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Search Results (261)

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Keywords = extremum

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21 pages, 2204 KB  
Article
Adhesion Control of High-Speed Train Based on Improved Nonlinear Kalman Filter
by Haotian Gan, Song Wang, Junqi Lu and Haoran Ou
Appl. Sci. 2025, 15(19), 10524; https://doi.org/10.3390/app151910524 - 29 Sep 2025
Viewed by 267
Abstract
In the operation of high-speed trains, the effective transmission of traction force heavily relies on the adhesion between the wheel and the rail. Excessive traction or braking force may exceed the adhesion limit, causing wheel creep or slide, which threatens both equipment and [...] Read more.
In the operation of high-speed trains, the effective transmission of traction force heavily relies on the adhesion between the wheel and the rail. Excessive traction or braking force may exceed the adhesion limit, causing wheel creep or slide, which threatens both equipment and safety. To address this, a state estimation method based on the SVD-ACKF (singular value decomposition adaptive cubature Kalman filter) is proposed for high-precision estimation of train speed. Combined with an extremum-seeking algorithm, a closed-loop adhesion control strategy is developed to maintain train operations near the maximum adhesion point. Simulation results show that the method ensures accurate tracking under varying rail conditions and noise, while the control algorithm maintains adhesion utilization above 90%, thereby meeting operational demands and enhancing railway safety. Full article
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23 pages, 17838 KB  
Article
Integrating Multi-Temporal Sentinel-1/2 Vegetation Signatures with Machine Learning for Enhanced Soil Salinity Mapping Accuracy in Coastal Irrigation Zones: A Case Study of the Yellow River Delta
by Junyong Zhang, Tao Liu, Wenjie Feng, Lijing Han, Rui Gao, Fei Wang, Shuang Ma, Dongrui Han, Zhuoran Zhang, Shuai Yan, Jie Yang, Jianfei Wang and Meng Wang
Agronomy 2025, 15(10), 2292; https://doi.org/10.3390/agronomy15102292 - 27 Sep 2025
Viewed by 286
Abstract
Soil salinization poses a severe threat to agricultural sustainability in the Yellow River Delta, where conventional spectral indices are limited by vegetation interference and seasonal dynamics in coastal saline-alkali landscapes. To address this, we developed an inversion framework integrating spectral indices and vegetation [...] Read more.
Soil salinization poses a severe threat to agricultural sustainability in the Yellow River Delta, where conventional spectral indices are limited by vegetation interference and seasonal dynamics in coastal saline-alkali landscapes. To address this, we developed an inversion framework integrating spectral indices and vegetation temporal features, combining multi-temporal Sentinel-2 optical data (January 2024–March 2025), Sentinel-1 SAR data, and terrain covariates. The framework employs Savitzky–Golay (SG) filtering to extract vegetation temporal indices—including NDVI temporal extremum and principal component features, capturing salt stress response mechanisms beyond single-temporal spectral indices. Based on 119 field samples and Variable Importance in Projection (VIP) feature selection, three ensemble models (XGBoost, CatBoost, LightGBM) were constructed under two strategies: single spectral features versus fused spectral and vegetation temporal features. The key results demonstrate the following: (1) The LightGBM model with fused features achieved optimal validation accuracy (R2 = 0.77, RMSE = 0.26 g/kg), outperforming single-feature models by 13% in R2. (2) SHAP analysis identified vegetation-related factors as key predictors, revealing a negative correlation between peak biomass and salinity accumulation, and the summer crop growth process affects soil salinization in the following spring. (3) The fused strategy reduced overestimation in low-salinity zones, enhanced model robustness, and significantly improved spatial gradient continuity. This study confirms that vegetation phenological features effectively mitigate agricultural interference (e.g., tillage-induced signal noise) and achieve high-resolution salinity mapping in areas where traditional spectral indices fail. The multi-temporal integration framework provides a replicable methodology for monitoring coastal salinization under complex land cover conditions. Full article
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17 pages, 1662 KB  
Article
The Relationship Between the Seismic Regime and Low-Frequency Variations in Meteorological Parameters Measured at a Network of Stations in Japan
by Alexey Lyubushin and Eugeny Rodionov
Atmosphere 2025, 16(10), 1129; https://doi.org/10.3390/atmos16101129 - 26 Sep 2025
Viewed by 299
Abstract
The relationship between the seismic regime and humidity, pressure, temperature, and wind speed measured at a network of stations on the Japanese islands was studied for the period from 1973 to 2025. For each of the parameters, weighted average time series were constructed [...] Read more.
