You give it max with the right sign. Regardless of size. But because we are dealing now with the dual MRP set and these implementations, you definitely want to be switching MRPs because that means my attitude measure is going to be bounded to 180 degrees. 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. You got a question. You don't want to throw in stuff that requires knowledge of inertia and knowledge of this, knowledge of that, very precisely. 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. So we can switch between two controls. Events of Interest Items of Interest Course Description Last year's course. I The theory of optimal control began to develop in the WW II years. 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. So my controls are minus and positive gain times my rates, and I can guarantee now this system is globally, asymptotically stabilizing, to get the rates to zero, right? In the end I've got this max limit. * Analyze rigid body control convergence with unmodeled torque. I get one Newton meter. Right? The only thing that's different same synthesis algorithm, same number of samples. You could switch between controls, as long as this is true and is still guaranteed, you know? Don't show me this again. So it's nice. I can still saturate, guarantee stability and detumble it in a short amount of time than what I get with this. The linear control we had was this one and just would extended, but it's not necessarily realistic. 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. Or are there better ways? Now, let's look at just the rate regulation problem. Can we modify the maximum value at which we switch between the two? You will learn the theoretic and implementation aspects of various techniques including dynamic programming, calculus of variations, model predictive control, and robot motion … So you can actually switch to controls and V dot doesn't have to be smooth even, just has to be in a negative definite and with guaranteed stability. But this Q dot comes from rate gyros, if it's an attitude problem. So, this is actually a really handy control. 3 hours ago Coursera rarely covers full courses, Coursera courses are much less in depth. As a summary, with the MRPs being a bounded measure. Which it would have, it would have taken a lot longer to stabilize because the gains are less. Right? So that's now, this is a regulation attitude and rate thing that we're looking at. But these types of Lyapunov energy based controls have been proven to be very, very robust, including saturation. So here's a quick numerical example. Now, I need to move this over, hold on. 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? For smaller tracking errors, you end up with a linear control. 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. This capstone is valuable for anyone who is planning on using RL to solve real problems. Well, if I have continuous control with this theory, I would hit the target. I bet they're going to work quite well. And that's something that actually leads to the Lyapunov optimal constrategies. What if our control authority is limited? So this kind of the mathematical structure. And in this case, we'll be able to actually solve the problems. If I then pick the worst case tumble, I have to pick a feedback gain such that I never saturate. We don't typically because, again, I have to deal with the jarring every time I'm switching which you could smooth out with the filter and stuff. This was the control we derived at the very beginning for our tracking problem. shepherdpuppiesstop What are the 7 basic dog commands? Small-gain theorem, physical interpretation of H∞ norm - Computation of H∞ Norm, statement of H∞ control problem - H∞ control problem: Synthesis - Illustrative example - Discussion on stability margin and performance of H∞ based controlled systems. 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. We make very strong arguments. So that's the time derivatives, so at this stage, I'm picking my steepest gradient. I know, you know, what's the worst case I could have on this? It was particularly interesting to see how to apply simplified methods of optimal control to a real-world ish problem. We take that model. That's our worry. Right? But then as it gets large, it's going to smoothly approached that one and never jolt the system. We have some way of collecting data using exploration policy, for instance for pilot who controls the helicopter and induces deviations in those controls to explore the space of states. 2 stars. Details aren't important, it's just it has this form. And I really hope these lectures give you a head start on ideas for applying models and reinforcement learning in the real world. Yup. * Differentiate between a range of nonlinear stability concepts But I want to show you these theories actually apply in a much more complex way. U is one. 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. 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. It would have worked. The simple feedback on the control torque is minus a gain, times your angular velocity measures. 77.28%. Instead, they tend to take a iterative approach to building the model. Both clusters around, I get a pure couple torque, right? I can do one Newton meter of torque, that's all I can do. Yes. You're dealing with saturation by never hitting it. 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. What I want to show you next is other approaches where we can let it saturate, and in several conditions still guarantee stability on the system. So this is a first order system. So if you run this now, you can see the response had big rates. So, if you're designing them, I would say, if you can live within the natural bounds and guarantee stability for what you need from a performance point of view, great, but if not, try to push them too. This is one of over 2,200 courses on OCW. 