scholarly journals Aggregation of a Class of Large-Scale, Interconnected, Nonlinear Dynamical Systems

2000 ◽  
Author(s):  
Swaroop Darbha ◽  
K. R. Rajagopal
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Large Scale ◽  
Nonlinear Dynamical Systems ◽  
Linear Time ◽  
Hierarchical Systems ◽  
Nonlinear Dynamical ◽  
Analysis And Design ◽  
Linear Time Invariant ◽  
Time Invariant

Abstract In a previous paper, we discussed the characteristics of a “meaningful” average of a collection of dynamical systems, and introduced as well as contructed a “meaningful” average that is not usually what is meant by an “ensemble” average. We also addressed the associated issue of the existence and construction of such an average for a class of interconnected, linear, time invariant dynamical systems. In this paper, we consider the issue of the construction of a meaningful average for a collection of a class of nonlinear dynamical systems. The construction of the meaningful average will involve integrating a nonlinear differential equation, of the same order as that of any member of the systems in the collection. Such an “average” dynamical system is not only attractive from a computational perspective, but also represents the macroscopic behavior of the interconnected dynamical systems. An average dynamical system can be used in the analysis and design of hierarchical systems.

scholarly journals Aggregation of a class of large-scale, interconnected, nonlinear dynamical systems

2001 ◽  
Vol 7 (4) ◽  
pp. 379-392 ◽  
Author(s):  
Swaroop Darbha ◽  
K. R. Rajagopal
Keyword(s):  
Dynamical System ◽  
Differential Equation ◽  
Dynamical Systems ◽  
Large Scale ◽  
Nonlinear Dynamical Systems ◽  
Initial Conditions ◽  
Hierarchical Systems ◽  
Nonlinear Dynamical ◽  
Analysis And Design ◽  
Computational Perspective

In this paper, the authors consider the issue of the construction of a meaningful average for a collection of nonlinear dynamical systems. Such a collection of dynamical systems may or may not have well defined ensemble averages as the existence of ensemble averages is predicated on the specification of appropriate initial conditions. A meaningful “average” dynamical system can represent the macroscopic behavior of the collection of systems and allow us to infer the behavior of such systems on an average. They can also prove to be very attractive from a computational perspective. An advantage to the construction of the meaningful average is that it involves integrating a nonlinear differential equation, of the same order as that of any member in the collection. An average dynamical system can be used in the analysis and design of hierarchical systems, and will allow one to capture approximately the response of any member of the collection.

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.

scholarly journals Complete Controllability of Impulsive Fractional Linear Time-Invariant Systems with Delay

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Xian-Feng Zhou ◽  
Song Liu ◽  
Wei Jiang
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Linear Time ◽  
Complete Controllability ◽  
Linear Time Invariant ◽  
Impulsive Fractional Differential Equations ◽  
Fractional Differential ◽  
Linear Time Invariant Systems ◽  
Time Invariant

Some flaws on impulsive fractional differential equations (systems) have been found. This paper is concerned with the complete controllability of impulsive fractional linear time-invariant dynamical systems with delay. The criteria on the controllability of the system, which is sufficient and necessary, are established by constructing suitable control inputs. Two examples are provided to illustrate the obtained results.

Carathe´odory Controllability Criterion for Nonlinear Dynamical Systems

1978 ◽  
Vol 100 (3) ◽  
pp. 209-213 ◽  
Author(s):  
G. Langholz ◽  
M. Sokolov
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Control Theory ◽  
Linear Systems ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical ◽  
Modern Control Theory ◽  
Open Question ◽  
Modern Control

The question of whether a system is controllable or not is of prime importance in modern control theory and has been actively researched in recent years. While it is a solved problem for linear systems, it is still an open question when dealing with bilinear and nonlinear systems. In this paper, a controllability criterion is established based on a theorem by Carathe´odory. By associating a given dynamical system with a certain Pfaffian equation, it is argued that the system is controllable (uncontrollable) if its associated Pfaffian form is nonintegrable (integrable).

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