[SOUND] Let us consider an example and compute the executions. Let's go back to the, Digital to analog, Converter, To DAC, Where the model for that particular block involved continuous differential equations and discrete differential equation with constraints. We can consider that as a system that has a state mh and tau h, where the dynamics are given by the following equations in this case. So this defines the form up. And then the conditions for flowing will be given by the flow set, which will be defined by all x such that the timer component belongs to 0, tau h*, where that is the threshold. And let's open it. It helps parsing the events. And every time there is a jump, there is the effect of updating memory with a current input to a particular block. And the timer will be reset to 0. So this will define G, which now will be a function of gamma, where gamma is the input and is given by vh. And this is whenever, because the conditions are only determined on the timer component, this is just a jump set that depends on x. And this will be when tau h = tau h*. So given any input, let's say there is a function of time for simplicity, Gamma function of time. So we do the reconstruction of the time domain, according to the previous video. An execution, Input pair, Has phi such that phi (0,0) in this case, since the input is not being a part of the flow in the jump set, then this is just no more than this interval, once we do the union. So this is union C and D. And this is the tau component, or the tau h component. So let's say this is the tau h, so subscript tau h. So that's where that particular signal can belong to. And the signal, the input component, which will be just the gamma component, if you will, which is already plus gamma. Gamma (0,0) is anything, it doesn't have any constraint from the first component. So let's assume for a minute that we're given, so pick theta h, the component in the tau h to be equal to 0. We can now build the entire whole arc corresponding to this particular initial condition of phi h (0,0). And the way we can do that, because this is 0, is to draw the signal as a function of t and j. So this would be the 0, the initial condition. And as we have the revolution of the system, the timer will count all the way to tau h *. Whenever that happens, it will reset to 0. But these initial piece will be this component for t, 0 with t in this range. Okay, so this is just the line that passes through 0 with a slope equal to 1. So then, we can actually go and define what would phi tau h be as a function of t and j. We can define our whole arc, and what's going to happen is that for the first here of the flow will be equal to t. And this is when t belongs to 0 to Th*, and j is equal to 0. So that's the first piece. As soon there is a reset of a timer, due to this event that maps that to 0, we will have an increase by 1 on j. And then the value would be reset to 0 for the timer, from where we will have now evolution. And it will be according to the line. So when the event occurs again, we will actually be at t to Th*. And from there, we're going to flow again. So, the way we can actually capture that here is by defining when t belongs to (Th*, 2Th*) and j = 1. We can define this as t minus the initial amount of time that has evolved. And we can continue doing this, What we will see that when t belongs to the interval j Th*, (j + 1), Th*, with j in the serial one and so on, but we'll have here t- j Th*, okay? And they will correspond to go in from 1 to 2 and then all the way to other values. So this, Is where the reset occurs. And so, The average time domain of this particular whole arc is given by, The following. So, is the union from 0 to Th* across 0, Th*, 2Th cross 1, and if we keep doing this, it will be jTh*, (j+1) Th* cross j. Okay, it turns out that we can define it for all j, so these can continue for all js. And we can define, To be complete, In this case because Th* is positive, it's a constant, then the higher tenement will be unbounded in both t and j directions. And certainly because its completely would be maximal, it would not be single because it's complete. But in the t direction, the amount of flow is not finite, it's infinite. It is nontrivial because it has these two points. We have at least two points in this interval, in this domain. It's not continuous because it doesn't just flow. It's not discrete because it doesn't just jump. It's a true hybrid execution for the system. And similarly, we can define the gamma point which is a sign. So basically we can say that this gamma tj will be equal to whatever the input is over this horizon, wherever the input is over this horizon, and so on, and come out with a general expression. Actually the general expression for the execution, the state part, is no more than what we just wrote here, which is t-j Th*. [MUSIC]