The relationship between the seismic regime and humidity, pressure, temperature, and wind speed measured at a network of stations on the Japanese islands was studied for the period from 1973 to 2025. For each of the parameters, weighted average time series were constructed using the principal component method and then subjected to wavelet decomposition. For wavelet decomposition levels, the amplitudes of the envelopes and the points of their local extrema were found and compared with the times at which earthquakes occurred. The problem of estimating an advanced measure of envelope extremum points relative to earthquake moments was considered using a model of interacting point processes. For a sequence of 213 strong earthquakes with a magnitude of at least 6.5, the same numbers for the largest local maxima and the smallest local minima were selected for the extrema of the envelope amplitude of each parameter. It turned out that the largest advance measures occurred for the seventh level of decomposition (the period from 16 to 32 days). Two advance mechanisms were identified: one mechanism is associated with the trigger effect of cyclones on seismicity, and the second is associated with the occurrence of atmospheric earthquake precursors. Full article
(This article belongs to the Section Meteorology)
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14 pages, 596 KB  
Article
Quantum Security Multi-Party Extremum Protocol with Greenberger–Horne–Zeilinger States
by Ping Wang, Gong-De Guo, Shu-Xin Lin, Xin Zhang and Song Lin
Technologies 2025, 13(10), 431; https://doi.org/10.3390/technologies13100431 - 24 Sep 2025
Viewed by 396
Abstract
Secure multi-party extremum, as a significant offshoot of secure multi-party computation, has extensive applications in various domains, including healthcare, financial transactions, market analysis, sports events, etc. Nevertheless, most existing secure multi-party extremum protocols rely on computational hard problems and are thus vulnerable to [...] Read more.
Secure multi-party extremum, as a significant offshoot of secure multi-party computation, has extensive applications in various domains, including healthcare, financial transactions, market analysis, sports events, etc. Nevertheless, most existing secure multi-party extremum protocols rely on computational hard problems and are thus vulnerable to quantum algorithms. This paper presents a quantum secure multi-party extremum protocol that is built upon the correlations of Greenberger–Horne–Zeilinger (GHZ) states. Within this protocol, multiple participants, with the aid of a semi-honest third party, can obtain the maximum and minimum values of their secret inputs. GHZ states act as the information carriers and are transmitted among the participants and the third party. Their unique correlations ensure the secure transmission of quantum particles. The analysis demonstrates that the proposed protocol is capable of not only warding off common external attacks but also resisting internal attacks launched by dishonest participants and the semi-honest third party. Moreover, the protocol boasts correctness and high scalability. Full article
(This article belongs to the Section Quantum Technologies)
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16 pages, 2496 KB  
Article
Surface Properties of Recombinant Pea Vicilin and Cupin-1.2 Solutions in 8M Urea
by Nikolay Isakov, Dmitry Angel, Mikhail Belousov, Giuseppe Loglio, Reinhard Miller, Anton Nizhnikov and Boris Noskov
Polymers 2025, 17(18), 2463; https://doi.org/10.3390/polym17182463 - 11 Sep 2025
Viewed by 409
Abstract
The kinetic dependencies of the surface pressure, the dilational dynamic surface elasticity and ellipsometric angles of cupin-1.2 and vicilin solutions in 8M urea were measured at different concentrations. The analysis of these kinetics dependencies and the obtained master curves allowed us to determine [...] Read more.
