[MUSIC] Given a discrete-time system, we can define the solutions or trajectories to the system from a particular initial condition for a given input. So as we defined in the previous slide, If we are given the functions G and h, and now the initial condition that we are calling x naught. And the input which is function of discrete time k, now, a solution, From x naught Under the effect, Of the input V is a function, Of discrete time, Which we will label as x. Sometimes one wants to use a different label like phi or psi. But to keep things simple we will somehow use a notation, use the same value or the same symbol as x. It needs to satisfy the following properties, x(k + 1) = G(x(k),v(k)). And this is for all k, in the space of discrete time where the input v is defined. So how do you check this? You basically, you guess a function x of discrete time. You plug it into the dynamics of this system which is x plus = G (x, v). And if for the given input v, this is satisfied then you obtain an answer to this being that that function is a solution. As we did in the previous slide, it was natrual to construct this function x essentially when k equals to 0 you have here g of x 0, v 0. And then you can compute x at 1 and then, you can compute x at 2 by putting k equal to 1 here and then you can compute x3 and so on. So the construction of these solutions is very simple, is very systematic and it can be done step by step. Now the reason that I say here where v is defined is because I might have an input that is not necessarily defined for all values of k in the set of discrete time. So this leads to the following definition, so given a function, Label, It as f, sub function, its domain, Is defined As domain of f, which collects, All of the points at which, f is defined. So what is this allowing us to do? This is allowing us to do the following. We can now realize that because this computation can be done whenever v is defined, that now I can intersect this. Set intersection with the domain of v, and this intersection here is capturing this particular property of where v is defined. This definition of a domain is a very general definition, I wrote it for a function f and we'll use it for many functions. But whenever you see dom of a function, then that collects all the values where the function is defined. This brings to another point that I would like to make which is, the space, For G and h. As you notice through the construction of our solution or the definition of solution that is right here, G is evaluated at pairs x, v. So if we know that x is a vector and v is a vector in a particular space then G will need to be defined on this basis for all these two vectors okay? So we will say that x belong to the Euclidean space of dimension n. Okay, when n is equal to 1 you get a scalar, when n is equal to 2 you get the plane and you can see. And let's say that v belongs to the space of dimension m. Similarly, if you have 1 input then m is equal to 1, if you have 2 inputs then m is equal to 2 and so on. So with that being said, then G will need to be defined in the entire space or in subset. Let's assume that's in the entire space of this state and cross with the space for the input. And because the output values of G will assign values of the state and the state belongs to Rn. Then the result, the space or the columna of G should be also Rn and similarly for h. Now, we never said what happens to the output, right? And naturally, a solution is just the state component but we can also add here how the output is being generated. And that output will be generated no more than evaluating the function h at x and v, k. Which in the context of defining the spaces, this suggests that I might have some space for the output, let's call it p. So in my system I can have for instance n the state variables in the vector x, m inputs and p outputs, okay? And those numbers can be totally different depending on the problem that you have. What does this mean? That if I have, let's say, a vehicle, I can probably control the gas, so the force, and the steering wheel, so I have two inputs. I might be measuring position x and y. I may also be measuring the angle of the vehicles. I may have three variables that will be x, okay? And then all that I care about those variables maybe is just the angle. Or if I have a GPS maybe itâ€™s just position x and y, okay? So the dimensions might all be different. [MUSIC]