We will now look at different representations of time, so when we normally think about time, we normally
think about continuous time.
But there's an important concept when we're dealing with dynamic systems called discrete time.
And we're going to have a look at that now.
The differential equations that have been shown so far, all have been in continuous time.
This just means that they're defined with respect to an independent variable time t, which varies continuously
and smoothly.
But for many practical purposes, especially in engineering, we only really need to know the state
of the system at this great point in time, or we only need to calculate or operate on that system in
this great point of time.
So basically, we only need to do different operations at TI, not T1, T2 all the way up.
And we represent this is Teekay minus one for the time set before Teekay for the current time and Teekay
plus one for the timestep in feature.
They can be a discrete instances in time, so there could be zero point one point two all the way up
in continuous increasing segments.
Or it could be something such as zero two four six eight alé up.
So this is what we call discrete time.
This is where we take time.
Values fix a discrete value to them and incrementally increase them with time.
So when we talk about this great time, we use this K notation here.
So K is an integer value representing different steps and times.
Every time we take a step in time, the time value actually increases by a constant amount to Kansi.
Between these values here is increasing by point.
One of the second between these values here is increasing by two seconds.
So again, this is discrete time.
This is breaking up continuous time into discrete time points.
So this is easier saying up on this graph here.
So here we have an example of continuous times.
You can see at times varying.
So he starts out at zero seconds and it continuously varies with time.
So this is the first Orlistat equations written with continuous time.
But we can break this equation up and represent in discrete time so he can see we have the dotted line,
here is the continuous time, but we have breaking up the response into discrete time.
So basically have discrete time steps.
K, so K not K one, K two all the way up.
And we evaluate the output of the state space and the d different time points.
We have this value here.
The value here, the value here and so forth.
So basically the time steps forward by this Delta T amount and so we can calculate the continuous time
just by being our energy.
K times is timestep.
So here we can have be one times DTI, here will be two times DETI all the way up and K being the timestep.
So when we represent things in this form here, we can represent the dynamic system in this discrete
time here.
So basically where we have t we basically replace it with T, OK, so you can see here we have T of
K being time.
So it's a time varying dynamic system.
We have the vector at this point in time.
We have the input vector at this distinct point in time.
But this function here does not calculate a state.
Right.
So it does not calculate the derivative.
Instead, it calculates the next point, the state at the next point in time.
So it's not a not a first order differential equation is now turned it into a recursive equation.
So it moves and steps.
It does not move in rates.
So this is the discrete time representation of a dynamic system.
This is the continuous time representation of a dynamic system.