Alexandre Didier

Last Name

Didier

First Name

Alexandre

ORCID

0000-0001-5396-8996

Organisational unit

09563 - Zeilinger, Melanie / Zeilinger, Melanie

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Publications 1 - 8 of 8

Safety, Stability and Performance in Learning-Based Control Using Predictive Control Methods
Didier, Alexandre (2025)
The optimal control problem is a fundamental concept in control theory. The aim is to minimize a cost function subject to the modeled dynamics of the system. Many applications in the domains of robotics, autonomous driving, aerospace and biomedical engineering among others fall within this framework. Beyond performance specifications, most real-world systems have physical or task-specific restrictions, which should be satisfied during operation. These restrictions can be formulated mathematically as constraints in the optimal control problem. Additionally, models are generally not accurate and thereby model uncertainties such as exogenous disturbances need to be considered, leading to a constrained robust optimal control problem (CROCP) formulation. Popular approaches to obtain approximate solutions to optimal control problems are given by model predictive control (MPC) and learning-based methods, such as, e.g., reinforcement learning. While MPC provides a principled solution approach when the system is subject to constraints, it can struggle in tasks with sparse cost signals which require a long prediction horizon. Learning-based methods can overcome these issues, however it can be difficult to ensure constraint satisfaction guarantees, especially when the system is subject to exogenous disturbances. Safety filters have recently emerged as a principled method to augment controllers with constraint satisfaction guarantees. The proposed inputs are projected onto the set of inputs for which constraints can be satisfied for all times, aiming to retain the performance obtained from the employed controller, while ensuring safe system operation. If the constraints on the system are ever violated during operation, methods should ideally be designed to ensure that the system recovers back to a safe region of operation. A framework that provides such guarantees is given by discrete-time control barrier functions (CBF). CBFs extend Lyapunov functions, allowing to certify invariance and stability of a set, rather than the origin, in the state space. However, similarly to Lyapunov functions, obtaining an explicit formulation of a CBF can be a challenging task. A key challenge is therefore to design control methods, that provide robust constraint satisfaction, ideally with recovery guarantees, while achieving a high performance. Such methods should introduce little conservativeness in terms of overly restricting the system from approaching constraints, while at the same time being computationally efficient. In this thesis, we consider approaches which aim to satisfy these requirements, either solving the CROCP directly or augmenting any controller with robust constraint satisfaction and recovery guarantees. In the first part of the thesis, we consider a specific cost function for the optimal control problem, i.e., we consider the generalized dynamic regret, a comparative performance metric. For systems subject to exogenous disturbances, the controller which achieves the optimal cost has access to all future disturbances in a non-causal fashion. As this is generally not implementable for dynamical systems, regret optimal control aims to minimize the cost difference to this surrogate benchmark controller. We propose a semi-definite program (SDP) for the generalized dynamic regret minimization problem for linear dynamical systems subject to additive disturbances with bounded energy. If explicit bounds on the disturbance are known at every time step, we modify the proposed method such that an improved bound on the incurred regret can be achieved. The optimization problem formulation enables integrating state and input constraints in the controller synthesis, allowing to compute a controller with closed-loop constraint satisfaction guarantees. In the second part of the thesis, we consider predictive control methods, which guarantee constraint satisfaction for nonlinear dynamical systems subject to disturbances. Additionally, we provide recovery mechanisms in case the constraints are violated during operation due to unexpected disturbances acting on the system. The proposed methods allow for the integration of a general control cost or they can be framed in terms of a safety filter, allowing to augment learning-based control methods. First, we consider how state measurements during online operation can be used to reduce conservativeness of predictive safety filters by improving uncertain model descriptions and improving its terminal set. Next, we provide a theoretical analysis of discrete-time CBFs allowing to analyze closed-loop behavior when the system is subject to disturbances and propose a robust MPC-based CBF formulation. We propose multiobjective approaches allowing to guarantee closed-loop stability through a Lyapunov and CBF decrease constraint, while optimizing a primary performance objective. Finally, we propose a computationally efficient approximation-based CBF formulation.
A System Level Approach to Regret Optimal Control
Didier, Alexandre; Sieber, Jerome; Zeilinger, Melanie N. (2022)
IEEE Control Systems Letters
We present an optimisation-based method for synthesising a dynamic regret optimal controller for linear systems with potentially adversarial disturbances and known or adversarial initial conditions. The dynamic regret is defined as the difference between the true incurred cost of the system and the cost which could have optimally been achieved under any input sequence having full knowledge of all future disturbances for a given disturbance energy. This problem formulation can be seen as an alternative to classical H2 - or H∞ -control. The proposed controller synthesis is based on the system level parametrisation, which allows reformulating the dynamic regret problem as a semi-definite problem. This yields a new framework that allows to consider structured dynamic regret problems, which have not yet been considered in the literature. For known pointwise ellipsoidal bounds on the disturbance, we show that the dynamic regret bound can be improved compared to using only a bounded energy assumption and that the optimal dynamic regret bound differs by at most a factor of 2/π from the computed solution. Furthermore, the proposed framework allows guaranteeing state and input constraint satisfaction.
Moving Horizon Estimation for Simultaneous Localization and Mapping with Robust Estimation Error Bounds
Trisovic, Jelena; Didier, Alexandre; Muntwiler, Simon; et al. (2025)
2025 European Control Conference (ECC)
