Iran University of Science and Technology, Tehran 16844, IRAN

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1 DECENTRALIZED DECOUPLED SLIDING-MODE CONTROL FOR TWO- DIMENSIONAL INVERTED PENDULUM USING NEURO-FUZZY MODELING Mohammad Farrokhi and Ali Ghanbarie Iran Universit of Science and Technolog, Tehran 6844, IRAN Abstract: In this paper, a new method has been proposed to control a two-dimensional inverted pendulum. The proposed method in this paper is as follows: First, the sstem of a two-dimensional inverted pendulum is divided into two subsstems using decentralized control theor. Then, using decoupling method, each subsstem is decoupled into two surfaces for appling sliding-mode control. Those portions of the controllers, which are directl dependent on the dnamic of the sstem and its parameters, are modeled with two neuro-fuzz (ANFIS) networks. Simulations show a high performance for the proposed method as compared to the eisting methods. Kewords: decentralized control, sliding-mode control, neuro-fuzz modeling. INTRODUCTION One-dimensional inverted pendulum is a nonlinear problem, which has been considered b man researchers (Omatu and Yashioka, 998; Magana and Holzapfel, 998; Nelson and Kraft, 994; Anderson, 989), most of which have used linearization theor in their control schemes. In general, the control of this sstem b classical methods is a difficult task (Lin and Sheu, 99). This is mainl because this is a nonlinear problem with two degrees of freedom (i.e. the angle of the inverted pendulum and the position of the cart), and onl one control input. When this problem is etended into a two-dimensional inverted pendulum (e.g. an inverted pendulum, whose maneuver is not restricted in a plane, and can move in three-dimensional space, and also its cart does not move along one ais, but in - plane), then the sstem becomes a MIMO and ver complicated nonlinear sstem. The control of this sstem, which is a more realistic model of the launched missile, is the subject of this paper. In order to solve this problem, if the multivariable classical control methods are used, then the model of the sstem must be linearized. But, because of the highl nonlinear behavior of the two-dimensional inverted pendulum in large deviation angles, these methods can control this sstem onl for small deviation angles. On the other hand, the use of nonlinear classical methods for a MIMO sstem with differential equation of order 8 can be etremel difficult. In contrast, the decentralized control theor has opened a horizon for controlling of complicated and nonlinear sstems (Ghanbarie, a). According to this theor, a complicated problem will be divided into a few simpler subsstems and each subsstem is controlled separatel. Therefore, instead of solving one complicated problem, a few simpler sub problems are solved. The use of this method simplifies the control of a two-dimensional inverted pendulum. In the proposed method, in this paper, each subsstem is further decoupled into two-sub subsstems. Then, with defining two sliding surface, which are related to each other, the sliding-mode control will be applied to it (Ji-Change and Ya-Hui, 998). In this wa, the problem of controlling a sstem of order 8 can be accomplished with sliding-mode controllers for sstems of order. To control each subsstem, dnamic-dependent portions and their parameters are modeled with two neuro-fuzz networks. This makes the controllers independent

2 from the sstem model. Moreover, due to the nature of the neuro-fuzz networks, it is possible to design an adaptive controller using on-line training mechanism for these networks. In the net section, the model of the tow-dimensional inverted pendulum will be given. In section III, the decentralized control theor and its application to the two-dimensional inverted pendulum will be briefl eplained. The design of the controller will be brought in section IV, followed b neuro-fuzz modeling in section V. Section VI shows the simulation results. The conclusion is given in section VII.. THE MODEL OF TWO-DIMENSIONAL INVERTED PENDULUM The sstem of a two-dimensional inverted pendulum consists of an inverted pendulum connected to a cart, and can move in the three dimensional space. That is, it can deviate from vertical direction (i.e. parallel to the z ais) towards both and directions. The cart also can move in - plane (Fig.). The dnamic of this sstem is as follows (Esmailie-Khatier, 994): & = f, u = f = ( ) [ ] [ α α& & β β& & ] (, u) = G ( ). h(, u) u = [ F F ] T T 8 T () where h is a 8 vector, influenced b the state vector of the sstem and the input vector u, and G is a 8 8 matri, which is a function of the state vector. If the deviation angle of the inverted pendulum from the z-ais is assumed to be γ, then α and β are the projections of γ on the -z and -z planes, respectivel, and α& and β & are the corresponding angular velocities. Also, and are the coordinates of the position of the cart in - plane, and & and & are the velocities of the cart along -ais and -ais, respectivel. F and F are the applied forces to the cart along and - ais, respectivel. After some matri manipulations, the dnamic of the sstem can be rewritten as follows: & = & = f & 3 = 4 & = f 4 & 5 = 6 & = f 6 & 7 = 8 & = f 8 ( ) + b ( ) F + v ( ) ( ) + b ( ) F + v ( ) ( ) + b ( ) F + v ( ) ( ) + b ( ) F + v ( ) F F F F () Fig.. The schematic diagram of a twodimensional inverted pendulum where, b f ( ) ( ), and v ij ( )( i =,and j =,) ij ij are nonlinear functions of the state variables (Ghanbarie, b). 3. THE USE OF THE DECENTRALIZED CONTROL THEORY IN TWO- DIMENSIONAL INVERTED PENDULUM What are mostl common in control theor are the centralized control methods: dada are received from the plant and all of them are sent to the controller. Then, the decision is made b the central control and the appropriate commands are sent to the plant. Here, the controller considers the plant as a whole. But in complicated sstems the design of the controllers can encounter difficulties. One of the methods to control these kinds of sstems is the decentralized control scheme. The main idea in this theor is the distribution of tasks. That is, the process under control is appropriatel divided into several sub processes. Then, the controllers are designed locall and the processing is made in a decentralized fashion. In other words, the local controllers generate the control commands. Obviousl, because of the interactions between subsstems on each other, the control commands, which are applied to the corresponding subsstems, affects other subsstems as well. Therefore, a third kind of disturbance, in addition to two well-known disturbances (i.e. the disturbances due to eternal signals and the disturbances due to the unmodelled dnamics), which is the effects of all subsstems on ever subsstem, is defined (Ghanbarie, a). If the dnamic equations of a two-dimensional inverted pendulum are linearized around =, then its state equations become diagonal. In other words, in the process of linearization, the decoupling occurs along the an -ais. It should be mentioned that the linearized model of the two-dimensional

3 inverted pendulum has not been used in this paper. The goal is onl to conclude from the above comments that considering the -ais as one subsstem and the -ais as the other can be appropriate because the interactions between these two subsstems are relativel small. Moreover, because of the similarities between the -ais and the -ais dnamics, one needs onl to design the controller for one ais. As a result, the MIMO sstem of order 8 is converted into two SIMO subsstems of order 4. Net, considering the disturbances of the third kind, one controller is designed for each subsstem. The closer the pendulum gets to the operating point ( = ), the smaller the magnitude of the third kind of disturbance (i.e. at = this disturbance is equal to zero and at the deviation angle equal to 9 degrees it has its maimum value. Therefore, eqs. () become & = (3) & = f( ) + b( ) F + d( t) (4) & 3 = 4 (5) & 4 = f ( ) + b ( ) F + d ( t) (6) & 5 = 6 (7) & 6 = f( ) + b( ) F + d( t) (8) & 7 = 8 (9) & = f + b F d t () 8 ( ) ( ) + ( ) In these equations ( t) ( i =, and j =,) d ij are the disturbances of the third kind. Later it will be shown how to convert each subsstem of order 4 into two sub subsstems of order using the decoupled sliding-mode method. The result is a simple design procedure and a controller with good performance. 4. DESIGN OF THE CONTROLLER The goal of the control is to bring the inverted pendulum to the vertical position (in the z-ais direction) while the cart is brought to the origin of the coordinates. The state equations (3)-(6) are for the subsstem of the -ais and the state equations (7)-() are for the subsstem of the -ais. Hence, there are two subsstems of order 4, which are not in canonical form and must be controlled independentl. 