September  2019, 9(3): 297-318. doi: 10.3934/naco.2019020

Identification of Hessian matrix in distributed gradient-based multi-agent coordination control systems

1. 

Department of Automatic Control, Lund University, Sweden

2. 

Research School of Engineering, The Australian National University, Canberra ACT, Australia

3. 

Graduate School of Informatics, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto 606-8501 Japan

* Corresponding author: Zhiyong Sun

Received  April 2018 Revised  April 2019 Published  May 2019

Multi-agent coordination control usually involves a potential function that encodes information of a global control task, while the control input for individual agents is often designed by a gradient-based control law. The property of Hessian matrix associated with a potential function plays an important role in the stability analysis of equilibrium points in gradient-based coordination control systems. Therefore, the identification of Hessian matrix in gradient-based multi-agent coordination systems becomes a key step in multi-agent equilibrium analysis. However, very often the identification of Hessian matrix via the entry-wise calculation is a very tedious task and can easily introduce calculation errors. In this paper we present some general and fast approaches for the identification of Hessian matrix based on matrix differentials and calculus rules, which can easily derive a compact form of Hessian matrix for multi-agent coordination systems. We also present several examples on Hessian identification for certain typical potential functions involving edge-tension distance functions and triangular-area functions, and illustrate their applications in the context of distributed coordination and formation control.

Citation: Zhiyong Sun, Toshiharu Sugie. Identification of Hessian matrix in distributed gradient-based multi-agent coordination control systems. Numerical Algebra, Control & Optimization, 2019, 9 (3) : 297-318. doi: 10.3934/naco.2019020
References:
[1] K. Abadir and J. Magnus, Matrix Algebra, Cambridge University Press, 2005. doi: 10.1017/CBO9780511810800. Google Scholar
[2]

P.-A. Absil and K. Kurdyka, On the stable equilibrium points of gradient systems, Systems & Control Letters, 55 (2006), 573-577. doi: 10.1016/j.sysconle.2006.01.002. Google Scholar

[3]

B. D. O. Anderson and U. Helmke, Counting critical formations on a line, SIAM Journal on Control and Optimization, 52 (2014), 219-242. doi: 10.1137/120890533. Google Scholar

[4]

B. D. O. AndersonZ. SunT. SugieS.-I. Azuma and K. Sakurama, Formation shape control with distance and area constraints, IFAC Journal of Systems and Control, 1 (2017), 2-12. Google Scholar

[5]

B. D. O. AndersonC. YuS. Dasgupta and A. S. Morse, Control of a three-coleader formation in the plane, Systems & Control Letters, 56 (2007), 573-578. doi: 10.1016/j.sysconle.2007.04.004. Google Scholar

[6]

B. D. O. Anderson, C. Yu, S. Dasgupta and T. H. Summers, Controlling four agent formations, in The 2nd IFAC Workshop on Distributed Estimation and Control in Networked Systems, (2010), 139–144, Available at: http://www.sciencedirect.com/science/article/pii/S1474667016309570.Google Scholar

[7]

R. B. Bapat, Graphs and Matrices, Springer, 27 (2010). doi: 10.1007/978-1-84882-981-7. Google Scholar

[8]

Y. CaoW. YuW. Ren and G. Chen, An overview of recent progress in the study of distributed multi-agent coordination, IEEE Transactions on Industrial Informatics, 9 (2013), 427-438. Google Scholar

[9]

X. Chen, Gradient flows for organizing multi-agent system, in Proc. of the 2014 American Control Conference, (2014), 5109–5114.Google Scholar

[10]

X. ChenM.-A. Belabbas and T. Basar, Global stabilization of triangulated formations, SIAM Journal on Control and Optimization, 55 (2017), 172-199. doi: 10.1137/15M105094X. Google Scholar

[11]

H. G. de MarinaB. Jayawardhana and M. Cao, Distributed rotational and translational maneuvering of rigid formations and their applications, IEEE Transactions on Robotics, 32 (2016), 684-697. Google Scholar

[12]

D. A. Harville, Matrix Algebra from a Statistician's Perspective, Taylor & Francis, 1998. doi: 10.1007/b98818. Google Scholar

[13]