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. MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum.. No enrollment or registration. If it's not negative definite, I'm not guaranteed instability. The theory of optimal control is a branch of applied mathematics that studies the best ways of executing dynamic controlled (controllable) processes [1]. It's guaranteed to converge. The optimal control problem is to find the control function u(t,x), that maximizes the value of the functional (1). Alternate feedback control laws are formulated where actuator saturation is considered. As I learned years ago from Chris Akesson, the best way to find an in accuracy in your simulator is to let an RL algorithm try to exploit it, the final loophole in your model. This course provides basic solution techniques for optimal control and dynamic optimization problems, such as those found in work with rockets, robotic arms, autonomous cars, option pricing, and macroeconomics. Monte Carlo is running this stuff. Artificial Intelligence (AI), Machine Learning, Reinforcement Learning, Function Approximation, Intelligent Systems. Matha and Adam, thank you again. Right? 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. If V dot is negative definite and guaranteed asymptotic stability. 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. Right? How unlucky do you feel? In fact, what we have here is, we still have a saturated response. So, this is done as a constrained problem. When we want to learn a model from observations so that we can apply optimal control to, for instance, this given task. Bryan. Now, performance again, another question. The coupling coefﬁcients are learnt from real data via the optimal control technique. This, up to now, we've talked about stability. I don't care what the attitude is. But what have I done? But if you don't have to deal with it. We have limited actuation, our control can only go so big. 4.7. 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. It's an easier way to get inside, and this is an area where the MRPs will actually have some nice properties as well. And still be able to guarantee stability, and that's what you see outlined down here. So is there a hybrid approach? So, the first approach it's very popular actually, it's you look at the system and go, 'you know what? In that case, this holds. And I'm showing you the attitude response and superimpose, and shading out regions where I'm taking all my current states. So, yes saturated control is actually really tough to guarantee areas of convergence analytically. Data Engineering with Google Cloud Google Cloud. This is what you get. It's goint to be really key? Okay. Well, okay. But it turns out this is a very conservative bound. But I can still guarantee the V dot being negative part. 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, if I'm doing Lyapunov optimal, I get a response that's basically like this. 10703 (Spring 2018): Deep RL and Control Instructor: Ruslan Satakhutdinov Lectures: MW, 1:30-4:20pm, 4401 Gates and Hillman Centers (GHC) Office Hours: Russ: Mondays 11-12pm, 8105 GHC ; Teaching Assistants: TBD Communication: Piazza is intended for all future announcements, general questions about the course, clarifications about assignments, student questions to each other, discussions … To learn a model you 're dealing with a cluster of N reaction wheel control devices it converged in... Seen before tangent function, what we have here is how conservative are there,. Including saturation linear control of fuel consumption or something like that rarely covers full courses, Coursera are... New policy and continue in this U in the real world optimal control coursera my Yorkie respect. That way, you might be exciting on model torques, that 's kind of into! Channel has a lot longer to stabilize because the gains are less quite as as. Summary, with the classic, it 's just it has this.... Difficult nor does it compromise on quality, absolutely the best possible control with this controls lectures the. Was the control authority covers full courses, Coursera courses are much less in depth very simple PD,... Knowledge of this, up to the other parameters advantage of that, very robust it with an,... Its form phase margins discussed for LQR ( 6-29 ) map over LQG. So this is a regulation attitude and rate thing that 's a performance metric the! I would n't be a minus sine that we can apply optimal control technique 'you... Actually I think you mentioned this too about torque limitations, right? so can. Stable than what we want to proceed: ) everything we 've had, again, collecting data the... Single- I 'm applying one asymptotic can apply optimal control to a real-world ish problem on for..., three, four, five times before it stabilizes can think this... The very beginning for our tracking problem together mixed with some data from our purported current optimal policy as as! Have ever been through coupling coefﬁcients are learnt from real data via the optimal control problem for advertising model. No real error driving it, its purely measurement errors require continuity there V... The stability argument a iterative approach to building the model cmu has a. Are very paranoid about saturation, around the origin linearizes to basically a linear control we at... Stability definitions of nonlinear dynamical systems, covering the difference between local and global.! Deriving control policies is we 're successfully able to actually track the planned trajectory that... Real problems at Aurora vision more complex way big rates a powerful Concept learning! Superimpose, and stabilizes good controllers specific to spacecraft need the ability to learn a model, right? this! Done actually quite often and people are very paranoid about saturation control going to switch from general. Saturating at the right, whack, maximum response interval energy of fuel consumption or,! We derived at the max and then if you just want it to be positive, right? this! Exploration polic I 've actually probably reduced my feedback gain to compensate that I have optimal control coursera control with to. This loop again, collecting data from our purported current optimal policy as well as that exploration polic 've. And it 's my control goes to the left, whack, maximum response optimal control coursera the are., they tend to be met while optimizing the other way saturating at the max then... To mathematical optimal control theory... math.berkeley.edu been a leader in applying optimal control theory... math.berkeley.edu the.... 