Control System Plots#

This module contains plotting functions for some of the common plots used in control system. Matplotlib will be required as an external dependency if the user wants the plots. To get only the numerical data of the plots, NumPy will be required as external dependency.

Pole-Zero Plot#

control_plots.pole_zero_plot(pole_color='blue', pole_markersize=10, zero_color='orange', zero_markersize=7, grid=True, show_axes=True, show=True, **kwargs)[source]#

Returns the Pole-Zero plot (also known as PZ Plot or PZ Map) of a system.

A Pole-Zero plot is a graphical representation of a system’s poles and zeros. It is plotted on a complex plane, with circular markers representing the system’s zeros and ‘x’ shaped markers representing the system’s poles.

Parameters

system : SISOLinearTimeInvariant type systems

The system for which the pole-zero plot is to be computed.

pole_color : str, tuple, optional

The color of the pole points on the plot. Default color is blue. The color can be provided as a matplotlib color string, or a 3-tuple of floats each in the 0-1 range.

pole_markersize : Number, optional

The size of the markers used to mark the poles in the plot. Default pole markersize is 10.

zero_color : str, tuple, optional

The color of the zero points on the plot. Default color is orange. The color can be provided as a matplotlib color string, or a 3-tuple of floats each in the 0-1 range.

zero_markersize : Number, optional

The size of the markers used to mark the zeros in the plot. Default zero markersize is 7.

grid : boolean, optional

If True, the plot will have a grid. Defaults to True.

show_axes : boolean, optional

If True, the coordinate axes will be shown. Defaults to False.

show : boolean, optional

If True, the plot will be displayed otherwise the equivalent matplotlib plot object will be returned. Defaults to True.

Examples

>>> from sympy.abc import s
>>> from sympy.physics.control.lti import TransferFunction
>>> from sympy.physics.control.control_plots import pole_zero_plot
>>> tf1 = TransferFunction(s**2 + 1, s**4 + 4*s**3 + 6*s**2 + 5*s + 2, s)
>>> pole_zero_plot(tf1)   

(png, hires.png, pdf)

../../../_images/control_plots-1.png

References

R652

https://en.wikipedia.org/wiki/Pole%E2%80%93zero_plot

control_plots.pole_zero_numerical_data()[source]#

Returns the numerical data of poles and zeros of the system. It is internally used by pole_zero_plot to get the data for plotting poles and zeros. Users can use this data to further analyse the dynamics of the system or plot using a different backend/plotting-module.

Parameters

system : SISOLinearTimeInvariant

The system for which the pole-zero data is to be computed.

Returns

tuple : (zeros, poles)

zeros = Zeros of the system. NumPy array of complex numbers. poles = Poles of the system. NumPy array of complex numbers.

Raises

NotImplementedError

When a SISO LTI system is not passed.

When time delay terms are present in the system.

ValueError

When more than one free symbol is present in the system. The only variable in the transfer function should be the variable of the Laplace transform.

Examples

>>> from sympy.abc import s
>>> from sympy.physics.control.lti import TransferFunction
>>> from sympy.physics.control.control_plots import pole_zero_numerical_data
>>> tf1 = TransferFunction(s**2 + 1, s**4 + 4*s**3 + 6*s**2 + 5*s + 2, s)
>>> pole_zero_numerical_data(tf1)   
([-0.+1.j  0.-1.j], [-2. +0.j        -0.5+0.8660254j -0.5-0.8660254j -1. +0.j       ])

See also

pole_zero_plot

Bode Plot#

control_plots.bode_plot(initial_exp=- 5, final_exp=5, grid=True, show_axes=False, show=True, freq_unit='rad/sec', phase_unit='rad', **kwargs)[source]#

Returns the Bode phase and magnitude plots of a continuous-time system.

