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Timm Faulwasser

Publications and source records attributed to Timm Faulwasser.

At least 55 records · Page 3Linked to original sources
arXiv2024

Near-optimal performance of stochastic economic MPC

This paper presents first results for near optimality in expectation of the closed-loop solutions for stochastic economic MPC. The approach relies on a recently developed turnpike property for stochastic optimal control problems at an optimal stationary process, combined with techniques for analyzing time-varying economic MPC schemes. We obtain near optimality in finite time as well as overtaking and average near optimality on infinite time horizons.

math.OC↗
arXiv2024

On the relationship between stochastic turnpike and dissipativity notions

In this paper, we introduce and study different dissipativity notions and different turnpike properties for discrete-time stochastic nonlinear optimal control problems. The proposed stochastic dissipativity notions extend the classic notion of Jan C. Willems to $L^r$ random variables and to probability measures. Our stochastic turnpike properties range from a formulation for random variables via turnpike phenomena in probability and in probability measures to the turnpike property for the moments. Moreover, we investigate how different metrics (such as Wasserstein or Lévy-Prokhorov) can be leveraged in the analysis. Our results are built upon stationarity concepts in distribution and in random variables and on the formulation of the stochastic optimal control problem as a finite-horizon Markov decision process. We investigate how the proposed dissipativity notions connect to the various stochastic turnpike properties and we work out the link between different forms of dissipativity.

math.OC↗
arXiv2024

A continuous-time fundamental lemma and its application in data-driven optimal control

Data-driven control of discrete-time and continuous-time systems is of tremendous research interest. In this paper, we explore data-driven optimal control of continuous-time linear systems using input-output data. Based on a density result, we rigorously derive error bounds for finite-order polynomial approximations of elements of the system behavior. To this end, we leverage a link between latent variables and flat outputs of controllable systems. Combined with a continuous-time counterpart of the fundamental lemma by Willems et al., we characterize the suboptimality resulting from polynomial approximations in data-driven linear-quadratic optimal control problems. Finally, we draw upon a numerical example to illustrate our results.

math.OC↗
arXiv2024

On Dissipativity of Cross-Entropy Loss in Training ResNets

The training of ResNets and neural ODEs can be formulated and analyzed from the perspective of optimal control. This paper proposes a dissipative formulation of the training of ResNets and neural ODEs for classification problems by including a variant of the cross-entropy as a regularization in the stage cost. Based on the dissipative formulation of the training, we prove that the trained ResNet exhibit the turnpike phenomenon. We then illustrate that the training exhibits the turnpike phenomenon by training on the two spirals and MNIST datasets. This can be used to find very shallow networks suitable for a given classification task.

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arXiv2024

Distributed Optimization for Energy Grids: A Tutorial on ADMM and ALADIN

The ongoing transition towards energy and power systems dominated by a large number of renewable power injections to the distribution grid poses substantial challenges for system operation, coordination, and control. Optimization-based methods for coordination and control are of substantial research interest in this context. Hence, this chapter provides a tutorial introduction of distributed optimization algorithms for energy systems with a large share of renewables. Specifically, we focus on the Alternating Direction Method of Multipliers (ADMM) and on the Augmented Lagrangian Alternating Direction Inexact Newton (ALADIN) method as both algorithms are frequently considered for coordination and control of power and energy systems. We discuss the application of ALADIN and ADMM to AC optimal power flow problems and to energy management problems. Moreover, we give an outlook on open problems.

math.OC↗
arXiv2024

Towards Efficient Aggregation of Storage Flexibilities in Power Grids

The increasing penetration of volatile renewables combined with increasing demands poses a challenge to modern power grids. Furthermore, distributed energy resources and flexible devices (electric vehicles, PV generation, ...) are becoming more widespread, making their aggregate usage for ancillary services interesting. However, accurately quantifying the aggregate flexibility of numerous flexible devices is known to be limited by the curse of dimensionality, i.e., it does not scale well computationally. This has led to the development of various approximation algorithms. In this study, we improve upon our previously proposed vertex-based inner approximation, extending it to more general storage devices. We demonstrate the efficacy and accuracy of the proposed method in a case study comparing our approach with an exact centralized control framework, where the flexibility of numerous electric vehicles is combined to reduce the peak load in a residential area.

math.OC↗
arXiv2024

Alleviating the Curse of Dimensionality in Minkowski Sum Approximations of Storage Flexibility

