Nonlinear Control: Hamilton Jacobi Bellman (HJB) and Dynamic Programming

This video discusses optimal nonlinear control using the Hamilton Jacobi Bellman (HJB) equation, and how to solve this using dynamic programming.
Citable link for this video: doi.org/10.52843/cassyni.4t5069
This is a lecture in a series on reinforcement learning, following the new Chapter 11 from the 2nd edition of our book "Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control" by Brunton and Kutz
Book Website: databookuw.com
Book PDF: databookuw.com/databook.pdf
Amazon: www.amazon.com/Data-Driven-Sc...
Brunton Website: eigensteve.com
This video was produced at the University of Washington

[@higasa24351]

This is the first time I've ever seen the explanation of HJB-DP in a intuitive and fashionable way, not by following the text book lines one by one. Thank you so much for the great talk.

[@charlescai6672]

Very good explanation to derivative of HJB equation. But there's a point I may have to add that I think there may be a typo in 'DERIVING HJB EQUATION': In dV/dt, minimizing the integral of L(x,u), the lower limits of integral should be t instead of 0. Only by the case, we can conclude in the second last equation that -L(x(t), u(t)) can be obtained from the time derivative of integral of function L(x,u)...

[@alanzhus2730]

Can't believe serious topic as this can have thousands of views hours after release. KZbin is really a magic place.

[@Eigensteve]

It's pretty wild to me how many people like hard math :)

[@roro4787]

@@Eigensteve we love math💕

[@ecologypig]

Excellent. Can see a lot of connections with Control and how the essence of Bellman equation are all over the place in different fields. Thanks Prof. Brunton!

[@ailsani8749]

I am a follower from his 'control bootcamp' series. Just trying to tell everyone new here that his video is life-saving.

[@mingyucai6559]

Clear tutorial. Thanks Prof. Steve. Keep following your steps.

[@sounghwanhwang5422]

One of the best lectures that I've ever seen!

[@hfkssadfrew]

Hey Steve, on 9:11 it should be integration from t to t_f, then that’s where the - comes from.

[@umarniazi7320]

Yes, you are right.

[@MBronstein]

But then shouldn’t there also be an integral going from t0 to t?

[@hfkssadfrew]

@@MBronstein I guess it is because such t can varying arbitrarily from t0 to tf. And the whole point is to analyze the derivative wrt anyway. so there is no need to derive another one from t0 to t.

[@MBronstein]

@@hfkssadfrew But the definition of V goes from t_0 to t_f.. So, we have V= integral of L going from t_O to t and from t to t_f +Q. Notice, if we take derivative now, we get -L from the first integral and +L from the second integral. I don't understand why we can just ignore the second integral

[@Eigensteve]

Good catch, thanks! I caught this in the 2nd edition book proofs, but not before the video...

[@amaarquadri]

Wow it's so cool that these concepts from reinforcement learning apply so perfectly to nonlinear control.

[@aiwithhamzanaeem]

Thanks Professor Steve, Finally I completed the playlist.

[@geonheelee4717]

A Great Lecture. I hope the next lecture will open asap. In particular, I'm interest in detailed relationship between RL and optimal control.

[@dmitryb4666]

thanks Steve for a great lecture; looking forward to more lectures on RL and non-linear control if possible with some simple examples. thank you very much!

[@ramanujanbose6785]

Steve I follow all of your lectures. Being a mechanical engineer I really got amazed by watching your turbulence lectures. I personally worked with CFD using scientific python and visualization and computation using python and published a couple of research articles. I'm very eager to work under your guidance in the field of CFD and Fluid dynamics using Machine learning specifically simulation and modelling of turbulence fluid flow field and explore the mysterious world of turbulence. How should I reach you for further communication?

[@tuptge]

More on non linear control please! Im trying to make up my mind on topics for my postgrad thesis!

[@boldirio]

Great as always Steve! I was wondering if you have any experience in transfer learning, specifically domain adaptation? If so it would be a cool topic to go through! /J

[@blitzkringe]

Please do more of this content. Thank you.

[@Eigensteve]

Glad you like it!

[@mohammadabdollahzadeh268]

Thanks dear steve for this wonderful tutorial I was wondering would it be ok if you solving an example for that?

[@InfernalPasquale]

Excellent communication

[@ronbackal]

Thanks! That is very interesting. I have the book Data driven science and engineering, which I want to get to sometime to learn more deeply

[@emmab5151]

Amazing!

[@rajanisingh1148]

@Eigensteve, Thanks for such a nice and interesting videos. I've seen all your videos on reinforcement learning. It would be really helpful if you could do a lecture on how dynamic games (either discrete or continuous time) can be solved using reinforcement learning with a walkthrough example. For now, the theoretical concepts on reinforcement learning are clear from your videos, but how it's actually implemented to solve problems is still unclear. Also if you can recommend some resource that would be bonus!

[@prantel1]

At 11:47 the bounds of the integral should be from “t” to “tf”; not from 0 to tf. If you make that change then the derivative of the integral wrt to t will be -L(.,.)

[@BalajiSankar]

Can you please tell how changing lower limit changes the sign?

[@BarDownBoys]

@@BalajiSankar I’m happy I can answer as I came here to ask the same question, and Behzad cleared it up for me. As behzad stated, it should be integral (t to tf). Then you agree that you can write this as negative the integral (tf to t). Then simply look at the fundamental theorem of calculus - the lower limit being a constant drops out and the upper limit being the variable you’re differentiation is with respect to just means that what’s inside (-L) is your output

[@kirar2004]

@@BarDownBoys Thanks

[@sounakmojumder5689]

thank you, I have a request if you can please upload a lecture on infinite horizon model predictive control......