4.. The Decoupled Sliding-Mode Control Method Consider the following four state equations, similar to the subsstems of the two-dimensional inverted pendulum: & = () ( ) + b ( ) u d ( t) & = f + () & = (3) 3 4 & 4 = f( ) + b( ) u+ d( t) (4) where = [ 3 4] T is the state vector, and f ( ), f ( ), b ( ), and b ( ) are nonlinear functions equal to f ij ( ) and b ij ( ) ( i =, and j =,), respectivel. Also, u is control input, and d ( t) and ( t) are disturbances of the d third kind. It is assumed that these disturbances have a higher limit: d ( t) D( t), d( t) D( t). Now, two sliding surfaces are defined as follows: s = c +, s = c3 + 4 (5) According to the sliding-mode control theor, the control laws can be defined as follows (Slotine and Li, 99): u = ˆ u k sat s b / ϕ k > D / b ( ) ˆ u = ˆ u u = ˆ u ( ( ) ) ( c f ( ) )/ b ( ) k sat ( s b ( ) / ϕ ) ( c f ( ) )/ b ( ) k > D / b ( ) = 4 (6) (7) Clearl, if u = u in Eqs. () and (4), onl states and along with the hper surface s will converge to zero. On the other hand, if u = u, onl states 3 and 4 along with the hper surface s will converge to zero. In other words, the sliding-mode controller is able to control either the inverted pendulum or the cart, while the goal is the simultaneous control of both the inverted pendulum and the cart. One method might be as follows: converting the above fourth order sstem into a canonical form and then appling the sliding-mode control theor. But, this conversion has some conditions (Khalil, 99). Moreover, its computations are complicated and laborious. Using the decoupled sliding-mode control scheme (Ji- Change and Ya-Hui, 998), one can emplo the sliding-mode theor without converting the sstem into canonical form. The main idea of the decoupled sliding-mode controller is as follows: The eisting fourth order sstem is divided into two subsstems A and B of order two. Subsstem A consists of the state variables and, and the sliding surface s ; subsstem B consists of the state variables 3 and 4, and the sliding surface s. The main goal of the controller is to guide the state variables of subsstem A to the surface s = such that and approach eponentiall to zero.

4 The secondar goal is the same thing for the state variables of subsstem B and the corresponding sliding surface s =. Since the main goal is to bring subsstem A into stable conditions, the information of subsstem B is considered as secondar data. These secondar data must be transferred via a mechanism to the primar data. For this reason, a dumm variable z is defined, which transfers the secondar data to the primar data. Therefore, the sliding surface s changes into s = c ( z ) +. Now with this modified s, the main goal changes from = and = into = z = and =, such that z is a function of s. Hence, the main goal and the secondar goal are linked together through the dumm variable z, and both of these goals will be controlled simultaneousl. The dumm variable z can be found as follows: According to the above statements ( z ) s = c + (8) 4and s can be define as before s = c (9) The control input is the sliding-mode control of subsstem A (Eq. (6)). Since in sliding-mode control theor it is assumed that u = u to control the entire sstem, the boundedness of can be assured with < z u < and z zu. In other words, the maimum absolute value of is alwas bounded. Here, z u is the upper limit of z. Therefore, z can be defined as follows: z = sat( s / ϕ z ) zu () < z < u Therefore, z is a decaing signal, since z u is less than one. The control action is accomplished as follows: The main object of Eq. (6) is to make and equal to zero according to the sliding-mode control theor. But when s, then z in Eq. (8). This causes Eq. (6) to appl an input such that z is decreased. When z is decreased, s will be decreased too. Moreover, when s, then 3 and 4, which satisfies the secondar goal as well. In summar, with the introduction of dumm variable z in s, the sstem is decoupled into two sliding surfaces. Then, with appropriate changes in u, s and s will converge simultaneousl to zero. The same procedure will be performed for the -direction. The overall structure of the proposed control sstem is shown in Fig.. A few points are worth to be mentioned here:. All nonlinear functions f ij ( ) and b ij ( ) ( i =,and j =, ), are independent of state variables 3 and 7.. All nonlinear functions b ij ( ) ( i =,and j =, ), are independent of state variables, 4, 6, and It can be shown that b is less affected b 5 than b, and f is less affected b 4, 5, 6, and 8 than b and. Similarl, it can be shown that b is less affected b than b 5, and f is less affected b,, 4, and 5 than b 6 and According to Eq. (6), k must be selected such that k > D / b ( ), where D is the upper limit of the third kind disturbance. 