Power variables

Type of cell Gas pressure (Torr) Calibration ratio (fiW/mm) Constant thermal power Variable thermal power... 

Figure 8.18. Comparative chromatogram at two different pHs near the optimum pH of 9.1. These data predicted some robustness issues toward mobile phase pH since it is close to that of the pKa of the APIs. A small change of pH would bring dramatic change to the elution order of some analytes. However, it should also be noted that pH is a powerful variable for fine-tuning the resolution of complex samples.

Fig. 5.3. The reduced nuclear and electronic stopping cross-sections as a function of em. The electronic stopping power variable, k, is dependent on the mass and atomic number of the ion and target...

The choice of electrode material is a powerful variable in optimizing selectivity in LCEC. In optical detection schemes, the detector cell functions merely as a passive container through which to direct flow. In electrocheiriis-... 

Remark 2.1 Van der Schaft and Schuhmacher call variables such as the power variables of the bond graph switch element Sw complementary variables in the sense that for the two of them an inequality holds and for all times at least one of them is strictly an equality. Systems in which mode switching is determined by complementarity conditions they call complementary systems [22]. In the case that both variables can be assumed to be nonnegative, the complementarity condition is often expressed as... 

For some systems with structural changes such as a clutch, storage elements may temporarily become dependent for the duration of a system mode. In such a case, a residual sink may be switched on that delivers a power variable so that the conjugate power variable vanishes and storage elements can keep integral causality. As their state variables jump to a new joint value, numerical integration has to be re-initialised at such a discrete event. 

It is sufficient to quantise the static relation between theenergy variable of a storage element and its output power variable. Consider, e.g. the linear 1-port C element in integral causality in Fig.2.24 and let qq(t) = floor(g(f)) the quantisation of its displacement q. The continuous constitutive equations of a C element with the capacitance C... 

Derivation of ARRs from a diagnostic bond graph then starts by summing power variables at all those junctions that have a BG sensor element in inverted causality attached to it. At first, these balances of power variables will contain unknown variables. They may be eliminated by following causal paths and by using constitutive equations of bond graph elements. The result may be a set of ARRs in closed symbolic form [cf. (4.2)] if nonlinear constitutive element equations permit necessary... 

The sum of power variables at the junctions in the bond graph of Fig. 4.2 read... 

Notice that element C Ci in derivative causality and the switch Sw in conductance causality do not allow a flow sensor Df / in inverted causality attached to junction 11. The non-inverted flow sensor Df / has been replaced by a virtual detector Df r for residual n and a modulated flow source MSf /. Virtual detectors are distinguished from detectors of power variables by an asterisk. The modulated flow source delivers a measured flow /. 

For ARRs in closed symbolic form, parameter sensitivities of ARR residuals can be obtained by symbolic differentiation. In case an explicit formulation of ARRs is not achievable, e.g. due to nonlinear algebraic loops, parameter sensitivities of ARR residuals can be numerically computed by using a sensitivity bond graph, in which bonds carry sensitivities of power variables [12-14], or by using incremental bond graphs, in which bonds carry variations of power variables [5]. In Chap. 5, incremental bond graphs are used for the determination of adaptive fault thresholds. 

The measured variables ji obtained from the faulty system model and the outputs yi of the non-faulty system model may be coupled by feeding the differences y,- - y,-into modulated sinks that deliver an output r, so that their input becomes zero. These sinks are termed residual sinks. The output r, of a residual sink is a power variable and is inserted into the balance of power variables at that junction in the non-faulty system model from which its co-variable y, is obtained. The outputs of the residual sinks are additional inputs into the non-faulty system model that force the faultless system to alter its behaviour so that it becomes identical to the one of the faulty system. If no fault is introduced into the real system model, then there are no differences and all values r, (f) are close to zero. Differences, however, lead to values r, (r) that remain distinguishably different from zero as long as a fault is effective, i.e. is not repaired. That is, the outputs of the residual sinks can serve as fault indicators. The balance of power variables at a junctions in the non-faulty system model connected to a residual sink becomes an ARR when unknowns have been eliminated and the output of the residual sink becomes the residual of that ARR. 

There is no need to set up sums of power variables at junctions to which a sensor has been attached and to eliminate unknown variables in order to obtain ARRs in symbolic form. The DAE system is of index 2 if... 

As to FDI robust with regard to parameter uncertainties, an approach based on so-called uncertain bond graphs in linear fractional transformation form (LFT) has been reported in the literature [8-10] for time-continuous models. In an uncertain bond graph, bonds carry power variables uncertain with regard to parameter variations... 

An incremental bond graph can be constructed in a systematic manner from the original bond graph of a switched LTI system by replacing an element that is due to parameter variations by its incremental element model. Equations for variations of power variables can be automatically derived in the same way as they are derived from an initial bond graph with nominal parameters. 

The underlying idea of incremental bond graphs is that if a parameter 0 of a component model varies, then both power variables at its port are perturbed due to its interaction with the ports of other elements in the model. That is, an effort (f) in a bond graph with nominal parameters becomes e t) = en t) + Ae t). The same holds for the conjugate power variable /(r). In incremental bond graphs, bonds carry the increments Aeif), Af(l) of power variables. In other words, they represent energy flows carrying the amount of power Ae t) Af(t). 

The power variable fc controlling the modulated source is an output variable of the original bond graph model. If fc has been obtained by measurements of the real system, then the contribution to the output of the incremental bond graph model of the C element may contain sensor noise. In any case, the outputs of the incremental bond graph of a bond graph element indicate a parameter variation. 

In ON-mode, the switch model simply reduces to a resistor with the small nominal ON-resistance The constitutive equation of a linear 1-port resistor in conductance causality then leads to an equation for the increments of the power variables. 

The power variable zj is an output variable of the original bond graph model with nominal parameters and as such it is a weighted sum of the inputs Uk(t) into the original bond graph in the case of a switched LTI system. 

Figure 5.8 displays the corresponding incremental bond graph. Again, the purpose of the auxiliary storage element C Q is just to resolve the causal conflict at junction O2. In the process of equation formulation, the capacitance Ca is set to zero. Summing variations of power variables at junctions li, O2, and is yields... 

In contrast to an original bond graph, the bonds of an incremental bond graph carry variations of power variables. Outputs of interest with regard to FDI are variations of ARR residuals. For a switched LTI system, these variations are a weighted sum of the... 

The next sections first recall standard least square optimisation. Subsequently, this technique is applied to ARR residuals assuming that they can be obtained in closed symbolic form so that their expressions can be used in an objective function to be minimised. If unknowns cannot be eliminated from the sum of power variables at junctions to which a detector has been attached, then the time evolution of ARR residuals can be obtained by either computing ARRs as outputs of a... 

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