontological and consequently also methodological mistakes in contemporary mainstream economics. Among them, the so-called ergodic axiom is play significant role. It is understandable that the real economy elaborated as formalized mental model looks like dynamic system on first sight. However, that is right only of dynamical systems in mathematical formalism. Economy that is in our understanding societal and/or collective economy is complex evolving organism. If we imagine such organism in the form of dynamical system that is as clear mathematical formalism, we are losing their crucial authentic character. The significant irredeemable attribute of societal economy is lying in his complex evolving network process character created by large population of people with different decision-making and complex realizing among them. Going from these imaginations the two entities in a question that is dynamical system with their ergodicity and societal economic organism as complex evolving network are qualitative very different ones. That is the reason why we cannot accede with endeavours to draw on living economy straitjacket of ergodic axiom. To articulate that cause by other words ergodic dynamical systems are applicable for physical and partly for chemical entities and only scarcely are fit for living organisms. On the other hand however, as clear method the ergodic dynamical system have good applying for didactical approaches in economics where helping in better understanding some types of complexities in dynamics. The purpose of that essay is to discuss problems around usability of ergodic dynamical system theory and methods in economics in the age of advanced ICT knowledge based society.

Four Quadrant Hybrid Control Oriented Dynamical System Model of Digital Displacement® Units

2018 ◽  
Author(s):  
Niels H. Pedersen ◽  
Per Johansen ◽  
Torben O. Andersen
Keyword(s):  
Dynamical System ◽  
System Theory ◽  
Dynamical Systems ◽  
Feedback Control ◽  
Hybrid Model ◽  
Dynamical System Theory ◽  
Modular Construction ◽  
Discrete Elements ◽  
Control Laws ◽  
Linear Differential

Proper feedback control of dynamical systems requires models that enables stability analysis, from where control laws may be established. Development of control oriented models for digital displacement (DD) fluid power units is complicated by the non-smooth behavior, which is considered the core reason for the greatly simplified state of the art control strategies for these machines. The DD unit comprises numerous pressure chambers in a modular construction, such that the power throughput is determined by the sequence of activated pressure chambers. The dynamics of each pressure chamber is governed by non-linear differential equations, while the binary input (active or inactive) is updated discretely as function of the shaft angle. Simple dynamical approximations based on continuous or discrete system theory is often inaccurate and is not applicable for such system when it is to operate in all four quadrants. Therefore, a method of applying hybrid dynamical system theory, comprising both continuous and discrete elements is proposed in this paper. The paper presents a physical oriented hybrid model accurately describing the machine dynamics. Since development of stabilizing control laws for hybrid dynamical systems is a complicated task, a simpler hybrid model only including the fundamental machine characteristics is beneficial. Therefore, a discussion and several proposals are made on how a simpler DD hybrid model may be established and used for feedback control development.

scholarly journals Tsallis Entropy of Fuzzy Dynamical Systems

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 264
Author(s):  
Dagmar Markechová
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Conditional Entropy ◽  
Tsallis Entropy ◽  
Positive Real ◽  
Fuzzy Dynamical System ◽  
Fuzzy Partitions ◽  
Definition Of ◽  
Mathematical Behavior ◽  
Fuzzy Dynamical Systems

This article deals with the mathematical modeling of Tsallis entropy in fuzzy dynamical systems. At first, the concepts of Tsallis entropy and Tsallis conditional entropy of order where is a positive real number not equal to 1, of fuzzy partitions are introduced and their mathematical behavior is described. As an important result, we showed that the Tsallis entropy of fuzzy partitions of order satisfies the property of sub-additivity. This property permits the definition of the Tsallis entropy of order of a fuzzy dynamical system. It was shown that Tsallis entropy is an invariant under isomorphisms of fuzzy dynamical systems; thus, we acquired a tool for distinguishing some non-isomorphic fuzzy dynamical systems. Finally, we formulated a version of the Kolmogorov–Sinai theorem on generators for the case of the Tsallis entropy of a fuzzy dynamical system. The obtained results extend the results provided by Markechová and Riečan in Entropy, 2016, 18, 157, which are particularized to the case of logical entropy.

scholarly journals Graph Labelings and Continuous Factors of Dynamical Systems

10.37236/2213 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Stephen M. Shea
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Topological Entropy ◽  
Directed Graphs ◽  
Sufficient Conditions ◽  
Perfect Graph ◽  
Shift Space ◽  
Finite Set ◽  
Vertex Set ◽  
Definition Of

A labeling of a graph is a function from the vertex set of the graph to some finite set.  Certain dynamical systems (such as topological Markov shifts) can be defined by directed graphs.  In these instances, a labeling of the graph defines a continuous, shift-commuting factor of the dynamical system.  We find sufficient conditions on the labeling to imply classification results for the factor dynamical system.  We define the topological entropy of a (directed or undirected) graph and its labelings in a way that is analogous to the definition of topological entropy for a shift space in symbolic dynamics.  We show, for example, if $G$ is a perfect graph, all proper $\chi(G)$-colorings of $G$ have the same entropy, where $\chi(G)$ is the chromatic number of $G$.

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