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Page 6 of 15 Fan et al. Complex Eng Syst 2023;3:5 I http://dx.doi.org/10.20517/ces.2023.04

∗
∗
Proof. The lemma can be proved by showing ( ) is a candidate Lyapunov function. We can find ( ) ≥ 0

in Equation (11), which implies that ( ) is a positive definite function. The derivative of ( ) along with
∗
∗

the th ATIS is given by
T
¤ ∗ ∗
( ) = (∇ ( )) ¤

T
∗
= (∇ ( )) [F ( ) + G ( ) ¯ ( ) + H ( ) ( )]. (20)

Substituting Equations (18) and (19) into Equation (20), we can rewrite it as
1 T −1 T
2
∗
∗
∗
∗
¤
( ) = − ( ) + ( ) + (∇ ( )) G ( ) G ( )∇ ( )

4
1 T −1 T
∗
∗
− (∇ ( )) G ( ) G ( )∇ ( )

2

2
1 1
1
2 ∗ − 2 T ∗

= − ( ( ) − ( )) − −

G ( )∇ ( )
. (21)

2 4
Observing Equation (21), we can obtain that ( ) < 0 holds under the condition ( ) > ( ) for all
2
¤ ∗
∗

≥ 1/2 and ≠ 0. Thus, the conditions are satisfied for Lyapunov local stability theory and the actual state of
each ATIS can realize desired tracking objectives under the feedback control strategy. The proof is completed.
Remark 1. It is worth mentioning that only when = 1, the feedback control is optimal. Then, we will show the
following theorem to verify the proposed control law can effectively establish the DTC strategy.
Theorem 1 Taking Equation (2) and the interconnected augmented tracking system Equation (5) into account,
there exist positive numbers , such that, for any > , the feedback control polices given by Equa-
∗
∗

tion (19) guarantee that the interconnected tracking system can maintain the asymptotic stability. In other
words, the control pair ¯ 1 ( 1 ), ¯ 2 ( 2 ), . . . , ¯ ( ) is the DTC strategy for the large-scale system.
Proof. Inspired by Lemma 1, we observe that ( ) is the Lyapunov function. Therefore, a composite Lya-
∗

punov function of ( ) is chosen as
∗

Õ
∗
L( ) = ( ), (22)

=1
where is a random positive constant. Taking the time derivative of L( ), we have

Õ
∗
L( ) = ( )
¤
¤

=1

Õ n T
∗
= (∇ ( )) [F ( ) + G ( ) ¯ ( ) + H ( ) ( )]

=1
o
T
¯
+ (∇ ( )) G ( ) ( ) . (23)
∗

q
2
2
Considering Equation (2), the mentioned inequality ( ) ≤ ( ) − ( ), where ( ) > ( ), and

Equation (21), the upper formula can be converted to

2
Õ 1 1
1
¤ 2 ∗ T − 2
L( ) ≤ − ( ) − ( ) + −
(∇ ( )) G ( )

=1 2 4

q

T − 1
Õ
2
∗
− (∇ ( )) G ( ) 2
( ) − ( ) . (24)

=1