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Learning optimal safety certificates for unknown nonlinear control systems
IF 1.8 Q3 AUTOMATION & CONTROL SYSTEMS Pub Date : 2025-03-01 DOI: 10.1016/j.ifacsc.2025.100300
Pouria Tooranjipour, Bahare Kiumarsi
This paper introduces a novel approach for designing safe optimal controllers that avoid destructive conflicts between safety and performance in a large domain of the system’s operation. Designing computationally tractable feedback controllers that respect safety for a given set is impossible in general. The best one can do in this case is to maximize the region contained in the safe set that respects both safety and optimality. To this end, our key contribution lies in constructing a safe optimal domain of attraction (DoA) that ensures optimal convergence of the system’s trajectories to the origin without violating safety. To accomplish this, we leverage the concept of the relaxed Hamilton–Jacobi–Bellman (HJB) equation, which allows us to learn the most permissive control barrier certificates (CBCs) with a maximum-volume conflict-free set by solving a tractable optimization problem. To enhance computational efficiency, we present an innovative sum-of-squares (SOS)-based algorithm, breaking down the optimization problem into smaller SOS programs at each iteration. To alleviate the need for the system model to solve these SOS optimizations, an SOS-based off-policy reinforcement learning (RL) method is presented. This off-policy learning approach enables the evaluation of a target policy distinct from the behavior policy used for data collection, ensuring safe exploration under mild assumptions. In the end, the simulation results are given to show the efficacy of the proposed method.
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引用次数: 0
A sparse approach to transfer function estimation via Least Absolute Shrinkage and Selection Operator
IF 1.8 Q3 AUTOMATION & CONTROL SYSTEMS Pub Date : 2025-02-18 DOI: 10.1016/j.ifacsc.2025.100299
S.K. Laha
Estimating transfer functions from sampled input–output data is a critical task in system identification. Traditional approaches, such as least square optimization, often result in dense parameter estimates, which can be less interpretable and computationally intensive. This paper introduces a novel method for transfer function estimation by applying the Least Absolute Shrinkage and Selection Operator (LASSO), which promotes sparsity in the identified coefficients. The proposed approach enables sparse identification of both the numerator and denominator coefficients of the transfer function. The efficacy of the method is demonstrated through numerical experiments and application to the estimation of the natural frequencies of a turbine blade from its impulse response. By leveraging LASSO, we achieve a more parsimonious and interpretable model that captures the essential dynamics of the system. The results highlight the advantages of sparse modelling in system identification and its potential for improving the understanding and prediction of complex mechanical systems.
{"title":"A sparse approach to transfer function estimation via Least Absolute Shrinkage and Selection Operator","authors":"S.K. Laha","doi":"10.1016/j.ifacsc.2025.100299","DOIUrl":"10.1016/j.ifacsc.2025.100299","url":null,"abstract":"<div><div>Estimating transfer functions from sampled input–output data is a critical task in system identification. Traditional approaches, such as least square optimization, often result in dense parameter estimates, which can be less interpretable and computationally intensive. This paper introduces a novel method for transfer function estimation by applying the Least Absolute Shrinkage and Selection Operator (LASSO), which promotes sparsity in the identified coefficients. The proposed approach enables sparse identification of both the numerator and denominator coefficients of the transfer function. The efficacy of the method is demonstrated through numerical experiments and application to the estimation of the natural frequencies of a turbine blade from its impulse response. By leveraging LASSO, we achieve a more parsimonious and interpretable model that captures the essential dynamics of the system. The results highlight the advantages of sparse modelling in system identification and its potential for improving the understanding and prediction of complex mechanical systems.</div></div>","PeriodicalId":29926,"journal":{"name":"IFAC Journal of Systems and Control","volume":"31 ","pages":"Article 100299"},"PeriodicalIF":1.8,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143445842","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Local vs regional neural air pollution forecasting models
IF 1.8 Q3 AUTOMATION & CONTROL SYSTEMS Pub Date : 2025-02-13 DOI: 10.1016/j.ifacsc.2025.100298
Matteo Sangiorgio, Giorgio Guariso
Selecting a suitable dataset to develop a data-based forecasting model is often problematic. This is particularly important in the case of air pollution, where concentration measures are scattered over large areas. On the one hand, the classical approach creates a single-station (local) forecasting model using only the data collected at the same station. This guarantees a training dataset that considers all the site’s specific characteristics. On the other hand, these data may be limited and not sufficient to develop a robust predictor. Thus, one may use data from other stations to complement the dataset or develop a unique model considering all the data available within a region/domain. While this approach may be prone to filtering high variations, it may consider information on peculiar episodes that have not occurred in the past to a specific station. This paper discusses the topic of air pollution forecasting using the example of several stations in the Padana Plain, Northern Italy. Local forecasting models are developed using LSTM neural networks for nitrogen dioxide and ozone and hourly data from 2010 to 2023 and then compared with regional models. All these models perform extremely well under various regression-based and classification-based performance indicators, except for a few sites with peculiar characteristics that can be considered at the border of the information domain.
