ELKOMNIKA, Vol., No., March 4, pp. 79 ~ 86 ISSN: 693-693, accredited A by DIKI, Decree No: 58/DIKI/Kep/3 DOI:.98/ELKOMNIKA.vi.59 79 Neural Networ Adaptive Control for X-Y Position Platform with Uncertainty Ye Xiaoping, Zhang Wenhui*, Fang Yamin Institute of echnology,lishui University, China-33, elp 578-75, Fax 578-75 Institute of Science,Lishui University, China -33, elp 578-75, Fax 578-75 *Corresponding author, e-mail: hit_zwh@6.com Abstract An improvement neural networ adaptive control strategy is put forward for X-Y position platform with uncertainty by the paper. Firstly, dynamics model of X-Y position platform is established. hen, RBF neural networ with good learning ability is used to approach non-linear system. he early period control accuracy of the problem is considered by the paper, because good precision in the early period is difficult to be obtained by neural networ controller, so PID controller is designed to compensate control. An improvement dynamic optimization adjustment algorithm of networ weights is designed to speed up the learning speed. Simulation results show that the control method is more effective to improve the control precision and real-time and has a good application value. Keywords: Neural networ; X-Y position platform; PID controller; Optimized learning algorithm. Introduction With the development of computers and advanced control technology, High-precision digital drive technology has become the mainstream in the development of the numerical control (NC) machine tools. he X-Y NC platform is a flat with two-dimensional space control system. It is not only able to complete the two-dimensional space plane processing, and can be used as the prototype of the NC machine tools (for example, robots and other large equipment). he NC platform has a certain degree of nonlinear and coupling, and therefore obtains higher control accuracy; it is difficult for the traditional PID control technology to meet the control requirements ref.[]-[4]. o eliminate the influence of these non-linear factors, all sorts of presented control strategies such as adaptive control, fuzzy control and neural networ control have been used on X-Y NC platform ref. [5]-[]. For these nonlinear uncertain systems, the adoption of adaptive control method can achieve better results, but the ascertainment of the linear parameters and the regression matrix require large number of calculations, which affect its application. As the neural networ control is not required to now the exact model of control object, at the same time the controller is simple and has learning ability, it is increasingly widely used in the field of space control. Now people have made some research results. Ref. []-[] present a fuzzy control scheme, but there are too much fuzzy rules, which will lead to the computational geometry multiply, and too little cannot guarantee the accuracy of control. Ref [3] presents X-Y platform control scheme of Elman neural networ, but there are disadvantages of large amount of calculation and time-consuming of using bac-propagation algorithm, it is difficult to guarantee real-time and need offline training. Ref [4] presents robust adaptive control for X-Y platform, which needs a certain degree of uncertainty on the system boundary, and it is not the best control method. For the above shortcomings, a radial basis function (RBF) neural networ adaptive PID control strategy is put forward for X-Y position platform with uncertainty. Firstly, dynamics model of X-Y position platform is established. hen, RBF neural networ with good learning ability is used to adaptive control non-linear system. he good control accuracy is difficult to be obtained in the early period of study, so PID controller is designed as a compensation controller. Dynamic optimization algorithm of networ weights is designed to speed up the learning speed and the adjustment velocity. Received July, 3; Revised January 8, 4; Accepted February 8, 4
