eigen_dec

Eigen decomposition of state-space model (either discrete or continuous time) defined by the A,B,C matrices. Eigen-states are organised in decreasing damping order or increased frequency order such that the truncation

A[:N,:N], B[:N,:], C[:,:N]

will retain the least N damped (or lower frequency) modes.

If the eigenvalues of A, eigs, are complex, the state-space is automatically convert into real by separating its real and imaginary part. This procedure retains the minimal number of states as only 2 equations are added for each pair of complex conj eigenvalues. Extra care is however required when truncating the system, so as to ensure that the chosen value of N does not retain the real part, but not the imaginary part, of a complex pair.

For this reason, the function also returns an optional output, Nlist, such that, for each N in Nlist, the truncation

A[:N,:N], B[:N,:], C[:,:N]

does guarantee that both the real and imaginary part of a complex conj pair is included in the truncated model. Note that if `order_by == None, the eigs and UR must be given in input and must be such that complex pairs are stored consecutively.

param A:

state-space matrix

param B:

state-space matrix

param C:

matrices of state-space model

param dlti:

specifies whether discrete (True) or continuous-time. This information is only required to order the eigenvalues in decreasing dmaping order

param N:

number of states to retain. If None, all states are retained

param eigs:

eigenvalues and right eigenvector of A matrix as given by: eigs,Ur=scipy.linalg.eig(A,b=None,left=False,right=True)

param Ur:

eigenvalues and right eigenvector of A matrix as given by: eigs,Ur=scipy.linalg.eig(A,b=None,left=False,right=True)

param Urinv:

inverse of Ur

param order_by={‘damp’:

order according to increasing damping (damp)

param ‘freq’:

order according to increasing damping (damp)

param ‘stab’}:

order according to increasing damping (damp)

param or decreasing frequency:

If None, the same order as eigs/UR is followed.

type or decreasing frequency:

freq) or decreasing damping (stab

param tol:

absolute tolerance used to identify complex conj pair of eigenvalues

param complex:

if true, the system is left in complex form

Returns: (Aproj,Bproj,Cproj): state-space matrices projected over the first N (or N+1

if N removes the imaginary part equations of a complex conj pair of eigenvalues) related to the least damped modes

Nlist: list of acceptable truncation values