In this section, our objective is to explicitly characterize the observability criteria of complex-valued -matrix impulsive system (1). First, a claim is presented : For complex matrix A, there exist scalar functions such that
From system (1) and Lemma 1, we can get the output
where . Rewrite (14) and (15)
It is easy to see, from Definition 2, that the observability of system (1) is equivalent to the estimation of from . We denote the block matrices as follows:
Now we present the sufficient and necessary condition for the observability of system (1).
Theorem 3 System (1) is observable on () if and only if complex block matrix defined in (17) is invertible.
Proof Multiplying both sides of (16), respectively, by and
from left and integrating with respect to t from to yield that
It is easy to see that the left-hand side of (18) depends on , . So, if is invertible, then the initial state is uniquely determined by the corresponding complex system output and input for .
Next we consider the necessary part. If the complex matrix is not invertible, then there exists a nonzero vector such that . Since () and are positive semidefinite matrices, we have
If let initial state , namely, each column of is , then
it follows from (16) and (17) that
which implies that . Thus by (16),
From Definition 2, system (1) is not observable on (). This contradicts the assumption of observability. This completes the proof. □
For system (1), when , , , , the complex impulsive system becomes a complex linear time-invariant impulsive system. We have a more concise result than Theorem 3. Denote
where , .
Theorem 4 If complex -matrix impulsive system (1) has complex constant coefficient matrices A, B, C, D, then the following conclusions hold.
If , then complex -matrix linear impulsive system (1) is observable on ().
Assume that , . If complex -matrix system (1) is observable, then .
Proof (i) If while complex -matrix system (1) is not observable, then by Theorem 3 the matrix is not invertible, which implies that there exists a nonzero vector satisfying . Since the matrices are non-negative definite, we obtain
This shows that
Clearly, when , we have . Differentiating (21) j times and evaluating the results at yield that
Hence we deduce that for . It follows that , which leads to a contradiction with the assumption that . The proof of part (i) is completed.
If otherwise, assume that complex impulsive system (1) is observable while , then there exists a vector satisfying which reduces from (20) to
where . From (23), (13) and the fact that , we obtain
So . Because , the matrix is not invertible. Hence complex -matrix impulsive system (1) is not observable from Theorem 3, and it contradicts the assumption of observability. This completes the proof. □