Bases: NonlinearIOSystem, LTI
State space representation for LTI input/output systems.
The StateSpace class is used to represent state-space realizations of linear time-invariant (LTI) systems:
where is the input, is the output, and is the state. State space systems are usually created with the ss factory function.
System matrices of the appropriate dimensions.
System timebase. 0 (default) indicates continuous time, True indicates discrete time with unspecified sampling time, positive number is discrete time with specified sampling time, None indicates unspecified timebase (either continuous or discrete time).
Number of input, output and state variables.
shapetuple
2-tuple of I/O system dimension, (noutputs, ninputs).
Names for the input, output, and state variables.
System name.
Notes
The main data members in the StateSpace class are the A, B, C, and D matrices. The class also keeps track of the number of states (i.e., the size of A).
A discrete-time system is created by specifying a nonzero ‘timebase’, dt when the system is constructed:
dt = 0: continuous-time system (default)
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 other system. Similarly, a system with timebase None can be combined with a system having any timebase; the result will have the timebase of the other system. The default value of dt can be changed by changing the value of config.defaults['control.default_dt'].
A state space system is callable and returns the value of the transfer function evaluated at a point in the complex plane. See StateSpace.__call__ for a more detailed description.
Subsystems corresponding to selected input/output pairs can be created by indexing the state space system:
subsys = sys[output_spec, input_spec]
The input and output specifications can be single integers, lists of integers, or slices. In addition, the strings representing the names of the signals can be used and will be replaced with the equivalent signal offsets. The subsystem is created by truncating the inputs and outputs, but leaving the full set of system states.
StateSpace instances have support for IPython HTML/LaTeX output, intended for pretty-printing in Jupyter notebooks. The HTML/LaTeX output can be configured using config.defaults['statesp.latex_num_format'] and config.defaults['statesp.latex_repr_type']. The HTML/LaTeX output is tailored for MathJax, as used in Jupyter, and may look odd when typeset by non-MathJax LaTeX systems.
config.defaults['statesp.latex_num_format'] is a format string fragment, specifically the part of the format string after ‘{:’ used to convert floating-point numbers to strings. By default it is ‘.3g’.
config.defaults['statesp.latex_repr_type'] must either be ‘partitioned’ or ‘separate’. If ‘partitioned’, the A, B, C, D matrices are shown as a single, partitioned matrix; if ‘separate’, the matrices are shown separately.
Attributes
Dynamics matrix.
Input matrix.
Output matrix.
Direct term.
System timebase.
List of labels for the input signals.
Number of system inputs.
Number of system outputs.
Number of system states.
List of labels for the output signals.
String representation format.
2-tuple of I/O system dimension, (noutputs, ninputs).
List of labels for the state signals.
Deprecated attribute; use nstates instead.
Methods
Evaluate system transfer function at point in complex plane.
Append a second model to the present model.
Evaluate bandwidth of an LTI system for a given dB drop.
Generate a Bode plot for the system.
Make a copy of an input/output system.
Natural frequency, damping ratio of system poles.
Return the zero-frequency ("DC") gain.
Compute the dynamics of the system.
Feedback interconnection between two LTI objects.
Find the index for an input given its name (None if not found).
Return list of indices matching input spec (None if not found).
Find the index for a output given its name (None if not found).
Return list of indices matching output spec (None if not found).
Find the index for a state given its name (None if not found).
Return list of indices matching state spec (None if not found).
Generate the forced response for the system.
(deprecated) Evaluate transfer function at complex frequencies.
Evaluate LTI system response at an array of frequencies.
Evaluate value of transfer function using Horner's method.
Generate the impulse response for the system.
Compute the initial condition response for a linear system.
Check to see if a system is a continuous-time system.
Check to see if a system is a discrete-time system.
Indicate if a linear time invariant (LTI) system is passive.
Check to see if a system is single input, single output.
Return the linear fractional transformation.
Linearize an input/output system at a given state and input.
Remove unobservable and uncontrollable states.
Generate a Nichols plot for the system.
Generate a Nyquist plot for the system.
Compute the output of the system.
Compute the poles of a state space system.
Return a list of a list of scipy.signal.lti objects.
Convert a continuous-time system to discrete time.
Set the number/names of the system inputs.
Set the number/names of the system outputs.
Set the number/names of the system states.
Laub's method to evaluate response at complex frequency.
Generate the step response for the system.
Convert to state space representation.
Convert to transfer function representation.
Update signal and system names for an I/O system.
Compute the zeros of a state space system.
Dynamics matrix.
Input matrix.
Output matrix.
Direct term.