Many real-world applications require the joint optimization of a large number of flexible devices over time. The flexibility of, e.g., multiple batteries, thermostatically controlled loads, or electric vehicles can be used to support grid operation and to reduce operation costs. Using piecewise constant power values, the flexibility of each device over $d$ time periods can be described as a polytopic subset in power space. The aggregated flexibility is given by the Minkowski sum of these polytopes. As the computation of Minkowski sums is in general demanding, several approximations have been proposed in the literature. Yet, their application potential is often objective-dependent and limited by the curse of dimensionality. We show that up to $2^d$ vertices of each polytope can be computed efficiently and that the convex hull of their sums provides a computationally efficient inner approximation of the Minkowski sum. Via an extensive simulation study, we illustrate that our approach outperforms ten state-of-the-art inner approximations in terms of computational complexity and accuracy for different objectives. Moreover, we propose an efficient disaggregation method applicable to any vertex-based approximation. The proposed methods provide an efficient means to aggregate and to disaggregate energy storages in quarter-hourly periods over an entire day with reasonable accuracy for aggregated cost and for peak power optimization.

math.OC↗
arXiv2024

Approximate Dynamic Programming with Feasibility Guarantees

Safe and economic operation of networked systems is often challenging. Optimization-based schemes are frequently considered, since they achieve near-optimality while ensuring safety via the explicit consideration of constraints. In applications, these schemes, however, often require solving large-scale optimization problems. Iterative techniques from distributed optimization are frequently proposed for complexity reduction. Yet, they achieve feasibility only asymptotically, which induces a substantial computational burden. This work presents an approximate dynamic programming scheme, which is guaranteed to deliver a feasible solution in "one shot", i.e., in one backward-forward iteration over all subproblems provided they are coupled by a tree structure. Our proposed scheme generalizes methods from seemingly disconnected domains such as power systems and optimal control. We demonstrate its efficacy for problems with nonconvex constraints via numerical examples from both domains.

math.OC↗
arXiv2023

Hidden regularity in singular optimal control of port-Hamiltonian systems

We study the problem of state transition on a finite time interval with minimal energy supply for linear port-Hamiltonian systems. While the cost functional of minimal energy supply is intrinsic to the port-Hamiltonian structure, the necessary conditions of optimality resulting from Pontryagin's maximum principle may yield singular arcs. The underlying reason is the linear dependence on the control, which makes the problem of determining the optimal control as a function of the state and the adjoint more complicated or even impossible. To resolve this issue, we fully characterize regularity of the (differential-algebraic) optimality system by using the interplay of the cost functional and the dynamics. In case of the optimality DAE being characterized by a regular matrix pencil, we fully determine the control on the singular arc. In case of singular matrix pencils of the optimality system, we propose an approach to compute rank-minimal quadratic perturbations of the objective such that the optimal control problem becomes regular. We illustrate the applicability of our results by a general second-order mechanical system and a discretized boundary-controlled heat equation.

math.OC↗
arXiv2023

The Dynamics of a Bicycle on a Pump Track -- First Results on Modeling and Optimal Control

We investigate the dynamics of a bicycle on an uneven mountain bike track split into straight sections with small jumps (kickers) and banked corners. A basic bike-rider model is proposed and used to derive equations of motion, which capture the possibilities to accelerate the bicycle without pedaling. Since this is a first approach to the problem, only corners connected by straight lines are considered to compute optimal riding strategies. The simulation is validated with experimental data obtained on a real pump track. It is demonstrated that the model effectively captures the longitudinal bike acceleration resulting from the relative vertical motion between the rider's upper body and the bicycle. Our numerical results are in good analogy with real rider's actions on similar tracks.

eess.SY↗
arXiv2023

On a Stochastic Fundamental Lemma and Its Use for Data-Driven Optimal Control

Data-driven control based on the fundamental lemma by Willems et al. is frequently considered for deterministic LTI systems subject to measurement noise. However, besides measurement noise, stochastic disturbances might also directly affect the dynamics. In this paper, we leverage Polynomial Chaos Expansions (PCE) to extend the deterministic fundamental lemma towards stochastic systems. This extension allows to predict future statistical distributions of the inputs and outputs for stochastic LTI systems in data-driven fashion, i.e., based on the knowledge of previously recorded input-output-disturbance data and of the disturbance distribution we perform data-driven uncertainty propagation. Finally, we analyze data-driven stochastic optimal control problems and we propose a conceptual framework for data-driven stochastic predictive control. Numerical examples illustrate the efficacy of the proposed concepts.

eess.SY↗
arXiv2023

Distributed Interior Point Methods for Optimization in Energy Networks

This note discusses an essentially decentralized interior point method, which is well suited for optimization problems arising in energy networks. Advantages of the proposed method are guaranteed and fast local convergence also for problems with non-convex constraints. Moreover, our method exhibits a small communication footprint and it achieves a comparably high solution accuracy with a limited number of iterations, whereby the local subproblems are of low computational complexity. We illustrate the performance of the proposed method on a problem from energy systems, i.e., we consider an optimal power flow problem with 708 buses.

