SVD decoupling case study (3): three controlled variable, one manipulated variable

Antonio Sala, UPV

Difficulty: *** ,       Relevance: ,      Duration: 12:34

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Summary:

This video continues with the case study on the intuition behind principal maneuvers and SVD decoupling. A basic review of preceding videos [dsvdintu1EN] (theory) up to [01:30], and video [dsvdintu2EN] (2mv, 1cv) up to [02:15] is provided, as well as a quick outline of a 3MV, 1CV example in the same line of video 2.

From [3:30], it presents a case with 3 sensors (temperatures of a piece of material) to be controlled with a single actuator (resistor). The static gain matrix is a column vector and, obviously, the three temperatures cannot be controlled to arbitrary references, so we will seek to adjust them by, say, “least squares” fitting.

In this case, the SVD of the DC gain matrix $G=US{V}^T$ results in $V=1$ (there are no input directions, since it is only a single input), and $U$ proportional to $G$, simply being $G∕norm(G)$. It is the “dual” case seen in the video [dsvdintu2EN], where $G$ was a row vector.

In this case, the meaning of the single principal maneuver and the pseudoinverse is that the least squares fit is achieved by controlling a “virtual” variable $y^{SVD}$ that is a weighted average of the temperatures, weighted proportionally to the DC gain.

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