Extremum Seeking Method And Its Applications In Automotive Control

Tez (Doktora) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 2011Thesis (PhD) -- İstanbul Technical University, Institute of Science and Technology, 2011Kontrol uygulamalarındaki ana yöntem, ele alınan bir sistemi belli bir çalışma noktasına veya referans yörüngesine oturtmaktır. Fakat bazı kontrol problemlerinde, arzu edilen sistem performansı ile o performansı sağlayacak sistem çalışma noktası arasındaki ilişki önceden bilinmemektedir. Örneğin sistemin çalışma noktası ile çıkışı arasında o şekilde bir ilişki olabilir ki, bu fonksiyonun bir ekstremumu olabilir ve amaç, sistem çıkışını bu ekstremum değere getirecek çalışma noktasının aranması olabilir. Sistemin çalışma noktası ile çıkışı arasındaki fonksiyonun belirsizliği, çıkışı maksimize (veya minimize) edecek çalışma noktasının bulunması için bir uyarlama algoritmasının kullanımını gerekli kılmaktadır. Bu problem Ekstremum Arama Algoritması (EAA) ile çözülebilmektedir. Bu algoritma, sistemin performans fonksiyonunun tamamen veya kısmen bilinmediği, zamanla değişebildiği, sistemin eğrisel olduğu, belirsizlik ve bozucular içerdiği durumlar için uygundur. Örneğin acil durum frenlemesinde ihtiyaç duyulduğu gibi, bilinmeyen yol koşullarında tekerlek ile yol arasındaki teker kuvvetlerinin maksimize edilmesi başa çıkılması gereken zor bir iştir. Yol sürtünme katsayısı genellikle önceden bilinmemektedir ve anlık olarak kestirimi zordur. ABS kontrol algoritması, bilinmeyen yol koşullarında teker frenleme kuvvetini maksimize edecek hidrolik fren basıncının optimum çalışma noktasını bulmalıdır. Optimum çalışma noktası seçimindeki bir yanlış karar, ya olabilecekten daha az frenleme kuvvetinin üretilmesine ya da tekerleklerin kilitlenmesine, böylece aracın kontrol edilebilirliğinin ortadan kalkmasına sebep olacaktır. Minimum durma mesafesi ancak tekerleklerin, tekerlek kuvveti-tekerlek kayma oranı eğrisinde en tepe noktasında çalışmaları durumunda gerçekleşir. Bu durumda tekerleklerin kilitlenmesi engellendiği için aracın yanal kararlılığı ve direksiyon ile yönlendirilebilirliği de iyileşecektir. Tezde önce, optimum tekerlek kayma değeri bilinmeden tekerlek kuvvetinin maksimize edilmesi için, tekerlek modeli parametrelerinin uyarlanması yöntemi ile entegre edilmiş bir Ekstremum Arama Algoritması (EAA) önerilmiştir. Bunun için bir çeyrek araç modeli ele alınmıştır. Literatürdeki çoğu ekstremum arama algoritmaları, optimum çalışma noktasını ararken amaç fonksiyonunun gerçek zamanlı olarak ölçümüne dayanmaktadır. Bu çalışmada önerilen kontrol algoritması, amaç fonksiyonunun anlık ölçümü gereksinimini ortadan kaldırarak onun yerine parametre uyarlamalı analitik bir yöntem geliştirmiştir. Kararlılık ve global maksimum noktasına yakınsama durumları, Lyapunov kararlılık analizi ile gösterilmiştir. Önerilen yaklaşımın etkinliğini göstermek için farklı yol koşullarında simulasyon çalışmaları yapılmıştır. İkinci olarak, boyuna frenleme yanında engelden kaçınma manevrasında olduğu gibi yanal hareketi de gözönüne alan EAA temelli bir ABS kontrol algoritması sunulmuştur. Bu algoritmada, yol sürtünme katsayısını kestirmeye gerek kalmadan, tekerlek ve yol arasındaki optimum kayma oranı anlık olarak aranmaktadır. Literatüre getirilen bir yenilik olarak, “tekerlek kuvveti”-“kayma oranı” karakteristik eğrisi üzerinde tekerleklerin çalışma bölgesini belirlemek için sürücü direksiyon girişi ABS frenleme prosedürüne eklenmiştir. Sadece boyuna frenleme durumunda algoritma, tekerleklerin çalışma bölgesini, kuvvet-kayma eğrisinin tepe noktası yakınında tutmaktadır. Eğer sürücü frenlemeye ek olarak yanal hareket de talep ederse, tekerleklerin çalışma bölgesi otomatik olarak değiştirilmekte ve böylece yanal tekerlek kuvvetleri arttırılarak aracın yanal kararlılığı iyileştirilmektedir. Gerçek bir araçtan alınan ölçümlerle doğrulanmış bir tam araç modeli kullanılarak yapılan simülasyonlar algoritmanın etkinliğini göstermektedir. Üçüncü olarak, bir paralel tip hibrid elektrikli araç (HEA) için enerji yönetimi stratejisi önerilmiştir. HEA’lar, daha verimli, daha az çevreyi kirleten araçlara gereksinim sonucunda geliştirilmiştir. Elektrikli araçlar parlak bir çözüm olsa da şu andaki kısa menzilleri ve uzun batarya şarj süreleri, yaygın kullanımlarını geleceğe ötelemektedir. HEA’lar bu doğrultuda kabul edilebilir bir ara çözüm