5. NEURO-FUZZY MODELING According to Eq. (6), each local controller needs f ( ) and b ( ) to control its own subsstem. In other words, the control sstem depends on f i ( ) and b i( ) ( i =, ) during the control process. In addition, because of the dependenc of the control sstem to the parameters of the dnamic of the sstem, the sstem makes errors when these parameters change. Moreover, because of the ideal dnamic model, the controller might encounter difficulties when it is used for a real sstem. Here, those sections of the controller, which are dependent on the dnamic of the sstem, are modeled with two ANFIS networks (Jh-Shing and Jang, 993). The advantages of this kind of modeling is as follows: Because of a good approimation, the modeling accurac is high. Due to the simplicit of these networks, their training is ver fast. The trained network has a ver fast on-line response. The controller is able to control the sstem without an need to the dnamic equations of the sstem. Because of the trainabilit nature of these networks, the controller can adapt itself to the changes in the dnamic of the sstem, b adding an on-line training mechanism. In this wa, the controller becomes adaptive and robust. Due to the eistence of the phsical f interpretation for ( ) ij

5 ( =,and j =,) i and the fuzz nature of the ANFIS network, one can use the epert knowledge in defining the fuzz rules in the corresponding networks. Hence, two ANFIS networks are constructed to model f i ( ) and b i( ) ( i =, ) for each local s s Fig.. The control sstem structure controller. The inputs to the ANFIS networks for f, f, b, and b are {, }, { 5, 6 }, { }, and { 5}, respectivel. The elimination of the less-effective state variables not onl does not deteriorate the accurac of the modeling, but also increases the accurac of the modeling due to the smaller and simpler structure of ANFIS networks. 5. Defining Fuzz Rules s SMC X_Subcontroller s SMC Y_Subcontroller In this paper the subtractive clustering method has been used to define fuzz rules (Chiu, 996). The subtractive clustering method is a scheme for etraction and classification of fuzz rules. The advantages of this method, as compared to similar methods, are (Chiu, 994): ) there is no need to determine the number of clusters beforehand, ) the compleit of the computations increases linearl with the dimension of the problem, which ields a higher computational speed, 3) the abilit to consider each data, not each section, as the candidate for the center of the cluster, 4) etraction of fewer rules along with higher performance. Using this method for 656 input-output data points, which have been obtained for f i ( ) and b i( ) ( i =, ) based on their maimum ecitation, 8 and 3 cluster centers have been obtained, respectivel. Therefore, f i ( ) ( i =, ) is modeled with eight F F S S T E M fuzz rules and b ( ) ( =, ) rules. i i with three fuzz 5. The Structure of the ANFIS Networks b with three f i with eight first order fuzz rules, for each local controller, are constructed using singleton fuzzifier, center average defuzzifier, and product inference engine. The membership functions of the input variables are gaussian. Two ANFIS networks, one for i( ) first order rules and one for ( ) 5.3 Training of the ANFIS Networks The trainable parameters of the ANFIS networks are the primar and the tall parameters of the fuzz rules. Primar parameters are the adjustable variables of the input membership functions (i.e. the center and the width of the gaussian membership functions) and the tall parameters are the coefficients of the linear equations of the tall part of the first order fuzz rules. These adjustable parameters in the ANFIS network for f i ( ) ( i =, ), for the primar and the tall parts of the fuzz rules, are 4 and 34, respectivel. Therefore, the total number of adjustable parameters for this network is 56. In the ANFIS network for b i( ) ( i =, ) the primar and the tall parts of the fuzz rules have 6 and 6 adjustable parameters, respectivel; hence, a total of adjustable parameters. The hbrid method, which is a combination of gradient descent and least square estimation, has been emploed to train the networks (Jh-Shing and Jang, 993). Therefore, this method has the advantages of both gradient descent and least square estimation schemes. Reduction of the search space dimensions and hence a faster convergence speed, and less possibilit of falling into local minima are two advantages of this method. The average errors of the ANFIS networks for f i ( ) and b i( ) after the completion of the training phase (45 and 4 5 epochs) are.856 and , respectivel. The reasons for such small errors are the proper structure of the ANFIS networks and the high performance of the hbrid training method. 