Control & Optimization

Regelung und Optimierung

One research focus of the Chair of Automatic Control is on the development of nonlinear and (model) predictive control schemes as well as path/trajectory planning for dynamical systems. An important aspect in this regard are real-time feasible algorithms and their embedded hardware implementation for highly dynamical applications.


Prof. Dr.-Ing. Knut Graichen
Tel.: +49 9131 85-27127
E-Mail | Homepage

Model predictive control

Model predictive control (MPC) is a modern control scheme that is particularly suitable for nonlinear systems (NMPC) with multiple control inputs and state/control constraints (e.g. actuator limits, security constraints, etc.). The main research focus at the institute is on the development of highly efficient and real-time feasible MPC schemes as well as the hardware portability of the corresponding algorithms, e.g. for dSPACE, PLC and ECU hardware. Besides the methodological research and algorithmic development, the application of these real-time MPC schemes on different mechatronic systems is subject of various industrial research projects.

The computational efficiency of the (nonlinear) MPC schemes is achieved by tailored suboptimal/real-time implementations that are based on the approximate solution of the underlying dynamic optimization problem with incremental refinement.

Part of the research was implemented within the open-source toolbox GRAMPC that implements an efficient (projected) gradient method with adaptive line search. GRAMPC allows for a convenient setup, tuning and utilization of the MPC scheme for highly dynamical systems with sampling times in the (sub)millisecond range or high-dimensional systems. GRAMPC contains an additional interface (GUI) to MATLAB/Simulink.

Current research extends the existing results to stochastic nonlinear systems (stochastic NMPC) as well as to modular, networked systems.


  • Distributed MPC for modular, networked systems: Link
  • Real-time NMPC-Toolbox GRAMPC: Link

MPC: Moving horizon optimization

Stochastic NMPC for collision avoidance with a probability of 99,9%

Real-time embedded optimization

Complex systems or highly dynamical systems such as mechatronic systems with sampling times in the (sub)millisecond range pose significant challenges for the real-time feasibility of optimization based methods, in particular in the presence of constraints. Focus at the institute therefore concerns memory and computation time efficient algorithms for online optimization and model predictive control that are suitable for hardware implementation with limited resources (embedded design).

The interfacing to standard soft- and hardware allows for a straightforward integration and simulation under Matlab/Simulink as well as the direct implementation on real-time hardware (e.g. dSPACE). The methods are used for efficient implementation on PLC and ECU level within several industrial projects and cooperations.


  • Real-time NMPC of a magnetic levitation system: Video
  • Interactive Java applet for NMPC of an overhead crane with state/control constraints: Java-Applet
  • PLC implementation of NMPC for an overhead crane (lab scale): Video

Connection to standard software

Embedded NMPC design on PLC

Nonlinear control methods

Besides the classical control task of stabilizing set points, the design of tracking controllers is of importance when the exact tracking of reference trajectories or set point changes are required. Typical examples are position changes in robotics (point-to-point control) or start-up and load changes in process engineering. In this regard, the research focus at the institute is on the development of nonlinear feedforward and feedback control schemes for nonlinear systems as well as for distributed/modular systems.

Feedforward controllers are often used in control engineering to realize transitions between set points or to track desired trajectories. Research in this field concerns inversion- and flatness-based methods for nonlinear systems as well as the design of optimal feedforward controllers.

A further research focus is on nonlinear control schemes, for instance, in the context of two-degrees-of-freedom (2dof) control schemes that are often employed in control engineering applications in order to separately design the tracking performance and robustness.


  • Control of a bionic kangaroo: Link
  • Trajectory planning for multiphysics systems: Link

Feedforward control scheme

Two-degrees-of-freedom control scheme

2dof control of a bionic kangaroo

Path planning

One focus of research at the Chair of Automatic Control is the field of motion planning of robots. Especially in dynamic environments, both the collision free and executability of movements must be ensured. Beside the calculation of a collision-free geometric space curve (path), a time profile must also be calculated, to ensure, that the motion can be performed by the robot. The coupling of path and time is called trajectory.

Usually the path planning problem can be divided into local and global. The main goal of the global planning is to find a feasible path from a start to a destination configuration. The algorithm has to calculate kinematically feasible and at the same time collision-free paths. For robot arms, the difficulty lies in the high number of degrees of freedom and the complex geometry, whereas in vehicles non-holonomic kinematics must be taken into account.

Afterwards, the local planning calculates the robot motion trajectory along the global path, including further constraints, such as maximum speeds and accelerations. In particular, optimization-based methods are being researched at the Chair of Automatic Control, as they are also used in model predictive control (MPC). Due to the complexity of the systems, real-time implementation of these methods on microcontrollers is a major challenge.


Globale Bahnplanung

Global path planning with optimal path in red