(Microsoft Word - 138-148\307\315\343\317 \346\344\346\321\307 \332\346\335) Design an Integral Sliding Mode Controller for Ahmed Khalaf Hamoudi *,**Department of Control and Systems Engineering *Email: (Received https://doi.org/10.22153/kej.2017.09.003 Abstract The goal of this paper is to design a robust controller for controlling a pendulum system. The control of nonlinear systems is a common problem The Sliding Mode Controller (SMC) is the best solution for controlling a nonlinear system. The classical from two phases. The first phase is the reaching phase and the second is the sl chattering phenomenon which is considered as a severe problem and undesirable property. It is a zigzag motion along the switching surface. In this paper, the chattering spite of SMC is a good method for controlling a nonlinear system but considered as undesired property. The Integral Sliding Mode controller (ISMC) Also, the ISMC is a good method for controlling a nonl be considered as an effective and powerful technique. In ISMC method, the reaching phaseis eliminated which considered a main part in designing classical S Mode Controller (CSMC),is the ability to make the systems asymptotically stable. The pendulum system testing the CSMC and ISMC. The results obtained comparied with the CSMC. Finally, MATLAB Keywords: Chattering phenomenon, Classical sliding mode controller, Integral sliding mode controller, Swi surface. 1. Introduction In last two decades, the methods for controlling nonlinear systems take a much interest from many researchers and as a result many methods were developed [1]. One of them is the Sliding Mode Control system (SMC) which is consider as an effective and robust control method that is used successfully in wide variety of systems. The most important property in using SMC is the ability of this controller to make the system insensitive to external disturb parameters uncertainty [2]. In spite of the SMC robustness and its better performance, sever from the problem of ''chattering phenomenon '', which is considered as drawback property. To reduce this chattering phenomenon in the SMC, many methods were developed Al-Khwarizmi Engineering Journal,Vol. 13, No. 1, P.P. 1 Design an Integral Sliding Mode Controller for a Nonlinear Ahmed Khalaf Hamoudi* Noora off Abdul Rahman *Department of Control and Systems Engineering/ University of Technology *Email: ahmed_khk22@yahoo.com **Email: rb_6_1981@yahoo.com Received 29 May 2016; accepted 19 September 2016) https://doi.org/10.22153/kej.2017.09.003 The goal of this paper is to design a robust controller for controlling a pendulum system. The control of nonlinear systems is a common problem that is facing the researchers in control systems design. ontroller (SMC) is the best solution for controlling a nonlinear system. The classical from two phases. The first phase is the reaching phase and the second is the sliding phase. The SMC suffers from the chattering phenomenon which is considered as a severe problem and undesirable property. It is a zigzag motion along the switching surface. In this paper, the chattering is reduced by using a saturation function instead spite of SMC is a good method for controlling a nonlinear system but it still suffers from long settling time desired property. The Integral Sliding Mode controller (ISMC) can be used to reduce the ISMC is a good method for controlling a nonlinear systems. ISMC is simple, has a high performance and can be considered as an effective and powerful technique. In ISMC method, the reaching phaseis eliminated which signing classical SMC. The important property of the ISMC as well as the ability to make the systems asymptotically stable. The pendulum system testing the CSMC and ISMC. The results obtained from the simulation showed the advantages of using the ISMC when MATLAB software package was adopted in this paper. Classical sliding mode controller, Integral sliding mode controller, Swi the methods for controlling nonlinear systems take a much interest from many researchers and as a result many [1]. One of them is the Sliding Mode Control system (SMC) which is an effective and robust control method used successfully in wide variety of The most important property in using SMC is the ability of this controller to make the external disturbance and In spite of the SMC e, but it is from the problem of ''chattering phenomenon '', which is considered as drawback chattering phenomenon were developed to overcome this problem. One of them is by using a boundary layer instead of signmum nonlinear part of controller [2]. proposed to use a fuzzy logic system sliding mode controller to get a new structure called sliding mode fuzzy controller [ other hands, some researchers the chattering by using a genetic algorithm [ Recently some researches, proposed to use the particle swarm optimization technique i reduce the drawbacks of the chattering