SLIDING MODE CONTROL Edited by Andrzej Bartoszewicz Sliding Mode Control Edited by Andrzej Bartoszewicz Published by InTech Janeza Trdine 9, 51000 Rijeka, Croatia Copyright © 2011 InTech All chapters are Open Access articles distributed under the Creative Commons Non Commercial Share Alike Attribution 3.0 license, which permits to copy, distribute, transmit, and adapt the work in any medium, so long as the original work is properly cited After this work has been published by InTech, authors have the right to republish it, in whole or part, in any publication of which they are the author, and to make other personal use of the work Any republication, referencing or personal use of the work must explicitly identify the original source Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher No responsibility is accepted for the accuracy of information contained in the published articles The publisher assumes no responsibility for any damage or injury to persons or property arising out of the use of any materials, instructions, methods or ideas contained in the book Publishing Process Manager Ivana Lorkovic Technical Editor Teodora Smiljanic Cover Designer Martina Sirotic Image Copyright Jenny Solomon, 2010 Used under license from Shutterstock.com First published March, 2011 Printed in India A free online edition of this book is available at www.intechopen.com Additional hard copies can be obtained from orders@intechweb.org Sliding Mode Control, Edited by Andrzej Bartoszewicz p cm ISBN 978-953-307-162-6 free online editions of InTech Books and Journals can be found at www.intechopen.com Contents Preface Part Chapter IX Sliding Mode Control in Power Electronics Sliding Mode Control and Fuzzy Sliding Mode Control for DC-DC Converters Kamel Ben Saad, Abdelaziz Sahbani and Mohamed Benrejeb Chapter Investigation of Single-Phase Inverter and Single-Phase Series Active Power Filter with Sliding Mode Control 25 Mariya Petkova, Mihail Antchev and Vanjo Gourgoulitsov Chapter Sliding Mode Control for Industrial Controllers Khalifa Al-Hosani, Vadim Utkin and Andrey Malinin Chapter The Synthetic Control of SMC and PI for Arc Welding/cutting Power Supply 77 Guo-Rong Zhu and Yong Kang Chapter Sliding Mode Control of Fuel Cell, Supercapacitors and Batteries Hybrid Sources for Vehicle Applications 87 M Y Ayad, M Becherif, A Aboubou and A Henni Chapter Sensorless First- and Second-Order Sliding-Mode Control of a Wind Turbine-Driven Doubly-Fed Induction Generator 109 Ana Susperregui, Gerardo Tapia and M Itsaso Martinez Part Chapter 45 Sliding Mode Control of Electric Drives 133 Sliding Mode Control Design for Induction Motors: An Input-Output Approach 135 John Cortés-Romero, Alberto Luviano-Juárez and Hebertt Sira-Ramírez VI Contents Chapter Cascade Sliding Mode Control of a Field Oriented Induction Motors with Varying Parameters 155 Abdellatif Reama, Fateh Mehazzem and Arben Cela Chapter Sliding Mode Control of DC Drives 167 B M Patre, V M Panchade and Ravindrakumar M Nagarale Chapter 10 Sliding Mode Position Controller for a Linear Switched Reluctance Actuator 181 António Espírito Santo, Maria Rosário Calado and Carlos Manuel Cabrita Chapter 11 Application of Sliding Mode Control to Friction Compensation of a Mini Voice Coil Motor 203 Shir-Kuan Lin, Ti-Chung Lee and Ching-Lung Tsai Part Sliding Mode Control of Robotic Systems 219 Chapter 12 Sliding Mode Control for Visual Servoing of Mobile Robots using a Generic Camera Héctor M Becerra and Carlos Sagüés 221 Chapter 13 Super-Twisting Sliding Mode in Motion Control Systems 237 Jorge Rivera, LuisGarcia, Christian Mora, Juan J Raygoza and Susana Ortega Chapter 14 Non-Adaptive Sliding Mode Controllers in Terms of Inertial Quasi-Velocities 255 Przemyslaw Herman and Krzysztof Kozlowski Part Selected Applications of Sliding Mode Control Chapter 15 Force/Motion Sliding Mode Control of Three Typical Mechanisms 281 Rong-Fong Fung and Chin-Fu Chang Chapter 16 Automatic Space Rendezvous and Docking using Second Order Sliding Mode Control 307 Christian Tournes, Yuri Shtessel and David Foreman Chapter 17 High Order Sliding Mode Control for Suppression of Nonlinear Dynamics in Mechanical Systems with Friction 331 Rogelio Hernandez Suarez, America