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Page 4 of 15 Fan et al. Complex Eng Syst 2023;3:5 I http://dx.doi.org/10.20517/ces.2023.04

will not reach zero at the same time. For this reason, ≥ ( )/ ( ) holds, where is also the
nonnegative constant.

In this paper, considering the nonlinear system Equation (1), a reference system is introduced as follows:

¤ ( ) = ( ( )), (3)

where ( ) ∈ R denotes the desired trajectory with (0) = 0, the function is locally Lipschitz continuous

satisfying (0) = 0. For the th subsystem, the trajectory tracking error can be defined as ( ) = ( ) − ( )
with (0) = 0. Thus, the dynamics of the tracking error is

¯
¤ ( ) = ( ( )) + ( ( )) ¯ ( ( )) + ( ( )) + ℎ ( ( )) ( ( )) − ( ( )). (4)
T T T
Noticing ( ) = ( ) + ( ), we define the augmented subsystem states as ( ) = [ ( ), ( )] ∈ R 2

T T
T
with (0) = 0 = [ , ] . Hence, the dynamic of the th ATIS based on Equations (1) and (3) can be
0 0
formulated as a concise form

¯
¤ ( ) = F ( ( )) + G ( ( )) ¯ ( ( )) + ( ( )) + H ( ( )) ( ( )), (5)
where F ( ( )) ∈ R , G ( ( )) ∈ R 2 × , and H ( ( )) ∈ R 2 × respectively. Specifically, they can be
2
expressed as

( ( ) + ( )) − ( ( ))
F ( ( )) = , (6)
( ( ))

( ( ) + ( ))
G ( ( )) = , (7)
0 ×

ℎ ( ( ) + ( ))
H ( ( )) = . (8)
0 ×
We aim to design a pair of decentralized control policies ¯ 1 , ¯ 2 , . . . , ¯ to ensure that large-scale system Equa-
tion (1) can track the desired object while being restricted by external disturbances. It means that as → +∞,
k ( ) − ( )k → 0. Meanwhile, it is noteworthy that the control pair ¯ 1 , ¯ 2 , . . . , ¯ should be pointed out
only as a corresponding controller with the local information. In what follows, it presents the DTC problem
by transforming it into the optimal controller design of ATISs by considering an appropriate discounted cost
function.

3. DTC DESIGN VIA OPTIMAL REGULATION
3.1. Optimal control and the HJI equations
In this section, the optimal DTC strategy of the ATIS with the disturbance rejection is elaborated. It is ad-
dressed by solving the HJI equation with a discounted cost function. Then, we consider the nominal part of
the augmented system Equation (5) as
¤ ( ) = F ( ( )) + G ( ( )) ( ( )) + H ( ( )) ( ( )). (9)

We assume that F +G +H is Lipschitz continuous on a set Ω ⊂ R , which is commonly used in the field
2
of adaptive critic control to ensure the existence and uniqueness of the solution for the differential equation.
Related to the th ATIS, we manage to minimize and maximize the discounted cost function as

¹
∞ n o
2 T
T
T
( 0 ) = − ( − ) ( ) ( ) + ( ( )) ( ( )) − ( ( )) ( ( )) d , (10)

0