Parallel Control for Continuous-Time Linear Systems: A Case Study

Qinglai Wei,,, Hongyang Li, Fei-Yue Wang,,

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, 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.

I. INTRODUCTION

OVER 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, 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 multioutput 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, 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 Management and Control for Complex Systems, Institute of Automation, Chinese Academy of Sciences, Beijing 100190, with the University of Chinese Academy of Sciences, Beijing 100049, 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, with the Institute of Systems Engineering, Macau University of Science and Technology, 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, 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(Artificial systems,computational experiments,parallel execution)theory [12] in control theory, where artificial systems (A) are used for modeling the physical systems, computational experiments (C) are used for analysis, evaluation and learning, 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 corresponding 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-physicalsocial systems (CPSS) which refer to the deep integration of social networks, information resources, physical space. In addition to the research objects,there are certain differences in core ideas,frameworks,mathematical descriptions,implementation methods, so on [17], [18]. Fig.2 demonstrates the architecture of parallel control and management for CPSS.The detailed description can be found in [19]?[21].

Fig.1. The basic structure of parallel systems [12].

Fig.2. The architecture of parallel control and management for CPSS [20], [21].

It is pointed out that parallel execution is an important and distinctive step to guarantee the performance of the control systems. The basic block diagram of parallel execution [10]is shown in Fig.3.

It is shown in Fig.3 that the parallel execution is established based on the parallel system theory. Based on the parallel execution between the artificial systems and physical systems,we can convert passive computer simulations to the active artificial systems, give full play to the role of artificial systems in management and control of physical systems.Many tasks,such as learning and training,experiment and evaluation,management and control,and so on,can be executed based on parallel execution.The parallel control theory is a hot research spot in resent study,and it has sparked a great deal of attention[22]?[25]. However, it is worth pointing out that the present parallel control methods focus on the artificial systems on the reconstruction of the system dynamics and the computational experiments focus on the performance evaluation with state feedback controllers. Furthermore, the properties analysis of the parallel control methods are scarce,which are necessary to guarantee the performance of the control laws.These motivate our research.

Fig.3. The basic structure of parallel execution [10].

In this paper, a new parallel control structure is developed for continuous-time linear systems. The main contribution of the method is to establish a new parallel control law, where the state and control input are both considered to construct the variation of the control, such that the system states are forced to converge to the equilibrium point and simultaneously analyze the performance of the parallel control laws. First,the basic structure of the parallel control is provided. The relationship between the parallel control and traditional feedback controls is presented and the advantages of the parallel control are explained.Second,considering the continuous-time linear systems, the expression of parallel controller is shown.Third, considering two situations including system controllable 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 provided 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 analyzed and the parallel controller design algorithms are presented. Simulation results are provided and discussed in Section IV. Some concluding remarks are given in Section V.

II. PROBLEM FORMULATIONS

In this section, the design ideas of the parallel control are presented. First, the basic structure of the parallel control is introduced and the comparisons between the parallel control and traditional control are illustrated, where the advantages of the parallel control are emphasized. Second, the problem formulations of the parallel control for continuous-time linear systems are presented.

A. Basic Structure of Parallel Control

In this subsection, the basic structure of parallel control is introduced. Consider the following systems

wherex ∈Rnis then-dimensional state vector,u ∈Rmis them-dimensional control vector, andf(x,u) is the system function. A new parallel control method is established. The structure of the parallel control is shown in Fig.4, the parallel control can be expressed by where the variation of the control is explicitly depended with the state and the current control.

In parallel control method,system(1)and parallel controller(2) are executed in parallel with information interaction. It is shown that the parallel control is not a traditional feedback control, where traditional control laws are function of the states, i.e., ˙x=f(x,K(x)) underu=K(x). However, it is worth pointing out that the parallel control in (2) can be transformed into a open-loop or a closed-loop control law.First, according to (2), there is a functionG, such thatu=G(x,t), which indicates the closed-loop control law. On the other hand, according to (1) and (2), letting augmented state variablezbez=xT,uTT, the system function can be written as

which establishes a closed-loop system by designing the control functiong. This is an obvious merit of the parallel control method. Second, if parallel control (2) is reduced to ˙u=g(u), then there exists a function ?G, such thatu= ?G.In this situation, the parallel control law is reduced to a openloop control law. Thus, the flexible structure is another merit of the parallel control. In the following, we focus on the parallel control method for continuous-time linear systems and properties of the parallel control method will be analyzed.

IV. NUMERICAL ANALYSIS

In this section, two simulations are employed to evaluate the effectiveness of the proposed method.

Example 1:In the first example, we consider the following linearized model of the power system [29], [30]

It is easy to derive that system (45) is controllable. The desired poles are?1,?1,?2,?2,?3,which have characteristic polynominal

According to Algorithm 1, we can obtain that

and the parallel controller can be expressed as

The initial state and control law arex0=00.100 andu0= 0.1, respectively. Then we can obtain the simulation results for the trajectories of the system states and control, which are shown in Figs.5 and 6,respectively.The trajectories of the system states are shown in Fig.5, the trajectory of the control law is shown in Fig.6.From the figures, we can see that the system is stable after control law is applied to system. Therefore, the correctness of the proposed method can be verified.

Example 2:In the second example, we consider the following uncontrollable system

Fig.5. System states in Example 1.

Fig.6. Control law in Example 1.

The poles of controllable subsystem are 1, 2, the pole of uncontrollable subsystem is?1. The desired poles are?1,?1,?2 and?2. According to Algorithm 2, we can obtain matricesCandDas

wherecis an arbitrary constant. And the parallel controller can be obtained as follows.

The initial state and control law arex0=0.5 1 1 andu0=0.1, respectively. Lettingc=?3, we can obtain the simulation results as Figs. 7 and 8.

The trajectories of the system states are shown in Fig.7,and the trajectory of the control law is shown in Fig.8. The correctness of the proposed control method can be demonstrated.

V. CONCLUSION

Fig.7. System states in Example 2.

Fig.8. Control law in Example 2.

This paper has concerned a new control, that is the parallel control method, to obtain the effective control law for continuous-time linear systems. Different from state feedback controller,this new parallel control law contains both the state and control input to construct the variation of the control.Considering two situations including system controllable and uncontrollable, respectively, analyses for linear continuoustime systems are given to verify the effectiveness of the proposed method. Simultaneously, the detailed controller design algorithms are also given and two simulation examples are provided which verify the effectiveness of the developed method. In the future, we will further analyze the parallel controller design problems for nonlinear systems.