Optimal Control
Ordinary differential equations (ODEs) and differential-algebraic equations (DAEs) are widely used to model control systems in engineering, natural sciences, and economy. Optimal control plays a central role in optimizing such systems and to operate them efficiently and safely. Basically, an optimal control problem aims to minimize or maximize a given objective function subject to a dynamic system, constraints on controls or states, and boundary conditions. Typical applications are path planning tasks for UGVs, UAS, or robots, which often need to be performed in real-time.
Our research focuses on the following topics:
- derivation of necessary and sufficient optimality conditions for DAE optimal control problems
- convergence analysis for discretized optimal control problems
- direct discretization methods, specifically direct shooting methods and full discretization (collocation) methods
- dynamic programming techniques
- mixed-integer and hybrid optimal control
- bilevel optimal control problems, e.g. dynamic scheduling and Stackelberg games
- collision avoidance techniques
In addition, model design and simulation tools are part of our portfolio as well.
Optimization
Efficient optimization methods are at the core of every direct discretization method for optimal control problems, but of course optimization problems also occur in many other disciplines. Depending on the origin of the problem, the optimization problem can be convex or non-convex, small and dense, or large-scale and sparse. Solution methods need to be tailored to such structures in order to work efficiently. Linear-quadratic optimization problems deserve particular attention as they play an important role in linear model-predictive control and in iterative solution methods where they occur as subproblems.
Our group is active in algorithm development and in the analysis of certain classes of optimization problems:
- sequential quadratic programming (SQP) for small and dense and large-scale and sparse problems
- interior-point methods (IP) for linear-quadratic and nonlinear problems
- semi-smooth Newton methods for discretized optimal control problems
- generalized Nash equilibrium problems
- proximal augmented Lagrangian methods
- parametric sensitivity analysis and real-time optimization
Model-predictive Control (MPC)
Model-predictive control is a powerful and versatile optimization-based control concept, which is able to obey control and state constraints. The basic idea is to solve a (discretized) optimal control problem for the current state on a prediction horizon and to apply the first control of the control sequence to the system. Then, the prediction horizon is shiftet in time and the procedure is repeated. This way, MPC constitutes a feedback control law. The computational bottleneck is typically the underlying optimizer, which needs to be fast and robust.
We use MPC extensively for path following and path planning tasks and investigate the following aspects:
- multi-step model-predictive control
- use of parametric optimization in model-predictive control
- imitation learning model-predictive control
- design of path following controllers using model-predictive control
- economic model-predictive control as a tool for path planning
- tailored QP and NLP methods and software packages for model-predictive control
Dynamic Inversion Control
Dynamic inversion is a control concept which is well suited for path following problems. We often use it in combination with model-predictive control as a low-level controller, which tracks the mpc-generated trajectories. Since dynamic inversion usually does not involve costly operations, it can work at a high frequency. The underlying idea is to invert the dynamics of a system in order to find the control input which corresponds to a given reference path. Then, feedback terms are added such that the error dynamics in case of deviations from the reference path become asympotically stable.
Automatic Control Architecture
The robot operating system ROS is used as a tool to realistically simulate and eventually deploy control procedures. Herein, ROS provides the communication layer and coordinates the exchange of data between the system (simulated or real) and the controller (e.g. a model-predictive controller). The interface to the Unreal Engine essentially takes the current state and visualizes the system accordingly. This concept is simple but effective. It is flexible and well-suited for multi-agent systems.
