Pub Date : 2026-04-01Epub Date: 2026-01-15DOI: 10.1016/j.automatica.2026.112838
Juan Javier Palacios Roman , Sigurdur Hafstein , Peter Giesl , Sebastiaan van den Eijnden , Stefania Andersen , W.P.M.H. Heemels
In this paper, we present a method for constructing continuous piecewise quadratic (CPQ) Lyapunov functions for continuous-time switched and conewise linear systems using linear programming (LP). Key in our approach is the formulation of effective sufficient conditions for the copositivity of matrices via diagonal dominance. This formulation consists of linear constraints and can be expressed as an LP. It is shown that the sufficient conditions are also necessary conditions for the existence of a Lyapunov function, given a sufficiently refined CPQ function. We provide an in-depth comparison between our new method and other computational methods in the literature, and provide extensive numerical experiments on various switched and conewise linear systems. In particular, we show that the proposed method is the most accurate of the LP-based methods for constructing Lyapunov functions and is a numerically competitive alternative to LMI-based methods.
{"title":"Constructing piecewise quadratic Lyapunov functions with linear programming for continuous-time switched and conewise linear systems","authors":"Juan Javier Palacios Roman , Sigurdur Hafstein , Peter Giesl , Sebastiaan van den Eijnden , Stefania Andersen , W.P.M.H. Heemels","doi":"10.1016/j.automatica.2026.112838","DOIUrl":"10.1016/j.automatica.2026.112838","url":null,"abstract":"<div><div>In this paper, we present a method for constructing continuous piecewise quadratic (CPQ) Lyapunov functions for continuous-time switched and conewise linear systems using linear programming (LP). Key in our approach is the formulation of effective sufficient conditions for the copositivity of matrices via diagonal dominance. This formulation consists of linear constraints and can be expressed as an LP. It is shown that the sufficient conditions are also necessary conditions for the existence of a Lyapunov function, given a sufficiently refined CPQ function. We provide an in-depth comparison between our new method and other computational methods in the literature, and provide extensive numerical experiments on various switched and conewise linear systems. In particular, we show that the proposed method is the most accurate of the LP-based methods for constructing Lyapunov functions and is a numerically competitive alternative to LMI-based methods.</div></div>","PeriodicalId":55413,"journal":{"name":"Automatica","volume":"186 ","pages":"Article 112838"},"PeriodicalIF":5.9,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145963137","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-04-01Epub Date: 2026-01-27DOI: 10.1016/j.automatica.2025.112767
Jens Püttschneider , Timm Faulwasser
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. Specifically, we consider a variant of the cross-entropy (label smoothing) as a loss function and as a regularization in the stage cost. Based on our dissipative formulation of the training, we prove that the training OCPs for ResNets and neural ODEs alike exhibit the turnpike phenomenon. We illustrate this finding with numerical results for the two spirals and MNIST datasets. Crucially, our training formulation ensures that the transformation of the data from input to output is achieved in the first layers. In the following layers, which constitute the turnpike, the data remains at an equilibrium state and therefore these layers do not contribute to the transformation learned. In principle, these layers can be pruned after training, resulting in a network with only the necessary number of layers thus simplifying tuning of hyperparameters.
{"title":"On dissipativity of cross-entropy loss in training ResNets — A turnpike towards architecture search","authors":"Jens Püttschneider , Timm Faulwasser","doi":"10.1016/j.automatica.2025.112767","DOIUrl":"10.1016/j.automatica.2025.112767","url":null,"abstract":"<div><div>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. Specifically, we consider a variant of the cross-entropy (label smoothing) as a loss function <em>and</em> as a regularization in the stage cost. Based on our dissipative formulation of the training, we prove that the training OCPs for ResNets and neural ODEs alike exhibit the turnpike phenomenon. We illustrate this finding with numerical results for the two spirals and MNIST datasets. Crucially, our training formulation ensures that the transformation of the data from input to output is achieved in the first layers. In the following layers, which constitute the turnpike, the data remains at an equilibrium state and therefore these layers do not contribute to the transformation learned. In principle, these layers can be pruned after training, resulting in a network with only the necessary number of layers thus simplifying tuning of hyperparameters.</div></div>","PeriodicalId":55413,"journal":{"name":"Automatica","volume":"186 ","pages":"Article 112767"},"PeriodicalIF":5.9,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146056059","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-04-01Epub Date: 2026-01-16DOI: 10.1016/j.automatica.2026.112827
Minghao Han , Kiwan Wong , Adrian Wing-Keung Law , Xunyuan Yin
In this work, we propose a meta-learning-based Koopman modeling and predictive control approach for nonlinear systems with parametric uncertainties. An adaptive deep meta-learning-based modeling approach, called Meta Adaptive Koopman Operator (MAKO), is proposed. Without knowledge of the parametric uncertainty, the proposed MAKO approach can learn a meta-model from a multi-modal dataset and efficiently adapt to new systems with previously unseen parameter settings by using online data. Based on the learned meta Koopman model, a predictive control scheme is developed, and the stability of the closed-loop system is ensured even in the presence of previously unseen parameter settings. Through extensive simulations, our proposed approach demonstrates superior performance in both modeling accuracy and control efficacy as compared to competitive baselines.