M. Ji and M. Egerstedt, Distributed coordination control of multi-agent systems while preserving connectedness, IEEE Transactions on Robotics, 23 (2007), 693-703. Google Scholar

[14]

H. Kawashima and M. Egerstedt, Manipulability of leader-follower networks with the rigid-link approximation, Automatica, 50 (2014), 695-706. doi: 10.1016/j.automatica.2013.11.041. Google Scholar

[15]

S. KnornZ. Chen and R. H. Middleton, Overview: Collective control of multiagent systems, IEEE Transactions on Control of Network Systems, 3 (2016), 334-347. doi: 10.1109/TCNS.2015.2468991. Google Scholar

[16]

L. KrickM. E. Broucke and B. A. Francis, Stabilisation of infinitesimally rigid formations of multi-robot networks, International Journal of Control, 82 (2009), 423-439. doi: 10.1080/00207170802108441. Google Scholar

[17] M. Mesbahi and M. Egerstedt, Graph Theoretic Methods in Multiagent Networks, Princeton University Press, 2010. doi: 10.1515/9781400835355. Google Scholar
[18]

R. Olfati-SaberJ. A. Fax and R. M. Murray, Consensus and cooperation in networked multi-agent systems, Proceedings of the IEEE, 95 (2007), 215-233. Google Scholar

[19]

K. SakuramaS. I. Azuma and T. Sugie, Distributed controllers for multi-agent coordination via gradient-flow approach, IEEE Transactions on Automatic Control, 60 (2015), 1471-1485. doi: 10.1109/TAC.2014.2374951. Google Scholar

[20]

K. SakuramaS. I. Azuma and T. Sugie, Multi-agent coordination to high-dimensional target subspaces, IEEE Transactions on Control of Network Systems, 5 (2018), 345-358. doi: 10.1109/TCNS.2016.2609638. Google Scholar

[21]

T. Sugie, B. D. O. Anderson, Z. Sun and H. Dong, On a hierarchical control strategy for multi-agent formation without reflection, in Proc. of the 2018 IEEE Conferences on Decision and Control, accepted, 2018. Google Scholar

[22]

T. H. Summers, C. Yu, B. D. O. Anderson and S. Dasgupta, Formation shape control: Global asymptotic stability of a four-agent formation, in Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference, (2009), 3002–3007.Google Scholar

[23]

Z. Sun, Cooperative Coordination and Formation Control for Multi-agent Systems, Springer, 2018. doi: 10.1007/978-3-319-74265-6_1. Google Scholar

[24]

Z. Sun, U. Helmke and B. D. O. Anderson, Rigid formation shape control in general dimensions: an invariance principle and open problems, in Proc. of the 2015 IEEE 54th Annual Conference on Decision and Control (CDC), IEEE, (2015), 6095–6100.Google Scholar

[25]

Z. Sun, S. Mou, B. D. O. Anderson and M. Cao, Exponential stability for formation control systems with generalized controllers: A unified approach, Systems & Control Letters, 93 (2016), 50–57, https://www.sciencedirect.com/science/article/pii/S016769111600061X.Google Scholar

[26]

Z. SunM.-C. ParkB. D. O. Anderson and H.-S. Ahn, Distributed stabilization control of rigid formations with prescribed orientation, Automatica, 78 (2017), 250-257. doi: 10.1016/j.automatica.2016.12.031. Google Scholar

[27]

Y.-P. Tian and Q. Wang, Global stabilization of rigid formations in the plane, Automatica, 49 (2013), 1436-1441. doi: 10.1016/j.automatica.2013.01.057. Google Scholar

[28]

M. H. TrinhV. H. PhamM.-C. ParkZ. SunB. D. O. Anderson and H.-S. Ahn, Comments on "global stabilization of rigid formations in the plane [automatica 49 (2013) 1436–1441]", Automatica, 77 (2017), 393-396. doi: 10.1016/j.automatica.2016.11.006. Google Scholar

[29]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Vol. 2, Springer Science & Business Media, 2003. Google Scholar

[30]

M. M. ZavlanosM. B. Egerstedt and G. J. Pappas, Graph-theoretic connectivity control of mobile robot networks, Proceedings of the IEEE, 99 (2011), 1525-1540. Google Scholar