phenomenon [5]. The advantage SMC is the reduction in order original system equation [6]. The designed control law in SMC can drive the system state towards the manifold surface and stay surface for all future time until re origin. Churn and We was first introduced the Khwarizmi Engineering Journal,Vol. 13, No. 1, P.P. 138- 147 (2017) onlinear System Rahman** The goal of this paper is to design a robust controller for controlling a pendulum facing the researchers in control systems design. ontroller (SMC) is the best solution for controlling a nonlinear system. The classical SMC consists iding phase. The SMC suffers from the chattering phenomenon which is considered as a severe problem and undesirable property. It is a zigzag motion along s reduced by using a saturation function instead of sign function. In still suffers from long settling time which is can be used to reduce the settling time. high performance and can be considered as an effective and powerful technique. In ISMC method, the reaching phaseis eliminated which as well as the Classical Sliding ability to make the systems asymptotically stable. The pendulum system was used for ed the advantages of using the ISMC when Classical sliding mode controller, Integral sliding mode controller, Switching . One of them is by using a signmum function in ]. Other researchers logic system with a sliding mode controller to get a new structure called sliding mode fuzzy controller [3]. On the suggest to reduce the chattering by using a genetic algorithm [4]. Recently some researches, proposed to use the technique in order to reduce the drawbacks of the chattering advantage of using the by one from the ]. The designed control law in SMC can drive the system state trajectory towards the manifold surface and stay in this for all future time until reaching the was first introduced the Al-Khwarizmi Engineering Journal Ahmed Khalaf Hamoudi Al-Khwarizmi Engineering Journal, Vol. 13, No. 1, P.P. 138- 147(2017) 139 Integral Sliding Mode (ISMC) [7], which is similar to the SMC since it is insensitive to external disturbance and parameters uncertainty [8]. The control law in the ISMC consists from two major parts, the first part is the nominal control which is responsible for the performance of the nominal system and the second part is the discontinuous control that is used to reject the external disturbance and parameters uncertainty [8]. In this paper the performance of pendulum system will be improved by using the ISMC and the results show high validity when using the proposed controller. 2. Classical Sliding Mode Controller (SMC) In modern control systems, the SMC is a common method for designing a robust controller technique. This controller was extremely used with nonlinear system since 1950, and it is extended to use with large various types of applications such as electrical servo drives, pendulum, ETV and others. The differential equation that is used to govern the sliding mode control has order less by one than the order of original system. The main drawback of the SMC is the ''chattering'' which is phenomenon of oscillations having a finite frequency and amplitude along the sliding surface. The problem of chattering phenomenon can be solved by using many methods as mentioned above in section 1. Sliding Mode Controller consists of two major phases [6]: A: Reaching phase: in this phase the state trajectories are oriented toward the switching surface S=0; hence, the sliding phase will be started at this instant as shown in Figure (1). B: Sliding phase: in this phase the state trajectory is enforced to stay on the switching surface and to move along this surface until reaching the origin in finite time as shown in Figure (1). Fig. 1. The two phases of the sliding control [2]. Fig. 2. The shape of sliding surface [2]. The control law is defined as: � � ���� ��� … �1 where, ��� is the equivalent control part which required to oriented the system state trajectory toward switching surface (� � 0) and ��� is the discontinuous control part to enforce the state trajectory to move along the switching surface towards the origin. The control ��� is defined as below [10]: ��� � ���� ������ … �2 where, k is a constant with positive value. Fig. 3. The shape of a signum function. By substituting equation (2) in (1), the control law can be rewritten as: � � ��� � ��� ������ … �3 The sliding surface characterized as: � � �� � �� ; � � 0 … �4 where, λ is a constant parameter with a positive value. Let the error and its derivatives defined as: �� � � � � ! and �"= �� � � where ! is the final position that can be considered as the desired position. Then, equation (4) will be rewritten as below: � � λ�� � �" … �5 for � � 1, equation (5) will be as: � � �� � �" the derivative of the sliding variable �� = ���+ ��" … �6 Ahmed Khalaf Hamoudi Al-Khwarizmi Engineering Journal, Vol. 13, No. 1, P.P. 138- 147(2017) 140 The main goal is to keep the ���, ' close to switching surface in phase plane. The general form of motion equation for any nonlinear system: �� = (�� � )�x � � +�x, t … �7 To satisfy the condition �� � 0 that the right side of equation (6) equal zero by selecting the discontinuous gain ��� as follows [6]: 0, )( ),( )( >             ∂ ∂ ∂ ∂ += ko xB txd x So x So koxk n … (8) In above control equation (3), the ������ function caused a chattering phenomenon. This chattering phenomenon is undesirable property appearing along the sliding surface, the major reason that caused this phenomenon is the ''���� function'' that is present in control equation (3). The Classical SMC is suffering from the chattering which is considered as a severe problem in SMC. The boundary layer function can be using to reduce the chattering. The �.' �� function is used instead of ���� �� function in control law. The �.' �� function that shown in figure (4) can be described as below [6]:      <− <<− >+ = )0/(1 )1/1(/ )0/(1 )/.( ϕ ϕϕ ϕ ϕ s ss s ssat … (9) Fig. 4. The /01 �/ function [2]. The ������ function in equation (3) is replaced by �.' �� function and the control law can be written as below: � � ��� � ��� �.' … �10 3. Integral Sliding Mode Controller (ISMC) Integral sliding mode control (ISMC) is a nonlinear robust controller, designed for controlling nonlinear systems [1]. The goal of the proposed ISMC is to eliminate the reaching phase by enforcing the state trajectory, which starts from any initial state, to be in sliding phase throughout the entire plant trajectory and to slide along the switching surface until reaching the origin. The difference between the ISMC approach and the classical sliding mode is that the order of motion equation in the ISMC is the same as the order of the original system, while in the classical sliding mode; the order is less by one from the original system [9]. The robustness of the system in the ISMC is guaranteed because in final trajectory the error and its derivatives reach zero value. Also in ISMC the system is insensitive to variations of system parameter and external disturbance. The main problem in this controller, as well as in classical SMC, is the chattering phenomenon in the control action; and to reduce this chattering may use some functions such as saturation, dead zone and inverse of tan instead of sign function which is usually used in classical sliding mode. The complete system with the ISMC is shown in the following figure: Fig. 5. The closed loop control system using ISMC. The procedure of designing the ISMC with any nonlinear system can be described as below: ��� � �2 � 3 ; 3�0 � ��2�0 … �11 The ��� is sliding surface, 3 is the integral term and �2 defined as in classical sliding mode control as described in equation (5). The integral term 3�0 is determined based on the initial condition �2�0 . The integral term z will be selected in order that the sliding variable ��� has a zero value and this make the system dynamic in the sliding mode from the initial instant of time. The derivative of the sliding variable ��� is given as: �� � 4 5 46 �� � 3� … �12 The second step is to describe the control law of the ISMC as: � � �7 � ��� … �13 The nominal part of controller �7 is used to maintain the nominal system dynamics with reference characteristics, where ��� is the second Ahmed Khalaf Hamoudi Al-Khwarizmi Engineering Journal, Vol. 13, No. 1, P.P. 138- 147(2017) 141 part of the controller that is used to reject the external disturbance and parameters uncertainty As in [9] and by substuting equation (7) in equation (12) ���4 5 46 8(�� � )�� �7�)�� ��� �+��, ' ] +3� … (14) where the discontinuous controller is defined as ��� � ���� ������ … �15 where, ��� is the same as presented in equation (8) with positive value. Therefore, equation (12) will be as bellow: � � �7 � ��� ������ To satisfy the rejection of the external disturbance and variation of system parameters, the Integral term is assumed to be as: 3��� 4:5 46 8(�� � )�� �7] … �16 By substituting the equation (16) in (14), we will get the equation as: ���4:5 46 8)�� ��� � +��, ' ] … �17 In the design of the ISMC, the sliding surface and control will be described as: ��� � �2 � 3 ; z�0 � ��<�0 ���� � ��2 � 3� ; � � �7 � ��� ������ … �18 Finally, when using the boundary layer, the equation of control law (18) rewritten as below: � � �7 � ��� �.' … �19 4. Plant Description Consider a second order of the pendulum system described by equation: ? � �. sin � C � � DEF � +�' G … �20 Where: the angular position of the rod with vertical axis and it is measure in (radian) and it consider as the controlled variable (output), � the angular velocity and it is measure in (radian/ second), T the torque applied at the end of the pendulum and it measure in (Newton. meter) and it is considering as the control input. and d(t) is the external disturbance applied to