Morales Diaz, Norberto Flores Guzman, Eliseo Hernandez Martinez and Hector Puebla 279 Contents Chapter 18 Control of ROVs using a Model-free 2nd-Order Sliding Mode Approach 347 Tomás Salgado-Jiménez, Luis G García-Valdovinos and Guillermo Delgado-Ramírez Chapter 19 Sliding Mode Control Applied to a Novel Linear Axis Actuated by Pneumatic Muscles Dominik Schindele and Harald Aschemann Chapter 20 Chapter 21 Part Adaptive Sliding Mode Control of Adhesion Force in Railway Rolling Stocks Jong Shik Kim, Sung Hwan Park, Jeong Ju Choi and Hiro-o Yamazaki 369 385 A Biomedical Application by Using Optimal Fuzzy Sliding-Mode Control Bor-Jiunn Wen 409 New Trends in the Theory of Sliding Mode Control 429 Chapter 22 Sliding Mode Control of Second Order Dynamic System with State Constraints 431 Aleksandra Nowacka-Leverton and Andrzej Bartoszewicz Chapter 23 Sliding Mode Control System for Improvement in Transient and Steady-state Response 449 Takao Sato, Nozomu Araki, Yasuo Konishi and Hiroyuki Ishigaki Chapter 24 A New Design for Noise-Induced Chattering Reduction in Sliding Mode Control 461 Min-Shin Chen and Ming-Lei Tseng Chapter 25 Multimodel Discrete Second Order Sliding Mode Control : Stability Analysis and Real Time Application on a Chemical Reactor 473 Mohamed Mihoub, Ahmed Said Nouri and Ridha Ben Abdennour Chapter 26 Two Dimensional Sliding Mode Control Hassan Adloo, S.Vahid Naghavi, Ahad Soltani Sarvestani and Erfan Shahriari Chapter 27 Sliding Mode Control Using Neural Networks 509 Muhammad Yasser, Marina Arifin and Takashi Yahagi Chapter 28 Sliding Mode Control Approach for Training On-line Neural Networks with Adaptive Learning Rate 523 Ademir Nied and José de Oliveira 491 VII Preface The theory of variable structure systems with sliding modes is currently one of the most important research topics within the control engineering domain Moreover, recently a number of important applications of the systems primarily in the field of power electronics, control of electric drives, robotics and position regulation of sophisticated mechanical systems have also been reported Therefore, the objective of this monograph is to present the most significant latest developments in the theory and engineering applications of the sliding mode control and to stimulate further research in this field The monograph consists of 28 chapters It begins with six contributions devoted to various significant issues in power electronics In the first chapter, Ben Saad et al propose, test and compare sliding mode and fuzzy sliding mode controllers for DC-DC converters In the second chapter, Petkova et al consider the operation of the singlephase inverter and single-phase active power filter and prove, both in simulations and laboratory experiments, the effectiveness of sliding mode controllers in these two applications Then, Al-Hosani et al also consider the design of DC-DC buck and boost converters They develop the sliding mode approach which implements – very common in industry – proportional integral derivative (PID) controllers The main idea of that chapter may be summarized as enforcing sliding mode such that the output converter voltage contains proportional, integral and derivative components with the predefined coefficients Chattering is then reduced through the use of multiphase power converter structure The proposed design methods are confirmed by means of computer simulations In the next chapter, Zhu and Kang consider arc welding/cutting power supply and propose a “synthetic” sliding mode and PI controller They propose to use the PI controller in the current loop and the sliding mode controller in the voltage loop The results are verified by experiments conducted on a 20 kW arc welding/cutting power source They show on one hand good dynamic performance of the system, and on the other decreased undesirable voltage overshoot Another contribution concerned with power electronics is the chapter by Ayad et al which presents sliding mode control of fuel cells, supercapacitors and battery hybrid sources for vehicle applications Then, the chapter by Susperregui presents and evaluates first-order and higher-order sensorless sliding mode control algorithms, for a doubly-fed induction