{"title":"MAKO: Meta-Adaptive Koopman Operators for learning-based model predictive control of parametrically uncertain nonlinear systems","authors":"Minghao Han , Kiwan Wong , Adrian Wing-Keung Law , Xunyuan Yin","doi":"10.1016/j.automatica.2026.112827","DOIUrl":"10.1016/j.automatica.2026.112827","url":null,"abstract":"<div><div>In this work, we propose a meta-learning-based Koopman modeling and predictive control approach for nonlinear systems with parametric uncertainties. An adaptive deep meta-learning-based modeling approach, called Meta Adaptive Koopman Operator (MAKO), is proposed. Without knowledge of the parametric uncertainty, the proposed MAKO approach can learn a meta-model from a multi-modal dataset and efficiently adapt to new systems with previously unseen parameter settings by using online data. Based on the learned meta Koopman model, a predictive control scheme is developed, and the stability of the closed-loop system is ensured even in the presence of previously unseen parameter settings. Through extensive simulations, our proposed approach demonstrates superior performance in both modeling accuracy and control efficacy as compared to competitive baselines.</div></div>","PeriodicalId":55413,"journal":{"name":"Automatica","volume":"186 ","pages":"Article 112827"},"PeriodicalIF":5.9,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145982058","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-04-01Epub Date: 2026-02-04DOI: 10.1016/j.automatica.2026.112851
Zhen Yang , Wangli He , Guanrong Chen
This paper investigates a distributed constrained optimization problem with privacy concerns over a multi-agent network, where each agent holds a private objective function and seeks a global optimizer subject to a global closed convex set of constraints in a distributed and privacy-preserving manner. To address this problem, a differentially private projected gradient tracking algorithm is proposed. The main idea is to introduce indirect projection in gradient tracking to handle the global constraints and to decompose the gradient tracking state into two sub-states, with the shared sub-state being perturbed by decaying Laplace noise, to establish differential privacy (DP). Two time scales and a lazy update rule are employed to facilitate the convergence analysis. It is demonstrated that the proposed algorithm simultaneously preserves linear convergence rates and -DP without requiring bounded gradients. Finally, numerical simulations are presented to verify the theoretical results.
{"title":"Differentially private projected gradient tracking for distributed constrained optimization","authors":"Zhen Yang , Wangli He , Guanrong Chen","doi":"10.1016/j.automatica.2026.112851","DOIUrl":"10.1016/j.automatica.2026.112851","url":null,"abstract":"<div><div>This paper investigates a distributed constrained optimization problem with privacy concerns over a multi-agent network, where each agent holds a private objective function and seeks a global optimizer subject to a global closed convex set of constraints in a distributed and privacy-preserving manner. To address this problem, a differentially private projected gradient tracking algorithm is proposed. The main idea is to introduce indirect projection in gradient tracking to handle the global constraints and to decompose the gradient tracking state into two sub-states, with the shared sub-state being perturbed by decaying Laplace noise, to establish differential privacy (DP). Two time scales and a lazy update rule are employed to facilitate the convergence analysis. It is demonstrated that the proposed algorithm simultaneously preserves linear convergence rates and <span><math><mi>ϵ</mi></math></span>-DP without requiring bounded gradients. Finally, numerical simulations are presented to verify the theoretical results.</div></div>","PeriodicalId":55413,"journal":{"name":"Automatica","volume":"186 ","pages":"Article 112851"},"PeriodicalIF":5.9,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146189716","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-04-01Epub Date: 2026-02-02DOI: 10.1016/j.automatica.2026.112855
Ivan Markovsky , Chris Verhoek , Roland Tóth
The most powerful unfalsified model (MPUM), i.e., the least complex exact model for the given data, is well established for linear time-invariant (LTI) systems. It has not been generalized for linear parameter-varying (LPV) systems. In order to do this, we define the notions of complexity for LPV systems with shifted-affine scheduling dependence. The MPUM leads to identifiability conditions and a method for exact LPV system identification. The method is based on lifting the LPV system to a higher dimensional space and LTI embedding in the lifted space. It is made rigorous by proving a formal connection between the parameters of the LTI embedding and the original LPV system.
