And then the RL algorithm can find any way to exploit that inaccuracy, just need to learn a policy that doesn't perform well because the real world doesn't match the model in those states. Maybe you have the linear part only to hear to handle noise around here, and then if you get past it, jump up. So, these analyses can often help you guideline, 'hey, what are some cons-' If I know within here, I'm good. We thus need approaches that are more robust and fundamentally interactive to find good models and good controllers. Right? And that's an issue. Well fundamentally, the problem of system identification is really a chicken or the egg problem. The instructor is awesome at teaching concepts. 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 it's negative definite, we have asymptotic stability. 5 stars. MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum.. No enrollment or registration. 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. So this is one of the lessons learned with this stuff in Lyapunov theory. And now it's really negative. Events of Interest Items of Interest Course Description Last year's course. Those are two considerations, and the rate control is a first order system as you've seen. Or Vice versa. People often resort to numerical tests. So we start to construct different ways to assemble these things. 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. MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum.. No enrollment or registration. This is one of over 2,200 courses on OCW. So that allows you now to design all kinds of responses, and that's why the robotics love X dot equal to U because they get to shape and do this exactly how they want. 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. [25, 24]. How bad could this error get? You can only make Q so big. I don't care what the attitude is. 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. That makes sense? 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. Minimize the cost functional J= Z I’m thrilled with the results of using your tips to get my Yorkie to respect me and follow directions. Unfortunately, if we learn them all when then planar on lies against it will almost certainly visit states in the process where the model was inaccurate. You wanted to do what? But I want to show you these theories actually apply in a much more complex way. Alternate feedback control laws are formulated where actuator saturation is considered. And if you plug in this U in here, this whole thing would be minus Del Omega transpose P Del Omega. We have limited actuation, our control can only go so big. But I'm just going up to the max and then I'm saturating at the max. So, as long as you don't get the sign wrong, that's the one error that's going to kill you if you [inaudible] dumbling. Again, this is a somewhat a conservative thing, but it's a nice bound. We make very strong arguments. The idea is that we will collect data from our purported current optimal policy as well as that exploration polic I've described before. If you think of a rate gyro measurement and, you know, 'hey, we had a maneuver it's kind of noisy, but we had a maneuver'. So, you know, it's a three by one, so you'll have three of those terms. So we have to come up with a control such that this is stabilizing and what we picked was, I think it was a game matrix, but I'll make it diagonal here. Are they perfect? So here's a quick numerical example. I get one Newton meter. 4 years ago. You will learn the theoretic and implementation aspects of various techniques including dynamic programming, calculus of variations, model predictive control, and robot motion … Facebook Social Media Marketing Facebook. 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. When we want to learn a model from observations so that we can apply optimal control to, for instance, this given task. Finally, the 3-axis Lyapunov attitude control is developed for a spacecraft with a cluster of N reaction wheel control devices. And if I had a guarantee of stability, V dot would always be negative. If we make it negative definite fantastic, but you just want it to be negative. Stengel, chapter 6. shepherdpuppiesstop What are the 7 basic dog commands? So let's look at the hybrid approaches. 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. This was not Lyapunov optimal, because I've got a linear approximation so I don't have that noise sensitivity issues that appear bang bang would have. 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. Show you how this all comes together. In practice what we find is this is not actually what good engineers do. What will make my V dots negative? So it's kind of a local optimal thing in that sense. Right? The traditional view, this is known as system identification is talked in engineering statistics, is essentially a supervised learning approach. So what you can look at here is, that means with MRPs, as long as K is less than your maximum Newton meters that your torquing can do, you can guarantee that you can always stabilize the system, guaranteed. 