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引用次数: 0
A top-down approach for climate change mitigation strategies
IF 1.8 Q3 AUTOMATION & CONTROL SYSTEMS Pub Date : 2025-02-03 DOI: 10.1016/j.ifacsc.2025.100297
Claudio Marchesi, Michele Francesco Arrighini, Laura Zecchi, Marialuisa Volta
This research examined the effects of various GHG reduction policies on climate change via optimization techniques using a top-down approach. The aim was to examine how different aspects of policies to reduce CO2 and CH4 emissions would affect changes in temperature compared to pre-industrial levels from 2025 to 2100. The proposed top-down approach allows for the investigation of several factors that may influence the results: (i) the objective function, (ii) the reduction pathway, and (iii) the starting point of the optimization. Two different objective functions were minimized: the overall sum of the temperature between 2025–2100 and the value at 2100. The results were also compared in terms of the reduction trajectories: two different emission trends were assumed: a gradual (gaussian) fall in emissions or a fast (exponential) decline, starting in 2025, in 2030, and in 2035. The mitigation of greenhouse gas (GHG) emissions was limited to a certain range of scenarios outlined by the Intergovernmental Panel on Climate Change (IPCC). These scenarios were determined by analyzing economic, social, and technical developments expected to occur in the next few decades. The analysis also included the interaction in global warming of air pollutant emission variations due to climate policies. The results revealed that exponential trajectories, depending on the initial year, can facilitate the stabilization of global temperatures below 1.5 °C. In contrast, gaussian trajectories were more likely to overtake this threshold if implementation is delayed beyond 2025.
{"title":"A top-down approach for climate change mitigation strategies","authors":"Claudio Marchesi,&nbsp;Michele Francesco Arrighini,&nbsp;Laura Zecchi,&nbsp;Marialuisa Volta","doi":"10.1016/j.ifacsc.2025.100297","DOIUrl":"10.1016/j.ifacsc.2025.100297","url":null,"abstract":"<div><div>This research examined the effects of various GHG reduction policies on climate change via optimization techniques using a top-down approach. The aim was to examine how different aspects of policies to reduce CO<sub>2</sub> and CH<sub>4</sub> emissions would affect changes in temperature compared to pre-industrial levels from 2025 to 2100. The proposed top-down approach allows for the investigation of several factors that may influence the results: (i) the objective function, (ii) the reduction pathway, and (iii) the starting point of the optimization. Two different objective functions were minimized: the overall sum of the temperature between 2025–2100 and the value at 2100. The results were also compared in terms of the reduction trajectories: two different emission trends were assumed: a gradual (gaussian) fall in emissions or a fast (exponential) decline, starting in 2025, in 2030, and in 2035. The mitigation of greenhouse gas (GHG) emissions was limited to a certain range of scenarios outlined by the Intergovernmental Panel on Climate Change (IPCC). These scenarios were determined by analyzing economic, social, and technical developments expected to occur in the next few decades. The analysis also included the interaction in global warming of air pollutant emission variations due to climate policies. The results revealed that exponential trajectories, depending on the initial year, can facilitate the stabilization of global temperatures below 1.5 °C. In contrast, gaussian trajectories were more likely to overtake this threshold if implementation is delayed beyond 2025.</div></div>","PeriodicalId":29926,"journal":{"name":"IFAC Journal of Systems and Control","volume":"31 ","pages":"Article 100297"},"PeriodicalIF":1.8,"publicationDate":"2025-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143270858","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A bilevel optimization approach for Balancing Markets with electric vehicle aggregators and smart charging
