supports HTML5 video. So you can include it or not. And the V dot becomes negative, and this is the vector. That's a different nonlinear phenomenon that often happens with spacecraft. Rated 4.9 out of five stars. This project will require you to implement both the environment to stimulate your problem, and a control agent with Neural Network function approximation. Which, of course, is guaranteed to be positive, right? You know, you come up with some bound and say, 'that's the worst tumble I have to deal with', right? We can do better. Drew Bagnell on System ID + Optimal Control 6m. Well, are you tracking something that's moving very slowly? You wanted to do what? So if this is- This is Q dot, Q was minus Q max, sine of Q dots. This point means hit it full on one way. Find materials for this course in the pages linked along the left. But this Q dot comes from rate gyros, if it's an attitude problem. We argued already this is insensitive completely to inertia modeling errors, because inertia doesn't appear. I find a really useful way to think about model base reinforcement learning is as a two-player game between the optimizer or RL algorithm, and the model learning and the world together. If you learn a low error model across iterations of interaction using a stable function approximator, and you generate policies using a good optimal control solver, you must achieve good performance. One is the worst attitude error you have. All right that one. But this control, if my rates go to infinity, my control goes to infinity. So, if you're driving this kind of assigned functions that mathematically optimal it made my V dot as negative as possible, but there's some really strong practical considerations to implement this in that. So, if I'm doing Lyapunov optimal, I get a response that's basically like this. But I'm just going up to the max and then I'm saturating at the max. At some point you're going to saturate where a thruster only be full on, there's nothing more that you can do, that's as big a torque as you can get. So, this is done as a constrained problem. If you can do that, that's great, but you're probably being overly conservative now with the gains you pick and how you can perform. Let's construct an optimal control problem for advertising costs model. But we liked the part that, 'hey, babe, big error', basically if you're tumbling to the right, you need to stop it and just give max torque to the left, to arrest it as quickly as possible. We combine them together using planning or optimal control synthesis algorithms, reinforcement learning algorithms, if you will. So, temporarily, our error measures actually increased with that gain function. Right? 3 stars. But it turns out this is a very conservative bound. One of the most amazing set of courses that I have ever been through. Show More Reviews. Let's pretend we only have one degree of freedom otherwise- There's a summation involved once. Which is very convenient when we're doing game design. Â© 2020 Coursera Inc. All rights reserved. In our case, the functional (1) could be the profits or the revenue of the company. Yup. 4.9 (520) 47k students. This doesn't care. I don't care what the attitude is. You could replace this whole thing with an A10 engine function if you wished, or other limits. That's a nice linear function. Now, I need to move this over, hold on. Will collect the same number of samples using this data set aggregation approach. Right? So, we can see now similar bounding arguments. Intermediate. I The theory of optimal control began to develop in the WW II years. People think of stability as somehow being tied to performance, those are two separate questions. Right? So if you run this now, you can see the response had big rates. Right? So, now we can look at what happens if we saturate. Alternate feedback control laws are formulated where actuator saturation is considered. 4.7. Question: how well do the large gain and phase margins discussed for LQR (6-29) map over to LQG? I know, you know, what's the worst case I could have on this? I mentioned briefly this idea of No-Regret learning. Lyapunov optimal really is defined as you've made your V dot as negative as possible. Right? How good are your rate gyros? * Analyze rigid body control convergence with unmodeled torque. People often resort to numerical tests. And it's good to torque at maximum capability. Anymore than 180 and I would switch to the other set. So that would have worked, but the key result is reduced performance. Reference tracking is tough too because your reference motion impacts, you know, it's my control going to be less than that? So it's nice. The result is both in theory and practice you can build statistical models that are dynamics with very low error, apply a good olptimizer or RL algorithm to that model and still get bad performance in the real world. CMU has been a leader in applying optimal control to animation and robotics. shepherdpuppiesstop What are the 7 basic dog commands? How bad could this error get? Okay, last time we discussed the key benefits of using a model, the ability to learn and simulation, and thus greatly reduce the expense and time of real-world interaction. Maybe you have the linear part only to hear to handle noise around here, and then if you get past it, jump up. Okay. Now, performance again, another question. After this course, you will be able to... To view this video please enable JavaScript, and consider upgrading to a web browser that Generally an Optimal Theory Problem would have a condition to be met while optimizing the other parameters. Torques and spacecraft is different. To top things off, I implemented a PD feedback controller to actually track the planned trajectory. And despite saturating, I am still actually converging, and working, and the rates, you know, the big one tumble rate took a long time to bring down, but once it all comes together, it all stabilizes nicely. The coupling coefﬁcients are learnt from real data via the optimal control technique. And the one that is saturated it applies this control, which gives a different Lyapunov rate function. The traditional view, this is known as system identification is talked in engineering statistics, is essentially a supervised learning approach. Jan 15: Introduction to the course. But, you know, then that limits how far you can go, or we'll find- we'll start here next time, there's other modified ways that we actually blend different behaviors with nice smooth response and saturated response. So, if you have little noise levels it's, you know, it scales with- if I measure intent to the minus 16 times again, it's going to ask me to torque to that direction but only by a little bit. 3 hours ago Coursera rarely covers full courses, Coursera courses are much less in depth. 