Extremum Seeking Research Articles

Brief communication: Real-time estimation of the optimal tip-speed ratio for controlling wind turbines with degraded blades

Abstract. Rotor performance is adversely affected by the wear and tear of blade surfaces caused, for example, by rain, snow, icing, dirt, bugs and aging. Blade surface degradation changes the aerodynamic properties of the rotor, which in turn changes the optimal tip-speed ratio (TSR) and the corresponding maximum power coefficient. Below the rated wind speed, if a turbine continues to operate at the manufacturer-designed optimal TSR, the rotor power could decrease more than necessary unless the optimal TSR is corrected to compensate for blade degradation or blade surfaces are restored. Re-tuning the tip-speed ratio can lead to an improvement in energy capture without blade repairs. In this work, we describe a real-time algorithm to re-tune the tip-speed ratio to its optimal but unknown value under blade degradation. The algorithm uses power measurements only and the Log-Power Proportional-Integral Extremum Seeking Control (LP-PIESC) strategy to re-tune the TSR. The algorithm is demonstrated in simulations to command the set-point TSR required by a generator speed control loop that maximizes power at below-rated wind speeds. Comparison of this solution with a baseline controller that uses the optimal TSR for a rotor with clean blades demonstrates improvements in energy capture between 0.5 % and 3.4 %, depending on the severity of blade degradation and the wind conditions. These results are obtained using the OpenFAST simulation tool, the National Renewable Energy Laboratory (NREL) 5 MW reference turbine and the NREL-developed Reference Open-Source Controller (ROSCO).

Prescribed-Time Extremum Seeking for Delays and PDEs Using Chirpy Probing

Prescribed-Time Extremum Seeking for Delays and PDEs Using Chirpy Probing

Experimental implementation of extremum-seeking control: Gas fuel efficiency at electrical generation under power requirements

A current challenge stands for operating the emergency power system (EPS) that involves the challenge of supplying sufficient electrical energy at same time the fuel consumption is minimum. A complication arises as power requirements change during the EPS operation which can be seen as uncertain load disturbances. In such a context, the extremum-seeking (ES) is alternative towards the efficient energy conversion when control goal involves fuel optimization along the time operation. Here, the dynamical response of a gas fueled power plant is identified via Hammerstein model. The model is realized such that an ES control is designed for automatically reaching an extreme in face to distinct (unmeasured and uncertain) power requirements. The ES control is designed and experimentally tested at an electrical generator at distinct power requirements. The results show the minimum gas fuel consumption remains in face to distinct power requirements.

Extremum Seeking-Based Radio Signal Strength Optimization Algorithm for Hoverable UAV Path Planning

For the safe autonomous operations of unmanned aerial vehicles (UAVs) and ground control stations (GCS), including autonomous battery replacement, wireless power transfer, and more, the precise landing of UAVs on GCS is essential. Accurate landing is only possible when the link capacity strength exceeds a certain threshold, but this is often disturbed due to complex terrain. To address this, we developed an extremum seeking (ES)-based radio signal strength optimization (RSSO) algorithm, ES-RSSO, designed to find the optimal positions of the UAV using radio communication signals. This ensures energy-efficient path planning while guaranteeing the minimum received signal strength indication (RSSI) capacity. This algorithm is particularly useful in obstacle-rich environments, where UAVs are limited in power resources. Simulation results demonstrate a 2.37% decrease in the mean, a 62.08% improvement in variance, and a 3.72% decrease in the integration strength of the link capacity when ES-RSSO is applied. These results confirm that the RADIO.rssi maintenance ability remains above a critical boundary level, supporting robust communication links and energy-efficient path planning. Throughout the study, we showed how, in many cases, simply moving the UAV a few meters can significantly improve the communication link.

Hybrid feedback control of PMSG-based WECS with multilevel flying-capacitor inverter enhanced by fractional order extremum seeking

This paper proposes an original hybrid control strategy for a three-phase multilevel flying capacitor inverter (FCI) used in wind energy conversion system (WECS). A state feedback law, computed using a Lyapunov function and an invariance principle, is designed to manage grid current control and capacitor voltage balancing, guaranteeing the asymptotic stability of the closed-loop system. Additionally, a fractional-order extremum seeking control (FOESC) algorithm for Maximum Power Point Tracking (MPPT) is introduced, which optimizes power capture by adjusting turbine speed from the DC side without requiring a detailed system model (model-free approach). The use of fractional-order calculus leads to improved convergence and more seamless performance while preserving the robust characteristics of the system. The proposed control strategies are validated through simulation, demonstrating improved transient response and stability under varying wind conditions and parameter uncertainties. These results highlight the potential of advanced control algorithms to enhance the efficiency and reliability of wind energy systems, contributing to global efforts to achieve sustainable energy goals.

