Bond graph illustration

The bond graph of the transport across the membrane is shown in Figure 14.2 by a two-port resistance R element. The basic element of the bond graph is the ideal energy bond transmitting power without loss. A bond graph illustrates the system components and their interconnections with arrows, which indicate the positive direction of power flow associated with the transport processes. All time-dependent processes and all dissipative transformations are localized conceptually as capacity and resistance elements. Two ideal junctions are used in the method the 0-junction is defined... 

Fio. 12. Graph illustrating the dependence of the logarithm of retention factor for aromatic hydrocarbons on the carbon load of octadecyl silica bonded phases prepared from Par-tisil with octadecyhrichlorosilane. Mobile phase methanol-water (70 30) eluitest A, benzene A, naphthalene , phenanthrene , anthracene O, pyrene. Reprinted with permission from Herndon t al. (70). 

Alternatively the network equations can be solved by the method of simultaneous equations which is illustrated here for the case of CaCrFs whose bond graph is shown in Fig. A3.1. 

Clearly, for FDI it is necessary that a system is structurally observable. As switches temporarily disconnect and reconnect model parts they change the structure of a hybrid system model. Consequently, control properties, i.e. structural observability and structural controllability as well as characteristics of the mathematical model derived fl om the bond graph, i.e. the number of state variables, or the index of a DAE system become system mode dependent. Chapters briefly addresses these issues by confining to switched LTI systems and provides some small illustrating examples. 

As fault indicators should be sensitive to real faults but insensitive to parameter uncertainties, adaptive system mode dependent thresholds are needed for FDI in hybrid systems robust with regard to parameter uncertainties. Chapter 5 demonstrates that incremental bond graph can serve this purpose for switched LTI systems. To that end some basics of incremental bond graphs are recalled. It is shown how parameter sensitivities of ARRs and ARR thresholds can be obtained. A small example illustrates the approach. 

Chapter demonstrates that ARR residuals cannot only serve as fault indicators but may also be used for system mode identification in online FDI. First, the general case of switched LTI systems is considered and it is assumed that ARRs can be expressed in explicit form relating known system inputs and inputs either obtained by measurements or by simulation of the real system behaviour. A small example illustrates bond graph based system mode identification using ARRs. 

In Chap. 8, fault scenarios in small switched power electronic systems are studied for illustration. The bond graph model-based approach to FDI in hybrid systems is, however, not limited to this kind of systems. 

In order to account for the abstraction of ideal, no power consuming switching in a bond graph with invariant causalities that holds for all system modes, Umarikar extended 0- and 1-junctions by allowing for more than one bond to impose an effort on a 0-junction and more than one bond imposing a flow on a 1-junction with the constraint that only one of the causality imposing bonds is active at a time instant [12, 13], These extensions are called switched power junctions and are not to be confused with controlled junctions to be referred to subsequently. Figure 2.2 illustrates the idea. 

The bond graph of a boost converter in Fig. 2.3 illustrates the use of these extended junctions. 

For illustration, consider the example of a buck converter displayed in Fig.2.19. As can be seen, in this bond graph in preferred integral causality, the inductor I L is in derivative causality. The bond graph captures the physical feasible system mode in which the two switches are off (m2 = 0 a mi =0). However, derivative causality at the I-element indicates that the inductor current is not observable in this mode. If one of the two switches is closed, its causality can be changed into resistive causality. As... 

For illustration of the bond graph-based test for structural observability consider the switched circuit in Fig. 3.4. Figure3.5 displays the bond graph of the switched network. 

The derivation of ARRs from a bond graph of a hybrid system model that holds for all system modes is illustrated by means of a simple network with one switch and elements with a linear constitutive equation displayed in Fig. 4.1. 

For illustration. Fig. 4.5 reproduces the diagnostic bond graph of the switched network in Fig. 4.1. 

As an example, the circuit with one switch in Fig. 4.1, is considered. To keep the illustration of the procedure short and simple it is assumed that only one parameter is uncertain. Accordingly, the incremental bond graph is obtained by replacing the element by its incremental model and by replacing the constant voltage source Se Vi by an effort source of value zero and by replacing detectors by dual virtual detectors for the variations of ARR residuals. 

Figure5.15 illustrates this situation assuming that measurement uncertainties are additive. A flow / = f + Af with a predicted part / and an uncertain part A/ due to measurement uncertainty is the output of a non-ideal sensor and an input into the diagnostic bond graph. The input / into the diagnostic bond graph results in an effort e = e + Ae that controls a modulated sink MSe where e denotes the predicted part and Ae the uncertain part. The output w = 5(e - - Ae )A0 of the modulated sink is an input into the incremental bond graph that is needed to compute the variation Ar of an ARR residual r.

In the following, first, sensitivity pseudo bond graphs are briefiy reviewed and are then used to obtain residual sensitivity functions needed for the previously presented least squares ARR residuals minimisation. The simple hybrid network in Fig.4.1 is used again for illustration of the approach. 

The previous chapters address various aspects of quantitative bond graph-based FDI and system mode identification for systems represented by a hybrid model. This chapter illustrates applications of the presented methods by means of a number of small case studies. The examples chosen are widely used switched power electronic systems. Various kinds of electronic power converters, e.g. buck- or boost converters, or DC to AC converters are used in a variety of applications such as DC power supplies for electronic equipment, battery chargers, motor drives, or high voltage direct current transmission line systems [1]. 

The systematic manual derivation of equations from a causal bond graph shall be illustrated by considering the simple circuit depicted in Fig. B.6. 

The book briefly recalls various bond graph representations of hybrid system models proposed in the literature. The development of hybrid models for the purpose of fault detection and isolation, in this book, makes use of conceptual nonideal switches representing devices for which it is justified to abstract their fast state transitions into instantaneous discrete state switches and accounts for stmctural model changes by special sources that are switched on or off at the advent of a discrete event. As other possible approaches, this approach has its pros and cons. For illustration, the presented method is applied in a number of elaborated case studies that consider fault scenarios for switched power electronic systems that are commonly used in a variety of applications. Power electronic systems have been chosen because they may be appropriately described by a hybrid model and are well suited for application of the presented bond graph model-based approach to fault detection and isolation. The approach, however, is not limited to this kind of systems. 

Abstract 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. 

The second illustrative example is the well-known voltage-driven separately excited DC motor that drives a mechanical load against an external moment (Fig. 4.10). Figure 4.11 shows a direct bond graph model. Like the previous example, this model also has two inputs and two outputs. That is, a transfer matrix H with four transfer functions Fij can be derived ... 

In [14], the author of this chapter briefly showed that incremental bond graphs can also be used to determine parameter sensitivities of the residuals of analytical redundancy relations (ARRs) used in model-based fault detection and isolation. This section elaborates this aspect and gives an illustration. 

For small linear models, the above steps can be carried out by hand as has been shown by means of the illustrating examples. In any case, no symbolic differentiation of transfer functions has to be performed. The use of an incremental bond graph means that the total differential of constitutive equations has already been taken. 

Abstract 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. 

Therefore, the bond graph of a single rigid body is as illustrated in Fig. 9.12. 

By way of example. Fig. 9.16 illustrates two types of constraints and the resulting bond graph in each case. 

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