This article can be considered as an English version of http://post.cdsl.kr/archives/30.
A review of our research has been presented as an online seminar at https://www.youtube.com/watch?v=Vj_vzt0T3cY&t=869s.
A review article summarizing the research of CDSL on heterogeneous multi-agent system:
Design of heterogeneous multi-agent system for distributed computation
Jin Gyu Lee, Hyungbo Shim
https://arxiv.org/abs/2101.00161
Trends in Nonlinear and Adaptive Control, Springer, 2022
Our research on multi-agent system began with the study of output feedback consensus. (Consensus means that the internal state of every dynamic agent converges to each other.) The contribution compared to the previous results is that only the outputs of every agents are exchanged.
Consensus of high-order linear systems using dynamic output feedback compensator: Low gain approach
J.H. Seo, H. Shim, and J. Back
Automatica, vol. 45, no. 11, pp. 2656-2664, 2009
http://dx.doi.org/10.1016/j.automatica.2009.07.022
Similar results can also be obtained when the communication network is switched. This is a simple application of the (time) averaging theory:
Consensus of output-coupled linear multi-agent systems under fast switching network: Averaging approach
H. Kim, H. Shim, J. Back, and J.H. Seo
Automatica, vol. 49, no. 1, pp. 267-272, 2013
http://dx.doi.org/10.1016/j.automatica.2012.09.025
While the above results deal with identical multi-agents, a natural question is whether the similar consensus can be achieved when each agent is not the same. In this case, it is easily imagined that the exact consensus is not possible, under the diffusive coupling, due to the mismatch between the vector fields of every agents. Then, existence of common internal model across the multi-agents becomes necessary. If there is not common internal model, then the local controller can embed one for every agents. Details are studied in:
Output consensus of heterogeneous uncertain linear multi-agent systems
H. Kim, H. Shim, and J.H. Seo
IEEE Trans. on Automatic Control, vol. 56, no. 1, pp. 200-206, 2011
http://dx.doi.org/10.1109/TAC.2010.2088710
Embedding internal model can only be done for engineering systems. In the case that it is not possible to install local controllers, exact consensus is hopeless. Instead, we studied approximate consensus, and found a novel phenomenon that the effect of “averaging vector fields” when heterogeneous agents are coupled with large coupling gains. This new finding firstly appeared in:
Practical consensus for heterogeneous linear time-varying multi-agent systems
J. Kim, J. Yang, J.S. Kim, and H. Shim
In Proc. of 12th Int. Conf. on Control, Automation and Systems (ICCAS), Jeju Island, Korea, 2012, pp. 23-28
Download: 12.ICCAS.Kim.PracCons
On robustness of synchronization in heterogeneous multi-agent systems
J. Kim, J. Yang, H. Shim, and J.S. Kim
In Proc. of 12th European Control Conf., Zurich, Switzerland, 2013, pp. 3821-3826
Download: 13.ECC.Kim.draft
and as a more formal form in:
Robustness of synchronization of heterogeneous agents by strong coupling and a large number of agents
Jaeyong Kim, Jongwook Yang, Hyungbo Shim, Jung-Su Kim, and Jin Heon Seo
IEEE Trans. on Automatic Control, vol. 61, no. 10, pp. 3096-3102, Oct. 2016
http://dx.doi.org/10.1109/TAC.2015.2498138
Later we realized that “averaging vector fields” (we now call it “blending vector fields” in order not to be confused with the classical averaging theory) has a potential power to be utilized in many engineering problems. While the heterogeneity of multi-agents had been considered as something unfavorable (recall the cases when the heterogeneity arises from uncertain parameters and external disturbances to individual agents), heterogeneity can somtimes be intentional (recall the cases when a big task is divided into small sub-tasks each of which is assigned to individual agents).
A well-known example is the problem of distributed optimization:
Initialization-free privacy-guaranteed distributed algorithm for economic dispatch problem
Hyeonjun Yun, Hyungbo Shim, and Hyo-Sung Ahn
Automatica, vol. 102, pp. 86-93, April 2019
https://doi.org/10.1016/j.automatica.2018.12.033
Distributed Algorithm for Economic Dispatch Problem with Separable Losses
Seungjoon Lee and Hyungbo Shim
IEEE Control Systems Letters, vol. 3, no. 3, pp. 685-690, 2019
https://doi.org/10.1109/LCSYS.2019.2916250
Since the above result is based on our unique blended dynamics approach, an immediate benefit is the plug-and-play operation (that is, a new agent can join or leave the network on-line and there is no need to re-initialize the algorithm). This feature comes from the fact that we are not relying on the average of the initial conditions, but on the average of the vector fields (rather, the initial conditions are forgotten as time tends to infinity).
