Mathematical systems theory, see, e.g.
- DFG project "Funnel MPC with application to the control of magnetic levitation systems" (Funnel MPC mit Anwendung in der Schweberegelung), (grant WO 2056/12-1); project staff: D. Dennstädt (Scholarship "Landesgraduiertenförderung" since 11/2020), Cooperation partner: Jun.-Prof. Dr. Thomas Berger, U Paderborn
- DFG project "System theory of partial differential-algebraic equations" (grant WO 2056/2-1; funding period: 2016/04 - 2020/10) → H. Gernandt and Dr. M. Schaller
- Heisenberg-professorship "Optimization-based Control" (DFG grant WO 2056/6-1; funding period: 2019/07 - 2022/06)
Dr. M. Wilson & Dr. M. Schaller
- DFG project "Temperature controlled retinal laser treatment" (Temperaturgeregelte Laserbestrahlung der Netzhaut des Auges); grant WO 2056/7-1; funding period: 2020/05 - 2022/04) → Dr. M. Wilson and Dr. M. Schaller
Retinal laser photocoagulation has become a standard tool for the treatment of various eye diseases. The main idea is to use intensive laser pulses resulting in a temperature increase and finally in coagulation of the retinal tissue. In current clinical practice, the treating physician chooses the laser power for the subsequent lesions according to the visibility of the previous lesions, which is a very cumbersome and time-consuming procedure and often leads to quite unsatisfactory results. The goal of this joint project together with the BMO Lübeck (Dr. Ralf Brinkmann) and LU Hannover (Prof. Dr.-Ing. Matthias A. Müller) is to develop closed-loop automatic control strategies, which guarantee an accurate realization of the desired treatment temperature predetermined by the ophthalmologist.
- Stability analysis of (sampled-data and/or networked) nonlinear (and non-holonomic) control system
This includes structure-exploiting techniques, e.g., using symmetries of mechanical systems represented by Lie groups. Such systems arise, among others, in robotics.
- Scientific network "Regulation of Mobile Robots using Model Predictive Control" funded by the DFG (grants WO 2056/1-1 and WO 2056/4-1). Funding periods: 2015/07 – 2016/12 and 2017/09 – 2019/02
Optimal control, the turnpike property, and dissipativity
- DFG project "Actively Deformable Cannulas for Stereotactic Neurosurgery" (Aktiv krümmbare Kanülen für die stereotaktische Neurochirurgie),
(grant WO 2056/11-1, funding period: 2021/10 - 09/ 2023) → P. Sauerteig
Today, stereotactic procedures such as biopsies of deep-seated pathologies and deep brain stimulation are performed with surgical instruments that were designed for a straight approach from the entry point on the skull to the deep-seated target point. In this interdisciplinary project - jointly proposed by physicians, engineers, and mathematicians - the feasibility of actively deformable cannulas for stereotactic neurosurgery is assessed. The technique will offer a multitude of alternative routes and planning possibilities to reach a target with considerably reduced risk by avoiding eloquent structures, i.e. fiber bundles of the pyramidal tract and blood vessels. read more
- Sensitivity and turnpike analysis → Dr. M. Schaller
Many finite and infinite-dimensional optimal control problems are subject to an inherent stability that can be exploited for control and numerical algorithms. One consequence of this stability is the so-called turnpike property, which in its simplest form describes the fact the solutions of a dynamic optimal control problem on a long time horizon reside close to a steady state for the majority of the time. Besides being an interesting structural property in itself, the turnpike property can be used to prove convergence and performance estimates for Model Predictive Control. Another consequence of this stability that is the exponential decay of discretization errors, which can be leveraged to formulate very efficient time and space discretization techniques for Model Predictive Control.
