Moritz Diehl

University of Freiburg | Albert-Ludwigs-Universität Freiburg · Department of Microsystems Engineering (IMTEK)

We present a method for the approximate propagation of mean and covariance of a probability distribution through ordinary differential equations (ODE) with discontinous right-hand side. For piecewise affine systems, a normalization of the propagated probability distribution at every time step allows us to analytically compute the expectation integr...

This paper develops high-accuracy methods for numerically solving optimal control problems subject to nonsmooth differential equations with set-valued step functions. A notable subclass of these systems are Filippov systems. The set-valued step functions are here written as the solution map of a linear program. Using the optimality conditions of th...

This paper presents an implementation of robust model predictive control (MPC) for collision-free reference trajectory tracking for mobile robots. The presented approach considers the robot motion to be subject to process noise bounded by ellipsoidal sets. In order to efficiently handle the evolution of the disturbance ellipsoids within the MPC, th...

VV4RTOS is an activity supported by the European Space Agency aimed at the development and validation of a framework for the verification and validation of spacecraft guidance, navigation, and control (GNC) systems based on embedded optimisation, tailored to handle different layers of abstraction, from guidance and control (G&C) requirements down t...

In this article, we propose a novel time-energy optimal control approach with applications in cooperative eco-driving of connected and automated vehicles (CAVs) in urban traffic networks. Safely approaching and departing signalized intersections requires the satisfaction of both spatial equality constraints determined by intersection locations and...

For the fast approximate solution of Mixed-Integer Non-Linear Programs (MINLPs) arising in the context of Mixed-Integer Optimal Control Problems (MIOCPs) a decomposition algorithm exists that solves a sequence of three comparatively less hard subproblems to determine an approximate MINLP solution. In this work, we propose a problem formulation for...

Dynamic soaring for UAVs is a flight technique that enables continuous, powerless periodic flight patterns in the presence of a wind gradient. However, sufficiently large wind gradients are uncommon over land, while at offshore locations the largest wind gradients are located close to the ocean surface, thereby limiting the scope of practical appli...

The Finite Elements with Switch Detection (FESD) is a high-accuracy method for the numerical simulation and solution of optimal control problems subject to discontinuous ODEs. In this article, we extend the FESD method [35] to the dynamic equations of multiple rigid bodies that exhibit state jumps due to impacts and Coulomb friction. This new metho...

Direct collocation methods are powerful tools to solve trajectory optimization problems in robotics. While their resulting trajectories tend to be dynamically accurate, they may also present large kinematic errors in the case of constrained mechanical systems, i.e., those whose state coordinates are subject to holonomic or nonholonomic constraints,...

This work presents a novel loss function for learning nonlinear Model Predictive Control policies via Imitation Learning. Standard approaches to Imitation Learning neglect information about the expert and generally adopt a loss function based on the distance between expert and learned controls. In this work, we present a loss based on the Q-functio...

This paper extends the Finite Elements with Switch Detection (FESD) method [Nurkanovi\'c et al., 2022] to optimal control problems with nonsmooth systems involving set-valued step functions. Logical relations and common nonsmooth functions within a dynamical system can be expressed using linear and nonlinear expressions of the components of the ste...

We revisit three classical numerical methods for solving unconstrained optimal control problems - Multiple Shooting (MS), Single Shooting (SS), and Differential Dynamic Programming (DDP) - and examine their local convergence behaviour. In particular, we show that all three methods converge with the same linear rate if a Gauss-Newton (GN) - or more...

We consider a stage-varying nonlinear model predictive control (NMPC) formulation and provide a stability result for the corresponding closed-loop system under the assumption that cost and constraints are progressively tightening. We illustrate the generality of the stage-varying formulation pointing out various approaches proposed in the literatur...

In recent years, nonlinear model predictive control (NMPC) has been extensively used for solving automotive motion control and planning tasks. In order to formulate the NMPC problem, different coordinate systems can be used with different advantages. We propose and compare formulations for the NMPC related optimization problem, involving a Cartesia...

Flexible robots may overcome the industry's major problems: safe human-robot collaboration and increased load-to-mass ratio. However, oscillations and high dimensional state space complicate the control of flexible robots. This work investigates nonlinear model predictive control (NMPC) of flexible robots -- for simultaneous planning and control --...

We present an approach for safe trajectory planning, where a strategic task related to autonomous racing is learned sample-efficient within a simulation environment. A high-level policy, represented as a neural network, outputs a reward specification that is used within the cost function of a parametric nonlinear model predictive controller (NMPC)....

In this paper we present AWEbox, a Python toolbox for modeling and optimal control of multi-aircraft systems for airborne wind energy (AWE). AWEbox provides an implementation of optimization-friendly multi-aircraft AWE dynamics for a wide range of system architectures and modeling options. It automatically formulates typical AWE optimal control pro...

In this paper we introduce an open-source software package written in C++ for efficiently finding solutions to quadratic programming problems with linear complementarity constraints. These problems arise in a wide range of applications in engineering and economics, and they are challenging to solve due to their structural violation of standard cons...