IF 1.8 Q3 AUTOMATION & CONTROL SYSTEMS Pub Date : 2025-01-30 DOI: 10.1016/j.ifacsc.2025.100296
Daniel Fernández Valderrama, Giulio Ferro, Luca Parodi, Michela Robba
Demand Response (DR) programs can help alleviate the management of the electrical distribution grid by reducing loads in specified areas. They can be enabled within the energy Balancing Market (BM). Aggregators can manage different customers providing flexibility. Recently, Electric Vehicles Aggregators (EVAs) have emerged as significant players in the BM because they can manage fleets of electric vehicles (EVs) in the distribution grid. This paper addresses a multi-objective optimization problem for a distribution power grid that includes EVs and smart charging parks. At the higher level, the Distribution System Operator (DSO) considers the characteristics of each BM actor to minimize costs. Meanwhile, EVAs focus on controlling EV charging at the lower level to maximize their profit. The optimization problems of EVAs and other actors are replaced by KKT (Karush–Kuhn–Tucker)​ conditions, which are embedded as constraints in the DSO decision problem. Moreover, the resulting bilinear terms (in the optimization problem constraints) are linearized to fasten the finding of an optimal solution. The overall optimization problem is a mixed-integer quadratic programming (MIQP) and has been applied to the IEEE 13-bus test benchmark. The results demonstrate a reduction of about 6% of power loss in the grid achieved by the developed model. Besides, the linearized model can afford a more discretized model due to the reduction of computational effort.
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引用次数: 0
New approach of series-PID controller design based on modern control theory: Simulations and real-time validation
IF 1.8 Q3 AUTOMATION & CONTROL SYSTEMS Pub Date : 2025-01-25 DOI: 10.1016/j.ifacsc.2025.100295
Vivek Kumar, Yogesh V. Hote
This paper proposed a novel approach to designing the series-proportional–integral derivative (series-PID) controller, which provides desired performance specifications. In this approach, the design of the series-PID controller is carried out by modern control theory, which is based on the Butterworth pattern of pole placement, and classical control theory, which is based on Krishnamurthi’s corollary on the Routh criterion. The uniqueness of the proposed approach in comparison with existing methods is that it comprises both classical and modern control theory for improving performance and robustness trade-off. The validation of the proposed control theory is carried out using numerical examples (Linear & Non-linear models). The results show that the performance is improved compared to the existing results. The main aim of the paper is that the proposed theory should be industrial-friendly. In view of this, the proposed theory is validated on the D.C. servo motor and power system problem of load frequency control. For these practical problems, comparisons are carried out with well-known control approaches, such as the internal model control approach proposed by various authors. Finally, the proposed approach has been implemented and validated on the hardware setup of the DC–DC buck converter (DDBCc). In numerical examples and practical problems, the efficacy of the proposed approach has been checked by robustness analysis and fragility analysis. Further, it has also been checked by determining various performance indices such as Integral Square Error (ISE), Integral Absolute Error (IAE), and Total Variations (TV).
