Balancing Inverted Pendulum by Angle Sensing Using Fuzzy Logic ...

Sensors & TransducersVolume 133, Issue 10October 2011www.sensorsportal.com ISSN 1726-5479Editors-in-Chief: professor Sergey Y. Yurish, tel.: +34 696067716, e-mail: editor@sensorsportal.comEditors for Western EuropeMeijer, Gerard C.M., Delft University of Technology, The NetherlandsFerrari, Vittorio, Universitá di Brescia, ItalyEditor for Eastern EuropeSachenko, Anatoly, Ternopil State Economic University, UkraineEditors for North AmericaDatskos, Panos G., Oak Ridge National Laboratory, USAFabien, J. 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Sensors & Transducers JournalContentsVolume 133Issue 10October 2011www.sensorsportal.com ISSN 1726-5479Research ArticlesA Soft Technique for Measuring Friction Force Using Neural NetworkSunan Huang, Kok Kiong Tan............................................................................................................ 1Neural Net Based Optimization of Wet Thermal Lateral Oxidation Rates“Moh’d Sami” Ashhab, Nabeel Abo Shaban and Abdulla N. Olimat ................................................. 8ECG Acquisition and Analysis System for Diagnosis of Heart DiseasesChannappa Bhyri, Satish T. Hamde, Laxman M. Waghmare ............................................................ 18Soft Computing Based PID Controller Tuning and Application to the Pulpand Paper IndustryB. Nagaraj, P. Vijayakumar ................................................................................................................ 30Greenhouse Environmental Control Using Optimized MIMO PID TechniqueFateh Bounaama and Belkacem Draoui ............................................................................................ 44Design and Real Time Implementation of CDM-PI Control System in a Conical Tank LiquidLevel ProcessP. K. Bhaba and S. Somasundaram .................................................................................................. 53Fuzzy Logic Applied to an Oven Temperature Control SystemNagabhushana Katte, Nagabhushan Raju Konduru, Bhaskar Pobbathi and Parvathi Sidaraddi...... 65Balancing Inverted Pendulum by Angle Sensing Using Fuzzy Logic Supervised PIDController Optimized by Genetic AlgorithmAshutosh K. Agarwal, Sanjeev Kumar ............................................................................................... 74A Novel Method for Gearbox Fault Detection Based on Biorthogonal B-spline WaveletGuangbin Zhang and Yunjian Ge....................................................................................................... 83A Multi-hop Topology Control Based on Inter-node Range Measurement for WirelessSensor Networks Node LocalizationAli Husein Alasiry, Shinji Ohyama ...................................................................................................... 95Distributed Point Source Technique in Modeling Surface-Breaking Crack in a MFL TestMehdi Kiyasatfar, Maqsud Golzan, Nader Pourmahmoud, Mehdi Eskandarzade............................. 108Authors are encouraged to submit article in MS Word (doc) and Acrobat (pdf) formats by e-mail: editor@sensorsportal.comPlease visit journal’s webpage with preparation instructions: http://www.sensorsportal.com/HTML/DIGEST/Submition.htmInternational Frequency Sensor Association (IFSA).

Sensors & Transducers Journal, Vol. 133, Issue 10, October 2011, pp. 74-82Sensors & TransducersISSN 1726-5479© 2011 by IFSAhttp://www.sensorsportal.comBalancing Inverted Pendulum by Angle SensingUsing Fuzzy Logic Supervised PID ControllerOptimized by Genetic Algorithm1Ashutosh K. AGARWAL, 2 Sanjeev KUMAR1Department of Electronics & Communication Engineering, Krishna Engineering College,Ghaziabad, 201009, IndiaTel.: +9199996454632Department of Mechanical Engineering, Ajay Kumar Garg Engineering College,Ghaziabad, 201007, IndiaTel. +919818187936E-mail: ashutosh@blueink.in, s.kumar@akgec.orgReceived: 2 July 2011 /Accepted: 25 October 2011 /Published: 31 October 2011Abstract: Genetic algorithms are robust search techniques based on the principles of evolution. Agenetic algorithm maintains a population of encoded solutions and guides the population towards theoptimum solution. This important property of genetic algorithm is used in this paper to stabilize theInverted pendulum system. This paper highlights the application and stability of inverted pendulumusing PID controller with fuzzy logic genetic algorithm supervisor . There are a large number of wellestablished search techniques in use within the information technology industry. We propose a methodto control inverted pendulum steady state error and overshoot using genetic algorithm technique.Copyright © 2011 IFSA.Keywords: Inverted pendulum, Robotic arm, Genetic algorithm, Fuzzy logic, PID controller.1. IntroductionThe inverted pendulum is the classic nonlinear control experiment. In addition to being the simplestunstable non-linear system imaginable, it also provides a simple model for rocket control. Surprisingly,this dynamics continues to appear in numerous other systems of interest. The inverted pendulumdriven by a lateral acceleration is clearly present in the dynamic balance of a skier racing down the74