[31] X.-D. Zhang, Matrix Analysis and Applications, Cambridge University Press, 2017. doi: 10.1017/9781108277587. Google Scholar
[32]

S. ZhaoD. V. DimarogonasZ. Sun and D. Bauso, A general approach to coordination control of mobile agents with motion constraints, IEEE Transactions on Automatic Control, 63 (2018), 1509-1516. doi: 10.1109/tac.2017.2750924. Google Scholar

show all references

References:
[1] K. Abadir and J. Magnus, Matrix Algebra, Cambridge University Press, 2005. doi: 10.1017/CBO9780511810800. Google Scholar
[2]

P.-A. Absil and K. Kurdyka, On the stable equilibrium points of gradient systems, Systems & Control Letters, 55 (2006), 573-577. doi: 10.1016/j.sysconle.2006.01.002. Google Scholar

[3]

B. D. O. Anderson and U. Helmke, Counting critical formations on a line, SIAM Journal on Control and Optimization, 52 (2014), 219-242. doi: 10.1137/120890533. Google Scholar

[4]

B. D. O. AndersonZ. SunT. SugieS.-I. Azuma and K. Sakurama, Formation shape control with distance and area constraints, IFAC Journal of Systems and Control, 1 (2017), 2-12. Google Scholar

[5]

B. D. O. AndersonC. YuS. Dasgupta and A. S. Morse, Control of a three-coleader formation in the plane, Systems & Control Letters, 56 (2007), 573-578. doi: 10.1016/j.sysconle.2007.04.004. Google Scholar

[6]

B. D. O. Anderson, C. Yu, S. Dasgupta and T. H. Summers, Controlling four agent formations, in The 2nd IFAC Workshop on Distributed Estimation and Control in Networked Systems, (2010), 139–144, Available at: http://www.sciencedirect.com/science/article/pii/S1474667016309570.Google Scholar

[7]

R. B. Bapat, Graphs and Matrices, Springer, 27 (2010). doi: 10.1007/978-1-84882-981-7. Google Scholar

[8]

Y. CaoW. YuW. Ren and G. Chen, An overview of recent progress in the study of distributed multi-agent coordination, IEEE Transactions on Industrial Informatics, 9 (2013), 427-438. Google Scholar

[9]

X. Chen, Gradient flows for organizing multi-agent system, in Proc. of the 2014 American Control Conference, (2014), 5109–5114.Google Scholar

[10]

X. ChenM.-A. Belabbas and T. Basar, Global stabilization of triangulated formations, SIAM Journal on Control and Optimization, 55 (2017), 172-199. doi: 10.1137/15M105094X. Google Scholar

[11]

H. G. de MarinaB. Jayawardhana and M. Cao, Distributed rotational and translational maneuvering of rigid formations and their applications, IEEE Transactions on Robotics, 32 (2016), 684-697. Google Scholar

[12]

D. A. Harville, Matrix Algebra from a Statistician's Perspective, Taylor & Francis, 1998. doi: 10.1007/b98818. Google Scholar

[13]

M. Ji and M. Egerstedt, Distributed coordination control of multi-agent systems while preserving connectedness, IEEE Transactions on Robotics, 23 (2007), 693-703. Google Scholar

[14]

H. Kawashima and M. Egerstedt, Manipulability of leader-follower networks with the rigid-link approximation, Automatica, 50 (2014), 695-706. doi: 10.1016/j.automatica.2013.11.041. Google Scholar

[15]

S. KnornZ. Chen and R. H. Middleton, Overview: Collective control of multiagent systems, IEEE Transactions on Control of Network Systems, 3 (2016), 334-347. doi: 10.1109/TCNS.2015.2468991. Google Scholar

[16]

L. KrickM. E. Broucke and B. A. Francis, Stabilisation of infinitesimally rigid formations of multi-robot networks, International Journal of Control, 82 (2009), 423-439. doi: 10.1080/00207170802108441. Google Scholar

[17] M. Mesbahi and M. Egerstedt, Graph Theoretic Methods in Multiagent Networks, Princeton University Press, 2010. doi: 10.1515/9781400835355. Google Scholar
[18]

R. Olfati-SaberJ. A. Fax and R. M. Murray, Consensus and cooperation in networked multi-agent systems, Proceedings of the IEEE, 95 (2007), 215-233. Google Scholar