{"title":"New approach of series-PID controller design based on modern control theory: Simulations and real-time validation","authors":"Vivek Kumar,&nbsp;Yogesh V. Hote","doi":"10.1016/j.ifacsc.2025.100295","DOIUrl":"10.1016/j.ifacsc.2025.100295","url":null,"abstract":"<div><div>This paper proposed a novel approach to designing the series-proportional–integral derivative (series-PID) controller, which provides desired performance specifications. In this approach, the design of the series-PID controller is carried out by modern control theory, which is based on the Butterworth pattern of pole placement, and classical control theory, which is based on Krishnamurthi’s corollary on the Routh criterion. The uniqueness of the proposed approach in comparison with existing methods is that it comprises both classical and modern control theory for improving performance and robustness trade-off. The validation of the proposed control theory is carried out using numerical examples (Linear &amp; Non-linear models). The results show that the performance is improved compared to the existing results. The main aim of the paper is that the proposed theory should be industrial-friendly. In view of this, the proposed theory is validated on the D.C. servo motor and power system problem of load frequency control. For these practical problems, comparisons are carried out with well-known control approaches, such as the internal model control approach proposed by various authors. Finally, the proposed approach has been implemented and validated on the hardware setup of the DC–DC buck converter (<span><math><mrow><mi>D</mi><mi>D</mi><mi>B</mi><msub><mrow><mi>C</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></math></span>). In numerical examples and practical problems, the efficacy of the proposed approach has been checked by robustness analysis and fragility analysis. Further, it has also been checked by determining various performance indices such as Integral Square Error (ISE), Integral Absolute Error (IAE), and Total Variations (TV).</div></div>","PeriodicalId":29926,"journal":{"name":"IFAC Journal of Systems and Control","volume":"31 ","pages":"Article 100295"},"PeriodicalIF":1.8,"publicationDate":"2025-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143141323","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Control-Equivalent-Turbulence-Input estimation method for unmanned helicopters
IF 1.8 Q3 AUTOMATION & CONTROL SYSTEMS Pub Date : 2025-01-04 DOI: 10.1016/j.ifacsc.2024.100293
Sergey Nazarov, Per-Olof Gutman
This paper describes a new time-domain estimation method of the Control Equivalent Turbulence Input (CETI). CETI simulates the effects of natural atmospheric turbulence on rotorcraft dynamics by additional control inputs. The identified linear model of helicopter dynamics is changed into a quasi-nonlinear model by adding trim data and nonlinear kinematic equations. An unscented Kalman filter is designed using the quasi-nonlinear helicopter model and scaled unscented transformation to estimate CETI in real-time. The method was first tested in simulations; then, results were obtained for flight. The proposed method is also compared with other methods used in practice.
{"title":"A Control-Equivalent-Turbulence-Input estimation method for unmanned helicopters","authors":"Sergey Nazarov,&nbsp;Per-Olof Gutman","doi":"10.1016/j.ifacsc.2024.100293","DOIUrl":"10.1016/j.ifacsc.2024.100293","url":null,"abstract":"<div><div>This paper describes a new time-domain estimation method of the Control Equivalent Turbulence Input (CETI). CETI simulates the effects of natural atmospheric turbulence on rotorcraft dynamics by additional control inputs. The identified linear model of helicopter dynamics is changed into a quasi-nonlinear model by adding trim data and nonlinear kinematic equations. An unscented Kalman filter is designed using the quasi-nonlinear helicopter model and scaled unscented transformation to estimate CETI in real-time. The method was first tested in simulations; then, results were obtained for flight. The proposed method is also compared with other methods used in practice.</div></div>","PeriodicalId":29926,"journal":{"name":"IFAC Journal of Systems and Control","volume":"31 ","pages":"Article 100293"},"PeriodicalIF":1.8,"publicationDate":"2025-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143141322","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
CHoKI-based MPC for blood glucose regulation in Artificial Pancreas
IF 1.8 Q3 AUTOMATION & CONTROL SYSTEMS Pub Date : 2025-01-01 DOI: 10.1016/j.ifacsc.2024.100294
Beatrice Sonzogni , José María Manzano , Marco Polver , Fabio Previdi , Antonio Ferramosca
This work presents a Model Predictive Control (MPC) for the artificial pancreas, which is able to autonomously manage basal insulin injections in type 1 diabetic patients. Specifically, the MPC goal is to maintain the patients’ blood glucose level inside the safe range of 70-180 mg/dL, acting on the insulin amount and respecting all the imposed constraints, taking into consideration also the Insulin On Board (IOB), to avoid excess of insulin infusion. MPC uses a model to make predictions of the system behavior. In this work, due to the complexity of the diabetes disease that complicates the identification of a general physiological model, a data-driven learning method is employed instead. The Componentwise Hölder Kinky Inference (CHoKI) method is adopted, to have a customized controller for each patient. For the data collection phase and also to test the proposed controller, the virtual patients of the FDA-accepted UVA/Padova simulator are exploited. The MPC is also tested on simulations with variability of the insulin sensitivity and with physical activity sessions. The final results are satisfying since the proposed controller is conservative and reduces the time in hypoglycemia (which is more dangerous) if compared to the outcomes obtained without the IOB constraints.
