damp

Natural frequency and damping ratio

Syntax

damp(sys)

[wn,zeta] = damp(sys)

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

Description

damp(sys) displays the damping ratio, natural frequency, and time constant of the poles of the linear model sys. For a discrete-time model, the table also includes the magnitude of each pole. The poles are sorted in increasing order of frequency values.

[wn,zeta] = damp(sys) returns the natural frequencies wn, and damping ratios zeta of the poles of sys.

[wn,zeta,p] = damp(sys) also returns the poles p of sys.

Examples

Display Natural Frequency, Damping Ratio, and Poles of Continuous-Time System

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

$$sys(s)=\frac{2s^2+5s+1}{s^3+2s-3}.$$

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 in the 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.

Display Natural Frequency, Damping Ratio, and Poles of Discrete-Time System

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.

Natural Frequency and Damping Ratio of Zero-Pole-Gain Model

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.

Compute Natural Frequency, Damping Ratio and Poles of a State-Space Model

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

$$A=\left[\begin{array}{cc}-2& -1\\ 1& -2\end{array}\right],B=\left[\begin{array}{cc}1& 1\\ 2& -1\end{array}\right],C=\left[\begin{array}{cc}1& 0\end{array}\right],D=[\begin{array}{cc}0& 1\end{array}].$$

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.

Input Arguments

sys — Linear dynamic system

dynamic system model

Linear dynamic system, specified as a SISO, or MIMO dynamic system model. Dynamic systems that you can use include:

• Continuous-time or discrete-time numeric LTI models, such as tf (Control System Toolbox), zpk (Control System Toolbox), or ss (Control System Toolbox) models.

• Generalized or uncertain LTI models such as genss (Control System Toolbox) or uss (Robust Control Toolbox) models. (Using uncertain models requires Robust Control Toolbox™ software.)

  damp assumes

  • current values of the tunable components for tunable control design blocks.

  • nominal model values for uncertain control design blocks.

Output Arguments

wn — Natural frequency of each pole

vector

Natural frequency of each pole of sys, returned as a vector sorted in ascending order of frequency values. Frequencies are expressed in units of the reciprocal of the TimeUnit property of sys.

If sys is a discrete-time model with specified sample time, wn contains the natural frequencies of the equivalent continuous-time poles. If the sample time is not specified, then damp assumes a sample time value of 1 and calculates wn accordingly. For more information, see Algorithms.

zeta — Damping ratio of each pole

vector

Damping ratios of each pole, returned as a vector sorted in the same order as wn.

If sys is a discrete-time model with specified sample time, zeta contains the damping ratios of the equivalent continuous-time poles. If the sample time is not specified, then damp assumes a sample time value of 1 and calculates zeta accordingly. For more information, see Algorithms.

p — Poles of the dynamic system model

vector

Poles of the dynamic system model, returned as a vector sorted in the same order as wn. p is the same as the output of pole(sys), except for the order. For more information on poles, see pole.

Algorithms

damp computes the natural frequency, time constant, and damping ratio of the system poles as defined in the following table:

	Continuous Time	Discrete Time with Sample Time Ts
Pole Location	$$s$$	$$z$$
Equivalent Continuous-Time Pole	$$\text{Not applicable}$$	$$s=\frac{ln(z)}{T_s}$$
Natural Frequency	$${\omega }_n=\left|s\right|$$	$${\omega }_n=\left|s\right|=\left|\frac{ln(z)}{T_s}\right|$$
Damping Ratio	$$\zeta =-cos(\angle s)$$	$$\begin{array}{lll}\zeta \hfill & =-cos(\angle s)\hfill & =-cos(\angle ln(z))\hfill \end{array}$$
Time Constant	$$\tau =\frac{1}{{\omega }_n\zeta }$$	$$\tau =\frac{1}{{\omega }_n\zeta }$$

If the sample time is not specified, then damp assumes a sample time value of 1 and calculates zeta accordingly.

Version History

Introduced before R2006a

R2023b: Output behavior changed

The damp function output behavior has changed when the system has a pole at zero or infinity.

The function now returns these values.

• Continuous time

  Pole Location	Natural Frequency	Damping Ratio
  0	0	–1
  Inf	Inf	–1

• Discrete time

  Pole Location	Natural Frequency	Damping Ratio
  0	Inf	+1
  1	0	–1
  Inf	Inf	–1

Numerically, there is no difference between stable and unstable infinite poles. Therefore, the function considers all infinite poles as unstable.

Previously, the function returned NaN for these cases.