Design and Implementation of an Inverted Pendulum Controller to be used as a Lab Setup

Transcription

1 Design and Implementation of an Inverted Pendulum Controller to be used as a Lab Setup Harsha Abeykoon, S.R.H. Mudunkotuwa, Malithi Gunawardana, Haroos Mohamed, Darshana Mannapperuma Department of Electrical Engineering, University of Moratuwa Sri Lanka 1 harsha@elect.mrt.ac.lk codexdj@gmail.com 3 sasareka@yahoo.com 4 haroos.mlhm@gmail.com 3 darshana.uom@gmail.com Abstract The Inverted Pendulum is one of the most challenging and important classical problems of Control Engineering. This paper presents a solution to this inherently unstable problem, with the pendulum being mounted on a cart that is moved by a belt spread between two pulleys. An optical encoder and a potentiometer are used to feedback the cart position and the pendulum angular displacement. The digital control of the system is developed using a microcontroller. The user can tune the parameters of the system through a user interface panel and observe the system response through a Computer Interface. The apparatus is intended to be used as a teaching tool to demonstrate the effects of Proportional (P), Integral (I), Derivative (D) controls separately as well as in combination, on control systems to engineering students and a practical was developed for this purpose. I. INTRODUCTION The Inverted Pendulum has been popular in the field of control engineering since it provides a good practice for prospective control engineers with features such as inherent instability. It is a non-linear system, yet can be approximated as a linear system if the operating range is small. The previous approaches toward solving this complicated problem involved three categories: 1. Simulation of inverted pendulum on a conveyor. Simulation of cart- mounted inverted pendulum 3. Implementation of cart-mounted inverted pendulum We have adopted the third approach in solving the Inverted Pendulum problem as presented in this paper. The apparatus involves a cart, able to move backwards and forwards, and a pendulum, hinged to the cart at the bottom of its length such that the pendulum can move in the same plane as the cart. That is, the pendulum mounted on the cart is free to fall along the cart's axis of motion. The pendulum is constrained to swing within predefined limits of ±45 degrees on either side of vertical axis. The system is to be controlled so that the pendulum remains balanced and upright, and is resistant to a impulse disturbance. The feedback signals are cart position and pendulum angle from the vertical axis. These signals are fed to the controller in which PID control is implemented. The required driving signals are then fed to the motor. The aim of the study is to stabilize the pendulum such that the position of the carriage on the track is controlled quickly and accurately and that the pendulum is always maintained tightly in its inverted position during such movements. The paper is arranged as follows: firstly the system modelling and control technique are discussed. Then the implementation with emphasis on mechanical model and electronic circuitry is presented. Finally the simulation results show the feasibility of the proposed technique II. METHODOLOGY The various stages of the work for accomplishing the task of controlling the Inverted Pendulum are as follows: Modeling the IP and linearizing the model for the operating range Understanding the inherit instability of the system Designing the PID controller and simulating it in MATLAB for proper tuning and verification and implementing with the PIC Finding optimum system parameters Designing and modifying the mechanical system Designing the electronic control system which consists of motor drivers, user interfaces, other interface and the central controller Integrating the subsystems together Implementing the controller on the physical IP model Testing the system and fine tuning the necessary parameters to improve the performance A. Mathematical model of Inverted Pendulum A complete theoretical model of the pendulum cart system can be developed using Lagrangian Dynamics where the equations of motion can be derived easily using Lagrange's equations [1]. Of course the parameters of the system should be measured accurately to be used in it. Refer Table 1 for the relevant parameters.