{"title":"CHoKI-based MPC for blood glucose regulation in Artificial Pancreas","authors":"Beatrice Sonzogni ,&nbsp;José María Manzano ,&nbsp;Marco Polver ,&nbsp;Fabio Previdi ,&nbsp;Antonio Ferramosca","doi":"10.1016/j.ifacsc.2024.100294","DOIUrl":"10.1016/j.ifacsc.2024.100294","url":null,"abstract":"<div><div>This work presents a Model Predictive Control (MPC) for the artificial pancreas, which is able to autonomously manage basal insulin injections in type 1 diabetic patients. Specifically, the MPC goal is to maintain the patients’ blood glucose level inside the safe range of 70-180 mg/dL, acting on the insulin amount and respecting all the imposed constraints, taking into consideration also the Insulin On Board (IOB), to avoid excess of insulin infusion. MPC uses a model to make predictions of the system behavior. In this work, due to the complexity of the diabetes disease that complicates the identification of a general physiological model, a data-driven learning method is employed instead. The Componentwise Hölder Kinky Inference (CHoKI) method is adopted, to have a customized controller for each patient. For the data collection phase and also to test the proposed controller, the virtual patients of the FDA-accepted UVA/Padova simulator are exploited. The MPC is also tested on simulations with variability of the insulin sensitivity and with physical activity sessions. The final results are satisfying since the proposed controller is conservative and reduces the time in hypoglycemia (which is more dangerous) if compared to the outcomes obtained without the IOB constraints.</div></div>","PeriodicalId":29926,"journal":{"name":"IFAC Journal of Systems and Control","volume":"31 ","pages":"Article 100294"},"PeriodicalIF":1.8,"publicationDate":"2025-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143141324","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Adaptive chaos control: A novel continuous-time approach for enhanced stability
IF 1.8 Q3 AUTOMATION & CONTROL SYSTEMS Pub Date : 2024-12-10 DOI: 10.1016/j.ifacsc.2024.100292
Muhammad Shafiq , Israr Ahmad
Stabilizing chaotic systems with robustness, speed, and smoothness remains a significant challenge due to issues like chattering and slow convergence associated with traditional control methods. This paper proposes a novel continuous-time adaptive robust control (CTARC) scheme to overcome these limitations and enhance the stabilization of uncertain chaotic systems. CTARC employs smooth control functions; specifically hyperbolic secant and inverse hyperbolic sine functions to eliminate chattering and achieve faster, more precise convergence to equilibrium. Unlike conventional controllers that simplify system dynamics by removing nonlinearities, this approach preserves them, thereby improving robustness against time-varying disturbances and model uncertainties. A Lyapunov-based stability analysis rigorously establishes the asymptotic stability of the proposed control strategy. Numerical simulations on the Shimizu–Morioka​ chaotic system and a memristor-based hyperchaotic system validate CTARC’s superiority in convergence speed, energy efficiency, and stability compared to existing adaptive methods. By reducing transient effects like overshoots and oscillations, the proposed scheme ensures smoother transitions and minimizes energy consumption, addressing critical limitations of traditional methods. These results highlight CTARC’s potential as a robust and energy-efficient solution for chaos stabilization and provide a foundation for future developments in complex system control.
