Documentation for the Bytronic Pendulum Control System

Transcription

1 Documentation for the Bytronic Pendulum Control Sytem Verion 3.0 The Courtyard Reddicap Trading Etate Sutton Coldfield Wet Midland B75 7BU ENGLAND Tel : +44 (0) Fax : +44 (0) ale@bytronic.co.uk Webite :

2

3 Copyright 00 Bytronic International Ltd. All Right Reerved. Documentation for the Bytronic. Thi manual, a well a the oftware decribed within, i furnihed under licene and may only be ued or copied in accordance with the term of uch licene. The content of thi manual i furnihed for information ue only, i ubject to change without notice and hould not be contrued a a commitment by Bytronic International Ltd. Bytronic International Ltd. aume no reponibility or liability for any error or inaccuracie that may appear in thi manual. Except a permitted by uch licene, no part of thi publication may be reproduced, tored in a retrieval ytem, or tranmitted in any form or by any mean, electronic, mechanical, recording or otherwie, without prior written permiion of Bytronic International Ltd. Bytronic acknowledge all copyright and trademark.

4 Table of Content Table of Content Getting Started.... Uing the PCS a an Analogue Control Sytem... Uing the PCS a a Digital Control Sytem...4 Overview: The Pendulum Control Problem.... Fundamental... Two mode: winging crane and inverted pendulum... Calculation and intability of y for inverted pendulum...3 Analogue and digital modelling...4 Labwork Labwork : Static and Dynamic Characteritic of the Apparatu Introduction Carriage Servo Effect of Servo Gain on Hyterei Tranient Repone of the Servo Subytem Dynamic Model of the Pendulum Modelling of the Carriage Servo Labwork : Analogue Control of an Inverted Pendulum Apparatu Introduction Meaurement of the Ma Poition Setting up the Pendulum Software Linear Control of the Pendulum Software Harmonic Control of the Pendulum Tranfer Function for the Inverted Pendulum Stabiliation uing Phae Lead Compenation...3. The Effect of Hyterei in the Servo Sub-Sytem Bytronic International Ltd

5 Table of Content Labwork 3: Fuy Control of the Inverted Pendulum Apparatu Introduction Decription of the Fuy Controller Fuification Rule Defuification Ue of the Fuy Controller Labwork 4: Direct Digital Controller Deign and Implementation: Inverted Pendulum Apparatu Introduction Deign of a DD Controller uing a Simplified Model...3. Deign Criteria and Sample Rate Selection...3. Pole Placement Implementing the Controller uing the PCS oftware Further work Labwork 5: Direct Digital Controller Deign and Implementation: Swinging Crane Apparatu Introduction Sample Time and Deign Objective Dead-Beat Controller Deign uing Simplified Plant Model Ue of Dead-Beat Controller Frequency Repone of Plant and Compenator Dead-Beat Controller uing Plant Model with Damping Ringing Pole Problem Correct Deign Approach to Avoid Ringing Pole Problem Bytronic International Ltd

6 Table of Content Appendice Appendix : Recalibrating the Angle θ Servo Potentiometer...4. Adjuting the Servo Potentiometer...4. Appendix : Reetting the X Poition Potentiometer...4. Appendix 3: Derivation of Plant Controller Appendix 4: Technical Specification Appendix 5: Header Pin-out Detail Connector : 6 Way IDC Microcomputer Header Appendix 6: Recommended Text for Block Diagram, Laplace and Z Tranform Appendix 7: MATLAB Solution for DDC Controller Inverted Pendulum Crane...4. Table and Diagram Table PCS Analogue Channel...5 Figure Linear Pendulum Controller Connection... Figure Inverted Pendulum Compenator...3, 3. Figure 3 Determination of Ma Poition... Figure 4 The Pendulum Control Module...3. Figure 5 Mounting of the X Poition Potentiometer...4. Bytronic International Ltd