2 Simplification, rearranging and substitution of these equations leads to the equations that describe the motion of the inverted pendulum where x(t) is the position of the cart, θ(t) is the angle of the pendulum with respect to the vertical direction. TABLE I SYSTEM PARAMETERS Symbol Description Value M Mass of the Cart 710 g M Mass of the Pendulum 00 g b Friction of the Cart (Assumed) 0.00 N/m/s L Length of pendulum to Center of Gravity 90 cm I Moment of Inertia (Pendulum) 0.16 Kgm r Radius of Pulley 4.34 cm t M Time Constant of motor 0.5 s K M Gain of Motor rad/s/v K F Gain of Feedback 1 V/rad/s F Force applied to the cart x Cart Position Coordinate θ Pendulum Angle with the vertical B. Simulation of Model Fig. 1 Control block diagram of the whole system The model described by the above equations was created in the MATLAB/Simulink environment as a standalone block []. The Simulink scheme is shown in figure. Simulation based tuning of PID controller proved helpful and was quicker than hit and trial tuning. Effects of tuning P, I and D values on system response, as simulated in Simulink [3], is included in Section 3 Results. This could help in understanding the fundamentals behind PID controls as well. Equations obtained are nonlinear, but since the goal of a control system would be to keep the pendulum upright the equations can be linearized assuming that θ represents a small angle from the vertical upward direction. By considering the Laplace Transformation of those we can obtain the transfer functions, which are shown below. The transfer function of the pendulum angle: (s) G a ml s θ(s) q U(s) b(i ml ) (M m)mgl bmgl [1] 3 s s s q q q The transfer function of the cart position: (I ml ) mgl s X(s) q q G b (s) [] U(s) 4 b(i ml ) 3 (M m)mgl bmgl s s s s q q q q [(M m)(i ml U(s) K G 1(s) E(s) ) (ml) The overall transfer function of the actuation mechanism: m (M m)r s τ s 1 m E (s) = Error Voltage (s) = Angular Position of the Pendulum X(s) = Cart Position of the Pendulum U(s) = Force on the Pendulum ] [3] Fig. MATLAB simulation block diagram C. Physical Implementation 1) Mechanical Modifications: Two side barriers are being constructed to prevent the possible mechanical damage (bending, etc) to the pendulum upon falling off to sideways when unstable. The height of the each barrier is calculated by considering the pendulum length. ) Feedback Network: An accurate feedback network is essential in stabilizing the Inverted Pendulum system. For this purpose the sensor need to have a fast response and be noiseless such that the information retrieved from the sensor accurately reflects the state of the system. In the case of Inverted Pendulum, there are four parameters that govern the inverted pendulum system and that could be used for feedback in order to determine the controls necessary to stabilize the system: - Pendulum Angle - Angular velocity of Pendulum - Displacement of the cart - Velocity of the cart We decided to go along with the existing feedback system that measures the Angular displacement of the Pendulum via a

3 Linear Potentiometer and the cart position via an Incremental rotary encoder. Pendulum Angle Sensor - Potentiometer A linear potentiometer was used to measure the angular displacement of the pendulum. The potentiometer is attached to the pivot of the pendulum above the cart such that when the pendulum tilts, it also rotates the potentiometer shaft attached to the pivot, and we are able to know its angle through the feedback. When gathering information from the Pendulum Angle sensor it is required it that produces a variable voltage output that can be sampled by the controller platform. The Potentiometer used in the project gives 0V to 5V voltage according to the angle. The microcontroller will be programmed to use its internal Analog-to-Digital Converter (ADC) to convert the voltage outputs of the sensors into a binary representation which then can be converted into a usable measurement. It may be possible to incorporate faster and more accurate external ADCs, but it will require a larger I/O interface and accurate timing to guarantee good readings. Since the ADC s results will be within the range of (0-55) and its input analogue voltages from the potentiometer are within the range (0-5) volts for the angles of (0-360 ), we can obtain the resolution of the potentiometer such as: Resolution = 360 /56 = / 1 bit Or = 5/56 = V/ 1 bit = / V Ideally, the potentiometer would have little friction; though practical potentiometers will have some friction, which could influence the dynamics of the pendulum falling. More friction would slow down the reaction of the pendulum to any of the forces exerted on it, making it easier to balance working for our benefit. Cart Postion Sensor - Incremental Rotary Encoder An incremental rotary encoder was used to get the position of the cart on the length of the track. This encoder produces 1000 pulses per one revolution. The Encoder used was a Voltage output type giving out three outputs. The following fig 3 shows the clock diagrams of the encoder. According to the above clock diagrams two outputs (A and B) are required to count the pulses and get the direction. When the rising edge of output A high and output B is low, direction of the rotation is Clockwise. When the rising edge of output A and output B is high, direction of the rotation is Counter Clockwise. Output Z is an Index pulse that can be used to count the number of rotations. CCS C compiler was used to program the microcontroller. As mentioned earlier the encoder generates 1000 ppr and our motor rotates about 3600rpm. Thus the encoder will generate about 60,000 pulses in a second and in order to ensure maximum accuracy of the position of the cart it is vital to count all the pulses without losing any of the pulses. Therefore, to determine the cart position using the incremental encoder, the interrupt service routines (ISR) in the PIC is used. As there are several ISR in the microcontroller, we had to identify which one of them to be used. By default, the port B s pin B0 was assigned to detect external interrupts. That was one of the choices we had. Also, another service routine was available to detect the change in port B (pin B4- B7). In both cases, each time that the particular situation occurs, the user specified coding is executed. The difference in two is that the Pin B0 only detects either the rising or falling edge on PIN B0, but not both, where as the latter ISR detects any edge on any pin. It is clear that the latter ones gives more attention to pulses coming in and will improve the accuracy and resolution of the position; so we selected the nd method. We had channels A, B as outputs from the encoder, we assigned pins from port B (PIN B4 & PIN B3) to read those pulses, as they were close (single pin header) to each other and available. Whenever a level change is detected at PIN B4, it will compare the values at PIN B4 & PIN B3. If PIN B4 is high and PIN B is low, a long data typed variable will be incremented by 1 (single pulse to clock wise direction), and if PIN B4 is high and PIN B3 is also high, the same variable will be decremented by 1 (single pulse to counter clock wise direction). So ultimately the variable can be used to determine the position of the cart if the initial position of the cart is known. Since the variable is handled and updated by the ISR, it s value can be read any time in the main function. Figure.6 Clock Diagrams of Encoder output Direction of rotation: Clockwise as viewed from the shaft Direction of rotation: Counter clockwise as viewed from the shaft Fig 3 Clock Diagrams of Encoder output

