ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Distributed MPC based consensus for single-integrator multi-agent systems Zhaomeng Cheng, Ming-Can Fan, Hai-Tao Zhang n School of Automation, Key Laboratory of Image Processing and Intelligent Control, State Key Laboratory of Digital Manufacturing Equipments and Technology, Huazhong University of Science and Technology, Wuhan 430074, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 19 March 2014 Received in revised form 6 February 2015 Accepted 21 March 2015 This paper was recommended for publication by Dr. Q.-G. Wang
This paper addresses model predictive control schemes for consensus in multi-agent systems (MASs) with discrete-time single-integrator dynamics under switching directed interaction graphs. The control horizon is extended to be greater than one which endows the closed-loop system with extra degree of freedom. We derive sufficient conditions on the sampling period and the interaction graph to achieve consensus by using the property of infinite products of stochastic matrices. Consensus can be achieved asymptotically if the sampling period is selected such that the interaction graph among agents has a directed spanning tree jointly. Significantly, if the interaction graph always has a spanning tree, one can select an arbitrary large sampling period to guarantee consensus. Finally, several simulations are conducted to illustrate the effectiveness of the theoretical results. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Model predictive control (MPC) Multi-agent systems (MASs) Consensus Sampled-data setting Distributed control
1. Introduction Collective behaviors of multi-agent systems (MASs) have been greatly studied in recent years due to the increasing interests in understanding animal group behaviors, such as flocking of birds, schooling of fish and swarming of bees [1,2], and also due to their wide applications in unmanned-air-vehicle formation control, flocking, distributed sensor networks, satellite clusters, robotic teams, etc. [3–13]. The research on such collective behaviors gives us much insight into the consensus problem [14], which is one of the fundamental research topics in the field of distributed control of the multi-agent systems. The basic idea of consensus is that all of the agents achieve an agreement upon a common value, called consensus point, by interacting with their neighbors (see [15] for a more general definition of “distributed”). Generally speaking, the consensus point depends on the agents' initial states and the interaction networks. These years have witnessed a large volume of extensive efforts devoted to consensus problems for networks of dynamic agents with different models and communication graphs. In [16], a consensus protocol was presented to solve agreement problems in a network of multiple agents with single-integrator continuous-time dynamics. Necessary and sufficient conditions were derived on the communication graphs and the control gains such that consensus was achieved.
n
Corresponding author. E-mail address: [email protected] (H.-T. Zhang).
Average-consensus problem was studied for directed networks of agents with both switching interaction graphs and time-delays [17,18]. Most of these control protocols are developed in continuous-time domain. However, in real industrial applications, agents are more likely to work in discrete-time domain or sampled-data formulation. Therefore, in these couple of years, discrete-time consensus for singleintegrator dynamics has been intensively investigated in [14,19–21]. Moreover, there are also some extra features taken into account in some recent works, such as time-delays [19,20], leader-following structure [22], and communication noises [23]. Model predictive control (MPC) is a form of control that the current control action is obtained by solving an online receding horizon optimization problem, and at each sampling instant, using the current state of the plant as the initial state for implementation [24]. The success of MPC in industry applications lies in its ability to handle multi-variable control problems naturally, to cope with hard constraints on controllers and states easily, and that the control update rates are relatively low such that there is plenty of time for the necessary on-line computations [25]. This method can be borrowed to solve the consensus problem in multi-agent systems in a distributed fashion as well, called distributed model predictive control (DMPC), which implies that each agent obtains its control input at each step via solving a finite-time optimal control problem involving the states of neighboring agents [26]. This paper deals with consensus problems for multiple agents with discrete-time single-integrator dynamics under time-varying communication graphs. The objective is to develop a DMPC protocol such that
http://dx.doi.org/10.1016/j.isatra.2015.03.011 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Cheng Z, et al. Distributed MPC based consensus for single-integrator multi-agent systems. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.03.011i