Parameters

system : SISOLinearTimeInvariant type

The LTI SISO system for which the Bode Plot is to be computed.

initial_exp : Number, optional

The initial exponent of 10 of the semilog plot. Defaults to -5.

final_exp : Number, optional

The final exponent of 10 of the semilog plot. Defaults to 5.

show : boolean, optional

If True, the plot will be displayed otherwise the equivalent matplotlib plot object will be returned. Defaults to True.

prec : int, optional

The decimal point precision for the point coordinate values. Defaults to 8.

grid : boolean, optional

If True, the plot will have a grid. Defaults to True.

show_axes : boolean, optional

If True, the coordinate axes will be shown. Defaults to False.

freq_unit : string, optional

User can choose between 'rad/sec' (radians/second) and 'Hz' (Hertz) as frequency units.

phase_unit : string, optional

User can choose between 'rad' (radians) and 'deg' (degree) as phase units.

Examples

>>> from sympy.abc import s
>>> from sympy.physics.control.lti import TransferFunction
>>> from sympy.physics.control.control_plots import bode_plot
>>> tf1 = TransferFunction(1*s**2 + 0.1*s + 7.5, 1*s**4 + 0.12*s**3 + 9*s**2, s)
>>> bode_plot(tf1, initial_exp=0.2, final_exp=0.7)   

(png, hires.png, pdf)

../../../_images/control_plots-2.png
control_plots.bode_magnitude_plot(initial_exp=- 5, final_exp=5, color='b', show_axes=False, grid=True, show=True, freq_unit='rad/sec', **kwargs)[source]#

Returns the Bode magnitude plot of a continuous-time system.

See bode_plot for all the parameters.

control_plots.bode_phase_plot(initial_exp=- 5, final_exp=5, color='b', show_axes=False, grid=True, show=True, freq_unit='rad/sec', phase_unit='rad', **kwargs)[source]#

Returns the Bode phase plot of a continuous-time system.

See bode_plot for all the parameters.

control_plots.bode_magnitude_numerical_data(initial_exp=- 5, final_exp=5, freq_unit='rad/sec', **kwargs)[source]#

Returns the numerical data of the Bode magnitude plot of the system. It is internally used by bode_magnitude_plot to get the data for plotting Bode magnitude plot. Users can use this data to further analyse the dynamics of the system or plot using a different backend/plotting-module.

Parameters

system : SISOLinearTimeInvariant

The system for which the data is to be computed.

initial_exp : Number, optional

The initial exponent of 10 of the semilog plot. Defaults to -5.

final_exp : Number, optional

The final exponent of 10 of the semilog plot. Defaults to 5.

freq_unit : string, optional

User can choose between 'rad/sec' (radians/second) and 'Hz' (Hertz) as frequency units.

Returns

tuple : (x, y)

x = x-axis values of the Bode magnitude plot. y = y-axis values of the Bode magnitude plot.

Raises

NotImplementedError

When a SISO LTI system is not passed.

When time delay terms are present in the system.

ValueError

When more than one free symbol is present in the system. The only variable in the transfer function should be the variable of the Laplace transform.

When incorrect frequency units are given as input.

Examples

>>> from sympy.abc import s
>>> from sympy.physics.control.lti import TransferFunction
>>> from sympy.physics.control.control_plots import bode_magnitude_numerical_data
>>> tf1 = TransferFunction(s**2 + 1, s**4 + 4*s**3 + 6*s**2 + 5*s + 2, s)
>>> bode_magnitude_numerical_data(tf1)   
([1e-05, 1.5148378120533502e-05,..., 68437.36188804005, 100000.0],
[-6.020599914256786, -6.0205999155219505,..., -193.4117304087953, -200.00000000260573])
control_plots.bode_phase_numerical_data(initial_exp=- 5, final_exp=5, freq_unit='rad/sec', phase_unit='rad', **kwargs)[source]#

Returns the numerical data of the Bode phase plot of the system. It is internally used by bode_phase_plot to get the data for plotting Bode phase plot. Users can use this data to further analyse the dynamics of the system or plot using a different backend/plotting-module.

