We have discussed in the past that most of the real world processes or dynamics can be represented as
differential equations.
Now, differential equations are inherently continuous time systems because they use a derivative of
a state with respect to time.
This makes them very easy to represent as continuous time processes.
So if we have a look at a continuous time invariant linear system as shown on this slide here, somehow
we want to represent this as a discrete time system.
This is because for engineering purposes, we don't really care about the system state and every single
time we only really care about the state at specific times in the future.
So we can represent this using a discrete system.
So if we look at a discrete time, time invariant linear system as shown here, we want to somehow convert
a continuous time system into a discrete time system.
And we're going to again look at our time invariant and linear systems.
So somehow we need a mapping process which can convert the A and B matrixes for the continuous time
system into the energy matrix for the discrete time system.
So this is what we're going to look at in this video, our continuous to screen time conversions.
So if we take our continuous time invariant linear system as shown here, we know that the solution
to this system is given by this equation here.
Now, all of this equation is saying that we can work out the state of the system at some time in the
future based on this matrix exponential equation and this integration here.
So this part of the equation here, we can see that it only depends on our system dynamics A and the
state at TI not so this is the free response to the system.
So this is what happens if we have no other inputs into the system.
This will be how the time this is.
This is how this is me rolls of time.
But if we have some input here, so if you have some input here, this gets added onto the system response
so we can separate the system and dynamics, the free response with the force response or the input
response to the system.
So this is integral here between Time Teno and the current time that we want T or what this equation
is doing is basically summing up the different impulse responses for all the time in between these two
time periods.
So if we have, for example, our input at time t not is going to have more of an effect than the input
at time t just because it acts over a longer time period.
So this is what this integral is doing, is just summing up all the different inputs for all the different
impulses over the time period in between T not and T.
So if we consider a discreet time step, so we have the current time, so let's say it is equal to take
the previous time, step is going to be out not which is going to be a T minus one, and we'll have
our discrete time set aside.
So Delta T is just a difference between the time at Teekay and the time at K minus one.
So this is just the difference between our two time points.
So if we would take these discrete time representations for T and T not and substitute them into our
above solution for the continuous time system here, we end up with this equation down here.
So this is basically saying we want to evaluate the system dynamics over a very small time interval.
Delta T, so we have Teekay minus one and we're working at the state for Altec.
And again, it looks very similar as a continuous time solution up here, except now we're going to
instead of using our T minus tonight, we're just going to use our ability and we're going to use that
K minus one and Otik.
So again, we have a free response to the system.
So what happens if we have no inputs and we just evolve the state of time and then we have the force
response?
And what happens if we adding a forcing input to the system?
So if we consider our equation from the last slide here, which is just the propagation of our current
state to our future state or from state, Teekay minus one to take, we can look at this equation here
and evaluate it with respect to this discrete system matrix matrix equation here.
So it is fairly easy to see this is a factor here, multiply by this matrix so we can easily create
this matrix, this matrix expansion into our F system matrix.
Looking at this side of the equation here.
We can do the same thing.
So if we group all of this together, multiply it by our control input, you we can see that this is
basically this matrix here.
So we can break it out into this here, Alef Matrix is our Matrix exponential of a Delta T g matrix.
A forcing response is just going to be our free response multiplied by the integral from zero to Delta
T of this part of the equation here.
Now, this part of the equation here can be simplified, so if our continuous time, Matrix A. is convertible,
so if Inverse A exists, then we can simplify this part down into this equation here so we don't have
to do the integral.
We can just assume that if a convertible, then this is the response.
So these equations are how we convert a continuous time system into a discrete time system, we can
use the and B matrix and then using a specified timestep, we can convert the system into a discrete
time system, working out alef energy matrix.
And if A is invariable, then we can use this identity here to help simplify the equation so we don't
have to do the integration.
So now you might be wondering about this matrix exponential and how do we actually calculate this matrix
exponential?
So this matrix exponential can be written as an infinite sum, and if we expand this out, you can see
that each of the eight is just going to be at the power of zero eight to power.
One plus eight is the power or two over two factorial and so on and so forth all the way up to infinity.
So this is a useful energy.
It allows us to make an approximation of what this infinite sum is going to be.
So depending on the matrix dynamics of A, we might be able to make a few approximations.
So if the matrix dynamics is appropriate, we can make a first order approximation.
So basically we can use a first order approximation in some cases so the exponential amte can be approximated
by the identity matrix plus eight times T..
So this is a first order approximation.
It just looks at these first two terms here.
So a T to the power of zero is just identity.
And then we just have a taste of how one is just able to it.
So this is where it comes out here.
Now, if the dynamics of A are such that this is a very bad approximation, well, then we'll have to
use a different approximation method to calculate this matrix.
Exponential.