[MUSIC] One of the problems that we can apply the network model to is a estimation problem over a network. The idea is that we have a system. It's a physical process which will define the physics of our CPS. And the information of the output of that system is only available at events corresponding to communication events. So we start with our physical component. Now we have a network, That it's digital. And when the information arrives of the measurements of the output to that physical component, we execute an estimation algorithm. What is estimation algorithm doing? It's generating an estimate of the state of this physical component. And that estimate we are going to label it as z hat. Now the network will have the parameters that we describe in our network model. This physical component will have the dynamics given by z dot and with some output. Okay, we're going to define them in a minute. And this estimation algorithm could be a discrete algorithm or it could be, actually, a more complicated algorithm. So let's say this is a discreet or continuous discreet, in which case it will have both a difference equation and a differential equation. So the problem is as follows. So given, A physical, System, z dot = Fp, let's say that it doesn't have any input, with an output that is a function of the state of that system. Design an estimation algorithm, Generating, An estimate, z hat of z such that when y is a measured over a network, z- z hat arrow converges to 0 as time gets large. So this is estimation problem over a network. For us, this measurement over a network would be modeled using these two parameters and the module that we derive in the previous video. The physics are already defined with a state. We will take a particular case of those dynamics. We will say that Fp is a linear map, and this will be capturing these dynamics. And then the question is, what is the algorithm that we will use? So we will consider the case, When Fp(z) = A times z. In this case, the system is no more than a linear timing variance system without any input. And we're going to say also that the output of this system is given by some matrix times the state, so this will correspond to a linear output as well. You can think about this as M picking some the components of the state z that are available because of the sensors you have for that particular system. So here's our particular choice. An algorithm for estimation, Is as follows. It's pretty natural that is as follows. Whenever you don't have any information, you make the estimate of the state z change according to these dynamics, okay? So we can write that z hat dot is change continuously is equal to Az hat, okay? Suddenly, z hat and z will not have an arrow to go to 0 unless you inject the information that you receive from the network. Just pick two different initial conditions, and pick an A that makes z grow away from the initial condition. However, at every time that I get information from the network, what I can do is to update z hat according to the information I have received, okay? And that information will be basically this value, y, which is equal to Mz at the instance where the network has transmitted information, okay? So the idea will be, when tau N e [0,TN* max], which corresponds to continuous change, z hat will be equal to a copy of the dynamics of my original system with z hat. However, when tau N = 0, which is an event, then what we're going to do is to reset z hat to the following expression. We are going to first use the current value of z hat. But to that we are going to add the information that we have received. So that information would be y, which has arrived. And we are going to compare it to what we know about y which is Mz hat. And that, call it output error would be multiplied by some gain, which in the case of a vector output will be a matrix and the state. So this is what arrives from the network. And this is local. And this is what we need to design. This is so called estimation gain to be designed. So how does the model look like? So, an outline of the model, Is as follows, events corresponds to when tau N = zero. What we're going to do is to update our timer back to a value in this range, to model the mechanism corresponding to the network events. So that counts from the network model. This variable will not change because it's the physical variable of the system, so z+ = z. While the other state that we have now is this z hat +, which we will be reset to z hat + L times y, which have arrive, minus Mz, okay? Notice that I'm not in need of having a memory state in this case. So actually I can remove the variable whenever I have these type of problems from the network. In the situation that I don't have an event, again, if we like to be right now open here, that's fine, And we'll be clear what we mean, then our timer would change according to the loss of the network. Our dynamical system will change according to its own dynamics. So that's a physical component. And then our estimator will change according to the dynamics that we attach during continuous change. It turns out that under some conditions on these two parameters, on this matrix M, and on this matrix A, that are giving from this system, there exists a gain matrix L that will actually guarantee that this quantity, which is estimation error goes to 0 for every initial condition of this system. In particular, it's saying that the estimate converges to the value of the state in the limit. With that estimate, then you can potentially use it to do some control, maybe over another network, if you were to have an input here. And the way to assign this matrix L will come clear in a future video. 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