arXiv subjects

Bart Besselink

Publications and source records attributed to Bart Besselink.

At least 19 recordsLinked to original sources

arXiv2026

Safety by Invariance, Liveness through Refinement: Heterogeneous Contract Framework for Co-Design of Layered Control

Real-world control systems must achieve long-horizon objectives (liveness) while respecting continuous-time safety constraints, a combination that motivates hierarchical layered control architectures (LCAs). Existing LCA research, however, lacks (i) a uniform specification language across discrete planning and continuous execution, (ii) formal guarantees that specifications are preserved when interconnecting subsystems at heterogeneous time scales, and (iii) compositional separation between layers, owing to reliance on naive input-filtering laws. This paper addresses all three gaps by importing the safety--liveness decomposition into a heterogeneous assume--guarantee framework: \emph{safety is enforced by invariance} at the continuous-time layer, while \emph{liveness is achieved through refinement} at the discrete-time layer, with inter-layer coordination formalized via vertical refinement and timing-compatibility conditions. We instantiate this contract with a novel LCA combining an MPC planner, an input-to-state stabilizing (ISS) low-level controller, and a reference-governor bridge, and validate it on a Hybrid Energy Storage System (HESS) comprising a battery and a supercapacitor.

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arXiv2026

Vertical Contracts for Safety Control

We propose a methodology that exploits the contract formalism to characterize the continuous-time safety control problem, which is often difficult to address, in terms of a discrete-time one, for which numerous efficient solution scheme exist. We construct contracts as pairs of assumptions and guarantees which are set-valued mappings that describe the safe boundaries within which the system must operate. By formalizing safety control as contract implementation, we develop a vertical hierarchy according to which we translate implementation from continuous to discrete time. We accomplish this by constructing a discrete-time system and a contract such that a solution to the continuous-time implementation problem can be characterized in terms of a solution to its discrete-time counterpart. We then use this characterization to construct a control input that establishes implementation in continuous time on the basis of the control sequence that achieves implementation in discrete time.

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arXiv2026

Comparison of Linear Systems Across Time Domains: Continuous-time vs. Discrete-time

We develop a formal framework for the behavioral comparison of linear systems across different time domains. We accomplish this by introducing the notion of system interpolation, which determines whether the input-state trajectories of a continuous-time system can be realized as piecewise polynomial interpolations of the input-state trajectories of a discrete-time system. In this context, a piecewise polynomial interpolation of a discrete-time signal is characterized as a continuous-time function that coincides with the discrete-time signal at given sampling instants and can be realized as a polynomial of a prescribed degree over intervals between these instants. By representing piecewise polynomial functions as linear combinations of shifted Legendre polynomials, we characterize system interpolation as a subspace inclusion that is completely in terms of system parameters. This therefore allows for a computationally efficient comparison of the input-state behavior of a continuous-time system with that of a discrete-time one. We then exploit this characterization to discretize a given continuous-time system into a discrete-time one. Lastly, given a control specification, we exploit system interpolation to synthesize controllers that ensure satisfaction at each given sampling instant, while they measure the extent of (possible) violation over intervals between these instants.

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arXiv2026

Convergence of energy-based learning in linear resistive networks

Energy-based learning algorithms are alternatives to backpropagation and are well-suited to distributed implementations in analog electronic devices. However, a rigorous theory of convergence is lacking. We make a first step in this direction by analysing a particular energybased learning algorithm, Contrastive Learning, applied to a network of linear adjustable resistors. It is shown that, in this setup, Contrastive Learning is equivalent to projected gradient descent on a convex function with Lipschitz continuous gradient, giving a guarantee of convergence of the algorithm for a range of stepsizes. This convergence result is then extended to a stochastic variant of Contrastive Learning.

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arXiv2025

Singular networks and ultrasensitive terminal behaviors

Negative conductance elements are key to shape the input-output behavior at the terminals of a network through localized positive feedback amplification. The balance of positive and negative differential conductances creates singularities at which rich, intrinsically nonlinear, and ultrasensitive terminal behaviors emerge. Motivated by neuromorphic engineering applications, in this note we extend a recently introduced nonlinear network graphical modeling framework to include negative conductance elements. We use this extended framework to define the class of singular networks and to characterize their ultra-sensitive input/output behaviors at given terminals. Our results are grounded in the Lyapunov-Schmidt reduction method, which is shown to fully characterize the singularities and bifurcations of the input-output behavior at the network terminals, including when the underlying input-output relation is not explicitly computable through other reduction methods.