7 Getting Started Getting Started The PCS conit of two eparate module linked by a connecting cable: the Carriage Module and the Control Module. The Carriage Module conit of a carriage that carrie a pivoted rod and ma. Thee are driven along a 500mm track by a dc ervo motor and toothed belt. The motor ha an integral tachometer. When tanding upright the carriage unit behave a an inverted pendulum. When turned upide down, the rod and ma repreent the lifting block of an overhead crane. In both mode the overall behaviour of the pendulum i a combination of linear and ocillatory behaviour. When the pendulum i upright thee two mode are coupled, and the behaviour of the pendulum i bet decribed by the poition of the pendulum bob. When the pendulum i turned upide down thee two mode are decoupled, and the behaviour of the pendulum i bet decribed by the poition of the carriage and the angle of the pendulum. The carriage poition and the attitude of the rod/ma aembly are meaured by potentiometer. The poition of the pendulum bob i calculated in hardware. The Control Module include a clear mimic diagram of the overall ytem and connection to the control/meaurement ignal are via eaily acceible 4mm colour coded ocket. The colour coding i a follow: Red terminal are analogue input and output. Blue terminal are monitoring point for feedback ignal. Yellow terminal are the compenator connection point. Green terminal are ground. The ha two mode of operation, namely tand-alone in analogue mode or linked to a computer in digital mode, a dicued in the remainder of thi manual. Thee control mode are quite ditinct. To begin, we recommend you tart with the impler analogue mode o a to gain undertanding of the control problem. Bytronic International Ltd.

8 Getting Started Uing the PCS a an Analogue Control Sytem The following procedure will allow a quick et-up of the PCS a firt a linear poitioning control ytem and then an inverted pendulum balancing ytem.. Poition the equipment on a dek with the carriage module in the inverted pendulum poition (the carriage to the bottom of the unit and the motor to your right).. Poition the et point potentiometer to the middle (0V). 3. With one of the 4mm lead upplied, connect the et point to the ervo amplifier input terminal a hown in Figure. 4. Turn on the power. Figure : Linear Poition Controller Connection 5. The ytem i now configured a a linear poition controller. The poition of the et point potentiometer will determine the poition of the carriage aembly along the x axi. 6. To achieve optimum poition control, the ervo amplifier gain and velocity feedback mut be calibrated on the control panel. Thi procedure i covered in detail in Labwork. Approximate value to get the ytem working are maximum for the ervo gain and three quarter maximum etting for velocity feedback. 7. Now the ervo ytem i et-up we may begin to balance the pendulum. Diconnect the ervo input from the et point and temporarily connect the ervo input to GND to prevent noie pick-up. Bytronic International Ltd.

9 Getting Started 8. To balance the pendulum the variable a, gain and the compenator all need to be deigned and calculated. The procedure to do thi i dicued in Labwork. Approximate value to get the ytem going are: a) Ma poitioned at the top of the rod. b) Set a to.7. c) Set gain to.. d) Switch the gain to negative. e) To allow quick et-up, preet compenator component are upplied with the PCS. Carefully plug thee module into the compenator yellow 4mm terminal a hown in Figure. Figure : Inverted Pendulum Compenator 9. Hold the pendulum upright in the centre of the track. Connect the controller output to the ervo input uing the 4mm connecting lead. Now, gently let go of the pendulum, and the pendulum hould balance in the centre of the track. 0 a) Nudge the balancing weight and ee the controller compenate and regain balance. b) Try adjuting the gain value from ero to 3.0 and find out it effect on the balancing operation. If the pendulum will not balance with the uggeted value, gently move the rod to the upright poition and the controller hould take over. If not, check all connection and etting, epecially the compenator a hown in Figure. Bytronic International Ltd.3

10 Getting Started Thi ection ha been a imple introduction into the operation of the PCS in it analogue control mode. Many of the ubtletie of thi control problem have been left out. To fully invetigate the et-up value and compenator deign, pleae refer to the Labwork. Uing the PCS a a Digital Control Sytem The PCS control module ignal are available for ue with any external controller via the 4mm connector. To facilitate PC digital control of the PCS, a ocket at the rear of the control module allow the meaurement ignal to be tranmitted directly to a Bytronic AD/DA Interface Board (MPIBM3) located in a PC expanion lot. In addition, the control ignal from the PC i alo routed via thi cable to drive the input of the carriage ervo amplifier that in turn control the motor. To achieve digital control of the PCS pleae follow the procedure outlined below-. Enure the power to the microcomputer ytem and the PCS are both off.. Intall the MPIBM3 card into a pare PC expanion lot and et both the ADC and DAC channel to operate on a ±0V bipolar operation, a decribed in the MPIBM3 manual. 3. Remove all 4mm cable that may be connected to the control panel. 4. The MPIBM3 card may be connected to the pendulum unit by either: a) Direct connection uing the MPIBM3 6 way ribbon cable into the rear of the control conole; or b) The MPIBM3 6-way cable may be connected to the MPIBM3A crew terminal board. Thi will allow more flexible connection of the analogue channel to the Control Module 4mm terminal. Note The MPIBM3A crew terminal board mut be purchaed eparately. 5. Power-up the PCS and then the microcomputer ytem. The carriage aembly hould poition itelf in the centre of the track and the Bytronic International Ltd.4