generator The algorithms not only aim at governing active and reactive power exchange between the doubly-fed induction generator stator and the grid, but also ensure the synchronization required for smooth connection of the generator stator to the grid Sliding mode systems are a feasible option not only for power converter control but also for electric drive regulation Therefore an important issue of induction motor control is X Preface addressed in the next two chapters The chapter by Cortes-Romero and Sira-Ramirez presents a combination of two control loops, one employing a discontinuous sliding mode controller and another one based on the combination of generalized proportional integral control and generalized proportional integral disturbance observer The authors of the chapter demonstrate – by experiments performed on an actual induction motor test bed with a voltage controlled brake – that the proposed combination results in robust position and tracking control of induction motors In the next chapter, written by Reama et al a new simple and easy to implement adaptive sliding mode scheme for speed and flux control of induction motor using online estimation of the rotor resistance and load torque are proposed The two chapters on control of induction motors are followed by a contribution of Patre and Panchade, which is concerned with a unified sliding mode approach to torque, position, current and speed regulation of DC drives Then the next chapter, by Santo et al., presents the design and implementation of a sliding mode position controller for a linear switched reluctance actuator devoted primarily for robotic applications The section devoted to the problem of electric drive control ends up with a chapter on friction compensation for a mini voice coil motors The chapter written by Lin et al., demonstrates that sliding mode control approach may reliably eliminate stick slip oscillations and reduce the steady state error This conclusion is drawn based on experimental results performed on a mini voice coil motor mounted on a compact camera module The next three chapters are concerned with selected issues in robotics The first of them, written by Becerra and Sagues proposes a robust controller for image-based visual servoing for differential drive mobile robots The second one, by Rivera et al., is devoted to the application of a higher order, namely super-twisting sliding mode controller for trajectory tracking of an under-actuated manipulator and also for induction motors Then Herman and Kozłowski consider rigid, serial manipulators and present an extensive survey of selected non-adaptive sliding mode controllers expressed in terms of the inertial quasi-velocities They also point out a number of advantages offered by sliding mode control schemes using inertial quasi-velocities The next seven chapters present successful applications of sliding mode control paradigm in other areas than power electronics, electric drives and robotics The section devoted to those applications begins with the chapter by Fung and Chang on sliding mode force and motion control of three very popular mechanisms, i.e slider-crank, quick-return and toggle mechanism Then Tournes et al propose a higher order sliding mode control scheme for automatic docking of space vehicles The issue of higher order sliding mode control is also considered in the chapter, by Suares et al In that contribution higher order sliding mode is successfully used to suppress nonlinear dynamics in physical plants with friction which is inevitable in all mechanical systems Higher order sliding mode approach is further considered in the chapter by Salgado-Jiménez et al on control of remotely operated vehicles which are nowadays indispensable in performing the inspection tasks and maintenance of numerous underwater structures, common in the oil industry, especially in deep and not easily accessible to humans waters That chapter demonstrates that sliding mode control is a viable option for controlling underwater vehicles which operate in a highly dynamic and uncertain environment often affected by waves and strong currents Another interesting and very well worked out application is described in the next chapter