This paper presents a robust moving horizon estimation (MHE) approach with provable estimation error bounds for solving the simultaneous localization and mapping (SLAM) problem. We derive sufficient conditions to guarantee robust stability in ego-state estimates and bounded errors in landmark position estimates, even under limited landmark visibility which directly affects overall system detectability. This is achieved by decoupling the MHE updates for the ego-state and landmark positions, enabling individual landmark updates only when the required detectability conditions are met. The decoupled MHE structure also allows for parallelization of landmark updates, improving computational efficiency. We discuss the key assumptions, including ego-state detectability and Lipschitz continuity of the landmark measurement model, with respect to typical SLAM sensor configurations, and introduce a streamlined method for the range measurement model. Simulation results validate the considered method, highlighting its efficacy and robustness to noise.
Generalised Regret Optimal Controller Synthesis for Constrained Systems
Didier, Alexandre; Zeilinger, Melanie N. (2023)
IFAC-PapersOnLine ~ 22nd IFAC World Congress
This paper presents a synthesis method for the generalised dynamic regret problem, comparing the performance of a strictly causal controller to the optimal non-causal controller under a weighted disturbance. This framework encompasses both the dynamic regret problem, considering the difference of the incurred costs, as well as the competitive ratio, which considers their ratio, and which have both been proposed as inherently adaptive alternatives to classical control methods. Furthermore, we extend the synthesis to the case of pointwise-in-time bounds on the disturbance and show that the optimal solution is no worse than the bounded energy optimal solution and is lower bounded by a constant factor, which is only dependent on the disturbance weight. The proposed optimisation-based synthesis allows considering systems subject to state and input constraints. Finally, we provide a numerical example which compares the synthesised controller performance to H2- and H∞-controllers.
Approximate Predictive Control Barrier Functions using Neural Networks: A Computationally Cheap and Permissive Safety Filter
Didier, Alexandre; Jacobs, Robin; Sieber, Jerome; et al. (2023)
2023 European Control Conference (ECC)
A predictive control barrier function (PCBF) based safety filter is a modular framework to verify safety of a control input by predicting a future trajectory. The approach relies on the solution of two optimization problems, first computing the minimal state constraint violation given the current state in the form of slacks on the constraint, and then computing the minimal deviation from a proposed input given the previously computed minimal slacks. This paper presents an approximation procedure that uses a neural network to approximate the optimal value function of the first optimization problem, which defines a control barrier function (CBF). By including this explicit approximation in a CBF-based safety filter formulation, the online computation becomes independent of the prediction horizon. It is shown that this approximation guarantees convergence to a neighborhood of the feasible set of the PCBF safety filter problem with zero constraint violation. The convergence result relies on a novel class K lower bound on the PCBF decrease and depends on the approximation error of the neural network. Lastly, we demonstrate our approach in simulation for an autonomous driving example and show that the proposed approximation leads to a significant decrease in computation time compared to the original approach.
Understanding the Differences in Foundation Models: Attention, State Space Models, and Recurrent Neural Networks
Sieber, Jerome; Amo Alonso, Carmen; Didier, Alexandre; et al. (2024)
Advances in Neural Information Processing Systems 37
Softmax attention is the principle backbone of foundation models for various artificial intelligence applications, yet its quadratic complexity in sequence length can limit its inference throughput in long-context settings. To address this challenge, alternative architectures such as linear attention, State Space Models (SSMs), and Recurrent Neural Networks (RNNs) have been considered as more efficient alternatives. While connections between these approaches exist, such models are commonly developed in isolation and there is a lack of theoretical understanding of the shared principles underpinning these architectures and their subtle differences, greatly influencing performance and scalability. In this paper, we introduce the Dynamical Systems Framework (DSF), which allows a principled investigation of all these architectures in a common representation. Our framework facilitates rigorous comparisons, providing new insights on the distinctive characteristics of each model class. For instance, we compare linear attention and selective SSMs, detailing their differences and conditions under which both are equivalent. We also provide principled comparisons between softmax attention and other model classes, discussing the theoretical conditions under which softmax attention can be approximated. Additionally, we substantiate these new insights with empirical validations and mathematical arguments. This shows the DSF's potential to guide the systematic development of future more efficient and scalable foundation models.
Predictive stability filters for nonlinear dynamical systems affected by disturbances
Didier, Alexandre; Zanelli, Andrea; Wabersich, Kim P.; et al. (2024)
IFAC-PapersOnLine ~ 8th IFAC Conference on Nonlinear Model Predictive Control NMPC 2024 Proceedings
Computationally efficient system level tube-MPC for uncertain systems
Sieber, Jerome; Didier, Alexandre; Zeilinger, Melanie N. (2025)
Automatica
Tube-based model predictive control (MPC) is one of the principal robust control techniques for constrained linear systems affected by additive disturbances. While tube-based methods with online-computed tubes have been successfully applied to systems with additive disturbances, their application to systems affected by additional model uncertainties is challenging. This paper proposes a tube-based MPC method – named filter-based system level tube-MPC (SLTMPC) – which overapproximates both types of uncertainties with an online optimized disturbance set, while simultaneously computing the tube controller online. For the first time, we provide rigorous closed-loop guarantees for receding horizon control of such a MPC method. These guarantees are obtained by virtue of a new terminal controller design and an online optimized terminal set. To reduce the computational complexity of the proposed method, we additionally introduce an asynchronous computation scheme that separates the optimization of the tube controller and the nominal trajectory. Finally, we provide a comprehensive numerical evaluation of the proposed methods to demonstrate their effectiveness.

Publications 1 - 8 of 8

Alexandre Didier

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