Why the Riccati Equation Is important for LQR Control

This Tech Talk looks at an optimal controller called linear quadratic regulator, or LQR, and shows why the Riccati equation plays such an important role in solving it efficiently.

The talk walks through three different ways that the LQR problem can be solved: an intuitive, but ultimately inefficient brute force way; a more efficient learning algorithm way; and then the most efficient approach, which is accomplished analytically using the algebraic Riccati equation.

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[Jair_inacio_Neto_Teixeira]

Brian is one of the greatest teachers on YouTube. I really appreciate his way of teaching complex topics.

[aammoojanhastam]

What my professor couldn't reach in a semester I learned in a 14 minutes YouTube video. As always informative and simply brilliant. Thank you Brian

[-ion]

Every now and then over several years, I have looked for and failed to find an explanation for the ARE solution of LQR that I manage to understand. This is it, finally. Thank you!

[JohnLangleyAkaDigeratus]

This is an absolutely excellent presentation.

Super complex. Somewhat esoteric for a lot of people, but the presentation and review and then seeing it implemented in matlab makes it seem so accessible!

Thank you!

[tongvanngoc_gv-t]

After reading a bulk of documents for hours, I decided to watch Brian's videos. It's always the most effective way to gain a deep understanding of the problem

[ga]

I've spent more than a few hours trying to understand how Riccati equations work after Steve Brunton mentioned their importance in his control playlist. I could never get my head around exactly what it was or how it helps solve dynamics problems. I can tell after watching this a few more times I'll be on my way. You really have a knack for explaining not just the method but the motivation in the simplest way. Thanks Brian.

[brennanukrainetz]

Amazing work. I'm an undergrad MecE student at Ualberta, and your lectures have been super helpful for intuition on what I'm learning. My textbook (Feedback Control of Dynamic Systems) is decent but sometimes they get lost in the weeds while missing the intuitive explanations. Thanks a bunch!

[mehmetkilic]

Honestly, I was waiting this topic for a long time :)

[SamChak]

Thanks, Brian. This is a very good introductory video on the Algebraic Riccati Equation. I hope there will be a continuation on this topic, such as solving the LQR problem or proving Lyapunov stability in Takagi–Sugeno Fuzzy Control Systems using the LMI Solvers from the Robust Control Toolbox.

[EddieSanchez]

Eres muy bueno explicando Brian, tus videos me han animado más y más a aprender acerca de teoría de control. Gracias!

[ft]

This is just perfect timing, since I have an exam on this kind of topics just this semester 😁😁
Will there be anything on Hamilton functions and general optimal control (dynamic programming, variational calculus, Bellman principle) too quite soon?

[venkateshnayak]

Ahhh you just read my mind. A few days back I was wondering, "Why hasn't Brain made a video on ARE" and kaboom! You surprise us again.

[hoolladd]

Could u make a video on Kalman filter & LQG?

[rspapero]

Thanks a lot. This video has dissolved any doubt.😊

[romaingrobety]

Hi all, I was desperately trying to do the same thing to make the discrete Ricatti algebraic equation appear for a cost function with the sum symbol, but I can't find any literature to help me understand how it appears from the cost function of a discrete system.
If anyone has any clues I'd be more than happy to help.

Many thanks for this exhaustive video, which has already enabled me to understand how the algebraic ricatti equation emerges from the cost function of a continuous system.

[ИльяОсокин-эд]

Hi Brian! Thank you for the video. I would be glad for a comment on a following question. In the derivation we introduce P, which stays outside of the integral. After that we set both terms under the integral to 0 by equating control to a product of a number of matrices and by demanding ARE to hold. However, it is not particularly clear for me, why does this exact value of P minimize J? Why couldn't there be a matrix P' that does not set the terms under the integral to zero at all times, but that minimizes x_0^T P x_0 somehow? I was not able to find this kind of derivation in the literature, everyone just applies HJB and that's it. I will be grateful for a clarification on that.

[hectorgautier]

Thanks for the content. I have some question if I may. I understand that at the beginning we want to find u to minimze J (because u is the only variable that we can tune). Then we make appears a term that has no influence on the value of J and that depends on the variable P. And from here I am loosing a bit the sense of the connection between our goal which is minimizing J with the transformation. We deduct that to minimize J we have to make in sort that the inside of the integral is zero, but to do so we constraint the shape of u with the variable P which has initially no impact on J. So now P has an importance because the expression of u depends on it, but P is outside of the integral so how can we know that making in sort that the inside of the integral is zero will really minimize J?
I admit I am a bit confuse, I think prefer the explanation through the Lyapunov function.

[oldcowbb]

my favourite thing is that the discrete riccati equation is analogous to value iteration

[enricofioresi]

Could someone explane me why "the state goes to zero as the time approaches infinity" (min 8:35)? I mean, if I have some non-zero reference state that I want to be reached, and the closed-loop system is stable, shouldn't x converge to the reference state? Thank you! :)

[trendyprimawijaya]

I still trying to figure out some points:

What probably motivates Riccati to expand the cost function?

Is it because he wants to explore the implicit relation between input u and state x (which ends up as u+R'B*Px) while maintaining the quadratic x terms independent in the resulting cost function?