Sensors & Transducers Journal, Vol. 133, Issue 10, October 2011, pp. 74-82slope. In much the same way, simple models for exploring bicycle and motorcycle dynamics includethe inverted pendulum as key subsystem, imposing strong constraints on the system performance [1].An inverted pendulum is a pendulum which has its mass above its pivot point. It is often implementedwith the pivot point mounted on a cart that can move horizontally and may be called a cart and pole.Whereas a normal pendulum is stable when hanging downwards, an inverted pendulum is inherentlyunstable, and must be actively balanced in order to remain upright, either by applying a torque at thepivot point or by moving the pivot point horizontally as part of a feedback system [2, 3].In this paper, we propose a tuning method of the PID gains by using fuzzy logic to stabilize theinverted pendulum. The tuning method proposed here is to replace the human operator by fuzzy logic.In order to get fast convergence and attain the better solution for a local minimum problem, we adoptthe genetic algorithm to find parameter included in the fuzzy logic [4].1.1 1.1. Problem FormulationIt is virtually impossible to balance a pendulum in the inverted position without applying someexternal force to the system. The Carriage Balanced Inverted Pendulum (CBIP) system, shown inFig. 1, allows this control force to be applied to the pendulum carriage.Fig. 1. Carriage Balanced Inverted Pendulum (CBIP) system.Inverted pendulum [5] and cart system adopted in this paper consists of a straight line rail, the cartwhich can move left and right on the rail without friction, the pendulum which is hinged on the centerof the top surface of the cart and can rotate around the pivot in the same vertical plane with the rail andthe driving unit as shown in Fig. 2. It means that joint between cart and pendulum is a revolute joint(1-dimensional motion). The outputs from the CBIP rig can be carriage position, carriage velocity,pendulum angle and pendulum angular velocity (only pendulum angle in our case). The pendulumangle is fed back to controller which controls the servo-motor, ensuring consistent and continuoustraction.The aim of the study is to stabilize the pendulum such that the position of the carriage on the track iscontrolled quickly and accurately and that the pendulum is always maintained tightly in its invertedposition during such movements. Using Lagrangian technique we obtain the transfer function [6-8].The transfer function can thus be simplified as:(s)U(s)qs3 b(I ml2) s2mls mgl(M m)s bgml,(1)75

Sensors & Transducers Journal, Vol. 133, Issue 10, October 2011, pp. 74-82where22q ( M m)(I ml ) ( ml) (s) Laplace transform of the Angular position of the pendulum.Fig. 2. A Cart.If we neglect the friction in the system, that is, we take the coefficient of friction b=0,then(s)U ( s)s2KAP2P1,(2)whereandK 1 P ( M m ) gA P ( M( M m)mgl2 m)(I ml ) ( ml)2Thus, the linearized approximation transfer function for the IP has been obtained.1.2. Actuation MechanismThe actuation mechanism consists of a movable cart (on rail), driven by a DC motor via a pulley andbelt [8]. So the overall transfer function of the actuation mechanism will depend upon the transferfunction of “the DC Motor” and “the Pulley, Belt & Cart”.After considering the contribution of DC Motor, pulley, belt and cart we get the transfer function,(s)E(s)s Ks( ms1)(A22P1)(3)76