8 ISSN: 693-693. Dynamic Equation of X-Y Position Platform he Figure shows the schematic diagram of X-Y position platform system. Figure. X-Y Position platform he motor dynamics is ignored by the paper, the inertia force friction and other disturbances are considered, the system dynamics model from each axis is obtained by ref. [4]. Dq Cq F () q Where,, q represent the velocity and acceleration of the system movement respectively. D is designed as the quality of the positioning platform; q is designed as the displacement of the screw carried by the slider; Cq is designed as the viscous friction; C is designed as the viscous friction coefficient, F are designed as the static friction and coulomb friction; is designed as the output torque of motor. 3. Designed of PID Controller Base on Neural Networ,then the above dynamic model can be written as If H Cq F () Dq H Augmented variable input method is used for the above dynamic model. Error vector is defined as eqd q, q d is the desired trajectory, unmolded dynamics isn't considered in the situation of the system (), the following controller(3) can guarantee the system stability. Dx ˆ( KeKe ) Hˆ (3) p d Where, K p K d is defined as feedbac gain matrix. However, the platform model is difficult to get accurately in practice; the ideal nominal model can be created. If the system nominal model is expressed as ˆD, Ĥ. hen the control law design is designed as : Dq ˆ ( KeKe ) Hˆ (4) p d ELKOMNIKA Vol., No., March 4: 79 86
ELKOMNIKA ISSN: 693-693 8 o put the control law (4) into the control law (5), it can be obtained: d p ˆ [ ] e K e K e D Dx H (5) Where, D DDˆ, H H Hˆ. From the above equation, as can be nown that the uncertainty of system modeling will lead to the degradation of control performance. he good learning ability of neural networ is considered to solve the nonlinear effects of X-Y NC platform; controller based on neutral networ is designed. A partial generalization of RBF networ is chosen to accelerate the learning speed and to avoid local minima value problems ref. [6]. According to the nonlinear dynamic model of the X-Y NC platform (), it can be obtained: Dx H f( xxx,, ) (6) he total control input contains PD and NN. he PID feedbac controller is designed as: K e K e (7) PD d p he RBF neural networ controller is designed as: M( x, x, x, ) (8) NN d d Where, : output hidden layer, b : node threshold. (9) PD NN he structure of control system is shown in Figure Figure. Neural networ PID control system Because the neural networ cannot fully complete study in the initial stage, the condition results in a bigger error. o solve this problem, PID feedbac controller participates in compensation control of neural networ, the combination controller ensure stability of system. PID feedbac controller mainly plays the leading role in the early stage of control, Along with learning of the neural networ, errors are decreased gradually, PID feedbac controller function get smaller and smaller gradually. Where, a variable learning law in Local generalization networ of RBF is used to accelerate the learning speed and convenient in applications in real-time. Neural Networ Adaptive Control for X-Y Position Platform with Uncertainly (Ye Xiaoping)
8 ISSN: 693-693 For the membership function of the hidden layer of RBF neural inverse mode approaching and neural controller to tae the Gaussian function, the hidden node output is: X cj j ( ) exp( ) () b j he output of the output layer is: m y ( ) W( ) ( ) ( i, ) () j ij j Where, W ( ): connection weights of hidden layer and output layer. ij he error signal of online learning is defined as: E y ( ) yd( ) y ( ) yd( ) () J W E W (3) ( ) ( i i ) According to function extreme value theory from ref [5]: the quadratic approximation of the performance indicators near the minima of J( W ) is: S W G W W H W W (4) ( ) ( ) ( ) S ( W ), Where, G : he minimum of the quadratic approximation function, H : positive definite Hessian matrix. he function E ( W ) i tae the first-order aylor polynomial in the near of W : W : minima of E ( W) E ( W ) ( W W ) E ( W ) () (5) i i i Where, Ei ( W) is defined as Ei ( W) to W on the gradient. o omit the higher order terms () of the above equation, putting it into(5), the following the equation can be obtained: i i (6) i J( W) [ E ( W ) ( W W ) E ( W )] he second approximation on the above equation is expanded as: J W E W W W E W J W (7) ( ) [ ( ) ( ) ( )] ( ) i i Where, :forgetting factor,. ELKOMNIKA Vol., No., March 4: 79 86