authored by Schindele and Aschemann They propose three types of sliding mode controllers (conventional, second-order and proxy) for a linear axis driven by four pneumatic muscles and verify performance of these controllers on a laboratory test rig Then Kim et al present adaptive sliding mode controller of adhesion force between the rail and the wheel in railway rolling stocks The section concerned with various applications of sliding mode control concludes with the chapter by Wen on optimal fuzzy sliding mode control of biochips and biochemical reactions Sliding Mode Control and Fuzzy Sliding Mode Control for DC-DC Converters 3.2.2 Simulation and experimental results The SMC is tested by simulation and experimentally using a dSAPCE control board The test bench was built as shown in Fig 10 and Fig 11 around: a Buck converter, a computer equipped with a dSPACE DS1104 with its connector panel, a DC voltage power supply, two load resistances Fig Photo of the studied Buck converter Fig Photo of the test bench The dSPACE DS1104 controller board is a prototyping system It is a real time hardware platform It can be programmed with MATLAB/SIMULINK software through a real time interface allowing the generation of a real time code Two ADC input channels of the DS1104, characterized by a 16 bits resolution, are used to acquire the Buck converter output voltage and the inductance current The control board generates a digital PWM signal which is used to control the switch of the Buck converter The proposed SMC was applied to a Buck converter characterized by the parameters given in the table Parameters Vin C L R Switching frequency Table Studied buck converter parameters Values 15 V 22 μF mH 10 Ω 10 kHz 10 Sliding Mode Control Voltage (V) 0 0.5 1.5 2.5 3.5 4.5 Time (s) -3 x 10 Fig Open loop responses of the buck converter by application of 16% PWM control signal Voltage (V) 0 0.5 1.5 2.5 3.5 4.5 Time (s) -3 x 10 Fig Application of the SMC to the studied Buck converter ( Vref = 5V ) 12 6.5 11.5 Voltage (V) Voltage (V) 11 10.5 10 9.5 4.5 8.5 5.5 3.5 0.042 0.046 0.05 Time (s) 0.054 (a) Reference voltage 10 V 0.058 0.042 0.046 0.05 Time (s) 0.054 0.058 (b) Reference voltage 5V Fig Experimental test robustness of the SMC for the variation of the load from 10 Ω to 15 Ω Sliding Mode Control and Fuzzy Sliding Mode Control for DC-DC Converters 11 Fig presents the simulated output voltage by application of a PWM control signal of 16% duty cycle The voltage response corresponds to a second order damped system response with an overshoot Fig presents the obtained result by application of the proposed SMC to the studied controller for a V voltage reference We can see clearly that the observed voltage overshoot obtained on the open-loop response disappeared by application of the SMC The SMC is tested experimentally for the case of the load variation Fig presents the obtained results for the case of the variation of the load resistance from 10 Ω to 15 Ω at 0.05s It is clear that this perturbation is quickly rejected because the output voltage attends the reference voltage The experimental test result for the case of the input voltage variation from 30 V to 20V, given in Fig 10, shows the robustness of the applied SMC 35 Vin V0 Voltage (V) 30 25 20 15 10 0.01 0.03 0.05 Time (s) 0.07 0.09 Fig 10 Experimental output voltage evolution by application of the SMC for the variation of the input voltage from 30V to 20V 3.3 SMC for Buck-Boost converter 3.3.1 Proposed SMC principle As for the Buck Converter, the Buck-Boost converter sliding surface and output voltage error are respectively defined by equations (8) and (9) k can be chosen so that the outer voltage loop is enough to guarantee a good regulation of the output voltage with a near zero steady-state error and low overshoot Without any high frequency, when the system is on the sliding surface, we have S = and S = (Hu et al, 2005; El Fadil et al, 2008) As the control signal applied to the switch is pulse width modulated, we have only to determine the equivalent control component By considering the mathematical model of the DC-DC Boost converter, at the study state the variation of the surface can be expressed as: S = ke = − kv0 = − k( − ueq