sunmaktadırlar. Hibrid bir elektrikli araçta, elektrokimyasal bir batarya ile güç verilen bir elektrikli motor (EM), fosil yakıt tarafından güç verilen içten yanmalı motor (İYM) ile birlikte kullanılmaktadır. Bunlar, yakıt tüketimi ve emisyonları azaltmadaki önemli potansiyelleri ile günümüzde en uygulanabilir teknoloji olarak görülmektedirler. Tezde verilen HEA enerji yönetim stratejisinin ana amacı, toplam verimi maksimize ederek yakıt tüketimini iyileştirmek ve bunu yaparken de sürücünün güç isteğini karşılamak, batarya şarj durumunu korumak ve İYM, EM güç kısıtları gibi çeşitli kısıtları göz önüne almaktır. Önerilen enerji yönetimi stratejisinde, ekstremum arama algoritması, toplam verimi maksimize edecek şekilde içten yanmalı motor ve elektrik motoru arasında optimum tork dağılımını belirlemektedir. Kontrol stratejisi üst seviye ve alt seviye olmak üzere iki seviyelidir: Üst seviyedeki karar verme kontrolcüsü aracın hangi modda çalışacağını tespit eder. Bu modlar: İçten yanmalı motor ve elektrik motorunun eşzamanlı çalışması, yalnızca elektrik motoru, yalnızca içten yanmalı motor, veya rejeneratif frenleme modlarıdır. İçten yanmalı motor ve elektrik motorunun eşzamanlı çalışması sırasında, bu iki enerji kaynağı arasındaki optimum enerji dağılımını ekstremum arama algoritması, toplam verimi maksimize edecek şekilde belirlemektedir. Böylece literatürde ilk defa bir ekstremum arama algoritması HEA kontrol problemine uyarlanmıştır. Önerilen kontrol algoritmasının performans değerlendirmesi için ayrıca bir dinamik programlama (DP) çözümü de elde edilmiştir. DP çözümü, ele alınan sürüş çevrimi ve sürüş koşulları için elde edilebilecek minimum yakıt tüketimini hesaplamaktadır. DP prosedürünü uygulamak için, bütün bir sürüş çevrimi ve sürüş koşulları önceden bilinmelidir. Gerçek bir araçta gelecekteki sürüş koşulları bilinmediği için DP gerçek zamanlı bir kontrolcü olarak kullanılamaz. Dinamik programlama çözümü gerçek zamanlı kontrol algoritmasının performansının değerlendirilmesi için kullanılmaktadır. Tezde önerilen kontrol algoritmasının etkinliğini göstermek için gerçekçi bir araç modeli kullanılarak çeşitli sürüş çevrimleri ile simülasyonlar yapılmıştır.The mainstream methodology in control applications is to regulate the considered system to known set points or reference trajectories. However, in some control problems, the relation between the system setpoint and a desired system performance is unknown a priori. One situation is that, the reference-to-output map has an extremum and the objective is to select the set point to keep the output at that extremum value. The uncertainty in the reference-to-output map makes it necessary to use an adaptation method to find the set point which maximizes (or minimizes) the output. This problem can be solved via the Extremum Seeking Algorithm (ESA). The algorithm fits problems that possess completely or partially unknown performance functions that may also change in time or that have nonlinear systems with structured or unstructured uncertainties and disturbances. For example, as needed in an emergency braking case, the maximization of the tire force between the tire contact patch and the road in the presence of unknown road conditions is a challenging task. The road friction coefficient is mostly unknown a priori and it is difficult to estimate it on-line. The ABS control algorithm should find the optimal set point of brake hydraulic pressure, which maximizes the wheel braking force subject to unknown and possibly changing road conditions. A misjudgment about the optimal set point choice may cause lower performance of braking via either less friction force generation or via blocking the tire rotation. The minimum stopping distance is ensured when the tires operate at the peak point of the braking force versus slip characteristic curve subject to unknown road conditions. In addition, lateral stability and steerability are also improved as locking of the wheels is prevented. In this thesis, firstly, an Extremum Seeking Algorithm (ESA) integrated with the adaptation of the tire model parameters is proposed for maximizing braking force without utilizing optimum slip value information. A quarter car vehicle model is considered in this section of the thesis. Most of the commonly used extremum seeking algorithms in the literature search for the optimal operating point in order