This study will investigate the distributed linear quadratic (LQ) control problem for discrete identical uncoupled multi-agent systems with a global performance index coupling the behaviour of the multiple agents. An existence condition to the optimal distributed LQ controller is given first. In general, such condition can be checked by solving a discrete algebraic Riccati equation through a numerical method. When the condition fails to hold, a suboptimal distributed controller design method is proposed for a class of LQ performance. The solution can be obtained by solving two local algebraic Riccati equations whose dimension is the same as a single agent. The stability condition is given in terms of the spectrum of a matrix representing the desired sparsity pattern of the distributed controller. Comparing to the centralised control, the computation and communication complexity is much lesser. Finally, the suboptimality is parameterised, and can be measured by solving a Lyapunov equation.

References

1. 1)
  - 25. Hendricks, E., Jannerup, O., Sørensen, P.H.: ‘Linear systems control: deterministic and stochastic methods’ (Springer-Verlag, Berlin, Heidelberg, 2008).
2. 2)
  - 10. Cao, Y.C., Ren, W.: ‘Optimal linear-consensus algorithms: an LQR perspective’, IEEE Trans. Syst. Man Cybern. B Cybern., 2009, 40, pp. 819–830.
3. 3)
  - 26. Lee, J.Y., Kim, J.S., Shim, H.: ‘Disc margins of the discrete-time LQR and its application to consensus problem’, Int. J. Syst. Sci., 2012, 43, pp. 1891–1900.
4. 4)
  - 24. Bourlès, H., Kosmidou, O.I.: ‘Gain and phase margins of the discrete-time guaranteed cost regulator’. Proc. of the 33rd IEEE Conf. on Decision and Control, Lake Buena Vista, FL, 1994, pp. 3837–3839.
5. 5)
  - 3. Liu, S., Xie, L.H., Zhang, H.S.: ‘Distributed consensus for multi-agent systems with delays and noises in transmission channels’, Automatica, 2011, 47, pp. 920–934.
6. 6)
  - 15. Wang, J., Xin, M.: ‘Multi-agent consensus algorithm with obstacle avoidance via optimal control approach’, Int. J. Control, 2010, 83, pp. 2606–2621.
7. 7)
  - 16. Wang, J., Xin, M.: ‘Distributed optimal cooperative tracking control of multiple autonomous robots’, Robot. Auton. Syst., 2012, 60, pp. 572–583.
8. 8)
  - 8. Dong, W.J.: ‘Distributed optimal control of multiple systems’, Int. J. Control, 2010, 83, pp. 2067–2079.
9. 9)
  - 23. Shaked, U.: ‘Guaranteed stability margins for the discrete-time linear quadratic optimal regulator’, IEEE Trans. Autom. Control, 1986, 31, pp. 162–165.
10. 10)
  - 22. Naidu, D.S.: ‘Optimal control systems (Electrical Engineering series)’ (CRC Press, 2002).
11. 11)
  - 7. Gupta, V., Hassibi, B., Murray, R.M.: ‘A suboptimal algorithm to synthesize control laws for a network of dynamic agents’, Int. J. Control, 2005, 78, pp. 1302–1313.
12. 12)
  - 27. Wieland, P., Kim, J.S., Frank, A.: ‘On topology and dynamics of consensus among linear high-order agents’, Int. J. Syst. Sci., 2011, 42, pp. 1831–1842.
13. 13)
  - 17. Zhang, F.F., Wang, W., Zhang, H.S.: ‘Design and analysis of distributed optimal controller for identical multi-agent systems’, Asian J. Control, 2015, 17, pp. 263–273.
14. 14)
  - 12. Deshpande, P., Menon, P., Edwards, C., et al: ‘Sub-optimal distributed control law with H2 performance for identical dynamically coupled linear systems’, IET Control Theory Appl., 2012, 6, pp. 2509–2517.
15. 15)
  - 2. Keviczky, T., Borrelli, F., Balas, G.J., et al: ‘Decentralized receding horizon control and coordination of autonomous vehicle formations’, IEEE Trans. Control Syst. Technol., 2008, 16, pp. 19–33.
16. 16)
  - 4. Olfati-Saber, R., Fax, J.A., Murray, R.M.: ‘Consensus and cooperation in networked multi-agent systems’, Proc. IEEE, 2007, 95, pp. 215–233.
17. 17)
  - 13. Zhang, F.F., Wang, W., Zhang, H.S.: ‘The design of distributed suboptimal controller for multi-agent systems’, Int. J. Robust Nonlinear Control, 2015, 25, pp. 2829–2842.
18. 18)
  - 6. Lewis, F.L., Zhang, H., Hengster-Movric, K., et al: ‘Cooperative control of multi-agent systems: optimal and adaptive design approaches’ (Springer-Verlag, London, 2014).
19. 19)
  - 11. Rogge, J, Suykens, J., Aeyels, D.: ‘Consensus over Ring Networks as a Quadratic Optimal Control Problem’. Proc. of the 4th IFAC Symp. on System, Structure and Control, Ancona, Italy, 2010, pp. 317–323.
20. 20)
  - 14. Borrelli, F., Keviezky, T.: ‘Distributed LQR design for identical dynamically decoupled systems’, IEEE Trans. Autom. Control, 2008, 53, pp. 1901–1912.
21. 21)
  - 21. Han, C.Y., Wang, W., Zhang, F.F.: ‘Distributed LQR control for scalar discrete-time uncoupled identical systems’. Proc. of 11th World Congress on Intelligent Control and Automation, Shenyang, China, 2014, pp. 4250–4255.
22. 22)
  - 20. Wang, W., Zhang, F.F., Han, C.Y.: ‘Distributed LQR control for discrete-time homogeneous systems’, Int. J. Syst. Sci., 2016, 47, pp. 3678–3687.
23. 23)
  - 19. Zhang, H.G., Feng, T., Liang, H.J., et al: ‘LQR-Based Optimal Distributed Cooperative Design for Linear Discrete-Time Multiagent Systems’, IEEE Trans. Neural Netw. Learn. Syst., 2017, 28, pp. 599–611.
24. 24)
  - 9. Deshpande, P., Menon, P., Edwards, C., et al: ‘A distributed control law with guaranteed LQR cost for identical dynamically coupled linear systems’. Proc. of the 2011 American Control Conf., San Francisco, CA, July 2011, pp. 5342–5347.
25. 25)
  - 28. Zhang, H., Lewis, F.L., Das, A.: ‘Optimal design for synchronization of cooperative systems: state feedback and output feedback’, IEEE Trans. Autom. Control, 2011, 56, pp. 1948–1952.
26. 26)
  - 18. Feng, T., Zhang, H.G., Luo, Y.H., et al: ‘Distributed LQR design for multi-agent systems on directed graph topologies’. Proc. of 2014 Int. Joint Conf. on Neural Networks, Beijing, China, 2014, pp. 2732–2737.
27. 27)
  - 5. Ren, W., Beard, R.W.: ‘Distributed consensus in multi-vehicle cooperative control: theory and applications’ (Springer, 2008).
28. 28)
  - 1. Borrelli, F., Keviczky, T., Balas, G.J., et al: ‘Hybrid decentralized control of large scale systems’, in Morari, M., Thiele, L. (Eds.):‘Hybrid systems: computation and control’ (Springer Verlag, New York, 2005), pp. 168–183.

Distributed linear–quadratic regulator control for discrete-time multi-agent systems

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