C iL − v0 ) RC and then from equation (20) and by considering the condition S = we have: (19) 12 Sliding Mode Control kv0 + v0 − ueq = iL RC C (20) From the state representation (7) we can write the following relation: v0 ( − ueq ueq − v − k) = ( v0 + in ) RC C L L (21) Then equivalent control component expression: ueq = − vin + vin + kL (CRk − L )( vref − v0 ) R v0 (22) 3.3.2 Simulation results The proposed SMC was applied by simulation to the studied Buck-Boost converter characterized by the parameters given in Table Fig 11 presents the studied converter open-loop voltage and current responses In Fig 12 the output voltages evolution obtained by application of the SMC are presented for a reference voltages Vref = −20V So the application of the SMC allowed the elimination of the overshoot observed for the open-loop response Fig 13 presents the control signal We can notice that it is strongly hatched This is a consequence of the chattering phenomenon PARAMETERS VALUES Vin 20 V C 22 μF L mH R 10 Ω SWITCHING FREQUENCY 10 kHz Table Studied Buck Boost converter parameters To test the robustness of the SMC, we consider now the variations of the load resistance and the input voltage Fig 14 presents the evolution of the output voltage and the current in the load for the case of a sudden change of the load resistance from 30Ω to 20Ω So by the application of the SMC, this perturbation was rejected in 10.10-3s and the output voltage attends the reference voltage after Fig 15 illustrates the sudden variation of the input voltage from 15V to 10V at 0.05s For such case we notice that the output voltage, presented by Fig 16, attends after the rejection of the perturbation the desired value −20V and the converter work as boost one 13 Sliding Mode Control and Fuzzy Sliding Mode Control for DC-DC Converters Voltage (V) -5 -10 -15 -20 -25 0.01 0.02 0.03 Time (s) 0.04 0.05 0.06 Fig 11 Output voltage evolution of the Buck-Boost converter obtained by open-loop control Vo ltag e (V) -5 -10 -15 -20 -25 0.01 0.02 0.03 0.04 0.05 0.06 Time (s) Fig 12 Output vvoltage evolution obtained by application of the SMC 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.01 0.02 0.03 Time (s) Fig 13 Control signal evolution 0.04 0.05 0.06 14 Sliding Mode Control Vo ltag e (V ) -5 -10 -15 -20 -25 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 Time (s) Fig 14 Output voltage evolution by application of the SMC for the case of load variation from 30Ω to 20Ω 20 Vo ltag e (V) 15 10 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.08 0.09 0.1 Time (s) Fig 15 Input voltage variation -5 Vo ltag e (V) -10 -15 -20 -25 -30 0.03 0.04 0.05 0.06 0.07 Time (s) Fig 16 Output voltage evolution by application of the SMC for the case of input voltage variation from 15 V to 10V Sliding Mode Control and Fuzzy Sliding Mode Control for DC-DC Converters 15 In order to overcome the problem of the chattering phenomena, it is possible to apply a high order SMC However, the obtained analytical expression of the control component can be so complicated In the following, we will propose to apply a FSMC to the studied converters in order to improve the robustness of the SMC and to overcome the chattering problem FSMC for DC- DC converters As SMC, Fuzzy Logic Control (FLC) is known to be robust Moreover, it is considered to be an alternative to the chattering problem FLC is an intelligent control complying with complex or uncertain systems Some researchers show that FLC is a general form of variable structure control Thus, some attempts have been made in order to integrate the SMC and FLC to a Fuzzy Sliding Mode Control (FSMC) However, the design of a fuzzy sliding mode controller for nonlinear system is a difficult problem There have been quite a lot of researches on the combination of sliding mode control with fuzzy logic control techniques for improving the robustness and the performances of nonlinear systems with uncertainty (Qiao et al, 03) We can distinguish two classes of control algorithms for FSMC The first class is the fuzzy boundary layer SMC where the signum function is replaced by a fuzzy map so that the control input switch in a smooth manner to the equivalent control component As consequence chattering