to maximize or minimize a given cost function which is measured on a real-time basis. The control algorithm introduced in this dissertation removes the on-line cost function measurement requirement and instead, an analytic approach with adaptive parameter tuning is developed along the ESA. Stability and reaching the global maximum operating point of the unknown cost function are proved using Lyapunov stability analysis. Simulation study for ABS control under different road pavement conditions is presented to illustrate the effectiveness of the proposed approach. Secondly, an ABS control algorithm based on ESA is presented for considering lateral motion in addition to the longitudinal emergency braking, such as the obstacle avoidance maneuvers, also. The optimum slip ratio between the tire contact patch and the road is searched online without having to estimate the road friction conditions. This is achieved by adapting the ESA as a self-optimization routine that seeks the peak point of the force-slip curve. As a novel addition to the literature, the proposed algorithm incorporates driver steering input information into the ABS braking procedure to determine the operating region of the tires on the “tire force”-“slip ratio” characteristic curve. The algorithm operates the tires near the peak point of the force-slip curve during straight line braking. When the driver demands lateral motion in addition to braking, the operating regions of the tires are modified automatically, for improving the lateral stability of the vehicle by increasing the tire lateral forces. Simulations with a full vehicle model validated with actual vehicle measurements show the effectiveness of the algorithm. Thirdly, an energy management strategy for a parallel type hybrid electric vehicle (HEV) is proposed. HEVs are developed in the need of more efficient, less polluting vehicles. Electric vehicles seem as a promising solution but for now, their short driving distance combined with the long recharging period for batteries postpones their widespread use to the future. HEVs offer an acceptable, intermediate solution. In a hybrid electric vehicle, an electric motor (EM) powered by an electrochemical battery is used along with the internal combustion engine (ICE) powered by fossil fuel. They appear to be one of the most viable technologies with significant potential to reduce fuel consumption and pollutant emissions. The main objective of the HEV energy management strategy given in the thesis is maximizing the powertrain efficiency and hence improving the fuel consumption while meeting the driver’s power demand, sustaining the battery state of charge and considering constraints such as engine and electric motor power limits. In the proposed energy management strategy, extremum seeking algorithm searches constantly optimum torque distribution between the internal combustion engine and electric motor for maximizing the powertrain efficiency. The control strategy has two levels of operation: the upper and lower levels. The upper level decision making controller chooses the vehicle operation mode such as the simultaneous use of the internal combustion engine and electric motor, use of only the electric motor, use of only the internal combustion engine, or regenerative braking. In the simultaneous use of the internal combustion engine and electric motor, the optimum energy distribution between these two sources of energy is determined via the extremum seeking algorithm that searches for maximum powertrain efficiency. In the literature, this is the first time an extremum seeking algorithm is applied to the HEV control problem. A dynamic programming (DP) solution is also obtained and used to form a benchmark for performance evaluation of the proposed method. DP solution gives the minimum obtainable fuel consumption in a considered driving cycle and driving conditions. In order to apply DP procedure, the whole driving cycle and driving conditions should be known in advance. Since future driving conditions are unknown in a real vehicle, DP cannot be utilized in a real time controller. The dynamic programming solution is used offline for performance evaluation of the real time control algorithm. Detailed simulations with various driving cycles and using a realistic vehicle model are presented to illustrate the effectiveness of the methodology.DoktoraPh

