Current Research Projects of the Group:
Project period: 2018-2020
Funding institution: DFG
Funding amount: 190.600,00 €
Funding code: GR 5293/1-1
Motion planning for nonlinear control systems is one of the most important problems of mathematical control theory because of its theoretical challenges and many practical applications. Many applied problems are related to dynamic environments, where the locations of targets and obstacles change with time, e.g., when an autonomous vehicle should follow a moving target avoiding collisions with other moving objects. From a theoretical point of view, the complexity of control design for such tasks increases significantly, and many classical methods developed for static environments are no longer applicable.
The goal of this project is to develop a general framework for stabilization and motion planning of nonholonomic systems governed by driftless control-affine systems. The main idea is to guarantee that the motion of the system steers along the approximated gradient flow of a certain potential function. Depending on the approximation method, two types of controls will be constructed: gradient-based and gradient-free. Gradient-based controls may explicitly depend on the derivatives of a potential function and can be used in situations where its analytical expression is known, for example, when complete information on the target and obstacles is available. However, in a variety of practical scenarios an explicit analytical expression of the potential function is partially or completely unknown, for example, when the target trajectory is unknown, or in extremum seeking problems. For such cases, it is planned to construct gradient-free controls.
Project period: 2018-2019
Funding institution: ERASMUS + National Agency "Higher Education", DAAD - Deutscher Akademischer Austauschdienst
Funding sum: 66.000,00 €
Funding code: 2018-1-DE01-KA107-003928
Dynamical systems: on the response of attractors to disturbances.
Stability and robustness of attractors of nonlinear infinite-dimensional systems under the influences of disturbances is studied. In this project, Professor Sergey Dashkovskiy and his ambitious postdoc Dr. Jochen Schmid are working together with the group of Professor Oleksiy Kapustyan from the Taras Shevchenko National University of Kyiv. The postdoc position is financed by the DFG for two years.
The notion of an attractor originates from the theory of dynamical systems. An attractor is a state towards which a dynamical system is moving as time proceeds. This means that a set of variables approaches the attractor and then stays in its neighborhood. Attractors provide information about the long-term behavior of a system. However, certain disturbances can destroy an attractor or change its properties. Such effects are studied qualitatively and quantitatively in this new research project.
Another question that is raised in this project is how couplings of two or more dynamical systems with attractors influence the existence and the properties of an attractor of the overall system. The research group of Professor Kapustyan is specialized in the theory of attractors of nonlinear systems. The research group of Professor Dashkovskiy is specialized in disturbances and interconnections of systems. The skills of both teams complement each other in an ideal way.
Assistant: Dr. Joachim Schmid
Project period: 2016 - 2019
Funding institution: DFG
Funding sum: 168.800,00 €
Funding code: DA 767/7-1
Robust stability and stabilization of control systems are fundamental and challenging problems in control theory and its applications. One of the milestones of stability theory is the theory of input-to-state stability (ISS), developed over the last two decades for ordinary differential equations and more general finite dimensional systems. This concept is especially useful for the analysis of robust stability of nonlinear systems, control design for nonlinear systems and the dynamics of interconnected systems.
However, in the area of infinite-dimensional systems the theory is far from being complete. In modern applications, a crucial role is played by distributed parameter systems (DPS), both linear and nonlinear. ISS theory for such systems is becoming increasingly popular in recent years, but it is still fragmentary and considerably less developed than the finite dimensional case. Furthermore, over the last decade novel methods for stabilization of infinite-dimensional systems have been proposed; most notably, a continuum backstepping method. These approaches yield ISS-based methods for the design of robust and adaptive controllers for linear and nonlinear distributed parameter systems. To obtain powerful methods for control, however, major steps in the understanding and development of these methods are still required.
In this project we are going to build a firm basis for the investigation of input-to-state stability and stabilization of distributed parameter systems. More specifically, our aims are:
1. To develop an ISS theory for linear and bilinear distributed parameter systems, including criteria for input-to-state stability and stabilizability of linear and bilinear DPS and sufficient conditions for robustness of ISS.
2. To obtain the infinite-dimensional counterparts of fundamental nonlinear results from ISS theory of finite-dimensional systems. In particular, Lyapunov characterizations of the ISS property, small-gain theorems for DPS and characterizations of ISS in terms of other stability properties.
3. To develop methods for robust stabilization of infinite-dimensional systems, namely, a robust version of continuum backstepping, finite-time robust stabilization of partial differential equations and design of ISS stabilizers for port-Hamiltonian systems. In order to obtain these aims expertise is required in ISS theory, functional analysis, semigroup theory, infinite-dimensional systems theory, partial differential equations, backstepping design and Lyapunov theory. Therefore we will work on this project as a team, consisting of three groups with complementary knowledge covering all of the above topics. It is the aim of the project to establish a solid basis for the long term development of ISS theory as a fundamental tool for a wide range of nonlinear infinite-dimensional systems.
Project period: 2017-2020
Funding institution: DFG
Funding sum: 177.400,00 €
Funding code: SCHO 1780/1-1
- Prof. Dr. Frank Allgöwer, University of Stuttgart, Germany
- Prof. Dr. Volodymyr Andriyevskyy, Kent State University, Ohio, USA
- Prof. Dr. Oliver Roth, Chair of Mathematics IV, University of Würzburg, Germany
Ensembles are dynamical systems with a large, possibly infinite, number of states and/or system parameters, which shall be steered collectively by suitable control variables. For example, one can think of simultaneously steered parameter-dependent control systems or swarms of robots. An investigation of control strategies for ensembles also contributes to a better understanding of finite and infinite networks of interconnected dynamical systems. This field of research is increasingly important for mathematical control theory. Many of the emerging problems are still unsolved. The analysis of ensembles of control systems is currently in an initial phase. In particular, a rigorous mathematical characterization of ensemble observability and controllability are not developed yet. These questions establish new connections between approximation theory, complex analysis, and control theory. The intention of the project is to develop theoretical foundations and efficient methods for the control of ensembles of linear systems.
Assistant: Kateryna Sapozhnikova, M. Sc.
Project period: 2016 - 2018
Funding institution: Ernst-Abbe-Stiftung, Jena
Systems with dynamics which depends on a prehistory of the solution over some past time interval belong to a class of nonlinear infinite dimensional systems with time delay. The non-linearity enters to the system already due to the maximization of solution, which is a nonlinear and nonsmooth map. The system can be considered as one with time varying delay, where the delay may change discontinuously. In this project we study the stability of these kinds of systems with respect to external disturbances. Approximation methods are also studied.
Assistant: Michael Schönlein, Dr.
Project period: 2016-2018
Funding institution: DFG
Funding sum: 171.700 (original amount)
Funding code: HE 1858/14-1
The main goal of the project is the simulation of large, possibly infinitely many, linear systems by means of a single open loop control or a single feedback regulator. The initial task is to develop comprehensive theoretical foundations for the control and observation of parameter-dependent linear systems, so-called linear ensembles.
Project A: Stability and Stabilization of Large Digital Networks
Project management: Prof. Dr. F. Wirth
Project period: 01/2008 - 12/2010
Project B: Observation and Control of Heterogeneous Dynamical Systems
Project management: Prof. Dr. U. Helmke, Prof. Dr. K. Schilling
Project period: 10/2007 - 09/2010