{"title":"The most powerful unfalsified linear parameter-varying model","authors":"Ivan Markovsky , Chris Verhoek , Roland Tóth","doi":"10.1016/j.automatica.2026.112855","DOIUrl":"10.1016/j.automatica.2026.112855","url":null,"abstract":"<div><div>The most powerful unfalsified model (MPUM), <em>i.e.</em>, the least complex exact model for the given data, is well established for linear time-invariant (LTI) systems. It has not been generalized for linear parameter-varying (LPV) systems. In order to do this, we define the notions of complexity for LPV systems with shifted-affine scheduling dependence. The MPUM leads to identifiability conditions and a method for exact LPV system identification. The method is based on lifting the LPV system to a higher dimensional space and LTI embedding in the lifted space. It is made rigorous by proving a formal connection between the parameters of the LTI embedding and the original LPV system.</div></div>","PeriodicalId":55413,"journal":{"name":"Automatica","volume":"186 ","pages":"Article 112855"},"PeriodicalIF":5.9,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146189718","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-04-01Epub Date: 2026-01-20DOI: 10.1016/j.automatica.2026.112829
Filippos Fotiadis , Kyriakos G. Vamvoudakis
In this paper, we consider the problem of input–output data-driven sensor selection for unknown cyber–physical systems (CPS). In particular, out of a large set of sensors available for use, we choose a subset of them that maximizes a metric of observability of the CPS. The considered observability metric is related to the system norm, which quantifies the average output energy of the selected sensors over a finite or an infinite horizon. However, its computation inherently requires knowledge of the unknown matrices of the system, so we draw connections from the reinforcement learning literature and design an input–output data-driven algorithm to compute it in a model-free manner. We then use the derived data-driven metric expression to choose the best sensors of the system in polynomial time, effectively obtaining a provably convergent model-free sensor selection process. Additionally, we show how the proposed data-driven approach can be exploited to select sensors that optimize volumetric measures of observability, while also noting its applicability to the dual problem of actuator selection. Simulations are performed to demonstrate the validity and effectiveness of the proposed approach.
{"title":"Input–output data-driven sensor selection for cyber-physical systems","authors":"Filippos Fotiadis , Kyriakos G. Vamvoudakis","doi":"10.1016/j.automatica.2026.112829","DOIUrl":"10.1016/j.automatica.2026.112829","url":null,"abstract":"<div><div>In this paper, we consider the problem of input–output data-driven sensor selection for unknown cyber–physical systems (CPS). In particular, out of a large set of sensors available for use, we choose a subset of them that maximizes a metric of observability of the CPS. The considered observability metric is related to the <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> system norm, which quantifies the average output energy of the selected sensors over a finite or an infinite horizon. However, its computation inherently requires knowledge of the unknown matrices of the system, so we draw connections from the reinforcement learning literature and design an input–output data-driven algorithm to compute it in a model-free manner. We then use the derived data-driven metric expression to choose the best sensors of the system in polynomial time, effectively obtaining a provably convergent model-free sensor selection process. Additionally, we show how the proposed data-driven approach can be exploited to select sensors that optimize volumetric measures of observability, while also noting its applicability to the dual problem of actuator selection. Simulations are performed to demonstrate the validity and effectiveness of the proposed approach.</div></div>","PeriodicalId":55413,"journal":{"name":"Automatica","volume":"186 ","pages":"Article 112829"},"PeriodicalIF":5.9,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146006438","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-04-01Epub Date: 2026-02-03DOI: 10.1016/j.automatica.2026.112846
Matti Noack , Davi Goncalves Accioli , Johann Reger , Jerome Jouffroy
In the paper “Innovative non-asymptotic and robust estimation method using auxiliary modulating dynamical systems” by J. Liu et al., the authors discuss state estimation for singular systems using the modulating function method, incorporating auxiliary systems for kernel calculation. This note provides additional context for their work by highlighting that the use of auxiliary systems within the modulating function framework has already been explored in earlier contributions.
