Analysis and control of nonlinear systems
Nonlinear systems arise, for instance, from first principles when deriving mathematical models for physical processes. An example for a nonlinear model are the equations describing the motion of a mechanical system. One particularity of these systems is that they typically exhibit nontrivial steadystate dynamics such as isolated periodic orbits or multiple isolated equilibria. The research at our institute focuses on the development of methods that cope with these particularities or that utilize them for control purposes.
 Contact Persons: Frank Allgöwer, Christian Ebenbauer, Jan Feiling, Victoria Grushkovskaya, Simon Michalowsky, Matthias A. Müller, Simon Niederländer, Anne Romer
Please find below all our recent research fields at the Institute for Systems Theory and Automatic Control referring to Nonlinear Systems.
Extremum Seeking Control
The stabilization of an a priori known steadystate behavior is a common control task. However, there are many applications where the desired steadystate behavior is unknown or changes over time. For a combustion engine, for example, a typical task is to find and stabilize an optimal operating point in order to maximize the efficiency or to minimize emissions. Due to the complex dynamics and changing operation conditions the optimal operating point of a combustion engine is often unknown or is steadily changing. Extremum seeking control is a modelfree control method to solve such kind of problems, i.e. it is a method to find and stabilize an a priori unknown optimal steadystate behavior without the need of detailed model information.
The research at the IST focuses on the development of methods to analyze and design extremum seeking systems, i.e. extremum seeking problems with constraints, vibrational stabilization, distributed extremum seeking problems for networked and multiagent systems or modelbased extremum seeking algorithms.
 Contact Persons: Christian Ebenbauer, Jan Feiling, Victoria Grushkovskaya, Simon Michalowsky
 Publications:
 HansBernd Dürr, Milos Stankovic, Christian Ebenbauer, KarlHenrik Johansson
Lie bracket approximation of extremum seeking systems
Automatica, 2013  HansBernd Dürr, Milos Stankovic, KarlHenrik Johansson, Christian Ebenbauer
Extremum seeking on submanifolds in the Euclidian space
Automatica, 2014  HansBernd Dürr, Miroslav Krstic, Alexander Scheinker, Christian Ebenbauer
Singularly Perturbed Lie Bracket Approximation
TAC, 2015  Simon Michalowsky, Christian Ebenbauer
Modelbased extremum seeking for a class of nonlinear systems
ACC, 2015
 Cooperations:
 Karl Henrik Johansson, KTH Royal Institute of Technology, Stockholm, Sweden
 Milos Stankovic, University of Belgrade, Belgrade, Serbia
 Miroslav Krstic, University of California San Diego (UCSD), San Diego (CA), USA
Normcontrollability of nonlinear systems
Controllability is one of the fundamental concepts in control theory. Usually, it is formulated as the ability to steer the state of a system from any point to any other point in any given time by an appropriate choice of the control input. For linear timeinvariant systems, controllability can be easily checked via necessary and sufficient matrix rank conditions. On the other hand, for general nonlinear control systems our understanding of pointtopoint controllability is much less complete, and even in those settings where controllability tests are available they are more difficult to apply.
Recently, we proposed a new notion called normcontrollability, which can be seen as a weaker/coarser version of the standard controllability. In particular, we analyse how the norm of the system state (or, more general, of some output) can be affected by applying inputs of different magnitude. Interestingly, this concept can also be seen as complementary to the well known concept of inputtostate stability (or related notions involving outputs).
Research at the IST focuses on the development of different Lyapunovlike conditions for a systems to be normcontrollable, the analysis of relations to the standard controllability, and the potential of the novel framework in different application areas.
 Contact Person: Frank Allgöwer, Matthias A. Müller
 Publications:
 M. A. Müller, D. Liberzon, and F. Allgöwer.
Normcontrollability of nonlinear systems.
IEEE Transactions on Automatic Control, vol. 60, no. 7, pp. 18251840, 2015.  M. A. Müller, D. Liberzon, and F. Allgöwer.
Normcontrollability, or how a nonlinear system responds to large inputs.
In Proc. of the 9th IFAC Symposium on Nonlinear Control Systems (NOLCOS), Toulouse, France, 2013, pp.104109.
 Cooperations:
 Daniel Liberzon, University of Illinois at UrbanaChampaign, USA
Convex Optimization via Dynamical Systems
We study and develop continuoustime algorithms to solve convex minimization problems. Our strategy is to pose the optimization problem at hand as one of "finding the zeros" of some (appropriate) operator via a dynamical system. Our objective is to synthesize algorithms that are asymptotically correct, i.e., whose solutions seek to solve the optimization as time evolves. To this end, we combine classical methods from optimization theory and dynamical systems to provide formal characterizations of the convergence and performance properties of the algorithms.
 Contact Person: Frank Allgöwer, Simon Niederländer
 Publications:
 S. K. Niederländer and J. Cortés. Distributed Coordination for Nonsmooth Convex Optimization via SaddlePoint Dynamics. SIAM Journal on Control and Optimization, submitted.

S. K. Niederländer, F. Allgöwer and J. Cortés. Exponentially Fast Distributed Coordination for Nonsmooth Convex Optimization. In Proc. of the 55th Conference on Decision and Control (CDC), Las Vegas, USA, 2016, pp. 10361041.
Submanifold Stabilization
In this research direction, we study control problems in which an embedded submanifold is ought to be stabilized. These problems include setpoint regulation (in which case the submanifold is a singleton), synchronization (in which case the submanifold is the span of the vector of ones), pattern generation (in which case the submanifold is a circle), and path following (in which case the submanifold is the image of a curve). We focus on developing constructive and graphical tools for this class of problems.
 Contact Person: Frank Allgöwer, Jan Maximilian Montenbruck
 Publications:
 JM Montenbruck, M Burger, F Allgower.
Compensating Drift Vector Fields with Gradient Vector Fields for Asymptotic Submanifold Stabilization.
To appear in IEEE TAC 61, 2016
 Cooperations:
 Murat Arcak, UC Berkeley, USA
Output Regulation for Rigid Body Systems
The theory of output regulation concerns a specific controller design method for nonlinear systems which are affected by a family of disturbances. The goal is to design a controller such that achieves asymptotic tracking a family of references for a system output while asymptotically rejecting the given family of disturbances. At the IST, we study output regulation problems for rigid body systems, which constitute one important class of mechanical models. The nontrivial geometry of the state space of rigid body systems leads to the presence of multiple isolated equilibria for smooth vector fields, which is a challenge for established methods. In this project, we develop new design tools to solve output regulation problems for nonlinear systems where the state space geometry enforces multiple isolated equilibria.
 Contact Person: Frank Allgöwer, Christian Ebenbauer, Gerd Simon Schmidt
 Publications:
 G. S. Schmidt, C. Ebenbauer, and F. Allgöwer.
A solution for a class of output regulation problems on SO(n).
In Proc. of the American Control Conference (ACC), Montreal, Canada, 2012, pp. 17731779.  G. S. Schmidt, S. Michalowsky, C. Ebenbauer, and F. Allgöwer.
Global output regulation for the rotational dynamics of a rigid body.
atAutomatisierungstechnik, Vol. 61, No. 8, pp. 567582, 2013.  G.S. Schmidt, C. Ebenbauer, F. Allgöwer.
Output Regulation for Control Systems on SE(n): A Separation Principle Based Approach.
Automatic Control, IEEE Transactions on, vol.59, no.11, pp.3057,3062, Nov. 2014.