where K K K K r( M m)FPSensors & Transducers Journal, Vol. 133, Issue 10, October 2011, pp. 74-82ME(s) = Laplace transform of the Error Voltage, and (s) Laplace transform of the Angular position of the pendulumNow taking feedback into consideration the over all system takes the form as shown in Fig. 3. Herer(s) is reference input signal (unit step signal) and c(s)is output signal.PID Controller Inverted Pendulum + (s) e(s)rE(s) Servomechanism c(s)Feedback H(s)Fig. 3. Inverted Pendulum system with position feedback.The overall transfer function with feedback is: ( s)c ( s)rG(s)1G(s)H ( s),(4) ( s)c ( s)r1.78s2( s 4.356)( s 6.356s10.31)(5)There is a pole on the right hand side of the origin. So the feedback alone can not make the systemstable [9].1.3. Controller for the Inverted PendulumNow we have two challenges for the upper described problem:1) Decrease the steady state error up to a maximum allowable range;2) Reduce the overshoot (or undershoot) and settling time.1.4. Parameters for Designing Fuzzy Supervised PID ControllerThe fuzzy PID supervisor described by Van Nauta Lemke and De-Zhao and Van Nauta Lemke andKrijgsman is based on a fuzzy supervisor and a PID controller [10, 11]. In their scheme the fuzzy77

Sensors & Transducers Journal, Vol. 133, Issue 10, October 2011, pp. 74-82supervisor has a rule base in which the fuzzy rules have consequents which address the gains of thePID controller. Hence, the fuzzy supervisor has three “outputs”: KP, KIand KDwhich result indifferent parameters of the PID controller each sampling instant:K'P'I[ kT] K K[ kT](6)PPK [ kT] K K[ kT](7)K'DII[ kT ] K K[ kT](8)DThe inputs of the fuzzy supervisor are: the error e [kT]and its first difference e[kT]. Or in otherwords the error and error rate are given as inputs to fuzzy system. We find that the error depends on K Ponly and the error is reducing with increase in value of K P . For K P 450, the steady state error is 2 %.First we will focus on the error only. We will design such a fuzzy logic supervisory system whichgives output in incremental form of K P for the given inputs e(t) and de/dt. The fuzzy supervisor willchange the value of K P so that the error is less than 2 %.Rewriting the equations:D2.55KDDamping ratio (9)2 2.55K 22.47PSettling time (on 2% basis)ts3.137 (10)KDError 22.47ess(11)2.55K 22.47P Maximum Overshoot = exp 10021 (12)The inverted pendulum system as in eq. (1) is analyzed at different stages. The open loop system(Servomechanism + Inverted Pendulum) has one pole on the right hand side, so the inverted pendulumis unstable in open loop. After introducing position feedback as in Fig. 3 and a feedback transferfunction H(s) =2.86, we get transfer function of the closed loop system. Still one pole is on the righthand side. Hence, this closed loop system is also unstable.[reference of same paper]2. Proposed SystemExisting system incorporates fuzzy controllers for stabilizing an inverted pendulum [9]. The error inthe system depends on K P only and the error is reducing with increase in value of K P , the steady stateerror being less than or equal to 2 % , for K P 450.This design has a fuzzy supervisory system whichgives output in incremental form of K P for the given inputs e(t) and de/dt [11].78

Sensors & Transducers Journal, Vol. 133, Issue 10, October 2011, pp. 74-82In the proposed system we have included a Genetic Algorithm system after the Fuzzy supervisor asshown in Fig. 4. Initially, fuzzy supervisor gives output in incremental form so that error is less than2 %, then the GA system finds the optimal solution in the range 264 K P 529, so that the error andthe overshoot both are minimum. In the proposed system coding of GA system has been done in C++language.Fig. 4. Fuzzy GA supervised PID Controller for IP System.2. Simulation ResultsAfter introducing fuzzy supervised PID-controller, we get a stable response of the system as shown inFig. 5. The fuzzy supervisor changes the value of K P so that the error is less than 2 % [11].Fig. 5. Response of inverted pendulum for an applied input.79