{"title":"Comments on “Innovative non-asymptotic and robust estimation method using auxiliary modulating dynamical systems” [Automatica 152 (2023) 110953]","authors":"Matti Noack , Davi Goncalves Accioli , Johann Reger , Jerome Jouffroy","doi":"10.1016/j.automatica.2026.112846","DOIUrl":"10.1016/j.automatica.2026.112846","url":null,"abstract":"<div><div>In the paper “Innovative non-asymptotic and robust estimation method using auxiliary modulating dynamical systems” by J. Liu et al., the authors discuss state estimation for singular systems using the modulating function method, incorporating auxiliary systems for kernel calculation. This note provides additional context for their work by highlighting that the use of auxiliary systems within the modulating function framework has already been explored in earlier contributions.</div></div>","PeriodicalId":55413,"journal":{"name":"Automatica","volume":"186 ","pages":"Article 112846"},"PeriodicalIF":5.9,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146109887","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-04-01Epub Date: 2026-01-21DOI: 10.1016/j.automatica.2026.112837
Rui Gao , Guang-Hong Yang , Tianyou Chai
This paper studies the problem of resilient distributed state estimation for a linear system using a set of agents connected through time-varying networks containing some Byzantine agents, where the Byzantine agents can arbitrarily manipulate the messages that they send or relay to other agents. First, an evaluation mechanism is introduced to assess the quality of the estimate generated by each agent at each step. Based on the quality evaluation and the multi-hop communication, a novel resilient distributed state estimation scheme is proposed, which enables each regular agent to evaluate the messages received from its time-varying multi-hop neighbors and discard the suspicious ones in real time. Then, to facilitate the stability analysis, the relationship between the estimates and the corresponding quality evaluations is established and the upper bound of the quality evaluations is derived. Specifically, the graph conditions under which all regular agents can estimate the state of the system exponentially fast at any given rate and even in finite time despite the existence of Byzantine agents are established. Different from the existing results requiring the time-varying networks to contain a sufficient amount of redundancy within bounded adjacent intervals, the proposed scheme can reduce the network redundancy requirement and allow the intervals over which such redundancy is preserved to grow over time. Finally, an example is simulated to validate the effectiveness of the proposed scheme.
{"title":"Quality evaluation-based resilient distributed state estimation over time-varying networks via multi-hop communication","authors":"Rui Gao , Guang-Hong Yang , Tianyou Chai","doi":"10.1016/j.automatica.2026.112837","DOIUrl":"10.1016/j.automatica.2026.112837","url":null,"abstract":"<div><div>This paper studies the problem of resilient distributed state estimation for a linear system using a set of agents connected through time-varying networks containing some Byzantine agents, where the Byzantine agents can arbitrarily manipulate the messages that they send or relay to other agents. First, an evaluation mechanism is introduced to assess the quality of the estimate generated by each agent at each step. Based on the quality evaluation and the multi-hop communication, a novel resilient distributed state estimation scheme is proposed, which enables each regular agent to evaluate the messages received from its time-varying multi-hop neighbors and discard the suspicious ones in real time. Then, to facilitate the stability analysis, the relationship between the estimates and the corresponding quality evaluations is established and the upper bound of the quality evaluations is derived. Specifically, the graph conditions under which all regular agents can estimate the state of the system exponentially fast at any given rate and even in finite time despite the existence of Byzantine agents are established. Different from the existing results requiring the time-varying networks to contain a sufficient amount of redundancy within bounded adjacent intervals, the proposed scheme can reduce the network redundancy requirement and allow the intervals over which such redundancy is preserved to grow over time. Finally, an example is simulated to validate the effectiveness of the proposed scheme.</div></div>","PeriodicalId":55413,"journal":{"name":"Automatica","volume":"186 ","pages":"Article 112837"},"PeriodicalIF":5.9,"publicationDate":"2026-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146014289","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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