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Given a CPS in terms of differential inclusion or equation with constraints,

Â and difference inclusions or equations with constraints,

Â we can now define the notion of solution or

Â execution to the systems using the following construction.

Â The first thing we want to define is the notion of hybrid time.

Â As we mentioned earlier,

Â we will use two parameters: the executions

Â will be parameterized by

Â one parameter which is t ordinary time taking

Â values on non-negative reals which is this set.

Â And this would be for the flows or continuous change of your trajectories or executions.

Â While every time there is an event,

Â we're going to define j as the counter of events and

Â that counter would take values in the set of naturals including a series.

Â An execution, now will be

Â for as a function,

Â we'll call it in general phi,

Â defined on a hybrid time domain.

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When you define a function,

Â you can specify the domain of the function as

Â dom of the function and it's co-domain in this case,

Â since we're thinking about CPS with states that take generic dimensions,

Â say n dimension, we will say that the co-domain is R_n,

Â could be much smaller than that.

Â You can make it tight.

Â This is the domain of the function.

Â This domain is what we're going to give some structure which is

Â what we call hybrid time domain.

Â This function for the domain such that the following holds.

Â For every element in the domain,

Â let's call them capital T and capital J,

Â be an element, define an element on this domain,

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we have that the domain of that function intersected with a box

Â of size ([0,T]x{0,1...J}) can

Â be written as the union of intervals of flow if we have non-zero length index by j.

Â Now, these intervals of flows will be given in terms of

Â a sequence of elements that I'm going to a label as t_j,t_j+1,

Â two consecutive elements where

Â t_j satisfy the following properties: t at j equal to zero is equal to zero,

Â t_1 is larger or equal than zero,

Â t_2 is larger or equal than t_1, and so on.

Â This will go to up to some t. This will be from j equal to zero.

Â And if we would like to get all the way to capital J,

Â this would be the final element in

Â which case the last element on the sequence will be t capital

Â J plus one,

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is some sequence.

Â In other words, we need to have an ordering of

Â the events of flow and jump and that ordering is given by this sequence.

Â Now, we can think about a particular execution.

Â Let's say for a problem where you have

Â a timer that the timer counts to a particular number,

Â let's say one, and then you get through set to zero, and so on.

Â In that situation, the example that we're thinking will lead to a function, phi,

Â that we'll define on t

Â given by the following possibilities.

Â If the timer says start at zero then counts linearly to one,

Â we will have this whole interval from t_0 equals zero to

Â t_1 equal one as a time where the system flows,

Â and this will define zero to one in this particular structure.

Â This will be for j equal to zero.

Â When the jump occurs,

Â what's going to happen is that j is going to be incremented.

Â We're going to go to another value of j,

Â at which we will have another interval of

Â flow for this particular system that we're thinking of that will go all the way to two.

Â And you can continue this structure and keep increasing your j

Â after one second of flow, so this continues.

Â This structure that you see right here is a hybrid time domain because it's

Â defined for a sequence given by j precisely for the t_j's.

Â And every time that there is a recent event,

Â the j is incremented and leads to a structure like this.

Â You can validate that by picking a particular element, t_j.

Â Let's say, this is t and this is j,

Â and you pick this particular element,

Â now you can write down this whole box that is left behind

Â as the union of intervals from zero to one for flow across zero,

Â union with one to two cross one for j equal one,

Â and then two to wherever capital T is across with two for a particular last piece.

Â Notice that this structure needs to

Â be possible whenever you pick an element in the domain.

Â And this allows you to now think about situations where

Â the end point of this time domain is open,

Â where, in that case,

Â you will not be able to pick the right most element because it's open.

Â But in any element on that domain,

Â you will actually be able to regulate this.

Â It allows you to have a particular trajectories that are not already defined at t equal

Â three j equal two maybe because of blob or

Â maybe because of other difficulty on extending the solution.

Â That's the structure of time.

Â It covers ordinary time when j is always to zero.

Â It covers discrete time,

Â not necessarily with the discretization when t is equal to zero.

Â In our case, covers the hybrid case.

Â What you can now think about is what kind of executions you could have to your system,

Â and it turns out that some type of executions can be

Â determined by just studying the hybrid time domain of execution.

Â