IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 7, NO. 4, JULY 2020 919 Parallel Control for Continuous-Time Linear Systems: A Case Study Qinglai Wei, Member, IEEE, Hongyang Li, and Fei-Yue Wang, Fellow, IEEE Abstract—In this paper, a new parallel controller is developed for continuous-time linear systems. The main contribution of the method is to establish a new parallel control law, where both state and control are considered as the input. The structure of the parallel control is provided, and the relationship between the parallel control and traditional feedback controls is presented. Considering the situations that the systems are controllable and incompletely controllable, the properties of the parallel control law are analyzed. The parallel controller design algorithms are given under the conditions that the systems are controllable and incompletely controllable. Finally, numerical simulations are carried out to demonstrate the effectiveness and applicability of the present method. Index Terms—Continuous-time linear systems, digital twin, parallel controller, parallel intelligence, parallel systems. I. I NTRODUCTION O VER the past decades, with the rapid development of science and technology, control theory and technology are playing increasingly important roles. The development of control theory has generally gone through three stages: classical control theory, modern control theory, and intelligent control theory [1]. Based on frequency domain analysis, the classical control theory mainly solves the control problems of single input single output linear time-invariant systems. Based on state space description, the modern control theory mainly solves the control problems of multi-input and multi- output systems. Comparing with classical control theory, the modern control theory is more suitable for the analysis of time-varying nonlinear systems. The typical modern control theory includes optimal control [2], adaptive control [3] and so on [4], [5]. With the increase of complexity and nonlinearity Manuscript received May 8, 2020; accepted June 9, 2020. This work was supported in part by the National Key Research and Development Program of China (2018AAA0101502, 2018YFB1702300) and the National Natural Science Foundation of China (61722312, 61533019, U1811463, 615330 17). Recommended by Associate Editor Jun Zhang. (Corresponding author: Qinglai Wei.) Citation: Q. L. Wei, H. Y. Li, and F.-Y. Wang, “Parallel control for continuous-time linear systems: A case study,” IEEE/CAA J. Autom. Sinica, vol. 7, no. 4, pp. 919-928, Jul. 2020. Q. L. Wei and H. Y. Li are with the State Key Laboratory of Manage- ment and Control for Complex Systems, Institute of Automation, Chinese Academy of Sciences, Beijing 100190, and with the University of Chinese Academy of Sciences, Beijing 100049, and also with Qingdao Academy of Intelligent Industries, Qingdao 266109, China (e-mail: qinglai.wei@ia.ac.cn; lihongyang2019@ia.ac.cn). F.-Y. Wang is with the State Key Laboratory of Management and Control for Complex Systems, Institute of Automation, Chinese Academy of Sciences, Beijing 100190, and with the Institute of Systems Engineering, Macau University of Science and Technology, and also with Qingdao Academy of Intelligent Industries, Qingdao 266109, China (e-mail: feiyue.wang@ia.ac.cn). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JAS.2020.1003216 of industrial control systems, the intelligent control theory, such as fuzzy control [6], neural network control [7], adaptive dynamic programming [8], [9], is attracted by researchers. Among these previous stages, most system control problems are analyzed by state feedback control methods in present study: we generally design state feedback controllers to form closed-loop systems, that is, the control laws are functions of the system states. However, the state feedback controllers have some disadvantages: 1) The traditional state feedback controllers are only related to the system states rather than the properties of the controllers and it causes that the control signals may change greatly with the system states, which brings great difficulty to the execution of the controllers. 2) The control signals are generated passively, and it is difficult to generate control signals under the condition that the system states have no changes or the system states cannot be obtained. 3) The structure of the state feedback controllers is onefold, which forces the system into a closed-loop one. It causes difficulties in performance improvements of the systems. Therefore, it is necessary to build a new type of controller to overcome the above problems. Parallel control theory, proposed by Wang [1], [10], [11], is an effective method to obtain the control laws of the control systems [12]-[16]. The basic structure of parallel systems is shown in Fig. 1. The basic idea of parallel control is expanding the practical problems into virtual space, then the control tasks can be realized by means of virtual-reality interaction. To be specific, parallel control is the application of ACP (Ar- tificial systems, computational experiments, parallel execution) theory [12] in control theory, where artificial systems (A) are used for modeling the physical systems, computational exper- iments (C) are used for analysis, evaluation and learning, and parallel executions (P) are utilized for control, management, and optimization. Comparing with parallel systems, a similar concept is digital twins. The parallel systems and digital twins manage and control systems which are difficult to analyze with mathematical models by establishing the virtual systems cor- responding to physical systems [17]. However, there are some differences between parallel systems and digital twins. The research objects of digital twins are cyber-physical systems (CPS) which are composed of information space and physical space. And parallel systems mainly focus on cyber-physical- social systems (CPSS) which refer to the deep integration of social networks, information resources, and physical space. In addition to the research objects, there are certain differences in core ideas, frameworks, mathematical descriptions, implemen-