Page 739 - Mechanical Engineers' Handbook (Volume 2)

730 State-Space Methods for Dynamic Systems Analysis

Table 2 (Continued)
(ii) New state matrix TAT 1

s 1
0 0 0
0 s k 1
s kr s ki
0 0
s ki s kr
s k 2
0 0 0
0 s n

VI. Nondiagonal Jordan canonical form
Transformation conditions
(i) A matrix has one repeated, real eigenvalue s k of multiplicity m (i.e., s k s k 1
s k m 1 ). All other eigenvalues are real and distinct.
(ii) Degeneracy d n rank(s k I A) 1. Simple degeneracy.
Transformation matrices
(i) q Tx, T 1 [t 1 t 2 .. . t n ]
where (a) t i , i 1, k 1 and i k m, n, are the linearly independent eigenvectors
corresponding to the real, distinct eigenvalues;
(b) t i , for i 1, k 1 and i k m, n, are taken to be equal or proportional to
any nonzero column of Adj(s i I A); and
1
(c) t i , i k, k m 1, are obtained by solution of the equation AT 1 T J,
where J is the Jordan canonical matrix given. Each t i is determined to within a
constant of proportionality.
(ii) New state matrix J TAT 1
0
s 1

0 s k 1 0 0

s 1 0
k
0 1 0
0 s k

s k m 1 0
0 0
0 s n
VII. Nondiagonal Jordan canonical form
Transformation conditions
(i) A matrix has one repeated, real eigenvalue of multiplicity m (i.e., s k s k 1
s k m 1 ). All other eigenvalues are real and distinct.
(ii) Degeneracy d n rank(s k I A) is between 1 and m. General degeneracy.