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.

Stability Criteria for Distributed Nonlinear Sampled-Data Systems Defined by Green’s Functions

1974 ◽  
Vol 96 (3) ◽  
pp. 315-321 ◽  
Author(s):  
G. Jumarie
Keyword(s):  
Stability Criterion ◽  
Feedback Gain ◽  
Continuous Systems ◽  
Mean Square ◽  
Data Systems ◽  
Sampled Data ◽  
Input Output ◽  
Output Stability ◽  
Lumped Parameter ◽  
The Mean

Sampled-data, nonlinear, distributed systems, which exhibit a structure similar to that of the standard closed loop with lumped parameter, are investigated from the viewpoint of their input-output stability. These systems are governed by operational equations involving discrete Laplace-Green kernels. Their feedback gains are bounded by upper and lower values which depend explicitly on the time and the distributed parameter. The main result is: an input-output stability theorem is given which applies both in L∞ (O, ∞) and L2 (O, ∞). This criterion, which may be considered as being an extension of the ≪circle criterion≫, involves the mean square value on the bounds of the feedback gain. Stability conditions for continuous systems are derived from this result. In the special case of systems with distributed periodical time-varying feedback gains, a stability criterion is given which applies in Marcinkiewicz space M2 (O, ∞). This result which involves the mean square value of the feedback gain is generally less restrictive than the L2 (O, ∞) stability criterion mentioned above.

Space Sampling Effects in Nonlinear Distributed System Stability

1981 ◽  
Vol 103 (1) ◽  
pp. 54-60 ◽  
Author(s):  
Guy Jumarie
Keyword(s):  
Distributed Control ◽  
Distributed System ◽  
Absolute Stability ◽  
Transfer Functions ◽  
Point Of View ◽  
System Stability ◽  
Distributed Parameter ◽  
Practical Point ◽  
Output Stability ◽  
Constant Coefficients

The purpose of this paper is to determine, from a practical point of view, the effects of a space sampling on the input-output stability, and the absolute stability of distributed parameter systems with distributed control. Three models are proposed which involve, respectively, transfer functions in space and time, Laplace-Green’s functions, and distributed transfer functions. It is shown that the first model is the most pessimistic, but in contrast is the easiest to use. While the first model applies to system with constant coefficients only, the other ones can be used with constant or space dependent coefficients.

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.

scholarly journals Vibroacoustic Assessment of an Innovative Composite Material for the Roof of a Coupe Car

2021 ◽  
Vol 11 (3) ◽  
pp. 1128
Author(s):  
Nunziante Cascone ◽  
Luca Caivano ◽  
Giuseppe D’Errico ◽  
Roberto Citarella
Keyword(s):  
Composite Material ◽  
Transfer Functions ◽  
Realistic Model ◽  
Point Of View ◽  
Damping Capacity ◽  
Geometry Structure ◽  
Fiberglass Composite ◽  
New Material ◽  
Static Rigidity ◽  
Innovative Material

The objective of this paper is the vibroacoustic evaluation of an innovative material for a sports car roof, aiming at replacing fiberglass composite materials. Such evaluation was carried out using numerical and experimental analysis techniques, with cross-comparison between the corresponding results. The innovative material under analysis is a composite material, with a thermoplastic polypropylene matrix and reinforcement made of cellulose fibers. In order to validate the virtual dynamic modeling of the new material, the inertance on different points of some sheets made of the material under analysis was evaluated by an in-house made experimental activity, performed in the CRF (Fiat Research Center) test room, and cross-compared with corresponding results from a numerical analysis performed with the MSC Nastran software. Then, a realistic model of the car roof of the Alfa Romeo 4C car, made with the new material, was implemented and analyzed from the vibroacoustic point of view. The mere switch to the new material, with no changes in the geometry/structure of the car roof, did not allow preserving the original values of static rigidity, dynamic rigidity, and configuration of modal shapes. For this reason, a geometric/structural optimization of the component was performed. Once the new geometry/structure was defined, a vibroacoustic analysis was carried out, checking for a possible coupling between the fluid cavity modes and the structure car body modes. Finally, the vibroacoustic transfer functions to the driver’s ear node were assessed, considering two different excitation points on the structure. The excellent damping capacity of the proposed material led to an improvement in the vibroacoustic transfer functions and to a reduction in the weight of the pavilion.

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