Controlled/manipulated variable selection: setpoint tracking, polyhedra (linprog) Matlab example (3)

Antonio Sala, UPV

Difficulty: **** ,       Relevance: ,      Duration: 18:40

*Enlace a Spanish version

Summary:

This video addresses the same setpoint tracking problem as in the previous videos [sacerf1EN] and [sacerf2EN] of this case study, which are briefly reviewed.

Here, we address how to determine if there is a $u$ within limits that verifies $Gu=v_i$, being $v_i$ each one of the four extreme vertices of the rectangle of desired incremental outputs. It is done with quadprog, justifying that since we are only interested in feasibility, the cost index can be anything, for example, minimize $|u{\parallel }^2$.

The second half of the video discusses a couple of observations about the geometry of polyhedra:

First, in order to approximate the pseudoinverse (scaling) problem, the $H$ matrix of the quadprog command should be $inv{(E_u)}^2$; thus, if the pseudoinverse/SVD solution is feasible, it would match the result of quadprog.

Second, by “zooming out” the desired output increment vertices until the problem is no longer feasible, we will obtain an excess power margin with an analogous interpretation to the minimum gain (above 1) of the SVD.

The final part of the video draws, in the output space, the polyhedra achievable with the available input increments and the desired one. Since the achievable is greater than the desired, the problem is feasible. The last few minutes of the video superimpose the results of the SVD approach and the multifaceted approach of the same problem, in order to get an intuitive idea of their similarities and differences.

*Link to my [ whole collection] of videos in English. Link to larger [ Colección completa] in Spanish.