An example of distributed optimization is the distributed least square solver. Our version of it appears in:
A distributed algorithm that finds almost best possible estimate under non-vanishing and time-varying measurement noise
Jin Gyu Lee and Hyungbo Shim
IEEE Control Systems Letters, vol. 4, no. 1, pp. 229-234, 2020
https://doi.org/10.1109/LCSYS.2019.2923475
Another simple but useful application of the blended dynamics approach is to figure out the number of participating agents in a network:
Distributed Algorithm for the Network Size Estimation: Blended Dynamics Approach
Donggil Lee, Seungjoon Lee, Taekyoo Kim, Hyungbo Shim
In Proc. of IEEE Conf. on Decision and Control, Miami Beach, USA, 2018
https://doi.org/10.1109/CDC.2018.8619676
The same philosophy also yields the distributed state estimation:
On distributed optimal Kalman-Bucy filtering by averaging dynamics of heterogeneous agents
Jaeyong Kim, Hyungbo Shim, and Jingbo Wu
In Proc. of IEEE 55th Conf. Decision and Control, Las Vegas, December, 2016
http://dx.doi.org/10.1109/CDC.2016.7799240
Distributed Luenberger observer design
Taekyoo Kim, Hyungbo Shim, and Dongil Dan Cho
In Proc. of IEEE 55th Conf. Decision and Control, Las Vegas, December, 2016
http://dx.doi.org/10.1109/CDC.2016.7799336
Completely Decentralized Design of Distributed Observer for Linear Systems
Taekyoo Kim, Chanhwa Lee, and Hyungbo Shim
IEEE Trans. on Automatic Control, Nov 2020
http://doi.org/10.1109/TAC.2019.2962360
The problem of distributed state estimation also leads to an algorithm for secure control systems, when combined with distributed optimization. In fact, an interesting idea of computing the ‘median’ of given data set in a distributed way appears in:
Fully Distributed Resilient State Estimation based on Distributed Median Solver
Jin Gyu Lee, Junsoo Kim, and Hyungbo Shim
IEEE Trans. on Automatic Control, Special Issue on Security and Privacy of Distributed Algorithms and Network Systems, Sept. 2020
https://doi.org/10.1109/TAC.2020.2989275
Based on the above idea, maximum, minimum, median, or even the second largest one can be found in a distributed manner:
Distributed Dynamic Quantile Solver With Plug-and-Play Operation
Jeong Mo Seong, Jeong Woo Kim, Seungjoon Lee, and Hyungbo Shim
IEEE Access, vol. 9, pp. 165517-165525, 2021
https://doi.org/10.1109/ACCESS.2021.3134655
On the other hand, when only the output is exchanged, we have to deal with the case of rank-deficient couplings. A general framework is established in:
A tool for analysis and synthesis of heterogeneous multi-agent systems under rank-deficient coupling
Jin Gyu Lee and Hyungbo Shim
Automatica, July 2020
Download the paper (Open Access) by https://authors.elsevier.com/sd/article/S0005109820301503
Handling rank-deficient case yields an important application of “output” coupled oscillators. The reason why this is important is that it may explain the group and emergent behavior of biological organs. The blended dynamics approach also ensures the robustness of this behavior; i.e., all agents need not operate well and some defective agents can co-exist in the network for proper operation. The details appear in:
Heterogeneous Van Der Pol Oscillators under Strong Coupling
Jin Gyu Lee, Hyungbo Shim
In Proc. of IEEE Conf. on Decision and Control, Miami Beach, USA, 2018
https://doi.org/10.1109/CDC.2018.8618901
Behavior of a network of heterogeneous Lienard systems under output coupling
Jin Gyu Lee and Hyungbo Shim
In Proc. of 11th IFAC Symposium on Nonlinear Control Systems (NOLCOS), Vienna, Austria, 4–6 September, 2019
https://doi.org/10.1016/j.ifacol.2019.11.788
On the other hand, determination of the level of large coupling gain may become difficult in some cases. In this case, one may rely on the nonlinear coupling gain, which is called funnel coupling:
Synchronization with prescribed transient behavior: Heterogeneous multi-agent systems under funnel coupling
J.G. Lee, S. Trenn, and H. Shim
Automatica, Volume 141, July 2022, 110276
https://doi.org/10.1016/j.automatica.2022.110276Edge-wise funnel output synchronization of heterogeneous agents with relative degree one
J.G. Lee, T. Berger, S. Trenn, and H. Shim
Automatica, volume 156, Oct. 2023, 111204
https://doi.org/10.1016/j.automatica.2023.111204
Stabilization of a linear system that has multi-channel shows an interesting behavior of self-organizing controllers, that is, identical controllers develops their own different control gains for a global goal:
Decentralized Design and Plug-and-Play Distributed Control for Linear Multi-Channel Systems
Taekyoo Kim, Donggil Lee, and H. Shim
to appear at IEEE Trans. on Automatic Control, 2023
https://doi.org/10.1109/TAC.2023.3293036
When the blended dynamics theorem is used for distributed optimization, each cost function need not be convex as long as their sum is convex. Also, PI(proportional-integral) coupling law is analyzed in:
Blended Dynamics Approach to Distributed Optimization: Sum Convexity and Convergence Rate
Seungjoon Lee and Hyungbo Shim
Automatica, Volume 141, July 2022, 110290
https://doi.org/10.1016/j.automatica.2022.110290
While the discussions are based on continuous-time communications, discrete-time communications with continuous-time plants can be analyzed based on the hybrid system theory:
A Design Method of Distributed Algorithms via Discrete-time Blended Dynamics Theorem
Jeong Woo Kim, Jin Gyu Lee, Donggil Lee, Hyungbo Shim
to appear at Automatica, 2023
preprint: https://arxiv.org/abs/2210.05142
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