SVD decoupling case study (2): one controlled variable, two manipulated variables

Antonio Sala, UPV

Difficulty: *** ,       Relevance: ,      Duration: 17:09

*Enlace a Spanish version

Summary:

This video discusses the concept of principal maneuvers and SVD decoupling in a ‘1 controlled output, 2 inputs’ process. To understand the “intuitive” meaning of what is proposed, it is illustrated with a physical example of heating a piece of material (a thermocouple gives the reading of the variable to be controlled) with both a resistor and a blower (cool air being blown by the latter).

Obviously, there are infinitely many solutions to $y=Gu$, and with appropriate scaling so that 1 means ‘100% effort up to saturation”, we would want to use, say, the one that minimizes $u_1^2+u_2^2$, i.e., the least-squares criterion.

Specifically, the singular value decomposition $G=US{V}^T$ of a matrix that is a row vector has $U=1$, $S=norm(G)$, and $V=G^T∕norm(G)$, i.e., a norm 1 scaling of the original matrix; directions orthogonal to $G$ form the null space. The pseudonverse is $G^T∕norm{(G)}^2$. If the theory is not familiar to you, check the video [dsvdintu1EN], prior to this one.

In fact, nothing is too interesting about SVD in a one-output process, in the sense that, actually, the input directions with non-zero effect are simply proportional to the gain of each actuator: this is just a first motivational toy example.

But the objective is to capture the intuition behind the fact that least squares solutions distribute the control effort proportionally to the gain and, therefore, we could think of the only controlled variable being controlled by a “virtual” actuator that moves the physical actuators in a coordinated manner, replicating the same actuator command, but multiplied by the gain.

The video [dsvdintu3EN] continues with the “dual” case of several outputs to be controlled with a single manipulated input.

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