Suppose one has a system which may be modeled as a differential or difference equation. How does one determine the order? For a system such as:

y’’’ + ay’’ + by’ + cy = u

I think most would agree this is a third order system. However, what about zero Eigenvalues in the characteristic polynomial?

y’’’ + ay’’ + by’ = u

Some may still argue this is a third order system because the model contains a third order derivative term. Others may argue the span of the model is (3 - 1 = 2) so they avoid zero Eigenvalues. However, what if your system input contains derivatives:

y’’’ + ay’’ + by’ = u’

Is this just an over differentiated second order system? If yes, then one is allowing the system input to determine the order of your system which seems dangerous.

Furthermore, how do these two systems differ?

A) y’’ + ay’ + by = u

B) y’’’ + ay’’ + by’ = u’

Where B is just A differentiated.

The same idea in my mind applies to difference equations.

y((k+4)T) - 1.9*y((k+1)T) = 0

Is this a (4-1=3) or a fourth order system? This is not homework, but it would help my understanding in a class I am taking. Thanks