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 and 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.
[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.

[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.

[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. 

[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.

[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.

[7]

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

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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. 

[9]

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

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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.

[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. 

[12]

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

[13]

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

[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.

[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.

[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.

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

[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.

[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.

[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.

[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.

[23]

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

[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.

[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.

[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.

[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.

[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.

[29]

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

[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. 

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

show all references

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

[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.

[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. 

[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.

[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.

[7]

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

[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. 

[9]

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

[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.

[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. 

[12]

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

[13]

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

[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.

[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.

[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.

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

[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.

[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.

[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.

[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.

[23]

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

[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.

[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.

[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.

[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.

[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.

[29]

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

[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. 

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

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