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Fan et al. Complex Eng Syst 2023;3:5 I http://dx.doi.org/10.20517/ces.2023.04 Page 13 of 15

1.4 0.2
ˆ u ∗ ˆ v ∗
1.2 1 1
ˆ u ∗ 2 0.15 ˆ v ∗ 2
1
0.8 0.1
0.6 0.05
Control inputs 0.4 Disturbance inputs 0
0.2
-0.2 0 -0.05
0.2
-0.4 -0.1 0
0
-0.6
-0.15
-0.2
-0.8 22 24 26 28 30
22 24 26 28 30
-1 -0.2
0 10 20 30 40 50 0 10 20 30 40 50
Time (s) Time (s)
Figure 4. Tracking control inputs and disturbance inputs for subsystem 1 and subsystem 2

1.5 1.5
e 11 e 21
1 e 12 1 e 22
0.5 0.5
Tracking errors -0.5 0 Tracking errors -0.5 0

0.2
-1
0 -1 0.2 0
-0.2 -0.2
-1.5 -0.4 -1.5 -0.4
24 26 28 30 32 22 24 26 28 30
-2 -2
-2.5 -2.5
0 10 20 30 40 50 0 10 20 30 40 50
Time (s) Time (s)
Figure 5. Tracking error trajectories for subsystem 1 and subsystem 2.

6. CONCLUSION
In this paper, the optimal DTC strategy for CT nonlinear large-scale systems with external disturbances is
proposed by employing the ADP algorithm. The approximate optimal control law of the ATISs can achieve
the trajectory tracking goal. Then, the establishment of the DTC strategy is derived by adding the appropriate
feedback gain, whose feasibility has been proved via the Lyapunov theory. Note that all the above-mentioned
results are investigated by considering a cost function with the discount. Then, only a series of single critic
networks are employed to solve HJI equations of ATISs, so that we acquire the approximate optimal control
law and the worst disturbance law. In addition, the stability term added in the weight updating process avoids
theselectionoftheinitialstablecontrolpolicy. Furthermore, thesimulationresultsaredisplayedforthespring-
mass-damper system to indicate the validity of the proposed DTC method. In the future, we will utilize more
advanced methods to deal with the DTC problem for nonaffine systems. Besides, we can also consider the
unmatched interconnected relationship for the DTC problem, which is a considerable direction of improved
research.

DECLARATIONS
Authors’ contributions
Made significant contributions to the conception and experiments: Fan W, Wang D
Made significant contributions to the writing: Fan W, Wang D
Made substantial contributions to the revision and translation: Liu A, Wang D