Linear Quadratic Regulator (LQR) Control for the Inverted Pendulum on a Cart [Control Bootcamp]

Here we design an optimal full-state feedback controller for the inverted pendulum on a cart example using the linear quadratic regulator (LQR). In Matlab, we find that this is a simple one-line command 'lqr'.

These lectures follow Chapter 8 from:
"Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control" by Brunton and Kutz

This video was produced at the University of Washington

[gabrielh]

Watching this at 2 am before sleeping became a habit for me. And it has worked, I wake up in the morning with a clear mind of what you just taught me.

[osamaahmadieh]

What is fascinating about this series of Lectures is that you always link the abstract mathematical quantities to an actual physical interpretation. This makes understanding the concepts much easier and familiar. Thank you for taking the time to make such helpful lectures! looking forward to watch the data driven control!. Thank you !

[yahugh]

I ostensibly learned all this in grad school...and then forgot it all in the intervening 15 years when I went off and worked on other things. But now that I'm back to designing control systems, this series of lectures has really helped me get the rust off my skills!

[manhhoang]

Teaching complicated problem in easy way. Thank you Professor!

[ashegofd]

You are a legend
I was thinking what I'm doing with control systems, poles, eigen values.... I m not getting the practical example
But hey you helped me
Thanks a bunch
I hope every enthusiastic student finds prof like you
Good job man 👍🏻

[alial-ghanimi]

I like the way that you have used to present the cost function. Using an intuitive explanation is the key of your unique strategy which amazed me the most. Well done.

[augustogomez]

Writing words backwards on the board is quite a job! Great video profe!

[Drone.Robotics]

Thank you Sir....I have seen the whole series and it have cleared lot of my concepts about control theory. Your videos are just great and your way of teaching complex things in simple manner is appreciable. Thanks Again.

[rohitn]

The line he said, "It's interesting (taking a small pause) and it's complicated "..This are the situation we are facing 😅😅😄👍🏾

[gayathrimenon]

Such a wonderful series!!! Thank you so much professor. Just wanted to ask you whether LQR requires any improvement in its performance. If so, then by combining it with any other controller can any improvement be brought?

[navsquid]

If you don't have MATLAB (like me), the Python control library mimics most of the commands from MATLAB. Just be sure to do things like ensure your B matrices are shaped like (-1, 1), etc.

[murat]

How can we add disturbance and measurement noise to simulink model.

[MrFiskur]

Thank you for the video, sir! Really a good explaination of LQR. I have question though. At 9:20 you compute the eigenvector to the most stable eigenvalue and mention that the most stabilizing directions are x_dot and theta_dot. So aggressive control on x_dot and theta_dot would really improve performance.
So my question is: Isn't it a good idea to have high values of the 2nd and 4th diagonal entries of the Q matrix, as these correspond to x_dot and theta_dot, and then have lower values at the 1st and 3rd entries?

[hariranga]

Hi Steve! That was a great introduction to LQR. Is there any chance you could share the MATLAB code for the inverted pendulum, I would be excited to see it work? Thanks!

[ajj]

Quick question, how do I change my matrixes A, B, C and D in my state space block in simulink so that k is taken into account of. I currently am simply replacing A by A-BK for my full state. It looks wrong

[ijbldsn]

Hi steve, it's a super video for LQR, but there is a one point that I can not understand, for the objective function J, why we don't use (X(t) - set point(t)) but we use X to minimize J?? We want x to get close to our set point @steve brunton

[comvnche]

In my MPC lecture eigenvalues where only said to be stable if they are <= 1

[cheriyanhomey]

Thank you sir, this video series was very helpful to get an introductory idea on control theory .It would be very nice if u could kindly provide us the sources from where we can get to know more about the mathematics behind these concepts
Thanks again for this wonderful series.

[matthewjames]

Amazing video. Is there a value of 'R' (the penalty of motor usage) for which the system can't find an ideal 'K' (linear feedback controller)? Intuitively, I can imagine if R was too large then the cart simply can't move fast enough to keep the pendulum up (in other words, lqr() can't make all real parts of the eigenvalues negative)

[amr.a-m]

Good video 7 effort. I hope you going to demonstrate the MCP with the real application supported by codes & Simulink on Matlab such as this please if available.