Least Squares Based Adaptive Control and Extremum Seeking with Active Vehicle Safety System Applications

On-line parameter estimation is one of the two key components of a typical adaptive control scheme, beside the particular control law to be used. Gradient and recursive least squares (RLS) based parameter estimation algorithms are the most widely used ones among others. Adaptive control studies in the literature mostly utilize gradient based parameter estimators for convenience in nonlinear analysis and Lyapunov analysis based constructive design. However, simulations and real-time experiments reveal that, compared to gradient based parameter estimators, RLS based parameter estimators, with proper selection of design parameters, exhibit better transient performance from the aspects of speed of convergence and robustness to measurement noise. One reason for the control theory researchers' preference of gradient algorithms to RLS ones is that there does not exist a well-established stability and convergence analysis framework for adaptive control schemes involving RLS based parameter estimation. Having this fact as one of the motivators, this thesis is on systematic design, formal stability and convergence analysis, and comparative numerical analysis of RLS parameter estimation based adaptive control schemes and extension of the same framework to adaptive extremum seeking, viz. adaptive search for (local) extremum points of a certain field. Extremum seeking designs apply to (i) finding locations of physical signal sources, (ii) minimum or maximum points of (vector) cost or potential functions for optimization, (iii) calculating optimal control parameters within a feedback control design. In this thesis, firstly, gradient and RLS based on-line parameter estimation schemes are comparatively analysed and a literature review on RLS estimation based adaptive control is provided. The comparative analysis is supported with a set of simulation examples exhibiting transient performance characteristics of RLS based parameter estimators, noting absence of such a detailed comparison study in the literature. The existing literature on RLS based adaptive control mostly follows the indirect adaptive control approach as opposed to the direct one, because of the difficulty in integrating an RLS based adaptive law within the direct approaches starting with a certain Lyapunov-like cost function to be driven to (a neighborhood) of zero. A formal constructive analysis framework for integration of RLS based estimation to direct adaptive control is proposed following the typical steps for gradient adaptive law based direct model reference adaptive control, but constructing a new Lyapunov-like function for the analysis. After illustration of the improved performance with RLS adaptive law via some simple numerical examples, the proposed RLS parameter estimation based direct adaptive control scheme is successfully applied to vehicle antilock braking system control and adaptive cruise control. The performance of the proposed scheme is numerically analysed and verified via Matlab/Simulink and CarSim based simulation tests. Similar to the direct adaptive control works, the extremum seeking approaches proposed in the literature commonly use gradient/Newton based search algorithms. As an alternative to these search algorithms, this thesis studies RLS based on-line estimation in extremum seeking aiming to enhance the transient performance compared to the existing gradient based extremum seeking. The proposed RLS estimation based extremum seeking approach is applied to active vehicle safety system control problems, including antilock braking system control and traction control, supported by Matlab/Simulink and CarSim based simulation results demonstrating the effectiveness of the proposed approach

Extremum seeking control of uncertain systems

Extremum seeking is used in control problems where the reference trajectory or reference set point of the system is not known but it is searched in real time in order to maximize or minimize a performance function representing the optimal behaviour of the system. In this paper, extremum seeking algorithm is applied to the systems with parametric uncertainties.Publisher's Versio

Robust control of brake systems with decoupled architecture

Modern brake systems have the tendency to decoupled brake system design involving electric and/or electrohydraulic brake actuators. In this thesis, a corresponding brake control architecture applicable for electric and automated vehicles is proposed and includes (i) base braking, (ii) brake blending and (iii) wheel slip control functions. Main focus has been given to the robustness of continuous wheel slip control during emergency braking in high and low road friction conditions. As the solution, several control laws were designed and experimentally validated during road tests. Results obtained for three vehicle prototypes with individual on-board and in-wheel electric motors and electrohydraulic brake-by-wire system present significant improvement in braking performance and ride quality compared to the conventional wheel slip control strategies.Moderne Bremssysteme tendieren zur entkoppelten Konstruktion mit involvierten elektrischen und/oder elektrohydraulischen Aktuatoren. In der vorliegenden Arbeit ist die entsprechende Bremsregelungsarchitektur für die elektrischen und automatisierten Fahrzeuge vorgeschlagen, die beinhaltet Funktionen zur (i) primären Bremsung, (ii) gemischten Bremsung und (iii) Radschlupfregelung. Der Schwerpunkt dieser Arbeit ist auf die Robustheit der kontinuierlichen Radschlupfregelung während einer Notbremsung bei hoher und niedriger Fahrbahnreibung gelegt. Als die Lösung sind mehrere Regelungsstrategien entwickelt und experimentell validiert. Die Ergebnisse für drei Fahrzeugprototypen mit individuellen Board- und Radnabemotoren und einem elektrohydraulischen Brake-by-Wire System demonstrieren wesentliche Verbesserung der Bremsleistung und Fahrqualität im Vergleich zu den konventionellen Strategien der Radschlupfregelung