4 Since the number of instructions in the ISR is not very long, the number of clock cycles interrupted is not significant and will not create a latency problem, hence will not affect the main algorithm. 3). Electronic Design Basically, the three main Circuits used were: Control circuit Board, motor driving Circuit to drive the DC motors according to the required acceleration and Interface circuit to display the output value and take to PID value for controlling. Control Circuit Board : This is the main circuit used to carry out the controlling algorithm with taking necessary inputs. A serial communication through MAX33 level converter is included on the board for computer interface and it is used to give P, I and D values and plot the angle measurement as a function of time. 5V DC power supply need to run the system can be given by a rechargeable battery or power supply unit. To maintain the constant voltage this circuit uses LM7805 voltage regulator so the circuit can handle Output Current up to 1A with up to 18V DC. Motor Drive Circuit In this project 75V, A brushed DC motor was used. This motor was drawing fairly high current during the PWM switching, with an average of A and peaks reaching up to 4 A and also since it takes high voltage, we designed motor drive for 6 A and 100V capable circuit keeping a safety margin. It is designed with MOSFETs because it has few advantages over BJTs and power transistors such as: fast switching and minimum power dissipations and is operated at 4 khz. The motor is controlled by an H-bridge which is driven by the PIC16F877A MOTOR CONTROL PWM MODULE. Therefore H-bridge was design with separate Circuit due to 75V DC and three 1V DC separate power supply requirement. This circuit consist with logic for proper control by PIC16F877A. Switching techniques are used to control the amount of power delivered to the motor by using pulse width modulator (PWM). The motor control circuit consist two main parts H-bridge Circuit and Logical Circuit. H-bridge circuit Design This circuit mainly contains four MOSFETs as shown in figure (14), N-channel IRF540N were used (Refer Appendix 3 for IRF540N Data sheet). It is 100V, A. Freewheeling diodes were added in order to protect the MOSFETs from back emf generated by the motor in the form of spikes. It can handle 4A current. Heat sinks hinged to the MOSFETs were also added to provide cooling for the MOSFETs. To Bias the MOSFET it is necessary to maintain the voltage between the Gate to Source (V GS ) so in the H-Bridge to achieve this we used different grounds for four MOSFETs therefore four 1V DC supply was used in this circuit. The direction of the current is controlled by the direction signal; two MOSFETs will be switched on at the same time while the other two remain off. To Bias the MOSFETs it is need 15V voltage but the signals from the microcontroller system are 0-5V so These signals converted to 0-15V to bias the MOSFETs using the Optocouples [4N35]. This operation controls the direction of rotation of the DC motor. On the other hand the PWM signal controls the speed of the DC motor. Pulse-width modulation control works by switching the power supplied to the motor on and off very rapidly. If the switching frequency is high enough, the output signal appears to be the average voltage of the PWM which is proportional to the duty cycle and the motor runs at a steady speed. By adjusting the duty cycle of the signal the average power can be varied, and hence the motor speed. In the logical circuit having three input from PIC16f877 to control the direction, delay and PWM signal. Speed of the motor is controlled from the Pulse Width Modulation (Pin RC) and direction is controlled from the signal of the direction pin (Pin RC3). When the direction of the motor is change, there should be a delay that should be higher than the time taken to discharge the capacitance of the MOSFETs. Otherwise the MOSFETs will burn due to switching MOSFETs are on simultaneously. So when the motor wants to change the direction, 5 microsecond delay is set from the pin C4. Through the logic circuit get out put as PWM clockwise and PWM counter clockwise. Those signals are sent to the H- bridge circuit. Interface Circuit User Interface circuit consist of 3 varistors, 6 push buttons, 3 indicators and finally the Liquid Crystal Display (LCD). Varistors are being used to vary Kp, Ki & Kd parameters of Fig 4 Logical and H-Bridge circuit

