If the order of the numerator is equal to the order of the denominator, it becomes more difficult to convert from a system transfer function to a state space model. This document shows how to do this for a 3rd order system. The technique easily generalizes to higher order.

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. A result that works in all cases is given below; the details are here.

Consider the third order differential transfer function:

We can convert this to a differential equation and solve for the highest order derivative of y:

Now we integrate twice (the reason for this will be apparent soon), and collect terms according to order of the integral (this includes bringing the first derivative of u to the left hand side):

Without an justification we choose y-b_{0}u as our first state variable

Looking at the right hand side of the differential equation
we note that y=q_{1} and we call the two integral terms q_{2}:

This isn't a valid state equation because it has "y" on the right side (recall that only state variables and inputs are allowed). We can get rid of it by noting:

so

This is our first state variable equation.

Now let's examine q_{2} and its derivative:

Again we note that y=q_{1}+b_{0}u and we call the integral
terms q_{3}:

This is our second state variable equation.

Now let's examine q_{3} and its derivative:

This is our third, and last, state variable equation.

Our state space model now becomes:

Here is a good reference that does the same derivations from a different perspective: http://www.ece.rutgers.edu/~gajic/psfiles/canonicalforms.pdf

© Copyright 2005 to 2019 Erik Cheever This page may be freely used for educational purposes.

Erik Cheever Department of Engineering Swarthmore College