A Space Function Approach to the Hyperstability and the Average Hyperstability of Distributed Systems With Space-Varying Linear Parts and Time-Varying Gains

1975 ◽  
Vol 97 (4) ◽  
pp. 345-353 ◽  
Author(s):  
G. Jumarie
Keyword(s):  
Distributed Systems ◽  
Transfer Function ◽  
Broad Class ◽  
Linear Part ◽  
Function Approach ◽  
Distributed Parameter ◽  
Time Varying ◽  
Function Concept ◽  
Distributed Transfer Function ◽  
Hyperstability Theory

We propose an extension of the Popov’s hyperstability theory which applies to a class of single-control distributed systems in which the linear part depends explicitely upon the distributed parameter, z. The nonlinearness of these systems is expressed by means of the control. The main features of our results are the following: (i) The hyperstability conditions that we obtain involve specific z-dependent functions which we can consider as being extensions of the transfer function concept; (ii) they also involve integrals with respect to the distributed parameter, which express an averaging effect of this latter. Then systems in which the admissible controls are defined via time-varying conditions are investigated. For such systems, we define the concept of “average hyperstability” in time, and average hyperstability conditions are given. Similar problems are solved for multi-control distributed systems. As an application we show how these results yield a broad class of absolute stability conditions for distributed systems: they are space averaging conditions and they may apply when other criteria are in-applicable. Three examples are given: the last one illustrates how a space-describing function approach can be used to determine the distributed transfer function of the system.

Port-Based Modeling of Distributed-Lumped Parameter Systems

2010 ◽  
Vol 164 ◽  
pp. 183-188
Author(s):  
Cezary Orlikowski ◽  
Rafał Hein
Keyword(s):  
Distributed Systems ◽  
Transfer Function ◽  
Distributed System ◽  
Function Method ◽  
Input Data ◽  
Distributed Parameter ◽  
Transfer Function Method ◽  
Lumped Parameter ◽  
Concise Representation ◽  
Distributed Transfer Function

This paper presents a uniform, port-based approach for modeling of both lumped and distributed parameter systems. Port-based model of the distributed system has been defined by application of bond graph methodology and distributed transfer function method (DTFM). The proposed approach combines versatility of port-based modeling and accuracy of distributed transfer function method. A concise representation of lumped-distributed systems has been obtained. The proposed method of modeling enables to formulate input data for computer analysis by application of DTFM.

Transfer Function Analysis of Non-Uniformly Distributed Parameter Systems

1993 ◽  
Author(s):  
Bingen Yang ◽  
Houfei Fang
Keyword(s):  
Distributed Systems ◽  
Transfer Function ◽  
State Space ◽  
Fundamental Matrix ◽  
Function Analysis ◽  
Distributed Parameter Systems ◽  
System Response ◽  
Distributed Parameter ◽  
Transfer Function Analysis ◽  
Function Formulation

Abstract This paper studies a transfer function formulation for general one-dimensional, non-uniformly distributed systems subject to arbitrary boundary conditions and external disturbances. The purpose is to provide an useful alternative for modeling and analysis of distributed parameter systems. In the development, the system equations of the non-uniform system are cast into a state space form in the Laplace transform domain. The system response and distributed transfer functions are derived in term of the fundamental matrix of the state space equation. Two approximate methods for evaluating the fundamental matrix are proposed. With the transfer function formulation, various dynamics and control problems for the non-uniformly distributed system can be conveniently addressed. The transfer function analysis is also applied to constrained/combined non-uniformly distributed systems.

