A fractional variational approach for modelling dissipative mechanical systems: continuous and discrete settings

Employing a phase space which includes the (Riemann-Liouville) fractional derivative of curves evolving on real space, we develop a restricted variational principle for Lagrangian systems yielding the so-called restricted fractional Euler-Lagrange equations (both in the continuous and discrete settings), which, as we show, are invariant under linear change of variables. This principle relies on a particular restriction upon the admissible variation of the curves. In the case of the half-derivative and mechanical Lagrangians, i.e. kinetic minus potential energy, the restricted fractional Euler-Lagrange equations model a dissipative system in both directions of time, summing up to a set of equations that is invariant under time reversal. Finally, we show that the discrete equations are a meaningful discretisation of the continuous ones

Variational multirate integration in discrete mechanics and optimal control

Systems with dynamics on different time scales have contradicting requirements on the integrator. These can be resolved with a multirate integration approach, where the system is split in parts which are integrated with different methods and time steps. This leads to computing time savings compared to a highly accurate simulation of the complete system. In this work, we benefit from these savings for optimal control problem simulations. Based on DMOC, which is a structure preserving simulation method for optimal control problems in mechanics, we develop an optimal control simulation method with a variational multirate integration scheme. Via an example system, we show convergence and the computing time behaviour of the multirate optimal control simulation method

Variational multirate integration in discrete mechanics and optimal control

Systems with dynamics on different time scales have contradicting requirements on the integrator. These can be resolved with a multirate integration approach, where the system is split in parts which are integrated with different methods and time steps. This leads to computing time savings compared to a highly accurate simulation of the complete system. In this work, we benefit from these savings for optimal control problem simulations. Based on DMOC, which is a structure preserving simulation method for optimal control problems in mechanics, we develop an optimal control simulation method with a variational multirate integration scheme. Via an example system, we show convergence and the computing time behaviour of the multirate optimal control simulation method

Reduced order model based multiobjective optimal control of fluids

In this article, we show how to combine reduced order modeling and multiobjective optimal control techniques in order to efficiently solve multiobjective optimal control problems constrained by PDEs. Using Galerkin projection and Proper Orthogonal Decomposition (POD), the underlying PDE is replaced by a system of ordinary differential equations, thereby drastically reducing the computational effort. We develop a gradient based algorithm by using an adjoint approach in combination with a scalarization method and the numerical results are illustrated with the example of the two-dimensional incompressible flow around a cylinder.

Multi-objective minimum time optimal control for low-thrust trajectory design

We propose a reachability approach for infinite and finite horizon multi-objective optimization problems for low-thrust spacecraft trajectory design. The main advantage of the proposed method is that the Pareto front can be efficiently constructed from the zero level set of the solution to a Hamilton-Jacobi-Bellman equation. We demonstrate the proposed method by applying it to a low-thrust spacecraft trajectory design problem. By deriving the analytic expression for the Hamiltonian and the optimal control policy, we are able to efficiently compute the backward reachable set and reconstruct the optimal trajectories. Furthermore, we show that any reconstructed trajectory will be guaranteed to be weakly Pareto optimal. The proposed method can be used as a benchmark for future research of applying reachability analysis to low-thrust spacecraft trajectory design

Verification of safety critical control policies using kernel methods

Hamilton-Jacobi reachability methods for safety-critical control have been well studied, but the safety guarantees derived rely on the accuracy of the numerical computation. Thus, it is crucial to understand and account for any inaccuracies that occur due to uncertainty in the underlying dynamics and environment as well as the induced numerical errors. To this end, we propose a framework for modeling the error of the value function inherent in Hamilton-Jacobi reachability using a Gaussian process. The derived safety controller can be used in conjuncture with arbitrary controllers to provide a safe hybrid control law. The marginal likelihood of the Gaussian process then provides a confidence metric used to determine switches between a least restrictive controller and a safety controller. We test both the prediction as well as the correction capabilities of the presented method in a classical pursuit-evasion example

Reduced order model based multiobjective optimal control of fluids

In this article, we show how to combine reduced order modeling and multiobjective optimal control techniques in order to effi- ciently solve multiobjective optimal control problems constrained by PDEs. Using Galerkin projection and Proper Orthogonal Decomposition (POD), the underlying PDE is replaced by a system of ordinary differential equations, thereby drastically reducing the computational effort. We develop a gradient based algorithm by using an adjoint approach in combination with a scalarization method and the numerical results are illustrated with the example of the two-dimensional incompressible flow around a cylinder

Second-order switching time optimization for switched dynamical systems

Switching time optimization arises in finite-horizon optimal control for switched systems where, given a sequence of continuous dynamics, one minimizes a cost function with respect to the switching times. We propose an efficient method for computing the optimal switching times for switched linear and nonlinear systems. A novel second-order optimization algorithm is introduced where, at each iteration, the dynamics are linearized over an underlying time grid to compute the cost function, the gradient and the Hessian efficiently. With the proposed method, the most expensive operations at each iteration are shared between the cost function and its derivatives, thereby greatly reducing the computational burden. We have implemented the algorithm in the Julia package SwitchTimeOpt, allowing users to easily solve switching time optimization problems. In the case of linear dynamics, many operations can be further simplified and benchmarks show that our approach is able to provide optimal solutions in just a few ms. In the case of nonlinear dynamics, our method provides optimal solutions with up to two orders of magnitude time reductions over state-of-the-art approaches

Symmetry and motion primitives in model predictive control

t Symmetries, e.g. rotational and translational invariances for the class of mechanical systems, allow to characterize solution trajectories of nonlinear dynamical systems. Thus, the restriction to symmetry-induced dynamics, e.g. by using the concept of motion primitives, may be considered as a quantization of the system. Symmetry exploitation is well-established in both motion planning and control. However, the linkage between the respective techniques to optimal control is not yet fully explored. In this manuscript, we want to lay the foundation for the usage of symmetries in Model Predictive Control (MPC). To this end, we investigate a mobile robot example in detail where our contribution is twofold: Firstly, we establish asymptotic stability of a desired set point w.r.t. the MPC closed loop, which is also demonstrated numerically by using motion primitives applied to the parallel parking scenario. Secondly, if the optimization criterion is not consistent with the symmetry action, we provide guidelines to rigorously derive stability guarantees based on symmetry exploitation