Industrial Applications of Genetic Algorithms:Space Shuttle Main Engine Condition Monitoring Using Genetic Algorithms and Radial Basis Function Neural Network

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FITNESS MEASURE

Reference [2] had reasoned that the relative significance of a neuron in the overall solution of Equation (4) could be associated with the size of weight assigned to it during the RBF training sequence. Subsequent work of [2] then found a fitness measure that promotes competition between neurons that are doing the same job and cooperation between those neurons supplying independent contributions to the approximation of f(x). As shown in [2], the competition component performs implicit niching; that is, niche sharing is actually accomplished without ever calculating a niche sharing function. The measure for doing this is stated below

where, wi is the weight given to the neuron in a single generation, E(|w.,i|) is the mean weight given to all the other neurons in the population, and β regulates the tradeoff between competition and cooperation (1 < β < 2).

In many function approximating simulations by the author, it was found that the assumption of a neuron’s assigned weight determining its importance cannot always be justified. For example, considering the series approximation of Equation 10.4, a single neuron with an assigned weight could be replaced by two neurons at the same position with each possessing half the weight of the original neuron. Indeed, function approximation simulations have shown that the aforementioned fitness measure of [2] could lead to overcrowding of the RBF centers and a subsequent poor network formulation. For these reasons, the fitness measure specific to this chapter was developed in such a manner that allowed both an explicit and implicit calculation of fitness sharing (or niching) with a concomitant competitive factor, all of which are independent of a neuron’s assigned weight. The following sub-sections detail the different aspects of the fitness measure.

A Neuron’s Utility

With regard to the training data, a neuron’s importance can be determined by the amount of data points for which it is actually contributing significant responses. If there are p training patterns, then the number of times an RBF neuron responds with a value greater than a preset utility factor (α) would be a direct indication of how much the neuron is doing to approximate f(x). Thus, a utility measure can be stated as:

It was found in subsequent studies that a value of α=0.5 worked best for most trial problems.

Overlap Measure

Simply awarding neurons for covering data will certainly lead to a crowding of neurons around the most dense portion of the data hull. Therefore, there needs to be a measure which will award neuron separation. This can be defined by understanding the nature of the RBF parameters. Using the center positions and widths of two neurons, a measure can be written in terms of an overlap parameter:

Here, E( ) defines the mean separation of neuron i from all other neurons for which j does not equal i. λ values greater than unity specify separation of internal regions with respect to the neurons widths, values equal to unity imply touching edges of the regions, and values less than 1 signify overlap of internal regions. It is obvious that such a measure would award those centers which sought to increase their respective λ’s.

The two foregoing measures can be written in collective form as follows:

The choice of the Wi weighting factors can be determined by an empirical study for the specific problem under consideration. The final section of this chapter subjects this measure to a highly nonlinear approximation problem.

APPLICATION TO SSME PLUME SPECTRAL ANALYSIS

For years, researchers at NASA’s Marshall Space Flight Center (MSFC) have been filming developmental tests of the SSME. Subsequent analysis of film, which involved major engine failure incidents, revealed a consistent feature common to many of the breakdowns. In eight of twenty-seven failure events observed, a visible discharge of some substance was seen in the plume prior to engine failure [9,10]. These discharges ranged from extreme plume flashes to small regional streaks. This discovery lead to the thought that if it is possible to visually detect anomalous events with the naked eye, then perhaps it would be possible to discover anomalous events below the “visible threshold” (i.e., infrared, ultraviolet, etc.) that would be indicative of an imminent failure. This would be tantamount to real-time engine health monitoring via the quantitative analysis of the SSME exhaust plume spectral data.

The idea of extracting information about a system from its electromagnetic (EM) emission is certainly not new. Consider a piece of nickel being held over an open flame; at some point in the heating process if the flame is hot enough, the molecular structure of the nickel will become so excited that it begins to radiate. As shown in Figure 10.3, this radiation, being electromagnetic in nature, can be represented as a spectrum of EM intensity versus transition wavelength. Therefore, with a spectrometric detector some distance away during the nickel experiment, it would be a simple task to identify the metal which is being burned.

Individual spectra for other elements vary in complexity. Some have a few atomic transitions (peaks) while others have many. Three germane points of importance to the current effort result from Figure 10.3 and the associated radiation physics: 1) every element has its own “spectral signature,” 2) the emission will contain atomic transitions at wavelengths which may not be part of the visible spectrum, and 3) the intensity of the emission is a function of the quantity of emitting matter present in addition to the system temperature and other quantum variables.

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