Control Bootcamp: Observability Example in Matlab (Part 2)

This video continues to explore observability in Matlab on the example system of an inverted pendulum on a cart. We look at the observability Gramian.

These lectures follow Chapters 1 & 3 from:

Machine learning control, by Duriez, Brunton, & Noack

This video was produced at the University of Washington

[JustAnotherUserski]

omg 0:32 I was wondering this ever since the inverted pendulum example came up and it was driving me crazy. Thanks for the great crash course and for answering our questions without us needing to ask them!

[saapman]

omg! you are the best controls lecturer ever! Thank you for these videos.❤

[gtorres]

You ignored the x state, but this was only possible because the first column of the A matrix is all zeros, right? There is no coupling between X and the other states.

[rogiervdw]

Excellent and fun lecture. But...
1. Under what conditions can you drop a state (in this case the cart position x) from a state-space system? Intuitively, I suppose that state shouldn't be coupled with any of the others, but how to test this?
2. Why would the level of observability (estimated using the determinant of the Gramian) in the pendulum down position be indicative of the level of observability of the pendulum up position? These are very different dynamics, right?

[chainetravail]

Have you made the error of trying to compute the gramian on an unstable system on purpose ?

[navsquid]

Doesn't partitioning the matrix like at 2:18 eliminate some of the dynamics from the linearized system, though? Seems like changing the system or ignoring some of the dynamics isn't a good tradeoff to make the system observable.

[VTdarkangel]

A question I have: would the PBH test work for observability? Would you get similar information from it as you would for controllability such as how many sensors would be needed to make the system observable?

[arkansassignalintegritycom]

Wait, I'm lost. I thought the inverted pendulum was stable. I thought we could pick either pendulum down or inverted as an equilibrium point we could linearize about? Why can't we calculate the gramian for inverted?

[viniciusfriasaleite]

What if the initial theta is unknown? In this case, looks like theta dot isn't sufficient to have observability of theta

[masoodroohi]

You did explain very intuitively what's the benefit of finding the most ctrb direction but what about the most obsv one? I don't get what's the point of knowing x is the most obsv state?

[mohab.ahmed.]

Can the controller be included as a part of system matrix (A-BK) hence the system is stable in top position then we can calculate the gramian, will this work around work ?