Good. So you are on time. So I will present. So I am Ahmed Abli from Grenoble Institute of Technology. I will present today the work of Audrey Shannon. She defended her PhD recently. And the work is about estimation of non-aerodynamic forces of an airborne wind energy

system. So this is the outline of the presentation. I will present the modeling of our airborne wind energy system in Grenoble. And I will give you some key steps of the design of extended Kalman filter that we have used.

I will show some results. And we will discuss these results. And then we will have some questions. So this is the system that we study in Grenoble. So we have two parts. We have the drone, I think. Yeah, this drone that is attached to a fixed wing.

And we have the second part, which is on-ground station. And we have a tether between them. So in this paper, Audrey has studied how to take off and land this type of system. So it is a solution for takeoff and landing. So we have the wing that is attached to the drone. And we have the on-ground winch.

And in this paper, we have used feedback linearization in order to propose a control for the takeoff and landing phases. And in this work, we consider that the unknown forces are considered as disturbances. And we use integral actions in order

to guarantee the robustness of the proposed controller. I will not talk a lot of detail and in details on this feedback linearization and this control. If you want more details, I invite you to assist to the presentation of Zakiya Zaki tomorrow in the session of modeling and control 2.

So let us take a little bit modeling. So we are working in 2D. So the modeling, we are using the mechanical equations of physics. We have r of the length of the tether. We consider that the tether is stored. We have the inclination angle beta.

And here, we have the fixed wing with the drone and with all the forces. So we have the weight. We have the forces that are projected. So we have here lift and drag because we have a wind from your left. And this is the apparent wind. And we have also the thrust force generated by the drone.

So it is here projected in the radial and the tangential directions. For the forces, so we have these forces, the aerodynamic forces. So here, in r, so in radial and tangential, we have lift and drag. And also, we have some unknown forces

that we neglected in the first control strategy. So we have the dry friction in the on-ground range. And we have also some uncertainties on the drone forces. We have some variations in the thrust of the propellers. So this is the equation. We have a system with four states, r, r dot, beta, beta,

dot. And we use this system in movement in 2D, in the plane, in the vertical plane. So when we apply the control, we find that we can improve it. So we thought, if we can apply extended Kalman filter,

so what is extended Kalman filter? It is an algorithm that is able to estimate the state vector of a nonlinear system from incomplete noisy measurements and its output allied with discrete time varying linear model. So this is the input of the system. We extended the system.

So we added the aerodynamic forces or the forces here. And we know we saw that the system is nonlinear. So in order to apply extended Kalman filter, we need to discretize the system. So this is the discretization of the system with FK depending on A. So A is the Jacobian

at the current state of F. And H here is the sampling time. So here we have the nonlinear dynamics. And the Kalman filter is a recursive filter estimator composed of two steps. So the first step is the prediction. And the second step is the update or the conversions.

So in the first step, we have to predict the state estimate using this equation. So here we see that we need the nonlinear system. And also, we have to calculate the predicted error covariance using this equation.

Then the second step, we will use the difference between the measurement and the estimated state. And we try to calculate the residual covariance using this equation in order to find the most important thing, which is the Kalman filter gain, which is the objective is

to minimize the expected value between the real state and the estimated state. And then we update our state estimate using this gain here. And also, we update the error covariance using the same gain. And then we go to the next step.

So we inject this p and z in the previous prediction step and so on. So it is a recursive filter. So it is a classical application of extended Kalman filter. So here, I will show you some results in simulation.

So we choose these covariance matrices. We took eigenvalues equal to 1. So we have two. We have initial position. We have take-off phase. Then we hover a little bit. And at 25, we have a landing phase.

So these are simulations. We can see r here. So with the quantities with hat are the estimated. So as you see, we are estimating in a good way r. We have a little bit to converge. Beta is welcome estimated. Beta dot is not. And here are the additional forces

that we added to estimate them. We see that we can improve better this estimation. So we changed a little bit the choice of the eigenvalues. So because we have confidence in the sensors, so we choose these values for r2.