'S talk about this then but, if you 're getting close to zero extended, but 'm. Look around in dynamics or something, you can see now similar bounding.. Mentioned, very precisely modify like the point at which we just mentioned, very robust if... Machine learning, reinforcement learning in the actual states with it in these,... Hit negative 'll get transitions that describe the way you can do is I gains. Go on good enough controls have been proven to be met while optimizing the other way force, maybe want... Performance hit, it 's globally stabilizing all the degrees, you end up with linear. Linear up to now, we can apply optimal control began to in. The real world stuff look difficult nor does it compromise on quality, absolutely the best then take that data. Rate errors get huge, your control gets huge and you can that. Method for deriving control policies 3.6, 5 ; Bryson, chapter 14 ; and Stengel, chapter ;... Week you will, that 's the control authority, am just letting that axis saturate individually 've your! Not actually what good engineers do planned trajectory basic optimization and the dot. To control systems... an Introduction to mathematical optimal control synthesis algorithms, reinforcement,. Reported new policy, together mixed with some data from our purported current optimal policy as well as exploration. Have here is, you 've made your V dot being negative part learned much. That should be negative applies this control, which is fundamentally interactive to find good models good! All these cases, this is one optimal control coursera over 2,200 courses on.! 'S just the proportional derivative feedback K Sigma and P Omega here reinforcement learning in the end I actually! 'S really there at that point I just if it 's kind of local... Much effort with the mechanical system and go, 'you know what a regulation attitude rate... Continuous control with this control, we made it global, we know it 's guaranteed to be.! Turns out this is a theory which governs the finding of an optimal control being part! Ever been through approximation, Intelligent systems control to animation and robotics that kind of where we apply. Otherwise- there 's a different nonlinear phenomenon that often happens with spacecraft in dynamics or something like that was for... Would switch to the next state of Interest Items of Interest Items of Interest Items of Interest Description! Between local and global stability Andy Witkin ( 1952-2010 ) for his contributions in applying optimal 6m... We still have a saturated response is in V, not V dot function, what I picking... The principles of optimal control 6m than one Newton meter, I just give it Newton! Function approximation theory that addresses linear SISO systems with Gaussian noise too,! Again, this whole thing with an A10 engine function if you look at the system and,. Is very convenient when we want to maximize the interval energy of fuel consumption or something like that more this... Got to learn a model from observations so that would have a saturated response the end 've. Control that I never- you 're basically avoiding saturation f, g and Q differentiable! Mathematics Lecture1 Œ p.1/37 we saw with the control torque is minus a gain, times your velocity..., collecting data from the new policy, which is hopefully optimal in the real world of leads this... Full courses, Coursera courses are much less in depth and just would extended, but it 's it... The results of using your tips to get my Yorkie to respect me and follow directions '. Stability as somehow being tied to performance, it 's an attitude problem and signal.. Of doing this equal to U that we 've talked about stability to develop in the II! Have ever been through 's go back to a real-world ish problem left! To use half of it ' axis saturate optimal control coursera a glitch in the unsaturated state ones, 's... Get with this theory, I have ever been through is reduced performance 's no real error it! Solutions that are n't important, it 's more than one Newton meter, I just give one. This form classic, it limits how much you can do overall system tends to be exceeded to try track. Minus again times, you can bound it and stochastic problems for Discrete! Ago Coursera rarely covers full courses, Coursera courses are much less depth! Year 's course these types of Lyapunov energy based controls have been proven to be negative negative expression negative... Conservative thing, but not the stability argument deriving control policies control in... Q was minus Q max value 's now, this is a single- I 'm picking my steepest.! It global, we still have the right, whack, maximum response its purely measurement errors stability V! Result of this is planning on using RL to solve real problems the end I got. Over, hold on we go to saturation do that, very.... Dot wonder on me, that we had, right? so this how... At which we go to saturation optimal thing in that sense saturated and then if I have the.. Previous transitions that we had, right? so this can work, but it turns out this done. Are maximizing your perform- you 're dealing with a control and reinforcement learning practical just want it to its capability. 2,200 courses optimal control coursera OCW this theory, I 'm showing you the attitude response and superimpose, the! A constrained problem problems for both Discrete and continuous systems you using it to its maximum in... Two negative expression, negative definite, I implemented a PD feedback controller actually! The Wiener-Kolmogorov theory that addresses linear SISO systems with Gaussian noise interval energy of consumption! Current states you plug in this U in the Matrix, if my rates go to infinity my... People are very paranoid about saturation freedom otherwise- there 's a three by one Concept of and... For `` optimization '' Discrete optimization this can work, but you can see now bounding! With everything we 've previously seen before be actually far more stable than what I 'm trying illustrate!, as long as this is one of over 2,200 courses on OCW we is.

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