**adaptive control** This is a topic that many people are looking for. **newyorkcityvoices.org** is a channel providing useful information about learning, life, digital marketing and online courses …. it will help you have an overview and solid multi-faceted knowledge . Today, ** newyorkcityvoices.org ** would like to introduce to you **Why Adaptive Control?**. Following along are instructions in the video below:

“This short video. I am going to explain why adaptive control theory is important for for real time applications. And what can it achieve more when we compare such controller fixed gain or robust controllers. Also i will show a formulation of standard adaptive control architecture.

So that you can directly implement your adaptive controller after watching this video. I will finalize my short talk by referring to my recent advanced adaptive control research. Let s get started. Why adaptive control you are seeing a helicopter a picture.

So here s an helicopter from our laboratory unmanned aerial research facility from georgia tech. We deal with such systems and these systems are basically nonlinear has time variations in their dynamics. And they have multi input multi output and hence. It has coupled loops is a control engineer.

The first thing we do is the model such systems and wild into modeling we resort to first principles of physics idolised assumptions model simplifications and at the end of the day the models that we have do not adequately capture the physical system from a control design point of view. We need to design a controller to achieve and maintain performance goals. Under such system uncertainties and of course uncertainty is not only result from the modeling phase for example. It can be a result of the mage for example.

Let s consider this video. So an aircraft is flying well. But in the middle of the flight. If the person of the right wing is off.

We need to design a controller such that it has to handle such unforeseen. Adverse conditions and by the way. This video is a game from our research facility at georgia tech. We deal with on certain systems and first approach is to design a fixed game controller.

No adaptation fixing controllers like robust controllers first of all requires an accurate system modeling information for example. Let s look at this graph. So we need to minimize a performance error in order to improve the system our achievement goal so in order to minimize this so enter so that we can achieve a better performance. However as you see from this graph in order to minimize this performance error.

We need to know the model more and more as you see that exist. A direct rate of performance with uncertainty and also if we look to left side of this graph. If we have high levels of certainty fixed. Key controllers may fail to achieve is un system performance.

And also as this being said. It is important to mention that such controllers are tuned to the worst case rather than the physical system meaning they cannot improve themselves they assume that the feature will be much like present. Ignoring environmental changes change in dynamics or structural damage as i showed you in the earlier video. What awaiting from this standpoint.

Basically adaptive controllers are advantageous because first of all they achieve a consistent performance asymptotically without excessively relying on system models. And they do not create performance for modeling accuracy again last check..

This a graph. Given below of this slide. Basically. As you see.

The adaptive controller. Basically achieves the same level of performance that their fixed game controller achieved without knowing the system model and with heart basically requiring modeling liquor. See and also more importantly it improves them in itself underpants foreseen adverse condition. It learns as it flies here s the standard adaptive control architecture that i am going to talk about in this presentation.

Very briefly for motivational purposes. Basically. It is a very common scheme. So called model reference adaptive controller.

Let s look at more closely. We have an answer to dynamical system. Given in the blue box. We have a controller and we have a reference system in this reference system given in a yellow box the driven by a command captures a desired overshoot desired system setting time so on and so forth so i want my system output blue output to behave as the reference system output this difference can m corresponds to a system error and system error drives.

The update low and update loss students to controller such that my uncertain system output will be behaved like the different system of a des the philosophy behind model returns adaptive control mathematically. Let s consider this. Uncertain. Dynamical system.

X. Dot. Equals to ax. Plus.

Bu. Plus. Delta. A d.

Multiplied by delta of x. And this is the simplest form and here are you know a you don t know b and you know the but you don t know delta delta is demanding your uncertainty here i am assuming i know the state vector x. And here u corresponds. The control.

Vector and b. Is the parameterize. As the known part x. Lambda.

Unknown. Part and lambda is basically control effectiveness or here..

I am. Assuming that the uncertainty delta of x. Parameter is w. Transpose sigma x.

Here w. Is the unknown set of ideal rates and a sigma is the my basis function so here i know of the basis. I know how uncertainty is captured by these big bases and if we don t know basically we can always resort the universal approximation tools such as neural network. So here there is no loss in general here is my reference system.

I want my system to behave like this linear system here a r is asymptotically stable or so called hurwitz and hence it satisfies the loop on allocution had br is the comment input matrix and c is the connect if you think about a second order reference system you need to design choose it in such a way that you have some settling time. You know rise time or worship. So on and so forth. This is your desired close with model that you would like to ultimately achieve here is my a control signal composed of a nominal controller u.

And and plus an adaptive control y. We have a nominal controller because for example. If you think about let s say you are going to design. A controller for f.

6316. Right. So. No one will allow you to remove the existing controller and put your fan setup your controller in the industry.

So you always have a nominal controller here um. You have a nominal controller you can have any nominal controller. Okay. Api pid.

So here i am assuming that it has k 1 x. Plus k. To cc is. The command.

Again you can design k. 1. And k. 2.

Based on a r equals to a plus d k. 1. It is your how you choose k 1. Sorry.

For the typo and you choose k. 2..

Based on v r. Equals to be multiplied by k 2. You all know these terms after that you need to design adaptive controller right you are not mounting a nominal controller with adaptive controller. And to do that let s write.

The closed loop. Dynamics. As x. Dot.

Equal to ar. X. Plus. V.

Rc. Plus t. Delta. X.

Ua. Plus. The unknown term. These are w.

Sigma w. N. Are unknown. You know sigma raises you know you and phenomenal control signal now inside the brackets adaptive controller can directly access these uncertainties illogical selection for adaptive controller is like that minus w hat sigma minus w hat you add x you back here w had a bolt rw hence corresponds to estimates of the original w.

O. Based. On this adaptive. Control.

Signal. Uh. Here. Is your first.

Read. Update. Goes. Given the last two lines basically.

These tune your w hat. So that a few controller minimizes the difference between your uncertain systems output and the reference system heart so that your system behaves as the actual reference system as t goes to infinity and these are gamma gamma sigma and gamma u and are basically positive definite adaptation games or learning crates if you choose them higher you learn faster basically and these to weight update law given the last two lines are not drive out of view..

Basically they are based on leap on all three things. That is the point of energy function candidate and when you minimize. When you take its derivative. With respect to closed loop.

Trajectories of ee w. Tilde. And other w. Tilde.

You and basically you have v. Dot. Equals to minus n. Transpose pd.

Which gives you leave on stability and by the barbel ats lemma. Basically you can say as t goes to infinity error goes to 0. Perfect. Now of course.

Ok oh. You can read oh this standard adaptive control formulation from my website. I include most of my papers. Very briefly so just check part for my conference or journal papers for a simple formulation.

But of course. There are caveats with the standard level control for example. How we can have robustness with respect to noise with respect to high frequency dynamical system content also how we can achieve robustness against our model dynamics and match disturbances so on and so forth and how we can achieve a guaranteed thrust transient response. Okay we are learning the system and we eventually drive this error to zero.

But during the learning phase. How we can achieve a guaranteed transient performance and robustness. So this is one of my current research topics. So i will encourage you to go to my webpage and check for these six recent papers.

Most of them published in 2012. And some of them are 2011 of course. This is a very hot topic. So i would encourage you to check the literature.

As well you can read the literature from my papers or you can just amazing websites color google check the robust adaptive control high performance or the field control literature standard model reference adaptive control is cool. But if you are dealing with safety critical systems. The ” ..

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