control.frequency_response(sysdata, omega=None, omega_limits=None, omega_num=None, Hz=None, squeeze=None)[source]

Frequency response of an LTI system at multiple angular frequencies.

In general the system may be multiple input, multiple output (MIMO), where m = sys.ninputs number of inputs and p = sys.noutputs number of outputs.

  • sysdata (LTI system or list of LTI systems) – Linear system(s) for which frequency response is computed.

  • omega (float or 1D array_like, optional) – A list of frequencies in radians/sec at which the system should be evaluated. The list can be either a Python list or a numpy array and will be sorted before evaluation. If None (default), a common set of frequencies that works across all given systems is computed.

  • omega_limits (array_like of two values, optional) – Limits to the range of frequencies, in rad/sec. Ignored if omega is provided, and auto-generated if omitted.

  • omega_num (int, optional) – Number of frequency samples to plot. Defaults to config.defaults[‘freqplot.number_of_samples’].

  • Hz (bool, optional) – If True, when computing frequency limits automatically set limits to full decades in Hz instead of rad/s. Omega is always returned in rad/sec.

  • squeeze (bool, optional) – If squeeze=True, remove single-dimensional entries from the shape of the output even if the system is not SISO. If squeeze=False, keep all indices (output, input and, if omega is array_like, frequency) even if the system is SISO. The default value can be set using config.defaults[‘control.squeeze_frequency_response’].


response – Frequency response data object representing the frequency response. This object can be assigned to a tuple using

mag, phase, omega = response

where mag is the magnitude (absolute value, not dB or log10) of the system frequency response, phase is the wrapped phase in radians of the system frequency response, and omega is the (sorted) frequencies at which the response was evaluated. If the system is SISO and squeeze is not False, magnitude and phase are 1D, indexed by frequency. If the system is not SISO or squeeze is False, the array is 3D, indexed by the output, input, and frequency. If squeeze is True then single-dimensional axes are removed.

Returns a list of FrequencyResponseData objects if sys is a list of systems.

Return type


See also

evalfr, bode_plot


  1. This function is a wrapper for StateSpace.frequency_response() and TransferFunction.frequency_response().

  2. You can also use the lower-level methods sys(s) or sys(z) to generate the frequency response for a single system.

  3. All frequency data should be given in rad/sec. If frequency limits are computed automatically, the Hz keyword can be used to ensure that limits are in factors of decades in Hz, so that Bode plots with Hz=True look better.

  4. The frequency response data can be plotted by calling the control_bode_plot() function or using the plot method of the FrequencyResponseData class.


>>> G =[[-1, -2], [3, -4]], [[5], [7]], [[6, 8]], [[9]])
>>> mag, phase, omega = ct.freqresp(G, [0.1, 1., 10.])


Add example with MIMO system

#>>> sys = rss(3, 2, 2) #>>> mag, phase, omega = freqresp(sys, [0.1, 1., 10.]) #>>> mag[0, 1, :] #array([ 55.43747231, 42.47766549, 1.97225895]) #>>> phase[1, 0, :] #array([-0.12611087, -1.14294316, 2.5764547 ]) #>>> # This is the magnitude of the frequency response from the 2nd #>>> # input to the 1st output, and the phase (in radians) of the #>>> # frequency response from the 1st input to the 2nd output, for #>>> # s = 0.1i, i, 10i.