The Order of a System Modeled as a Differential or Difference Equation

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

I’m not finding a clear definition of order online. However, if you let x = y’, your second system becomes x’’ + ax’ + bx = u, and it’s pretty obviously second order.

For ordinary differential equations, I’ve always seen “order” mean the highest derivative present. Another way to look at it is how many additional pieces of information do you need to completely determine the solution. You need one additional restriction to determine each of the arbitrary constants of integration, and the number of integrations you need “to do” is equal to the number of derivatives taken in the equation.

You can see this in The difference between your A and B. Any y = C[sub]1[/sub] and u = C[sub]2[/sub] will satisfy B but only y = C and u = bC for any C will satisfy the second.