[MUSIC] Now if we want to consider a concrete example where these linear type of form appears naturally when we model the system, module of some constant that might lead us to a an assigned system actually. But, let's consider the model of a temperature. In a room. So given a room. We denote its temperature. As T. This can be any real number. And the device. That. Can change T. Is given by a heater. A similar analysis could be done for the case of a cooler. The idea for us will be to determine how this temperature changes over time. So that automatically leads to a derivative of temperature with respect to time, which will basically boil down to a differential equation. A simple model of such a variation of temperature is given by the following linear affine differential equation. The constant a. Is the decay rate. Of a temperature. It actually characterizes objects, walls, and all sort of things present in the room. And will give us a rough approximation of how rapidly that temperature changes. This parameter here, Tr will correspond to the information of temperature outside a room. And how that affects the room that we are interested in, its temperature. Since we're considering a heating type of scenario, we would potentially consider this signal, u, as either zero, or off whenever the heater is off, or one, or on whenever the heater is on. So whenever it's off, we will say that u = 0. And whenever it's on, we say that u = 1. Therefore, this is somehow the effect or the amount of temperature change that the heater can introduce into the room. This is what is called the heater capacity. So, we have a temperature model. This model can be written as a physical model, given by z dot = Fp, (z,u) with the following assignments. Z is the state in our model, therefore will equal to T, u is equal to the input itself that is also labeled as u. And the function is given by this entire expression that we have right here. Which notice that because we are using z, these Ts actually should be z. Coming back to the model in the previous video, this -a will correspond to the ap matrix, in this case a scalar. This T Delta will correspond to the BP. And as I said, we have an additional constant that generates what is called an affine linear system. So if someone gives you an initial state, in this case initial temperature and the values of these constants, then we can predict forward in time what the values of the temperature will be for a given input profile, okay? So, when the input is equal to zero. Then. The model we obtain. T dot minus aT plus this constant here. I would like to talk about this particular model, because it is simple to analyze. And it will allow us to essentially figure out where the temperature will go to. So consider for a second, the fact that we actually reach a particular value of the temperature that is not infinity. And we call that value of the temperature, the steady state temperature. So, in a steady state. The temperature. Is constant. And equal to. A temperature that I'm going to label as Tss, ss for steady state. Then. Since it's constant, what we actually obtain is that the derivative of that state is equal to zero. So therefore, we satisfy the following equation. From that equation, we can actually solve for the value of the steady state. So this told us that the temperature as a function of time will converge to this particular value if there is a steady state. It turns out that because we said that the temperature is actually governed with a decay rate that is positive, we can always guarantee that this steady state exists and is actually equal to this value. And moreover, we can breakdown from this expression using the general formula from the previous video. The expression of T, temperature, as a function of time. [MUSIC]