is reduced As an example of this kind of control a FSMC proposed by PALM is adopted The second class consists of the set of fuzzy control algorithms which approximate the input-output map of traditional sliding mode control (Alouani, 1995) 4.1 Fuzzy boundary layer SMC For the first class of FSMC we present in the following the method proposed by PALM in (Palm, 1992) e S ep d0 dsn P( e p , e p ) E o ep e Fig 17 Distances dsn and d Let us consider a second order system and as an example the sliding surface defined as follows (Sahbani et al, 2008): S = EY T (23) 16 Sliding Mode Control where E = [ e e ] and Y = [ k 1] with k a constant gain and e the output system error The distance between the trajectory error and the sliding surface dsn is defined as follows: dsn = ep + kep (24) + k2 dsn is the normal distance between the point P( ep , ep ) and the sliding surface Such distance is illustrated graphically in Fig 17 for an arbitrary point P( e p , e p ) Let H ( ep , e p ) be the intersection point of the switching line and its perpendicular passing through the point P( e p , e p ) is defined as the distance between the point H ( ep , e p ) and the origin O The distance is expressed as follows: = 2 E − dsn (25) The presented FSMC has as inputs the two distances dsn and The output signal is the control increment ΔU ( k ) which is used to update the control signal defined as follows: U ( k ) = ΔU ( k ) + U ( k − 1) (26) The control law is equivalent to an integral action allowing a steady state error The presented FSMC is a Mamdani fuzzy inference system composed by a fuzzification block, a rule base bloc and a defuzzification block Trapezoidal and triangular membership functions, denoted by N (Negative), Z (Zero) and P (Positive), are used for dsn The same shape of membership functions denoted by Z (Zero), PS (Positive small ) and PB (Positive Big) are used for dsn and membership functions are presented respectively in Fig 18 and Fig 19 in the normalized domain [ −1 1] for dsn and [ 1] for -1 N P Z 0.5 dsn 0.5 Fig 18 dsn membership functions For the output signal of the proposed FSMC, fives triangular membership functions, denoted by NB (Negative Big), NM (Negative Middle), Z (Zero), PM (Positive Middle), PB (Positive Big) are used for the output signal Δd , Fig 20 The rule base is given by table Sliding Mode Control and Fuzzy Sliding Mode Control for DC-DC Converters PS Z 0.2 17 PB 0.4 0.6 0.8 d0 Fig 19 membership functions dsn N P Z PS Z NS PS PB PS NB PB Z PB NS NB Z PS Table Rule base of the proposed FSMC NB NS -1 -0.5 0.5 PB ΔU Fig 20 ΔU membership functions 4.2 Fuzzy Lyapunov function SMC The second class of FSMC uses the surface S and its variation S to define the changes on the control signal The aim of this kind of FSMC is to insure the Lyapunov stability condition SS < Let us consider the sliding surface S The proposed fuzzy sliding mode controller forces the derivative of the Lyapunov function to be negative definite So, the rule base table is established to satisfy the inequality (17) 18 Sliding Mode Control Intuitively, suppose that S > and S > , the duty cycle must increase Also, if S < and S < the duty cycle must decrease Thus, the surface S and its variation S are the inputs of the proposed controller The output signal is the control increment ΔU ( k ) which is used to update the control law As for the Fuzzy boundary layer SMC the control signal is defined by equation (26) The proposed Fuzzy Sliding Mode Controller is a Sugeno fuzzy controller which is a special case of Mamdani fuzzy inference system Only the antecedent part of the Sugeno controller has the “fuzzyness”, the consequent part is a crisp function In the Sugeno fuzzy controller, the output is obtained through weighted average of consequents As the proposed approach have to be implemented in practice, such choice can be motivated by the fact that Sugeno fuzzy controller is less time consuming than the Mamdani one Trapezoidal and triangular membership functions, denoted by N (Negative), Z (Zero) and P (Positive), were used for both the surface and the surface change They are respectively presented in Fig 21 and Fig 22 in the normalized domain [ −1 1] For the output signals, fives