Objective function algebraic

FIG. 8-46 Diagram for selection of optimization techniques with algebraic constraints and objective function. 

As in algebraic models, the error term accounts for the measurement error as well as for all model inadequacies. In dynamic systems we have the additional complexity that the error terms may be autocorrelated and in such cases several modifications to the objective function should be performed. Details are provided in Chapter 8. 

The objective function is a suitable measure of the overall departure of the model calculated values from the measurements. For an individual measurement the departure from the model calculated value is represented by the residual e,. For example, the i,h residual of an explicit algebraic model is... 

The choice of the objective function is very important, as it dictates not only the values of the parameters but also their statistical properties. We may encounter two broad estimation cases. Explicit estimation refers to situations where the output vector is expressed as an explicit function of the input vector and the parameters. Implicit estimation refers to algebraic models in which output and input vector are related through an implicit function. 

In this chapter we are focusing on a particular technique, the Gauss-Newton method, for the estimation of the unknown parameters that appear in a model described by a set of algebraic equations. Namely, it is assumed that both the structure of the mathematical model and the objective function to be minimized are known. In mathematical terms, we are given the model... 

Having the smoothed values of the state variables at each sampling point and having estimated analytically the time derivatives, n we have transformed the problem to a usual nonlinear regression problem for algebraic models. The parameter vector is obtained by minimizing the following LS objective function... 

Under certain conditions we may have some prior information about the parameter values. This information is often summarized by assuming that each parameter is distributed normally with a given mean and a small or large variance depending on how trustworthy our prior estimate is. The Bayesian objective function, SB(k), that should be minimized for algebraic equation models is... 

All the algebraic and geometric methods for optimization presented so far work when either there is no experimental error or it is smaller than the usual absolute differences obtained when the objective functions for two neighboring points are subtracted. When this is not the case, the direct search and gradient methods can cause one to go in circles, and the geometric method may cause the region containing the maximum to be eliminated from further consideration. 

In all the problems discussed, the response surface was unknown and could be approximated only by making some tests. If everything in the objective function is known and can be expressed algebraically and the variables are continuous, a number of other techniques such as linear programming17-1 19 can be used. These will not be discussed here because this is usually not the case for plant designs. 

The ingredients of formulating optimization problems include a mathematical model of the system, an objective function that quantifies a criterion to be extremized, variables that can serve as decisions, and, optionally, inequality constraints on the system. When represented in algebraic form, the general formulation of discrete/continu-ous optimization problems can be written as the following mixed integer optimization problem ... 

Finally in this chapter, an alternative approach for nonlinear dynamic data reconciliation, using nonlinear programming techniques, is discussed. This formulation involves the optimization of an objective function through the adjustment of estimate functions constrained by differential and algebraic equalities and inequalities and thus requires efficient and novel solution techniques. 

F = objective function, g = algebraic inequality constraint vector, c = algebraic equality constraint vector,... 

As the plant to be optimized considers a process operating at steady state, then the variation of the phase concentrations with time is zero. For this reason, the mathematical model that describes the plant is a set of ordinary differential equations, as the phase concentrations depend only on the module axial position. In the tanks, the concentrations are constant. The differential-algebraic nonlinear optimization (DNLP) problem PI to be solved includes the ordinary differential equations that represent the mass balances for the phases in the membrane module. The objective function to be maximized is the amount of metal processed FeC , where Fe is the effluent flow rate whose Cr(VI) concentration after dilution from wastewaters is C . The problem has the following form ... 

The dynamic models of chemical processes are represented by differential-algebraic equations (DAEs). Equation (2) and (3) define such a system. Equations (4), (5) and (6) are the path constraints on the state variables, control variables and algebraic variables respectively, while equation (7) represents the initial condition of the state variables. Obj is a scalar objective function at final time, tj. ... 

Perform algebraic manipulations to express the objective function in terms of variables that are not in the basis, i.e., are equal to zero. This determines the value of the objective function for the variables in the basis. 

The model equation is written according to eq. (10.22) y = f (x.p). thus non-linear regression strictly speaking is valid for algebraic models, but it can be applied to differential models, as well. For differential models, the solution (y) is obtained numerically from the model equation. The objective function to be minimized by regression is defined by ... 

But often we need to define our own functions. To solve algebraic equations or optimization problems, for example, the user needs to provide a function that can be called by the solver to evaluate the nonlinear equations or the objective function. To define a function in Octave, /(x) = for example, the command structure is... 

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