nichols

Nichols response of dynamic system

Syntax

[mag,phase,wout] = nichols(sys)

[mag,phase,wout] = nichols(sys,w)

nichols(___)

Description

[mag,phase,wout] = nichols(sys) computes the frequency response of dynamic system model sys and returns the magnitude and phase of the response at each frequency in the vector wout. The function automatically determines frequencies in wout based on system dynamics.

[mag,phase,wout] = nichols(sys,w) returns the response data at the frequencies specified by w. You can specify a frequency range or a vector of frequencies.

nichols(___) plots a Nichols chart of the frequency response of sys. The plot displays the magnitude (in dB) and phase (in degrees) of the system response as a function of frequency. Use ngrid to superimpose Nichols chart grid lines on an existing SISO Nichols chart. For more plot customization options, use nicholsplot.

• To plot responses for multiple dynamic systems on the same plot, you can specify sys as a comma-separated list of models. For example, nichols(sys1,sys2,sys3) plots the responses for three models on the same plot.

• To specify a color, line style, and marker for each system in the plot, specify a LineSpec value for each system. For example, nichols(sys1,LineSpec1,sys2,LineSpec2) plots two models and specifies their plot style. For more information on specifying a LineSpec value, see nicholsplot.

Examples

Nichols Response with Nichols Grid Lines

Plot the Nichols response with Nichols grid lines for the following system:

$$H(s)=\frac{-4s^4+48s^3-18s^2+250s+600}{s^4+30s^3+282s^2+525s+60}.$$

H = tf([-4 48 -18 250 600],[1 30 282 525 60]);
nichols(H)
ngrid

Nichols Plot at Specified Frequencies

Create a Nichols plot over a specified frequency range. Use this approach when you want to focus on the dynamics in a particular range of frequencies.

H = tf([-0.1,-2.4,-181,-1950],[1,3.3,990,2600]);
nichols(H,{1,100})

The cell array {1,100} specifies the minimum and maximum frequency values in the Nichols plot. When you provide frequency bounds in this way, the function selects intermediate points for frequency response data.

Alternatively, specify a vector of frequency points to use for evaluating and plotting the frequency response.

w = 1:0.5:100;
nichols(H,w,'.-')

nichols plots the frequency response at the specified frequencies only.

Nichols Plot of Several Dynamic Systems

Compare the frequency response of a continuous-time system to an equivalent discretized system on the same Nichols plot.

Create continuous-time and discrete-time dynamic systems.

H = tf([1 0.1 7.5],[1 0.12 9 0 0]);
Hd = c2d(H,0.5,'zoh');

Create a Nichols plot that displays both systems.

nichols(H,Hd)

Nichols Plot with Specified Line Attributes

Specify the line style, color, or marker for each system in a Nichols plot using the LineSpec input argument.

H = tf([1 0.1 7.5],[1 0.12 9 0 0]);
Hd = c2d(H,0.5,'zoh');
nichols(H,'r',Hd,'b--')

The first LineSpec, 'r', specifies a solid red line for the response of H. The second LineSpec, 'b--', specifies a dashed blue line for the response of Hd.

Nichols Response Magnitude and Phase Data

Compute the magnitude and phase of the frequency response of a SISO system.

If you do not specify frequencies, nichols chooses frequencies based on the system dynamics and returns them in the third output argument.

H = tf([1 0.1 7.5],[1 0.12 9 0 0]);
[mag,phase,wout] = nichols(H);

Because H is a SISO model, the first two dimensions of mag and phase are both 1. The third dimension is the number of frequencies in wout.

size(mag)

ans = 1×3

     1     1   110

length(wout)

ans = 
110

Thus, each entry along the third dimension of mag gives the magnitude of the response at the corresponding frequency in wout.

Nichols Plot of MIMO System

For this example, create a 2-output, 3-input system.

rng(0,'twister');
H = rss(4,2,3);

For this system, nichols plots the frequency responses of each I/O channel in a separate plot in a single figure.

nichols(H)

Compute the magnitude and phase of these responses at 20 frequencies between 1 and 10 radians.

w = logspace(0,1,20);
[mag,phase] = nichols(H,w);

mag and phase are three-dimensional arrays, in which the first two dimensions correspond to the output and input dimensions of H, and the third dimension is the number of frequencies. For instance, examine the dimensions of mag.

size(mag)

ans = 1×3

     2     3    20

Thus, for example, mag(1,3,10) is the magnitude of the response from the third input to the first output, computed at the 10th frequency in w. Similarly, phase(1,3,10) contains the phase of the same response.