[19]

K. SakuramaS. I. Azuma and T. Sugie, Distributed controllers for multi-agent coordination via gradient-flow approach, IEEE Transactions on Automatic Control, 60 (2015), 1471-1485. doi: 10.1109/TAC.2014.2374951. Google Scholar

[20]

K. SakuramaS. I. Azuma and T. Sugie, Multi-agent coordination to high-dimensional target subspaces, IEEE Transactions on Control of Network Systems, 5 (2018), 345-358. doi: 10.1109/TCNS.2016.2609638. Google Scholar

[21]

T. Sugie, B. D. O. Anderson, Z. Sun and H. Dong, On a hierarchical control strategy for multi-agent formation without reflection, in Proc. of the 2018 IEEE Conferences on Decision and Control, accepted, 2018. Google Scholar

[22]

T. H. Summers, C. Yu, B. D. O. Anderson and S. Dasgupta, Formation shape control: Global asymptotic stability of a four-agent formation, in Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference, (2009), 3002–3007.Google Scholar

[23]

Z. Sun, Cooperative Coordination and Formation Control for Multi-agent Systems, Springer, 2018. doi: 10.1007/978-3-319-74265-6_1. Google Scholar

[24]

Z. Sun, U. Helmke and B. D. O. Anderson, Rigid formation shape control in general dimensions: an invariance principle and open problems, in Proc. of the 2015 IEEE 54th Annual Conference on Decision and Control (CDC), IEEE, (2015), 6095–6100.Google Scholar

[25]

Z. Sun, S. Mou, B. D. O. Anderson and M. Cao, Exponential stability for formation control systems with generalized controllers: A unified approach, Systems & Control Letters, 93 (2016), 50–57, https://www.sciencedirect.com/science/article/pii/S016769111600061X.Google Scholar

[26]

Z. SunM.-C. ParkB. D. O. Anderson and H.-S. Ahn, Distributed stabilization control of rigid formations with prescribed orientation, Automatica, 78 (2017), 250-257. doi: 10.1016/j.automatica.2016.12.031. Google Scholar

[27]

Y.-P. Tian and Q. Wang, Global stabilization of rigid formations in the plane, Automatica, 49 (2013), 1436-1441. doi: 10.1016/j.automatica.2013.01.057. Google Scholar

[28]

M. H. TrinhV. H. PhamM.-C. ParkZ. SunB. D. O. Anderson and H.-S. Ahn, Comments on "global stabilization of rigid formations in the plane [automatica 49 (2013) 1436–1441]", Automatica, 77 (2017), 393-396. doi: 10.1016/j.automatica.2016.11.006. Google Scholar

[29]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Vol. 2, Springer Science & Business Media, 2003. Google Scholar

[30]

M. M. ZavlanosM. B. Egerstedt and G. J. Pappas, Graph-theoretic connectivity control of mobile robot networks, Proceedings of the IEEE, 99 (2011), 1525-1540. Google Scholar

[31] X.-D. Zhang, Matrix Analysis and Applications, Cambridge University Press, 2017. doi: 10.1017/9781108277587. Google Scholar
[32]

S. ZhaoD. V. DimarogonasZ. Sun and D. Bauso, A general approach to coordination control of mobile agents with motion constraints, IEEE Transactions on Automatic Control, 63 (2018), 1509-1516. doi: 10.1109/tac.2017.2750924. Google Scholar

[1]

Giulia Cavagnari, Antonio Marigonda, Benedetto Piccoli. Optimal synchronization problem for a multi-agent system. Networks & Heterogeneous Media, 2017, 12 (2) : 277-295. doi: 10.3934/nhm.2017012

[2]

Rui Li, Yingjing Shi. Finite-time optimal consensus control for second-order multi-agent systems. Journal of Industrial & Management Optimization, 2014, 10 (3) : 929-943. doi: 10.3934/jimo.2014.10.929

[3]

Daniela Saxenhuber, Ronny Ramlau. A gradient-based method for atmospheric tomography. Inverse Problems & Imaging, 2016, 10 (3) : 781-805. doi: 10.3934/ipi.2016021

[4]