The kinetic dependencies of the surface pressure, the dilational dynamic surface elasticity and ellipsometric angles of cupin-1.2 and vicilin solutions in 8M urea were measured at different concentrations. The analysis of these kinetics dependencies and the obtained master curves allowed us to determine a few adsorption steps in the investigated systems and showed that the master curves are individual characteristics of the protein for a given solvent. At the same time, the shape of these curves can be different for adsorbed and spread layers of plant proteins indicating different structures of these layers. The dependencies of the dynamic surface elasticity on surface pressure are non-monotonic, unlike the corresponding results for most of the solutions of the investigated plant proteins. The extremums of these dependencies can be connected to the formation of the distal region of the surface layer in agreement with the theory for the surface viscoelasticity of polymer solutions. Full article
(This article belongs to the Section Polymer Analysis and Characterization)
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19 pages, 2638 KB  
Article
Analysis of High–Low Runoff Encounters Between the Water Source and Receiving Areas in the Xinyang Urban Water Supply Project
by Jian Qi, Fengshou Yan, Qingqing Tian, Chaoqiang Yang, Yu Tian, Xin Li, Lei Guo, Qianfang Ma and Yunfei Ma
Water 2025, 17(17), 2618; https://doi.org/10.3390/w17172618 - 4 Sep 2025
Viewed by 1014
Abstract
The construction of the Xinyang Urban Water Supply Project, centered on the Chushandian Reservoir, required a thorough investigation of high–low runoff encounters between the water source and receiving areas to optimize water allocation and operational scheduling. Based on the hydrological stations at Changtaiguan [...] Read more.
The construction of the Xinyang Urban Water Supply Project, centered on the Chushandian Reservoir, required a thorough investigation of high–low runoff encounters between the water source and receiving areas to optimize water allocation and operational scheduling. Based on the hydrological stations at Changtaiguan (CTG) on the main stream of the Huaihe River (HR) in the water source area and Miaowan (MW) on the main stream of the Honghe River in the receiving area, the trends and abrupt change characteristics of monthly runoff from 2014 to 2024 were analyzed using methods such as extremum symmetry mode decomposition (ESMD) and heuristic segmentation, with spatial encounter patterns determined using Copula functions. The results indicate that (1) the runoff in the water source area showed a quasi-6.05-month periodic characteristic on a monthly scale, while the runoff in the receiving area exhibited a quasi-6.72-month periodic characteristic on a monthly scale; (2) the water source area experienced runoff mutation in August 2015 (extreme drought) and June 2024 (extreme precipitation), with the receiving area responding 7 months earlier than the water source area, revealing differences in system vulnerability; (3) synchronous hydrological states were significantly more likely to occur (51.2%) compared with asynchronous conditions (25.2%), with the highest probability of “concurrent drought” (19.8%) and a high-risk “normal water source—receiving area drought” combination (14.1%). These findings provide theoretical and technical support for the optimized scheduling of the Chushandian Reservoir, improving the resilience and adaptability of the Xinyang Urban Water Supply Project to climate fluctuations and extreme hydrological events. Full article
(This article belongs to the Section Hydrology)
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23 pages, 3138 KB  
Article
Design of Organic Rankine Cycle Recovering Multi-Grade Waste Heat from a Two-Stroke Marine Engine
by Jinfeng Feng, Yuncheng Gu, Shengjun Han, Xunhu Zhao, Yujun Tang, Sipeng Zhu, Hao Yuan and Guihua Wang
J. Mar. Sci. Eng. 2025, 13(9), 1679; https://doi.org/10.3390/jmse13091679 - 1 Sep 2025
Viewed by 984
Abstract
Waste heat recovery using Organic Rankine Cycle (ORC) systems shows significant potential for reducing CO2 emissions from ships. This study designs and analyzes ORC systems for recovering multi-grade waste heat from the exhaust gas, jacket water, and scavenging air of a marine [...] Read more.
