Distributed event-triggered scheduling in networked interconnected systems with sparse connections
Introduction
An interconnected system is usually referred to a large-scale system, which consists of a group of coupling subsystems. In the past decade, interconnected systems have received significant attention since they are deemed as a powerful tool for describing and modeling many practical coupling processes, such as electricity power grids, chemical reaction systems, manufacturing systems, transportation networks, robotic systems and so on [1]. One main impellent factor of these pervasive applications is the booming development of network technology, by which information flows flexibly and the system maintenance becomes easy, in addition to some other appealing merits [2], [3], [4], [5], [6], [7], [8].
To achieve the successful design of an interconnected system, a series of control stations are often deployed so that each subsystem can be effectively and efficiently monitored and controlled. Traditionally, such a design procedure is achieved in a decentralized manner, namely, each control station can only sense output measurements from its own subsystem and then implement the control input to its own subsystem [9]. Such a decentralized configuration promises several benefits including relaxed computational burden and design complexity, but meanwhile it inevitably leads to deteriorated system performance and limited application scopes in networked environments [10]. As a matter of fact, with the advent of communication networks, control stations can be geographically distributed and remotely scheduled. Information of each local control station is allowed to be exchanged among its neighbors using a shared communication network. In this way, more feedback information can be used by each control station, which implies that improved control performance can be expected for the overall interconnected system. For example, in [11] the stabilization problem of interconnected systems is considered using partial information exchange, where one control station can get access to its neighbors’s information through a distributed control network. Readers are referred to the surveys [10], [12] for more recent results on decentralized control and distributed control.
Note that in the existing results on distributed control, the coupling connections among the distributed control stations are usually described by a fixed weighted graph, and thus are predetermined. In other words, the coupling structure of spatially distributed controllers is given a prior and remains unchanged during the entire design and implementation process. However, from an engineering viewpoint, it may not be easy to determine whether or not one control station should retrieve information from another since there must exist a tradeoff between the installation cost and variation in system performance, which is not known in advance. Therefore, it is necessary to schedule the coupling network connections for networked interconnected systems with due consideration of some required system performance. So far, there are several results reported in the literature for tackling this issue, see, for example, [13], [14], [15], [16], [17], [18], [19]. In particular, with a prescribed amount of control performance degradation compared to the centralized control scenario, an iterative algorithm is presented in [13] to minimize the number of the required feedback channels by using a weighted l1 minimization. An l1 norm based iterative algorithm is proposed in [19] to promote sparsity in the state feedback gain matrix by decomposing the coupled terms in matrix inequalities. As stated in [19], the issue of structured scheduling is quite challenging even if the structure is predetermined since this is intrinsically a non-convex problem. In [16], a weighted l1 norm based algorithm is presented to relax the combinatorial optimization problem and a suboptimal solution is obtained by resolving iterative convex optimization problems. It is noted that one critical issue in the (weighted) l1 norm based iterative approach is how to determine the initial points, which usually has a great effect on seeking a convergent solution. Another issue is that although the relaxed (weighted) l1 norm based method is proposed to minimize the number of nonzero entries of the controller gain matrix [19], [20], there is still no explicit explanation on the sparse structure of the gain matrix since no topological structure is concerned in the studied linear systems.
Motivated by the discussions presented above, in this paper, we are concerned with κ-sparse distributed event-triggered scheduling for networked interconnected systems, where the control station of each subsystem could retrieve information from other control stations, but the number of the coupling connections is upper bounded by a prescribed integer κ. A distributed event-triggered transmission scheme (ETTS) is proposed for network transmissions from subsystems to control stations. This issue of event-triggered scheduling under cardinality constraint is known as NP hard since the problem is nonlinear and non-convex [21], [22]. In this paper, we will propose a mixed-integer programming based approach to constrain the number of nonzero blocks in a block matrix, which is used to represent the coupling connections in a networked interconnected system. Based on that, an explicit κ sparse distributed scheduling algorithm for the networked interconnected system will be presented by utilizing the Lyapunov functional method and some matrix transformations. Compared with the weighted l1 norm based iterative method [13], [16], the iterative computations with appropriate initial points are no longer needed. In addition, the number of coupling connections can be preassigned in advance by the proposed approach, while the l1 norm based optimization method [13] could only yield a scheduling matrix with as many zero blocks as possible.
The organization of this paper is as follows. Section 2 presents the distributed event-triggered scheduling framework for a networked interconnected system with limited coupling connections, and formulates the κ sparse distributed event-triggered scheduling problem for the system. A sufficient condition that guarantees the exponential stability of the event-triggered closed-loop system is provided in Section 3. Section 4 proposes a mixed-integer programming based approach to impose cardinality constraints on block matrices. Based on that, an explicit κ-sparse distributed scheduling algorithm is presented in the section. The proposed approach is applied to a 3-machine power system in Section 5. Section 6 provides a conclusion of the paper.
