control.StateSpace¶

class
control.
StateSpace
(A, B, C, D[, dt])¶ A class for representing statespace models
The StateSpace class is used to represent statespace realizations of linear timeinvariant (LTI) systems:
 dx/dt = A x + B u
y = C x + D u
where u is the input, y is the output, and x is the state.
The main data members are the A, B, C, and D matrices. The class also keeps track of the number of states (i.e., the size of A). The data format used to store state space matrices is set using the value of config.defaults[‘use_numpy_matrix’]. If True (default), the state space elements are stored as numpy.matrix objects; otherwise they are numpy.ndarray objects. The
use_numpy_matrix()
function can be used to set the storage type.A discrete time system is created by specifying a nonzero ‘timebase’, dt when the 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 any timebase; the result will have the timebase of the latter system. The default value of dt can be changed by changing the value of
control.config.defaults['control.default_dt']
.StateSpace instances have support for IPython LaTeX output, intended for prettyprinting in Jupyter notebooks. The LaTeX output can be configured using control.config.defaults[‘statesp.latex_num_format’] and control.config.defaults[‘statesp.latex_repr_type’]. The LaTeX output is tailored for MathJax, as used in Jupyter, and may look odd when typeset by nonMathJax LaTeX systems.
control.config.defaults[‘statesp.latex_num_format’] is a format string fragment, specifically the part of the format string after ‘{:’ used to convert floatingpoint numbers to strings. By default it is ‘.3g’.
control.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.

__init__
(*args, **kwargs)¶ StateSpace(A, B, C, D[, dt])
Construct a state space object.
The default constructor is StateSpace(A, B, C, D), where A, B, C, D are matrices or equivalent objects. To create a discrete time system, use StateSpace(A, B, C, D, dt) where dt is the sampling time (or True for unspecified sampling time). To call the copy constructor, call StateSpace(sys), where sys is a StateSpace object.
The remove_useless_states keyword can be used to scan the A, B, and C matrices for rows or columns of zeros. If the zeros are such that a particular state has no effect on the inputoutput dynamics, then that state is removed from the A, B, and C matrices. If not specified, the value is read from config.defaults[‘statesp.remove_useless_states’] (default = False).
Methods
__init__
(*args, **kwargs)StateSpace(A, B, C, D[, dt])
append
(other)Append a second model to the present model.
damp
()Natural frequency, damping ratio of system poles
dcgain
([warn_infinite])Return the zerofrequency gain
dynamics
(t, x[, u])Compute the dynamics of the system
feedback
([other, sign])Feedback interconnection between two LTI systems.
freqresp
(omega)(deprecated) Evaluate transfer function at complex frequencies.
frequency_response
(omega[, squeeze])Evaluate the linear timeinvariant system at an array of angular frequencies.
horner
(x[, warn_infinite])Evaluate system’s transfer function at complex frequency using Laub’s or Horner’s method.
isctime
([strict])Check to see if a system is a continuoustime system
isdtime
([strict])Check to see if a system is a discretetime system
issiso
()Check to see if a system is single input, single output
lft
(other[, nu, ny])Return the Linear Fractional Transformation.
minreal
([tol])Calculate a minimal realization, removes unobservable and uncontrollable states
output
(t, x[, u])Compute the output of the system
pole
()Compute the poles of a state space system.
returnScipySignalLTI
([strict])Return a list of a list of
scipy.signal.lti
objects.sample
(Ts[, method, alpha, prewarp_frequency])Convert a continuous time system to discrete time
slycot_laub
(x)Evaluate system’s transfer function at complex frequency using Laub’s method from Slycot.
zero
()Compute the zeros of a state space system.
Attributes
inputs
outputs
states

append
(other)¶ Append a second model to the present model.
The second model is converted to statespace if necessary, inputs and outputs are appended and their order is preserved

damp
()¶ Natural frequency, damping ratio of system poles
 Returns
wn (array) – Natural frequencies for each system pole
zeta (array) – Damping ratio for each system pole
poles (array) – Array of system poles

dcgain
(warn_infinite=False)¶ Return the zerofrequency gain
The zerofrequency gain of a continuoustime statespace system is given by:
and of a discretetime statespace system by:
 Parameters
warn_infinite (bool, optional) – By default, don’t issue a warning message if the zerofrequency gain is infinite. Setting warn_infinite to generate the warning message.
 Returns
gain – 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 zerofrequency (or DC) gain, or inf, if the frequency response is singular.
For real valued systems, the empty imaginary part of the complex zerofrequency response is discarded and a real array or scalar is returned.
 Return type
(noutputs, ninputs) ndarray or scalar

dynamics
(t, x, u=None)¶ Compute the dynamics of the system
Given input u and state x, returns the dynamics of the statespace system. If the system is continuous, returns the time derivative dx/dt
dx/dt = A x + B u
where A and B are the statespace matrices of the system. If the system is discretetime, 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 timeinvariant. It is included so that the dynamics can be passed to most numerical integrators, such asscipy.integrate.solve_ivp()
and for consistency withIOSystem
systems. Parameters
t (float (ignored)) – time
x (array_like) – current state
u (array_like (optional)) – input, zero if omitted
 Returns
dx/dt or x[t+dt]
 Return type
ndarray

feedback
(other=1, sign= 1)¶ Feedback interconnection between two LTI systems.