ELKOMNIKA ISSN: 693-693 83 In order to export recursive learning process easily, put (4) into J ( W ) and the following the equation can be obtained: J W E W G E W W W E W ( ) ( ) ( )( ) ( ) i i i ( ) [ ( ) ( )]( ) i i W W H E W E W W W (8) According to(4), it can be obtained: S ( W) G ( W W ) H ( W W ) S ( W) G ( W W ) H ( W W ) ( W W ) H ( W W ) ( W W ) H ( W W ) (9) Mae J ( W) S ( W) hen E ( W ) G G ( W W ) H ( W W ) () i H H Ei ( W) Ei ( W) () E ( W ) E ( W ) H ( W W ) () i i Using the matrix inverse theorem and according to (), as can be obtained: ( ( ) ( ) / ) i i H H H E W E W H (3) E ( W ) H E ( W ) (4) i i In (), the same multiplied by H on both sides, H H, and put it into (3), after collated, the final learning algorithm of neural networ weights can be obtained: W W H E ( W ) E ( W )/ (5) i i So, the algorithm of the RBF neural networ weights is designed as E ( W ) E / W i i O () PD / W O () ( ) (6) PD j Neural Networ Adaptive Control for X-Y Position Platform with Uncertainly (Ye Xiaoping)
84 ISSN: 693-693 4. Simulation and Analysis In order to illustrate the effectiveness of the control algorithm, the paper designs the dynamic model parameters as: D 6, C 3, F.sin t he desired trajectory of the X-Y platform is: X : xd cos. t ; Y : xd sin. t. PD controller gain: K diag (,) ; K diag (3,3). p d Learning factor:.6. he initial value of the X-Y platform movement: ; X : x, x. Y : x 4, x he simulation results are shown in the follow figures. Where Figure 3 and Figure 4 is the trajectory tracing curve and the speed tracing error curve in this paper program Figure5 is control input of X-Y platform..5 desired position of X real position of X 4 3 desired position of Y real position of Y Position tracing of X/m.5 -.5 - -.5 - Position tracing of Y/m - -.5 5 5 (a) Position tracing of X - 5 5 (b) Position tracing of Y Figure 3. Position tracing of X-Y platform According to the Figure3, as can be learned that the design of RBF neural networ PID controller is able to fast-trac the expected angle of trajectory in a relatively short period of time, but also to achieve better tracing of angular velocity. It turns out that in the initial phase of the control process, the RBF neural networ is still in the learning period, now neural networ cannot approach system model, in this time, neural networ controller together with the conventional PD feedbac controller meet the tracing error of the joint angle. With the study of neural networ, better control effect can be obtained. According to the figure 4, the good velocity tracing can be achieved in 5s. From figure 5, control torques is small, and relatively smooth. Simulation shows the proposed control method is effective, and has better engineering value. ELKOMNIKA Vol., No., March 4: 79 86
ELKOMNIKA ISSN: 693-693 85 3 Velocity tracing of X/m.s - - desired velocity of X real velocity of X Velocity tracing of Y/m - - - -3 desired velocity of Y real velocity of Y - 5 5 (a) Velocity tracing of X -4 5 5 (b) Velocity tracing of Y Figure 4. Velocity tracing of X-Y platform 6 4 Control input X/N.m 4 - -4 4 6 8 (a) Control input of X Control input Y/N.m 3-4 6 8 (b) Control input of Y Figure 5. Control input of X-Y platform 5. Conclusion rajectory tracing control problems X - Y NC platform system with uncertainty are studied by the paper, an improvement neural networ PID control strategy is proposed. ) An integrated controller is designed. he controller integrates neural networ controller with PID controller, the good control precision can be achieved in the initial learning phase of neural networ, because of compensation affection of PID controller; ) Improvement optimization learning algorithm is designed, the algorithm can ensure the online real-time adjustment of the weights of networ; 3) he control mechanism is analyzed in detail, the simulation proves the validity of the scheme. he improvement neural networ control strategy can achieve good control effect, and has high engineering application value for X-Y NC control platform system with uncertainty. Acnowledgments Project supported by Zhejiang Provincial Natural Science Emphasis Foundation of China (No.LZF), Zhejiang Provincial Natural Science Foundation (No.Y4F3) and (No. LY3F), Zhejiang Provincial Education Department Science Research Project (No.Y33). Neural Networ Adaptive Control for X-Y Position Platform with Uncertainly (Ye Xiaoping)
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