Inverted Pendulum control with pole assignment, LQR and multiple layers sliding mode control

Transcription

1 J. Basic. Appl. Sci. Res., 3(1s) , , TetRoad Publication ISSN Journal of Basic and Applied Scientific Research Inverted Pendulu control with pole assignent, LQR and ultiple layers sliding ode control Haed Ebrahii Mollabashi 1, Marziyeh Raabpoor, Soroush Rastegarpour 3 1,3 MSc student in Departent of Electrical Engineering, Islaic Azad University (IAU), South Tehran Branch, Tehran, Iran. MSc student in Departent of Electrical Engineering, Islaic Azad University (IAU), science and Research, Tehran, Iran. Received: June Accepted: July ABSTRACT Inverted pendulu syste is one of popular and iportant laboratory odels for teaching control syste engineering. This paper presents a ultiple layers sliding ode controller and pole assignent controller and LQR for inverted pendulu syste. In ultiple layers sliding ode control, firstly, the given syste is divided into several subsystes. Then, one subsyste is selected to construct the first layer sliding ode surface and it is used to construct the second layer sliding ode surface with the sliding ode surface of another subsyste. This process continues till all states of all subsystes are included. The controller is designed according to this ultiple layers structure. For optiization of sliding surfaces constants are used genetic algorith. KEYWORD: Pendulu, LQR, sliding ode I. INTRODUCTION The inverted pendulu syste is a standard proble in the area of control systes. They are often useful to deonstrate concepts in linear control such as the stabilization of unstable systes. The sliding ode controller is a powerful nonlinear controller, which has been developed and applied to feedback control systes for the last three decades [1]. For under actuated systes, designing a conventional sliding ode surface is not appropriate, because the paraeters of the sliding ode surface can't be obtained directly according to the Hurwitz condition. Many papers on the control of inverted pendulu(s) systes have been published. Nieann [1] designed a u controller for a double inverted pendulus syste. In [], Oatu presented two PID controllers for single and double inverted pendulus systes, whose paraeters were regulated by genetic algorith (GA) and neural network (NN), respectively. Yi [3] proposed a fuzzy controller for this class of under-actuated systes. The structure characteristic of a class of under actuated systes, such as inverted pendulu(s) systes, is that they can be divided into several subsystes. According to this structure characteristic of inverted pendulu(s) systes, Mon and Lin [5] presented a hierarchical sliding ode controller (HSMC) for a double inverted pendulus syste and a single inverted pendulu syste, respectively, whose paraeters were regulated by fuzzy logic. But both of the only guaranteed the second level sliding ode surface was stable. Ebrahii [4] give the stability analysis for under-actuated systes with ore subsystes. For optiization of response is used genetic algorith. (GA) iproved the coefficient of sliding ode surface. In this paper, three ethods are used for control of linear odel of Inverted Pendulu, pole assignent, LQR and ultiple layers sliding ode control II. MODELING A scheatic of the inverted pendulu is shown in Figure 1. Self-balancing robots heavy cranes lifting containers in shipyards rockets and issiles future transport systes like: Segway s and etpacks real eaples of inverted pendulu syste. For the analysis of syste dynaic equations, Newton s second law of otion was applied [7]. Fig. 1: inverted pendulu * Corresponding Author: Haed Ebrahii Mollabashi, MSc student in Departent of Electrical Engineering, Islaic Azad University (IAU), South Tehran Branch, Tehran, Iran. haed_ebrahii@aol.co 363

2 Mollabashi et al.,013 These equations are obtained by equating Horizontal Forces Cart, F M M F N b M F N l cos l cos ( M ) b l cos l sin F (1) And these equations are obtained by equating Forces acting Perpendicular to the Pendulu FN N p sin N cos g sin l cos ( I l ) gl sin l cos () I. M I. pl sin Nl cos I. p sin N cos l Nonlinear for of Inverted Pendulu syste can be depicted by: 1 f b u f b u 1 4 is the cart position, 4 is the velocity of the cart, 3 (3) is the pendulu angle with respect to the vertical line; is the angle velocity of the pendulu with respect to the vertical line, and the epressions of f 1, f, b 1 and b are given in [1] We linearize the syste about the unstable equilibriu : ( ) I l g l l ( M ) b l F The linearization of the cart-pendulu syste around the upright position is: A Bu y C D b ( I l ) gl 0 0 I ( M ) Ml I ( M ) Ml A bl gl ( M ) 0 0 I ( M ) Ml I ( M ) Ml ( 4 ) B C 0 ( I l ) I ( M ) M l 0 l I ( M ) M l (5 ) 364

