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Show PROCESS MODELING - HEART AND SOUL OF A PEMS Process modeling methods generally fall into two broad classes: first -principle and regression modeling methods. Models developed by these methods have been used in PEMS. Both methods have advantages and disadvantages. This section compares and contrasts these two methods, concentrating on the high order, multivariate, nonlinear regression modeling. FIRST-PRINCIPLES MODELS First-principles modeling techniques describe the process dynamics via the physical equations designed to approximate the system. Examples include kinetics, thermodynamics, and fluid-flow equations. These models are typically expressed in the form of a set of algebraic expressions and/or coupled partial differential equations with coefficients fitted to the data of the system. First-principle models have been used to create PEMS for natural gas fired turbines and reciprocating internal combustion. LINEAR REGRESSION MODELS The most widely known form of regression is linear regression-fitting a straight line or a plane to a data set. The simplest form of linear regression minimizes the sum squared error of the model output versus the plant output. That is, given a set of ouUJut values y ={y 1 (t),y2(t), yn(t)} and input data values x ={xl(t),x2(t), xm(t)} sampled at times t = {l,2,N}, linear regression minimizes the sum-squared error, E, of the predicted values p(x) and the output values y, where: N E = L (y - p(X))2 Equation .l t=1 For a linear system, the predicted outputs, p(x), are a linear sum of a matrix of coefficients A times the inputs: m P· = L A·x· + b· 1 IJ J I j=l or, in matrix notation, 5 |