Combined emergency braking and turning of articulated heavy vehicles

‘Slip control’ braking has been shown to reduce the emergency stopping distance of an experimental heavy goods vehicle by up to 19%, compared to conventional electronic/anti-lock braking systems (EBS). However, little regard has been given to the impact of slip control braking on the vehicle’s directional dynamics. This paper uses validated computer models to show that slip control could severely degrade directional performance during emergency braking. A modified slip control strategy, ‘attenuated slip demand’ (ASD) control, is proposed in order to rectify this. Results from simulations of vehicle performance are presented for combined braking and cornering manoeuvres with EBS and slip control braking with and without ASD control. The ASD controller enables slip control braking to provide directional performance comparable with conventional EBS while maintaining a substantial stopping distance advantage. The controller is easily tuned to work across a wide range of different operating conditions.This work was funded by the Engineering and Physical Sciences Research Council and the Cambridge Vehicle Dynamics Consortium (CVDC). At the time of writing, the members of the CVDC included Anthony Best Dynamics, Camcon, Denby Transport, Firestone, Goodyear, Haldex, SIMPACK, HORIBA MIRA, SDC Trailers, Tinsley Bridge, Tridec, Volvo Trucks and Wincanton

Optimal abs frenleme kontrolcüsünün geliştirilmesi

Günümüzde taşıtlarda kullanılan ticari ABS sistemleri, frenleme sırasında tekerleklerin dönme hızlarını ölçmekte, tekerleklerin yavaşlama ivmesi belli bir kritik eşik değerini aştığında fren basıncına müdahale ederek tekerleklerin kilitlenme tehlikesi olmadan yavaşlamalarını sağlamaktadır. Bu sistem, kilitlenmeyi önlemekle birlikte tekerleklerin maksimum frenleme potansiyelini tam olarak ortaya çıkaramamaktadır. Frenleme performansını iyileştirecek çeşitli ABS algoritmaları akademik düzeyde geliştirilmeye devam edilmektedir. Bu süreçteki en önemli zorluk, tekerleklerin farklı yol koşullarında farklı karakteristiklerde davranış göstermesidir. Maksimum frenleme performansını sağlayacak bir ABS algoritması yol koşullarını göz önüne almalı fakat yol koşullarının önceden ölçümü ya da kestiriminin zorluğu, bu amacın önünde bir engel olarak durmaktadır. Bu projede, yol koşulları bilgisine ihtiyaç duymadan optimum frenleme performansını sağlayan bir ABS kontrol algoritması geliştirilmiş ve bu algoritma deneysel bir sisteme uygulanmıştır. Geliştirilen algoritmanın temeli literatürde Ekstremum Arama Algoritması (EAA) olarak adlandırılan yöntemdir. Bu algoritma, bir sistemin optimum çalışma noktasının önceden bilinmediği durumlarda kullanılabilmektedir. Örneğin acil durum frenlemesinde ihtiyaç duyulduğu gibi bilinmeyen yol koşullarında tekerlek ile yol arasındaki frenleme kuvvetlerinin maksimize edilmesi problemi EAA’nın uygulama alanına girmektedir. Projede EAA temelli bir ABS kontrol algoritması geliştirilmiştir. Literatüre getirilen bir yenilik olarak, arama algoritmasında kullanılan parametreler uyarlamalı hale getirilerek Uyarlamalı Ekstremum Arama Algoritması tabanlı ABS kontrolcüsü geliştirilmiştir. Bu şekilde algoritmanın hızlı bir şekilde optimum çalışma noktasını bulması, bulduktan sonra ise parametrelerin uyarlanarak optimum nokta etrafında düşük genlikli salınımlar ile frenleme performansının önemli ölçüde iyileştirilmesi hedeflenmiştir. Ayrıca literatürde önerilen ABS kontrol algoritmalarının hemen hemen hepsi sadece bilgisayarda gerçekleştirilen simulasyonlar ile doğrulanmakta, gerçek bir sistemde gerçek zamanlı uygulanabilirlikleri soru işareti olarak kalmaktadır. Bu projede bir ABS deney düzeneği satın alınmış ve algoritmanın gerçek zamanlı testleri gerçekleştirilmiştir.Commercial ABS systems used in vehicles today measure rotational speed of the wheels. When the wheel deceleration value exceeds acertain critical threshold, the ABS intervenes to the brake pressure to slow down the wheel without the danger of locking. While this system prevents locking of the wheels, it does not provide full braking potential of the tires. The most important challenge in this process is that the wheels show different characteristics in different road conditions. For maximum braking performance, an ABS algorithm must take into account current road conditions, but the difficulty of pre-measurement or estimation of road conditions stands as an obstacle on this goal.In this project, an ABS control algorithm was developed and implemented in an experimental setup. The algorithm provides optimal braking performance without any road condition information. The basis of the algorithm is Extremum Seeking Algorithm (ESA). ESA can be used for the situations where the optimum operating point of a system is not known in advance. For example in emergency braking cases, the problem of maximizing the braking force between the wheel and the road in an unknown road condition is an example of the application area of the ESA. In this project, an ESA-based ABS control algorithm was developed. As a novelty to the literature, by making the parameters of the search algorithm adaptive, an Adaptive Extremum Seeking Algorithm based ABS controller was proposed. In this way, the algorithm finds quickly the optimum operating point, and after finding it, by adapting the parameters, braking with low-amplitude oscillations around the optimum operating point is accomplished. In addition, an ABS experimental setup was purchased and real time tests of the algorithm were conducted in this setup.TÜBİTA