Z. Cheng et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
the states of all agents reach an agreement eventually. There are some literatures related to the topic of this paper in recent years. In [27], a DMPC scheme was proposed to solve the consensus problem for multi-agent systems with single- and double-integrator dynamics and input constraints. However, it was assumed that the control horizon equals the prediction control, which reduced the degree of freedom in controller design. After that, Zhang et al. [28,29] proposed a DMPC protocol to increase the consensus convergence rate in MASs under some special communication networks. In [30], the authors considered a class of single-integrator consensus problems using a DMPC approach and gave the corresponding sufficient conditions to guarantee consensus. However, for simplicity, the control horizon length was set to be one, which reduces the design degree of freedom. Motivated by the aforementioned literatures, we focus on developing a niched DMPC protocol to solve distributed consensus problems for dynamical agents described by discrete-time single-integrator model. In summary, the contribution of this paper is threefold. First, the control horizon can be arbitrarily picked from one to prediction horizon, which increases the degree of freedom in controller design. Second, in contrast to [14,19–21], the sampling period can be selected as an arbitrarily large positive real number if the switching interaction network has a directed spanning tree. Thus, the communication and sampling cost can be substantially saved. Third, the interaction network is assumed to be directed and dynamically switching. It is worth mentioning that, the cost function cannot be adopted as a Lyapunov function with control horizon larger than one, which is challenging in controller design and stability analysis of DMPC. Therefore, we have used matrix analysis theory together with Cauchy theorem instead for analysis, and in this way, we provide a new mathematical tool for DMPC analysis. The rest of this paper is organized as follows. In Section 2, we present some concepts in graph theory and summarize some results on matrix theory in some previous works. A problem formulation is also given in the end of the section. In Section 3, we propose a DMPC protocol for single-integrator MASs with a “window” parameter and derive sufficient conditions for the proposed protocol to solve a consensus problem. Afterwards, simulations are given in Section 4 to illustrate the effectiveness of the theoretical results. Finally, conclusion remarks are drawn in Section 5. Notations: Let R þ ; N and N þ be the sets of positive real numbers, nonnegative integers and positive integers, respectively. col½x1 ; …; xn ≔½xT1 ; …; xTn T with vectors xi A Rm~ ði ¼ 1; …; nÞ. A vector ξ A Rm 4 0 if and only if all of its m entries are all positive. Matrices In and 0nn represent the identity and zero matrices with dimension n, respectively, and 1n ; 0n respectively denote the n 1 vectors with all entries being 1 and 0. The matrix diagfA1 ; A2 ; …; An g is a block diagonal matrix with diagonal entries being A1 ; A2 ; …; An . The symbol J J denotes the Euclidean norm and J n J 2Q ≔nT Q n where n is a column vector and Q is a positive definite matrix. The notations J J and J J 1 denote the Euclidean and the infinity norms, respectively. The notation nðk þ tj kÞ indicates the prediction value n at the ðk þ tÞ-th moment on the basis of the currently available information at k-th moment.
2. Problem description and preliminaries It is natural to model interaction among agents by directed graphs. Consider a group consisting of n agents. The communication network among agents is illustrated by a digraph (directed graph) denoted as G ¼ fV; E; Ag, where V≔f1; 2; …; ng and E D V V are the node set and the edge set, respectively. If ðj; iÞ A E, we say that agent j is neighbor to agent i and the j-th agent transmits instantaneously its state to the i-th agent. The set of neighbors to the node iA V is
N i ¼ fj A V : ðj; iÞ A Eg and j N i j , i A V, is the valency or degree of i-th node. A ¼ ½aij A Rnn is the adjacency matrix defined as aij ¼ 1 if ðj; iÞ A E; aij ¼ 0 otherwise. Self-loops are not considered in this paper, i.e., aii ¼ 0. The graph G is undirected if and only if aij ¼ aji , which implies that A is a symmetric matrix. We assume that the time-varying topology switches every T seconds, which is called the sampling period. Denote by GðkÞ and N i ðkÞ the graph G and the set N i at the moment kT, k A N. GðkÞ always has a directed spanning tree if and only if there is an agent i such that there is a path from agent i to any other agent at any time kT [31]. A collection of graphs fGð0Þ; …; Gðm 1Þg jointly has a directed spanning 1 tree if the graph fV; ⋃m k ¼ 0 EðkÞg has a directed spanning tree. GðkÞ jointly has a directed spanning tree across the time interval ½kT; ðk þ m 1ÞT if the collection of graphs fGðkÞ; …; Gðk þ m 1Þg jointly has a directed spanning tree. The Laplacian matrix LðkÞ ¼ ½lij ðkÞ associated with graph GðkÞ is defined as follows:
lij ðkÞ ¼
8
Contents lists available at ScienceDirect
