We will now have a look at different mathematical ways of representing different states, based systems
in the previous video we have seen, there are two different ways of representing the time domain of
the system.
We can break the system down into a continuous time system or either a discrete time system.
Now, these are the different ways of representing time.
We can also break them down into different subcategories as well for the continuous time system.
We can also break it down into a non-linear or general system, or we can also break it down into a
linear system.
We can do the same thing with the discrete system.
We can break it down into a nonlinear or the general system, and we can also break it down into a linear
system.
So now let's have a close look at each of these different systems and what they actually mean.
So let's start by looking at the continuous non-linear or general time system as shown here.
The continuous non-linear time model can be represented by a general function here that can take any
form, it can be anything, it can be linear, it can be non-linear is just any general function that
can be a input of time because it can be time varying.
It can be a function of the input of the current state.
X can be input of the current control input.
You and the general function F here outputs the continuous time state writes.
So this function here is a very general way of representing that this function can be any function with
these inputs that produces the right output.
So now let's have a look at this continuous time, linear system category.
The continuous time linear model is also, again, a function of the current state x the current input
you.
It is also time Verinata is a function of T and it produces an output of the state rights.
So X as a function of T.
So this is why is a continuous time system.
Now this model here is a linear model.
So we make the assumption that the output of the system, state rights is a linear combination of the
states and the inputs based on this matrix multiplications here.
So we have this state matrix A and we have the state vector X, we have this input matrix B and the
input vector you.
So the state rates here can be a function of this whole operation here.
So anything that can be represented as a matrix multiplication of the states and a matrix multiplication
of the inputs can be represented as a linear model.
Now, each of these coefficients in The Matrix can also be time varying because you can see they're
a function of T and the same thing for the control matrix B, but any system that can be represented
in this form can be considered as a continuous time, linear, dynamic system.
So those are the two continuous time representations.
Now let's have a look at the discrete time representation starting off with the discrete non-linear
model shown here.
So we already know that this is great time model, instead of calculating a rate, it calculates the
next state.
So you can see here this is a general function.
If that can be any function that can take the input of the current time, the current state and the
current input.
And this function here calculates the next state for the discrete time, step one in the future.
Now, this is a very generous representation because it is the most generous representation, any function
that takes in the current time state and input or one or more of these and outputs the next time in
this great time is a general non-linear model for this great time.
And again, we can break the discrete time system down into a more restrictive category of being the
linear, discrete time system as shown here.
And again, the linear model is just a way of representing the function in a more structured form,
so a linear system again is any system that is a modification of this vector and the input vector as
a matrix multiplication system here is a linear system, is a combination of this F matrix multiplied
by the state plus this G matrix multiplied by the input.
And again, this is a linear operation because of the matrix times, a vector and again a matrix times
the vector here.
Now each of these matrix F and G, they can vary with time because we have this subscript K here.
You can see that for every discrete time.
Update this whole function here and this whole function here can change with time.
But as you can see, this is linear function.
It is a function of the states and the inputs.
That's a linear multiplication and outputs the next state for the next timestep.
So these are all the different ways we can represent states based systems.
So, again, it can be in a continuous time domain or convenience, great demand time domain.
The continuous systems calculate the state rates as seen here, the extent of the derivatives with respect
to time, the discrete time systems just calculate the next time for the next discrete point where you
can break it down into linear and general systems.
So you can say here we have a linear systems here.
So the state vector and the input vector are multiplied by a matrix.
Or we can have the general or nonlinear functions here where we just have a general function that can
be anything, including a linear function.
Now can also break it down into time varying and time invariant.
So time varying systems means that the function can change if time is a function of time, or we can
see the linear systems, the matrix and B here change over time.
Same thing with the F and G in a time varying systems.
It just means that the function is not a function of time.
So this function F does not have the time.
Input here is always a constant function and the same thing for the linear system.
So you can see the linear system here.
The A and B matrix are fixed for all time and the same thing for this great time system, F and G do
not change the constants.
So this just means that this is a time invariant system here.
The system dynamics do not change with time.
They just change with the state and inputs.