Optimal control aims to prescribe the motion of a dynamical system in such a way that a certain optimality criterion is achieved. The research focus lies in the development of efficient numerical schemes for the solution of optimal control problems that are based on structure-preserving integration. In particular, optimal control methods are designed for the treatment of multi-body systems as well as for complex systems with certain substructures for which hierarchical approaches are developed. Further aspects of research interest are the development of numerical methods using inherent properties of the dynamical system such as symmetries or invariant objects, multi-objective optimization and model predictive approaches for optimal control problems and the optimal control of hybrid systems. Besides the investigation of theoretical aspects regarding accuracy and convergence of the numerical schemes, their performance is validated by means of problems from different fields of applications, e.g., mechatronic systems, biomechanics and astrodynamics.

• Discrete mechanics and optimal control (DMOC)
• Exploiting symmetry in optimal control
• Primitives and multi-objective model predictive control
• Turnpikes in optimal control
• Safety-Critical control of High Dimensional Systems
• Optimal control of multirate dynamical systems
• Optimal control of hybrid systems and mixed-integer problems
• Optimal control approach to deep learning

For the solution of nonlinear optimal control problem, we developed the numerical method DMOC (Discrete Mechanics and Optimal Control). DMOC is based on the discretization of the variational structure of the mechanical system directly in contrast to other methods like, e.g., shooting, multiple shooting, or collocation methods, relying on a direct integration of the associated ordinary differential equations or on its fulfillment at certain grid points. In the context of variational integrators, the discretization of the Lagrange-d’Alembert principle leads to structure-preserving time-stepping equations which serve as equality constraints for the resulting finite dimensional nonlinear optimization problem that can be solved by standard nonlinear optimization techniques like sequential quadratic programming.

There are several benefits of this optimal control scheme compared to standard methods: 1. An implementation on configuration level only, rather than on configuration-momentum or configuration-velocity level, leads to computational savings. 2. The benefits of variational integrators are inherited into the optimal control context, e.g., in the presence of symmetry groups in the continuous dynamical system, also along the discrete trajectory the change in momentum maps is consistent. 3. The approximating scheme of the adjoint equations resulting from the necessary optimality conditions is again symplectic with the same order of approximation as for the state equations due to the symplecticity of the state discretization. For the treatment of holonomic constraints, as present for example in multi-body dynamics, the approach is extended for constrained mechanical systems (DMOCC).

During the last years DMOC(C) has been successfully applied to problems of trajectory planning for vehicle dynamics, space mission design, robotics and biomechanics.

Most optimal control techniques provide only locally optimal solutions and in general, a good initial guess is required. On the other hand, the use of global methods leads to high computational cost since a search over the whole phase space has to be performed. The main goal is to compute an approximate globally optimal solution which can serve as a good initial guess for iterative nonlinear control optimization.

Already for many applications of trajectory optimization in space mission design invariant manifolds have been proven to provide good initial guesses for a local optimization. This concept is extended to that effect that the space of trajectories is quantized by representative small pieces of energy efficient solution trajectories which can be combined in various ways. Candidates for these so-called motion primitives can be obtained by the inherent dynamical properties of the system under consideration, such as relative equilibria induced by symmetries and invariant (un)stable manifolds of hyperbolic fixed points. Additionally, short controlled maneuvers are required to arrange these trajectories in sequences. All pre-computed trajectories can be stored in a library which can then be used, e.g., for a roadmap algorithm to find the optimal sequence for the control problem. It is demonstrated by simple multi-body systems that an initial guess constructed in this way leads to more energy-efficient optimal solutions compared to an initial guess that is generated without exploiting system structures.

Model predictive control is a prominent approach to construct a feedback control loop
for dynamical systems. Due to real-time constraints, solving model-based optimal control problems in a very short amount of time can be challenging. In particular when considering nonlinear problems with multiple conflicting objective functions the subproblems defined on the prediction horizons are too expensive to solve in real time. The basic idea is to use motion primitives which can be considered as a modular construction system to design suitable solution trajectories. They encode structural system properties and can be assembled in various ways when exploiting system symmetries. This enables to solve multi-objective model predictive control problems in real time: In an offline phase, we build a library of Pareto optimal solutions from which we then obtain a valid compromise solution in the online phase according to a decision maker’s preference. In the online phase the symmetries in the dynamical system and the corresponding multi-objective optimal control problem are exploited to reduce the number of problems that has to be solved. This method has been successfully applied for the multi-objective model predictive control of an autonomous driving electric vehicle, also taking uncertainty into account.

Hybrid systems comprise continuous dynamics as well as discrete events. For example, structural changes in the system (discrete events) require a switch between different mathematical descriptions and models (continuous dynamics). Thus, the switching law encoding the switching times and the sequence of modes may serve as additional design parameters. Several approaches have been developed for the numerical solution of hybrid systems such as switching time optimization for continuous and time-discretized systems, two layers approaches which handle continuous dynamics and switching times of the discrete events on different layers as well as optimal control methods for mixed-integer problems allowing to determine not only the optimal switching times but also the optimal sequence of discrete events.