Add two LTI systems (parallel connection).
Evaluate system transfer function at point in complex plane.
Returns the value of the system’s transfer function at a point x in the complex plane, where x is s for continuous-time systems and z for discrete-time systems.
See LTI.__call__ for details.
Examples
>>> G = ct.ss([[-1, -2], [3, -4]], [[5], [7]], [[6, 8]], [[9]]) >>> fresp = G(1j) # evaluate at s = 1j
Array style access
Multiply two LTI objects (serial connection).
Negate a state space system.
Power of a state space system
Right add two LTI systems (parallel connection).
Right multiply two LTI objects (serial connection).
Right subtract two LTI systems.
Division by state space system
Subtract two LTI systems.
Division of state space systems by TFs, FRDs, scalars, and arrays
Append a second model to the present model.
The second model is converted to state-space if necessary, inputs and outputs are appended and their order is preserved.
StateSpace or TransferFunction
System to be appended.
StateSpace
System model with other appended to self.
Evaluate bandwidth of an LTI system for a given dB drop.
Evaluate the first frequency that the response magnitude is lower than DC gain by dbdrop dB.
A strictly negative scalar in dB (default = -3) defines the amount of gain drop for deciding bandwidth.
The first frequency (rad/time-unit) where the gain drops below dbdrop of the dc gain of the system, or nan if the system has infinite dc gain, inf if the gain does not drop for all frequency.
If sys is not an SISO LTI instance.
If dbdrop is not a negative scalar.
Generate a Bode plot for the system.
See bode_plot for more information.
Make a copy of an input/output system.
A copy of the system is made, with a new name. The name keyword can be used to specify a specific name for the system. If no name is given and use_prefix_suffix is True, the name is constructed by prepending config.defaults['iosys.duplicate_system_name_prefix'] and appending config.defaults['iosys.duplicate_system_name_suffix']. Otherwise, a generic system name of the form ‘sys[<id>]’ is used, where ‘<id>’ is based on an internal counter.
Name of the newly created system.
If True and name is None, set the name of the new system to the name of the original system with prefix config.defaults['duplicate_system_name_prefix'] and suffix config.defaults['duplicate_system_name_suffix'].
InputOutputSystem
Natural frequency, damping ratio of system poles.
Natural frequency for each system pole.
Damping ratio for each system pole.
System pole locations.
Return the zero-frequency (“DC”) gain.
The zero-frequency gain of a continuous-time state-space system is given by:
and of a discrete-time state-space system by:
By default, don’t issue a warning message if the zero-frequency gain is infinite. Setting warn_infinite to generate the warning message.
Array or scalar value for SISO systems, depending on config.defaults['control.squeeze_frequency_response']. The value of the array elements or the scalar is either the zero-frequency (or DC) gain, or inf, if the frequency response is singular.
For real valued systems, the empty imaginary part of the complex zero-frequency response is discarded and a real array or scalar is returned.
System timebase.
Compute the dynamics of the system.
Given input u and state x, returns the dynamics of the state-space system. If the system is continuous, returns the time derivative dx/dt
dx/dt = A x + B u
where A and B are the state-space matrices of the system. If the system is discrete time, returns the next value of x:
x[t+dt] = A x[t] + B u[t]
The inputs x and u must be of the correct length for the system.
The first argument t is ignored because StateSpace systems are time-invariant. It is included so that the dynamics can be passed to numerical integrators, such as scipy.integrate.solve_ivp and for consistency with InputOutputSystem models.
Time.
Current state.
Input, zero if omitted.
Feedback interconnection between two LTI objects.
InputOutputSystem
System in the feedback path.
Gain to use in feedback path. Defaults to -1.
Find the index for an input given its name (None if not found).
Signal name for the desired input.
Index of the named input.
Return list of indices matching input spec (None if not found).
List of signal specifications for the desired inputs. A signal can be described by its name or by a slice-like description of the form ‘start:end` where ‘start’ and ‘end’ are signal names. If either is omitted, it is taken as the first or last signal, respectively.
List of indices for the specified inputs.
Find the index for a output given its name (None if not found).
Signal name for the desired output.
Index of the named output.
Return list of indices matching output spec (None if not found).
List of signal specifications for the desired outputs. A signal can be described by its name or by a slice-like description of the form ‘start:end` where ‘start’ and ‘end’ are signal names. If either is omitted, it is taken as the first or last signal, respectively.
List of indices for the specified outputs.
Find the index for a state given its name (None if not found).
Signal name for the desired state.
Index of the named state.
Return list of indices matching state spec (None if not found).
