Control Theory State-Space method with derivative input

[Chacabucogod]

Hi,

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

x₁=y-β₀u
x₂=y'-β₁u-β₀u' 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?

 

[donpacino]

note: this is not mine...
http://lpsa.swarthmore.edu/Representations/SysRepTransformations/TF2SS.html

Consider the third order differential transfer function:

We start by multiplying by Z(s)/Z(s) and then solving for Y(s) and U(s) in terms of Z(s). We also convert back to a differential equation.

We can now choose z and its first two derivatives as our state variables

Now we just need to form the output

Unfortunately, the third derivative of z is not a state variable or an input, so this is not a valid output equation. However, we can represent the term as a sum of state variables and outputs:

and

From these results we can easily form the state space model:

In this case, the order of the numerator of the transfer function was less than that of the denominator. If they are equal, the process is somewhat more complex.