{"title":"Adaptive chaos control: A novel continuous-time approach for enhanced stability","authors":"Muhammad Shafiq ,&nbsp;Israr Ahmad","doi":"10.1016/j.ifacsc.2024.100292","DOIUrl":"10.1016/j.ifacsc.2024.100292","url":null,"abstract":"<div><div>Stabilizing chaotic systems with robustness, speed, and smoothness remains a significant challenge due to issues like chattering and slow convergence associated with traditional control methods. This paper proposes a novel continuous-time adaptive robust control (CTARC) scheme to overcome these limitations and enhance the stabilization of uncertain chaotic systems. CTARC employs smooth control functions; specifically hyperbolic secant and inverse hyperbolic sine functions to eliminate chattering and achieve faster, more precise convergence to equilibrium. Unlike conventional controllers that simplify system dynamics by removing nonlinearities, this approach preserves them, thereby improving robustness against time-varying disturbances and model uncertainties. A Lyapunov-based stability analysis rigorously establishes the asymptotic stability of the proposed control strategy. Numerical simulations on the Shimizu–Morioka​ chaotic system and a memristor-based hyperchaotic system validate CTARC’s superiority in convergence speed, energy efficiency, and stability compared to existing adaptive methods. By reducing transient effects like overshoots and oscillations, the proposed scheme ensures smoother transitions and minimizes energy consumption, addressing critical limitations of traditional methods. These results highlight CTARC’s potential as a robust and energy-efficient solution for chaos stabilization and provide a foundation for future developments in complex system control.</div></div>","PeriodicalId":29926,"journal":{"name":"IFAC Journal of Systems and Control","volume":"31 ","pages":"Article 100292"},"PeriodicalIF":1.8,"publicationDate":"2024-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143141459","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the turnpike to design of deep neural networks: Explicit depth bounds 通往深度神经网络设计之路显式深度边界
IF 1.8 Q3 AUTOMATION & CONTROL SYSTEMS Pub Date : 2024-11-02 DOI: 10.1016/j.ifacsc.2024.100290
Timm Faulwasser , Arne-Jens Hempel , Stefan Streif
It is well-known that the training of Deep Neural Networks (DNN) can be formalized in the language of optimal control. In this context, this paper leverages classical turnpike properties of optimal control problems to attempt a quantifiable answer to the question of how many layers should be considered in a DNN. The underlying assumption is that the number of neurons per layer—i.e., the width of the DNN—is kept constant. Pursuing a different route than the classical analysis of approximation properties of sigmoidal functions, we prove explicit bounds on the required depths of DNNs based on asymptotic reachability assumptions and a dissipativity-inducing choice of the regularization terms in the training problem. Numerical results obtained for the two spiral task data set for classification indicate that the proposed constructive estimates can provide non-conservative depth bounds.
众所周知,深度神经网络(DNN)的训练可以用最优控制语言来形式化。在此背景下,本文利用最优控制问题的经典岔道特性,尝试对 DNN 应考虑多少层这一问题给出可量化的答案。基本假设是每层神经元的数量,即 DNN 的宽度保持不变。我们采用了与经典的西格玛函数逼近特性分析不同的方法,基于渐近可达性假设和训练问题中正则化项的耗散诱导选择,证明了 DNN 所需深度的明确界限。针对两个螺旋任务分类数据集获得的数值结果表明,所提出的构造性估计可以提供非保守的深度边界。
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IFAC Journal of Systems and Control
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