Parameters

system : SISOLinearTimeInvariant

The system for which the Bode phase plot data is to be computed.

initial_exp : Number, optional

The initial exponent of 10 of the semilog plot. Defaults to -5.

final_exp : Number, optional

The final exponent of 10 of the semilog plot. Defaults to 5.

freq_unit : string, optional

User can choose between 'rad/sec' (radians/second) and ‘``’Hz’`` (Hertz) as frequency units.

phase_unit : string, optional

User can choose between 'rad' (radians) and 'deg' (degree) as phase units.

Returns

tuple : (x, y)

x = x-axis values of the Bode phase plot. y = y-axis values of the Bode phase plot.

Raises

NotImplementedError

When a SISO LTI system is not passed.

When time delay terms are present in the system.

ValueError

When more than one free symbol is present in the system. The only variable in the transfer function should be the variable of the Laplace transform.

When incorrect frequency or phase units are given as input.

Examples

>>> from sympy.abc import s
>>> from sympy.physics.control.lti import TransferFunction
>>> from sympy.physics.control.control_plots import bode_phase_numerical_data
>>> tf1 = TransferFunction(s**2 + 1, s**4 + 4*s**3 + 6*s**2 + 5*s + 2, s)
>>> bode_phase_numerical_data(tf1)   
([1e-05, 1.4472354033813751e-05, 2.035581932165858e-05,..., 47577.3248186011, 67884.09326036123, 100000.0],
[-2.5000000000291665e-05, -3.6180885085e-05, -5.08895483066e-05,...,-3.1415085799262523, -3.14155265358979])

Impulse-Response Plot#

control_plots.impulse_response_plot(color='b', prec=8, lower_limit=0, upper_limit=10, show_axes=False, grid=True, show=True, **kwargs)[source]#

Returns the unit impulse response (Input is the Dirac-Delta Function) of a continuous-time system.

Parameters

system : SISOLinearTimeInvariant type

The LTI SISO system for which the Impulse Response is to be computed.

color : str, tuple, optional

The color of the line. Default is Blue.

show : boolean, optional

If True, the plot will be displayed otherwise the equivalent matplotlib plot object will be returned. Defaults to True.

lower_limit : Number, optional

The lower limit of the plot range. Defaults to 0.

upper_limit : Number, optional

The upper limit of the plot range. Defaults to 10.

prec : int, optional

The decimal point precision for the point coordinate values. Defaults to 8.

show_axes : boolean, optional

If True, the coordinate axes will be shown. Defaults to False.

grid : boolean, optional

If True, the plot will have a grid. Defaults to True.

Examples

>>> from sympy.abc import s
>>> from sympy.physics.control.lti import TransferFunction
>>> from sympy.physics.control.control_plots import impulse_response_plot
>>> tf1 = TransferFunction(8*s**2 + 18*s + 32, s**3 + 6*s**2 + 14*s + 24, s)
>>> impulse_response_plot(tf1)   

(png, hires.png, pdf)

../../../_images/control_plots-3.png

References

R653

https://www.mathworks.com/help/control/ref/lti.impulse.html

control_plots.impulse_response_numerical_data(prec=8, lower_limit=0, upper_limit=10, **kwargs)[source]#

Returns the numerical values of the points in the impulse response plot of a SISO continuous-time system. By default, adaptive sampling is used. If the user wants to instead get an uniformly sampled response, then adaptive kwarg should be passed False and nb_of_points must be passed as additional kwargs. Refer to the parameters of class sympy.plotting.plot.LineOver1DRangeSeries for more details.

Parameters

system : SISOLinearTimeInvariant

The system for which the impulse response data is to be computed.

prec : int, optional

The decimal point precision for the point coordinate values. Defaults to 8.

lower_limit : Number, optional

The lower limit of the plot range. Defaults to 0.

upper_limit : Number, optional

The upper limit of the plot range. Defaults to 10.

kwargs :

Additional keyword arguments are passed to the underlying sympy.plotting.plot.LineOver1DRangeSeries class.

