A Spectral Property of a Graph Matrix and Its Application to the Leader-Following Consensus of Discrete-Time Multiagent Systems

The leader-following consensus problem for linear discrete-time multiagent systems has been studied in the literature using <inline-formula><tex-math notation="LaTeX">$H_{\infty }$</tex-math></inline-formula> Riccati inequality design and <inline-formula><tex-math notation="LaTeX">$H_{2}$</tex-math></inline-formula> Riccati equation design methods, respectively. These two methods lead to a solvability condition in terms of an inequality that relies on a scaling gain and a fixed weighting vector called nominal weighting vector. In this paper, we further study this problem by proposing a more general class of distributed state feedback control laws, which not only depends on a scaling gain, but also a set of weighting vectors. We first establish a spectral property of a weighted graph matrix, which will be instrumental in solving the problem. Then, we present a solvability condition based on a modified algebraic Riccati equation, which is somehow more general than <inline-formula><tex-math notation="LaTeX">$H_{\infty }$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$H_{2}$</tex-math></inline-formula> Riccati methods, in that the solvability condition can be made satisfied by tuning the weighting vector. We show that our solvability condition is also necessary for single-input follower systems and is always satisfied for leader systems without exponentially growing modes. Moreover, in case the solvability condition is not satisfied under the nominal weighting vector, we show that our solvability condition can always be made satisfied by choosing a weighting vector other than the nominal one for multiagent systems over acyclic digraphs.

[1]  Lihua Xie,et al.  Network Topology and Communication Data Rate for Consensusability of Discrete-Time Multi-Agent Systems , 2011, IEEE Transactions on Automatic Control.

[2]  Wei Xing Zheng,et al.  Exponential Synchronization of Complex Networks of Linear Systems and Nonlinear Oscillators: A Unified Analysis , 2015, IEEE Transactions on Neural Networks and Learning Systems.

[3]  Brian D. O. Anderson,et al.  On leaderless and leader-following consensus for interacting clusters of second-order multi-agent systems , 2016, Autom..

[4]  Jie Huang,et al.  Stability of a Class of Linear Switching Systems with Applications to Two Consensus Problems , 2011, IEEE Transactions on Automatic Control.

[5]  T. Katayama On the matrix Riccati equation for linear systems with random gain , 1976 .

[6]  Jiangping Hu,et al.  Leader-following coordination of multi-agent systems with coupling time delays , 2007, 0705.0401.

[7]  Wei Xing Zheng,et al.  Consensus of multiple second-order vehicles with a time-varying reference signal under directed topology , 2011, Autom..

[8]  John N. Tsitsiklis,et al.  Parallel and distributed computation , 1989 .

[9]  Sezai Emre Tuna,et al.  Synchronizing linear systems via partial-state coupling , 2008, Autom..

[10]  Daizhan Cheng,et al.  Leader-following consensus of multi-agent systems under fixed and switching topologies , 2010, Syst. Control. Lett..

[11]  Randal W. Beard,et al.  Consensus seeking in multiagent systems under dynamically changing interaction topologies , 2005, IEEE Transactions on Automatic Control.

[12]  Jie Lin,et al.  Coordination of groups of mobile autonomous agents using nearest neighbor rules , 2003, IEEE Trans. Autom. Control..

[13]  Wei Ren,et al.  Synchronization of coupled harmonic oscillators with local interaction , 2008, Autom..

[14]  Richard M. Murray,et al.  Consensus problems in networks of agents with switching topology and time-delays , 2004, IEEE Transactions on Automatic Control.

[15]  S. E. Tuna LQR-based coupling gain for synchronization of linear systems , 2008, 0801.3390.

[16]  Wei Ren On Consensus Algorithms for Double-Integrator Dynamics , 2008, IEEE Trans. Autom. Control..

[17]  Jie Huang,et al.  Leader-following consensus for linear discrete-time multi-agent systems subject to static networks , 2017, 2017 36th Chinese Control Conference (CCC).

[18]  Bruno Sinopoli,et al.  Foundations of Control and Estimation Over Lossy Networks , 2007, Proceedings of the IEEE.

[19]  Guanrong Chen,et al.  Consensus of Discrete-Time Linear Multi-Agent Systems with Observer-Type Protocols , 2011, ArXiv.

[20]  Frank L. Lewis,et al.  Optimal Design for Synchronization of Cooperative Systems: State Feedback, Observer and Output Feedback , 2011, IEEE Transactions on Automatic Control.

[21]  Frank L. Lewis,et al.  Synchronization of discrete-time multi-agent systems on graphs using Riccati design , 2012, Autom..