Detection and tracking of ocean layers using an AUV with UKF based extremum seeking control in the Baltic Sea

Adaptive sampling and situational awareness are key features of modern autonomous underwater vehicles (AUVs) since data quality can be improved while operation time and cost can be reduced. An example for adaptive sampling in the marine environmental context is thermocline detection and tracking. The thermocline as horizontal ocean layer separates warm and cold water and is a key feature in many marine disciplines. For example, it influences the distribution and exchange of nutrients and is a habitat for many organisms. In this paper we use an unscented Kalman Filter (UKF) based extremum seeking control (ESC) to find and follow ocean layers such as the thermocline. Computer simulations and real-world tests show that the method is able to find and track non-trivial real-world ocean layers with sensors subject to hysteresis and delay effects.

Model-free stabilization via Extremum Seeking using a cost neural estimator

In this paper, a fully model-free architecture for vertical stabilization of thermonuclear plasmas in tokamak experimental reactors is presented. For the first time, an Extremum Seeking control algorithm is combined with neural networks to estimate the Lyapunov function to be minimized, resulting in a fully data-driven control architecture. The performance of different neural networks are compared. Specifically, Multilayer Perceptrons and Extreme Learning Machines are considered. The proposed architecture is tested in simulation to show that it can counteract relevant plasma disturbances, resulting in a significant improvement in terms of the achievable operative space compared to the Extremum Seeking algorithm, which still relies on model-based cost estimator.

Initialization-free Lie-bracket Extremum Seeking

Stability results for extremum seeking control in Rn have predominantly been restricted to local or, at best, semi-global practical stability. Extending semi-global stability results of extremum-seeking systems to unbounded sets of initial conditions often demands a stringent global Lipschitz condition on the cost function, which is rarely satisfied by practical applications. In this paper, we address this challenge by leveraging tools from higher-order averaging theory. In particular, we establish a novel second-order averaging result with global (practical) stability implications. By leveraging this result, we characterize sufficient conditions on cost functions under which uniform global (i.e., under any initialization) practical asymptotic stability can be established for a class of extremum-seeking systems acting on static maps. Our sufficient conditions include the case when the gradient of the cost function, rather than the cost function itself, satisfies a global Lipschitz condition, which covers quadratic cost functions. Our results are also applicable to vector fields that are not necessarily Lipschitz continuous at the origin, opening the door to non-smooth Lie-bracket ES dynamics. We illustrate all the results via different analytical and/or numerical examples.

A Study on Personal Fit Control Design based on Extremum Seeking Approach

A Study on Personal Fit Control Design based on Extremum Seeking Approach

Perturbation-based Extremum Seeking Control Design for a Class of Nonlinear Systems

In this paper, we try to address the problem of output (performance) function by applying the Perturbation-based Extremum Seeking Control (PESC) approach to the observer of the Single-Input Single-Output (SISO) nonlinear systems. In this work, PESC is applied so that the performance function can reach its maximum value. We apply tow controllers design that will take care of maximizing of the cost function. First controller is designed in the availability of full and unknown variables which are fed to the objective function, and the second controller is designed when the full state availability is removed, and these variables become known by applying the High Gain Observer (HGO) model to estimate the system variables. The construction of a seeking algorithm is used to drive the system variables and the observer output to the desired set-points that maximize the value of an objective (performance) function. In addition, Lyapunov's stability theorem and the perturbation theory including the averaging method are used in the design of the extremum seeking controller structure to check the stability of the system. Finally, the simulation results show the performance of the proposed procedures.

A stochastic approach for the solution of single and multi–objective optimisation problems of biological processes in sequencing batch reactor

This paper investigates the impact of implementing single and multi-optimisation solutions on the biological treatment process in a sequencing batch reactor (SBR). The research is based on a case study of the water resource recovery facility (WRRF) in Swarzewo, Northern Poland. The paper introduces the adaptive extremum seeking control (ESC) method for dissolved oxygen (DO) concentration control and places it in a layered control structure. Further, it presents the introduction of an optimisation layer for the structure and parameters of the SBR cycle, through the synthesis of stochastic methods: single-objective optimisation (SOO) using a genetic algorithm (GA) and multi-objective optimisation (MOO) using the NSGA-II algorithm. The results were compared to a classical approach with fixed cycle parameters. The paper shows the advantages of optimising cycle parameters, including the number of phases as well as the DO value, on the process flow. These control structures underwent simulation tests in the MATLAB environment with the Simba package. The biochemical processes occurring in the reactor are based on the Activated Sludge Model No. 2d (ASM2d). The optimising control system demonstrates tangible improvements in operational efficiency and significant reductions in electrical energy consumption, highlighting the effectiveness of the proposed methodologies. © 2017 Elsevier Inc. All rights reserved.

