# damp

Natural frequency and damping ratio

## Syntax

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

## Description

example

````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.```

example

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

example

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

## Examples

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

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

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 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:

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

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:

`$A=\left[\begin{array}{cc}-2& -1\\ 1& -2\end{array}\right],\phantom{\rule{1em}{0ex}}B=\left[\begin{array}{cc}1& 1\\ 2& -1\end{array}\right],\phantom{\rule{1em}{0ex}}C=\left[\begin{array}{cc}1& 0\end{array}\right],\phantom{\rule{1em}{0ex}}D=\left[\phantom{\rule{0.1em}{0ex}}\begin{array}{cc}0& 1\end{array}\right].$`

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

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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

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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.

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.

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 TimeDiscrete Time with Sample Time Ts
Pole Location

`$s$`

`$z$`

Equivalent Continuous-Time Pole

`$s=\frac{ln\left(z\right)}{{T}_{s}}$`

Natural Frequency

`${\omega }_{n}=|s|$`

`${\omega }_{n}=|s|=|\frac{ln\left(z\right)}{{T}_{s}}|$`

Damping Ratio

`$\zeta =-cos\left(\angle s\right)$`

`$\begin{array}{lll}\zeta \hfill & =-cos\left(\angle s\right)\hfill & =-cos\left(\angle ln\left(z\right)\right)\hfill \end{array}$`

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.