11 Getting Started microcomputer power-up a normal. If not, power down immediately and check all the connection. 6. Run the PCS oftware. When uing the Bytronic MPIBM3 card with the PCS the analogue channel uage i given in Table. MPIBM3 ANALOGUE Function INPUT CHANNEL 0 Poition Y Poition X Slider Poition 3 Angle theta 4 Tachometer Feedback (dx/dt) MPIBM3 ANALOGUE Function OUTPUT CHANNEL 0 Carriage Servo Input Table : PCS Analogue Channel When connected to the MPIBM3 card thee ADC input channel and DAC output channel can be viewed directly uing the Internal Interface Card oftware. Pleae refer to the manual upplied with the MPIBM3 interface card. Bytronic International Ltd.5

12 Overview: The Pendulum Control Problem Overview: The Pendulum Control Problem Fundamental The poition of the pendulum bob, y, the poition of the carriage, x, and the angle of the pendulum, θ, are related by the equation: y x + L inθ (equation ) where L i the effective length of the pendulum, the ditance between the pivot and the centre of ma of the combined pendulum and bob. Figure 3: Determination of Ma Poition Bytronic International Ltd.

13 Overview: The Pendulum Control Problem Thi tell u that in any mode the dynamic of the poition of the pendulum bob, y, i a combination of linear dynamic, x, and ocillatory dynamic, L inθ. Linear and ocillatory quantitie poe quite different dynamic propertie. Linear behaviour with error reduction require feedback. In feedback, ocillatory behaviour will either be amplified or damped. To analye thi repone need a frequency repone tet. A uch the pendulum i a very difficult control problem. Two mode: winging crane and inverted pendulum The pendulum provide two control problem: inverted pendulum (upright, bae on ground), and winging crane (turned over, pendulum hanging). The table behaviour of the pendulum in the two cae i fundamentally different. Conider the two variable x and θ. In the cae of the inverted crane x and θ can be varied independently and the crane i till table: move x, and θ will return to ero, change θ and x will not be affected. By contrat, with the pendulum inverted and table, any mall change θ in θ require a adjutment x in x, a x ha to be adjuted to keep the pendulum upright. Variation in x are dependent on θ. However, the revere i alo true: variation in θ are dependent entirely on x. Any adjutment x in x require a mall change θ in θ. Thi mean that for the inverted pendulum it i not meaningful to talk about x and θ a independent variable. Thi ha the conequence that the decription and dimenion of the control problem of the pendulum are different in the two mode. In the crane mode x and θ are independent variable and o the poition of the pendulum i decribed by ( x, θ ). Thi i two-dimenional. By contrat in the inverted pendulum mode x and θ are not independent variable. Any attempt to balance the inverted pendulum in term of x and θ will have to take into account all the mode of interaction between x and θ a well a the value of x and θ themelve. However, we only conider the table control problem. So long a the pendulum i in balance we can talk about a ingle independent variable, the poition of the pendulum bob or y. Thu the appropriate variable for control for the inverted pendulum i y alone. Thi i one-dimenional. Bytronic International Ltd.

