Continuous inverse model

Given a causal bond graph of a hybrid model, the question is how ARRs containing only known system inputs, outputs, system parameters, and information about the system mode can be derived. This section considers three methods that have been reported in the literature. One approach is the so-called causality inversion method [2-4], It has been introduced for causal bond graphs of continuous time models but can also be applied to bond graphs of hybrid models [5, Chap. 7]. 

If ARRs can be obtained in closed symbolic form, parameter sensitivities can be determined by symbolic differentiation with respect to parameters. If this is not possible, parameter sensitivities of ARRs can be computed numerically by using either a sensitivity bond graph [1 ] or an incremental bond graph [5, 6]. Incremental bond graphs were initially introduced for the purpose of frequency domain sensitivity analysis of LTI models. Furthermore, they have also proven useful for the determination of parameter sensitivities of state variables and output variables, transfer functions of the direct model as well as of the inverse model, and for the determination of ARR residuals from continuous time models [7, Chap. 4]. In this chapter, the incremental bond graph approach is applied to systems described by switched LTI systems. 

In any case, FDI is a prerequisite also for this task and bond graph based ARR residual generation can provide the information needed by fault diagnosis. Chapter 11 of reference [11] addresses fault tolerant control of systems represented by continuous time model and related issues such as system inversion. In [12], a bond graph approach to diagnosis and FTC has been recently presented and applied to an intelligent autonomous vehicle. FTC of hybrid systems has been considered for instance... 

Despite the work of Overton and Meyer, it was to be many years before structure-activity relationships were explored further. In 1939 Ferguson [10] postulated that the toxic dose of a chemical is a constant fraction of its aqueous solubility hence toxicity should increase as aqueous solubility decreases. Because aqueous solubility and oil-water partition coefficient are inversely related, it follows that toxicity should increase with partition coefficient. Although this has been found to be true up to a point, it does not continue ad infinitum. Toxicity (and indeed, any biological response) generally increases initially with partition coefficient, but then tends to fall again. This can be explained simply as a reluctance of very hydrophobic chemicals to leave a lipid phase and enter the next aqueous biophase [11]. An example of this is shown by a QSAR that models toxicity of barbiturates to the mouse [12] ... 

The mathematical basis for the exponential series method is Eq. (5.3), the use of which has recently been criticized by Phillips and Lyke.(19) Based on their analysis of the one-sided Laplace transform of model excited-state distribution functions, it is concluded that a small, finite series of decay constants cannot be used to represent a continuous distribution. Livesey and Brouchon(20) described a method of analysis using pulse fluorometry which determines a distribution using a maximum entropy method. Similarly to Phillips and Lyke, they viewed the determination of the distribution function as a problem related to the inversion of the Laplace transform of the distribution function convoluted with the excitation pulse. Since Laplace transform inversion is very sensitive to errors in experimental data,(21) physically and nonphysically realistic distributions can result from the same data. The latter technique provides for the exclusion of nonrealistic trial solutions and the determination of a physically realistic solution. These authors noted that this technique should be easily extendable to data from phase-modulation fluorometry. 

Various aspects of fluorophore emission at surfaces have been investigated, particularly within the past two decades. For nondissipative surfaces (e.g., bare glass), the lifetime(14) and the inversely related total radiated power 15 for a single emission dipole, modeled as a continuous classical oscillator, have been calculated as functions of orientation and distance from the surface. The radiated intensity emitted from a continuous dipole oscillator has been calculated as a function of observation angle, dipole orientation, and distance.(16 21)... 

Other models are based upon the immiscibility of polymer blends described above, and they model the system as Newtonian drops of the dispersed polymer with concentration (pi in a Newtonian medium of the second polymer with concentration (p2 = — (pi. There exists some concentration, cpu = cp2i — 1, at which phase inversion takes place that is, at snfficiently high concentration, the droplet phase suddenly becomes continuous, and the second phase forms droplets. The phase inversion concentration has been shown to correlate with the viscosity ratio, A. = r i/r 2, and the intrinsic viscosity for at least a dozen polymer alloys and blends ... 

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