Page 747 - Mechanical Engineers' Handbook (Volume 2)

738 State-Space Methods for Dynamic Systems Analysis

Table 4 (Continued)
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(z i I F); 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 F)]
dz m 1
z z k
(ii) New state matrix TFT 1

z 1
0 0 0
0 z k 1
z k
0 0 0
0 z k
z k m
0 0 0
0 z n
V. Near-normal canonical form
Transformation conditions
(i) F matrix has one pair of complex-conjugate eigenvalues, z k , z k 1
kr
k
z z jz ki
z k 1 z jz ki
kr
(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(z i I F); and
(c) v k v kr jv ki , v k 1 v kr jv ki
are the complex-conjugate eigenvectors corresponding to z k and z k 1 ,
respectively.
(ii) New state matrix TFT 1

z 1
0 0 0
0 z k 1
z kr z ki
0 0
z ki z kr
z k 2
0 0 0
0 z n