14 Overview: The Pendulum Control Problem Calculation and intability of y for inverted pendulum We hall now conider the inverted pendulum mode alone for a moment. In the inverted pendulum mode, o long a the pendulum i balanced, the behaviour of θ i mall angle (θ maller than about 5 or o) and we can apply the mall angle approximation: in θ θ (with θ meaured in radian). Subtituting into equation we obtain: y x + Lθ (equation ). A L i fixed in any one control application, thi mean we can quickly calculate y in term of x and θ, which are meaurable. Hardware analogue control require that thi calculation i performed in analogue term, and in fact the Control Module perform thi calculation in analogue voltage. If V x i a voltage repreenting x, and V θ a voltage repreenting θ, then V θ can be caled by a factor a (uing an op-amp with a variable reitor to change the multiplication factor) and then ummed with V x uing a umming junction to give a voltage V y repreenting y on the ame cale a V x. Thu: Vy Vx + av θ, a voltage implementation of equation, with the factor a caled to repreent the value of L (ee Figure 4 on page 3.). Thi i the method ued to give the voltage repreenting y which i available from junction L on the Control Module. Note that though thi voltage um i exact, the value V y i an approximated repreentation of y, becaue equation i an approximation. In the inverted pendulum mode, the pendulum i fundamentally untable. Thi can be een with a imple gain model. Suppoe we imply amplify the error r y of diplacement from the et-point by a gain g a the control ignal for the poition of the pendulum carriage, x, enuring that the ignal i 80 out of phae. A block diagram can repreent the ytem a follow: Bytronic International Ltd.3

15 Overview: The Pendulum Control Problem With G C g. (Y i the meaured value, R the et point, E R Y the error, G C i the controller gain, G p the plant gain and X the output). Thu X Gc ( R Y ). Subtituting our gain function into thi equation we obtain x g( r y). (Here capital letter repreent function in the block diagram; lower cae letter repreent real variable). Taking r a ero for implicity, we ee x will move in the ame direction a y (negative gain; poitive feedback a minu time a minu equal a plu), amplified by an amount g. Thinking linearly, for the pendulum to balance, clearly we require g : the carriage mut move at leat a far a the bob or the pendulum will fall over. Thinking harmonically, for ocillation to be damped (or at leat not be amplified) we require g. Thu, the only poible value of g that can balance the pendulum i preciely. Clearly, thi i impoible to attain in practice, and the lightet deviation will, a we will ee, lead to intability. The dynamic of the inverted pendulum are thu fundamentally untable. Analogue and digital modelling A a control ytem, the pendulum may be repreented by the cloed-loop diagram: Where G C i the controller gain (the controller), G p i the plant gain (the pendulum and aociated hardware), R i the et-point, E R Y the error ignal, X the output ignal and Y the meaured value. Thi i a cloed loop control ytem. Bytronic International Ltd.4

16 Overview: The Pendulum Control Problem For any block in a cloed loop ytem the ignal tranformation or tranfer function, H, of a gain i defined in term of the output ignal, O, and the input ignal, I, by: Thu: O H. I Y G p and X X G c. E Defining the tranfer function of the entire feedback loop a F and uing the block diagram relation for a feedback loop we obtain: F Y R GCG p + G G c p (for further explanation refer to a textbook on block diagram). Thi relation can be rewritten a: G C G p F F. Thee relation will be ued later. For the inverted pendulum, the ignal are a in the previou ection: R r, X x and Y y. For the crane, R (r,0), X ( x, θ ) (output to pendulum) and Y ( x, θ ) (input from pendulum). In particular, θ i under control becaue it value will tend to ero (damped by friction; to fixed point defined by gravity). The fundamental teaching point of the i the relation between the linear and ocillatory mode of behaviour and the conequence of thi for control. The Labwork dicu the relation between the linear and ocillatory mode through variou poible control trategie, and two technique are ued to tackle the relation between the linear and ocillatory mode. Thee are Laplace tranform and Z tranform. Laplace tranform are a method of analying continuou ytem and a uch are particularly ueful for dicuion of analogue control. Z tranform analye dicrete ytem and a uch are particularly ueful for developing Direct Digital Control. The Laplace and Z tranform are related and reference i made to thi in the Labwork. Only outline dicuion of the role and propertie of block diagram, Laplace tranform and Z tranform will be given here. For further detail, a reference uch a thoe given in the Appendix 6 will be required. Bytronic International Ltd.5