This paper proposes and simulates vertical airborne wind energy (AWE) farms based on multi-aircraft systems with high power density (PD) per ground area. These farms consist of many independently ground located systems that are flying at the same inclination angle, but with different tether lengths, such that all aircraft fly in a large planar elli...

We present a novel optimization-based method for parameter estimation of a time-varying dynamic linear system. This method optimizes the likelihood of the parameters given measured data using an optimization algorithm tailored to the structure of this maximum likelihood estimation problem. Some parameters of the covariance of process and measuremen...

Mechanical systems are usually modeled by second-order Ordinary Differential Equations (ODE) which take the form $\ddot{q} = f(t, q, \dot{q})$. While simulation methods tailored to these equations have been studied, using them in direct optimal control methods is rare. Indeed, the standard approach is to perform a state augmentation, adding the vel...

In many industrial applications a workpiece is continuously fed through a heating zone in order to reach a desired temperature to obtain specific material properties. Many examples of such distributed parameter systems exist in heavy industry and also in furniture production such processes can be found. In this paper, a real-time capable model for...

Efficient integrators with sensitivity propagation are an essential ingredient for the numerical solution of optimal control problems. This paper gives an overview on the acados integrators, their Python interface and presents a workflow that allows using them with their sensitivities within a nonlinear programming (NLP) solver interfaced by CasADi...

Model-based control requires an accurate model of the system dynamics for precisely and safely controlling the robot in complex and dynamic environments. Moreover, in presence of variations in the operating conditions, the model should be continuously refined to compensate for dynamics changes. In this paper, we propose a self-supervised learning a...

This paper presents a modeling approach of an industrial heating process where a stripe-shaped workpiece is heated up to a specific temperature by applying hot air through a nozzle. The workpiece is moving through the heating zone and is considered to be of infinite length. The speed of the substrate is varying over time. The derived model is suppo...

In this paper we propose a formulation for approximate constrained nonlinear output-feedback stochastic model predictive control. Starting from the ideal but intractable stochastic optimal control problem (OCP), which involves the optimization over output-dependent policies, we use linearization with respect to the uncertainty to derive a tractable...

The future utility-scale deployment of airborne wind energy technologies requires the development of large-scale multi-megawatt systems. This study aims at quantifying the interaction between the atmospheric boundary layer (ABL) and large-scale airborne wind energy systems operating in a farm. To that end, we present a virtual flight simulator comb...

This paper introduces Finite Elements with Switch Detection (FESD), a numerical discretization method for nonsmooth differential equations. We regard the Filippov convexification of these systems and a transformation into dynamic complementarity systems introduced by Stewart. FESD is based on solving of nonlinear complementarity problems and able t...

In this paper, we propose a Feasible Sequential Linear Programming (FSLP) algorithm applied to time-optimal control problems (TOCP) obtained through direct multiple shooting discretization. This method is motivated by TOCP with nonlinear constraints which arise in motion planning of mechatronic systems. The algorithm applies a trust-region globaliz...

We present an efficient transcription method for highly oscillatory optimal control problems. For these problems, the optimal state trajectory consists of fast oscillations that change slowly over the time horizon. Out of a large number of oscillations, we only simulate a subset to approximate the slow change by constructing a semi-explicit differe...

We present a formulation for both implicit and explicit dual model predictive control with chance constraints. The formulation is applicable to systems that are affine in the state and disturbances, but possibly nonlinear in the controls. Awareness of uncertainty and dual control effect is achieved by including the covariance of a Kalman Filter sta...

This paper proposes an optimization-based approach to predict trajectories of autonomous race cars. We assume that the observed trajectory is the result of an optimization problem that trades off path progress against acceleration and jerk smoothness, and which is restricted by constraints. The algorithm predicts a trajectory by solving a parameter...

This letter introduces the open source software package for Nonsmooth Numerical Optimal Control (NOS-NOC). It is a modular tool based on CasADi [Andersson et al., 2019], IPOPT [W\"achter and Biegler, 2006] and MATLAB, for numerically solving Optimal Control Problems (OCP) with piecewise smooth systems (PSS). It relies on the recently introduced Fin...

This article regards numerical optimal control of a class of hybrid systems with hysteresis using solely techniques from nonlinear optimization, without any integer variables. Hysteresis is a rate independent memory effect which often results in severe nonsmoothness in the dynamics. These systems are not simply Piecewise Smooth Systems (PSS); they...

The tunnel-following nonlinear model predictive control (NMPC) scheme allows to exploit acceptable deviations around a path reference. This is done by using convex-over-nonlinear functions as objective and constraints in the underlying optimal control problem (OCP). The convex-over-nonlinear structure is exploited by algorithms such as the generali...

We present an efficient transcription method for highly oscillatory optimal control problems. For these problems, the optimal state trajectory consists of fast oscillations that change slowly over the time horizon. Out of a large number of oscillations, we only simulate a subset to approximate the slow change by constructing a semi-explicit differe...

