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I'm reading Ogata's Modern Control Engineering, and when he talks about the representation of a differential equation in state space he divides the method in two. The first one is when the input of the differential equation involves no derivative term, for example:

x'(t)+x(t)=u(t)

The next step is doing it with a differential equation that has inputs that have derivatives. For example:

x'(t)+x(t)=u(t)+u'(t)

He then mention that the state varibles will be

x_{1}=y-β_{0}u

x_{2}=y'-β_{1}u-β_{0}u' and so on...

I've tried finding a reason for this and the nearest I've come is the following PDF, which has errors:

http://www.ece.rutgers.edu/~gajic/psfiles/canonicalforms.pdf

Anybody got an idea how that can be derived?

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# Control Theory State-Space method with derivative input

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