Waste heat recovery using Organic Rankine Cycle (ORC) systems shows significant potential for reducing CO2 emissions from ships. This study designs and analyzes ORC systems for recovering multi-grade waste heat from the exhaust gas, jacket water, and scavenging air of a marine two-stroke diesel engine. A thermodynamic model is developed to investigate the effects of working fluid preheating temperature, evaporation pressure, and heat source conditions on system performance. Results show that appropriately increasing the preheating temperature of the working fluid can enhance power output. For hydrocarbons with higher critical temperatures, power output exhibits an extremum as preheating temperature increases, while for fluids with lower critical temperatures, power output increases continuously until the evaporation pressure limit is reached. Increasing evaporation pressure decreases power output but improves thermal efficiency, with a corresponding increase in heat transfer and exergy loss rates in the exhaust gas preheater. Additionally, the temperature of the heat source has an important effect on the energy and exergy balance distribution and power output of the ORC. For every 10 K rise in exhaust temperature, the bottoming cycle power output of cyclohexane increases by approximately 12.3%. This study provides theoretical support for efficient marine waste heat recovery and working fluid selection. Full article
(This article belongs to the Section Ocean Engineering)
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14 pages, 7467 KB  
Proceeding Paper
Entropy-Based Optimization in Chaotic Image Encryption Algorithms with Implementation of Artificial Intelligence
by Hristina Stoycheva and Georgi Mihalev
Eng. Proc. 2025, 104(1), 16; https://doi.org/10.3390/engproc2025104016 - 25 Aug 2025
Viewed by 368
Abstract
This paper addresses the challenge of determining optimal parameters in chaotic systems used for image encryption algorithms based on chaos theory. A baseline algorithm employing a third-order Lorenz chaotic system is examined, incorporating core procedures such as permutation (shuffling) and diffusion. Graphical results [...] Read more.
This paper addresses the challenge of determining optimal parameters in chaotic systems used for image encryption algorithms based on chaos theory. A baseline algorithm employing a third-order Lorenz chaotic system is examined, incorporating core procedures such as permutation (shuffling) and diffusion. Graphical results are presented to illustrate the variation of image entropy in relation to changes in system parameters. The analysis reveals a distinct region in the parameter space where entropy reaches its highest values. Based on these observations, an optimality criterion is formulated, defining an objective function that captures the entropy’s sensitivity to two key system parameters, including the bifurcation parameter. A complex objective function is derived, and the optimization problem is solved using a modified version of the Price algorithm enhanced with artificial intelligence techniques. The proposed modification demonstrates superior performance in locating the global extremum of the objective function, resulting in enhanced security of the encrypted image. Numerical and graphical results for various images are provided, along with a comparative analysis between the standard and the modified Price method. Full article
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19 pages, 1287 KB  
Article
Extremum-Seeking Control for a Robotic Leg Prosthesis with Sensory Feedback
by Ming Pi
Sensors 2025, 25(16), 4975; https://doi.org/10.3390/s25164975 - 12 Aug 2025
Viewed by 586
Abstract
By sensing changes in the contact force between the leg and level ground, humans can perceive their walking speed and adjust leg stiffness to accommodate walking terrains. To realize this natural regulation mechanism on the lower limb amputee, noninvasive functional electrical stimulation (nFES) [...] Read more.
