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 of industrial control systems, the intelligent control theory, for continuous-time linear systems. The main contribution of the such as fuzzy control [6], neural network control [7], adaptive method is to establish a new parallel control law, where both dynamic programming [8], [9], is attracted by researchers. state and control are considered as the input. The structure of the parallel control is provided, and the relationship between the Among these previous stages, most system control problems parallel control and traditional feedback controls is presented. are analyzed by state feedback control methods in present Considering the situations that the systems are controllable and study: we generally design state feedback controllers to form incompletely controllable, the properties of the parallel control closed-loop systems, that is, the control laws are functions of law are analyzed. The parallel controller design algorithms are the system states. However, the state feedback controllers have given under the conditions that the systems are controllable and incompletely controllable. Finally, numerical simulations are some disadvantages: carried out to demonstrate the effectiveness and applicability of 1) The traditional state feedback controllers are only related the present method. to the system states rather than the properties of the controllers Index Terms—Continuous-time linear systems, digital twin, and it causes that the control signals may change greatly with parallel controller, parallel intelligence, parallel systems. the system states, which brings great difficulty to the execution of the controllers. 2) The control signals are generated passively, and it is I. INTRODUCTION difficult to generate control signals under the condition that VER the past decades, with the rapid development of the system states have no changes or the system states cannot O science and technology, control theory and technology be obtained. are playing increasingly important roles. The development 3) The structure of the state feedback controllers is onefold, of control theory has generally gone through three stages: which forces the system into a closed-loop one. It causes classical control theory, modern control theory, and intelligent difficulties in performance improvements of the systems. control theory [1]. Based on frequency domain analysis, the Therefore, it is necessary to build a new type of controller classical control theory mainly solves the control problems to overcome the above problems. of single input single output linear time-invariant systems. Parallel control theory, proposed by Wang [1], [10], [11], is Based on state space description, the modern control theory an effective method to obtain the control laws of the control mainly solves the control problems of multi-input and multi- systems [12]¡[16]. The basic structure of parallel systems is output systems. Comparing with classical control theory, the shown in Fig. 1. The basic idea of parallel control is expanding modern control theory is more suitable for the analysis of the practical problems into virtual space, then the control tasks time-varying nonlinear systems. The typical modern control can be realized by means of virtual-reality interaction. theory includes optimal control [2], adaptive control [3] and so To be specific, parallel control is the application of ACP (Ar- on [4], [5]. With the increase of complexity and nonlinearity tificial systems, computational experiments, parallel execution) theory [12] in control theory, where artificial systems (A) are Manuscript received May 8, 2020; accepted June 9, 2020. This work was supported in part by the National Key Research and Development Program used for modeling the physical systems, computational exper- of China (2018AAA0101502, 2018YFB1702300) and the National Natural iments (C) are used for analysis, evaluation and learning, and Science Foundation of China (61722312, 61533019, U1811463, 615330 parallel executions (P) are utilized for control, management, 17). Recommended by Associate Editor Jun Zhang. (Corresponding author: Qinglai Wei.) and optimization. Comparing with parallel systems, a similar Citation: Q. L. Wei, H. Y. Li, and F.-Y. Wang, “Parallel control for concept is digital twins. The parallel systems and digital twins continuous-time linear systems: A case study,” IEEE/CAA J. Autom. Sinica, manage and control systems which are difficult to analyze with vol. 7, no. 4, pp. 919¡928, Jul. 2020. Q. L. Wei and H. Y. Li are with the State Key Laboratory of Manage- mathematical models by establishing the virtual systems cor- ment and Control for Complex Systems, Institute of Automation, Chinese responding to physical systems [17]. However, there are some Academy of Sciences, Beijing 100190, and with the University of Chinese differences between parallel systems and digital twins. The Academy of Sciences, Beijing 100049, and also with Qingdao Academy of Intelligent Industries, Qingdao 266109, China (e-mail: [email protected]; research objects of digital twins are cyber-physical systems [email protected]). (CPS) which are composed of information space and physical F.-Y. Wang is with the State Key Laboratory of Management and Control space. And parallel systems mainly focus on cyber-physical- for Complex Systems, Institute of Automation, Chinese Academy of Sciences, Beijing 100190, and with the Institute of Systems Engineering, Macau social systems (CPSS) which refer to the deep integration of University of Science and Technology, and also with Qingdao Academy of social networks, information resources, and physical space. In Intelligent Industries, Qingdao 266109, China (e-mail: [email protected]). addition to the research objects, there are certain differences in Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. core ideas, frameworks, mathematical descriptions, implemen- Digital Object Identifier 10.1109/JAS.2020.1003216 920 IEEE/CAA JOURNAL OF AUTOMATICA SINICA, VOL. 7, NO. 4, JULY 2020 Fig. 1. The basic structure of parallel systems [12]. Fig. 2. The architecture of parallel control and management for CPSS [20], [21]. tation methods, and so on [17], [18]. Fig. 2 demonstrates the [22]¡[25]. However, it is worth pointing out that the present architecture of parallel control and management for CPSS. The parallel control methods focus on the artificial systems on the detailed description can be found in [19]¡[21]. reconstruction of the system dynamics and the computational It is pointed out that parallel execution is an important and experiments focus on the performance evaluation with state distinctive step to guarantee the performance of the control feedback controllers. Furthermore, the properties analysis of systems. The basic block diagram of parallel execution [10] the parallel control methods are scarce, which are necessary to is shown in Fig. 3. guarantee the performance of the control laws. These motivate It is shown in Fig. 3 that the parallel execution is established our research. based on the parallel system theory. Based on the parallel In this paper, a new parallel control structure is developed execution between the artificial systems and physical systems, for continuous-time linear systems. The main contribution of we can convert passive computer simulations to the active the method is to establish a new parallel control law, where artificial systems, and give full play to the role of artificial the state and control input are both considered to construct systems in management and control of physical systems. Many the variation of the control, such that the system states are tasks, such as learning and training, experiment and evaluation, forced to converge to the equilibrium point and simultaneously management and control, and so on, can be executed based on analyze the performance of the parallel control laws. First, parallel execution. The parallel control theory is a hot research the basic structure of the parallel control is provided. The spot in resent study, and it has sparked a great deal of attention relationship between the parallel control and traditional feed- WEI et al.: PARALLEL CONTROL FOR CONTINUOUS-TIME LINEAR SYSTEMS: A CASE STUDY 921 Fig. 3. The basic structure of parallel execution [10]. back controls is presented and the advantages of the parallel where the variation of the control is explicitly depended with control are explained. Second, considering the continuous-time the state and the current control. linear systems, the expression of parallel controller is shown. Third, considering two situations including system control- lable and incompletely controllable (uncontrollable in brief), respectively, the properties of the parallel control method are analyzed. The detailed controller design algorithms are also given under the conditions that the systems are controllable and uncontrollable. Next, two simulation examples are pro- vided which verify the effectiveness of the developed method and the conclusion is finally drawn. The rest of this paper is organized as follows. In Section II, the structure of parallel controller is introduced and the controller design problem is formulated. In Section III, the existence of parallel controller is