Nichols Plot of Model with Complex Coefficients

Create a Nichols plot of a model with complex coefficients and a model with real coefficients on the same plot.

rng(0)
A = [-3.50,-1.25-0.25i;2,0];
B = [1;0];
C = [-0.75-0.5i,0.625-0.125i];
D = 0.5;
Gc = ss(A,B,C,D);
Gr = rss(7);
nichols(Gc,Gr)
legend('Complex-coefficient model','Real-coefficient model','Location','southwest')

ans = 
  Legend (Complex-coefficient model, Real-coefficient model) with properties:

         String: {'Complex-coefficient model'  'Real-coefficient model'}
       Location: 'southwest'
    Orientation: 'vertical'
       FontSize: 9
       Position: [0.1491 0.1564 0.3785 0.0789]
          Units: 'normalized'

  Use GET to show all properties

For models with complex coefficients, nichols shows a contour comprised of both positive and negative frequencies. For models with real coefficients, the plot shows only positive frequencies, even when complex-coefficient models are present. You can click the curve to further examine which section and values correspond to positive and negative frequencies.

Input Arguments

sys — Dynamic system

dynamic system model | model array

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

• Continuous-time or discrete-time numeric LTI models, such as tf, zpk, or ss models.

• Sparse state-space models, such as sparss or mechss models. Frequency grid w must be specified for sparse models.

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

  • For tunable control design blocks, the function evaluates the model at its current value to plot the response.

  • For uncertain control design blocks, the function plots the nominal value and random samples of the model.

• Frequency-response data models such as frd models. For such models, the function plots the response at the frequencies defined in the model.

• Identified LTI models, such as idtf (System Identification Toolbox), idss (System Identification Toolbox), or idproc (System Identification Toolbox) models. Using identified models requires System Identification Toolbox™ software.

If sys is an array of models, the plot shows responses of all models in the array on the same axes.

w — Frequencies

{wmin,wmax} | vector | []

Frequencies at which to compute and plot frequency response, specified as the cell array {wmin,wmax} or as a vector of frequency values.

• If w is a cell array of the form {wmin,wmax}, then the function computes the response at frequencies ranging between wmin and wmax.

• If w is a vector of frequencies, then the function computes the response at each specified frequency. For example, use logspace to generate a row vector with logarithmically spaced frequency values. The vector w can contain both positive and negative frequencies.

• [] — Automatically select frequencies based on system dynamics.

For models with complex coefficients, if you specify a frequency range of [w_min,w_max] for your plot, then the plot shows a contour comprised of both positive frequencies [w_min,w_max] and negative frequencies [–w_max,–w_min].

Specify frequencies in units of rad/TimeUnit, where TimeUnit is the TimeUnit property of the model.

Output Arguments

mag — Magnitude of system response

3-D array

Magnitude of the system response in absolute units, returned as a 3-D array. The dimensions of this array are (number of system outputs) × (number of system inputs) × (number of frequency points).

• For SISO systems, mag(1,1,k) gives the magnitude of the response at the kth frequency in w or wout. For an example, see Nichols Response Magnitude and Phase Data.

• For MIMO systems, mag(i,j,k) gives the magnitude of the response at the kth frequency from the jth input to the ith output. For an example, see Nichols Plot of MIMO System.

To convert the magnitude from absolute units to decibels, use:

magdb = 20*log10(mag)

phase — Phase of system response

3-D array

Phase of the system response in degrees, returned as a 3-D array. The dimensions of this array are (number of outputs)-by-(number of inputs)-by-(number of frequency points).

• For SISO systems, phase(1,1,k) gives the phase of the response at the kth frequency in w or wout. For an example, see Nichols Response Magnitude and Phase Data.

• For MIMO systems, phase(i,j,k) gives the phase of the response at the kth frequency from the jth input to the ith output. For an example, see Nichols Plot of MIMO System.

wout — Frequencies

vector

Frequencies at which the function returns the system response, returned as a column vector. The function chooses the frequency values based on the model dynamics, unless you specify frequencies using the input argument w.

wout also contains negative frequency values for models with complex coefficients.

Frequency values are in radians/TimeUnit, where TimeUnit is the value of the TimeUnit property of sys.

Version History

Introduced before R2006a