Variance minimum

Assuming the minimum variance per unit length of a packed column is (2dp), then... 

Constraint parameter In constrained minimum variance controller... 

Appendix Converting a Constrained Minimum Variance Controller to a PIP... 

The MPC controller that minimizes the variance of the output (minimum variance controller) during a setpoint change corresponds to the controller setting w = 1, A = 0, and no bounds on the input. The response for this controller design for m = 2 and p = 4 is given in Figure E16.3 by the solid line. 

Comparison of the system behavior using three different model predictive controllers (a) minimum variance, (b) input constraint, (c) input penalty. 

The input for most chemical processes is normally constrained, (e.g., a valve ranges between 0 and 100 percent open). An unconstrained minimum variance controller might not be able to achieve the desired input trajectory for the response. The controller design should take the process input constraints into account. The results of a simulated setpoint change for such a controller with bounds of —40 and 40 for the input and controller parameters w = 1 and A = 0 is given by the dashed line in Figure El6.3. 

If it is assumed that the measurement errors are normally distributed, the resolution of problem (5.3) gives maximum likelihood estimates of process variables, so they are minimum variance and unbiased estimators. 

A serial elimination algorithm was first proposed by Ripps (1965) and extended later by Nogita (1972). This approach eliminates one measuring element at a time from the set of measurements and each time checks the value of a test function, subsequently choosing the consistent set of data with the minimum variance. In this case, after a new measurement has been deleted, the test function and the variance for the resulting system have to be recomputed when the number of suspect measurements is increased, this may become a laborious solution. 

If the predicted values x / i and Et/t-i are already computed, the minimum variance estimates of the states are obtained as the solution of the minimization problem... 

If the errors are normally distributed, the OLS estimates are the maximum likelihood estimates of 9 and the estimates are unbiased and efficient (minimum variance estimates) in the statistical sense. However, if there are outliers in the data, the underlying distribution is not normal and the OLS will be biased. To solve this problem, a more robust estimation methods is needed. 

Implementation of a covariance structure into this numerical scheme is described in Tarantola and Valette (1982). In essence, an a priori covariance structure is assumed for the whole set of observations and parameters, which should be tightened by iterative refinements since we are still dealing with a minimum variance estimate. 

For this linear model, then, we simply find values of x, and x2 that maximize A. We do not need estimates of the parameters to define this minimum variance design. For nonlinear models, such as Eq. (40), this is not true. The... 

In the last decade several other multivariable controllers have been proposed. We will briefly discuss two of the most popular in the sections below. Other multivariable controllers that will not be discussed but are worthy of mention are minimum variance controllers (see Bergh and MacGregor, lEC Research, Vol. 26, 1987, p. 1558) and extended horizon controllers (see Ydstie, Kershenbaum, and Sargent, AIChE J., Vol. 31, 1985, p. 1771). 

The system is not strictly ternary, because of the limited solubility of AI2O3 in diopside, but we will neglect this minor complication here. Because the maximum number of phases in the system is three (diopside plus plagioclase mixtures plus liquid) and the components also number three, the minimum variance of the system is 1, at constant P. 

The sample size and sample allocation scheme is obtained in one of two ways. Either the cost is fixed and the variance of the mean is minimized or the variance is fixed and the cost is minimized. The first approach is used when the budget for sampling and analysis is determined in advance. The objective in this case is to use that budget to obtain an estimate with maximum precision (equivalently minimum variance). The second approach is used when the required precision of the estimator is specified in advance. Then the objective is to derive an estimator with the desired level of precision at the lowest possible cost. 

Minimum Variance. Fixed Cost. The mathematical problem is... 

Unbiased and Minimum-Variance Unbiased Estimation, Particularly for Variances... 

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