control.step_response(sys, T=None, X0=0, input=None, output=None, T_num=None, transpose=False, return_x=False, squeeze=None, params=None)[source]

Compute the step response for a linear system.

If the system has multiple inputs and/or multiple outputs, the step response is computed for each input/output pair, with all other inputs set to zero. Optionally, a single input and/or single output can be selected, in which case all other inputs are set to 0 and all other outputs are ignored.

For information on the shape of parameters T, X0 and return values T, yout, see Time series data.

  • sys (StateSpace or TransferFunction) – LTI system to simulate

  • T (array_like or float, optional) – Time vector, or simulation time duration if a number. If T is not provided, an attempt is made to create it automatically from the dynamics of sys. If sys is continuous-time, the time increment dt is chosen small enough to show the fastest mode, and the simulation time period tfinal long enough to show the slowest mode, excluding poles at the origin and pole-zero cancellations. If this results in too many time steps (>5000), dt is reduced. If sys is discrete-time, only tfinal is computed, and final is reduced if it requires too many simulation steps.

  • X0 (array_like or float, optional) – Initial condition (default = 0). This can be used for a nonlinear system where the origin is not an equilibrium point.

  • input (int, optional) – Only compute the step response for the listed input. If not specified, the step responses for each independent input are computed (as separate traces).

  • output (int, optional) – Only report the step response for the listed output. If not specified, all outputs are reported.

  • params (dict, optional) – If system is a nonlinear I/O system, set parameter values.

  • T_num (int, optional) – Number of time steps to use in simulation if T is not provided as an array (autocomputed if not given); ignored if sys is discrete-time.

  • transpose (bool, optional) – If True, transpose all input and output arrays (for backward compatibility with MATLAB and scipy.signal.lsim()). Default value is False.

  • return_x (bool, optional) – If True, return the state vector when assigning to a tuple (default = False). See forced_response() for more details.

  • squeeze (bool, optional) – By default, if a system is single-input, single-output (SISO) then the output response is returned as a 1D array (indexed by time). If squeeze=True, remove single-dimensional entries from the shape of the output even if the system is not SISO. If squeeze=False, keep the output as a 3D array (indexed by the output, input, and time) even if the system is SISO. The default value can be set using config.defaults[‘control.squeeze_time_response’].


results – Time response represented as a TimeResponseData object containing the following properties:

  • time (array): Time values of the output.

  • outputs (array): Response of the system. If the system is SISO and squeeze is not True, the array is 1D (indexed by time). If the system is not SISO or squeeze is False, the array is 3D (indexed by the output, trace, and time).

  • states (array): Time evolution of the state vector, represented as either a 2D array indexed by state and time (if SISO) or a 3D array indexed by state, trace, and time. Not affected by squeeze.

  • inputs (array): Input(s) to the system, indexed in the same manner as outputs.

The return value of the system can also be accessed by assigning the function to a tuple of length 2 (time, output) or of length 3 (time, output, state) if return_x is True.

Return type



This function uses the forced_response function with the input set to a unit step.


>>> G = ct.rss(4)
>>> T, yout = ct.step_response(G)