6. SIMULATION RESULTS In this section the simulation results of the proposed controller, which is performed on the model of a two-dimensional inverted pendulum, are presented. The initial values of the state variables are as follows:

6 β ( ) = 85 deg, α& ( ) = ( ) = 78 deg, β& () = () = m, & () = m/s, () = 3 m, & () = m/s, α rad/s, rad/s, Ever local controller controls its corresponding sub sstem. While the inverted pendulum comes to the vertical position, the cart reaches the origin of the coordinates. Figs. 3 and 4 show the results. 8 6 Despite the large initial values for angles o o ( α( ) = 85 and β ( ) = 78 ) the proposed controller is able to bring the pendulum to the vertical position. Also, the responses have acceptable overshoot and undershoot. It should be mentioned, however, that such large initial values for angles are not practical because the initial amount of forces would be also ver large in order to bring the pendulum to the vertical position. These initial 3 5 α (deg.) 4 d tim e (s ec.) -5 β (deg.) Fig. 3. Simulation results of α and β d k b tim e(sec.) tim e (s ec.) 9 8 (m) k b tim e(sec.) Fig. 5. Inequalit (6) for each local controller (m) Fig. 4. Simulation results of and values for angles have been onl considered to show the performance limitations of the proposed method. According to the inequalit in Eqs. (6), for the disturbances of the third kind D < b ( ) where D is the upper limit of d. Fig. 5 shows d, d, k b( ), and k b( ). As it can be observed from these graphs, the above inequalit is alwas valid. Therefore, it can be concluded that

7 despite large initial deviation of the inverted pendulum from the vertical position, the disturbance of the third kind is situated in such a range that the local controllers are able to overcome that. Moreover, Fig. 5 shows that the disturbance of the third kind decreases as the inverted pendulum reaches the vertical position and is equal to zero at the balanced position. As a comparison with other methods, the proposed method in (Asgarie-Raad, 998) is able to control the same pendulum as in this paper onl for much smaller initial values of the inverted pendulum with larger overshoots and undershoots. 6. Appling Eternal Disturbances The above simulations are repeated with eternal disturbances applied to both and directions. This disturbance, which is shown in Fig. 6, has relativel large amplitude and has been created using several sinusoidal waveforms with different frequenc and amplitude along with random coefficients. The same initial values have been considered for α and β as before. The simulation results are shown in Fig. 7. As it is clear from the graphs, the proposed controller can bring the inverted pendulum in the vertical position and hold it there, despite large amount of eternal disturbances and large initial value for angles. 6. On-line Change of the Parameters of Inverted Pendulum As it was mentioned before, the two-dimensional inverted pendulum is the model of a balanced missile. Due to the consumption of the fuel, the mass of a launched missile decreases continuousl. In addition, the gravit acceleration changes according to the following equation as the missile is distancing from the surface of the earth: d s g = g () d where g is equal to 9.8 m/s, d s is the radius of earth, and d is the distance of the missile from the center of the earth. Moreover, some portions of the missiles, which are not needed anmore, are separated from it. This brings some sudden changes in the mass of the missile. These changes are simulated in the two-dimensional inverted pendulum as on-line. For this reason, it is assumed that the mass of the cart changes in two steps as follows:.8 kg t <.8 s M =.6 kg.8 t < 5 s ().kg t > 5 s In fact, here the mass of the cart has been decreased b 5% and 78.5% in the first and the second steps, respectivel. To simulate the change of the mass due the reduction of the mass of the fuel, the mass of the inverted pendulum has been changed according to the following equation:.5 t m =.5 kg () where t is time in s. And finall the gravit acceleration changes as follows: 9.8 g = m/s (3) ( +. t) where t is also time in s. The same initial values have eternal disturbance(n) Fig. 6. Applied eternal disturbance to each ais α (deg.) β (deg.) (m) (m) Fig. 7. Simulation results in the presence of noise

