Converting Between System Representations

Converting Between System Representations

 State space to transfer function [ Zeros at infinity]  Transfer function to state space  State space to zero/pole and transfer function to zero/pole  Pole/zero to state space and pole/zero to transfer function

A dynamic system is most commonly described in one of three ways: by a set of state-space equations and the corresponding matrices, by a transfer function with the numerator and denominator polynomials, or by a list of poles and zeros and the associated gain. From time to time, it is useful to convert between these various representations. Matlab can do these conversions quickly and easily.

State space to transfer function: To begin with, suppose you have a set of state-space equations and you would like to convert them to the equivalent transfer function. This is done using the command

[num,den] = ss2tf(A,B,C,D,iu) This command creates the numerator and denominator of the transfer function for the iu'th input. Note that in most of the systems considered in this tutorial, there is only one input and therefore the "iu" term does not need to be included. For example, suppose you had the following set of state equations:

 m=1000  b=50  u=500 If you want to change this to a transfer function, just run the following m-file:

A = [0 1 0 -0.05];

B = [0; 0.001];

C = [0 1];

D = 0;

[num,den]=ss2tf(A,B,C,D) Matlab should return the following to the command window: num =

0 0.0010 0 den =

1.0000 0.0500 0 This is the way Matlab presents 0.001s + 0 ------------------ s^2 + 0.05s + 0

You now have the transfer function that describes the system. As you can see this command is pretty easy to use. Here are some notes about ss2tf:

 The numerator, num, will have as many rows as there are outputs (or rows in the matrix C).  The numerator and denominator are returned in descending powers of s  Care must be taken to check the numerator and denominator, as zeros at infinity may produce erroneous transfer functions.

Zeros at infinity This last point needs some further explanation. We say that a system has zeros at infinity if the limit as s->infinity of the value of the transfer function is equal to zero; this happens whenever you have more poles than zeros. You will see this in the root locus plot as asymptotes which go to infinity (the number of asymptotes is equal to the number of zeros at infinity). Matlab sometimes computes these zeros at infinity as being large finite numbers.When this happens, some of the coefficients in the numerator that are supposed to be zero end up being very small numbers. It may not seem like a big deal, but it can cause errors when trying to use the transfer function later on. You should always check your transfer function, and if numbers that are 0.0000 show up that are supposed to be zero, rewrite the numerator into Matlab to compensate.

A good example of this is given by the following set of state equations:

If you enter this into Matlab using the following m-file: A = [0 1 0 0 0 -0.1818 2.6727 0 0 0 0 1 0 -4.545 31.1818 0];

B = [0 1.8182 0 4.5455];

C = [1 0 0 0 0 0 1 0];

D = [0 0];

[num,den]=ss2tf(A,B,C,D)

You should get the following transfer function: num =

0 0.0000 1.8182 -0.0000 -44.5460 0 0.0000 4.5455 0.0000 0

den =

1.0000 0.1818 -31.1818 -4.4541 0

If you look at the numerator, the first and last element of each row are 0, while the second and fourth element in each row are 0.0000. If you look closer at each of these elements, you will find that they are not zero, but in fact some very small number. To see this, enter any of the following commands into the Matlab command window: num(1,2), num(1,4), num(2,2) or num(2,4). You should get something similar to the following as an answer: 7.1054e-15, -6.2172e-15, 1.2434e-14, or 4.4409e-15. Look at the roots of the numerator polynomials using roots(num(1,:)) and you will see the roots of the numerator which are almost at infinity but not quite.

This numerical inconsistency can be eliminated by adding the following line after the ss2tf command to get rid of the numbers that are not supposed to be there:

num = [num(1,3) 0 num(1,5) num(2,3) 0 num(2,5)];

Now all of the small numbers have been replaced with zeros. Always make sure to look at your transfer function and understand what it means before you use it in the design process.

Transfer function to state space: The reverse of the command ss2tf is the tf2ss command, which converts a transfer function of a system into state-space form. The command is issued like this: