Digital Double-loop PID Controller for Inverted Pendulum

Transcription

1 Sensors & Transducers, Vol. 156, ssue 9, September 013, pp Sensors & Transducers 013 by FSA Digital Double-loop PD Controller for nverted Pendulum Ming Li College of Mathematics & Computer Science, Wuhan Textile University, Wuhan, , China Tel.: Received: 5 June 013 /Accepted: 5 August 013 /Published: 30 September 013 Abstract: nverted pendulum system is a complicated, unstable and multivariable nonlinear system. n order to control the angle and displacement of inverted pendulum system effectively, a novel double-loop digital PD control strategy is presented in this paper. Based on impulse transfer function, the model of the single linear inverted pendulum system is divided into two parts according to the controlled parameters. The inner control loop that is formed by the digital PD feedback control can control the angle of the pendulum, while in order to control the cart displacement, the digital PD series control is adopted to form the outer control loop. The simulation results show the digital control strategy is very effective to single inverted pendulum and when the sampling period is selected as 50 ms, the performance of the digital control system is similar to that of the analog control system. Copyright 013 FSA. Keywords: nverted pendulum, Double-loop PD control, Digital control, Sampling period. 1. ntroduction n the development of control theory, the correctness and feasibility of the theory must be validated by a typical object. The inverted pendulum system, which can reflect some problems in control systems such as nonlinearity, robustness, stabilization, and following control, etc., is an ideal platform to carry out these experiments. The rocket's flight control can be simplified as an inverted pendulum system, so as the step robot control and the satellite attitude control. Therefore it has remarkably theoretical and practical significances to research how to control inverted pendulum systems. nverted pendulum is a kind of complicated, unstable and multivariable nonlinear system. There are many types of inverted pendulum systems. According to the shapes of the inverted pendulum, there have linear type and ring type etc.; according to stage numbers, there have single, double and multistage inverted pendulum systems; but according to the types of control electric motor, there have single motor, multi motors etc. [1-3]. From 1960 s, there are increasingly interests to research inverted pendulum systems. L. F. Sun et al. [3] introduced some control methods for inverted pendulum and holded that intelligent control method is better than linear control method. Some researchers [4-8] have developed some control strategies for the cart-pole inverted pendulum system using fuzzy logic controller. Siuka et al. [9] deals with the application of energy based control methods for the inverted pendulum on a cart model and presented a swing up controller. There have been some studies [10-1] on PD methods for the purpose of balance control of inverted pendulum system. Also, some other people [13, 14] have researched the adaptive control problems of inverted pendulum systems. Aguilar-banez et al. [15] present 34 Article number P_1364

2 Sensors & Transducers, Vol. 156, ssue 9, September 013, pp an output feedback stabilization method for the nverted Pendulum Cart system around its unstable equilibrium point, and adapt an observer based controller devoted to render the closed-loop system to the origin. Mladenov [16] had used two Neural Network controllers to swing a pendulum attached to a cart from an initial downwards position to an upright position. But this works are based on statespace representation, were focused on the analog PD controller which can t be implemented by the computer program easily. n this paper, under considering there exits the cart s friction, in order to control the displacement of the cart of the inverted pendulum system meanwhile guaranteeing the pendulum verticality, the model of the single inverted pendulum system is divided into two parts based on impulse transfer function. A novel double-loop digital PD control strategy and how to choose its sampling period are researched in this paper. Finally Use matlab/simulink to verify the performance of the presented control strategy.. Model of Linear nverted Pendulum System The single inverted pendulum system consists of a cart and a pendulum, as Fig. 1 (a) shows. The transfer function of the linear inverted pendulum is: Xs () Gs () Fs () ml g ( ml ) s 4 3 ( M Mml m) s b( ml ) s ( M m) mlgs bmlgs () The control goal of the linear inverted pendulum is to make the horizontal displacement of the cart controllable while guaranteeing the pendulum verticality. This problem is a type of classic control systems with one input (external force F) and two outputs (angle of pendulum and car displacement x). n order to control two output parameters effectively, we must establish the mathematical model of the linear inverted pendulum. According to formula (), we can get two transfer functions for two controllable parameters. () s G1 () s Fs () mls M Mml m s b ml s M m m s bm (3) 3 ( ) ( ) ( ) lg lg X () s mlg ( ml ) s G () s (4) () s mls (a) (b) 3. PD Controller The PD controller consists of three components that are proportion, integral and differential. The mathematical expressions are Fig. 1. Single inverted pendulum system. By ignoring the air fiction and some minor fictions, the force diagrams of the cart and the pendulum are shown as Fig. 1 (b), where M is the cart mass, x is the cart position, F is the force impacted on the cart, b is the cart friction coefficient, m is the pendulum mass, L is the pendulum length, is the included angle between the pendulum and the vertical direction, and is the pendulum rotary inertia. The dynamic models of the single inverted pendulum system are nonlinear differential equations. f selecting the working points 0 0, x 0 0 and linearizing the system, we have the simplified mathematical models of the inverted pendulum system. They are: ( ml ) mlx mlg ( M mx ) bx ml F (1) 1 de( t) ut () KP et () etdt () TD T dt U() s 1 K Ds () KP (1 Ts D ) KP KDs Es () Ts s where, (5) K P K, KD KPTD (6) T f T denotes the sampling period, we can conclude the positional PD controller as k, uk Kek K e j K ek ek ( ) ' ' ' P ( ) ( ) D ( ) ( 1) j0 where T TD KD K K, K K TK, K K T T T ' ' ' P P P D P (7) (8) 35

3 Sensors & Transducers, Vol. 156, ssue 9, September 013, pp According to Z transformation, we can get the impulse transfer function of the digital PD controller U( z) Dz ( ) Ez ( ) ( K K K ) z ( K K ) zk z z ' ' ' ' ' ' P D P D D (9) The block diagram of single-loop analog PD control system is shown in Fig. that can only control the displacement x of the car. and two outputs, we can divide the mathematical model into two parts, as shown in equations (3) and (4). When we use the double-loop PD controller, the inner loop controls the pendulum angle and the outer loop controls the cart horizontal position. Fig. 3 shows the block diagram of the analog double-loop PD control system. n order to simplify the computation, two PD controllers are all designed as PD controllers. An amplifier G is placed in the inner loop in order to suppress interference and unit negative feedback is used in the outer loop in order to obtain better following performance. X r (s) PD controller G 1 (s) X(s) θ(s) G () Xr(s) θ(s) X(s) PD1 Amplifier G G 1 (s) G (s) PD Fig.. Single-loop PD control system for single inverted pendulum. Fig. 3. Analog double-loop PD control system for inverted pendulum. 4. Double-loop Digital PD Controller As to single-loop PD control system, three components, the proportion, integral, differential, are located in the forward channel of control system. But as to double-loop PD control system, a PD controller and the object can form an inner loop that can make the unstable object stable; in the forward channel of the control system, another PD controller and the object form an outer loop that can make the control system have the expected performance [17]. Because the single inverted pendulum has one input Although the analog double-loop PD controllers can make the performance of the inverted pendulum system good, they can t be implemented by computer easily and their development cost is very expensive. Therefore we must design a digital controller that can make the performance of the inverted pendulum control system meet the control requirements. Based on the analog double-loop PD control system, we can get the control strategy of the digital double-loop controller using sampling control theory that is illustrated in Fig. 4. Xr(s) θ(s) Digital PD1 Amplifier G ZOH G 1 (s) G (s) X(s) Digital PD Fig. 4. Structure of digital double-loop PD control system for inverted pendulum. The sampling period T is a very important parameter in digital control system. We must choose the sampling period according to the performance requirements and the development cost of the digital control systems. When the sampling period is larger, the demands for the operation speed of computers are lower, so it is helpful to cut down the cost of digital controllers, but too large sampling period will degrade the performance of control systems. While the sampling period is smaller, the performance of digital control systems will be improved, but the demands for the operation speed of computers are higher, which lead to increase the development cost of digital control systems. Furthermore, excessive sampling period has little help to improve the control performance. Therefore the sampling period T must be chosen correctly. n this paper, we suppose the inner loop and the outer loop in the digital doubleloop control system have the same sampling period. 36

