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.Discretetime state space system are implemented by using the ‘dt’ instance variable and setting it to the sampling period. If ‘dt’ is not None, then it must match whenever two state space systems are combined. Setting dt = 0 specifies a continuous system, while leaving dt = None means the system timebase is not specified. If ‘dt’ is set to True, the system will be treated as a discrete time system with unspecified sampling time. The default value of ‘dt’ is None and can be changed by changing the value of
control.config.defaults['statesp.default_dt']
.
__init__
(*args, **kw)¶ 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.
Methods
__init__
(*args, **kw)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
()Return the zerofrequency gain
evalfr
(omega)Evaluate a SS system’s transfer function at a single frequency.
feedback
([other, sign])Feedback interconnection between two LTI systems.
freqresp
(omega)Evaluate the system’s transfer function at a list of frequencies
horner
(s)Evaluate the systems’s transfer function for a complex variable
True if and only if the system has no dynamics, that is, if A and B are zero.
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
pole
()Compute the poles of a state space system.
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
zero
()Compute the zeros of a state space system.

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
()¶ Return the zerofrequency gain
The zerofrequency gain of a continuoustime statespace system is given by:
and of a discretetime statespace system by:
 Returns
gain – An array of shape (outputs,inputs); the array will either be the zerofrequency (or DC) gain, or, if the frequency response is singular, the array will be filled with np.nan.
 Return type
ndarray

evalfr
(omega)¶ Evaluate a SS system’s transfer function at a single frequency.
self._evalfr(omega) returns the value of the transfer function matrix with input value s = i * omega.

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

freqresp
(omega)¶ Evaluate the system’s transfer function at a list of frequencies
Reports the frequency response of the system,
G(j*omega) = mag*exp(j*phase)
for continuous time. For discrete time systems, the response is evaluated around the unit circle such that
G(exp(j*omega*dt)) = mag*exp(j*phase).
 Parameters
omega (array_like) – A list of frequencies in radians/sec at which the system should be evaluated. The list can be either a python list or a numpy array and will be sorted before evaluation.
 Returns
mag ((self.outputs, self.inputs, len(omega)) ndarray) – The magnitude (absolute value, not dB or log10) of the system frequency response.
phase ((self.outputs, self.inputs, len(omega)) ndarray) – The wrapped phase in radians of the system frequency response.
omega (ndarray) – The list of sorted frequencies at which the response was evaluated.

horner
(s)¶ Evaluate the systems’s transfer function for a complex variable
Returns a matrix of values evaluated at complex variable s.

is_static_gain
()¶ True if and only if the system has no dynamics, that is, if A and B are zero.

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

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

returnScipySignalLTI
()¶ 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.

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')

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