16-745: Optimal Control and Reinforcement Learning Spring 2019, TT 3-4:20 NSH 3002 Instructor: Chris Atkeson, cga at cmu TA: Preeti Sar, psar1 at andrew, Office hours Tuesday 7 NSH 4508. Optimal Control Theory Emanuel Todorov University of California San Diego Optimal control theory is a mature mathematical discipline with numerous applications in both science and engineering. And that's all of course, if you have unconstrained control- If unconstrained control, you know, U minus k Sigma, minus P that, to make it as negative as possible, you make those gains infinite. Save Control Systems ... An Introduction to Mathematical Optimal Control Theory ... math.berkeley.edu. All of those transitions are aggregated together with everything we've previously seen before. But what I'm showing here too, is a single- I'm doing a linear feedback, that's one saturation function. And we went through this process already, we said, 'hey, we can make this kinetic energy', then a bunch of math later, this is your work energy principle that the rates times to control effort has to be equal to your power equation. Welcome! We're still going to build on the Lyapunov optimal control or the Lyapunov of control theory, so we build the Lyapunov function, take its derivative, we have to prove it's going to be negative, semi definite at least, right? 0.49%. And we have rate control we want to talk about and we also have the attitude, you know, the rate and attitude or just rate control. So that's the control that we can implement, this is very much a bang bang. If you look at the control authority, am actually saturating all this time. Instead, they tend to take a iterative approach to building the model. What we have here, right? You can have different forms as long as it's negative, that's all that Lyapunov theory requires, there's no smoothness requirements on this one, at least here. Right? You will learn the theoretic and implementation aspects of various techniques including dynamic programming, calculus of variations, model predictive control, and robot motion … This years emphasis is pre-computation. Then we'll hand this off to an optimal control synthesis approach or a planner or reinforcement learning algorithm, if you will, and the result will be a new policy, a new purported optimal policy. It turns out if you run this loop ,Stefan Roth demonstrated that this kind of interactive approach, which turns out too much more closely match the style used by expert engineers, works really well in practice and it can provide stronger theoretical guarantees. To be successful in this course, you will need to have completed Courses 1, 2, and 3 of this Specialization or the equivalent. If you just wanted to see it here, I have the max. Thanks Prof Schaub, that was a wonder of a course! Those are two considerations, and the rate control is a first order system as you've seen. And then, you either hit positive, or hit negative. Reinforcement learning is a body of theory and algorithms for optimal decision making developed within the machine learning and operations research communities in the last twenty-five years, and which have separately become important in psychology and neuroscience. The main result of this period was the Wiener-Kolmogorov theory that addresses linear SISO systems with Gaussian noise. Or are you tracking something that's spinning very quickly? Yes. It's a different kind of a way to look at it. If it's negative definite, we have asymptotic stability. Reinforcement learning is a new body of theory and techniques for optimal control that has been developed in the last twenty years primarily within the machine learning and operations research communities, and which have separately become important in psychology and neuroscience. 16.29%. You know, Epsilon off when you're hitting it hard. So, you get a linear response coupled with the nonlinear ones, that's a saturated function. So, Kevin what was your approach then. It's a really simple V dot and it gives you huge amounts of freedom in how you want to design and shape that response and deal with saturation. You got a question. Bryan. You know, if you go look at your V dot function, what was its form? The method is largely due to the work of Lev Pontryagin and Richard Bellman in the 1950s, after contributions to calculus of variations by Edward J. McShane. If you look at this function, if V dot has to become negative, if your control authority is larger, the maximum control authority is larger than all these other terms combined. We have limited actuation, our control can only go so big. But that assumes you can really implement this control. I can do one Newton meter of torque, that's all I can do. That's what you're doing. We really want to learn a model that's good where the RL algorithm or optimal control synthesis algorithm, is likely to visit as it learns a policy. So this USI, that is the... Actually, that should be U max I believe, that shouldn't be USI, that's a typo. We make very strong arguments. Our framework can be extended in different ways. An Introduction to Optimal Control Applied to Disease Models Suzanne Lenhart University of Tennessee, Knoxville Departments of Mathematics Lecture1 Œ p.1/37. And we said, if we made K less than U max, I could guarantee this would always stabilize. Yes. Finally, we look at alternate feedback control