Page 756 - Mechanical Engineers' Handbook (Volume 2)

6 Controllability and Observability 747

matrix for continuous-time systems or in the transformed G matrix for discrete-time systems. 4
The presence of a zero row in either of these matrices would indicate that the inputs are not
coupled to and cannot control the corresponding mode. Algebraic controllability conditions
for systems with repeated eigenvalues are given by Palm. 6
The controllability conditions for LTI systems in general are stated in terms of the matrix
P , referred to as the controllability matrix in Section 3, and are summarized in Table 8. The
c
n nr controllability matrix for a MIMO system is defined by
P [BAB A n 1 B] (63)
c
for continuous-time systems and by
P [GFG F n 1 G] (64)
c
for discrete-time systems. The condition for complete state controllability is simply that the
matrix P has rank n. The controllable canonical form for SISO systems, described in Section
c
3, derives its name from the fact that transformation to that form is possible if and only if

Table 8 Controllability Conditions for Linear Dynamic Systems
Continuous Time Discrete Time
Time-Invariant System
Necessary and efficient (i) rank(BAB A n 1 B) (i) rank(GFG F n 1 G)
condition for state rank (BAB A n r B) rank (GFG F n r G)
controllability n n
or or
T
(ii) det(P c P ) 0 (ii) det(P c P ) 0
T
c c
Necessary condition for rank(BA) n rank(GF) n
state controllability
Necessary and sufficient (i) rank(CP c ) p (i) rank(CP c ) p
condition for output or or
T
controllability (ii) det(CP c P c T C ) 0 (ii) det(CP c P c T C ) 0
T
Time-Varying System:
Time Interval of Interest [t 0 , t 1 ]or [k 0 , k 1 ]
Necessary and sufficient (i) W c (t 1 , t 0 ) is positive definite (i) W c (k 1 , k 0 ) is positive definite
condition for state or or
controllability (ii) Zero is not an eigenvalue of (ii) Zero is not an eigenvalue of
W c (t 1 , t 0 )or W c (k 1 , k 0 )or
(iii) W c (t 1 , t 0 ) 0 (iii) W c (k 1 , k 0 ) 0
where W c (t 1 , t 0 ) where W c (k 1 , k 0 )
k 1
t 1

T
T
(t , )B( )B ( ) (k , k)G(k)G (k)
1 1
t 0
k k 0
T
(t , ) d (k , k)
T
1
1
Necessary and sufficient det W y (t 1 , t 0 ) 0 det W y (k 1 , k 0 ) 0
condition for output where W y (t 1 , t 0 ) where W y (k 1 , k 0 )
controllability
k 1
t 1

C( ) (t , )B( )B ( ) C(k) (k , k)B(k)
T
1
1
t 0 k k 0
T
T
(t , )C ( ) d B (k) (k , k)C (k)
T
T
T
1 1