Page 738 - Mechanical Engineers' Handbook (Volume 2)

3 State-Variable Selection and Canonical Forms 729

Table 2 (Continued)
IV. Normal or diagonal 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) m. Full degeneracy.
Transformation matrices
(i) q Tx, T 1 M [v 1 v 2 v n ]
where (a) v i , i 1, k 1 nad i k m, n, are the linearly independent eigenvectors
corresponding to the real, distinct eigenvalues;
(b) v i , 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
(c) v i , i k, k m 1, are the m linearly independent eigenvectors
corresponding to the repeated eigenvalue. They are equal or proportional to the
nonzero linearly independent columns of
d m 1 [Adj(sI A)]
ds m 1
s s k
(ii) New state matrix TAT 1

s 1
0 0 0
0 s k 1
s k
0 0 0
0 s k
s k m
0 0 0
0 s n

V. Near-normal canonical form
Transformation conditions
(i) A matrix has one pair of complex-conjugate eigenvalues, s k , s k 1
s s js ki
kr
k
kr
s k 1 s js ki
(ii) All other eigenvalues are real and distinct.
Transformation matrices
(i) q Tx, T 1 [v 1 v k 1 v kr v ki v k 2 v n ]
where (a) v i , i 1, k 1 and i k 2, n are the linearly independent eigenvectors
corresponding to the real, distinct eigenvalues;
(b) v i for i 1, .. ., n are taken to be equal or proportional to any nonzero
column of Adj(s i I A); and
(c) v k v kr jv ki , v k 1 v kr jv ki
are the complex-conjugate eigenvectors corresponding to s k and s k 1 ,
respectively.