normalized singletons denoted by NB (Negative Big), NM (Negative Middle), Z (Zero), PM (Positive Middle), PB (Positive Big) are used for the output signal ΔU , Fig 23 The normalized control surface of the proposed FSMC, corresponding to the Rule Base given in table 5, is presented in Fig 24 Such surface shows clearly the nonlinear characteristic of the proposed fuzzy control law N Z P -1 -0,5 0,5 S Fig 21 Surface S membership functions N Z -1 -0,2 0,2 • S Fig 22 Surface change S membership functions P 19 Sliding Mode Control and Fuzzy Sliding Mode Control for DC-DC Converters Fig 23 Output singletons S N P P Z PM PB Z NM Z PM N S Z NB NM Z Table Rule base of the proposed FSMC ΔU 0.5 -0.5 -1 0.5 • S 0.5 0 -0.5 -0.5 -1 -1 S Fig 24 FSMC control surface In the following this second class of FSMC will be applied to the Buck and Buck Boost converters 4.3 Application of the Fuzzy Lyapunov function SMC to Buck and Buck-Boost converters The proposed Fuzzy Lyapunov function SMC is applied to the Buck and Buck-Boost converter to prove the efficiency of the proposed control law The obtained results are compared to the classical SMC As a fuzzy control, the main advantage of the FSMC is that it is not based on an analytical study 20 Sliding Mode Control 4.3.1 Application of the Fuzzy Lyapunov function SMC to Buck Fig 25 presents the simulated output voltage and output current evolutions by application of the proposed FSMC for a reference voltage Vref = 5V The obtained result is similar to the one obtained by SMC As the SMC, the SMC is tested experimentally for the case of the load variation Fig.26 presents the obtained results for the case of the variation of the load resistance from 10 Ω to 15 Ω at 0.05s The perturbation is rejected and the output voltage attends the reference voltage Moreover, the amplitudes of oscillations are smaller than those obtained by application of the SMC As for the SMC, the experimental test result for the case of the input voltage variation from 30 V to 20V, given in Fig.27, shows the robustness of the applied FSMC for this kind of variation Voltage (V) 0 0.5 1.5 2.5 3.5 4.5 Time (s) -3 x 10 Fig 25 Output voltage evolution by application of the SMC to the studied Buck converter ( Vref = 5V ) 12 11.5 6.5 10.5 5.5 Voltage (V) Voltage (V) 11 10 9.5 4.5 8.5 3.5 0.042 0.046 0.05 Time (s) 0.054 (a) Reference voltage 10 V 0.058 0.042 0.046 0.05 Time (s) 0.054 0.058 (b) Reference voltage V Fig 26 Experimental test robustness of the FSMC for the variation of the load from 10 Ω to 15 Ω Sliding Mode Control and Fuzzy Sliding Mode Control for DC-DC Converters 21 35 Vin V0 Voltage (V) 30 25 20 15 10 0.01 0.03 0.05 Time (s) 0.07 0.09 Fig 27 Experimental output voltage evolution by application of the FSMC for the variation of the input voltage from 30V to 20V 4.3.2 Application of the Fuzzy Lyapunov function SMC to Buck-Boost converter The proposed control is now applied to the studied Buck-Boost converter Fig 28 presents the simulated control signal obtained by application of the proposed FSMC By comparison with the control signal obtained by application of the SMC and presented in Fig 13, we notice that the control signal is smooth So the chattering phenomenon obtained by application of the FSMC disappeared 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.01 0.02 0.03 0.04 0.05 Time (s) Fig 28 Control signal by application of the FSMC Fig 29 presents the studied converter the output voltages evolution for Vref = −20V The obtained result is better than the one obtained by open-loop control However, by comparing it with the output voltage presented by Fig 12 we can notice a small oscillation Fig 30 presents the evolution of the output voltage and the current in the load for a change of the load resistance from 30Ω to 20Ω By application of the FSMC, this perturbation was rejected and the output voltage attends the reference voltage after 30 10-3 s For the case of a variation of the input voltage from 15V to 10V at 0.05s the output voltage, presented by Fig 31, attends after the rejection of the perturbation the desired reference value −20V 22 Sliding Mode Control Voltag e (V) -5 -10 -15 -20 -25 -30 0.01 0.02 0.03 0.04 0.05 0.06 Time (s) Fig 29 Output