math.OC↗
arXiv2023

An essentially decentralized interior point method for control

Distributed and decentralized optimization are key for the control of networked systems. Application examples include distributed model predictive control and distributed sensing or estimation. Non-linear systems, however, lead to problems with non-convex constraints for which classical decentralized optimization algorithms lack convergence guarantees. Moreover, classical decentralized algorithms usually exhibit only linear convergence. This paper presents an essentially decentralized primal-dual interior point method with convergence guarantees for non-convex problems at a superlinear rate. We show that the proposed method works reliably on a numerical example from power systems. Our results indicate that the proposed method outperforms ADMM in terms of computation time and computational complexity of the subproblems.

math.OC↗
arXiv2023

Distributionally robust uncertainty quantification via data-driven stochastic optimal control

This paper studies optimal control problems of unknown linear systems subject to stochastic disturbances of uncertain distribution. Uncertainty about the stochastic disturbances is usually described via ambiguity sets of probability measures or distributions. Typically, stochastic optimal control requires knowledge of underlying dynamics and is as such challenging. Relying on a stochastic fundamental lemma from data-driven control and on the framework of polynomial chaos expansions, we propose an approach to reformulate distributionally robust optimal control problems with ambiguity sets as uncertain conic programs in a finite-dimensional vector space. We show how to construct these programs from previously recorded data and how to relax the uncertain conic program to numerically tractable convex programs via appropriate sampling of the underlying distributions. The efficacy of our method is illustrated via a numerical example.

eess.SY↗
arXiv2023

Cooperative Distributed MPC via Decentralized Real-Time Optimization: Implementation Results for Robot Formations

Distributed model predictive control (DMPC) is a flexible and scalable feedback control method applicable to a wide range of systems. While the stability analysis of DMPC is quite well understood, there exist only limited implementation results for realistic applications involving distributed computation and networked communication. This article approaches formation control of mobile robots via a cooperative DMPC scheme. We discuss the implementation via decentralized optimization algorithms. To this end, we combine the alternating direction method of multipliers with decentralized sequential quadratic programming to solve the underlying optimal control problem in a decentralized fashion with nominal convergence guarantees. Our approach only requires coupled subsystems to communicate and does not rely on a central coordinator. Our experimental results showcase the efficacy of DMPC for formation control and they demonstrate the real-time feasibility of the considered algorithms.

eess.SY↗
arXiv2023

Behavioral Theory for Stochastic Systems? A Data-driven Journey from Willems to Wiener and Back Again

The fundamental lemma by Jan C. Willems and co-workers, which is deeply rooted in behavioral systems theory, has become one of the supporting pillars of the recent progress on data-driven control and system analysis. This tutorial-style paper combines recent insights into stochastic and descriptor-system formulations of the lemma to further extend and broaden the formal basis for behavioral theory of stochastic linear systems. We show that series expansions -- in particular Polynomial Chaos Expansions (PCE) of $L^2$-random variables, which date back to Norbert Wiener's seminal work -- enable equivalent behavioral characterizations of linear stochastic systems. Specifically, we prove that under mild assumptions the behavior of the dynamics of the $L^2$-random variables is equivalent to the behavior of the dynamics of the series expansion coefficients and that it entails the behavior composed of sampled realization trajectories. We also illustrate the short-comings of the behavior associated to the time-evolution of the statistical moments. The paper culminates in the formulation of the stochastic fundamental lemma for linear (descriptor) systems, which in turn enables numerically tractable formulations of data-driven stochastic optimal control combining Hankel matrices in realization data (i.e. in measurements) with PCE concepts.

math.OC↗
arXiv2023

Quantized Deep Path-following Control on a Microcontroller

Model predictive Path-Following Control (MPFC) is a viable option for motion systems in many application domains. However, despite considerable progress on tailored numerical methods for predictive control, the real-time implementation of predictive control and MPFC on small-scale autonomous platforms with low-cost embedded hardware remains challenging. While usual stabilizing MPC formulations lead to static feedback laws, the MPFC feedback turns out to be dynamic as the path parameter acts as an internal controller variable. In this paper, we leverage deep learning to implement predictive path-following control on microcontrollers. We show that deep neural networks can approximate the dynamic MPFC feedback law accurately. Moreover, we illustrate and tackle the challenges that arise if the target platform employs limited precision arithmetic. Specifically, we draw upon a post-stabilization with an additional feedback law to attenuate undesired quantization effects. Simulation examples underpin the efficacy of the proposed approach.

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