Page 758 - Mechanical Engineers' Handbook (Volume 2)

6 Controllability and Observability 749

More general observability conditions for LTI systems are stated in terms of a matrix
P referred to as the observability matrix and are summarized in Table 9. The np n
0
observability matrix is defined by

C
CA
P (65)
0
CA n 1
for continuous-time systems and by

C
CF
P (66)
0
CF n 1
for discrete-time systems. The condition for complete observability is simply that the matrix
P have rank n. The observable canonical form for SISO systems, described in Section 3,
0
derives its name from the fact that transformation to that form is possible if and only if the
system is observable. The transformation matrix T in Tables 2 and 4, required to transform
the state-space equations for a SISO system to the observable canonical form, exists if and
only if the n n matrix P is nonsingular, that is, the system is observable. Equivalent
0
observability conditions which are simpler to evaluate than the one stated previously are also
listed in Table 9 along with a necessary (but not sufficient) condition for complete observ-
ability. It should be noted that the observability conditions are independent of time for time-

Table 9 Observability Conditions for Linear Dynamic Systems
Continuous Time Discrete Time
Time-Invariant System
T
T n 1
T n 1
T
Necessary and (i) rank[C T A C T (A ) C ] (i) rank[C T F C T (F ) C ]
T
T
T
T
sufficient rank[C T A C T (A ) C ] rank[C T F C T (F ) C ]
T n p
T n p
T
T
condition for n n
observability or or
T P 0 ) 0 T P 0 ) 0
(ii) det(P 0 (ii) det(P 0
T
T
Necessary rank(C A ) n rank(C F ) n
T
T
condition
Time-Varying Systems:
Necessary and Observable at t 0 if and only if there Observable at k 0 if and only if there
sufficient exists a finite time t 1 , t 1 t 0 such that exists a finite time k 1 , k 1 k 0 such
condition for (i) W 0 (t 1 , t 0 ) is positive definite or that
observability (ii) zero is not an eigenvalue of W 0 (t 1 , (i) W 0 (k 1 , k 0 ) is positive definite or
t 0 )or (ii) zero is not an eigenvalue of W 0 (k 1 ,
(iii) W 0 (t 1 , t 0 ) 0 k 0 )or
where (iii) W 0 (k 1 , k 0 ) 0
W 0 (t 1 , t 0 ) where
W 0 (k 1 , k 0 )
k 1

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