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 You don't want to throw in stuff that requires knowledge of inertia and knowledge of this, knowledge of that, very precisely. This whole feels unsaturated control as well. 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. In fact, what we have here is, we still have a saturated response. And that's an issue. [25, 24]. Yes. This was the control we derived at the very beginning for our tracking problem. It impacts performance, but not the stability argument. So that would have worked, but the key result is reduced performance. Yes, sir. So I should have had six, but I want five. I can still saturate, guarantee stability and detumble it in a short amount of time than what I get with this. In this example, we begin by collecting data from expert demonstration of this nose-in funnel, and we collect a variety of examples of it doing this task. First, we cover stability definitions of nonlinear dynamical systems, covering the difference between local and global stability. I thank you for your attention. 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. What we haven't addressed is actuation limitations. So let's look at the hybrid approaches. So this kind of the mathematical structure. So, for a classic unsaturated control, just a PD, we proved actually, we didn't have to feedback compensate for the Omega tilde I Omega, that term vanished because it was Omega transpose, Omega tilde I Omega. 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. If V dot is negative definite and guaranteed asymptotic stability. 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. We have X dot equal to U that we haven't. You will learn the theoretic and implementation aspects of various techniques including dynamic programming, calculus of variations, model predictive control, and robot motion … Right? So, if you add a little bit of Epsilon. Let's see why. So, we can see now similar bounding arguments. This one would not be Lyapunov optimal because you're not making V dot as negative as possible. You can do it for one of the degrees, you can do it for all the degrees individually with this approach. Now you're a little bit- or actually I think I got it backwards. I mean, do we need to do the control to be continuous or-. 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. All of that stuff. So if you run this now, you can see the response had big rates. But it turns out this is a very conservative bound. For optimal control problem, we consider a cost function: J(x;u) = Z T 0 f 0(x(t);u(t)) dt + g(x(T);u(T)) and the aim is to nd a control u which realize the control problem and minimize J. J. Loh eac (BCAM) An introduction to optimal control problem 06-07/08/2014 2 / 41. I used him to help while taking my controls class. 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. How do we handle this? Then we apply the class of optimal control algorithms we talked about in the last lecture to try to generate a policy. 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. A glitch in The Matrix, if you will. Or are there better ways? Those are two considerations, and the rate control is a first order system as you've seen. Okay. The simple feedback on the control torque is minus a gain, times your angular velocity measures. But we can't do that because we have limited actuation. 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. After this course, you will be able to... The whole course is really good. Drew Bagnell on System ID + Optimal Control 6m. The optimal control problem is to find the control function u(t,x), that maximizes the value of the functional (1). That's a different nonlinear phenomenon that often happens with spacecraft. There's lots of ways you can do this in the end because if you look at... Let's play with some ideas here. And there's also actually a related homework problem you currently working on that kind of leads into this spirit a little bit. And that's something that actually leads to the Lyapunov optimal constrategies. Right? Yup. Right? 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. Reference tracking is tough too because your reference motion impacts, you know, it's my control going to be less than that? 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 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. So that can be one. Right?So this would work, but there's a performance hit, it limits how much you can do. 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. Can we modify the maximum value at which we switch between the two? It can't go more than some one meter per second or something. This is necessary condition for stability, but they're not necessary, you know, there's not an if and only if and this kind of stuff. We made it global, we made it asymptotic, we made it robust, if we have external unmodeled disturbances. And if I had a guarantee of stability, V dot would always be negative. Okay. 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. If I then pick the worst case tumble, I have to pick a feedback gain such that I never saturate. Thanks Prof Schaub, that was a wonder of a course! 3 stars. U is one. 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 am using the classic, it's just the proportional derivative feedback K Sigma and P Omega here. We take that model. 5 stars. If it's negative definite, we have asymptotic stability. 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. Show you how this all comes together. And that's going to guarantee that you're always negative definite. supports HTML5 video. www.coursera.org. Torques and spacecraft is different. So that's what we want to look at next. We'll fit them all, again, I'm showing your fitting linear regression. This just gives you a control performance. And so, UUS means map control authority U in the unsaturated state. It only feeds back on the sign of the radar. Susan Murphy on RL in Mobile Health 7m. So this USI, that is the... Actually, that should be U max I believe, that shouldn't be USI, that's a typo. They were under sampled during data collection procedure. To be successful in this course, you will need to have completed Courses 1, 2, and 3 of this Specialization or the equivalent. Yes. Welcome! The worst error is one, we can take advantage of that in some cases and come up with bounds. I can guarantee stability. This is what you get. To view this video please enable JavaScript, and consider upgrading to a web browser that So you could, if you wanted to, you could go up to here and then and from here go on good enough. I The theory of optimal control began to develop in the WW II years. We have limited actuation, our control can only go so big. It tumbled actually, one, two, three, four, five times before it stabilizes. Right? End of time. That's our worry. We argued already this is insensitive completely to inertia modeling errors, because inertia doesn't appear. Further, a control law is presented that perfectly linearizes the closed loop dynamics in terms of quaternions and MRPs. Explore 100% online Degrees and Certificates on Coursera. The coupling coefficients are learnt from real data via the optimal control technique. But what have I done? 1 practice exercise. So now you can see here that U has to compensate for this, and then add a term that makes this thing negative, semi definite at least, right? So, this is done as a constrained problem. Alternate feedback control laws are formulated where actuator saturation is considered. Right? So, let's go back to a really general dynamical system, and then we'll specialize it for the spacecraft again. The only thing that's different same synthesis algorithm, same number of samples. When we want to learn a model from observations so that we can apply optimal control to, for instance, this given task. 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. But the consequence is, you've reduced your performance, you've reduced your gains. I can't even draw the Gaussian noise too much, but it will do some weird stuff. Minimize the cost functional J= Zt You know, you come up with some bound and say, 'that's the worst tumble I have to deal with', right? 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. I want to bring my rates to zero. You could switch between controls, as long as this is true and is still guaranteed, you know? Now, I need to move this over, hold on. If you just measured half the rate that you actually have, it may take double the time to converge, but you're still guarantee it will converge cause that's often the issue. 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. So this is one of the lessons learned with this stuff in Lyapunov theory. So now we're going to switch from a general mechanical system and apply this to specific to spacecraft. Don't show me this again. This just says, 'are you positive or negative?' What else could be an issue? 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. ) is given by α∗(t) = ˆ 1 if 0 ≤ t≤ t∗ 0 if t∗