List of signal specifications for the desired states. A signal can be described by its name or by a slice-like description of the form ‘start:end` where ‘start’ and ‘end’ are signal names. If either is omitted, it is taken as the first or last signal, respectively.
List of indices for the specified states..
Generate the forced response for the system.
See forced_response for more information.
(deprecated) Evaluate transfer function at complex frequencies.
Evaluate LTI system response at an array of frequencies.
See frequency_response for more detailed information.
Evaluate value of transfer function using Horner’s method.
Evaluates sys(x) where x is a complex number s for continuous-time systems and z for discrete-time systems. Expects inputs and outputs to be formatted correctly. Use sys(x) for a more user-friendly interface.
Complex frequency at which the transfer function is evaluated.
If True (default), generate a warning if x is a pole.
Notes
Attempts to use Laub’s method from Slycot library, with a fall-back to Python code.
Generate the impulse response for the system.
See impulse_response for more information.
Compute the initial condition response for a linear system.
If the system has multiple outputs (MIMO), optionally, one output may be selected. If no selection is made for the output, all outputs are given.
For information on the shape of parameters T, X0 and return values T, yout, see Time series data conventions.
I/O system(s) for which initial response is computed.
Time vector, or simulation time duration if a number (time vector is auto-computed if not given; see step_response for more detail).
Initial condition (default = 0). Numbers are converted to constant arrays with the correct shape.
Index of the output that will be used in this simulation. Set to None to not trim outputs.
Number of time steps to use in simulation if timepts is not provided as an array (auto-computed if not given); ignored if the system is discrete time.
If system is a nonlinear I/O system, set parameter values.
If True, transpose all input and output arrays (for backward compatibility with MATLAB and scipy.signal.lsim). Default value is False.
If True, return the state vector when assigning to a tuple (default = False). See forced_response for more details.
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 2D array (indexed by the output number and time) even if the system is SISO. The default value can be set using config.defaults['control.squeeze_time_response'].
TimeResponseData or TimeResponseList
Time response represented as a TimeResponseData object or list of TimeResponseData objects. See forced_response for additional information.
Notes
This function uses the forced_response function with the input set to zero.
Examples
>>> G = ct.rss(4) >>> T, yout = ct.initial_response(G)
List of labels for the input signals.
Check to see if a system is a continuous-time system.
If strict is True, make sure that timebase is not None. Default is False.
Check to see if a system is a discrete-time system.
If strict is True, make sure that timebase is not None. Default is False.
Indicate if a linear time invariant (LTI) system is passive.
See ispassive for details.
Check to see if a system is single input, single output.
Return the linear fractional transformation.
A definition of the LFT operator can be found in Appendix A.7, page 512 in [1]. An alternative definition can be found here: https://www.mathworks.com/help/control/ref/lft.html
StateSpace
The lower LTI system.
Dimension of (plant) measurement output.
Dimension of (plant) control input.
StateSpace
References
[1]S. Skogestad, Multivariable Feedback Control. Second edition, 2005.
Linearize an input/output system at a given state and input.
Return the linearization of an input/output system at a given operating point (or state and input value) as a StateSpace system. See linearize for complete documentation.
Remove unobservable and uncontrollable states.
Calculate a minimal realization for a state space system, removing all unobservable and/or uncontrollable states.
Tolerance for determining whether states are unobservable or uncontrollable.
Generate a Nichols plot for the system.
See nichols_plot for more information.
Number of system inputs.
Number of system outputs.
Number of system states.
Generate a Nyquist plot for the system.
See nyquist_plot for more information.
Compute the output of the system.
Given input u and state x, returns the output y of the state-space system:
y = C x + D u
where A and B are the state-space matrices of the system.
The first argument t is ignored because StateSpace systems are time-invariant. It is included so that the dynamics can be passed to most numerical integrators, such as SciPy’s integrate.solve_ivp and for consistency with InputOutputSystem models.
The inputs x and u must be of the correct length for the system.
Time.
Current state.
Input (zero if omitted).
List of labels for the output signals.
Compute the poles of a state space system.
String representation format.
Format used in creating the representation for the system:
‘info’ : <IOSystemType sysname: [inputs] -> [outputs]>
‘eval’ : system specific, loadable representation
‘latex’ : HTML/LaTeX representation of the object
The default representation for an input/output is set to ‘eval’. This value can be changed for an individual system by setting the repr_format parameter when the system is created or by setting the repr_format property after system creation. Set config.defaults['iosys.repr_format'] to change for all I/O systems or use the repr_format parameter/attribute for a single system.
Return a list of a list of scipy.signal.lti objects.
