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Page 2 of 15 Fan et al. Complex Eng Syst 2023;3:5 I http://dx.doi.org/10.20517/ces.2023.04

1. INTRODUCTION
For large-scale nonlinear interconnected systems, which are considered as nonlinear plants consisting of many
interconnected subsystems, decentralized control has become a research hotspot in the last few decades [1–4] .
Compared with the centralized control, the decentralized control has the advantages of simplifying the struc-
ture and reducing the computation burden of the controller. Besides, the local controller only depends on
the information of the local subsystem. Meanwhile, with the development of science and technology, inter-
connected engineering applications have become increasingly complex, such as robotic systems [5] and power
systems [6,7] . In [8–10] , we found that the decentralized control of the large-scale system was connected with the
optimal control of the isolated subsystems, which means the optimal control method [11–14] can be adopted
to achieve the design purpose of the decentralized controllers. However, the optimal control of the nonlin-
ear system often needs to solve the Hamilton-Jacobi-Bellman (HJB) or Hamilton-Jacobi-Isaacs (HJI) equation,
which can be solved by using the adaptive dynamic programming (ADP) method [15,16] . Besides, in [13] , Wang
et al. investigated the latest intelligent critic framework for advanced optimal control. In [14] , the optimal
feedback stabilization problem was discussed with discounted guaranteed cost for nonlinear systems. It fol-
lows that the interconnection plays a significant role in designing the controller. Hence, it can be classified
as decentralized and distributed control schemes. There is a certain distinction between decentralized control
and distributed control. For decentralized control, each sub-controller only uses local information and the
interconnection among subsystems can be assumed to be weak in nature. Compared with the decentralized
control, the distributed control [17–19] can be introduced to improve the performance of the subsystems when
the interconnections among subsystems become strong. In [20] , the distributed optimal observer was devised
to assess the nonlinear leader state for all followers. In [21] , the distributed control was developed by means of
online reinforcement learning for interconnected systems with exploration.

It is worth mentioning that the ADP algorithm has been extensively employed for dealing with various opti-
mal regulation problems and tracking problems [22–24] , which will achieve the goal, that is, the actual signal can
track the reference signal under the noisy and the uncertain environment. In [25] , Ha et al. proposed a novel
cost function to explore the evaluation framework of the optimal tracking control problem. Then, aimed at
complicated control systems, it is necessary to consider decentralized tracking control (DTC) problems [26–29] .
The DTC systems can be transformed into the the nominal augmented tracking isolated subsystems (ATISs),
which are composed of the tracking error and the reference signal. In [26] , Qu et al. proposed a novel formu-
lation consisting of a steady-state controller and a modified optimal feedback controller of the DTC strategy.
Besides, the asymptotic DTC was realized by introducing two integral bounded functions in [27] . In [28] , Liu et
al. proposed a finite-time DTC method for a class of nonstrict feedback interconnected systems with distur-
bances. Moreover, the adaptive fuzzy output-feedback DTC design was investigated for switched large-scale
systems in [29] .

Game theory is a discipline that implements corresponding strategies. It contains cooperative and noncooper-
ative types, that is, zero-sum (ZS) games and non-ZS games. In particular, ZS games have been widely applied
in many fields [30–33] . The object of the ZS game is to derive the Nash equilibrium of nonliner systems, which
makes the cost function optimized. In [31] , the finite-horizon H-infinity state estimator design was studied for
periodic neural networks over multiple fading channels. The noncooperative control problem was formulated
as a two-player ZS game in [32] . In [33] , Wang et al. investigated the stability of the general value iteration al-
gorithm for ZS games. At the same time, we can also combine the ZS problem with the tracking problem to
make the system more stable while achieving the trajectory tracking. In [34] , Zhang et al. developed an online
model-free integral reinforcement learning algorithm for solving the H-infinity optimal tracking problem for
completely unknown systems. In [35] , a general bounded 2 gain tracking policy was introduced with a dis-
counted function. In [36] , Hou et al. proposed an action-disturbance-critic neural network frame to realize the
iterative dual heuristic programming algorithm.