By sensing changes in the contact force between the leg and level ground, humans can perceive their walking speed and adjust leg stiffness to accommodate walking terrains. To realize this natural regulation mechanism on the lower limb amputee, noninvasive functional electrical stimulation (nFES) was used to assist the subject in sensing the change in contact force between the leg and level ground, allowing for the adjustment of control parameters in the prosthetic leg. The cost function was designed to combine the tracking errors of the joints and changes in the stimulating current. For different walking terrains, an extremum-seeking control (ESC) method was employed to search for suitable control parameters in real time by monitoring the changes in the cost function. The stability of the proposed controller with extremum-seeking dynamics was demonstrated. The experimental results demonstrated that the extremum-seeking method effectively adjusted the control parameters of the prosthetic leg in response to changes in the cost function. Full article
(This article belongs to the Section Sensors and Robotics)
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16 pages, 1572 KB  
Article
Application of ANN in the Performance Evaluation of Composite Recycled Mortar
by Shichao Zhao, Yaohua Liu, Geng Xu, Hao Zhang, Feng Liu and Binglei Wang
Buildings 2025, 15(15), 2752; https://doi.org/10.3390/buildings15152752 - 4 Aug 2025
Viewed by 407
Abstract
To promote the large-scale utilization of construction and industrial solid waste in engineering, this study focuses on developing accurate prediction and optimization methods for the unconfined compressive strength (UCS) of composite recycled mortar. Innovatively incorporating three types of recycled powder (RP)—recycled clay brick [...] Read more.
To promote the large-scale utilization of construction and industrial solid waste in engineering, this study focuses on developing accurate prediction and optimization methods for the unconfined compressive strength (UCS) of composite recycled mortar. Innovatively incorporating three types of recycled powder (RP)—recycled clay brick powder (RCBS), recycled concrete powder (RCBP), and recycled gypsum powder (RCGP)—we systematically investigated the effects of RP type, replacement rate, and curing period on mortar UCS. The core objective and novelty lie in establishing and comparing three artificial intelligence models for high-precision UCS prediction. Furthermore, leveraging GA-BP’s functional extremum optimization theory, we determined the optimal UCS alongside its corresponding mix proportion and curing scheme, with experimental validation of the solution reliability. Key findings include the following: (1) Increasing total RP content significantly reduces mortar UCS; the maximum UCS is achieved with a 1:1 blend ratio of RCBP:RCGP, while a 20% RCBS replacement rate and extended curing periods markedly enhance strength. (2) Among the prediction models, GA-BP demonstrates superior performance, significantly outperforming BP models with both single and double hidden layer. (3) The functional extremum optimization results exhibit high consistency with experimental validation, showing a relative error below 10%, confirming the method’s effectiveness and engineering applicability. Full article
(This article belongs to the Section Building Materials, and Repair & Renovation)
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20 pages, 5160 KB  
Article
A PV Battery Charging System Based on Extremum-Seeking Control and a Series Resonant Converter with Capacitive Galvanic Isolation
by Abdulhakeem Alsaleem and Abdulrahman Alduraibi
Appl. Sci. 2025, 15(15), 8281; https://doi.org/10.3390/app15158281 - 25 Jul 2025
Cited by 1 | Viewed by 352
Abstract
This paper presents a standalone system that utilizes a capacitive isolated series resonant converter using an extremum-seeking control algorithm to extract the maximum power from PV panels. While resonant converters have been used for battery charging applications, series resonant converters that utilize capacitive [...] Read more.
This paper presents a standalone system that utilizes a capacitive isolated series resonant converter using an extremum-seeking control algorithm to extract the maximum power from PV panels. While resonant converters have been used for battery charging applications, series resonant converters that utilize capacitive galvanic isolation have not been sufficiently explored, and their design considerations for battery charging have not been established. In addition, extremum-seeking control algorithms have been explored for maximum power point tracking using PWM converters, but not using PFM converters such as resonant converters. This paper lays out the advantages of using an extremum-seeking-based control algorithm with resonant converters, specifically series resonant converters, and it presents simulation results of a 200 W standalone battery charging system to validate the stated benefits. Full article
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18 pages, 2456 KB  
Article
The Characterization of the Mechanical Harmonic Oscillator Extremum Envelope Shape According to Different Friction Types
by João C. Fernandes
Axioms 2025, 14(8), 554; https://doi.org/10.3390/axioms14080554 - 23 Jul 2025
Cited by 1 | Viewed by 252
Abstract
To characterize a phenomenological model of a mechanical oscillator, it is important to know the properties of the envelope of the three main physical motion variables: deviation from equilibrium, velocity, and acceleration. Experimental data show that friction forces restrict the shape of these [...] Read more.
