control.StateSpace

class control.StateSpace(A, B, C, D[, dt])

A class for representing state-space models

The StateSpace class is used to represent state-space realizations of linear time-invariant (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).

Discrete-time 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.

__init__(*args)

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) StateSpace(A, B, C, D[, dt])
append(other) Append a second model to the present model.
damp()
dcgain() Return the zero-frequency 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 func.
horner(s) Evaluate the systems’s transfer function for a complex variable
isctime([strict]) Check to see if a system is a continuous-time system
isdtime([strict]) Check to see if a system is a discrete-time system
issiso()
minreal([tol]) 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.
sample(Ts[, method, alpha]) 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 state-space if necessary, inputs and outputs are appended and their order is preserved

dcgain()

Return the zero-frequency gain

The zero-frequency gain of a continuous-time state-space system is given by:

and of a discrete-time state-space system by:

Returns:gain – An array of shape (outputs,inputs); the array will either be the zero-frequency (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 func. at a list of freqs, omega.

mag, phase, omega = self.freqresp(omega)

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).
omega: 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 (The magnitude (absolute value, not dB or log10) of the system) – frequency response.
  • phase (The wrapped phase in radians of the system frequency) – response.
  • omega (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.

isctime(strict=False)

Check to see if a system is a continuous-time system

Parameters:
  • sys (LTI system) – System to be checked
  • strict (bool (default = False)) – If strict is True, make sure that timebase is not None
isdtime(strict=False)

Check to see if a system is a discrete-time system

Parameters:strict (bool (default = False)) – If strict is True, make sure that timebase is not None
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 signal.scipy.lti object corresponding to the transfer function from the 6th input to the 4th output.

sample(Ts, method='zoh', alpha=None)

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.

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: zero-order 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
Returns:

sysd – Discrete time system, with sampling rate Ts

Return type:

StateSpace system

Notes

Uses the command ‘cont2discrete’ from scipy.signal

Examples

>>> sys = StateSpace(0, 1, 1, 0)
>>> sysd = sys.sample(0.5, method='bilinear')
zero()

Compute the zeros of a state space system.