3 J. Basic. Appl. Sci. Res., 3(1s) , 013 D 0 0 III. MULTIPLE LAYERS SLIDING MODE CONTROL DESIGN Fro [4-5] we can design ultiple layers sliding ode control for inverted pendulu. For SIMO under actuated echanical systes, the atheatical odel can be translated into the following for: 1 f1 b1u (6) f bu Where X [ 1,,... ] T the state variables and u is the input of the syste, f n n and b n are bounded noinal. As the Multiple layers sliding structure has been shown in Figure. First layer can be defined by [4-5] Fig. : sliding ode surfaces s c c the equivalent law control obtain fro 1 0 c f1 c1 0 c1 1 c c1 c ( f1 b1 u ) u eq (1) (7) c b1 s The second layer defined as s c3 3 s1 and s3 c4 4 s so the total U eq is: u c. c f e q ( i ) 1 c b u sw(i) is the switch control law for every layer sliding surface, u sw(i) can iproved the response tie.[4] The switch control law is defined as: 0 i 1 i u sw ( i ) sgn( s ) / den( i) i 1 1 ( 8 ) (9) Where is a positive constant, 1 d en ( i) c b ( c. b.sgn ( s )) (10) 1 1 c i and ɳ are constants which have the sae sign but this paraeters and other sliding surface can't be obtain according Hurwitz condition so for this reason and optiizations, using genetic algorith. U u u (11) s i eq i sw i 365

4 Mollabashi et al.,013 IV. The GA Optiization Genetic algoriths, introduced by Holland [6], are based on the idea of engendering new solutions fro parent solutions, eploying echaniss inspired by genetics. In the following tet the controller will be optiized by genetic algorith. The optiization paraeters are c i. The nuber of variables is 8. The fitness function is shown as forula. Evaluate the fitness value of each output. n T T ( R u qu) (1) i0 i i V. POLE ASSIGNMENT CONTROL DESIGN Pole placeent ethod is one of the classic control theories and has an advantage in syste control for desired perforance. Theoretically pole placeent is to set the desired pole location and to ove the pole location of the syste to that desired pole location to get the desired syste response. For using pole placeent control design assuptions are: 1- The syste is copletely state controllable - The state variables are easurable and are available for feedback. 3- Control input is unconstrained. The open loop poles are placed at 0, , -7.0, and fro the iaginary ais. Now we want to ove these poles to -8, -6, -1, and -1 fro the ais. To ove to the desired pole location we have to fine feedback gains. K is fine feedback gain. VI. SIMULATION RESULTS M ass of the cart 1kg ass of the pendulu 1kg b friction of the cart 0.1N//sec l length of the pendulu 0.1 i inertia of the pendulu kg. F force applied to the cart kg. / s g gravity 9.8 / s θ Vertical pendulu angle in degree 1 is cart position velocity of the cart, 3 Vertical pendulu angle and 4 angle velocity of the pendulu 0, 0, pi / 3, 0 Initial values are: With GA, constant is found: c= [ ] Fig. 3: Tie response of carts positions (ultiple layers sliding ode control). 366

5 J. Basic. Appl. Sci. Res., 3(1s) , 013 Fig. 4: Tie response of pendulus angles (ultiple layers sliding ode control). Fig. 5: sliding surfaces Fig. 6: Control output Fig. 7: Tie response of carts positions and pendulus angles (Pole assignent control) 367

6 Mollabashi et al.,013 Fig. 8: Tie response of carts positions and pendulus angles (LQR) VII. CONCLUSION Multiple layers sliding ode control (MLSMC), LQR and Pole assignent control ethod has been proposed in this paper. Siulation results were presented. In LQR ethod, SISO syste has to be considered and the wagon (cart) position will be forgotten while in the ulti-layer sliding ode ethod, cart position & pendulu are controlled siultaneously. Siulation results have shown that the all sliding surfaces are asyptotically stable. This approach has acceptable control effort. REFERENCES 1. Nieann and J.K. Poulsen, Analysis and design of controllers for a double inverted pendulu, Proceedings of Aerican Control Conference, vol. 4, pp , June 003. T. Fuinaka, Y. Kishida, M.Yoshioka, and S. Oatu, Stabilization of double inverted pendulu with self-tuning neuro-pid, Proceedings of the IEEE-INNS-ENNS International Joint Conference on Neural Networks, vol. 4, pp , July T. Sugie J. Yi, N. Yubazaki, and K. Hirota, Stabilization Control of Series-Type Double Inverted Pendulu Systes Using the SIRMs Dynaically Connected Fuzzy Inference Model, Artificial Intelligence in Engineering, vol. 15, pp , H. Ebrahii, A. Shahansoorian, S. Rastegarpour, A. H. Mazinan, New approach to control of ball and bea syste and optiization with genetic algorith, Life Sci J, Ch. Lin, and Y. Mon, Decoupling control by hierarchical fuzzy sliding-ode controller, IEEE Transactions on Control Systes Technology, vol. 13, pp , July Xiucheng Dong, Zhang Zhang, Jiagui Tao, Design of Fuzzy Neural Network Controller Optiized by GA for Ball and Plate Syste, Sith International Conference on Fuzzy Systes and Knowledge Discovery, M. Moghaddas, M. R. Dastran, N. Changizi, N. Khoori, Design of Optial PID Controller for Inverted PenduluUsing Genetic Algorith, International Journal of Innovation, Manageent and Technology, Vol. 3, No. 4, August