Janick V. Frasch

A parallel quadratic programming method for dynamic optimization problems

Quadratic programming problems (QPs) that arise from dynamic optimization problems typically exhibit a very particular structure. We address the ubiquitous case where these QPs are strictly convex and propose a dual Newton strategy that exploits the block-bandedness similarly to an interior-point method. Still, the proposed method features warmstarting capabilities of active-set methods. We give details for an efficient implementation, including tailored numerical linear algebra, step size computation, parallelization, and infeasibility handling. We prove convergence of the algorithm for the considered problem class. A numerical study based on the open-source implementation qpDUNES shows that the algorithm outperforms both well-established general purpose QP solvers as well as state-of-the-art tailored control QP solvers significantly on the considered benchmark problems.

Optimal Control of Formula 1 Race Cars in a VDrift Based Virtual Environment

Abstract Control of autonomous vehicles and providing recommendations to drivers in real time are challenging tasks from an algorithmic point of view. To include realistic effects, such as nonlinear tire dynamics, at least medium-sized mathematical models need to be considered. Yet, fast feedback is of utmost importance. Existing Nonlinear Model Predictive Control (NMPC) algorithms need to be enhanced to comply with these two contradictory requirements. As the testing of algorithms in an automatic driving context is cumbersome and expensive, we propose a virtual testbed for NMPC of driving cars. We use the open source race simulator VDrift as virtual real world, in which algorithms need to cope with the mismatch between the detailed physical model in the simulator and a coarser approximative model used for NMPC. We present the general framework of this virtual environment and an optimal control problem based on a medium-sized ordinary differential equation model and a generic and flexible parameterization of the track constraint. We discuss one possible algorithmic approach to the task of minimum time driving including gear shifts and give preliminary open loop numerical results for a Porsche on Germanys Formula One racing circuit Hockenheimring. This can be used as a reference against which other (closed loop) solutions can be compared in the future.

Mixed-Level Iteration Schemes for Nonlinear Model Predictive Control

Abstract In general, numerical schemes for nonlinear model predictive control (NMPC) require the (approximate) solution of a nonlinear program in each sample for feedback generation. Thus, the application of NMPC to processes that need fast feedback poses a major computational challenge. Recently, new multi–level iteration schemes have been proposed, extending the well–known idea of real–time iterations. These algorithms take into account different time scales inherent in the dynamic model by updating the data of the feedback–generating quadratic program (QP), i.e., Hessians and Jacobians, gradients, and constraint residuals, on different levels. In this contribution we consider new mixed–level updates of the QP data, which interval–wise apply different update levels. In particular, we apply higher–level updates more frequently on the first intervals of the control horizon, given their importance in the context of model predictive control in general. Targeting at modern computers with multi–core processing units, we describe an efficient parallel implementation of the mixed–level iteration approach and apply it to a benchmark problem from automotive engineering.

Towards time-optimal race car driving using nonlinear MPC in real-time

This paper addresses the real-time control of autonomous vehicles under a minimum traveling time objective. Control inputs for the vehicle are computed from a nonlinear model predictive control (MPC) scheme. The time-optimal objective is reformulated such that it can be tackled by existing efficient algorithms for real-time nonlinear MPC that build on the generalized Gauss-Newton method. We numerically validate our approach in simulations and present a real-world hardware setup of miniature race cars that is used for an experimental comparison of different approaches.

A new quadratic programming strategy for efficient sparsity exploitation in SQP-based nonlinear MPC and MHE

Abstract A large class of algorithms for nonlinear model predictive control (MPC) and moving horizon estimation (MHE) is based on sequential quadratic programming and thus requires the solution of a sparse structured quadratic program (QP) at each sampling time. We propose a novel algorithm based on a dual two-level approach involving a nonsmooth version of Newtons method that aims at combining sparsity exploitation features of an interior point method with warm-starting capabilities of an active-set method. We address algorithmic details and present the open-source implementation qpDUNES. The effectiveness of the solver in combination with the ACADO Code Generation tool for nonlinear MPC is demonstrated based on set of benchmark problems, showing significant performance increases compared to the established condensing-based approach, particularly for problems with long prediction horizons.

A distributed method for convex quadratic programming problems arising in optimal control of distributed systems

We propose a distributed algorithm for strictly convex quadratic programming (QP) problems with a generic coupling topology. The coupling constraints are dualized via Lagrangian relaxation. This allows for a distributed evaluation of the non-smooth dual function and its derivatives. We propose to use both the gradient and the curvature information within a non-smooth variant of Newtons method to find the optimal dual variables. Our novel approach is designed such that the large Newton system never needs to be formed. Instead, we employ an iterative method to solve the Newton system in a distributed manner. The effectiveness of the method is demonstrated on an academic optimal control problem. A comparison with state-of-the-art first order dual methods is given.

Model Predictive Control of Autonomous Vehicles

The control of autonomous vehicles is a challenging task that requires advanced control schemes. Nonlinear Model Predictive Control (NMPC) and Moving Horizon Estimation (MHE) are optimization-based control and estimation techniques that are able to deal with highly nonlinear, constrained, unstable and fast dynamic systems. In this chapter, these techniques are detailed, a descriptive nonlinear model is derived and the performance of the proposed control scheme is demonstrated in simulations of an obstacle avoidance scenario on a low-fricion icy road.

MIP formulations for flowshop scheduling with limited buffers

We focus on MIP-formulations for flowshop scheduling problems of the kind Fm|lwt|γ, with the restriction lwt indicating that jobs are allowed to wait on a fixed limited number of buffers between machine levels. Most of the models discussed in literature only consider permutation schedules, i.e., schedules in which jobs are processed in identical order on all machines. As these are not necessarily optimal in the general case, there is a need for models which are not restricted in this way. In this paper, we try to fill this gap by presenting a new model which allows overtaking of jobs between different machine levels. We introduce position-tracking variables, variables that describe the paths of the jobs between the positions on succeeding machine levels, and allow for a special branching strategy exploiting the particular structure of this model. In order to exemplify our models applicability to various objectives, we consider three different objective functions. In particular, we discuss the minimization of the makespan, the sum of completion times, and the number of strand interruptions, an objective function which is highly important in steel industry. For all of these we present specific improvements to the formulation, yielding reasonable computation times on instances of practically relevant size and setting.