lcapy.statespace.StateSpace

class lcapy.statespace.StateSpace(A, B, C, D, u=None, y=None, x=None, x0=None)

Bases: StateSpaceBase

Continuous-time linear time-invariant state space model.

Methods

characteristic_polynomial()	Characteristic polynomial (aka system polynomial).
discretize([method, alpha])	Convert to a discrete-time state space approximation.
from_circuit(cct[, node_voltages, ...])
generalized_bilinear_transform([alpha])
output_equations()	System of output equations:
state_equations()	System of first-order differential state equations:

Attributes

G	System transfer functions.
H	X(s) / U(s)
Lambda	Diagonal matrix of eigenvalues.
M	Modal matrix (eigenvectors of A).
P	Characteristic polynomial (aka system polynomial).
Phi	s-domain state transition matrix.
U	Laplace transform of input vector.
Wc	Controllability gramian matrix.
Wo	Observability gramian matrix.
Wr	Reachability gramian matrix.
X	Laplace transform of state-variable vector.
Y	Laplace transform of output vector.
controllability_gramian	Controllability gramian matrix.
dotx	Time derivative of state variable vector.
g	System impulse responses.
h	ILT{X(s) / U(s)}
observability_gramian	Observability gramian matrix.
phi	State transition matrix.
reachability_gramian	Reachability gramian matrix.
u	Input vector.
x	State variable vector.
x0	State variable initial value vector.
y	Output vector.

property G

System transfer functions. For a SISO system, use G[0].

property H

X(s) / U(s)

property Lambda

Diagonal matrix of eigenvalues.

property M

Modal matrix (eigenvectors of A).

property P

Characteristic polynomial (aka system polynomial).

lambda(s) = |s * I - A|

property Phi

s-domain state transition matrix.

property U

Laplace transform of input vector.

property Wc

Controllability gramian matrix.

property Wo

Observability gramian matrix.

property Wr

Reachability gramian matrix.

property X

Laplace transform of state-variable vector.

property Y

Laplace transform of output vector.

characteristic_polynomial()

Characteristic polynomial (aka system polynomial).

lambda(s) = |s * I - A|

property controllability_gramian

Controllability gramian matrix.

discretize(method='bilinear', alpha=0.5)

Convert to a discrete-time state space approximation.

The default method is ‘bilinear’. Other methods are ‘forward_euler’, ‘backward_euler’, and ‘gbf’. The latter has a parameter alpha.

property dotx

Time derivative of state variable vector.

property g

System impulse responses.

property h

ILT{X(s) / U(s)}

property observability_gramian

Observability gramian matrix.

output_equations()

System of output equations:

y(t) = C x(t) + D u(t)

where y is the output vector, x is the state vector and u is the input vector.

property phi

State transition matrix.

property reachability_gramian

Reachability gramian matrix. This is equivalent to the controllability gramian matrix for a linear time independent system.

state_equations()

System of first-order differential state equations:

dotx(t) = A x(t) + B u(t)

where x is the state vector and u is the input vector.

property u

Input vector.

property x

State variable vector.

property x0

State variable initial value vector.

property y

Output vector.