Transformation: Transfer Function ↔ State Space

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Contents

• State Space to Transfer Function
• Transfer Function to State Space
  • Controllable Canonical Form
  • Observable Canonical Form
  • Other forms

Two of the most powerful (and common) ways to represent systems are the transfer function form and the state space form.  This page describes how to transform a transfer function to a state space representation, and vice versa.  Converting from state space form to a transfer function is straightforward because the transfer function form is unique.  Converting from transfer function to state space is more involved, largely because there are many state space forms to describe a system.

State Space to Transfer Function

Consider the state space system:

Now, take the Laplace Transform (with zero initial conditions since we are finding a transfer function):

We want to solve for the ratio of Y(s) to U(s), so we need so remove Q(s) from the output equation.  We start by solving the state equation for Q(s)

The matrix Φ(s) is called the state transition matrix.  Now we put this into the output equation

Now we can solve for the transfer function:

Note that although there are many state space representations of a given system, all of those representations will result in the same transfer function (i.e., the transfer function of a system is unique; the state space representation is not).

>> % First define state space system
>> A=[0 1 0; 0 0 1; -3 -4 -2];
>> B=[0; 0; 1];
>> C=[5 1 0];
>> [n,d]=ss2tf(A,B,C,D)

n =
         0         0    1.0000    5.0000
d =
    1.0000    2.0000    4.0000    3.0000

>> mySys_tf=tf(n,d)
Transfer function:
       s + 5
----------------------
 s^3 + 2 s^2 + 4 s + 3

Transfer Function to State Space

Recall that state space models of systems are not unique; a system has many state space representations.  Therefore we will develop a few methods for creating state space models of systems.

Before we look at procedures for converting from a transfer function to a state space model of a system, let's first examine going from a differential equation to state space.  We'll do this first with a simple system, then move to a more complex system that will demonstrate the usefulness of a standard technique.

First we start with an example demonstrating a simple way of converting from a single differential equation to state space, followed by a conversion from transfer function to state space.

If we try this method on a slightly more complicated system, we find that it initially fails (though we can succeed with a little cleverness).

The process described in the previous example can be generalized to systems with higher order input derivatives but unfortunately gets increasingly difficult as the order of the derivative increases.  When the order of derivatives is equal on both sides, the process becomes much more difficult (and the variable "D" is no longer equal to zero).  Clearly more straightforward techniques are necessary.  Two are outlined below, one generates a state space method known as the "controllable canonical form" and the other generates the "observable canonical form (the meaning of these terms derives from Control Theory but are not important to us).

Controllable Canonical Form (CCF)

Probably the most straightforward method for converting from the transfer function of a system to a state space model is to generate a model in "controllable canonical form."  This term comes from Control Theory but its exact meaning is not important to us.  To see how this method of generating a state space model works, 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

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.  For a general n^th order transfer function:

the controllable canonical state space model form is

Observable Canonical Form (OCF)

Another commonly used state variable form is the "observable canonical form."  This term comes from Control Theory but its exact meaning is not important to us.  To understand how this method works consider a third order system with 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:

Choose the output as our first state variable

Looking at the right hand side of the differential equation we note that y=q₁ and we call the two integral terms q₂:

so

This is our first state variable equation. 

Now let's examine q₂ and its derivative:

Again we note that y=q₁ and we call the integral terms q₃:

so

This is our second state variable equation. 

Now let's examine q₃ and its derivative:

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

Our state space model now becomes:

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.  For a general n^th order transfer function:

the observable canonical state space model form is

Other forms

There are many other forms that are possible.  For example MATLAB uses a variant of the controllability canonical form in its "ss2tf" function.

References