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Fan et al. Complex Eng Syst 2023;3:5 Complex Engineering
DOI: 10.20517/ces.2023.04 Systems

Research Article Open Access

Decentralized tracking control design based on intelli-
gent critic for an interconnected spring-mass-damper
system

Wenqian Fan 1,2,3,4 , Ao Liu 1,2,3,4 , Ding Wang 1,2,3,4

1 Faculty of Information Technology, Beijing University of Technology, Beijing 100124, China.
2 Beijing Key Laboratory of Computational Intelligence and Intelligent System, Beijing University of Technology, Beijing 100124,
China.
3 Beijing Institute of Artificial Intelligence, Beijing University of Technology, Beijing 100124, China.
4 Beijing Laboratory of Smart Environmental Protection, Beijing University of Technology, Beijing 100124, China.

Correspondence to: Prof. Ding Wang, Faculty of Information Technology, Beijing University of Technology, 100 Pingleyuan,
Chaoyang District, Beijing 100124, China. E-mail: dingwang@bjut.edu.cn

How to cite this article: Fan W, Liu A, Wang D. Decentralized tracking control design based on intelligent critic for an interconnected
spring-mass-damper system. Complex Eng Syst 2023;3:5. http://dx.doi.org/10.20517/ces.2023.04

Received: 17 Feb 2023 First Decision: 6 Mar 2023 Revised: 14 Mar 2023 Accepted: 20 Mar 2023 Published: 30 Mar
2023
Academic Editor: Yurong Liu Copy Editor: Fanglin Lan Production Editor: Fanglin Lan

Abstract
In this paper, the decentralized tracking control (DTC) problem is investigated for a class of continuous-time nonlinear
systems with external disturbances. First, the DTC problem is resolved by converting it into the optimal tracking
controller design for augmented tracking isolated subsystems (ATISs). A cost function with a discount is taken into
consideration. Then, in the case of external disturbances, the DTC scheme is effectively constructed via adding the
appropriate feedback gain to each ATIS. In addition, utilizing the approximation property of the neural network, the
critic network is constructed to solve the Hamilton-Jacobi-Isaacs equation, which can derive the optimal tracking
control law and the worst disturbance law. Moreover, the updating rule is improved during the process of weight
learning, which removes the requirement for initial admission control. Finally, through the interconnected spring-
mass-damper system, a simulation example is given to verify the availability of the DTC scheme.

Keywords: Adaptive dynamic programming, discounted cost function, decentralized tracking control, disturbance
rejection, interconnected spring-mass-damper systems, neural networks, optimal control

© The Author(s) 2023. Open Access This article is licensed under a Creative Commons Attribution 4.0
International License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, shar-
ing, adaptation, distribution and reproduction in any medium or format, for any purpose, even commercially, as long as you
give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate
if changes were made.

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