To characterize a phenomenological model of a mechanical oscillator, it is important to know the properties of the envelope of the three main physical motion variables: deviation from equilibrium, velocity, and acceleration. Experimental data show that friction forces restrict the shape of these functions. A linear, exponential, or more abrupt decay can be observed depending on the different physical systems and conditions. This paper aimed to contribute to clarifying the role that some types of friction forces play in these shapes. Three types of friction—constant sliding friction, pressure drag proportional to the square of velocity, and friction drag proportional to velocity—were considered to characterize the line connecting the maxima and minima of displacement for a generic mechanical harmonic oscillator. The ordinary differential equation (ODE), describing the harmonic oscillator simultaneously containing the three types of dissipative forces (constant, viscous, and quadratic), was numerically solved to obtain energy dissipation, and the extrema of both displacement and velocity. The differential equation ruling the behavior of the amplitude, as a function of the friction force coefficients, was obtained from energy considerations. Solving this equation, we obtained analytical functions, parametrized by the force coefficients that describe the oscillator tail. A comparison between these functions and the predicted oscillator ODE extrema was made, and the results were in agreement for all the situations tested. Information from the velocity extrema and nulls was enough to obtain a second function that rules completely the ODE solution. The correlations obtained allow for the reverse operation: from the identified extremum data, it was possible to identify univocally the three friction coefficients fitting used in the model. Motion equations were solved, and some physical properties, namely energy conservation and work of friction forces, were revisited. Full article
(This article belongs to the Section Mathematical Physics)
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43 pages, 190510 KB  
Article
From Viewing to Structure: A Computational Framework for Modeling and Visualizing Visual Exploration
by Kuan-Chen Chen, Chang-Franw Lee, Teng-Wen Chang, Cheng-Gang Wang and Jia-Rong Li
Appl. Sci. 2025, 15(14), 7900; https://doi.org/10.3390/app15147900 - 15 Jul 2025
Viewed by 534
Abstract
This study proposes a computational framework that transforms eye-tracking analysis from statistical description to cognitive structure modeling, aiming to reveal the organizational features embedded in the viewing process. Using the designers’ observation of a traditional Chinese landscape painting as an example, the study [...] Read more.
This study proposes a computational framework that transforms eye-tracking analysis from statistical description to cognitive structure modeling, aiming to reveal the organizational features embedded in the viewing process. Using the designers’ observation of a traditional Chinese landscape painting as an example, the study draws on the goal-oriented nature of design thinking to suggest that such visual exploration may exhibit latent structural tendencies, reflected in patterns of fixation and transition. Rather than focusing on traditional fixation hotspots, our four-dimensional framework (Region, Relation, Weight, Time) treats viewing behavior as structured cognitive networks. To operationalize this framework, we developed a data-driven computational approach that integrates fixation coordinate transformation, K-means clustering, extremum point detection, and linear interpolation. These techniques identify regions of concentrated visual attention and define their spatial boundaries, allowing for the modeling of inter-regional relationships and cognitive organization among visual areas. An adaptive buffer zone method is further employed to quantify the strength of connections between regions and to delineate potential visual nodes and transition pathways. Three design-trained participants were invited to observe the same painting while performing a think-aloud task, with one participant selected for the detailed demonstration of the analytical process. The framework’s applicability across different viewers was validated through consistent structural patterns observed across all three participants, while simultaneously revealing individual differences in their visual exploration strategies. These findings demonstrate that the proposed framework provides a replicable and generalizable method for systematically analyzing viewing behavior across individuals, enabling rapid identification of both common patterns and individual differences in visual exploration. This approach opens new possibilities for discovering structural organization within visual exploration data and analyzing goal-directed viewing behaviors. Although this study focuses on method demonstration, it proposes a preliminary hypothesis that designers’ gaze structures are significantly more clustered and hierarchically organized than those of novices, providing a foundation for future confirmatory testing. Full article
(This article belongs to the Special Issue New Insights into Computer Vision and Graphics)
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39 pages, 16838 KB  
Article
Control of Nonlinear Systems Using Fuzzy Techniques Based on Incremental State Models of the Variable Type Employing the “Extremum Seeking” Optimizer
by Basil Mohammed Al-Hadithi and Gilberth André Loja Acuña
Appl. Sci. 2025, 15(14), 7791; https://doi.org/10.3390/app15147791 - 11 Jul 2025
Viewed by 435
Abstract
This work presents the design of a control algorithm based on an augmented incremental state-space model, emphasizing its compatibility with Takagi–Sugeno (T–S) fuzzy models for nonlinear systems. The methodology integrates key components such as incremental modeling, fuzzy system identification, discrete Linear Quadratic Regulator [...] Read more.