[@G12GilbertProduction]

This Hilbert space is include in f(x(k),u(k) * (x(0),y(k)-0) or outside the x(k) - (without double equation)?

[@qiguosun129]

Great Lecture, could you think about discusing HJB with variational inequality? thanks!

[@Connect.2source]

Is there any way I can learn from you in more detail? Any programs you offer by chance? Thanks so much!!

[@djredrover]

it would be lovely if you could do a MATLAB demo of an ONC using HJB for a hovercraft/drone with full 6-DOF model.

[@nitishabordoloi3987]

Hello Steve, can you please comment on the necessity of terminal cost in the performance index

[@SRIMANTASANTRA]

Lovely, Professor Steve

[@Eigensteve]

Thanks!

[@Silva98122]

In general, if DP algorithm depends on discretization and interpolation in continuous state space and input space when solving a discrete time, finite time optimal control problem, does it yield a suboptimal solution?

[@vietanhle6321]

Good instructor

[@chinamatt]

Hi Steve, thanks for the lecture. At the beginning, should the differential equation be dx/dt = f(x,u,t)? As in the derivation of the HJB equation, the subsitution of dx/dt to f(x,u) is made.

[@Eigensteve]

Yes, good call

[@mamadoupathely9369]

yeah it must be dx/dt = f(x,u)

[@leventguvenc917]

Very nice video. In deriving the HJB equation, the lower limit of the integral should be t instead of 0.

[@jameschen2308]

I was about to say

[@user-en1kz2ew6n]

I second that!

[@amirhosseinafkhami2606]

Hi Dr. Brunton, thanks for your excellent lecture. Do you have any good code examples of solving the HJB equation for non-linear systems? And what resources do you suggest for getting more depth into this field?

[@Eigensteve]

I don't have a good recent code... way back in grad school I remember solving these numerically as a two point boundary value problem... but all of that code is deprecated. Will look into a better example

[@amirhosseinafkhami2606]

@@Eigensteve Actually, I took a look into chapter 11 of your book, but unfortunately, unlike other chapters, I did not find any sample code in it. I think it would be great if an example code for solving the HJB of a non-linear system was added to the book! This could be a great complementary to this chapter! Thank you so much again for making such great contents

[@Eigensteve]

@@amirhosseinafkhami2606 Totally agree, but this will need to wait for an updated version. Definitely in the works though.

[@amirhosseinafkhami2606]

@@Eigensteve I look forward to the updated version of the book then

[@CoffeeVector]

In the equation, dx/dt = f(x(t), u(t), t), why is there an extra dt at the end?

[@qejacwa]

This is a fantastic video on the derivation. However, there are quite some typos in the video. Hopefully, Steve can correct them. For example, the lower limit in the integral is supposed to be t instead of 0 in the derivation of HJB equation.

[@sai4007]

Yep, without this correction -L(x, u) derivation doesn't make sense

[@hw1875]

16:58, shoud the V at RHS of Discrete time HJB be associated with n, not n-1? Because cost to go (from k to n) should be equal to current cost plus cost to go (from k+1 to n)

[@julienriou4511]

that's weird not to talk about Pontryagin Maximum Principle in an introduction to optiaml control

[@Eigensteve]

That's a great point. There are a lot of things conspicuously missing from these intro lectures. A lot of it is that I'm still learning more about these topics myself. Maybe a topic for another day!

[@justinting1422]

What's the purpose of the terminal cost? It just disappears when you take the time derivative at 9:22, since it's just a constant, so it shouldn't affect the trajectory of u(t). Also, isn't the cost of the final state already taken into account in the integral, since it integrates all the way to tf anyway?

[@sechristen]

The terminal cost term will appear as a boundary condition in the PDE that HJB gives us, as V(x(t_f),t_f,t_f)=Q(x(t_f),t_f). The terminal cost cannot be taken inside the integral (without breaking all the other math by including delta functions as valid cost functions). The formulas in the video are derived with the idea of a fixed tf, so if t_f doesn't vary the final cost function will probably look like "After attempting to control the dynamical system, did it end where I wanted it to? eg Q=(x(t_f)-x_target)^2"

[@qiangli4022]

actor-critic seems to be categorized as a model-free rl in other literatures.

[@clairecheung5388]

The lower bound of the integral for V(x(t),t,t_f) should be t instead of 0.

[@RGDot422]

Why d ( integral ( L(x,u)dt )/dt = - L(x,u)?... Specifically, why is the negative sign?

[@batoolalhashemi1167]

Please give us some examples to more understanding

[@peasant12345]

7:10 the bellman opt must include Q(x(t),t)

[@matouspikous7489]

min(L) != -min(-L), I don't know how to cancel these minus signs.

[@CupuycA]

1:35 mistake in the equation

[@wikipiggy0.0]

the derivation is not clear. maybe it is due to the typos metioned in other comments I find it hard to follow

[@CupuycA]

9:15 it's not obvious, that the operators min and d/dt commute. In general this of course is not true.

[@matouspikous7489]

I think there shouldn't be the minimum. V is just what is in the minimum. You do the calculations and then, you say that some V* is the optimal, which has the minimum in the equation.

[@canis_mjr]

Шикарный ролик (нет) пример где? Идею прдзода понятнати примитивна, как наипрактике жто применить?

[@__--JY-Moe--__]

👍I don't know why I see super mario Bros!! I love Calculus though!! this goes well, with my jacobian meshing geometries! Rosey the Robot was so over worked! X0-Xn= Cello...ha..ha..💫

[@ctrlaltdebug]

Your trajectory x(t) is not a function.

[@KHMakerD]

Lol solving PDE’s is heinous by definition 😂😂

[@beaglesnlove580]

Crap ur 100x better than this horrible professor I had who was teaching hjb equation without any background.