Returns

tuple : (x, y)

x = Time-axis values of the points in the impulse response. NumPy array. y = Amplitude-axis values of the points in the impulse response. NumPy array.

Raises

NotImplementedError

When a SISO LTI system is not passed.

When time delay terms are present in the system.

ValueError

When more than one free symbol is present in the system. The only variable in the transfer function should be the variable of the Laplace transform.

When lower_limit parameter is less than 0.

Examples

>>> from sympy.abc import s
>>> from sympy.physics.control.lti import TransferFunction
>>> from sympy.physics.control.control_plots import impulse_response_numerical_data
>>> tf1 = TransferFunction(s, s**2 + 5*s + 8, s)
>>> impulse_response_numerical_data(tf1)   
([0.0, 0.06616480200395854,... , 9.854500743565858, 10.0],
[0.9999999799999999, 0.7042848373025861,...,7.170748906965121e-13, -5.1901263495547205e-12])

Step-Response Plot#

control_plots.step_response_plot(color='b', prec=8, lower_limit=0, upper_limit=10, show_axes=False, grid=True, show=True, **kwargs)[source]#

Returns the unit step response of a continuous-time system. It is the response of the system when the input signal is a step function.

Parameters

system : SISOLinearTimeInvariant type

The LTI SISO system for which the Step Response is to be computed.

color : str, tuple, optional

The color of the line. Default is Blue.

show : boolean, optional

If True, the plot will be displayed otherwise the equivalent matplotlib plot object will be returned. Defaults to True.

lower_limit : Number, optional

The lower limit of the plot range. Defaults to 0.

upper_limit : Number, optional

The upper limit of the plot range. Defaults to 10.

prec : int, optional

The decimal point precision for the point coordinate values. Defaults to 8.

show_axes : boolean, optional

If True, the coordinate axes will be shown. Defaults to False.

grid : boolean, optional

If True, the plot will have a grid. Defaults to True.

Examples

>>> from sympy.abc import s
>>> from sympy.physics.control.lti import TransferFunction
>>> from sympy.physics.control.control_plots import step_response_plot
>>> tf1 = TransferFunction(8*s**2 + 18*s + 32, s**3 + 6*s**2 + 14*s + 24, s)
>>> step_response_plot(tf1)   

(png, hires.png, pdf)

../../../_images/control_plots-4.png

References

R654

https://www.mathworks.com/help/control/ref/lti.step.html

control_plots.step_response_numerical_data(prec=8, lower_limit=0, upper_limit=10, **kwargs)[source]#

Returns the numerical values of the points in the step response plot of a SISO continuous-time system. By default, adaptive sampling is used. If the user wants to instead get an uniformly sampled response, then adaptive kwarg should be passed False and nb_of_points must be passed as additional kwargs. Refer to the parameters of class sympy.plotting.plot.LineOver1DRangeSeries for more details.

Parameters

system : SISOLinearTimeInvariant

The system for which the unit step response data is to be computed.

prec : int, optional

The decimal point precision for the point coordinate values. Defaults to 8.

lower_limit : Number, optional

The lower limit of the plot range. Defaults to 0.

upper_limit : Number, optional

The upper limit of the plot range. Defaults to 10.

kwargs :

Additional keyword arguments are passed to the underlying sympy.plotting.plot.LineOver1DRangeSeries class.

Returns

tuple : (x, y)

x = Time-axis values of the points in the step response. NumPy array. y = Amplitude-axis values of the points in the step response. NumPy array.

Raises

NotImplementedError

When a SISO LTI system is not passed.

When time delay terms are present in the system.

ValueError

When more than one free symbol is present in the system. The only variable in the transfer function should be the variable of the Laplace transform.

When lower_limit parameter is less than 0.