An Enhanced Extremum Seeking-Based Energy Management Strategy with Equivalent State for Hybridized-Electric Tramway-Powered by Fuel Cell–Battery–Supercapacitors

This article proposes a novel real-time optimization-based energy management strategy (EMS) for proton membrane exchange fuel cell (PEMFC)-battery-supercapacitors-driven hybridized-electric tramways (HETs). The proposed algorithm is derived based on an enhanced extremum seeking (ES) algorithm, with a new equivalent state-of-charge (SOC) and a new adaptive co-state introduced. Thereby, optimized reference power for each power source can be distributed appropriately when using three components. The workability and prominent of the proposed technique are demonstrated through comparative simulations with fuzzy-rule-based EMS (FEMS) and equivalent consumption minimization strategy (ECMS) in two case studies: with and without considering the supercapacitors, as an important factor in the EMS design to stabilize the SOC of energy storage devices (ESDs). Briefly, under the proposed ES-based method, the PEMFC power can be regulated such that high-efficiency can be performed, approximately by 46.7%. Subsequently, the hydrogen consumption is reduced about 31.2% compared to a comparative fuzzy-based EMS. Besides, the supplements’ SOCs at the end of a driving cycle are also regulated to be equal to the initial ones.

Multi-row extremum seeking for wind farm power maximization

This paper presents results from wind tunnel experiments to evaluate power gains from wake steering via yaw control. An experimental scaled wind farm with 12 turbines in an aligned rectangular array is used. Wake steering is performed by yawing turbines using a closed-loop algorithm termed the Log-of-Power Proportional Integral Extremum Seeking Control (LP-PIESC). Two configurations are considered. In the first configuration, the turbines in the first two upstream rows are controlled. In the second case, yaw control is applied to the turbines in the first upstream row and the third row. For both cases, uncontrolled turbines have no yaw misalignment. The results show that by independent parallel maximization of the power sum of a reduced number of turbines, it is possible to obtain a close approximation of the true maximum power. The data shows that the LP-PIESC algorithm can converge relatively fast compared to traditional ESC algorithms.

Model Guided Extremum Seeking and Active Disturbance Rejection Control for Efficiency Real-Time Optimization of PEMFC System

Model Guided Extremum Seeking and Active Disturbance Rejection Control for Efficiency Real-Time Optimization of PEMFC System

Multi-USV cooperative oil spill source seeking via extremum seeking combined with information fusion

This paper addresses the issue of cooperative source seeking for oil spill pollution by multiple unmanned surface vehicles (USVs), where the oil spill plume propagation is modeled by a linear advection–diffusion equation in two-dimensional space. The main objective is to construct a cooperative source seeking scheme for a group of USVs in the framework of extremum seeking combined with information fusion to solve such cooperative source seeking problem. To do this, both a gradient-based control law and a formation control law are included in the cooperative source seeking scheme. The gradient-based control law that is constructed by extremum-seeking control combined with information fusion drives the multiple USVs to approach the oil spill source, where the gradient is estimated online via the extremum-seeking control and the fused concentration value that is derived via a linear H∞ filter from the local concentration measurements at the USV’s current position. In the source seeking process, the USVs driven by the formation control law based on the virtual structure method maintain a preset circular formation. The convergence performance of the cooperative source seeking scheme is analyzed rigorously by the Lyapunov method. Finally, extensive numerical simulations are included to validate the theoretical algorithm.

Fast Newton-like extremum seeking with asymptotic convergence guarantees

Extremum seeking denotes control approaches to steer the input of a dynamical system towards the optimizer of an associated cost function. The main strength of those schemes is to achieve their objective without any prior knowledge of the mathematical expression of the cost, nor the value of its gradient. However, those algorithms typically suffer from a slow convergence speed. In this paper, we propose a novel class of Newton-like extremum seeking systems separating the gradient and Hessian estimation from the update of the cost input towards the optimizer. The stability properties of the proposed schemes, including the ability of some of them to enforce the asymptotic convergence of the cost inputs to the optimizer, are proved. Their performances are also examined via some numerical examples. The latter will demonstrate the ability of the presented schemes to significantly reduce the convergence time compared with existing Newton-like schemes.

