Modeling and Control Design for the Ball and Plate System

See discussions stats and author profiles for this publication at httpswwwresearchgatenetpublication4071069 Modeling and control design for the ball and plate system Conference Paper January 2004 DOI 101109ICIT20031290810 Source IEEE Xplore CITATIONS 52 READS 6257 3 authors including Amor Chowdhury University of Maribor 100 PUBLICATIONS 334 CITATIONS SEE PROFILE Rajko Svečko University of Maribor 66 PUBLICATIONS 378 CITATIONS SEE PROFILE All content following this page was uploaded by Amor Chowdhury on 02 March 2016 The user has requested enhancement of the downloaded file Modeling and Control design for the ball and plate system Andrej Knuplei Amor Chowdhury Rajko SveEko University of Maribor Faculty of Electrical Engineering and Computer Science Smetanova 17 SI2000 Maribor Slovenia andrej Imuplezunimbsi Article is describing a controller synthesis for two dimensional electromechanical system of the ball and plate intended for a study of the system dynamic and laboratory experiments with different proceeding of the control based on the classical and modern control theory The system includes the quadratic metal plate which is movably fixed in the center Its inclination can be changeable in two rectangular directions For the inclination of the plate a servodrive with a controller and two stepping motors are used 151 The control problem of the described system is to hold the freely rolling ball in the specific position on the plate For measuring the ball position the intelligent video system composed from CCD camera picture framing interface and program equipment for real time picture processing are used According to the characteristics of the object dynamic a discrete lead compensator was chosen as a controller 12 Performing experiments up to the sample of a ball on the plate the modern method of experimenting called Rapid Prototypingu 161 has been chosen The main aim was to pass over the phase of simulation up to the real object experiments with the least changes possible 1 Introduction The system of the ball and plate is a twodimensional electromechanical system intended for a study of the dynamic systems and laboratory experiments with different proceeding of the control based on the classical and modem control theory The system includes the quadratic metal plate which is movably fixed in the center Its inclination can be changeable in two rectangular directions For the inclination of the plate a servodrive with a controller and stepping motor is used SI The control problem of the described system is to hold the freely rolling ball in the specific position or in the movement on specific trajectoy on the plate For measuring the ball position the intelligent video system composed from CCD camera pictureframing interface and program equipment for processing of the picture in real time is being used The interface to control the stepping motors and the pictureframing interface are being made as standard ISA cards which enable the system control with the personal computer PC The connection between the PC and the ball and plate system is enabled through the MATLAB program package Since the characteristics of actuators and sensors demand work in the realtime the program RealTime Toolbox that allocate with help of additional sorter the processor time to the operating system and to the program performing in the real time have been added The pictureframing interface receives the analogous video signal which digitizes with A D converter and saves as datasheet into storage A computer determines from the received data the position of the ball and calculates the difference between reference and real value on which basis the controller generates a signal that is needed for the drivinggear of the stepping motors According to the characteristics of the system dynamic a discrete phaselead controller has been chosen 2 Performing experiments up to the sample of a ball on the plate the modern method of experimenting called Rapid Prototypinga has been chosen The main aim was to pass over the phase of simulation up to the real object experiments with the least changes possible 2 Modeling and linearization of the object The starting point of modeling is the Lagranges equation d a w awaw 1 dt aii aqi aqj The model has four degrees of freedom two at ball moving and two at plate inclination The two coordinates regarding the plate are marked as x and y and the plate inclination is marked as a and p The plate inclination is caused by two rotational torques M and M having an effect on the plate Figure 1 With derivation of the EulerLagranges equation we get the system of nonlinear differential equations who are mathematically describing system dynamic By the simplification it has been considered that The ball never loses contact with the plate Theres no slithering between the ball and the plate All friction forces and rotational moments are being neglected IClT 2003 Maribor Slovenia 0780378520103151700 02003 IEEE 1064 The plate angle and the surface limitations are not taken into consideration Beside above mentioned it has been also considered the fact that none step could be lost using the stepping motor which means that skipping between steps doesnt occur and that an amplitude of plate moment doesnt affect the position of the rotor Figure 1 The scheme of the ball on the plate system The proportion between nonlinear term for centrifugal force and the activity of the gravity force in the specific direction depends on maximum angular velocity of the inclined plate In our case we can obtain the constant angular velocity with the amplitude which depends on driving frequency of the stepping motor Since the proportion is about 125 using the 400 Hz stepping motor frequency the centrifugal force could be neglected The position of the plate in the stable position is horizontal which means that both angles of inclination are equal to 0 If angles of inclination deviate up to fY we can approximate the sinuous function with sinx I x and for further analysis we can use the simplified model of the mechanical part of the system 3 Controller design A starting point is the linear object model 4 for which we wish to establish the controller with lead compensator structure The transfer function of the closedloop system with the lead compensator structure is When choosing or defming controller parameters the first condition is stability and thats why the analysis of stability Ts with Rouths criterion has been done I On the basis