# 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