Distributed control and optimization
- Participation in the project “Distributed Optimization and Control of Networked Energy Systems” (OptNetSys; chinesisch-deutsches Zentrum für Wissenschaftsförderung in der Förderlinie Mobilitätsprogramm: grant M-0007)
Information will be available soon (delayed project start due to COVID-19)
- Subproject "Distributed control of smart grids" (grant 05M18SIA) within the BMBF project KONSENS (Konsistente Optimierung uNd Stabilisierung Elektrischer NetzwerkSysteme), see also https://konsens.github.io/ → P. Sauerteig
The energy transition comes along with more and more volatile generators (renewable energy sources), flexible consumers (e-mobility) as well as novel energy storage devices (batteries, heat accumulators). Thus, new challenges have to be faced on the one hand, but also new opportunities can be explored on the other hand. Currently, potential flexibilities cannot be exploited within the traditional half-automated system management by grid operators. New mathematical methods and concepts are required to handle the interaction from the transmission grid through the distribution grid to the microgrid in a fully automated fashion.
The goal of this project is to develop mathematical optimization methods to exploit flexibilities on all levels of the electricity grid within the power flow optimization while ensuring security and stability. To this end, it is essential to focus on the core elements to generate mathematically manageable models.
- Frame Theory Dr. F. Philipp
Bases are systems of vectors that can be used to split vectors into their coordinates. This procedure is called analysis. Conversely, vectors can be reconstructed from its coordinates, which is called synthesis. The concept of frames is similar, but frames aren't bases in general - usually, they are overcomplete. Roughly speaking, a frame is a system of vectors that allows for stable analysis and synthesis, but the synthesis is not unique as in the case of bases. However, it is this redundancy which makes frames very flexible in design and, which is crucial, highly resilient to noise and losses. Therefore, they are often prefered over bases. In Frame Theory, frames are investigated as abstract objects, but also very specific types of systems are analyzed with repsect to conditions under which they are frames.
- Time-Frequency Analysis Dr. F. Philipp
The Fourier transform yields information about the frequency distribution of a time-variant signal f. However, it does not give any information about the frequencies of the signal in certain time-intervals. For this reason, one introduces the so-called Short-time Fourier transform (STFT), which depends on a so-called window function g and maps signals f to functions depending on both time and frequency. An important question in time-frequency analysis is whether signals can be stably recovered from samples of their STFT. This would be the case if the system of the corresponding time-frequency shifts (TFS) of the function g constituted a basis. Unfortunately, the Balian-Low theorem shows that this is not possible for functions that are sufficiently smooth in both time and frequency. However, the TFS of such a "nice" function might constitute a frame and a wide area of time-frequency analysis is dedicated to this kind of frames, called Garbor frames. The theory of Garbor frames is embedded in a part of functional analysis investigating function spaces such as modulation spaces and Bargmann spaces, several transforms and much more.
Data-driven modelling and control of dynamical systems
- Carl-Zeiss-Stiftung: DeepTurb – Deep Learning in and of Turbulence → Dr. F. Philipp
The application of machine learning (ML) techniques in the analysis of experimental measurements and numerical simulations of turbulence opens unique possibilities to analyse complex and comprehensive data sets by new physical criteria and thus to gather a deeper understanding of the fundamental transport processes in such flows. Our project aims at new effective modeling strategies of turbulent superstructures in extended turbulent convection flows – gradually evolving large-scale patterns – by means of machine learning techniques. We plan to accelerate the analysis in optical flow measurements, develop low-dimensional reduced models that can predict the coarse-scale dynamics, and extend the mathematical foundations of ML applications to obtain a more efficient prediction of these processes.
- Finite-data error bounds for Koopman-based prediction and control, F. Nüske, S. Peitz, F. Philipp, M. Schaller, K. Worthmann, in: arXiv:2108.07102, 2021
The Koopman operator has become an essential tool for data-driven approximation of dynamical (control) systems in recent years, e.g., via extended dynamic mode decomposition. Despite its popularity, convergence results and, in particular, error bounds are still quite scarce. In this paper, we derive probabilistic bounds for the approximation error and the prediction error depending on the number of training data points; for both ordinary and stochastic differential equations. Moreover, we extend our analysis to nonlinear control-affine systems using either ergodic trajectories or i.i.d. samples. Here, we exploit the linearity of the Koopman generator to obtain a bilinear system and, thus, circumvent the curse of dimensionality since we do not autonomize the system by augmenting the state by the control inputs. To the best of our knowledge, this is the first finite-data error analysis in the stochastic and/or control setting. Finally, we demonstrate the effectiveness of the proposed approach by comparing it with state-of-the-art techniques showing its superiority whenever state and control are coupled.