Shunted piezoelectric patch adaptive vibration absorber set to maximise electric power absorption: A comparison between parallel and series RL-shunts

This paper presents a comparative study on two types of shunted piezoelectric patch adaptive vibration absorbers, which can be bonded in batches on thin-walled structures to control the broad-band resonant response of low order flexural modes. The self-contained control units are formed by a thin piezoelectric patch connected either to a series or to a parallel resistive-inductive (RL) network. The two components of the shunt are adapted online in such a way as to maximise the time-averaged electric power absorbed by the shunt, which corresponds to minimising the time-averaged total flexural response of the hosting structure. The study considers a practical demonstrator formed by a thin rectangular plate with five piezoelectric patches connected to digital RL-shunts, which is excited in bending by a stationary white noise point force. The aim of the paper is to contrast the tuning and vibration control properties of the series and parallel RL shunts. More specifically, the online implementation of a recursive two-paths tuning strategy along R-constant and L-constant paths with an extremum seeking gradient search algorithm is analysed first for the two types of shunts. Then, the vibration control performance produced by the two types of shunts is investigated for the case where they are tuned to control the resonant response of the first flexural mode of the plate.

Coordinated energy-efficient walking assistance for paraplegic patients by using the exoskeleton-walker system

Overground walking can be achieved for patients with gait impairments by using the lower limb exoskeleton robots. Since it is a challenge to keep balance for patients with insufficient upper body strength, a robotic walker is necessary to assist with the walking balance. However, since the walking pattern varies over time, controlling the robotic walker to follow the walking of the human-exoskeleton system in coordination is a critical issue. Inappropriate control strategy leads to the unnecessary energy cost of the human-exoskeleton-walker (HEW) system and also results in the bad coordination between the human-exoskeleton system and the robotic walker. In this paper, we proposed a Coordinated Energy-Efficient Control (CEEC) approach for the HEW system, which is based on the extremum seeking control algorithm and the coordinated motion planning strategy. First, the extremum seeking control algorithm is used to find the optimal supporting force of the support joint in real time to maximize the energy efficiency of the human-exoskeleton system. Second, the appropriate reference joint angles for wheels of the robotic walker can be generated by the coordinated motion planning strategy, causing the good coordination between the human-exoskeleton system and the robotic walker. The proposed approach has been tested on the HEW simulation model, and the experimental results indicate that the coordinated energy-efficient walking can be achieved with the proposed approach, which is increased by 60.16% compared to the conventional passive robotic walker.

Parameter identification of PEMFC steady-state model based on p-dimensional extremum seeking via simplex tuning optimization method

For proton exchange membrane fuel cells (PEMFCs), accurate physical modeling and parameter identification play significant roles in simulation and control. To accurately and quickly identify unknown parameters in the model, a p-dimensional extremum seeking via simplex tuning optimization method is proposed in this paper. Different from the meta-heuristic algorithm, this method is a traditional algorithm, which can effectively estimate the unknown parameters of PEMFC model by constructing multidimensional space to find the extremum of the objective function and searching for the optimal value. The effectiveness of the proposed algorithm is verified by the estimation experiments of 7 unknown parameters in two groups of cases. The simulation and experimental data of the fuel cell model agree well in all cases. Also, through the study of four commercial stacks and the comparison of other excellent meta-heuristic algorithms, the potential of the proposed method in various operation scenarios is demonstrated, and the proposed algorithm has good performance in precision, convergence speed, and stability. Moreover, the accuracy of the model, the validity, and the robustness of parameter identification are further verified by new fitting and verification experiments. The proposed algorithm is a promising way to the most optimization problems in the engineering field.

Controlling a vehicle braking and longitudinal acceleration using a seeking control approach

Traditional methods for tracking the paths of driverless vehicles use plant models to determine the corresponding control laws. Due to the intricate interactions between the road and the tires, time-varying characteristics, and unidentified disturbances. It is challenging to create an accurate vehicle model. As a result, data-driven controllers, which are independent of a predetermined plant model are becoming more and more well-liked. This work implements adaptive cruise control (ACC) by employing a control approach called extremum seeking technique (EST), which is a model-free control (MFC), to control a vehicle braking and longitudinal acceleration. The main aim here is to create an ego vehicle that travels at a specific speed with maintaining a secure space with respect to a guide vehicle. A car including an ACC technique called ego car, exploits radar to determine relative velocity and relative space relating to the guiding car. The ACC technique is considered to keep maintain a relatively secure space or a preferred cruising velocity concerning the guiding vehicle. The developed model succeeded to determine the relative velocity and relative space according for the ego car to another guiding car with acceleration not more than ±2 m/s2 and spacing error less than 6 m.