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So we've looked at differential equations and we've also looked at states based representation of the

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different dynamic systems.

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So the next question is, how can you put these differential equations, interstates, space form so

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that we can carry out the different analyses on the systems?

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So we know that first, all the different equations make up the state equations.

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So this is the state equations here or in state spaced form, the state variables of the system are

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the dependent variables of the state equation.

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So that's this X here.

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So how can we represent different ordinary differential equations in states based form?

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So we're going to use the example of Newton's equations of motion.

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So force is equal to mass times acceleration of Iran in this form here.

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So our force is equal to mass times acceleration, which is the second derivative of the position.

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So how does the position exactly evolve with time for a given force, which is also a function of time?

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So we have to somehow put this equation here into this space form here.

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But you can see at the moment there's not a direct, obvious way to convert this form into this form

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here as this has second derivatives here, whereas this is only a first order differential equation.

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So somehow we have to convert this form into the state space representation form.

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So the problem is to convert Newton's equations of motion, which is a second order idea here, into

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a first order states based representation of the system that has the same dynamics.

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So what we can do is look at the dependent variable, as you can see that this variable here X is a

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position and is a function of the acceleration, which is a second derivative.

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So this is why a second order.

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Odey But we can break this down into we have position, velocity and acceleration, or we can write

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them in different locations of X, V and a so X for position velocity, i.e. for acceleration.

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But it should be noted that this X here is not the state vector.

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This is the for the actual position X.

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So what we can do is I can using this notation here by breaking it down into the different derivatives

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so we can form a state right vector here.

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So X not being Vedat and X so they don't bang the acceleration x the velocity.

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So this is what is calculated by the state function here.

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We also have the state vector, so X not position X for the state vector is going to be a function of

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velocity and time.

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So these are the dependent variables of the equation, velocity and diversity and position.

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We also have the input vector being f the force as a function of time so we can write the second order

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Odia here down in state space form here.

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So this is a nonlinear space form, but the state of the system of velocity and position, state rates

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then are going to be the velocity, rate or acceleration and the position rate or velocity.

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So this year is a state space representation of the second order.

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Odia.

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So we can say in general, when we turn a higher order early into a first Orlistat space representation,

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the different states of the states face representation are all the lower derivatives of the.

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So in our case, when we looked at the equations in motion, we have the acceleration, velocity and

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position.

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So velocity and position, the lower states derivatives become the states of the dynamic system.