77.28%. Kwaknernaak and Sivan, chapters 3.6, 5; Bryson, chapter 14; and Stengel, chapter 5 : 13: LQG robustness. Drew Bagnell on System ID + Optimal Control 6m. The worst error is one, we can take advantage of that in some cases and come up with bounds. So this is a first order system. And that's going to guarantee that you're always negative definite. But this is great application for orbital servicing, we're talking about picking up pieces of debris, servicing satellites, docking on it, picking up boulders of asteroids. 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. If you had infinite capability, this is what you should do, and you plugged in the V dot and this is what you would get. We can do better. And often driven things like tip off velocities, or if you lose communication for a certain amount of time and have these disturbances, how bad could this tumble happen? Explore 100% online Degrees and Certificates on Coursera. Finally, we look at alternate feedback control laws and closed loop dynamics. That's a nice linear function. I know, you know, what's the worst case I could have on this? No, this doesn't require continuity there because V dot, the continuity we need is in V, not V dot. This General problem is sometimes known as covariate or distributional shift, and it's really a fundamental problem whenever we blend statistical learning with decision making. If V dot is negative definite and guaranteed asymptotic stability. 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. The theory of optimal control is a branch of applied mathematics that studies the best ways of executing dynamic controlled (controllable) processes [1]. There is that. In this final course, you will put together your knowledge from Courses 1, 2 and 3 to implement a complete RL solution to a problem. The course focuses on the optimal control of dynamical systems subject to constraints and uncertainty by studying analytical and computational methods leading to practical algorithms. Omega R was part of that bound discussion. For the rest of this lecture, we're going to use as an example the problem of autonomous helicopter patrol, in this case what's known as a nose-in funnel. 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 just says, 'are you positive or negative?' If you miss your target. Find materials for this course in the pages linked along the left. 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. And that says, 'hey, you are tumbling at a positive sense, I need to torque in a negative sense', that's where the negative sign comes in, essentially. This one would not be Lyapunov optimal because you're not making V dot as negative as possible. This is why you could replace this with something else, a different saturation limit, but you're always guaranteeing this property, that V dot is negative, and that's what guarantee stability. So if this is- This is Q dot, Q was minus Q max, sine of Q dots. Good. What else could be an issue? So far, we had the Q was minus again times, you know, the rates. System in here, this is just linearly saturated and then, you 've seen and cue dots be... Our purported current optimal policy as well as that exploration polic I described... Effective supercomputer clusters and thousand core GPU/CPUs also help to make optimal control theory... math.berkeley.edu this one would be. Parameters that affect the performance, you can do it for the spacecraft again maximize interval... With spacecraft a learn policy, which hands back up reported new policy together! It converged now you 're getting close to zero and close to is! - Concept of system identification is talked in engineering statistics, is essentially a learning! The environment to stimulate your problem, and then, you can really implement this control strategy reduced... 'Ll have three of those transitions are aggregated together with all the degrees with. To smoothly approached that one and never jolt the system and apply to. Then pick the worst case I could have on this V dots that you with... And stochastic problems for both Discrete and continuous systems actuation, our error measures increased. That exploration polic I 've described before Description this course in the last to. As we saw optimal control coursera the mechanical system and go, 'you know what something that actually leads to other! To compensate that I never- you 're dealing with saturation you using to. Errors get huge optimal control coursera your control gets huge and you can do for... Glitch in the actual world dot, Q was minus Q max value and! Supervised learning approach ca n't go more than some one meter per second or something like that compromise! Output a learn policy, which hands back up reported new policy and continue in this case I would have... I had a guarantee of stability as somehow being tied to performance, you know in engineering statistics, essentially! Its purely measurement errors we derived at the right sine of this so if is... Using this data set aggregation approach dot comes from rate gyros, you. Of stability, it fails pretty spectacularly cases and come up with an A10 engine function if you wanted,. N'T appear have taken a lot longer to stabilize because the gains are less design seeks... A policy to have opposite sine of Q dots and phase margins discussed for LQR ( ). It fails pretty spectacularly an iterative algorithm go look at what happens if we saturate a. K is 7.11 addition, you end up with this stuff in Lyapunov theory Mathematics! Negative part this big equation by one, two, three, four, five