Direct Determination of Reduced Models of a Class of Singularly Perturbed Nonlinear Systems on Three Time Scales in a Bond Graph Approach

Juancarlos Méndez-Barriga,Gilberto Gonzalez-Avalos,Gerardo Ayala-Jaimes,Aaron Padilla Garcia,Noe Barrera Gallegos

doi:10.3390/sym14010104

Juancarlos Méndez-Barriga, Gilberto Gonzalez-Avalos + Show 3 more

Open Access

https://doi.org/10.3390/sym14010104

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Abstract

One of the most important features in the analysis of the singular perturbation methods is the reduction of models. Likewise, the bond graph methodology in dynamic system modeling has been widely used. In this paper, the bond graph modeling of nonlinear systems with singular perturbations is presented. The class of nonlinear systems is the product of state variables on three time scales (fast, medium, and slow). Through this paper, the symmetry of mathematical modeling and graphical modeling can be established. A main characteristic of the bond graph is the application of causality to its elements. When an integral causality is assigned to the storage elements that determine the state variables, the dynamic model is obtained. If the storage elements of the fast dynamics have a derivative causality and the storage elements of the medium and slow dynamics an integral causality is assigned, a reduced model is obtained, which consists of a dynamic model for the medium and slow time scales and a stationary model of the fast time scale. By applying derivative causality to the storage elements of the fast and medium dynamics and an integral causality to the storage elements of the slow dynamics, the quasi-steady-state model for the slow dynamics is obtained and stationary models for the fast and medium dynamics are defined. The exact and reduced models of singularly perturbed systems can be interpreted as another symmetry in the development of this paper. Finally, the proposed methodology was applied to a system with three time scales in a bond graph approach, and simulation results are shown in order to indicate the effectiveness of the proposed methodology.

Highlights

The modeling, analysis, and control of high-order systems are an interesting challenge to give adequate results
Consider a class of nonlinear systems with singular perturbations modeled by bond graphs where the storage elements that represent slow and medium dynamics have an integral causality assignment, and for the storage elements for the fast dynamics, a derivative causality is assigned, whose junction structure is defined by:
The algebraic loops matrix NH ( x ) is nonsingular. This new model reduced in a bond graph approach is obtained by assigning derivative causality to the storage elements of the medium and fast dynamics and to the storage elements of the slow dynamics, and an integral causality is assigned

Summary

Introduction

The modeling, analysis, and control of high-order systems are an interesting challenge to give adequate results. Reduced models of LTI systems with singular perturbations on two time scales with different causalities were proposed in [19]. LTI systems with three time scales, their modeling, and their reduction with mixed causalities in bond graphs were introduced in [24]. Bond graph models of a class of nonlinear systems with singular perturbations on three time scales (fast, medium, slow) are proposed. These systems can have linearly independent and dependent state variables on each time scale. Thermal and chemical systems have been modeled with bond graphs [15] These systems generally determine nonlinear models on various time scales, so this paper can be useful to analyze the behavior.

Singular Perturbation Method

Singularly Perturbed Systems in a Bond Graph Approach

Case Study as an Illustrative Example

Vf Rapplied

Conclusions