Analysis and control of nonlinear systems

In this research area at the IST, we strive to develop analysis and controller design methods that can cope with the particularities 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 steady-state 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.


Please find below all our recent research fields at the Institute for Systems Theory and Automatic Control referring to Nonlinear Systems.


Control and Analysis of Nonlinear Systems

Data-based systems analysis and controller design

In this research direction, we seek to determine dissipation inequalities of an input-output system, which allow for the application of well-known feedback theorems for designing feedback controllers. In practice, a mathematical model of the input-output system is often undisclosed while only input-output data are available. The input-output data are in this context obtained from (numerical) experiments in which one applies probing signals to a system and measures the corresponding output. We study methods to determine dissipation inequalities of the underlying system from available input-output data in storage as well as methods for iteratively conducting experiments to determine the respective properties.


  • J. M. Montenbruck and F. Allgöwer.
    Some Problems Arising in Controller Design from Big Data via Input-Output Methods. In Proc. of the 55th Conference on Decision and Control (CDC), 2016.

  • A. Romer, J. M. Montenbruck and F. Allgöwer.
    Determining Dissipation Inequalities from Input-Output Samples. In Proc. 20th IFAC World Congress, 2017.

  • A. Romer, J. M. Montenbruck and F. Allgöwer.
    Sampling Strategies for Data-driven Inference of Passivity Properties. In Proc. of the 56th Conference on Decision and Control (CDC), 2017.

Extremum Seeking Control

The stabilization of an a priori known steady-state behavior is a common control task. However, there are many applications where the desired steady-state 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 model-free control method to solve such kind of problems, i.e. it is a method to find and stabilize an a priori unknown optimal steady-state behavior without the need of detailed model information.
Oscillator synchronisation via extremum seeking evolving on a torus
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 multi-agent systems or model-based extremum seeking algorithms. 


  • 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

Norm-controllability 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 time-invariant 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 point-to-point 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 norm-controllability, 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 input-to-state stability (or related notions involving outputs).

Research at the IST focuses on the development of different Lyapunov-like conditions for a systems to be norm-controllable, the analysis of relations to the standard controllability, and the potential of the novel framework in different application areas.


  • Publications:
  • M. A. Müller, D. Liberzon, and F. Allgöwer.
    Norm-controllability of nonlinear systems.
    IEEE Transactions on Automatic Control, vol. 60, no. 7, pp. 1825-1840, 2015.
  • M. A. Müller, D. Liberzon, and F. Allgöwer.
    Norm-controllability, 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.104-109.
  • Cooperations:
  • Daniel Liberzon, University of Illinois at Urbana-Champaign, USA

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.


  • 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.


  • 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. 1773-1779. 
  • G. S. Schmidt, S. Michalowsky, C. Ebenbauer, and F. Allgöwer.
    Global output regulation for the rotational dynamics of a rigid body.
    at-Automatisierungstechnik, Vol. 61, No. 8, pp. 567-582, 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.
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