This letter introduces the NOnSmooth Numerical Optimal Control (NOSNOC) open-source software package. It is a modular MATLAB tool based on CasADi and IPOPT for numerically solving Optimal Control Problems (OCP) with piecewise smooth systems (PSS). The tool supports: 1) automatic reformulation of systems with state jumps into PSS (via the time-freez...

This letter regards numerical optimal control of a class of hybrid systems with hysteresis using solely techniques from nonlinear optimization, without any integer variables. Hysteresis is a rate independent memory effect which often results in severe nonsmoothness in the dynamics. These systems are not simply Piecewise Smooth Systems (PSS); they a...

Direct collocation methods are powerful tools to solve trajectory optimization problems in robotics. While their resulting trajectories tend to be dynamically accurate, they may also present large kinematic errors in the case of constrained mechanical systems, i.e., those whose state coordinates are subject to holonomic or nonholonomic constraints,...

We propose a formulation for approximate constrained nonlinear output-feedback stochastic model predictive control. Starting from the ideal but intractable stochastic optimal control problem (OCP), which involves the optimization over output-dependent policies, we use linearization with respect to the uncertainty to derive a tractable approximation...

This article presents a systematic approach to realize highly dynamic control strategies for the air path of diesel engines. It is based on grey-box models of individual air path components, designed to be applied in nonlinear model predictive control (NMPC). Specifically, they are suited for algorithmic differentiation and gradient-based optimizat...

This paper introduces the quadratically-constrained quadratic programming (QCQP) framework recently added in HPIPM alongside the original quadratic-programming (QP) framework. The aim of the new framework is unchanged, namely providing the building blocks to efficiently and reliably solve (more general classes of) optimal control problems (OCP). Th...

In this paper, an asymptotic stability proof for a class of methods for inexact nonlinear model predictive control is presented. General Q-linearly convergent online optimization methods are considered and an asymptotic stability result is derived for the setting where a limited number of iterations of the optimizer are carried out per sampling tim...

The future utility-scale deployment of airborne wind energy technologies requires the development of large-scale multi-megawatt systems. This study aims at quantifying the interaction between the atmospheric boundary layer (ABL) and large-scale airborne wind energy systems operating in a farm. To that end, we present a virtual flight simulator comb...

This paper introduces a novel reformulation and numerical methods for optimal control of complementarity Lagrangian systems with state jumps. The solutions of the reformulated system have jump discontinuities in the first time derivative instead of the trajectory itself, which is easier to handle theoretically and numerically. We cover not only the...

This paper presents the acados software package, a collection of solvers for fast embedded optimization intended for fast embedded applications. Its interfaces to higher-level languages make it useful for quickly designing an optimization-based control algorithm by putting together different algorithmic components that can be readily connected and...

We provide an overview of a class of iterative convex approximation methods for nonlinear optimization problems with convex-over-nonlinear substructure. These problems are characterized by outer convexities on the one hand, and nonlinear, generally nonconvex, but differentiable functions on the other hand. All methods from this class use only first...

Moving Horizon Estimation (MHE) is an optimization-based approach to nonlinear state estimation where the state estimate is obtained as the solution of a nonlinear optimization problem. Especially for large-scale nonlinear systems, the computational burden associated with the numerical solution of nonlinear optimization problems poses a major chall...

Mathematical programs with complementarity constraints are notoriously difficult to solve due to their nonconvexity and lack of constraint qualifications in every feasible point. This work focuses on the subclass of quadratic programs with linear complementarity constraints. A novel approach to solving a penalty reformulation using sequential conve...

This work presents the results of experimental operation of a solar‐driven climate system using mixed‐integer nonlinear model predictive control (MPC). The system is installed in a university building and consists of two solar thermal collector fields, an adsorption cooling machine with different operation modes, a stratified hot water storage with...

This paper presents a decentralized algorithm for non-convex optimization over tree-structured networks. We assume that each node of this network can solve small-scale optimization problems and communicate approximate value functions with its neighbors based on a novel multi-sweep communication protocol. In contrast to existing parallelizable optim...

Two complementary simulators aimed at the dynamic analysis of airborne wind energy systems based on multi-aircraft congurations are presented. The rst model considers a train of stacked aircraft linked among them by two inelastic and massless tethers with no aerodynamic drag. The architecture of the mechanical system in the second simulator is cong...

Airborne wind energy systems convert wind energy into electricity using tethered flying devices, typically flexible kites or aircraft. Replacing the tower and foundation of conventional wind turbines can substantially reduce the material use and, consequently, the cost of energy, while providing access to wind at higher altitudes. Because the fligh...

Solving the nonlinear optimal control problem (OCP) arising from Economic Nonlinear model predictive control (ENMPC) reliably in real time is still a challenge. In this chapter, we present an extended multi-level iteration scheme (MLI) and discuss the interplay of exact Hessian and Hessian approximations. We show the importance of proper use of reg...

Mathematical programs with complementarity constraints are notoriously difficult to solve due to their nonconvexity and lack of constraint qualifications in every feasible point. In this work we focus on the subclass of quadratic programs with linear complementarity constraints. We introduce a novel approach to solving a penalty reformulation using...