Brendan Pass. Multi-marginal optimal transport and multi-agent matching problems: Uniqueness and structure of solutions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1623-1639. doi: 10.3934/dcds.2014.34.1623

[5]

Zhongkui Li, Zhisheng Duan, Guanrong Chen. Consensus of discrete-time linear multi-agent systems with observer-type protocols. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 489-505. doi: 10.3934/dcdsb.2011.16.489

[6]

Yibo Zhang, Jinfeng Gao, Jia Ren, Huijiao Wang. A type of new consensus protocol for two-dimension multi-agent systems. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 345-357. doi: 10.3934/naco.2017022

[7]

Tyrone E. Duncan. Some partially observed multi-agent linear exponential quadratic stochastic differential games. Evolution Equations & Control Theory, 2018, 7 (4) : 587-597. doi: 10.3934/eect.2018028

[8]

Hong Man, Yibin Yu, Yuebang He, Hui Huang. Design of one type of linear network prediction controller for multi-agent system. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 727-734. doi: 10.3934/dcdss.2019047

[9]

Gianluca D'Antonio, Paul Macklin, Luigi Preziosi. An agent-based model for elasto-plastic mechanical interactions between cells, basement membrane and extracellular matrix. Mathematical Biosciences & Engineering, 2013, 10 (1) : 75-101. doi: 10.3934/mbe.2013.10.75

[10]

Dieter Armbruster, Christian Ringhofer, Andrea Thatcher. A kinetic model for an agent based market simulation. Networks & Heterogeneous Media, 2015, 10 (3) : 527-542. doi: 10.3934/nhm.2015.10.527

[11]

Holly Gaff. Preliminary analysis of an agent-based model for a tick-borne disease. Mathematical Biosciences & Engineering, 2011, 8 (2) : 463-473. doi: 10.3934/mbe.2011.8.463

[12]

Raymond Ching Man Chan, Henry Ying Kei Lau. An AIS-based optimal control framework for longevity and task achievement of multi-robot systems. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 45-56. doi: 10.3934/naco.2012.2.45

[13]

Zhengshan Dong, Jianli Chen, Wenxing Zhu. Homotopy method for matrix rank minimization based on the matrix hard thresholding method. Numerical Algebra, Control & Optimization, 2019, 9 (2) : 211-224. doi: 10.3934/naco.2019015

[14]

El-Sayed M.E. Mostafa. A nonlinear conjugate gradient method for a special class of matrix optimization problems. Journal of Industrial & Management Optimization, 2014, 10 (3) : 883-903. doi: 10.3934/jimo.2014.10.883

[15]

Yigui Ou, Yuanwen Liu. A memory gradient method based on the nonmonotone technique. Journal of Industrial & Management Optimization, 2017, 13 (2) : 857-872. doi: 10.3934/jimo.2016050

[16]

Anthony M. Bloch, Peter E. Crouch, Nikolaj Nordkvist, Amit K. Sanyal. Embedded geodesic problems and optimal control for matrix Lie groups. Journal of Geometric Mechanics, 2011, 3 (2) : 197-223. doi: 10.3934/jgm.2011.3.197

[17]

Xiaming Chen. Kernel-based online gradient descent using distributed approach. Mathematical Foundations of Computing, 2019, 2 (1) : 1-9. doi: 10.3934/mfc.2019001

[18]

Behrouz Kheirfam. Multi-parametric sensitivity analysis of the constraint matrix in piecewise linear fractional programming. Journal of Industrial & Management Optimization, 2010, 6 (2) : 347-361. doi: 10.3934/jimo.2010.6.347

[19]

Sie Long Kek, Mohd Ismail Abd Aziz, Kok Lay Teo. A gradient algorithm for optimal control problems with model-reality differences. Numerical Algebra, Control & Optimization, 2015, 5 (3) : 251-266. doi: 10.3934/naco.2015.5.251

[20]

Jian Zhao, Fang Deng, Jian Jia, Chunmeng Wu, Haibo Li, Yuan Shi, Shunli Zhang. A new face feature point matrix based on geometric features and illumination models for facial attraction analysis. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1065-1072. doi: 10.3934/dcdss.2019073

 Impact Factor: 

Metrics

  • PDF downloads (34)
  • HTML views (230)
  • Cited by (0)

Other articles
by authors

[Back to Top]