5 the controller. They are interfaced through the A/D module s channel 0- of the microcontroller. The push buttons are in 3x sets, where on set is used to increment, decrement & Enter the values of PID constants and the other set is used to issue START, STOP & RESET commands for the system. Switch s microcontroller side is grounded through a pull up resistor to prevent floating and hence creating unnecessary fluctuations. The LCD is used to view the parameters (Kp,Ki,Kd) systems states ( Working.. ) and messages( Please Enter ). States are also shown by the blinking LEDs on the panel as At work RED; Ready - GREEN. 3) User Interface: A user interface was designed, so that the user can carry out following functions; - Set, view and amend the values for K P, K I, K d - Reset the system - Indicate the system status - Give Starting command - Check the communicate status with the PC (RS 3) 16 x Parallel LCD with backlight is used to view all the changes being made (before starting) and will also display the status at operation. Input devices used are varistors & push buttons. In addition several indicator LEDs are used. All the instructions on entering values and other functions can also be viewed on the LCD. 4) Computer Interface It is more convenient to use a computer interface in the IP setup since it serves the requirement of varying the P, I and D values and observing the behaviour of the control theory through a single interface by the plot and the facility to compare the result with MatLab simulation result was arranged so user can get the good understanding of the controller behaviour. For the communication between the computer interface and the setup, serial communication, RF communication or Bluetooth technology can be used. The control circuit of the setup is designed to facilitate with serial communication. But, due to financial restrictions the RF unit required for RF communication is not installed in the setup. Controlling of the setup can be easily done through this computer interface. User can vary the P, I and D values respectively in text box provided in the interface if the user enter the wrong value the warning message will appear.the user can feed PID value to the control program by simply clicking the Send button. The Start button starts executing the control program and start to read the data from the controller and plot in the practical variation graph, A graph is plotted with the pendulum angle against time at the same time it is plot the practical simulation impulse response for respective P, I, D value So, the user can clearly see how the pendulum arm is behaving and how much time it takes to get balanced. By comparing these graphs which were plotted for different P, I and D values, one can get a better understanding about the effect of P, I and D values on the balancing of the pendulum arm, while the Stop button terminates the program and close the serial port. Optionally Clear button is providing to reset the plot graph and PID value. 5) System Integration Finally all the subsystems mechanical, electronic, controller and feedback network were gathered to produce the IP system. The individual modules (blocks) were solo tested, and their functionality according to specifications was also verified. Experimental test were fully performed and proved to be a success. D. Flowchart of algorithm. The overall flowchart of the Microcontroller algorithm is shown below [7]. III. Fig. 5 Flow chart of Algorithm RESULTS AND DISCUSSION The implementation of PID control method is done by adjusting the value of gain K p, K i, K d in order to get the best response of the system. Effects of tuning P, I and D values on system response, as simulated in Simulink, are summarized below. Fig 6 Effects of tuning P, I and D on response

6 Using the suitable value of K P =0, K I =50, K D =1, the pendulum s angle is satisfactorily achieved as shown in the following figures Figure 7 and Figure 8. The step response of the system is shown in the following Figure. The step starting time is 1 s and the step value is 100. Sample time is 0.5 s. A. Impulse Response The impulse response of the system is shown in the following figure. The settling time is 0.74 s. IV. CONCLUSION As a conclusion, the simulation results demonstrate that the inverted pendulum system can be stabilized successfully by implementing the PID control method. Furthermore, the simulation based tuning of PID controller proves that the apparatus can be used as a teaching tool to teach the characteristics of Proportional (P), Integral (I), Derivative (D), PI, PD and PID controls and their effects on control systems. Fig 7 Impulse Response for Pendulum s angle for K P =0, K I =50, K D =1 B. Step Response ACKNOWLEDGMENT We wish to acknowledge Dr. Harsha Abeykoon our project supervisor and to Dr. J.P. Karunadasa, Head of the Department of Electrical Engineering, University of Moratuwa for giving us technical advice and for guiding and inspiring us throughout the project to make all these project works possible. We are also thankful to all the academic and non academic staff of the Department of Electrical Engineering, University of Moratuwa for all the assistance given to make this project a success. REFERENCES [1] Williams, James H. Jr., Fundamentals of Applied Dynamics, John Wiley & Sons, New York, [] [] Harold Klee, Simulation of Dynamic Systems with Matlab and Simulink, University of Central Florida, Orlando, USA, pp , [3] [3] F C Teng, Real time control using Matlab Simulink, IEEE, pp , 000 [4] [4] Power Semiconductor Applications, DC Motor Control, Philips Semiconductors, February006. [5] [5] Martin P. Bates, Programming 8-bit Microcontrollers in C, 007 [6] [6] D. Ibrahim : Advanced PIC Microcontroller Projects in C, nd ed. Addison-Oxford, 008. [7] [7] John Charais, Ruan Lourens, Software PID Control using Microchip PIC, Microchip Technology.Inc, 004. Fig 8 Step Response for Pendulum s angle for K P =0, K I =50, K D =1