Notation. Throughout the paper, a superscript ‘T’ represents matrix transposition. A matrix P > 0 means that the matrix P is positive definite. I is employed to denote an identity matrix with appropriate dimension. (*) is utilized in a symmetric matrix to represent the terms that could be deduced by symmetry. card( · ) denotes the number of nonzero elements in a set or the number of nonzero blocks in a block matrix, and cardod(K) stands for the number of nonzero off-diagonal blocks in block matrix K. For a given real matrix ‖K‖max ≔ max i,j|Kij|.
Section snippets
Framework
As shown in Fig. 1, we consider an interconnected system composed of p subsystems of the following formwhere and denote the state and the control input of subsystem i, respectively, and Ai, Bi and Gij are constant matrices with appropriate dimensions. Subsystem i physically interacts with subsystem j through Gij and Gji. If subsystem j has no effect on the evolution of subsystem i, then coupling matrix Gij will become
Exponential stability analysis
In this section, we will establish a sufficient condition which guarantees the exponential stability of closed-loop controlled system (9). Theorem 1 Given a scalar σ > 0 and a κ-sparse distributed controller (7), with ETTS (6), the closed-loop system (9) is exponentially stable with decay rate σ, if there exist real matrices P > 0, Q > 0, R > 0 and Y with appropriate dimensions such thatwhere
κ-sparse distributed controller design
Based upon the stability analysis result obtained in the preceding section, we are going to design a κ-sparse distributed event-triggered controller in the form of (7) such that the closed-loop controlled system (9) is exponentially stable with decay rate σ.
It is noted that according to Theorem 1, one can obtain a full structure controller in the form of by some matrix manipulations. Here, a block matrix K is with full structure implies that all of the blocks in matrix K could be
An illustrative example
In this section, we will give an example to illustrate the effectiveness of the proposed method presented in this paper.
Consider a power system, which is composed of three interconnected machines. For more dynamics and modeling of the power system, we refer to references [46], [47]. The system can be represented by the interconnection of three subsystems, each of which is described in the form ofwhere is the
Conclusion
In this paper, a mixed-integer programming based approach has been proposed for an interconnected system to schedule a κ-sparse distributed state feedback controller, where the number of the coupling connections among distributed control stations is upper bounded by κ. For efficient utilization of resources, a distributed event-triggered transmission scheme has been introduced for the networked interconnected system. A sufficient condition has been presented to guarantee the exponential
Declarations of interest
None
Acknowledgment
This work was supported in part by the National Natural Science Foundation of China under Grant 61603232, the Shanxi Science and Technology Department of China under Grant 201701D221100, and the Australian Research Council under Grant DP120104986.
Yanpeng Guan was born in Shanxi, China, in 1984. He received the B.Sc. degree in mathematics from Changchun Normal University, in 2005, the M.Sc. degree in control theory and control engineering from Hangzhou Dianzi University, in 2010, and the Ph.D. degree in computer engineering from Central Queensland University, Australia, in 2014, respectively. He was a Visiting Fellow with the School of Computing, Engineering and Mathematics, Western Sydney University, Australia, from January 2018 to
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Yanpeng Guan was born in Shanxi, China, in 1984. He received the B.Sc. degree in mathematics from Changchun Normal University, in 2005, the M.Sc. degree in control theory and control engineering from Hangzhou Dianzi University, in 2010, and the Ph.D. degree in computer engineering from Central Queensland University, Australia, in 2014, respectively. He was a Visiting Fellow with the School of Computing, Engineering and Mathematics, Western Sydney University, Australia, from January 2018 to January 2019. He is currently a Associate Professor with the Department of Automation, Shanxi University. His research interests are in networked control systems, event-triggered control, distributed control, and sparse structure design.
Guonan Ping was born in Shanxi, China, in 1994. She received the B.Sc. degree in automation from Shanxi University 2016. She is currently pursuing her M.Sc. degree in Shanxi University. Her research interests are in networked control systems, event-triggered control, and distributed control.
Wei Xing Zheng received the Ph.D. degree in Electrical Engineering from Southeast University, Nanjing, China in 1989. Over the years Dr. Zheng has held various faculty/research/visiting positions at Southeast University, Nanjing, China; Imperial College of Science, Technology and Medicine, London, UK; University of Western Australia, Perth, Australia; Curtin University of Technology, Perth, Australia; Munich University of Technology, Munich, Germany; University of Virginia, Charlottesville, VA, USA; and University of California at Davis, Davis, CA, USA. Currently he holds the rank of Distinguished Professor at Western Sydney University, Sydney, Australia. Dr. Zheng has been an Associate Editor of several flagship journals, including Automatica, IEEE Transactions on Automatic Control, IEEE Transactions on Cybernetics, IEEE Transactions on Neural Networks and Learning Systems and so on. He is a Fellow of IEEE and a Clarivate Analytics Highly Cited Researcher.
Huijuan Yao was born in Shanxi, China, in 1987. She received the B.Sc. degree in control technology and instruments from Taiyuan University of Science and Technology in 2010, the M.Sc. degree in instrument science and technology from Chongqing University in 2013, respectively. She was a Visiting Fellow with the School of Computing, Engineering and Mathematics, Western Sydney University, Australia, from October 2018 to January 2019. She is currently a Lecture with the Department of Automation, Shanxi University. His research interests are in networked control systems and their applications.