freqresp
(omega)¶ (deprecated) Evaluate transfer function at complex frequencies.

frequency_response
(omega, squeeze=None)¶ Evaluate the linear timeinvariant system at an array of angular frequencies.
Reports the frequency response of the system,
G(j*omega) = mag*exp(j*phase)
for continuous time systems. For discrete time systems, the response is evaluated around the unit circle such that
G(exp(j*omega*dt)) = mag*exp(j*phase).
In general the system may be multiple input, multiple output (MIMO), where m = self.ninputs number of inputs and p = self.noutputs number of outputs.
 Parameters
omega (float or 1D array_like) – A list, tuple, array, or scalar value of frequencies in radians/sec at which the system will be evaluated.
squeeze (bool, optional) – If squeeze=True, remove singledimensional 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’].
 Returns
mag (ndarray) – The magnitude (absolute value, not dB or log10) of the system frequency response. If the system is SISO and squeeze is not True, the array is 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 singledimensional axes are removed.phase (ndarray) – The wrapped phase in radians of the system frequency response.
omega (ndarray) – The (sorted) frequencies at which the response was evaluated.

horner
(x, warn_infinite=True)¶ Evaluate system’s transfer function at complex frequency using Laub’s or Horner’s method.
Evaluates sys(x) where x is s for continuoustime systems and z for discretetime systems.
Expects inputs and outputs to be formatted correctly. Use
sys(x)
for a more userfriendly interface. Parameters
x (complex array_like or complex) – Complex frequencies
 Returns
output – Frequency response
 Return type
(self.noutputs, self.ninputs, len(x)) complex ndarray
Notes
Attempts to use Laub’s method from Slycot library, with a fallback to python code.

isctime
(strict=False)¶ Check to see if a system is a continuoustime system
 Parameters
sys (LTI system) – System to be checked
strict (bool, optional) – If strict is True, make sure that timebase is not None. Default is False.

isdtime
(strict=False)¶ Check to see if a system is a discretetime system
 Parameters
strict (bool, optional) – If strict is True, make sure that timebase is not None. Default is False.

issiso
()¶ Check to see if a system is single input, single output

lft
(other, nu= 1, ny= 1)¶ Return the Linear Fractional Transformation.
A definition of the LFT operator can be found in Appendix A.7, page 512 in the 2nd Edition, Multivariable Feedback Control by Sigurd Skogestad.
An alternative definition can be found here: https://www.mathworks.com/help/control/ref/lft.html
 Parameters
other (LTI) – The lower LTI system
ny (int, optional) – Dimension of (plant) measurement output.
nu (int, optional) – Dimension of (plant) control input.

minreal
(tol=0.0)¶ Calculate a minimal realization, removes unobservable and uncontrollable states

output
(t, x, u=None)¶ Compute the output of the system
Given input u and state x, returns the output y of the statespace system:
y = C x + D u
where A and B are the statespace matrices of the system.
The first argument t is ignored because
StateSpace
systems are timeinvariant. It is included so that the dynamics can be passed to most numerical integrators, such as scipy’s integrate.solve_ivp and for consistency withIOSystem
systems.The inputs x and u must be of the correct length for the system.
 Parameters
t (float (ignored)) – time
x (array_like) – current state
u (array_like (optional)) – input (zero if omitted)
 Returns
y
 Return type
ndarray

pole
()¶ Compute the poles of a state space system.

returnScipySignalLTI
(strict=True)¶ 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. Parameters
strict (bool, optional) –
 True (default):
The timebase ssobject.dt cannot be None; it must be continuous (0) or discrete (True or > 0).
 False:
If ssobject.dt is None, continuous time
scipy.signal.lti
objects are returned.
 Returns
out – continuous time (inheriting from
scipy.signal.lti
) or discrete time (inheriting fromscipy.signal.dlti
) SISO objects Return type
list of list of
scipy.signal.StateSpace

sample
(Ts, method='zoh', alpha=None, prewarp_frequency=None)¶ Convert a continuous time system to discrete time
Creates a discretetime system from a continuoustime system by sampling. Multiple methods of conversion are supported.
 Parameters
Ts (float) – Sampling period
method ({"gbt", "bilinear", "euler", "backward_diff", "zoh"}) –
Which method to use:
gbt: generalized bilinear transformation
bilinear: Tustin’s approximation (“gbt” with alpha=0.5)
euler: Euler (or forward differencing) method (“gbt” with alpha=0)
backward_diff: Backwards differencing (“gbt” with alpha=1.0)
zoh: zeroorder hold (default)
alpha (float within [0, 1]) – The generalized bilinear transformation weighting parameter, which should only be specified with method=”gbt”, and is ignored otherwise
prewarp_frequency (float within [0, infinity)) – 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.
 Returns
sysd – Discrete time system, with sampling rate Ts
 Return type
Notes
Uses
scipy.signal.cont2discrete()
Examples
>>> sys = StateSpace(0, 1, 1, 0) >>> sysd = sys.sample(0.5, method='bilinear')

slycot_laub
(x)¶ Evaluate system’s 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 userfriendly interface. Parameters
x (complex array_like or complex) – Complex frequency
 Returns
output – Frequency response
 Return type
(number_outputs, number_inputs, len(x)) complex ndarray

zero
()¶ Compute the zeros of a state space system.