17 Overview: The Pendulum Control Problem Each technique i generally aociated with a ditinguihing variable. A lower cae i generally ued a the ditinguihing variable of Laplace tranform, and Z tranform are generally aociated with a lower cae. In each cae, when we aociate the variable with the control ytem we write the ytem in term of that variable: or Analyi of the ytem can then take place in term of the block diagram relation and the propertie of the tranformation. If then (block diagram relation) G + 0. ( ) c X E + 0. ( ) o (propertie of the Laplace tranformation; repreent differentiation with repect to time) dx de X 0.0 E + 0. dt dt +.. Bytronic International Ltd.6

18 Overview: The Pendulum Control Problem Thi controller function i ued in Labwork. Similarly for Z tranform, if G c ( ) 3.7( 0.73)( 0.98) ( )( 0.5) then (block diagram relation) X E 3.7( ) ) ( o (propertie of Z tranformation; equation index) repreent unit drop of difference x.5x + 0.5x 3.7( e.694e 0.699). i+ i+ i i+ i+ + Shifting indice, we can then derive the output from previou output and error value. Thi controller function i ued in Labwork 5. In each cae, we have taken a ytem and created a repreentation of that ytem. Derivation of G C and G p in term of and are included in the Labwork for both the crane and the inverted pendulum mode. Dynamic analyed in the particular repreentation will generate valid olution for the original ytem o long a we tay within the parameter of that repreentation. The Laplace tranform require, for example, ero initial condition. The Laplace condition mean in practice that we have to hold the pendulum at the tart of analogue control to enure balance. If the pendulum i not upright the controller can ometime achieve balance; but thi i not guaranteed. The Z tranform require decription of the complete ytem, in particular thi mean that the ero order hold mut be included in the decription of the ytem. The conequence of the need to include the ero order hold are dicued below. In term of the mathematic, the Laplace and Z tranform are cloely related, and the Z tranform i uually calculated from the Laplace tranform. Direct method for doing thi are given in the Labwork. Conceptually, nonethele, the proce i rather more complicated. In moving from an to a tranform we are moving from one repreentation, and aociated condition, to another repreentation, and aociated condition. In other word, we are loing one contraint (the requirement for ero initial condition, that the pendulum be held upright), and impoing another. The new contraint i that we mut model a complete decription of the ytem. Bytronic International Ltd.7

19 Overview: The Pendulum Control Problem For mot ytem, we approach dicrete ( ) control a an approximation to continuou ( ) control: we ramp the ample time up a high a poible and aume everything i mooth, and continuou. We are implicitly making a move into the Z repreentation, finding a controller, and moving back into the Laplace repreentation. Thi movement i valid o long a there are no propertie of the ytem which preclude returning from the dicrete model to the continuou. For table, linear ytem, there are no uch propertie, and thi approach i valid and i often done without any regard to the underlying proce. For harmonic ytem, that i ytem with ocillatory dynamic, thi i not the cae. To undertand why, we mut conider the ADC (Analogue-to-Digital Converter). The ADC i the electronic unit which convert analogue electrical ignal to digital number for ue by the computer. The ADC necearily ha a ample time. Between thee ample intant the ignal i aumed to be tatic. The true ignal, of coure, could move ignificantly within that ample interval, and the computer would never know. The ADC thu introduce a ero order hold into the ytem: For a ytem with ocillatory dynamic, even a table one, the ero order hold cannot be neglected, becaue the error introduced within a ingle ample interval may be enough for the ytem to become untable. Indeed, any ample interval of any ie, any delay at all, may be enough for the ytem to become untable. Thi i becaue any ocillatory ytem will amplify ome ignal and damp other. In term of the Z tranform, the ero order hold affect the radiu of convergence of the tranform.. Bytronic International Ltd.8

20 Overview: The Pendulum Control Problem Within the radiu of convergence the tranform i table; outide it i untable; and the ero order hold affect thi. A practical example of the mathematic may help. In Labwork we derive a Laplace controller for the inverted pendulum of Thi can be implemented in hardware, and work. Tranforming thi controller into a differential equation and olving, a would be required in oftware, we obtain a controller of the form: de t ( E) e 00 dt + lower order term. dt 00t It i immediately apparent that the e dt term will explode for any dicrete time period (any range of t to be integrated over where the interval i non ero). Thu the ero order hold cannot be ignored when defining the Z tranform. Bytronic International Ltd.9

21 Labwork LABWORKS The following labwork begin with familiariation of the Pendulum Control Unit and it dynamic characteritic. A the labwork progre, they cover control of the pendulum control unit by both analogue and direct digital control method. In the labwork, variou connection need to be made on the control conole. To eae undertanding, we have referred to the 4mm terminal and the control knob a hown in Figure 4. Bytronic International Ltd 3.