8 been used for the inverted pendulum as before. Fig. 8 shows the simulation results. As the graphs show, despite considerable changes in the parameters of the sstem, the proposed control method can bring the inverted pendulum to the vertical position. This can be an indication of the high degree of the robustness of the controller. The run time of the simulations is ver fast, even in the MATLAB environment. A personal computer with Pentium II 4 processor and 96 MB of RAM performed a 5 second simulation in just 6 seconds. Hence, one can use the proposed method as an on-line controller, without an need to write the required codes in low level programming languages. Angle (deg.) Position(m) : Alpha : Betta : : Fig. 8. Simulation results for the case of on-line changes of the parameters of the sstem 7. CONCLUSION A new method, based on snthesis of the decentralized control theor and the decoupled sliding-mode method, was presented for controlling two-dimensional inverted pendulum. First, the dnamic equation of the sstem with order 8 was divided into two sub sstems with order 4. Second, two sliding surfaces have been assigned to each sub sstem, and the sliding-mode control was performed using an intermediate variable. In this wa, a nonlinear dnamic sstem was controlled in a simple fashion. Then, the dnamic-dependent portions of each controller were modeled with two ANFIS networks, with high degree accurac. This made the controllers independent of the sstem parameters. Moreover, the inverted pendulum was controlled with on-line changes in the parameters of the sstem. In addition, due to the parallel processing nature of ANFIS networks, the simulations were performed much faster than the cases with no such networks. The simulation results show that the performed control method is robust against eternal noises too. Also, theoreticall the controller is able to bring the inverted pendulum from large initial deviations to the vertical position. 8. REFERENCES Anderson C. W. (989). Learning to control an inverted pendulum using neural networks. In: IEEE Control Sstem Magazine, vol. 9, pp Asgarie-Raad A. (998), Intelligent control of twodimensional inverted pendulum. Master Thesis, Sharif Universit of Technolog. Chiu S. (996). Method and software for fuzz classification rules b subtractive clustering. In: Biennial Conference of the North American, NAFIPS, pp Chiu S. (994). A cluster estimation method with etension to fuzz model identification. In: Proc. Of the Third IEEE Conf. On Fuzz Sstems, vol., pp Esmailie-Khatier M. (994), Construction and control of a two-dimensional inverted pendulum. Master Thesis, Sharif Universit of Technolog. Ghanbarie A. (a). An outlook to the hierarchical control methods. Master Project, Iran Universit of Science and Technolog. Ghanbarie A. (b). Neuro-fuzz control of twodimensional inverted pendulums. Master Thesis, Iran Universit of Science and Technolog. Ji-Change L. and K. Ya-Hui (998). Decoupled fuzz sliding-mode control. In: IEEE Trans. Fuzz Sstems, vol. 6, pp Jh-Shing and R. Jang (993). ANFIS: adaptivenetwork-based fuzz inference sstem. In: IEEE Trans. Sstem, Man and Cbernetics, vol. 3, pp Khalil H. K. (99), Nonlinear Sstems, New York: Macmillan. Lin C. E. and Y. R. Sheu (99). A hbrid-control approach for pendulum-cart control. In: IEEE Trans. Industrial Electronics, vol. 39, pp Magana M. E. and F. Holzapfel (998). Fuzz logic control of an inverted pendulum with vision feedback, In: IEEE Trans. on Education, vol. 4, pp Nelson J. and L.G. Kraft (994). Real-time control of an inverted pendulum sstem using complementar neural network and optimal techniques. In: American Control Conference, vol. 3, pp Omatu S. and M. Yashioka (998). Stabilit of inverted pendulum b neuro-pid control with genetic algorithm. In: Proc. IEEE Intern. Cong. On Computational Intelligence, vol. 3, pp Slotine J. E. and W. Li (99). Applied Nonlinear Control, New Jerc: Prentice Hall.