4 Sensors & Transducers, Vol. 156, ssue 9, September 013, pp Simulation Analysis n order to compare and analyze the control results of these control strategies for linear inverted pendulum, Matlab/Simulink [18] is used to simulate. Suppose the parameters of the linear inverted pendulum are: M=1 kg, m=1 kg, L=0.6 m, b=0.1 N/m.s, =0.03 kg.m. From equations (), (3) and (4), we can get: X() s 0.1s 3 Gs () 4 3 F( s) 0.15s 0.01s 6s 0.3s () s 0.3s G1 () s 3 Fs ( ) 0.15s 0.01s 6s 0.3 X( s) 0.1s 3 () G s () s 0.3s n order to obtain the control results of digital double-loop PD control system, set the sampling period T as 100 ms, 90 ms, 80 ms, 50 ms, 10 ms and 5 ms respectively. According to equation (8), the parameters of the digital PD controller will be obtained directly from the analog PD controller. The results are shown in Table 1. Table 1. Parameters of the digital PD controller. Sampling Digital PD1 Digital PD period T K P1 K D1 K P K D 100 ms ms ms ms ms ms The analog single-loop PD control system, illustrated in Fig., can be simulated by Simulink. The simulation results show that no matter how to modify the parameters K P, K and K D of the controller, the control system is always unstable, which proves the single-loop PD controller isn t fit for the control of this type of inverted pendulum system. When simulation of analog double-loop PD control system for the linear inverted pendulum, we set the parameters of the control system as Kp 1 0.1, KD 1 0.1, Kp 1.65, KD and G= -3. Then the step response of the control system can be get in Fig. 5, from which we can conclude that for the pendulum angle, the overshoot is 1.33 % and the setting time is 3.1 s ( %); as to the cart position, the overshoot is % and the setting time is 6.15 s ( %). The results show the analog double-loop PD controller is effective to control the linear inverted pendulum system. When T is set as 100 ms, the simulation result is shown in Fig. 6 (a). While T is set as 90 ms and 80 ms, the results can be seen in Fig. 6 (b). The Fig. 6 (c) is the simulation results when T is set as 50 ms, 10 ms and 5 ms. From the simulation results, we can get the performance indexes of the digital double-loop PD control system as Table shows. Table. Performance indexes of the digital double-loop control system. Cart position x Pendulum angle Sampling period Overshoot time shoot time Setting Over- Setting 100 ms Unstable Unstable Unstable Unstable 90 ms 85.5 % s % 40.3 s 80 ms 50.6 % 5.9 s % 4.64 s 50 ms 38. % 6.05 s 8.3 %.4 s 10 ms 38.5 % 6.13 s 60.7 %.46 s 5 ms 38.5 % 6.14 s 59 %.46 s Fig. 5. Step response of analog double-loop PD control system. From Table, we can conclude that when T=100 ms the digital control system is unstable; When T=90 ms the control system become stable, but the performances are very bad which means the control system has little practical value; when T=80 ms, the system performances is improved greatly. When T is between 5 ms and 50 ms, we can see that the performances of the digital control system have little change and are very similar to those of the analog double-loop control system. The simulation results further prove that when the analog control system is transformed into the digital control system, the system performances will degrade. So we must select the sampling time T correctly when we use digital control system. As to the single inverted pendulum, the controlled 37

5 Sensors & Transducers, Vol. 156, ssue 9, September 013, pp parameter changes very quickly, so it is better that the sampling time T is set as 50 ms. Finally we can conclude that the digital double-loop PD controller is a good solution to control the single inverted pendulum system. (a) 6. Conclusions The inverted pendulum system is a complicated, unstable and multivariable nonlinear system that can reflect many typical problems in control systems, so researching inverted pendulum control system has remarkably theoretical and practical significance. Because the digital controller can be easily implemented by the computer program, it is a cheap controller and has very high practical value. The research results in this paper show: the single-loop PD controller is not suitable to control the single inverted pendulum that is unstable and has one input and multiple outputs. As to this type of problems, the controlled objects can be divided into different parts according to the numbers of the controlled parameters and also different mathematical models should be established. The digital double-loop PD controller is a good solution for the single inverted pendulum with one input and two outputs. For the controlled parameter changes very quickly, it is a good way that the sampling time T is set as 50 ms in order that the digital control system has good performance and cheaper cost. References (b) (c) Fig. 6. Step responses of digital PD controller. [1]. H. M. Liu, C.-F. Zhang, Stability Control and realization of single link rotary inverted pendulum on LQR controller, Journal of Central South University (Science and Technology), Vol. 43, ssue 9, 01, pp []. H. Ashrafiuon, A. M. Whitman, Closed-loop dynamic analysis of a rotary inverted pendulum for control design, Journal of Dynamic Systems, Measurement and Control, Transactions of the ASME, Vol. 134, ssue, 01. [3]. L. F. Sun, H. Kong, C. G. Liu, Overview of the control of the inverted pendulum system, Machine Tool & Hydraulics, Vol. 36, 008, pp , 313. [4]. S. Kizir, Z. Bingul, C. Oysu, Fuzzy control of a real time inverted pendulum system, Journal of ntelligent & Fuzzy Systems, Vol. 1, 010, pp [5]. M. K. B. M., Nor, S. Okubo, Fuzzy servo control of an inverted pendulum system, Artificial Life and Robotics, Vol. 17, ssue, 01, pp [6]. C. H. Huang, W. J. Wang, C. H. Chiu, Design and implementation of fuzzy control on a two-wheel inverted pendulum, EEE Transactions on ndustrial Electronics, Vol. 58, ssue 7, 011, pp [7]. D. Wang, S. Wu, L. Zhang, et al., Study of inverted pendulum robot using fuzzy servo control method, nternational Journal of Advanced Robotic Systems, 01. [8]. H. L. Bui, D. T. Tran, N. L. Vu, Optimal fuzzy control of an inverted pendulum, Journal of Vibration and Control, Vol. 18, ssue 14, 01, pp [9]. A. Siuka, M. Schöberl, Applications of energy based control methods for the inverted pendulum on a cart, Robotics and Autonomous Systems, Vol. 57, 009, pp [10]. A. Ghosh, T. R. Krishnan, B. Subudhi, Robust proportional-integral-derivative compensation of an 38

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