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The inputs that will be provided to the system will be defined over discrete time,

Â K. Let's call the input to the system,

Â V. So this V is a function of K,

Â and a different case might have different values.

Â So let's say that this is the value of K equals zero.

Â This is the value of K equals 1.

Â This is K equals 2, K equals 3, K equals 4.

Â What we expect to

Â have is an output but to also define a discrete time instance.

Â You may notice that's K. And the first value of

Â the output will also depend on what is called

Â the initial state of the discrete time system.

Â This state is typically a vector that captures

Â all the meaningful variables of this system in order to describe these change over time.

Â So given an initial state and initial input,

Â so this will be V at zero in initial state,

Â then you can actually compute what the initial value would be.

Â Or in some situations,

Â you can actually assign an arbitrary,

Â depends on the type of model you're thinking of.

Â Let's say that the output of these is these Greek symbol, theta.

Â Now, in order to compute the new value at time equal 1 of that output,

Â then we will need to use the information

Â of the input at time equals 0 of the initial state.

Â So let's say that this value right here is the result of using these values

Â here along with this value here.

Â This is what provides this information here.

Â And you can keep doing this for every K. So you can

Â actually generate a signal of the output that will

Â depend on the previous value of this state and the previous value of the input,

Â and you can continue these typically over all K. So this case,

Â remember, take value in that set.

Â The question is, what is the most efficient or

Â a good way to actually relate this input-output signals?

Â And the way that we're discussing this course is

Â a difference equation model.

Â In a difference equation model,

Â we are given a function.

Â 4:51

and the relationship between the state, the input,

Â and the output is given by equations,

Â where the value of the state after

Â each iteration of time K is a function of G. So this will be G,

Â and using the current value of X and the current value of the input.

Â So if this is my current state and this is my current input,

Â I will actually use the upper script

Â plus to say or denote what the new value of the state will be.

Â So this will correspond to the value

Â after the first iteration or the second iteration and so on.

Â While the output of these machine might also depend on some or this model,

Â might also depend on the current value of the state and

Â maybe the current value of the input.

Â So some notation here, this X plus,

Â so the plus denotes

Â new value of X,

Â and H here will

Â also correspond to some output function.

Â 6:35

So we can either actually have them here,

Â so give them functions H,

Â and H, we will have a model uniquely defined by this.

Â So this is my discrete time model.

Â Now, one of the things that you probably want to realize is that

Â the function can be thought of

Â a iterative map on the value of the states and of the input.

Â So, in the same way that we did this construction here,

Â now we can go back and pretty much solve for what

Â is the evolution of the state over time K. So this is what's happening inside this box.

Â So if this is K equals 0,

Â and this is 1,

Â and this is 2, and this is 3, and this is 4,

Â and we keep doing this,

Â assume that for this model,

Â the initial state is this value.

Â So it's the value at zero.

Â Then according to this law,

Â the value of the state of time K equals to 1 will be given by G,

Â evaluated at the value of the state at time equals 0

Â and of the value of the input or time equals 0.

Â So then this value,

Â whatever this expression gives,

Â will be worth let's say,

Â assigns this value right here for X.

Â And, now, we can compute the same way G2,

Â which will be now for X2,

Â which will be now X at 1, V at 1.

Â Now from this equation,

Â we already know that X1 is equals to this,

Â so then this will enter here,

Â and we'll end up with G of G of

Â X0 V0 comma V1.

Â And you can keep doing this construction to build the different values of

Â X for the time K.

Â