This work presents the design of a control algorithm based on an augmented incremental state-space model, emphasizing its compatibility with Takagi–Sugeno (T–S) fuzzy models for nonlinear systems. The methodology integrates key components such as incremental modeling, fuzzy system identification, discrete Linear Quadratic Regulator (LQR) design, and state observer implementation. To optimize controller performance, the Extremum Seeking Control (ESC) technique is employed for the automatic tuning of LQR gains, minimizing a predefined cost function. The control strategy is formulated within a generalized framework that evolves from conventional discrete fuzzy models to a higher-order incremental-N state-space representation. The simulation results on a nonlinear multivariable thermal mixing tank system validate the effectiveness of the proposed approach under reference tracking and various disturbance scenarios, including ramp, parabolic, and higher-order polynomial signals. The main contribution of this work is that the proposed scheme achieves zero steady-state error for reference inputs and disturbances up to order N−1 by employing the incremental-N formulation. Furthermore, the system exhibits robustness against input and load disturbances, as well as measurement noise. Remarkably, the ESC algorithm maintains its effectiveness even when noise is present in the system output. Additionally, the proposed incremental-N model is applicable to fast dynamic systems, provided that the system dynamics are accurately identified and the model is discretized using a suitable sampling rate. This makes the approach particularly relevant for control applications in electrical systems, where handling high-order reference signals and disturbances is critical. The incremental formulation, thus, offers a practical and effective framework for achieving high-performance control in both slow and fast nonlinear multivariable processes. Full article
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15 pages, 1239 KB  
Article
Extremum Seeking for the First Derivative of Nonlinear Maps with Constant Delays via a Time-Delay Approach
by Jianzhong Li, Hongye Su and Yang Zhu
Mathematics 2025, 13(13), 2196; https://doi.org/10.3390/math13132196 - 4 Jul 2025
Viewed by 291
Abstract
This paper introduces an extremum seeking (ES) scheme for the unknown map’s first derivative by tailoring a demodulation signal in which the closed-loop system is subject to constant transmission delays. Unlike most publications that manage delays using predictor-based methods, we are concerned with [...] Read more.
This paper introduces an extremum seeking (ES) scheme for the unknown map’s first derivative by tailoring a demodulation signal in which the closed-loop system is subject to constant transmission delays. Unlike most publications that manage delays using predictor-based methods, we are concerned with the delay-robustness of the introduced ES system via the newly developed time-delay approach. The original ES system is transformed to a nonlinear retarded-type plant with disturbances and the stability condition in the form of linear matrix inequalities is achieved. When the related bounds of the nonlinear map are not known, a rigorous practical stability proof is provided. Second, and more importantly, under the availability of prior knowledge about the nonlinear map, we are able to provide a quantitative calculation on the maximum allowable delay, the upper bound of the dither period, and the ultimate seeking error. Numerical examples are offered to exemplify the effectiveness of the proposed method. Full article
(This article belongs to the Section E2: Control Theory and Mechanics)
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