Page 737 - Mechanical Engineers' Handbook (Volume 2)

728 State-Space Methods for Dynamic Systems Analysis

Table 2 Transformation Matrices for Continuous-Time State-Space Canonical Forms
State-space equations (SISO, LTI system) Characteristic equation
˙ x(t) Ax(t) Bu(t) det(sI A)
n
y(t) Cx(t) Du(t) s n 1 s n 1 s 0
1
0
I. Controllable canonical form
Transformation conditions
(i) P c [BAB A n 1 B] must be nonsingular
Transformation matrices
1 1 1
(i) q Tx, T R P c (ii) New state matrix TAT
where
1 2 n 1 0 0 0
1
0

1
3
2
R 1 0 0 I n 1
1 n 1 0 0 0
0 1 n 1
II. Observable canonical form
Transformation conditions

C
CA
(i) P must be nonsingular
0
CA n 1
Transformation matrices
(ii) New state matrix TAT 1
(i) q Tx, T RP 0
where
1 2 n 1 0 00 0 0
1
1

0
2
3
1
R I n 1
n 1 1 0 n 1
1 0 0 0
III. Normal or diagonal Jordan canonical form
Transformation conditions
(i) A matrix has only distinct, real eiganvalues s i , i 1, .. ., n
Transformation matrices
(i) q Tx, T 1 M [v 1 v 2 v n ]
where (a) v i are the n linearly independent eigenvectors corresponding to s i and
(b) v i are taken to be equal or proportional to any nonzero columd of Adj(s i I A)
(ii) New state matrix TAT 1
s 1 s 2 0
0
s n