Page 746 - Mechanical Engineers' Handbook (Volume 2)

4 Solution of System Equations 737

Table 4 Transformation Matrices for Discrete-Time State-Space Canonical Forms
State-space equations (SISO system) Characteristic equation
x(k 1) Fx(k) Gu(k) det(zI F)
n
y(k) Cx(k) Du(k) z n 1 z n 1 1 z 0 0
I. Controllable canonical form
Transformation conditions
(i) P c [GFG F n 1 G] must be nonsingular
Transformation matrices
1 1 (ii) New state matrix TFT 1
(i) q Tx, T R P c
where
1 2 n 1 0 0 0 0
1

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

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

3
2
1
R 1 0 I n 1
1 n 1 0 0 0 n 1
III. Normal or diagonal Jordan canonical form
Transformation conditions
(i) F matrix has only distinct, real eiganvalues z 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 z i and
(b) v i are taken to be equal or proportional to any nonzero column of Adj(z i I F)
(ii) New state matrix TFT 1
z 1 z 2 0

0 z n
IV. Normal or diagonal Jordan canonical form
Transformation conditions
(i) F matrix has one repeated, real eigenvalue z k of multiplicity m (i.e., z k z k 1 .. .
z k m 1 ). All other eigenvalues are real and distinct.
(ii) Degeneracy d n rank(z k I F) m. Full degeneracy.