ISA Transactions journal homepage: www.elsevier.com/locate/isatrans
Distributed MPC based consensus for single-integrator multi-agent systems Zhaomeng Cheng, Ming-Can Fan, Hai-Tao Zhang n School of Automation, Key Laboratory of Image Processing and Intelligent Control, State Key Laboratory of Digital Manufacturing Equipments and Technology, Huazhong University of Science and Technology, Wuhan 430074, PR China
art ic l e i nf o
a b s t r a c t
Article history: Received 19 March 2014 Received in revised form 6 February 2015 Accepted 21 March 2015 This paper was recommended for publication by Dr. Q.-G. Wang
This paper addresses model predictive control schemes for consensus in multi-agent systems (MASs) with discrete-time single-integrator dynamics under switching directed interaction graphs. The control horizon is extended to be greater than one which endows the closed-loop system with extra degree of freedom. We derive sufficient conditions on the sampling period and the interaction graph to achieve consensus by using the property of infinite products of stochastic matrices. Consensus can be achieved asymptotically if the sampling period is selected such that the interaction graph among agents has a directed spanning tree jointly. Significantly, if the interaction graph always has a spanning tree, one can select an arbitrary large sampling period to guarantee consensus. Finally, several simulations are conducted to illustrate the effectiveness of the theoretical results. & 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Model predictive control (MPC) Multi-agent systems (MASs) Consensus Sampled-data setting Distributed control
1. Introduction Collective behaviors of multi-agent systems (MASs) have been greatly studied in recent years due to the increasing interests in understanding animal group behaviors, such as flocking of birds, schooling of fish and swarming of bees [1,2], and also due to their wide applications in unmanned-air-vehicle formation control, flocking, distributed sensor networks, satellite clusters, robotic teams, etc. [3–13]. The research on such collective behaviors gives us much insight into the consensus problem [14], which is one of the fundamental research topics in the field of distributed control of the multi-agent systems. The basic idea of consensus is that all of the agents achieve an agreement upon a common value, called consensus point, by interacting with their neighbors (see [15] for a more general definition of “distributed”). Generally speaking, the consensus point depends on the agents' initial states and the interaction networks. These years have witnessed a large volume of extensive efforts devoted to consensus problems for networks of dynamic agents with different models and communication graphs. In [16], a consensus protocol was presented to solve agreement problems in a network of multiple agents with single-integrator continuous-time dynamics. Necessary and sufficient conditions were derived on the communication graphs and the control gains such that consensus was achieved.
n
Corresponding author. E-mail address: [email protected] (H.-T. Zhang).
Average-consensus problem was studied for directed networks of agents with both switching interaction graphs and time-delays [17,18]. Most of these control protocols are developed in continuous-time domain. However, in real industrial applications, agents are more likely to work in discrete-time domain or sampled-data formulation. Therefore, in these couple of years, discrete-time consensus for singleintegrator dynamics has been intensively investigated in [14,19–21]. Moreover, there are also some extra features taken into account in some recent works, such as time-delays [19,20], leader-following structure [22], and communication noises [23]. Model predictive control (MPC) is a form of control that the current control action is obtained by solving an online receding horizon optimization problem, and at each sampling instant, using the current state of the plant as the initial state for implementation [24]. The success of MPC in industry applications lies in its ability to handle multi-variable control problems naturally, to cope with hard constraints on controllers and states easily, and that the control update rates are relatively low such that there is plenty of time for the necessary on-line computations [25]. This method can be borrowed to solve the consensus problem in multi-agent systems in a distributed fashion as well, called distributed model predictive control (DMPC), which implies that each agent obtains its control input at each step via solving a finite-time optimal control problem involving the states of neighboring agents [26]. This paper deals with consensus problems for multiple agents with discrete-time single-integrator dynamics under time-varying communication graphs. The objective is to develop a DMPC protocol such that
http://dx.doi.org/10.1016/j.isatra.2015.03.011 0019-0578/& 2015 ISA. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: Cheng Z, et al. Distributed MPC based consensus for single-integrator multi-agent systems. ISA Transactions (2015), http://dx.doi.org/10.1016/j.isatra.2015.03.011i