The property of optimal solutions of an optimal control problem to remain most of the time near an optimal steady state is known as the turnpike property. The characteristic features of the turnpike property are that it appears in case of different initial and final conditions when the solution is defined on a large time interval and that the time spent close to the steady state grows when the horizon increases. This property can be observed in many applications from economy to biology and mechanics. The knowledge of the turnpike property permits not only to characterize the asymptotic behaviour of solutions but also to develop more efficient numerical methods. It was recently discovered that this property can be generalized to a manifold turnpike property, where the set of attraction for optimal solutions is not reduced to points but can be a more general manifold. In particular, it was recently shown that mechanical systems with a certain kind of symmetries admit a generalized turnpike property toward a symmetry-induced manifold called trim manifold. The turnpike in this case has a more general form of convergence of optimal solutions of the original problem to solutions of some reduced problem on the trim manifold. We study this new variant of the turnpike property in relation to the dissipativity of the underlying optimal control problem and also possible extensions to more general systems, including non mechanical system.

We study a range of methodologies focused on the efficient and safe control of complex, high-dimensional systems. Conventional optimal control methods reliant on dynamic programming are often not tractable to applications that involve a large number of states or control variables. Safety-critical applications, especially in the presence of uncertainty, add additional constraints and complexity, further necessitating the need for approximation and/or state reduction techniques.

Drawing on research from Hamilton-Jacobi (HJ) reachability analysis with state constraints  (DOI, LibRef), Reinforcement Learning (DOI), and Multi-Objective Optimization (LibRef, LibRef), we develop new methods for efficient control of autonomous systems backed by rigorous safety guarantees. Though often theoretical in nature, our research has been applied to a variety of application areas, including Airborne Wind Energy and Spacecraft Trajectory Design.

The optimal control of systems, which exhibit dynamics and/or controls on different time scales, remains a challenging topic today. The simulation and control of such systems requires a very small discrete step in simulation in order to guarantee accurate resolution of the fast degrees of freedom and account for fast changing controls. For standard direct transcription methods this results in a large number of optimization variables and constraints and thus introduces great computational challenges.  

For this purpose, we have been working on formulating, validating, and extending a Multirate Discrete Mechanics and Optimal Control (Multirate DMOC) approach (LibRef). This method is an extension of single rate DMOC (LibRef,LibRef)  and as such relies on a discrete version of the Lagrange-d’Alembert principle to derive the description of the system dynamics. The resulting discrete-time model inherits the conservation properties of the continuous time model such as preservation of symplecticity and symmetries in the Lagrangian and allow accurate representation of energy and/or momenta for exponentially long times (Hairer, Lubich, Wanner, Geometric Numerical Integration DOI)

Most importantly by using a multirate model formulation  (LibRef) , Multirate DMOC allows slow and fast changing states and controls to be treated on separate macro and micro time grids, each with an appropriate time step, without the need to decouple the equations of motion. In this way and with the help of the structure-preserving variational model formulation the method achieves high fidelity resolution of both slow and fast dynamics. Furthermore, the use of a coarser macro grid reduces the number of instants at which slow changing dynamics and controls have to be computed, reducing the number of degrees of freedom and constraints in the optimization problem as well as providing a sparse structure for the Jacobian of the constraints   (LibRef,LibRef,LibRef)

The approach was first introduced in 2017 by (LibRef) and applied for the control of flexible spacecraft by (LibRef) . Most recently it has been used to formulate a multirate model predictive control scheme (ArXiv) . Model Predictive Control (MPC) is an online method for optimal control of systems with constraints on inputs and states, subject to model and measurement uncertainty. A key requirement for its successful implementation is the acquisition of a feasible trajectory within each step of the receding horizon control law in order to guarantee stability and recursive feasibility of the scheme (DOI). However, as mentioned earlier, the small discrete time step required for multirate systems reduces the available time to compute a solution while simultaneously increasing the size of the optimization problem solved online and thus the computational time for its solution. In this context Multirate DMOC was successfully used to develop a novel multirate nonlinear MPC scheme which addresses these issues and provides high fidelity solution at a greatly reduced computational cost as presented by  (ArXiv).

Deep Learning proved to be efficient in empirical learning from data. However, it still misses understanding of theoretical properties and behavior, and thus, may lead sometimes to significant errors. It was recently noted that a class of deep learning algorithms based on residual network (ResNet) can be seen as a discretized optimal control problem. Indeed, the propagation through layers of the given network coincides with the Euler integration scheme of some control system and the loss function which is minimized in deep learning corresponds to a discretized cost function.  Therefore, there exists a continuous counterpart corresponding to some given deep learning algorithm which takes form of an optimal control problems. Well developed analytical and numerical methods of optimal control theory can be used for construction of new deep learning architectures with predictable behaviour. In our work we use the optimal control approach to construct stable algorithms. On one hand, the continuous counterpart of a deep learning problem is set in such a way that it has stable solutions. On the other hand, the numerical methods should preserve the properties of these solutions. In this context, we analyse the symplectic integrators as well as their influence on the convergence properties and conservation of the stability properties of the analytic solutions. The choice of symplectic integrators is based on their role in the numerical analysis of general optimal control problems.