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arXiv2025

On Differential Controllability and Observability Functions

Differential balancing theory for nonlinear model reduction relies on differential controllability and observability functions. In this paper, we further investigate them from two different perspectives. First, we establish novel connections between these differential energy functions and their incremental counterparts by assuming the existence of the corresponding optimal state feedback for each controllability function. Specifically, an upper bound on the incremental controllability/observability function is provided by the corresponding differential energy function. Conversely, an upper bound on the differential controllability function can be estimated from the incremental controllability function. Furthermore, the differential observability function can be constructed from the incremental observability function. Second, we explore the positive definiteness of the differential controllability/observability function in the context of controllability/observability and stability.

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arXiv2025

Efficient Reduction of Interconnected Subsystem Models using Abstracted Environments

We present two frameworks for structure-preserving model order reduction of interconnected subsystems, improving tractability of the reduction methods while ensuring stability and accuracy bounds of the reduced interconnected model. Instead of reducing each subsystem independently, we take a low-order abstraction of its environment into account to better capture the dynamics relevant to the external input-output behaviour of the interconnected system, thereby increasing accuracy of the reduced interconnected model. This approach significantly reduces the computational costs of reduction by abstracting instead of fully retaining the environment. The two frameworks differ in how they generate these abstracted environments: one abstracts the environment as a whole, whereas the other abstracts each individual subsystem. By relating low-level errors introduced by reduction and abstraction to the resulting high-level error on the interconnected system, we are able to translate high-level accuracy requirements (on the reduced interconnected system) to low-level specifications (on abstraction and reduction errors) using techniques from robust performance analysis. By adhering to these low-level specifications, restricting the introduced low-level errors, both frameworks automatically guarantee the accuracy and stability of the reduced interconnected system. We demonstrate the effectiveness of both frameworks by applying them to a structural dynamics model of a two-stroke wafer stage, achieving improved accuracy and/or greater reduction compared to an existing method from literature.

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arXiv2024

Abstracted Model Reduction: A General Framework for Efficient Interconnected System Reduction

This paper introduces the concept of abstracted model reduction: a framework to improve the tractability of structure-preserving methods for the complexity reduction of interconnected system models. To effectively reduce high-order, interconnected models, it is usually not sufficient to consider the subsystems separately. Instead, structure-preserving reduction methods should be employed, which consider the interconnected dynamics to select which subsystem dynamics to retain in reduction. However, structure-preserving methods are often not computationally tractable. To overcome this issue, we propose to connect each subsystem model to a low-order abstraction of its environment to reduce it both effectively and efficiently. By means of a high-fidelity structural-dynamics model from the lithography industry, we show, on the one hand, significantly increased accuracy with respect to standard subsystem reduction and, on the other hand, similar accuracy to direct application of expensive structure-preserving methods, while significantly reducing computational cost. Furthermore, we formulate a systematic approach to automatically determine sufficient abstraction and reduction orders to preserve stability and guarantee a given frequency-dependent error specification. We apply this approach to the lithography equipment use case and show that the environment model can indeed be reduced by over 80\% without significant loss in the accuracy of the reduced interconnected model.

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arXiv2024

Learning in memristive electrical circuits

Memristors are nonlinear two-terminal circuit elements whose resistance at a given time depends on past electrical stimuli. Recently, networks of memristors have received attention in neuromorphic computing since they can be used as a tool to perform linear algebraic operations, like matrix-vector multiplication, directly in hardware. In this paper, the aim is to resolve two fundamental questions pertaining to a specific, but relevant, class of memristive circuits called crossbar arrays. In particular, we show (1) how the resistance values of the memristors at a given time can be determined from external (voltage and current) measurements, and (2) how the resistances can be steered to desired values by applying suitable external voltages to the network. The results will be applied to solve a prototypical learning problem, namely linear least squares, by applying and measuring voltages and currents in a suitable memristive circuit.