Z. Cheng et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
the states of all agents reach an agreement eventually. There are some literatures related to the topic of this paper in recent years. In [27], a DMPC scheme was proposed to solve the consensus problem for multi-agent systems with single- and double-integrator dynamics and input constraints. However, it was assumed that the control horizon equals the prediction control, which reduced the degree of freedom in controller design. After that, Zhang et al. [28,29] proposed a DMPC protocol to increase the consensus convergence rate in MASs under some special communication networks. In [30], the authors considered a class of single-integrator consensus problems using a DMPC approach and gave the corresponding sufficient conditions to guarantee consensus. However, for simplicity, the control horizon length was set to be one, which reduces the design degree of freedom. Motivated by the aforementioned literatures, we focus on developing a niched DMPC protocol to solve distributed consensus problems for dynamical agents described by discrete-time single-integrator model. In summary, the contribution of this paper is threefold. First, the control horizon can be arbitrarily picked from one to prediction horizon, which increases the degree of freedom in controller design. Second, in contrast to [14,19–21], the sampling period can be selected as an arbitrarily large positive real number if the switching interaction network has a directed spanning tree. Thus, the communication and sampling cost can be substantially saved. Third, the interaction network is assumed to be directed and dynamically switching. It is worth mentioning that, the cost function cannot be adopted as a Lyapunov function with control horizon larger than one, which is challenging in controller design and stability analysis of DMPC. Therefore, we have used matrix analysis theory together with Cauchy theorem instead for analysis, and in this way, we provide a new mathematical tool for DMPC analysis. The rest of this paper is organized as follows. In Section 2, we present some concepts in graph theory and summarize some results on matrix theory in some previous works. A problem formulation is also given in the end of the section. In Section 3, we propose a DMPC protocol for single-integrator MASs with a “window” parameter and derive sufficient conditions for the proposed protocol to solve a consensus problem. Afterwards, simulations are given in Section 4 to illustrate the effectiveness of the theoretical results. Finally, conclusion remarks are drawn in Section 5. Notations: Let R þ ; N and N þ be the sets of positive real numbers, nonnegative integers and positive integers, respectively. col½x1 ; …; xn ≔½xT1 ; …; xTn T with vectors xi A Rm~ ði ¼ 1; …; nÞ. A vector ξ A Rm 4 0 if and only if all of its m entries are all positive. Matrices In and 0nn represent the identity and zero matrices with dimension n, respectively, and 1n ; 0n respectively denote the n 1 vectors with all entries being 1 and 0. The matrix diagfA1 ; A2 ; …; An g is a block diagonal matrix with diagonal entries being A1 ; A2 ; …; An . The symbol J J denotes the Euclidean norm and J n J 2Q ≔nT Q n where n is a column vector and Q is a positive definite matrix. The notations J J and J J 1 denote the Euclidean and the infinity norms, respectively. The notation nðk þ tj kÞ indicates the prediction value n at the ðk þ tÞ-th moment on the basis of the currently available information at k-th moment.
2. Problem description and preliminaries It is natural to model interaction among agents by directed graphs. Consider a group consisting of n agents. The communication network among agents is illustrated by a digraph (directed graph) denoted as G ¼ fV; E; Ag, where V≔f1; 2; …; ng and E D V V are the node set and the edge set, respectively. If ðj; iÞ A E, we say that agent j is neighbor to agent i and the j-th agent transmits instantaneously its state to the i-th agent. The set of neighbors to the node iA V is
N i ¼ fj A V : ðj; iÞ A Eg and j N i j , i A V, is the valency or degree of i-th node. A ¼ ½aij A Rnn is the adjacency matrix defined as aij ¼ 1 if ðj; iÞ A E; aij ¼ 0 otherwise. Self-loops are not considered in this paper, i.e., aii ¼ 0. The graph G is undirected if and only if aij ¼ aji , which implies that A is a symmetric matrix. We assume that the time-varying topology switches every T seconds, which is called the sampling period. Denote by GðkÞ and N i ðkÞ the graph G and the set N i at the moment kT, k A N. GðkÞ always has a directed spanning tree if and only if there is an agent i such that there is a path from agent i to any other agent at any time kT [31]. A collection of graphs fGð0Þ; …; Gðm 1Þg jointly has a directed spanning 1 tree if the graph fV; ⋃m k ¼ 0 EðkÞg has a directed spanning tree. GðkÞ jointly has a directed spanning tree across the time interval ½kT; ðk þ m 1ÞT if the collection of graphs fGðkÞ; …; Gðk þ m 1Þg jointly has a directed spanning tree. The Laplacian matrix LðkÞ ¼ ½lij ðkÞ associated with graph GðkÞ is defined as follows:
lij ðkÞ ¼
8
