For this example, consider the following continuous-time transfer function:

Create the continuous-time transfer function.

sys = tf([2,5,1],[1,0,2,-3]);

Display the natural frequencies, damping ratios, time constants, and poles of sys.

damp(sys)

                                                                       
         Pole              Damping       Frequency      Time Constant  
                                       (rad/seconds)      (seconds)    
                                                                       
  1.00e+00                -1.00e+00       1.00e+00        -1.00e+00    
 -5.00e-01 + 1.66e+00i     2.89e-01       1.73e+00         2.00e+00    
 -5.00e-01 - 1.66e+00i     2.89e-01       1.73e+00         2.00e+00    

The poles of sys contain an unstable pole and a pair of complex conjugates that lie int he left-half of the s-plane. The corresponding damping ratio for the unstable pole is -1, which is called a driving force instead of a damping force since it increases the oscillations of the system, driving the system to instability.

For this example, consider the following discrete-time transfer function with a sample time of 0.01 seconds:

$$sys(z)=\frac{5z^2+3z+1}{z^3+6z^2+4z+4}$$.

Create the discrete-time transfer function.

sys = tf([5 3 1],[1 6 4 4],0.01)

sys =
 
     5 z^2 + 3 z + 1
  ---------------------
  z^3 + 6 z^2 + 4 z + 4
 
Sample time: 0.01 seconds
Discrete-time transfer function.

Display information about the poles of sys using the damp command.

damp(sys)

                                                                                    
         Pole             Magnitude     Damping       Frequency      Time Constant  
                                                    (rad/seconds)      (seconds)    
                                                                                    
 -3.02e-01 + 8.06e-01i     8.61e-01     7.74e-02       1.93e+02         6.68e-02    
 -3.02e-01 - 8.06e-01i     8.61e-01     7.74e-02       1.93e+02         6.68e-02    
 -5.40e+00                 5.40e+00    -4.73e-01       3.57e+02        -5.93e-03    

The Magnitude column displays the discrete-time pole magnitudes. The Damping, Frequency, and Time Constant columns display values calculated using the equivalent continuous-time poles.

For this example, create a discrete-time zero-pole-gain model with two outputs and one input. Use sample time of 0.1 seconds.

sys = zpk({0;-0.5},{0.3;[0.1+1i,0.1-1i]},[1;2],0.1)

sys =
 
  From input to output...
          z
   1:  -------
       (z-0.3)
 
            2 (z+0.5)
   2:  -------------------
       (z^2 - 0.2z + 1.01)
 
Sample time: 0.1 seconds
Discrete-time zero/pole/gain model.

Compute the natural frequency and damping ratio of the zero-pole-gain model sys.

[wn,zeta] = damp(sys)

wn = 3×1

   12.0397
   14.7114
   14.7114

zeta = 3×1

    1.0000
   -0.0034
   -0.0034

Each entry in wn and zeta corresponds to combined number of I/Os in sys. zeta is ordered in increasing order of natural frequency values in wn.

For this example, compute the natural frequencies, damping ratio and poles of the following state-space model:

Create the state-space model using the state-space matrices.

A = [-2 -1;1 -2];
B = [1 1;2 -1];
C = [1 0];
D = [0 1];
sys = ss(A,B,C,D);

Use damp to compute the natural frequencies, damping ratio and poles of sys.

[wn,zeta,p] = damp(sys)

wn = 2×1

    2.2361
    2.2361

zeta = 2×1

    0.8944
    0.8944

p = 2×1 complex

  -2.0000 + 1.0000i
  -2.0000 - 1.0000i

The poles of sys are complex conjugates lying in the left half of the s-plane. The corresponding damping ratio is less than 1. Hence, sys is an underdamped system.