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Consensus for First- and Second-Order Discrete-Time Multi-agent Systems with Delays Based on Model Predictive Control Schemes

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Abstract

The problem of consensus for first- and second-order discrete-time multi-agent systems with delays is concerned in this paper. In particular, we apply the decentralized model predictive control schemes to multi-agent systems with bounded time delays, by which the input constraints can be taken into account. First, we establish the stability properties of time-delayed systems under the connectivity assumption and the strict convexity update rule based on the concept of enlarged system. Then, decentralized model predictive control schemes are proposed to solve the consensus problem based on time-varying prediction horizon length, which gives better performance than the existing method with time invariant prediction horizon length. Finally, simulation results illustrate the effectiveness of the proposed methods.

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Abbreviations

\({\mathbb {R}}\) :

The set of real numbers

\({\mathbb {N}}\) :

The set of natural numbers

\(\subset \) :

Strict set inclusion

\(\subseteq \) :

Nonstrict set inclusion

\(X\) :

A finite-dimensional Euclidean space

\(K\) :

A subset of a finite-dimensional Euclidean space

\({\mathcal {B}}(K,c)\,\) :

The set of points in \(X\) whose distance to \(K\) is strictly smaller than \(c\)

\(2^X\) :

The collection of all closed subsets of \(X\)

\({\mathcal {N}}\) :

The set of multi-agent nodes

\({\mathcal {A}}\) :

The set of edges

\(w\) :

Weight of edge

\(( {m,l})\in {\mathcal {A}}\) :

A directed arc from node \(m\) to node \(l\)

\(G=( {{\mathcal {N}},{\mathcal {A}},w})\) :

A weighted directed graph

\(N_i ( G)\) :

The set of neighbors of the node \(i\)

\(N_{\mathcal {L}} ( G)\) :

The set of these nodes \(m\in {\mathcal {N}}\backslash {\mathcal {L}}\) for which there is \(l\in {\mathcal {L}}\) such that \(m\in N_l ( G)\)

\(( {{\mathcal {N}},\cup _{k\in {\mathcal {I}}} {\mathcal {A}}(k)})\) :

The union of graphs

\(\overline{P_1 P_2 } \) :

The segment joining \(P_1 ,P_2 \in X\)

\(\left| {\overline{P_1 P_2 } } \right| \) :

Segment length

\(r_{P_1 P_2 } \) :

The straight line passing through \(P_1 ,P_2 \in X\)

\(s_{P_1 P_2 } \) :

The straight half-line starting from \(P_1 \) and passing through \(P_2 \)

\(T=\left\{ {P_1 ,\ldots ,P_M } \right\} \) :

An ordered sequence of \(M\) points

\(x_i (k)\) :

State of agent \(i\) at time step \(k\)

\(x(k)\) :

State of the multi-agent system

\({\mathcal {X}}\subseteq {\mathbb {R}}^q\) :

State space of each agent

\({\mathcal {X}}^n\subseteq {\mathbb {R}}^{qn}\) :

State space of the multi-agent system

\(h\) :

The bound of delays

\(x_{i+\tau n} (k)=x_i (k-\tau )\) :

Noncomputing agent \(i+\tau n\)

\({{\tilde{\mathcal {N}}}}\) :

All the nodes of enlarged multi-agent system

\({\tilde{x}}(k)\) :

State of the enlarged multi-agent system

\({\mathcal {X}}^{hn}\subseteq {\mathbb {R}}^{qhn}\) :

State space of enlarged multi-agent system

\(\Phi \) :

The set of equilibrium points

\(\phi \in \Phi \) :

Equilibrium point

\({\varvec{V}}:{\mathcal {X}}^n\mapsto ( {2^{\mathcal {X}}})^n\) :

Set-valued function

\(\hbox {Co}( {\cup _{j\in {\tilde{\mathcal {N}}}} \left\{ {x_j (k)} \right\} })\) :

The convex hull of states \(\left\{ {x_1 ,\ldots ,x_n ,\ldots ,x_{hn} } \right\} \)

\(\hbox {card}( {a_i (k)})\) :

The cardinal number of the set \(a_i (k)\)

\(\hbox {Ci}( {\left\{ {y_1 ,y_2 } \right\} })\) :

The relative interior of \(\hbox {Co}( {\left\{ {y_1 ,y_2 } \right\} })\)

\(N\) :

Prediction horizon length

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Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (61304095 & 51204115), the Natural Science Foundation of Jiangsu Province (BK20130317), the Jiangsu Planned Projects for Postdoctoral Research Funds (1302103B), and the Suzhou Science and Technology Program (SGZ2013135).

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Authors and Affiliations

  1. School of Computer Science and Technology, Soochow University, Suzhou, 215006, Jiangsu, People’s Republic of China

    Zhaozhun Zhong

  2. School of Mechanical and Electric Engineering, Soochow University, Suzhou, 215006, Jiangsu, People’s Republic of China

    Lining Sun

  3. Department of Automation, Shanghai Jiaotong University, Shanghai, 200240, People’s Republic of China

    Jingcheng Wang

  4. Department of Industry Development, Zhejiang Province Development Planning and Research Institute, Hangzhou, 310012, Zhejiang, People’s Republic of China

    Penghong Lv

  5. School of Computer Engineering, Suzhou Vocational University, Suzhou, 215104, Jiangsu, People’s Republic of China

    Hongjing Zheng

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  1. Zhaozhun Zhong
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  2. Lining Sun
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Correspondence to Zhaozhun Zhong.

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Zhong, Z., Sun, L., Wang, J. et al. Consensus for First- and Second-Order Discrete-Time Multi-agent Systems with Delays Based on Model Predictive Control Schemes. Circuits Syst Signal Process 34, 127–152 (2015). https://doi.org/10.1007/s00034-014-9850-1

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