Examples

>>> from sympy.abc import s
>>> from sympy.physics.control.lti import TransferFunction
>>> from sympy.physics.control.control_plots import step_response_numerical_data
>>> tf1 = TransferFunction(s, s**2 + 5*s + 8, s)
>>> step_response_numerical_data(tf1)   
([0.0, 0.025413462339411542, 0.0484508722725343, ... , 9.670250533855183, 9.844291913708725, 10.0],
[0.0, 0.023844582399907256, 0.042894276802320226, ..., 6.828770759094287e-12, 6.456457160755703e-12])

Ramp-Response Plot#

control_plots.ramp_response_plot(slope=1, color='b', prec=8, lower_limit=0, upper_limit=10, show_axes=False, grid=True, show=True, **kwargs)[source]#

Returns the ramp response of a continuous-time system.

Ramp function is defined as the straight line passing through origin (\(f(x) = mx\)). The slope of the ramp function can be varied by the user and the default value is 1.

Parameters

system : SISOLinearTimeInvariant type

The LTI SISO system for which the Ramp Response is to be computed.

slope : Number, optional

The slope of the input ramp function. Defaults to 1.

color : str, tuple, optional

The color of the line. Default is Blue.

show : boolean, optional

If True, the plot will be displayed otherwise the equivalent matplotlib plot object will be returned. Defaults to True.

lower_limit : Number, optional

The lower limit of the plot range. Defaults to 0.

upper_limit : Number, optional

The upper limit of the plot range. Defaults to 10.

prec : int, optional

The decimal point precision for the point coordinate values. Defaults to 8.

show_axes : boolean, optional

If True, the coordinate axes will be shown. Defaults to False.

grid : boolean, optional

If True, the plot will have a grid. Defaults to True.

Examples

>>> from sympy.abc import s
>>> from sympy.physics.control.lti import TransferFunction
>>> from sympy.physics.control.control_plots import ramp_response_plot
>>> tf1 = TransferFunction(s, (s+4)*(s+8), s)
>>> ramp_response_plot(tf1, upper_limit=2)   

(png, hires.png, pdf)

../../../_images/control_plots-5.png

References

R655

https://en.wikipedia.org/wiki/Ramp_function

control_plots.ramp_response_numerical_data(slope=1, prec=8, lower_limit=0, upper_limit=10, **kwargs)[source]#

Returns the numerical values of the points in the ramp response plot of a SISO continuous-time system. By default, adaptive sampling is used. If the user wants to instead get an uniformly sampled response, then adaptive kwarg should be passed False and nb_of_points must be passed as additional kwargs. Refer to the parameters of class sympy.plotting.plot.LineOver1DRangeSeries for more details.

Parameters

system : SISOLinearTimeInvariant

The system for which the ramp response data is to be computed.

slope : Number, optional

The slope of the input ramp function. Defaults to 1.

prec : int, optional

The decimal point precision for the point coordinate values. Defaults to 8.

lower_limit : Number, optional

The lower limit of the plot range. Defaults to 0.

upper_limit : Number, optional

The upper limit of the plot range. Defaults to 10.

kwargs :

Additional keyword arguments are passed to the underlying sympy.plotting.plot.LineOver1DRangeSeries class.

Returns

tuple : (x, y)

x = Time-axis values of the points in the ramp response plot. NumPy array. y = Amplitude-axis values of the points in the ramp response plot. NumPy array.

Raises

NotImplementedError

When a SISO LTI system is not passed.

When time delay terms are present in the system.

ValueError

When more than one free symbol is present in the system. The only variable in the transfer function should be the variable of the Laplace transform.

When lower_limit parameter is less than 0.

When slope is negative.

Examples

>>> from sympy.abc import s
>>> from sympy.physics.control.lti import TransferFunction
>>> from sympy.physics.control.control_plots import ramp_response_numerical_data
>>> tf1 = TransferFunction(s, s**2 + 5*s + 8, s)
>>> ramp_response_numerical_data(tf1)   
(([0.0, 0.12166980856813935,..., 9.861246379582118, 10.0],
[1.4504508011325967e-09, 0.006046440489058766,..., 0.12499999999568202, 0.12499999999661349]))