For instance,
>>> out = ssobject.returnScipySignalLTI() >>> out[3][5]
is a scipy.signal.lti object corresponding to the transfer function from the 6th input to the 4th output.
The timebase ssobject.dt cannot be None; it must be continuous (0) or discrete (True or > 0).
If ssobject.dt is None, continuous-time scipy.signal.lti objects are returned.
scipy.signal.StateSpace
Continuous time (inheriting from scipy.signal.lti) or discrete time (inheriting from scipy.signal.dlti) SISO objects.
Convert a continuous-time system to discrete time.
Creates a discrete-time system from a continuous-time system by sampling. Multiple methods of conversion are supported.
Sampling period.
Method to use for sampling:
‘gbt’: generalized bilinear transformation
‘backward_diff’: Backwards difference (‘gbt’ with alpha=1.0)
‘bilinear’ (or ‘tustin’): Tustin’s approximation (‘gbt’ with alpha=0.5)
‘euler’: Euler (or forward difference) method (‘gbt’ with alpha=0)
‘zoh’: zero-order hold (default)
The generalized bilinear transformation weighting parameter, which should only be specified with method=’gbt’, and is ignored otherwise.
The frequency [rad/s] at which to match with the input continuous-time system’s magnitude and phase (the gain = 1 crossover frequency, for example). Should only be specified with method = ‘bilinear’ or ‘gbt’ with alpha = 0.5 and ignored otherwise.
Set the name of the sampled system. If not specified and if copy_names is False, a generic name ‘sys[id]’ is generated with a unique integer id. If copy_names is True, the new system name is determined by adding the prefix and suffix strings in config.defaults['iosys.sampled_system_name_prefix'] and config.defaults['iosys.sampled_system_name_suffix'], with the default being to add the suffix ‘$sampled’.
If True, copy the names of the input signals, output signals, and states to the sampled system.
StateSpace
Discrete-time system, with sampling rate Ts.
Description of the system inputs. If not specified, the original system inputs are used. See InputOutputSystem for more information.
Description of the system outputs. Same format as inputs.
Description of the system states. Same format as inputs.
Notes
Uses scipy.signal.cont2discrete.
Examples
>>> G = ct.ss(0, 1, 1, 0) >>> sysd = G.sample(0.5, method='bilinear')
Set the number/names of the system inputs.
Description of the system inputs. This can be given as an integer count or as a list of strings that name the individual signals. If an integer count is specified, the names of the signal will be of the form ‘u[i]’ (where the prefix ‘u’ can be changed using the optional prefix parameter).
If inputs is an integer, create the names of the states using the given prefix (default = ‘u’). The names of the input will be of the form ‘prefix[i]’.
Set the number/names of the system outputs.
Description of the system outputs. This can be given as an integer count or as a list of strings that name the individual signals. If an integer count is specified, the names of the signal will be of the form ‘y[i]’ (where the prefix ‘y’ can be changed using the optional prefix parameter).
If outputs is an integer, create the names of the states using the given prefix (default = ‘y’). The names of the input will be of the form ‘prefix[i]’.
Set the number/names of the system states.
Description of the system states. This can be given as an integer count or as a list of strings that name the individual signals. If an integer count is specified, the names of the signal will be of the form ‘x[i]’ (where the prefix ‘x’ can be changed using the optional prefix parameter).
If states is an integer, create the names of the states using the given prefix (default = ‘x’). The names of the input will be of the form ‘prefix[i]’.
2-tuple of I/O system dimension, (noutputs, ninputs).
Laub’s method to evaluate response at complex frequency.
Evaluate transfer function at complex frequency using Laub’s method from Slycot. Expects inputs and outputs to be formatted correctly. Use sys(x) for a more user-friendly interface.
Complex frequency.
Frequency response.
List of labels for the state signals.
Deprecated attribute; use nstates instead.
The state attribute was used to store the number of states for : a state space system. It is no longer used. If you need to access the number of states, use nstates.
Generate the step response for the system.
See step_response for more information.
Convert to state space representation.
See ss for details.
Convert to transfer function representation.
See tf for details.
Update signal and system names for an I/O system.
New system name.
List of strings that name the individual input signals. If given as an integer or None, signal names default to the form ‘u[i]’. See InputOutputSystem for more information.
Description of output signals; defaults to ‘y[i]’.
Description of system states; defaults to ‘x[i]’.
Set the prefix for input signals. Default = ‘u’.
Set the prefix for output signals. Default = ‘y’.
Set the prefix for state signals. Default = ‘x’.
Compute the zeros of a state space system.
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