Development of a vehicle dynamics controller for obstacle avoidance

As roads become busier and automotive technology improves, there is considerable potential for driver assistance systems to improve the safety of road users. Longitudinal collision warning and collision avoidance systems are starting to appear on production cars to assist drivers when required to stop in an emergency. Many luxury cars are also equipped with stability augmentation systems that prevent the car from spinning out of control during aggressive lateral manoeuvres. Combining these concepts, there is a natural progression to systems that could assist in aiding or performing lateral collision avoidance manoeuvres. A successful automatic lateral collision avoidance system would require convergent development of many fields of technology, from sensors and instrumentation to aid environmental awareness through to improvements in driver vehicle interfaces so that a degree of control can be smoothly and safely transferred between the driver and vehicle computer. A fundamental requirement of any collision avoidance system is determination of a feasible path that avoids obstacles and a means of causing the vehicle to follow that trajectory. This research focuses on feasible trajectory generation and development of an automatic obstacle avoidance controller that integrates steering and braking action. A controller is developed to cause a specially modified car (a Mercedes `S' class with steer-by-wire and brake-by-wire capability) to perform an ISO 3888-2 emergency obstacle avoidance manoeuvre. A nonlinear two-track vehicle model is developed and used to derive optimal controller parameters using a series of simulations. Feedforward and feedback control is used to track a feasible reference trajectory. The feedforward control loops use inverse models of the vehicle dynamics. The feedback control loops are implemented as linear proportional controllers with a force allocation matrix used to apportion braking effort between redundant actuators. Two trajectory generation routines are developed: a geometric method, for steering a vehicle at its physical limits; and an optimal method, which integrates steering and braking action to make full use of available traction. The optimal trajectory is obtained using a multi-stage convex optimisation procedure. The overall controller performance is validated by simulation using a complex proprietary model of the vehicle that is reported to have been validated and calibrated against experimental data over several years of use in an industrial environment

A global optimal control methodology and its application to a mobile robot model

A global optimal control algorithm is developed and applied to an omni-directional mobile robot model. The aim is to search and find the most intense signal source among other signal sources in the operation region of the robot. In other words, the control problem is to find the global extremum point when there are local extremas. The locations of the signal sources are unknown and it is assumed that the signal magnitudes are maximum at the sources and their magnitudes are decreasing away from the sources. The distribution characteristics of the signals are unknown, i.e. the gradients of the signal distribution functions are unknown. The control algorithm also doesn't need any position measurement of the robot itself. Only the signal magnitude should be measured via a sensor mounted on the robot. The simulation study shows the performance of the controller.Publisher's Versio

3-D Velocity Regulation for Nonholonomic Source Seeking Without Position Measurement

We consider a three-dimensional problem of steering a nonholonomic vehicle to seek an unknown source of a spatially distributed signal field without any position measurement. In the literature, there exists an extremum seeking-based strategy under a constant forward velocity and tunable pitch and yaw velocities. Obviously, the vehicle with a constant forward velocity may exhibit certain overshoots in the seeking process and can not slow down even it approaches the source. To resolve this undesired behavior, this paper proposes a regulation strategy for the forward velocity along with the pitch and yaw velocities. Under such a strategy, the vehicle slows down near the source and stays within a small area as if it comes to a full stop, and controllers for angular velocities become succinct. We prove the local exponential convergence via the averaging technique. Finally, the theoretical results are illustrated with simulations.Comment: submitted to IEEE TCST;12 pages, 10 figure