laws and closed loop dynamics. So, when I do this response, I'm taking the time scale that was 300 seconds and I'm showing you roughly 100 seconds worth here zoomed in. It's guaranteed to converge. I can guarantee stability. This was the control we derived at the very beginning for our tracking problem. This optimal control problem was originally posed by Meditch.3The objective is to attain a soft landing on moon during vertical descent from an initial altitude and velocity above the lunar surface. So that's the time derivatives, so at this stage, I'm picking my steepest gradient. So with this I could have settled in 30 seconds, now I'm going to settle in 30 minutes and I did that by bringing down my gains, all the gains, so the control requirements never flat-lined, you know, they never hit that limit. This is one of over 2,200 courses on OCW. The optimal control problem is to find the control function u(t,x), that maximizes the value of the functional (1). Here I've got the hybrid solution. If I then pick the worst case tumble, I have to pick a feedback gain such that I never saturate. And the nice thing is, with this control, I can stil, l if you plug that cue in here, you can still guarantee that V dot is always negative. We take a cost function. All of that stuff. The University of Melbourne. COURSE. But we can't do that because we have limited actuation. 77.28%. Right?So this would work, but there's a performance hit, it limits how much you can do. This capstone will let you see how each component---problem formulation, algorithm selection, parameter selection and representation design---fits together into a complete solution, and how to make appropriate choices when deploying RL in the real world. We thus need approaches that are more robust and fundamentally interactive to find good models and good controllers. So let's look at the hybrid approaches. Explore 100% online Degrees and Certificates on Coursera. Right? Good. If you're looking for something more in depth, edX tends to be better Right? And if you plug in this U in here, this whole thing would be minus Del Omega transpose P Del Omega. U is one. And that's why these bounds that we have, what I'm trying to illustrate here is how conservative are there. So, if we do one of them, basically it says you're using a linear response, until you saturate, and then you saturate. You still have the analytic guarantee, unless you invoke other fancy math. It was particularly interesting to see how to apply simplified methods of optimal control to a real-world ish problem. Learned so much and still want to proceed :). But we're going to deal with the control solutions that aren't just continuous. It's a very simple bound where we take advantage of the boundedness of attitude errors, and the MRP description that gives us a very elegant- the worst error is one in MRP space at least, right? It's goint to be really key? This optimal control problem was originally posed by Meditch.3The objective is to attain a soft landing on moon during vertical descent from an initial altitude and velocity above the lunar surface. If V dot is negative definite and guaranteed asymptotic stability. It's a powerful concept worth learning about and it's very useful in any context which seems game like. Further, a control law is presented that perfectly linearizes the closed loop dynamics in terms of quaternions and MRPs. The instructor is awesome at teaching concepts. 1 practice exercise. Yes. The worst error is one, we can take advantage of that in some cases and come up with bounds. Don't show me this again. Facebook Social Media Marketing Facebook. But then you deal with a discrete jump and you control authority that might, you know, excite on unmodeled dynamics, that's probably my biggest concern I would have. If you want minus six, you wouldn't give plus five, you would give minus five, the closest neighbor with the right sign. Yup. We really don't know what the inertia mass properties are, and you're picking it up and you want to first to stabilize yourself. You could actually just, you know, this kind of a control would also be saturated control and would be asymptotically destabilizing, which is not Lyapunov optimal. If we make it negative definite fantastic, but you just want it to be negative. The goal is to understand the space of options, to later enable you to choose which parameter you will investigate in-depth for your agent. And then, you look at the corresponding V dots that you get with the classic Lyapunov functions we had last time. And am using the classic, it's just the proportional derivative feedback K Sigma and P Omega here. And now it's really negative. If K times sigma is always less than the maximum control authority, you can cut- you can guarantee, you can come up with a control, U, that's going to make this V dot negative, and therefore, guarantee stability. So you can see here, I'm grossly violating actually that one condition I had and say, 'hey, if this were less than one, I would be guaranteed, analytically, this would always be completely stabilizing and V dots would always be negative' and life is good, but that's not the case here. Milestone 3: Identify Key Performance Parameters. We fit them all and it off to the synthesis, which hands back up reported new policy and continue in this loop. What I want to illustrate here though is I picked gains, this K is 7.11. So I should have had six, but I want five. And then we run this loop again, collecting data from the new policy, together mixed with some data from the exploration policy. That comes out of that controls. It's an easier way to get inside, and this is an area where the MRPs will actually have some nice properties as well. So now we're going to switch from a general mechanical system and apply this to specific to spacecraft. This just says, 'are you positive or negative?' And put in the contrapositive, to not get good real world performance using a No-Regret learning algorithm, the class of algorithm we're talking about for learning, a good optimizer against that all it must