Control Theory of Ensembles of Dynamical Systems

• Applicant: Professor Dr.-Ing. Frank Allgöwer
• Subject Area: Automation, Mechatronics, Control Systems, Intelligent Technical Systems, Robotics
• Term: from 2016 to 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 295903128

• Final Report Year: 2021

Final Report Abstract

Populations of identical dynamical systems which admit a certain degree of heterogeneity, known as ensembles, arise in various applications such as cell biology, quantum physics, multi-agent systems etc. These systems are subject to the restriction that they can only be controlled or observed at the popula- tion level, i.e. through the application of a broadcast signal and through the observation of aggregated population outputs respectively. The understanding of this critical combination of heterogeneity and in- teraction at population level only, is of fundamental importance. To this end, we studied state estimation and control of ensembles. In the first part of the project, we studied ensemble observability of dynamical systems using a measure- theoretic approach. We considered discrete linear ensembles and studied the problem of state recon- struction from the samples of the output distributions. We derived conditions on the number of output samples required for unique reconstruction of the initial state of the ensemble. Further, we proposed a clustered least squares observer for practical state estimation from output snapshots with measure- ment noise. These results were extended to ensemble of interconnected linear systems. The second part was centered around exploring the connections between discrete ensemble observability and multitarget tracking. We considered specific theoretical aspects of multitarget tracking related to the approach of symmetric measurement equations. Our work provided a systematic account on the introduction of symmetric measurement equations based on the idea of associating polynomials to discrete measures. Finally, we studied controllability of linear and nonlinear ensembles. We first considered a class of parameter-dependent linear ensembles and introduced a novel moment-based approach to ensemble con- trollability. Thereby, we showed how this approach leads to simple arguments for proving or disproving ensemble controllability in terms of the structure of the moments’ dynamics. Next we studied a class of oscillatory nonlinear systems, namely biological oscillators, and derived population-level feedback laws to control the distribution of the system to a desired distribution on their periodic orbit, thereby achieving cell synchronization. This gains importance in the light of understanding the cause and cure of various diseases such as Alzheimer’s, Parkinson’s, sleep disorder, cardio-vascular diseases etc., which are associ- ated with malfunction of biological oscillators. We also conducted a proof-of-concept study to verify the applicability of the proposed controller to real cell populations.

Publications

• A moment-based approach to ensemble controllability of linear systems. Systems & Control Letters, vol. 98, pp. 49–56, 2016
  Zeng and F. Allgöwer
  
• Ensemble Observability of Dynamical Systems. Ph.D. dissertation, 2016
  S. Zeng
  
• On the ensemble observability of dynamical systems. In: Proc. 22nd International Symposium on the Mathematical Theory of Networks and Systems, pp. 685–688, (Minneapolis, MN, USA), 2016
  S. Zeng and F. Allgöwer
  
• On the moment dynamics of discrete measures. In: Proc. 55th Conference on Decision and Control. (CDC), pp. 4901–4906, (Las Vegas, NV, USA), 2016
  S. Zeng and F. Allgöwer
  
• State estimation of interconnected ensembles with anonymized outputs. IFAC-PapersOnLine, vol. 49, no. 22, pp. 109–114, 2016
  S. Zeng, H. Ishii and F. Allgöwer
  
• Sampled observability and state estimation of linear discrete ensembles. IEEE Transactions on Automatic Control, vol. 62, no. 5, pp. 2406–2418, 2017
  S. Zeng, H. Ishii and F. Allgöwer
  
• Broadcast control of oscillating cell populations. EMBL Symposium: Biological Oscillators: Design, Mechanism, Function, (Heidelberg, Germany), 2018
  K. Kuritz and F. Allgöwer
  
• Ensemble control for cell cycle synchronization of heterogeneous cell populations. IFAC-PapersOnLine, vol. 51, no. 19, pp. 44–47, 2018
  K. Kuritz, D. Imig, M. Dyck and F. Allgöwer
  
• Passivity-based ensemble control for cell cycle synchronization. In Emerging Applications of Control and Systems Theory, pp. 1–13, Springer, 2018
  K. Kuritz, W. Halter and F. Allgöwer
  
• Ensemble Controllability of Cellular Oscillators. IEEE Control Systems Letters, vol. 3, no. 2, pp. 296–301, 2019
  K. Kuritz, S. Zeng and F. Allgöwer
  
• Analysis and Control of Cellular Ensembles. Exploiting dimensionality reduction in single-cell data and models. Ph.D. dissertation, 2020
  K. Kuritz