0:01

Now that we have a model of the CPS that is given by a combination of

Â differential equations and inclusions

Â with difference equations and inclusions on potentially constrains,

Â we can now think about the definition of an execution or a solution.

Â Let's derive a formal definition for that.

Â Let's recap, a CPS is given by the combination of differential inclusions or equations.

Â I will stick to the inclusions case,

Â so we remember the general formalism,

Â but you can replace that by any equality sign.

Â And I will consider the case of no inputs.

Â I would make a remark about the case of inputs shortly.

Â Those are the differential inclusions with constraints for the continuous change.

Â Then, whenever the events occur,

Â we're going to capture these as a condition of the state on

Â subset of the state to space, the jump set.

Â Remember that this is the two conditions defining that.

Â This is the flow set.

Â This is the jump set.

Â This is the jump map.

Â And this is the flow map.

Â This will define our CPS without inputs for the time being.

Â The notion of time was already defined.

Â We're going to use hybrid time.

Â We're going to have t and we're going to have j.

Â We have these pairs, t,j's,

Â that define our notion of time.

Â And at every value of t and j in a particular domain,

Â which we call the hybrid time domain,

Â which has a particular structure as I already defined,

Â we can characterize what a solution or an execution to the system would be.

Â Let's think about this informally for a minute.

Â The basic idea is that

Â during the continuous motion or the continuous evolution, so during flows,

Â solutions are governed by

Â x in F of x, x in C. This

Â boils down to what is called a differential inclusion.

Â In the case of an equality,

Â will be stressed an autonomous differential equation with a constrain.

Â During the jumps or at jumps,

Â solutions are governed by

Â the following difference equation or inclusion with constraints.

Â It is natural to expect that if I have an interval flow,

Â then I will be looking at intervals, let's say,

Â of the form t_j,

Â t_j plus one with

Â non-zero length over which

Â I will need to have this condition being satisfied.

Â And I would say that the trajectory derivative

Â is in the set F, and its value is in the flow set.

Â You need to satisfy this condition which is here.

Â I need to satisfy this other condition which is here.

Â 5:36

In such a case, what we need to now guarantee is that at the jump time,

Â which is t and j,

Â and after the jump time, which is t and j plus one,

Â this expression is being satisfied.

Â Basically, what we are saying is that the trajectory at

Â t,j plus one belongs to G,

Â while at t,j belongs

Â to D. That is the structure that we're looking for.

Â And in order to define what a solution is,

Â we need to put further properties on the function.

Â We are given hybrid time.

Â We're given trajectories given by hybrid arcs.

Â We're going to, in general,

Â define as a functional phi which has a domain or definition phi

Â and its values take from where to the space of the system would be.

Â Let's assume that x here is an element in R_n.

Â I grab a hybrid arc where this domain is a hybrid time domain,

Â has the main structure that was introduced in the previous video.

Â And remember that we need to be able to take derivatives of

Â this function every time we fix the j so that

Â this particular property is satisfied, okay?

Â How can we formalize these properties?

Â This is an informal definition.

Â Now, I'm going to write down here the formal definition.

Â A hybrid arc is a solution or

Â execution to the CPS

Â 8:04

if the following properties hold.

Â And we're going to have three properties here.

Â We're going to write two here and then the other one here.

Â The first property is that the initial value of the hybrid arc

Â belongs to the original for operation.

Â The system can evolve in C or D.

Â We will guarantee that

Â these initial value of that function is where I can flow or where I can jump.

Â Now at times, this set C is not necessarily

Â a close set and we would like to allow the trajectory to flow into the flow set,

Â in which case, we will be able to add there a bar and close the set.

Â Every time that there is an interval of flow,

Â like I say we find it here,

Â what I would like to say is that this interval which I define

Â here as i to the j having known empty interior or non-zero length,

Â I will have the property that on these I_j's with such a property,

Â I will have the derivative of the function of phi with respect to

Â time belongs to F and t,jl.

Â I'm more aware that I will have that its value belongs to

Â C. That will be the condition that captures

Â this property that I have

Â essentially a solution to the constrained differential inclusion.

Â And as we describe here for these elements t and j,

Â such that I have a jump so therefore,

Â t and t,j plus one is also an element in the domain,

Â I will need to have the property that if t and j,

Â and t,j plus one in the domain of phi,

Â we have that whenever we jump,

Â the new value belongs

Â to the image of a jump map evaluated when I jump and suddenly,

Â this value needs to belong to D. There are,

Â basically, three conditions that

Â these arcs that are hybrid will need to satisfy in order to have a solution.

Â Those conditions are written down here informally.

Â There are subtleties about whether

Â these should hold on the open or on the close interval,

Â and you can look at the references for that with those details.

Â But the main point is that every time that there is an interval of flow,

Â we should be able to take a derivative with respect to time and satisfy

Â the differential constrain with the inclusion or the equation.

Â And every time that there is a jump,

Â we should be able to guarantee that the value of the solution after the jump is given by

Â an element of the solution where the jump map is evaluated at,

Â and whenever the jump occurs, where in the jump set.

Â Now, we can consider different cases of CPS,

Â and I will provide one shortly,

Â where you can actually compute specifically and

Â analytically the solution to a system and then write down these arcs.

Â In many other cases,

Â you can't do that but you can have a generator of these executions numerically that will

Â actually give you the approximated execution

Â to your system from a given initial condition.

Â