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

2 T
T
where ∈ R 2 ×2 , ∈ R × are both positive definite matrices. Herein, we let − ( ) ( ) =

q
2 2 2
( ) and ( ) ≤ ( ) − ( ), where ( ) > ( ). It is worth noting that this inequality is

employed to prove the feasibility of Theorem 1. Then, Equation (10) can be equivalent to
¹
∞ n T o
2
( 0 ) = − ( − ) ( ) + ( ( )) ( ( )) d . (11)

0
If Equation (11) is continuously differentiable, the nonlinear Lyapunov equation is the infinitely small form of
Equation (11). The Lyapunov equation is as follows:
T
T
2
( ) + ( ) ( ) − ( ) + (∇ ( )) [F ( ) + G ( ) ( ) + H ( ) ( )] = 0. (12)

Define the Hamiltonian of the ith ATIS for the optimization problem as
2 T T
( , , , ∇ ( )) = ( ) + ( ) ( ) − ( ) + (∇ ( ))

× [F ( ) + G ( ) ( ) + H ( ) ( )]. (13)
To acquire the saddle point solution { , }, the local optimal cost function need to satisfy the following Nash
∗
∗

condition
( 0 ) = min max ( 0 ). (14)
∗

Then, the optimal cost function ( ) is derived via solving the local HJI equation in the following:
∗

min max ( , , , ∇ ( )) = 0. (15)
∗

Due to the saddle point solution { , } satisfies the extremum theorem, the optimal tracking control law and
∗
∗

the worst disturbance law can be computed by
1 −1 T
∗
( ) = − G ( )∇ ( ), (16)
∗

2
1 T
∗
∗
( ) = H ( )∇ ( ). (17)

2 2
Substituting the optimal tracking control strategy Equation (16) into Equation (15), the HJI equation for the
th ATIS becomes
−1 T
T
∗
∗ T ∗ 2 ∗ 1 (∇ ( )) G ( ) G ( )∇ ( ) = 0. (18)
∗
(∇ ( )) [F ( ) + H ( ) ( )] + ( ) − ( ) −

4
3.2. Establishment of the DTC strategy design
In the following, we present the DTC strategy by adding the feedback gain to the interconnected system Equa-
tion (5). Herein, the following lemma is given by
Lemma 1 Considering the ATIS Equation (9), the feedback control
∗ (19)
¯ ( ) = ( )

can ensure the ATISs are asymptotically stable as long as ≥ 1/2, which makes the tracking error approach
to zero.