Bond Graph Modelling of Engineering Systems

This chapter introduces the reader to the concept-oriented approach to modeling that clearly separates ideal concepts from the physical components of a system when modeling its dynamic behavior for a specific problem context. This is done from a port-based point of view for which the domain-independent bond graph notation is used, which has been misinterpreted over and over, due to the paradigm shift that concept-oriented modeling in terms of ports requires. For that reason, the grammar and semantics of the graphical language of bond graphs are first defined without making any connection to the physical modeling concepts it is used for. In order to get a first impression of how bond graphs can represent models, an existing model is transformed into bond graphs as the transformation steps also give a good impression of how this notation provides immediate feedback on modeling decisions during actual modeling. Next, physical systems modeling in terms of bond graphs is discussed as well as the importance of the role of energy and power that is built into the semantics and grammar of bond graphs. It is emphasized that, just like circuit diagrams, bond graphs are a topological representation of the conceptual structure and should not be confused with spatial structure. By means of a discussion of some examples of such confusions it is explained why bond graphs have a slow acceptance rate in some scientific communities.

Model reduction refers to reducing the complexity of a given model to achieve a balance between model simplicity and accuracy. This chapter presents a set of model reduction techniques that are particularly amenable to bond graph models due to the common energy-based nature of these techniques and the bond graph language. Three techniques are presented that are developed with model order reduction, model partitioning, and simultaneous order and structure reduction in mind. Each technique utilizes a different energy-based metric that can be easily calculated from a bond graph model. These underlying metrics are presented first, followed by the algorithms, each with a simple illustrative example, as well as summaries of larger case studies performed with those algorithms to highlight their benefits. All three techniques are applicable to nonlinear models in differential–algebraic form, are realization preserving in the sense that the original meanings of the states and parameters are preserved, are trajectory dependent and thus explicitly take the specific inputs and parameter values into account, and can reduce models directly at the bond graph level.

Diagnosis of uncertain systems has been the subject of several recent research works (Djeziri et al. Proceeding of the 2007 American Control Conference 3017–3022, 2007; Han et al. 15th IFAC World Congress 1887–1892, 2002; Henry and Zolghari Control Engineering Practice 14:1081–1097, 2006; Hsing-Chia and Hui-Kuo Engineering Applications of Artificial Intelligence 17:919–930, 2004; Ploix Ph.D. de I.N.P.L, C.R.A.N 1998; Yan and Edwards Automatica 43:1605–1614, 2007). This interest is reflected by the fact that physical systems are complex and non-stationary and require more security and performance. The bond graph model in LFT form allows the generation of analytical redundancy relations (ARRs) composed of two completely separated parts: a nominal part, which represents the residuals, and an uncertain part which serves for both the calculation of adaptive thresholds and sensitivity analysis.

Incremental true bond graphs are used for a matrix-based determination of first-order parameter sensitivities of transfer functions, of residuals of analytical redundancy relations, and of the transfer matrix of the inverse model of a linear multiple-input–multiple-output system given that the latter exists. Existing software can be used for this approach for the derivation of equations from a bond graph and from its associated incremental bond graph and for building the necessary matrices in symbolic form. Parameter sensitivities of transfer functions are obtained by multiplication of matrix entries. Symbolic differentiation of transfer functions is not needed. The approach is illustrated by means of hand derivation of results for small well-known examples.

A bond graph method is used to examine qualitative aspects of a class of unstable under-actuated mechanical systems. It is shown that torque actuation leads to an unstabilisable system, whereas velocity actuation gives a controllable system which has, however, a right-half plane zero. The fundamental limitations theory of feedback control when a system has a right-half plane zero and a right-half plane pole is used to evaluate the desirable physical properties of coaxially coupled inverted pendula. An experimental system which approximates such a system is used to illustrate and validate the approach.