voltage evolution obtained by application of the FSMC Voltage (V) -5 -10 -15 -20 -25 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 Time (s) Fig 30 Output voltage evolution by application of the SMC for the case of load variation from 30Ω to 20Ω -5 Voltage (V) -10 -15 -20 -25 -30 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Time (s) Fig 31 Output voltage evolution by application of the SMC for the case of input voltage variation from 15 V to 10V Sliding Mode Control and Fuzzy Sliding Mode Control for DC-DC Converters 23 By comparing the robustness test results obtained by application of SMC with those obtained by FSMC we can notice that SMC allows a faster rejection of the perturbation than SMC for the case of the studied Buck-Boost converter Conclusion In this chapter, Sliding Mode Control (SMC) and Fuzzy Sliding Mode Control (FSMC) for Buck, Boost and Buck-Boost converters are proposed, tested and compared SMC is suitable for switched mode DC-DC converters Moreover, such control approach leads to good results Two classical SMC are proposed respectively for Buck and Buck-Boost converters The obtained simulation and practical results confirm the robustness of this control technique The extension of SMC into FSMC aims to improve the SMC robustness and to overcome the chattering problem Two classes of FSMC are presented in this chapter The first class of FSMC aimed to reduce chattering by changing the nonlinear component control by a fuzzy function The second class of FSMC is based on a fuzzy control insuring the Lyapunov function stability Then, a Fuzzy Lypunov based SMC is developed and applied to the Buck and Buck-Boost converters FSMC is not based on a rigorous analytical study as SMC Thus, the same FSMC can be applied to Buck and Buck-Boost converters In addition, FSMC allows the reduction of chattering for the case of the Buck-Boost converter thanks to the fuzzy control surface which allows a smooth and continuous control signal However, the obtained results are nearly similar to those obtained by SMC References Ahmed M.; Kuisma M.; Tolsa K & Silventoinen P (2003) Standard procedure for modelling the basic three converters (Buck, Boost, and Buck-boost) with PID algorithm applied, Proceedings of the 13th International Symposium on Electrical Apparatus and Technologies (SIELA), May, 2003, Plovdive Alouani, A.T (1995) Sliding mode control synthesis using fuzzy logic, Proceedings of the American Control Conference on systems Man and cybernetics, Vol 2, Seattle, Washington, June 1995, pp 1528-1532 Bandyopadhyay, B & Janardhanan, B (2005) Discrete-time sliding mode control, Springer, Berlin, 2005, ISBN-10 3-540-28140-1 Ben Saad, K.; Sahbani, A & Benrejeb M (2008) Design Procedure and Implementation of a Robust Fuzzy Sliding Mode Controller for Buck Converters, International Review of Automatic Control, Vol.1, No.3, pp 303-310, September 2008, ISSN 1974-6059 El Fadil, H ; Giri F., & Ouadi, H (2008) Reducing Chattering Phenomenon in Sliding Mode Control of Buck-Boost Power Converters Proceedings of the IEEE International Symposium on Industrial Electronics, Cambridge,2008 Hu Z.B.; Zhang B.; Du G.P.; Zhong L & Deng W.H (2005) Fast transient three-level converters with sliding-mode control, Proceedings of the applied Power Electronics Conference and Exposition (APEC), Vol 3, pp 1436 - 1440, March, 2005 Kandel, A & Gideon, L (1993) Fuzzy control systems, CRC Press, New York, 1993 Middlebrook, R.D & Cuk, S., (1976) A General Unified Approach to Modelling Switching Power Stages, Proceedings of the IEEE Power Electronics Specialists Conference, pp 18-34, 1976 ... follows: (3) Sliding Mode Control and Fuzzy Sliding Mode Control for DC-DC Converters ⎧ A( d ) = dA1 + (1 − d ) A2 ⎪B( d ) = dB + (1 − d )B ⎪ ⎨ ⎪C ( d ) = C + (1 − d )C ⎪E( d ) = dE1 + (1 − d )E2... Bartoszewicz Institute of Automatic Control, Technical University of Łódź Poland XI Part Sliding Mode Control in Power Electronics Sliding Mode Control and Fuzzy Sliding Mode Control for DC-DC Converters... Tsai Part Sliding Mode Control of Robotic Systems 219 Chapter 12 Sliding Mode Control for Visual Servoing of Mobile Robots using a Generic Camera Héctor M Becerra and Carlos Sagüés 2 21 Chapter 13