be the case that at some point you failed to build a low error model of how the world works. Within the control authorities quickly going to be exceeded to try to track it. Glenn L. Murphy Chair of Engineering, Professor, To view this video please enable JavaScript, and consider upgrading to a web browser that, Optional Review: Unconstrained Attitude Control. I can't even draw the Gaussian noise too much, but it will do some weird stuff. www.coursera.org. So we're doing U is equal to minus Sigma, minus P Omega, it's unsaturated. In these examples, are we applying the previous controls we said? So, let's see. Control of Nonlinear Spacecraft Attitude Motion, Spacecraft Dynamics and Control Specialization, Construction Engineering and Management Certificate, Machine Learning for Analytics Certificate, Innovation Management & Entrepreneurship Certificate, Sustainabaility and Development Certificate, Spatial Data Analysis and Visualization Certificate, Master's of Innovation & Entrepreneurship. Right? We'll fit them all, again, I'm showing your fitting linear regression. supports HTML5 video, This course trains you in the skills needed to program specific orientation and achieve precise aiming goals for spacecraft moving through three dimensional space. So I need this to have opposite sine of Q dot. 5 stars. So, that's the way you can bound it. If you modify like the point at which we go to saturation. So, let's start with this briefly. You can only make Q so big. What else could be an issue? Goal: Introduce course. If you'd come up with a control and say, 'you know what? To view this video please enable JavaScript, and consider upgrading to a web browser that, Let's Review: Non-linear Approximation with Neural Networks, Drew Bagnell on System ID + Optimal Control. How do we handle this? Well, if I have continuous control with this theory, I would hit the target. And then I am saturating. The attitude and rate control gets a lot more complicated and sometimes we can come up with conservative bounds for stability, but they're conservative as you will see. Here we're always arguing the performance, no, the stability is arguments the same, the performance will be different if you measure the wrong Omegas. Or are there better ways? And, well, actually, let's talk about this then. You want something that's really robust, and this simple rate control allows you to prove amazing levels of robustness on how to stabilize these kind of tumbling. But, you know, the tangent function, or the A10 function, around the origin linearizes to basically a linear response. I don't like using max force, maybe I want to use half of it'. So we use a very simple PD control, we know it's globally stabilizing all the one asymptotic. So that's kind of where we can think of this. So just like that modified one, this basically gives me the control up to the point that I've reached my saturation limit, and then I'm enforcing a hard saturation limit where I'm not getting the six Newton meters, but I'll give you the max of five. Monte Carlo is running this stuff. To our best knowledge, the only work of applying optimal control to computer vision and image processing is by Kimia et al. By the end of this course, you will be able to: You can see initial conditions, were going to be having large attitude, large rates, principle inertia's three different gains on P, one on K, maximum torque is one, just an easy number. Alternate feedback control laws are formulated where actuator saturation is considered. This is interesting. There's no real error driving it, its purely measurement errors. If you have X amount of control authority, are you using it to its maximum capability in this case? Like, we switching between saturation and then a linear bar in the middle? Optimal control is an extension of the calculus of variations, and is a mathematical optimization method for deriving control policies. So then the question is, what do you make Q such that J, which is our cost function here, V dot, which is our cost function J, make it as negative as possible? Matha and Adam, thank you again. After the duel as wincrange with billericay town young teens were selling sterling silver natural onyx rose cut cz pendant pear shape 1 3/16 inch long xxx And that's an issue. Well, it's basically, you know, this could be at worst one so K, essentially Newton meters, tells you right away with this gain, 180 off, you would ask for k Newton meters. I’m thrilled with the results of using your tips to get my Yorkie to respect me and follow directions. You are maximizing your perform- you're making your-. There's lots of ways you can do this in the end because if you look at... Let's play with some ideas here. Susan Murphy on RL in Mobile Health 7m. And as we saw with the mechanical system, you can do this in a variety of ways. So that's what we want to look at next. Are they perfect? It's at this instant what control solution will make my Lyapunov rate as strongly negative as possible, so I'm coming down as steep as I can. In the second lecture, I'll give you a quick introduction to the process of learning models within model-based reinforcement learning with an emphasis on approaches you might find practical. Right? How much effort with the with the proportional feedback required of the control system in here? If you do a bang bang. So it's kind of cool because we can pick then, okay, for small responses, this is the slope I want, that's stiffness that I want for disturbance rejection, for closely performance considerations. But you can see from the performance, it behaves extremely well, still, and stabilizes. Showing 452 total results for "optimization" Discrete Optimization. 4 stars. Now, why does this go wrong? Â© 2020 Coursera Inc. All rights reserved. In the homework's actually, there's the one with the robotic system that you just- it's just X dot equal to U, and you're trying to track something, and then you have a maximum speed that you can control. ) is given by α∗(t) = ˆ 1 if 0 ≤ t≤ t∗ 0 if t∗

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