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,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’,’freq’,’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