### Systems theory for complex & large-scale dynamical systems

Current projects in the area of complex dynamical systems are centered around the development of computational, data-driven/machine learning approaches for control and estimation problems, as well as the problems of compressing data and extracting various different spatio-temporal structures in complex and large-scale dynamical systems.

- S. Zeng, Iterative optimal control syntheses illustrated on the Brockett integrator,
*Proc. IFAC Symposium on Nonlinear Control Systems*, 2019. - S. Zeng, Sample-based Population Observers,
*Automatica*, 2019. - S. Zeng, On sample-based computations of invariant sets,
*Nonlinear Dynamics*, 2018. - S. Zeng, Observability measures for nonlinear systems,
*Proc. 57th IEEE Conference on Decision and Control*, 2018. - S. Zeng, On systems theoretic aspects of Koopman operator theoretic frameworks,
*Proc. 57th IEEE Conference on Decision and Control*, 2018.

### Populations of dynamical systems

A prototype for a complex dynamical system is given by an ensemble, which is a collection of a large number of nearly identical dynamical systems that can only be controlled and observed as a whole. This restriction that typically stems from practical circumstances leads us to consider a mathematical description involving probability distributions.

A typical example that illustrates the need for considering such system classes is given by cell populations. Cells in the population cannot be influenced individually but only through the application of a common stimulus that is the same for all cells in the population. Likewise, for the observation of dynamical processes within cells, typical measurement devices only provide statistical data about the population *as a whole*.

Besides this example from cell biology, many problems related to ensembles have been emerging in different fields ranging from quantum physics, and process engineering to robotics swarms. The common theme of the seemingly different problems is indeed the consideration of populations of copies of one system, as well as the premise of interaction on the population-level only. Yet, despite the increasing interest for these novel problems, our understanding of the basic aspects in the control theory of ensembles is still very limited.

#### Ensemble Observability of Dynamical Systems

Consider the problem of estimating a two-dimensional distribution from knowledge of the underlying process (e.g., a rotation in state space) and the time-evolution of a one-dimensional marginal distribution. The consideration of problems like this led us to introduce the novel yet quite fundamental notion of ensemble observability.

Our study revealed an inherent connection between the observability problem in systems theory (1960s) and the classical X-Ray tomography problem (1960s), which is suggested in the following side-by-side comparison:

- S. Zeng, S. Waldherr, C. Ebenbauer, F. Allgöwer, Ensemble Observability of Linear Systems, IEEE Transactions on Automatic Control 61 (6), pp. 1452-1465, 2015.
- S. Zeng, H. Ishii, F. Allgöwer, Sampled Observability and State Estimation of Linear Discrete Ensembles, IEEE Transactions on Automatic Control 62 (5), pp. 2406 – 2418, 2016.
- S. Zeng, Ensemble Observability of Dynamical Systems,

Ph.D. dissertation, Logos Verlag, Berlin, 2016.

##### Sample-based / Non-parametric Population Observers

More recently, we were able to establish a first formulation of an observer for populations by means of a purely sample-based description. The example on the right-hand side shows the asymptotic tracking process of a population observer. There the state of a population of double integrators (blue) is tracked by the observer (red), which has only the distribution of the first component (position) at hand.

- S. Zeng, Sample-based Population Observers, Automatica, vol. 101, pp. 166-174, 2019.

#### Ensemble Controllability of Dynamical Systems

Consider the problem of steering a whole one-parameter family of systems, such as

by applying the exact same input signal to all the system in the family. Is it possible to control such an infinite-dimensional system using finite-dimensional control, e.g. to steer the systems to have the terminal state at 1 uniformly? Though highly non-obvious, the answer is yes:

- S. Zeng, W. Zhang, J.-S. Li, On the Computation of Control Inputs for Linear Ensembles, to appear in Proc. 2018 American Control Conference.
- S. Zeng, J.-S. Li, An Operator Theoretic Approach to Linear Ensemble Control, submitted to IEEE Transactions on Automatic Control, 2018.

At the Systems Theory Lab, we investigate basic principles of population systems and contribute to the long-term goal of establishing a systems theoretic foundation for problems centered around the class of ensembles. To this end, we study the fundamental concepts of controllability and observability of ensembles, and develop methods which are needed for the novel framework.

### Periodic Signal Compressors

A quite interesting type of compressors is given by periodic compressors which compress a multidimensional (periodic) signal into a scalar signal by periodically sweeping through the scalar components of the signal:

This type of compression scheme for lossless *causal* compression of

periodic signals was proposed and studied in our recent paper

- S. Zeng, J. M. Montenbruck, F. Allgöwer, Periodic Signal Compressors, Proc. 20th World Congress of the International Federation of Automatic Control, pp. 6465 – 6470, 2017.

In contrast to the vast majority of traditional compression schemes in signal processing, periodic compressors are *causal*, which is crucial in view of incorporating compressors into a feedback control loop.

In studying this type of periodic compressor in a discrete-time settings, we found novel connections to and interpretations of basic number-theoretical results such as the Chinese remainder theorem, as well as a novel interpretation of these in terms of a discrete analogue of the well-known result on the denseness of linear flows on the (continuous) torus:

In a continuous-time setting, we are studying problems of the following type:

### Structured optimal feedback laws

Controllers arising from some underlying general compression / expansion architecture such as the one shown on the left-hand side are often highly structured. In some cases, these specific structured controllers even turn out to be optimal with respect to some natural cost functional.

A study of this general phenomenon is presented in our paper:

- S. Zeng, and F. Allgöwer. Structured optimal feedback in multi-agent systems: A static output feedback perspective. Automatica, vol. 76, 2017.

Roughly speaking, we highlight how the analysis and control of large-scale or complex systems can be significantly simplified by exclusively focusing on the quantities that are crucial to a considered optimal control task.