of stability criterion it is possible to specify the areas in which controller parameters could be situated gain KR controller zero Sand parameter a that defmes the controller poles position Controller gain Kn is bounded with the eauation Considering the expression 10 and conditions yKGiw and a 0 l then a next expression have to hold 8 From the expression 8 it follows that the minimum time constant of controller zero is y a y 6 0 5 5 7 7 P g s m a Ka in j g s m p I K J Figure 2 The block scheme of the linear object model Associating the linear models of electrical and mechanical subsystems we obtain the transfer function describing the linear object model Figure 2 The zero of the controller can so be set on the closed interval 0 To specify the controllers pole position an interval in which is situated should be found According to Shannons theorem the sampling frequency should be at least twice higher than the highest frequency of the sampled signal When time of sampling is 40 ms we obtain 10 1 ri0042008 5 From the expression 10 we calculate the value a To fulfill the condition 7 a next expression must be true Since parameters6 and y are positive from where it follows that a can assume the highest value y a y 6 0 12 13 7 a I s 1065 We obtain an interval in which we search the value fof parameter a 6 Y 0 6 aaa 3 a 14 The area of changing parameter a and consequently positioning controllers pole depends on chosen controller zero 6 We obtain two straight lines Figure 3 the valid area is on the interval where the straight line a lies above the straight line G The next step is to calculate the controller gain KRmo f a S where the system is still stable over the area a a a We obtain a plane defining the maximum controller gain at chosen values a and 6 Figure 4 I I I Figure 3 Representation of an area in which a can be in motion according to the controller zero position The system is stable for all values a d and KR situated in the space OfaSK For optimization of parameters the criterion index of ISE Integral of Square Error has been used 41 The characteristic of ISE is because of squaring to increase an effect of bigger signals and to reduce an effect of smaller error signals Thats why the controlled systems for which we optimize the controller according to this criterion are quick and have pretty large gradient at the stepping change of reference which is the consequence of high square error value in the beginning of transient Figure 4 Maximum allowed controller gain according to the chosen controller zero and pole position When searching the controller that would be optimal in the ISE sense its calculation needed to be done in the most points possible inside spaceFK SaSS The Brute Forcea method performed with Matlab program has been used for searching the criterion index minimum Figure 5 represents a chart of the criterion index in dependence on KR and parameter a Figure 5 Chart of the criterion index in dependence on Kn and parameter a The final result of an algorithm is a controller transfer function at the minimum ISE index value 15 147935 209967 z 06095 K 4 Experiment In the experimental phase the so called Rapid Prototyping method 6 has been used Rapid Prototyping of control systems is generally referred to as methods and tools to solve control problems fast and efficiently This includes theoretical methods of control engineering from designing a system model and system analysis to the design of a control strategy test and optimization of the control strategy in a simulation environment automatic code generation for a realtime system operating a test stand and verification and optimization of the controller with the target object on the test stand The experimentsimulations part has been performed with MatlabSirnulink program package and for the real object experimentations the additional Realtime Toolbox has been used The Realtime Toolbox is a tool package associating the Matlah and the Simulink with the outside world to attain the data from environment in real time their immediate treatment with the Matlab and the Simulink orders and models and then sending them back to the outside world The setting of performing concept a driver is simply called by inputoutput device orders without knowing its details 1066 The Realtime Toolbox translates the Simulink diagram Figure 6 into a program code and then by using a specific object translator into an executive cod 67 h e w 0 I Figure 6 Controlled system scheme with MATLAB SIMULINK program package for one coordinate I 1 A o 5 3s I i Figure 7 Response on a step reference signal 8Ww The simulation and actual controlled system results obtained are in accordance with appointed demands and expectations Figure 7 and 8 The stationary error is a consequence of an object nonlinearity irregular ball geometry and of a sensor resolution limitation 5 Conclusion Choosing the controller structure is an important question broached already in the first beginning of the control design Since an object already has a double pole in the coordinate starting point an additional pole would only make the stability circumstances worst the usage of PID regulator is not recommendable When choosing between the compensating controllers a lead controller is relevant while a lag controller does not stabilize the system The usage of the lead controller has been established to be the best solution The next question is How to specify or calculate the coefficients of the chosen controller On the basis of the controlled system stability analysis with Rouths criterion and in combination with ISE criterion index a method of the compensation controller coefficient optimization has been derived and it has been confirmed as a successful tool to resolve a problem described References D DonlagiC T ZoriE R SveEko Diskrefni regulacijski sistemi Univerza v Mariboru TehniSka fakulteta Elektrotehnika raEunalniStvo in infonnatika Maribor 1992 B ZupanEiE Zvezni regulacijski sistemi 1 del Univerza v Ljubljani Fakulteta za elektrotehniko 1996 D Matko Diskretni regulacijski sisfemi Univerza Edvarda Kardelja v Ljubljani Fakulteta za elektrotehniko 1986 A Chowdbury Robustna sinteza regulacijskih sistemov z upoitevanjem performaninih kriterijev doktorska disertacija Univerza v Mariboru Fakulteta za elektrotehniko raEunalniStvo in informatiko 2001 HUMUSOFT CE151 BallPlate Apparatus Users Manual HUMUSOFT Real Time Toolbox for use with MATLAB Users Manual MatlabSimulink User Guide Figure 8 Response on a sinus reference signal 1067 View publication stats