the system. A common problem in real plant is the presence of an external disturbance and parameters uncertainty. The nominal value of parameter a=10, b=1 and c=10, The uncertainties values of above parameters are H. � 0, HC � ∓0.5, HD � ∓5. Table 1, The maximum and minimum pendulum parameters values) Parameter value Minimal value Maximal Value a 10 10 b 0.5 1.5 c 5 15 Fig. 6. The simple pendulum [4]. The error of the pendulum system can be described in the state equation as: Let the error �� = � ! and �"K � Where, ! is the desired position which is the equilibrium point. Equation above can be rewritten as: ��� = ��" … �21 ��" � �. ∗ sin Where, . � H. ∓ .7, C � HC ∓ C7, D � HD ∓ .D7 �� = ��� + ��" … �23 By substituting equation (21), (22) and (2) in equation (23), ��� � �2 � Case (A): Design the Classical sliding mode controller (CSMC) for pendulum system The design of CSMC controller is written as in equation (3): � � 1 c 8. sinE�� � !G � C�" ] � ��� ������ … �25 when the boundary layer is used, the control law rewritten as described in equation �10 : � � 1 D 8. ���E�� � !G � C�"] � ��� ������ … �26 Ahmed Khalaf Hamoudi Al-Khwarizmi Engineering Journal, Vol. 13, No. 1, P.P. Case (B): Design the Integral sliding mode controller (ISMC) for pendulum system The Sliding variable is written according to equation (18) as: ��� � �2 � 3 ; 3�0 � ��2�0 And �2 � �� � �" And the error equation for the pendulum system was described as: �? � �D�� � D"�� , D� , D" � 0 the values D� and D" are assigned depending on the required characteristics of plant dynamics. In ISMC design, the derivative of integral term is described as below: 3��D� ∗ � � D" ∗ � � � �" The nominal control is described as: Finally, the control law is written according to equation (18) as: when the boundary layer is used, the above equation will be rewritten as: 5. The Simulation Results Case (A): The classical sliding controller (CSMC) with a /\]^ function Fig. 7. The error _`vs. time of the C Khwarizmi Engineering Journal, Vol. 13, No. 1, P.P. 142 sliding mode system The Sliding variable is written according to … �27 … �28 And the error equation for the pendulum system … �29 assigned depending on dynamics. design, the derivative of integral term is ... �30 �7 � 1 D 8. ���E�� � ө!G � C�" � D�� � D"� ]� Finally, the control law is written according to � � 1 D 8. ���E�� � ө!G � C�" � D�� � D � ��� ������ used, the above � � 1 D 8. ���E�� � ө!G � C�" � D�� � D" � ��� �.' liding mode function CSMC. Fig. 8. The derivative error x CSMC. Fig. 9. The control action u vs. time of the Fig. 10. The sliding variable S classical SMC . Khwarizmi Engineering Journal, Vol. 13, No. 1, P.P. 138- 147(2017) … �31 D"��] … �32 "��] … �33 2x vs. time of the vs. time of the CSMC. S vs. time of the Ahmed Khalaf Hamoudi Al-Khwarizmi Engineering Journal, Vol. 13, No. 1, P.P. Fig. 11. The plot of the phase plane between _b of the classical SMC. Case (B): The classical sliding mode controller (CSMC) with boundary layer Fig. 12. The error _` vs. time of the CSMC Fig. 13. The derivative error _b vs. time of the CSMC. Khwarizmi Engineering Journal, Vol. 13, No. 1, P.P. 143 The plot of the phase plane between _` and lassical sliding mode with boundary layer vs. time of the CSMC. vs. time of the Fig. 14. The control action c vs. time of the CSMC. Fig. 15. The plot of sliding variable S vs. time of the CSMC. Fig. 16. The plot of phase plane between the error _` and the derivative error _b of the CSMC. Khwarizmi Engineering Journal, Vol. 13, No. 1, P.P. 138- 147(2017) vs. time of the CSMC. . The plot of sliding variable S vs. time of the . The plot of phase plane between the error of the CSMC. Ahmed Khalaf Hamoudi Al-Khwarizmi Engineering Journal, Vol. 13, No. 1, P.P. Case (C): The integral sliding controller with a /\]^ function Fig. 17. The error d` vs. time of the ISMC. Fig. 18. The derivative error _bvs. time of ISMC. Fig. 19. The control action c vs. time of the ISMC. Khwarizmi Engineering Journal, Vol. 13, No. 1, P.P. 144 liding mode vs. time of the ISMC. vs. time of the vs. time of the ISMC. Fig. 20. The plot of sliding variable S vs. time of the ISMC. Fig. 21. The plot of phase plane between the error _` and the derivative error _b of the ISMC. Fig. 22. The plot of the Sliding variable S and the Integral term e vs. time of the ISMC Khwarizmi Engineering Journal, Vol. 13, No. 1, P.P. 138- 147(2017) The plot of sliding variable S vs. time of the The plot of phase plane between the error of the ISMC. The plot of the Sliding variable S and the vs. time of the ISMC Ahmed Khalaf Hamoudi Al-Khwarizmi Engineering Journal, Vol. 13, No. 1, P.P. Case (D): The integral sliding mode controller with boundary layer Fig. 23. The error _` vs. time of the ISMC. Fig. 24. The derivative error _b vs. time ISMC. Fig. 25. The control action c vs. time of the ISMC. Khwarizmi Engineering Journal, Vol. 13, No. 1, P.P. 145 