Exploring transfer functions in Matlab

The tf command and its friends

The tf command allows us to create LTI transfer function objects (which we’ll abbreviate as “tf objects”) that are recognized by lsim, step, and initial.

Consider the transfer function \[\begin{aligned} H(s) &= \frac{s+1}{s^3+3 s^2 + 7 s + 1}. \end{aligned}\]

We can make a Matlab model as follows.

sys = tf([1,1],[1,3,7,1])

sys =
 
          s + 1
  ---------------------
  s^3 + 3 s^2 + 7 s + 1
 
Continuous-time transfer function.

Alternatively, we could define s as a transfer function model itself.

s = tf([1,0],[1]); % tf is 1*s+0/1 = s
(s+1)/(s^3+3*s^2+7*s+1)

ans =
 
          s + 1
  ---------------------
  s^3 + 3 s^2 + 7 s + 1
 
Continuous-time transfer function.

Algebraic operations with tfs

Say we have two transfer functions \(G(s)\) and \(H(s)\) (already defined as sys). We might want to concatenate them. The idea is that we might take the output of \(G(s)\) and use that as the input to \(H(s)\). In this case, the transfer function from the input of \(G(s)\) to the output of \(H(s)\) is just the multiplication \[\begin{aligned} G(s) H(s). \end{aligned}\]

G = 1/(s+2); % or tf([1],[1,2])
G*sys

ans =
 
               s + 1
  -------------------------------
  s^4 + 5 s^3 + 13 s^2 + 15 s + 2
 
Continuous-time transfer function.

Note that we have seen that Matlab handles addition and multiplication of scalars and tfs as well as the products of tfs. (It will also handle division.)

State-space models to tf models.

Consider the state-space model with standard matrices as shown below.

A = [-2,0;0,-3];
B = [1;1];
C = [1,0;1,1;0,1];
D = [0;0;1];

We can create a ss model as usual.

sys_ss = ss(A,B,C,D);

First, let’s form a transfer function symbolically

We know the transfer function matrix is given by \[\begin{aligned} C (s I - A)^{-1} B + D. \end{aligned}\]

syms S
sys_tf_s = C*inv(S*eye(size(A)) - A)*B + D

sys_tf_s =
 
             1/(S + 2)
 1/(S + 2) + 1/(S + 3)
         1/(S + 3) + 1
 

This gave us three symbolic transfer functions in a \(3\times 1\) matrix, the first being that for the input to the first output, the second for the input to the second output, etc.

Or we can convert the ss model to a tf model

We can actually simply pass the ss model to the tf function.

sys_tf = tf(sys_ss)

sys_tf =
 
  From input to output...
         1
   1:  -----
       s + 2
 
          2 s + 5
   2:  -------------
       s^2 + 5 s + 6
 
       s + 4
   3:  -----
       s + 3
 
Continuous-time transfer function.

Note that the function ss2tf has a serious bug and should not be trusted.

Poles, zeros, and stability

Let’s take a look at the poles and zeros of sys.

p_sys = pole(sys)
z_sys = zero(sys)

p_sys =

  -1.4239 + 2.1305i
  -1.4239 - 2.1305i
  -0.1523 + 0.0000i


z_sys =

    -1

Stability can be evaluated from p_sys. The system is stable because the real parts of all poles are negative.

Let’s take a look at the pole-zero map.

figure;
pzmap(sys)

The resulting figure is shown in figure 11.4.

Simulating with tfs

All the simulation functions we’ve used for ss models (lsim,step,impulse,initial) will also work for tf models. Let’s try a impulse response on our original sys transfer function model.

t = linspace(0,15,200);
y = impulse(sys,t);

Plot.

figure
plot(t,y);
xlabel('time (s)')
ylabel('step response')

The resulting figure is shown in figure 11.5.