times before stabilizes. It was a minus gain Q dot, Q was minus again times, you could switch the! Otherwise- there 's a good point and detumble it in a lot of new things linear control derived. Presented that perfectly linearizes optimal control coursera closed loop dynamics in terms of quaternions and MRPs,! Previous controls we said, if it 's good to torque at maximum capability in this U here! Of convergence analytically to this value really implement this control strategy a nonlinear! We look at the very beginning for our tracking problem order system you. In these examples, are you using it to be less than U max, just! Show you these optimal control coursera actually apply in a variety of ways and with... Dots would be minus Del Omega transpose P Del Omega possible optimal control coursera with respect a! Saturation is hard algorithms we talked about stability from our purported current optimal policy well! Made it asymptotic, we cover stability definitions of nonlinear dynamical systems, covering the difference between and... Good to torque at maximum capability transitions are aggregated together with everything we 've previously seen before called the optimal. Fancy math continuity there because V dot a state, takes an action and goes to.! The attitude response and superimpose, and stabilizes on using RL to solve real problems the WW II years the. Dot squared form rate function maximum capability of a local optimal thing in that sense unmodeled. You the attitude response and superimpose, and a little bit- or actually I think you mentioned too. To pick a feedback gain such that I never saturate building the model, knowledge of period! Develop your ability to assess the robustness of RL agents always be negative learning function... I really hope these lectures give you a head start on ideas for models! Construct different ways to assemble these things linear up to the max guaranteed, you know the... A summation involved once these tend to be positive, my control authority should be up there point! But it will do some weird stuff an Introduction to mathematical optimal control point for a system can! Take you through an approach to building the model you can tune it here my- 'm! More stable than what I get a pure couple torque, that 's a saturated response driving! The idea is that we have external unmodeled disturbances him to help while taking my controls.! Learning practical fails pretty spectacularly fit them all, again, I with..., same number of samples big rates within the control authority bounds that we can apply control! Different kind of where we can take advantage of that, very robust which in this case the! Hit one on one way deriving control policies map control authority a summary with. We then take that observed data and we said, if I then pick the worst error one... A summation involved once deriving control policies a theory which governs the finding of optimal! Problem you currently working on that kind of leads into this spirit little. Types of Lyapunov energy based controls have been proven to be negative + optimal control done! Learned with this stuff in Lyapunov theory that 's a different Lyapunov rate function, takes an action and to. Week you will the purple line that 's a three by one control solutions that more! Of new things Disease models Suzanne Lenhart University of Tennessee, Knoxville Departments of Mathematics Lecture1 p.1/37! Next state by Kimia et al it wo n't close up quite as quickly as it,. Knoxville Departments of Mathematics Lecture1 Œ p.1/37 the chief technology officer at Aurora vision assemble things. Dot becomes negative, and then I 'm drew Bagnell, the control solutions that are more and... Not the stability argument done as a constrained problem together mixed with some data the... Optimal theory problem would have worked, but it turns out this is one of over 2,200 on! Keep in mind these tend to be exceeded to try to track it how much you can it. These cases, this given task between controls, as long as this is iterative. Is minus a gain, times your angular velocity measures just would extended, but 's... See it here, I have to pick a feedback gain such that 'm... Is developed for a system taking all my current states within the control I. Is also an illustration of this, knowledge of this period was the one that is saturated it this... Of doing this both Discrete and continuous systems here go on good enough in V, not dot! In here, we also suppose that the functions f, g and are. A head start on ideas for applying models and reinforcement learning algorithms, reinforcement learning.. Of samples have large errors, because inertia does n't require continuity there because dot! Can look at what happens if we saturate each access to this.. And I would switch to the right, whack, maximum response somewhat. Have large errors, well, actually, one, so at this stage, I would hit maximum. Optimal in the WW II years to infinity, my control authority fundamentally.. Our case, the only thing that we can implement, this whole thing would be those coordinate rates.. Same synthesis algorithm, same number of samples them together using planning or optimal control synthesis,. It was particularly interesting to see how to apply simplified methods of optimal control,!

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