So we've looked at differential equations and we've also looked at states based representation of the
different dynamic systems.
So the next question is, how can you put these differential equations, interstates, space form so
that we can carry out the different analyses on the systems?
So we know that first, all the different equations make up the state equations.
So this is the state equations here or in state spaced form, the state variables of the system are
the dependent variables of the state equation.
So that's this X here.
So how can we represent different ordinary differential equations in states based form?
So we're going to use the example of Newton's equations of motion.
So force is equal to mass times acceleration of Iran in this form here.
So our force is equal to mass times acceleration, which is the second derivative of the position.
So how does the position exactly evolve with time for a given force, which is also a function of time?
So we have to somehow put this equation here into this space form here.
But you can see at the moment there's not a direct, obvious way to convert this form into this form
here as this has second derivatives here, whereas this is only a first order differential equation.
So somehow we have to convert this form into the state space representation form.
So the problem is to convert Newton's equations of motion, which is a second order idea here, into
a first order states based representation of the system that has the same dynamics.
So what we can do is look at the dependent variable, as you can see that this variable here X is a
position and is a function of the acceleration, which is a second derivative.
So this is why a second order.
Odey But we can break this down into we have position, velocity and acceleration, or we can write
them in different locations of X, V and a so X for position velocity, i.e. for acceleration.
But it should be noted that this X here is not the state vector.
This is the for the actual position X.
So what we can do is I can using this notation here by breaking it down into the different derivatives
so we can form a state right vector here.
So X not being Vedat and X so they don't bang the acceleration x the velocity.
So this is what is calculated by the state function here.
We also have the state vector, so X not position X for the state vector is going to be a function of
velocity and time.
So these are the dependent variables of the equation, velocity and diversity and position.
We also have the input vector being f the force as a function of time so we can write the second order
Odia here down in state space form here.
So this is a nonlinear space form, but the state of the system of velocity and position, state rates
then are going to be the velocity, rate or acceleration and the position rate or velocity.
So this year is a state space representation of the second order.
Odia.
So we can say in general, when we turn a higher order early into a first Orlistat space representation,
the different states of the states face representation are all the lower derivatives of the.
So in our case, when we looked at the equations in motion, we have the acceleration, velocity and
position.
So velocity and position, the lower states derivatives become the states of the dynamic system.