ntegral sliding mode vs. time of the ISMC. vs. time of the of the ISMC. Fig. 26. The Sliding variable f vs. time of the ISMC. Fig. 27. The plot of phase plane between the error _` and the derivative error _b of the ISMC. Fig. 28. The plot of the Sliding variable Integral term g vs. time of the ISMC. Khwarizmi Engineering Journal, Vol. 13, No. 1, P.P. 138- 147(2017) vs. time of the ISMC. The plot of phase plane between the error of the ISMC. The plot of the Sliding variable f and the vs. time of the ISMC. Ahmed Khalaf Hamoudi Al-Khwarizmi Engineering Journal, Vol. 13, No. 1, P.P. 138- 147(2017) 146 6. Discussion In this work, SMC and ISMC have been considered for controlling the position of the pendulum system. The results of SMC and ISMC have been included in this work to show the proprties of each controllers with the presence of the external disturbance and paramrters uncertinty. Each of the above controllers has the ablility to make the system asymptotically stable under the effect of external disturbance and the parameters uncertainty by making the error and the derivative of error equal to zero value as shown in figures (7), (8), (17) and (18). Both the SMC and ISMC are suffering from the chattering problem because of the effect of ������ function as shown clearly in figures (9) and (19), this problem is solved by using the boundary layer as shown in figures (14) and (16). When using the ������ function, the state trajectory in the CSMC hits the switching surface vertically as shown in figures (11) and (9), and this caused a chattering phenomenun, while in using the boundary layer the state trajectory hits the sliding surface in arc shape as shown in figures (16) and (14). The same thing was apear clearly when using the ISMC as shown in figures (21), (19), (27) and (25). The results show that the effect of the external disturbance and the parameters uncertainty of the dynamic system is cancelled by using the ISMC as shown in figures (25) and (27). In figure (27), the error (��) and the dervitive of error ��" reaches the origin in final trajectory, which means that its values equal to zero. 7. Conclusion The most important improvement of using ISMC is the reducing of the settling time response of system comparing with the CSMC as shown in figures (12) and (23). In the CSMC the settling time is 6.5 sec as shown in figure (12), where in the ISMC the settling time is reduced to 0.8 sec as shown in figure (23). The ISMC consists of two parts, The first part is nominal control that is used to control the nominal system dynamics while the second part is the discontinous control which is used to reject the pertubaration term (the perterbation term consists of the external disturbance and parameters uncertainty).In using the ISMC, the pendulum system is is preseented by the nominal model from the from thr first instsnt . this nominal model is not effected by the perturbation term. The results shows that the CSMC and ISMC can be considered as a rubust controller, since it can give a good response even with the presence of disturbance and parameters uncertainty as shown in figures (14) and (25). From figures (11), (16), (21) and (27), it is seen clearly that the CSMC and ISMC are able to make the system asymptotically stable. 8. References [1] C. A. Yfoulis, A. Muir, and P. E. Wellstead, “A New Approach for Estimating Controllable and Recoverable Regions for Systems with State and Control Constraints,” International Journal of Robust and Nonlinear Control, Vol. 12, No. 7, pp. 561-589, 2002. [2] V. I. Utkin, J. Guldner, and J. Shi, "Sliding Mode Control in Electromechanical Systems", CRC Press. Taylor & Francis Group, 2009. [3] A. K. Hamoudi, ''Design and Simulation of Sliding Mode Fuzzy Controller for Nonlinear'', Journal of Engineering, College of Engineering, University of Baghdad, Vol. 22, No. 103, pp. 66-76, 2016. [4] A. K. Hamoudi, ''Sliding Mode Control for Non linear system best on Genetic Algorithm'', Journal of Engineering and technology , Vol. 32, No. 11A, pp. 2745-2759, 2014. [5] Z. Chen, W. Meng, Z. Wang and J. Zhang, ''Sliding Mode Variable Structure Control Based on Particle Swarm Optimization'', Second International Symposium on Intelligent Information Technology Application, Taiyuan, China, pp. 692-696, 2008. [6] H. Lee and V. I. Utkin, “Chattering Suppression Methods in Sliding Mode Control Systems,” Annual Reviews in Control, Vol. 31, No. 2, pp. 179-188, 2007. [7] T. L. Chern and Y. C. Wu, “Design of Integral Variable Structure Controller and Application to Electrohydraulic Velocity Servo Systems”, Vol. 138, No. 5, pp. 439-444, 1991. [8] S. Mondal and C. Mahanta, “Adaptive Integral higher order Sliding Mode Controller for Uncertain Systems,” Journal of ControlTheory and Applications, Vol. 11, No. 1, pp. 61-68, 2013. [9] S. A. Al-Samarraie, A. S. Badri and M. H. 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