The python-control library uses a set of standard conventions for the way that different types of standard information used by the library.
LTI system representation¶
Linear time invariant (LTI) systems are represented in python-control in state space, transfer function, or frequency response data (FRD) form. Most functions in the toolbox will operate on any of these data types and functions for converting between compatible types is provided.
State space systems¶
StateSpace class is used to represent state-space realizations
of linear time-invariant (LTI) systems:
where u is the input, y is the output, and x is the state.
To create a state space system, use the
sys = StateSpace(A, B, C, D)
TransferFunction class is used to represent input/output
where n is generally greater than or equal to m (for a proper transfer function).
To create a transfer function, use the
sys = TransferFunction(num, den)
FRD (frequency response data) systems¶
FrequencyResponseData (FRD) class is used to represent systems in
frequency response data form.
The main data members are omega and fresp, where omega is a 1D array with the frequency points of the response, and fresp is a 3D array, with the first dimension corresponding to the output index of the FRD, the second dimension corresponding to the input index, and the 3rd dimension corresponding to the frequency points in omega.
FRD systems have a somewhat more limited set of functions that are available, although all of the standard algebraic manipulations can be performed.
Discrete time systems¶
A discrete time system is created by specifying a nonzero ‘timebase’, dt. The timebase argument can be given when a system is constructed:
dt = 0: continuous time system (default)
dt > 0: discrete time system with sampling period ‘dt’
dt = True: discrete time with unspecified sampling period
dt = None: no timebase specified
Systems must have compatible timebases in order to be combined. A discrete time
system with unspecified sampling time (dt = True) can be combined with a system
having a specified sampling time; the result will be a discrete time system with the sample time of the latter
system. Similarly, a system with timebase None can be combined with a system having a specified
timebase; the result will have the timebase of the latter system. For continuous
time systems, the
sample_system() function or the
can be used to create a discrete time system from a continuous time system.
See Utility functions and conversions. The default value of ‘dt’ can be changed by
changing the value of
Conversion between representations¶
LTI systems can be converted between representations either by calling the
constructor for the desired data type using the original system as the sole
argument or using the explicit conversion functions
Time series data¶
A variety of functions in the library return time series data: sequences of
values that change over time. A common set of conventions is used for
returning such data: columns represent different points in time, rows are
different components (e.g., inputs, outputs or states). For return
arguments, an array of times is given as the first returned argument,
followed by one or more arrays of variable values. This convention is used
throughout the library, for example in the functions
The convention used by python-control is different from the convention used in the scipy.signal library. In Scipy’s convention the meaning of rows and columns is interchanged. Thus, all 2D values must be transposed when they are used with functions from scipy.signal.
Arguments can be arrays, matrices, or nested lists.
Return values are arrays (not matrices).
The time vector is a 1D array with shape (n, ):
T = [t1, t2, t3, ..., tn ]
Input, state, and output all follow the same convention. Columns are different points in time, rows are different components:
U = [[u1(t1), u1(t2), u1(t3), ..., u1(tn)] [u2(t1), u2(t2), u2(t3), ..., u2(tn)] ... ... [ui(t1), ui(t2), ui(t3), ..., ui(tn)]] Same for X, Y
So, U[:,2] is the system’s input at the third point in time; and U or U[1,:] is the sequence of values for the system’s second input.
When there is only one row, a 1D object is accepted or returned, which adds convenience for SISO systems:
The initial conditions are either 1D, or 2D with shape (j, 1):
X0 = [[x1] [x2] ... ... [xj]]
Functions that return time responses (e.g.,
step_response()) return a
TimeResponseData object that contains the data for the time
response. These data can be accessed via the
sys = rss(4, 1, 1) response = step_response(sys) plot(response.time, response.outputs)
The dimensions of the response properties depend on the function being
called and whether the system is SISO or MIMO. In addition, some time
response function can return multiple “traces” (input/output pairs),
such as the
step_response() function applied to a MIMO system,
which will compute the step response for each input/output pair. See
TimeResponseData for more details.
The time response functions can also be assigned to a tuple, which extracts the time and output (and optionally the state, if the return_x keyword is used). This allows simple commands for plotting:
t, y = step_response(sys) plot(t, y)
The output of a MIMO system can be plotted like this:
t, y = forced_response(sys, t, u) plot(t, y, label='y_0') plot(t, y, label='y_1')
The convention also works well with the state space form of linear
D is the feedthrough matrix (2D array) of a linear system,
U is its input (array), then the feedthrough part of the system’s
response, can be computed like this:
ft = D @ U
Package configuration parameters¶
The python-control library can be customized to allow for different default values for selected parameters. This includes the ability to set the style for various types of plots and establishing the underlying representation for state space matrices.
To set the default value of a configuration variable, set the appropriate element of the control.config.defaults dictionary:
control.config.defaults['module.parameter'] = value
The ~control.config.set_defaults function can also be used to set multiple configuration parameters at the same time:
control.config.set_defaults('module', param1=val1, param2=val2, ...]
Finally, there are also functions available set collections of variables based on standard configurations.
Selected variables that can be configured, along with their default values:
freqplot.dB (False): Bode plot magnitude plotted in dB (otherwise powers of 10)
freqplot.deg (True): Bode plot phase plotted in degrees (otherwise radians)
freqplot.Hz (False): Bode plot frequency plotted in Hertz (otherwise rad/sec)
freqplot.grid (True): Include grids for magnitude and phase plots
freqplot.number_of_samples (1000): Number of frequency points in Bode plots
freqplot.feature_periphery_decade (1.0): How many decades to include in the frequency range on both sides of features (poles, zeros).
statesp.use_numpy_matrix (True): set the return type for state space matrices to numpy.matrix (verus numpy.ndarray)
statesp.default_dt and xferfcn.default_dt (None): set the default value of dt when constructing new LTI systems
statesp.remove_useless_states (True): remove states that have no effect on the input-output dynamics of the system
Additional parameter variables are documented in individual functions
Functions that can be used to set standard configurations:
Reset configuration values to their default (initial) values.
Use Feedback Systems (FBS) compatible settings.
Use MATLAB compatible configuration settings.
Turn on/off use of Numpy matrix class for state space operations.
Sets the defaults to whatever they were in a given release.