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). 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 pretty-printing 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 non-MathJax 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 floating-point 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 input-output 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 zero-frequency 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 time-invariant 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 continuous-time system

isdtime([strict])

Check to see if a system is a discrete-time 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 state-space 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 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:

Parameters

warn_infinite (bool, optional) – By default, don’t issue a warning message if the zero-frequency 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 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.

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 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 most numerical integrators, such as scipy.integrate.solve_ivp() and for consistency with IOSystem 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 time-invariant 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 single-dimensional 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 single-dimensional 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 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.

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 fall-back to python code.

isctime(strict=False)

Check to see if a system is a continuous-time 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 discrete-time 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 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 IOSystem 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 from scipy.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 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

  • 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

StateSpace

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 user-friendly 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.