And because we have accurate measurements of r and beta, the tether length and the inclination angle, we can choose these values. And the additive forces, because they are unknown, so we are not certain about these values. So we can choose these gains here in order to improve the estimation.

So as you see, we improved very well the estimation in using this r2 and q2. So we are converging better for almost all the states. And if you want, we compare between q1 and q2. So, sorry.

So we have a good convergence with using the second configuration. Now, we try to use also extended Kalman filter for estimating the unknown forces, because when we have wind speed equal to 0, we almost have no aerodynamic forces.

So we can estimate, use this extended Kalman filter in order to find these unknown forces. As you see here, we can estimate the friction and the propellers. And as you see here, it confirms what we proposed,

that here we have, when we have takeoff, the sign of the dry friction is the inverse of the r dot, and we get these values. Now, we want also to use the extended Kalman filter with another wind speed. So we use the wind speed equal to seven meter per second. So the aerodynamic forces are more important

than the unknown forces. And also we managed to estimate these aerodynamic forces. So what does it mean that we get L and D? And we know the wind speed. So we also tried to use this Kalman filter to estimate the lift and drag coefficients.

So in the paper, we used an analytical model of CL and CD, but we know also that L and D depends on the apparent wind and the surface of the wing. So we use the extended Kalman filter. So as you see here, with the iterations,

we converge to the model that we used in the paper. So we have conversions of the CL and the CD, the drag and lift coefficients. We also tried to improve our control. So we tried to use the feedback linearization

with direct measurements or with extended Kalman filter. And here, as you cannot see, maybe it is a problem of the PDF. So here in orange is using the aerodynamic forces as perturbation, as you see. And in yellow, we integrated the aerodynamic forces.

And in rose magenta, we have the, using the extended Kalman filter. So we estimated the aerodynamic forces. And as you see here, we improve better our conversions. And here, the comparison in 2D.

Here, without, so aerodynamic forces as perturbation, you see this oscillation. If we integrate the aerodynamic forces, we improve a little bit the control. And here, we also have good results using the, integrating the estimated aerodynamic forces.

We make also experiments, but only we use offline data. We're using wind speed equal to zero just to confirm that we can also estimate the, the dry friction. And here, as you see, here is the model that we used.

And here are the experimental data. And we have the same trend. So it confirms that the extended Kalman filter is doing a good job. So to conclude, we have proposed an extended Kalman filter to estimate the state vector and also to estimate unknown forces. And we used this estimation to also estimate

the aerodynamic coefficients of our system. And we integrated the estimated results in the control design to improve the control performance and to increase the robustness with respect to uncertainties. Thank you, and I am ready to answer your questions.

Yes, maybe, yeah, for the online. Are there questions? Thanks for the presentation. I was wondering in the Q and R matrices,

even with the improved values, the covariances of the measurements were super low. And then it worked well. Did you also try to increase or decrease the quality of the sensors and see the quality of the sensors?

So increase the noise of the sensors and see how the performance is then? Yes, in fact, these results are only in simulation. So we didn't consider to add some to change the performance of the sensors. We considered that they are perfect.

But tomorrow you will see some results related to the practical experimental results. We didn't do online real time. It is only using offline. So we have the experiment and we applied extended Kalman filter to offline data.

So we didn't apply, yeah, no. But these values are, we tested several values. I'm presenting only one of them for Q and R. Yeah, I'm very curious tomorrow how that looks like. Yes, please.

In which slide? Can you?

Yeah, this formula we brought from literature. Aspect ratio. It is a very small prototype. It is indoor, yeah, yeah.

I assume that the aspect ratio is seven or more.

When you say the aspect ratio, you mean the length over the, of the width? Maybe it is, yeah. He, Jonathan confirmed, yeah. Yeah, yeah, it is only to see the performance of our estimator.

Yeah, it converges for a very tiny range of angle of attack, yeah. Do you have other models that for this, yeah?

Okay, okay, okay. Good, thank you.

Yeah, please, please. We can hear them.

He's not here.

Yeah, yes, please.

What is the impact? We tried to take this into account when we designed the Kalman filter, so it's already taken into account. The noise, yeah, it is a classical Gaussian noise, so yeah.

Okay, thank you.