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arXiv2024

String stability and guaranteed safety via funnel cruise control for vehicle platoons

We study decentralized control strategies for platoons of autonomous vehicles with heterogeneous and nonlinear dynamics. Based on ideas from funnel control, we present a novel decentralized control algorithm which is able to guarantee a safety distance between any two vehicles, a good traffic flow and it achieves string stability of the controlled platoon. We illustrate the performance of the controller by simulations of two extreme scenarios.

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arXiv2024

Kron reduction of nonlinear networks

Kron reduction is concerned with the elimination of interior nodes of physical network systems such as linear resistor electrical circuits. In this paper it is shown how this can be extended to networks with nonlinear static relations between the variables associated to the edges of the underlying directed graph.

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arXiv2024

Modular Redesign of Mechatronic Systems: Formulation of Module Specifications Guaranteeing System Dynamics Specifications

Complex mechatronic systems are typically composed of interconnected modules, often developed by independent teams. This development process challenges the verification of system specifications before all modules are integrated. To address this challenge, a modular redesign framework is proposed in this paper. Herein, first, allowed changes in the dynamics (represented by frequency response functions (FRFs)) of the redesigned system are defined with respect to the original system model, which already satisfies system specifications. Second, these allowed changes in the overall system dynamics (or system redesign specifications) are automatically translated to dynamics (FRF) specifications on module level that, when satisfied, guarantee overall system dynamics (FRF) specifications. This modularity in specification management supports local analysis and verification of module design changes, enabling design teams to work in parallel without the need to iteratively rebuild the system model to check fulfilment of system FRF specifications. A modular redesign process results that shortens time-to-market and decreases redesign costs. The framework's effectiveness is demonstrated through three examples of increasing complexity, highlighting its potential to enable modular mechatronic system (re)design.

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arXiv2023

Comparison of Non-deterministic Linear Systems by $(γ,δ)$-Similarity

We introduce $(γ,δ)$-similarity, a notion of system comparison that measures to what extent two stable linear dynamical systems behave similarly in an input-output sense. This behavioral similarity is characterized by measuring the sensitivity of the difference between the two output trajectories in terms of the external inputs to the two potentially non-deterministic systems. As such, $(γ,δ)$-similarity is a notion that characterizes \emph{approximation} of input-output behavior, whereas existing notions of simulation target equivalence. Next, as this approximation is specified in terms of the $L_2$ signal norm, $(γ,δ)$-similarity allows for integration with existing methods for analysis and synthesis of control systems, in particular, robust control techniques. We characterize the notion of $(γ,δ)$-similarity as a linear matrix inequality feasibility problem and derive its interpretation in terms of transfer matrices. Our study on the compositional properties of $(γ,δ)$-similarity shows that the notion is preserved through series and feedback interconnections. This highlights its potential application in compositional reasoning, namely abstraction and modular synthesis of large-scale interconnected dynamical systems. We further illustrate our results in an electrical network example.

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arXiv2023

Passivity-Preserving, Balancing-Based Model Reduction for Interconnected Systems

This paper proposes a balancing-based model reduction approach for an interconnection of passive dynamic subsystems. This approach preserves the passivity and stability of both the subsystems and the interconnected system. Hereto, one Linear Matrix Inequality (LMI) per subsystem and a single Lyapunov equation for the entire interconnected system needs to be solved, the latter of which warrants the relevance of the reduction of the subsystems for the accurate reduction of the interconnected system, while preserving the modularity of the reduction approach. In a numerical example from structural dynamics, the presented approach displays superior accuracy with respect to an approach in which the individual subsystems are reduced independently.

