# Control Theory State-Space method with derivative input

1. Mar 29, 2015

### 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?

2. Mar 30, 2015

### 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.

Last edited by a moderator: Apr 19, 2017