An Augmented State Formulation for Modeling and Analysis of Multibody Distributed Dynamic Systems

2014 ◽  
Vol 81 (5) ◽  
Author(s):  
K. Noh ◽  
B. Yang
Keyword(s):  
Distributed Systems ◽  
Transfer Function ◽  
Dynamic Systems ◽  
Analytical Method ◽  
Dynamic Problems ◽  
Modeling And Analysis ◽  
Closed Form Solutions ◽  
Time Domains ◽  
Augmented State ◽  
Distributed Transfer Function

Multibody distributed dynamic systems are seen in many engineering applications. Developed in this investigation is a new analytical method for a class of branched multibody distributed systems, which is called the augmented distributed transfer function (DTFM). This method adopts an augmented state formulation to describe the interactions among multiple distributed and lumped bodies, which resolves the problems with conventional transfer function methods in modeling and analysis of multibody distributed systems. As can be seen, the augmented DTFM, without the need for orthogonal system eigenfunctions, produces exact and closed-form solutions of various dynamic problems, in both frequency and time domains.

scholarly journals On the Design of Hyperstable Feedback Controllers for a Class of Parameterized Nonlinearities. Two Application Examples for Controlling Epidemic Models

2019 ◽  
Vol 16 (15) ◽  
pp. 2689 ◽  
Author(s):  
Manuel De la Sen
Keyword(s):  
Transfer Function ◽  
Closed Loop ◽  
Equilibrium Points ◽  
Linear Part ◽  
Type Inequality ◽  
Epidemic Models ◽  
Time Varying ◽  
Input Output ◽  
Controlled System ◽  
Feed Forward

This paper studies the hyperstability and the asymptotic hyperstability of a single-input single-output controlled dynamic system whose feed-forward input-output dynamics is nonlinear and eventually time-varying consisting of a linear nominal part, a linear incremental perturbed part and a nonlinear and eventually time-varying one. The nominal linear part is described by a positive real transfer function while the linear perturbation is defined by a stable transfer function. The nonlinear and time-varying disturbance is, in general, unstructured but it is upper-bounded by the combination of three additive absolute terms depending on the input, output and input-output product, respectively. The non-linear time-varying feedback controller is any member belonging to a general class which satisfies an integral Popov’s-type inequality. This problem statement allows the study of the conditions guaranteeing the robust stability properties under a variety of the controllers designed for the controlled system and controller disturbances. In this way, set of robust hyperstability and asymptotic hyperstability of the closed-loop system are given based on the fact that the input-output energy of the feed-forward controlled system is positive and bounded for all time and any given initial conditions and controls satisfying Popov’s inequality. The importance of those hyperstability and asymptotic hyperstability properties rely on the fact that they are related to global closed-loop stability, or respectively, global closed-loop asymptotic stability of the same uncontrolled feed-forward dynamics subject to a great number of controllers under the only condition that that they satisfy such a Popov’s-type inequality. It is well-known the relevance of vaccination and treatment controls for Public Health Management at the levels of prevention and healing. Therefore, two application examples concerning the linearization of known epidemic models and their appropriate vaccination and/or treatment controls on the susceptible and infectious, respectively, are discussed in detail with the main objective in mind of being able of achieving a fast convergence of the state- trajectory solutions to the disease- free equilibrium points under a wide class of control laws under deviations of the equilibrium amounts of such populations.

Input-Output Stability Criteria for a Broad Class of Stochastic Nonlinear Distributed Systems Defined by Green’s Function

1975 ◽  
Vol 97 (1) ◽  
pp. 83-91 ◽  
Author(s):  
G. Jumarie
Keyword(s):  
Distributed Systems ◽  
Transfer Functions ◽  
Broad Class ◽  
Linear Part ◽  
Point Of View ◽  
Feedback Gain ◽  
Stability Criteria ◽  
Nonlinear Feedback ◽  
Input Output ◽  
Output Stability

This paper deals with the input-output stability of a class of nonlinear distributed systems defined by their Laplace-Green’s functions, or similarly from a practical point of view, by their distributed transfer functions. The time dependent nonlinear feedback element is distributed and bounded by two limiting gains which depend explicitly upon the distributed parameter. These systems are disturbed by a state and space dependent Gaussian noise which is added to the input of their linear components. This noise depends explicitly upon the output of the system via its own nonlinear feedback gain. Some input-output stability criteria are stated, which can be considered as being stochastic distributed versions of the circle criterion available for deterministic lumped parameter systems. They involve the stochastic mean square norm and they are expressed in term of the relative positions, in the complex plane, of a circle which depends upon the nonlinearities and the variance of the noise on the one hand; and a locus which may be interpreted as being the Nyquist locus of the linear part on the other hand.

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