This chapter is concerned with the design of mechatronic systems on dynamic and energy criteria. Compared to the traditional trial–error–correction approach a methodology is presented that drastically decreases the number of simulation iterations and ensures more relevant solutions with respect to the specifications. Moreover, early in the design stages, this methodology enables to check if the design problem is well posed before any simulation. This verification is possible according to the structural analysis concept that points out the characteristic properties of the design models independently of the parameter numerical values. Also, the methodology is based on model inversion that uses straightforwardly the information written in the specifications. Finally, because of its ability to represent multidisciplinary physical systems, to acausally describe a model and to easily undertake a structural analysis, and to visualize the results of this analysis, the bond graph language is well dedicated to this methodology. In this chapter topics like design model validity, specifications validity, structural analysis, technological component specifications, selection and validation, and open-loop control determination will be discussed.

Diagnostic search strategy is based on knowledge representation, which is developed from a fundamental understanding of the system. For physical systems, causal or model-based knowledge may be broadly represented in qualitative or quantitative form. In quantitative models, this understanding is expressed in terms of mathematical and/or functional relationships between the inputs and outputs of the system. Bond graphs are an excellent means for causal knowledge representation. In this context, some of the recent analytical model-based fault detection and isolation (FDI) procedures are compiled in this chapter.

This chapter surveys the bond graph modeling of rotary electric machinery. The discussion includes the DC- and AC-machines commonly found in industrial applications ranging from a few hundreds of watts up to megawatts, i.e., brushed DC-machines in all their connection types as well as the synchronous and the induction machine. Most of the presentation adopts the electrical drive point of view, but the generator operation is also addressed in some cases, as this simply implies reversing the power flow in certain bonds, at least in the model world. Also discussed is a variety of machines used in low-power drives, like the permanent magnet (PM) synchronous, brushless DC, synchronous reluctance, PM stepper, and switched reluctance motors. Different modeling techniques are illustrated when surveying the different models. First, most of the bond graphs are derived from equivalent electric circuits with inductances representing the magnetic phenomena. Later, in order to explicitly show this domain, two further approaches are employed: one that, re-using the previous bond graphs, “opens” the I-elements to expose the magnetics and the other that starts the modeling process from scratch. In this later case, again two alternatives are presented, the first one interconnects components defined from constitutive relationships and the second one derives the models from energy conservation properties of ideal coupling fields. The chapter closes with simulation results obtained using the models developed along with it. The intention in writing this chapter has been to give to the reader a comprehensive overview of the subject, to offer a compendium of useful models to the practitioner, and, simultaneously, to provide methodological tools to help applied researchers to successfully develop their own models.

The use of multi-bond graphs (MBGs) has an increasing importance in the development of large mechanical systems, called multi-body systems (MBS), composed of a finite number of rigid bodies interconnected by kinematical constraints. The constitutive relationships of multi-bond resistors, transformers, and gyrators give way to zero-order causal paths (ZCPs) whose most important peculiarity is that their associated topological loops involve more than one direction. Two different methods are used to solve the ZCPs. With the first one, Lagrange multipliers are introduced by means of new flows and efforts as break variables of causal paths, adding constraint equations. With the second one, break variables are used directly to open the ZCPs. The procedure used solves the problem and implies the presence of new variables and constraint equations. Several algorithms have been developed to obtain the set of equations. The result is a set of differential–algebraic equations (DAEs) solved using a backward differential formulae (BDF) numerical method. An application to multi-body systems with a combination of classes of ZCPs will be shown.

Fuel cells are environmentally friendly futuristic power sources. They involve multiple energy domains and hence bond graph method is suitable for their modelling. A true bond graph model of a solid oxide fuel cell is presented in this chapter. This model is based on the concepts of network thermodynamics, in which the couplings between the various energy domains are represented in a unified manner. The simulations indicate that the model captures all the essential dynamics of the fuel cell and therefore is useful for control theoretic analysis.

Automating the modeling process of mechatronics systems can be achieved by the use of a two-step process. First, a systematic modeling technique for modeling systems with components in the mechanical, electric, hydraulic, thermal domains and second, the use of software to automate the process. The chapter presents the modeling process using block diagrams and bond graph methods so that a common understanding develops between the readers used to block diagrams and those used to bond graph methods. It guides the reader to automate the creation of computer models and then to computer simulation using software tools such as CAMPG, MATLAB, SIMULINK, and SYSQUAKE. Using an automated process, it guides the reader to perform simulation in the time and also in the frequency domain. Applications for nonlinear and complex mechatronics systems are presented.