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arXiv2023

Mode Selection for Component Mode Synthesis with Guaranteed Assembly Accuracy

In this work, a modular approach is introduced to select the most important eigenmodes for each component of a composed structural dynamics system to obtain the required accuracy of the reduced-order assembly model. To enable the use of models of complex (structural) dynamical systems in engineering practice, e.g., in a design, optimization and/or control context, the complexity of the models needs to be reduced. When the model consist of an assembly of multiple interconnected structural components, component mode synthesis is often the preferred model reduction method. The standard approach to component mode synthesis for such system is to select the eigenmodes of a component that are most important to accurately model the dynamic behavior of this component in a certain frequency range of interest. However, often, a more relevant goal is to obtain, in this frequency range, an accurate model of the assembly. In the proposed approach, accuracy requirements on the level of the assembly are translated to accuracy requirements on component level, by employing techniques from the field of systems and control. With these component-level requirements, the eigenmodes that are most important to accurately model the dynamic behavior of the assembly can be selected in a modular fashion. We demonstrate with two structural dynamics benchmark systems that this method based on assembly accuracy allows for a computationally efficient selection of eigenmodes that 1) guarantees satisfaction of the assembly accuracy requirements and 2) results in most cases in reduced-order models of significantly lower order with respect to the industrial standard approach in which component eigenmodes are selected using a frequency criterion.

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arXiv2023

Translating Assembly Accuracy Requirements to Cut-Off Frequencies for Component Mode Synthesis

One of the most popular methods for reducing the complexity of assemblies of finite element models in the field of structural dynamics is component mode synthesis. A main challenge of component mode synthesis is balancing model complexity and model accuracy, because it is difficult to predict how component reduction influences assembly model accuracy. This work introduces an approach that allows for the translation of assembly model accuracy requirements in the frequency domain to the automatic selection of the cut-off frequencies for the model-order reduction (MOR) of components. The approach is based on a mathematical approach for MOR for coupled linear systems in the field of systems and control. We show how this approach is also applicable to structural dynamics models. We demonstrate the use of this approach in the scope of component mode synthesis (CMS) methods with the aim to reduce the complexity of component models while guaranteeing accuracy requirements of the assembly model. The proposed approach is illustrated on a mechanical, three-component structural dynamics system for which reduced-order models are computed that are reduced further compared to reduction using standard methods. This results in lower simulation cost, while maintaining the required accuracy.

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arXiv2023

Modular Model Reduction of Interconnected Systems: A Top-Down Approach

Models of complex systems often consist of multiple interconnected subsystem/component models that are developed by multi-disciplinary teams of engineers or scientists. To ensure that such interconnected models can be applied for the purpose of simulation and/or control, a reduced-order model for the interconnected dynamics is needed. In the scope of this paper, we pursue this goal by subsystem reduction to warrant modularity of the reduction approach. Clearly, by reducing the complexity of the subsystem models, not only the accuracy of the subsystem models is affected, but, consequently, also the accuracy of the interconnected model. It is typically difficult to predict a priori how the interconnected model accuracy is affected precisely by the subsystem reduction. In this work, we address this challenge by introducing a top-down approach which enables the translation of given accuracy requirements of the interconnected system model to accuracy requirements at subsystem model level, by using mathematical tools from robust performance analysis. This allows for the independent reduction of subsystem models while guaranteeing the desired accuracy of the interconnected model. In addition, we show how this top-down approach can be used to significantly reduce the interconnected model in an illustrative structural dynamics case study.

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arXiv2022

Contract Composition for Dynamical Control Systems: Definition and Verification using Linear Programming

Designing large-scale control systems to satisfy complex specifications is hard in practice, as most formal methods are limited to systems of modest size. Contract theory has been proposed as a modular alternative to formal methods in control, in which specifications are defined by assumptions on the input to a component and guarantees on its output. However, current contract-based methods for control systems either prescribe guarantees on the state of the system, going against the spirit of contract theory, or can only support rudimentary compositions. In this paper, we present a contract-based modular framework for discrete-time dynamical control systems. We extend the definition of contracts by allowing the assumption on the input at a time $k$ to depend on outputs up to time $k-1$, which is essential when considering the feedback connection of an unregulated dynamical system and a controller. We also define contract composition for arbitrary interconnection topologies, under the pretence of well-posedness, and prove that this notion supports modular design, analysis and verification. This is done using graph theory methods, and specifically using the notions of topological ordering and backward-reachable nodes. Lastly, we use $k$-induction to present an algorithm for verifying vertical contracts, which are claims of the form "the conjugation of given component-level contracts is a stronger specification than a given contract on the integrated system". These algorithms are based on linear programming, and scale linearly with the number of components in the interconnected network. A numerical example is provided to demonstrate the scalability of the presented approach, as well as the modularity achieved by using it.

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