Minimum constrained

Constraint parameter In constrained minimum variance controller... 

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

The bxact penal ty-function11 idea was to devize a penalty function which has an unconstrained local minimum exactly coinciding with the constrained minimum of the original problem (3.1). This goal seems to have been consciously sought independently by Fletcher 53 and Pietrzykowski 54. but the idea was already implicit in the work of Arrow and Solow 55 and Zangwill 56. ... 

The energy variations along the intrinsic reaction path (IRP) and the ADMP trajectory are reported in Figures 2 and 3, respectively. We want to underline that the IRP corresponds to a unique and constrained minimum energy path and does not contain any time information, whereas the ADMP profile is computed along a trajectory issued by a dynamics simulation. Therefore, any direct (i.e. point-to-point) comparison is misleading. Nevertheless, their global differences can be discussed. 

The necessary conditions for a constrained minimum are given by the following theorem ... 

Sometimes we must find a maximum or a minimum value of a function subject to some condition, which is called a constraint. Such an extremum is called a constrained maximum or a constrained minimum. Generally, a constrained maximum is smaller than the unconstrained maximum of the function, and a constrained minimum is larger than the unconstrained minimum of the function. Consider the following example ... 

I EXERCISE 7.31 Find the constrained minimum of Exercise 7.30 using... 

One reasonable hypothesis is to suppose that the quadratic approximation is satisfactory in a limited neighborhood of x . If this hypothesis is true, the constrained minimum of the quadratic function is a point that decreases the original function. 

The necessary conditions to get the constrained minimum of (3.111) are obtained by searching for the stationary point of the Lagrange function ... 

Please note that the one-dimensional search in the direction predicted by Newton s method can be considered as a constrained minimum search in a space where the metric is given by (3.68). 

Therefore, the Levenberg-Marquardt method solves the problem of the constrained minimum for the specific value of d, for which the relation (3.112), in which the solution d of the system (3.122) is placed, is verified. This means that d is function of d obtained for a specific y. By varying y, a new constrained minimization problem is solved with a different d. 

This is the lowest point on the paraboloid that also lies in the plane x + y = 6. As shown in Figure 5.8, the constrained minimum is higher than the global minimum of the unconstrained function. 

Bergh, L. G., and J. F, MacGregor, Constrained Minimum Variance Controllers Internal Model Structure and Robustness Properties, Ind. Eng. Chem. Res., 26,1558 (1987). 

---