Sensors & Transducers Journal, Vol. 133, Issue 10, October 2011, pp. 74-82The values of KPobtained from fuzzy system is fed into GA system. The optimal value of K P isobtained on the basis of cost functionCost function = 0.8 + 0.2 overshoot(in this case) (13)eSSFigs. 6 and 7 are the step responses for K GA , where K GA is the value of K P obtained from GA system.Fig. 6. Step Response of the System at K P =456.Fig. 7. Step Response of the System at K P =410.3. ConclusionAs depicted in Fig. 8. the results from fuzzy system have been improved by GA system. The GAsystem finds such a value of K P for each input, so that the cost function is decreased for each input.80

Sensors & Transducers Journal, Vol. 133, Issue 10, October 2011, pp. 74-82Here we have given more weightage to error than overshoot. So the output values of K P are closer toupper limit (K P =529). If more weightage is given to overshoot, then the output values of K P will becloser to lower limit (K P =264).Comparison of cost functionCost Function2.22.182.162.142.122.12.082.062.042.0221 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21Data PointsFig. 8. Comparison of Cost Function.KfuzzifiedKgaAs we can see in Fig. 8, the proposed controller gives the satisfactory response for the rule base offuzzy logic, and the results are further improved by the use of GA supervisor after Fuzzy logicSupervisor. Hence for the given design requirement the above simulated controller is satisfactory. Wealso can conclude that the system approaches towards stability with time.AcknowledgmentThe authors are grateful to the management of Krishna Engineering College for giving all the facilitiesrequired to carry out the work. We thank Prof S. B. Bajpayee and Prof T. Ramanujam for theirencouragement.References[1]. Hauser John, Saccon Alessandro, and Frezza Ruggero, On the Driven Inverted Pendulum, in Proc. of the44 th IEEE Conference on Decision and Control, and the European Control Conference, 2005,pp. 6176-6180.[2]. Furuta K., M. Yamakita M and S. Kobayashi, Swing-up control of inverted pendulum using pseudo-statefeedback, in Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems andControl Engineering November, 206, no.6, 1992, pp. 263-269.[3]. A. Ohsumi and T. Izumikawa, Nonlinear Control of Swing-up and Stabilization of an Inverted pedndulum,in Proceedings of the 34 th Conference on Decision & Control, 1995, pp. 3873-3880.[4]. Wang Q., Spronck Pieter, and Tracht Rudolf, An overview of Genetic Algorithms Applied to ControlEngineering Problems, in Proc. of the International Conference on Machine Learning and Cybernetics,2003, pp. 1651 – 1656.[5]. Kui Fu Chen, Standing Human - an Inverted Pendulum, Lat. Am. J. Phys. Educ., Vol. 2, No. 3, Sept. 2008.[6]. W. Zhong and H. Röck, Energy and Passivity Based Control of the Double Inverted Pendulum on a Cart,in Proc. of the IEEE Conference on Control Applications, 2001.[7]. Samatthachai Panya, Taworn Benjanarasuth, Songmoung Nundrakwang, Jongkol Ngamwiwit and NoriyukiKomine, Hybrid Controller for Inverted Pendulum System, in Proc. of the IEEE International Symposium81

Sensors & Transducers Journal, Vol. 133, Issue 10, October 2011, pp. 74-82on Communications and Information Technologies (ISCIT’ 2008), 2008, pp. 385-388-u.[8]. A. S. Shiriaev, A. Friesel, J. Perram, On Stabilization of Rotational Modes of an Inverted Pendulum, inProceedings of the 39 th IEEE Conference on Decision and Control, Sydney, Australia, December, 2000,pp. 5047-5052.[9]. Yamakita M., and K. Furuta, VSS Adaptive Control based on nonlinear model for TITech Pendulum, inProceedings of the IEEE International Conference on Industrial Electronics (IECON’ 92), San Diego,USA, 1992, pp. 1488-1493.[10]. Balázs KULCSÁR, LQG/LTR Controller Design for an Aircraft Model, Periodica Polytechnica Ser.Transp. Eng., Vol. 28, No. 1–2, Nov. 2000, pp. 131–142.[11]. Ashutosh K., Agarwal, A., Novel Method to Balance Inverted Pendulum by Angle Sensing Using FuzzyLogic supervised PID Controller, Sensors & Transducers, Vol. 126, Issue 3, March 2011, pp. 92-100.___________________2011 Copyright ©, International Frequency Sensor Association (IFSA). All rights reserved.(http://www.sensorsportal.com)82

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