Frequency response function

Frequency diagrams allow you to analyze the frequency response of the equivalent linear system. The frequency response function describes the relationship between the inputs (control variables) and the outputs (observer variables) of a system in the frequency domain. A representation of the response function has the form:

where, Y(s) and U(s) represent the Laplace transforms of the output and the input vectors respectively, and G(s) represents the transfer function that defines the relationship between the input and the output.

For more information, see the appendix on linear algebra.

The frequency response is characterized by the magnitude of the system response, typically in decibels (dB) or as a decimal, and the phase in degrees, versus the frequency in radians/sec (rad/s) or Hertz (Hz).

The Bode magnitude plot is the graph of the function |G(s=jω)| of frequency ω (with j being the imaginary unit). Usually the ω-axis of the magnitude plot is logarithmic and the magnitude is given in decibels, i.e. 20log10|G|.

The Bode phase plot is the graph of the phase, commonly expressed in degrees, of the response function arg(G(s=jω)) as a function of ω. The phase is plotted on the same logarithmic ω-axis as the magnitude plot, but the value is plotted on a linear vertical axis.

To generate a Bode plot which is obtained at a linearization time, select the control and observer variables and push the Create diagram button.

Figure 34: Main Bode plot generation page

Control and observer variable selection

If you select one control variable and one observer variable, you can generate a frequency response plot that describes the relationship between the selected control variable and the selected observer variable.

If you select multiple control variables and one observer variable, you can generate as many frequency response plots as there are control variables selected. The plots are on the same graph.

If you select one control variable and multiple observer variables, you can generate as many frequency response plots as there are observer variables selected. The plots are on the same graph.

You cannot select several control variables and several observer variables.

Parameters of the diagram

To calculate the magnitude of the frequency response function, an evaluation of the function H(s) is performed at several frequency values.

The calculation and the plot differ depending on the selected scale type:

• If you select decibel, the magnitude of the gain is computed as:

    and the y-axis of the plot is set to decimal.
  

• If you select linear, the magnitude of the gain is computed as:

    and the y-axis of the plot is set to linear
  

• If you select logarithmic, the magnitude of the gain is computed as:

    and the y-axis of the plot is set to logarithmic
  

Start frequency, Final frequency and Number of points allow you to define the frequency range and its discretization for the calculation.

Bode diagram

Once you create a Bode plot, a tabulation is created with both the magnitude and the phase curves.

Figure 35: Bode diagram tab

You can adjust the parameters that are used to evaluate the frequency response function. When you change any of the parameters, the Update bode diagram button is activated. Click the button to refresh the plot.

Figure 36: Bode diagram settings group

Note

You can rename the tab title by double-clicking on the text. By default, the title is Diagram_n where n is the index of the tab.

Click the button to create a new Bode plot. The main Bode plot creation page appears.

The Bode diagram settings are contextual to the currently-selected tab.

When you close the window, all your modifications are saved.

See frequency response function partial decomposition for more information on Decomposition settings parameters.