22 Labwork Figure 4: The Pendulum Control Module Bytronic International Ltd 3.

23 Labwork Labwork : Static and Dynamic Characteritic of the Apparatu PCS, two voltmeter, ignal generator, ocillocope or PCS oftware and Analogue I/O card (MPIBM3). Introduction In thi experiment you will:-. Familiarie yourelf with the PCS rig.. Examine the characteritic of the carriage ervo. Tune it gain and the degree of velocity feedback to optimie it performance. Obtain a dynamic model of the ervo. 3. Examine the dynamic characteritic of the pendulum and hence model it dynamically. Carriage Servo Place the rig in the inverted pendulum poition and uncrew the pendulum rod. Diconnect all the link on the Control Module. Connect the output of the et-point potentiometer (ocket A) to the input of the carriage ervo (ocket H). Poition the et-point potentiometer (P) o that it i at it mid-range. Adjut the ervo gain potentiometer (P3) and the velocity feedback potentiometer (P4) o that they are about midrange. Switch on the power to the apparatu. Bytronic International Ltd 3.3

24 Labwork Move the et-point potentiometer to and fro and oberve the carriage change poition. Uing the two voltmeter, meaure the et-point voltage and the carriage poition voltage, V x (ocket J or J). Increae the etpoint uniformly from 0 to +0V and then back down to 0V and finally back to 0V in tep of V. Plot a graph of the ervo poition againt etpoint voltage. Make ure you alway move the pot in the required ene. If you move it too far one way, move the et-point potentiometer back and then approach the required value lowly. a) What i the ueful linear range of the carriage ervo? b) What i the hyterei of the carriage ervo expreed in volt? c) What i the enitivity of the carriage ervo expreed in volt/volt? The non-linear behaviour (clipping characteritic) ha been introduced deliberately to prevent the carriage hitting the end top too hard and damaging the rig. Effect of Servo Gain on Hyterei Diconnect the ervo input (ocket H) from the et-point potentiometer and connect the ervo input to ground. Puh the carriage to the right with your hand and gently releae it. Make a note of the carriage poition voltage. Repeat, puhing the carriage in the oppoite direction and letting go again. Hence calculate the hyterei of the carriage ervo in volt. Compare thi figure with that obtained in b above. Reduce the ervo gain pot to about 5% of it range and repeat the tet. Finally put the ervo gain to maximum and meaure the carriage hyterei once more. What concluion do you draw about the effect of ervo gain on the amount of hyterei preent in the carriage ervo? Can you explain the reaon for your obervation? Tranient repone of the Servo Subytem Reduce the velocity feedback to ero. Ue a ignal generator to apply a +V quare wave of frequency H to the input of the carriage ervo. Oberve the carriage poition on an ocillocope (ocket J). Adjut the ervo gain until the repone how a mall overhoot of about 5% (damping ratio of about 0.7). Now increae the ervo gain to a maximum and adjut the velocity feedback pot (P4) o that the repone i imilar to the one oberved previouly. Bytronic International Ltd 3.4

25 Labwork Which of thee mode of operation i preferable? Explain your reaon. From now on leave the gain of the carriage ervo at a maximum and the amount of velocity feedback at the value that produced a tep repone with a light overhoot. Dynamic Model of the Pendulum In thi part of the experiment we will model the dynamic behaviour of the pendulum. We hall do thi by oberving the tranient repone of the ytem from an initial value. Before fitting the pendulum rod into the carriage, poition the ma at the end of the rod. Etimate the poition of the centre of ma of the rod/ma aembly by trying to balance it on your finger. (The centre of ma i located jut below the bottom of the ma). We hall call thi length the effective pendulum length L. Screw the pendulum rod firmly into the carriage. Tip the rig upide down into the crane poition. Connect the pendulum angle ignal, V θ, to the ocillocope. Alternatively the PCS oftware can be ued to log V θ. Connect the Control Module to the computer a decribed in Getting Started. In the PCS oftware elect Derive Characteritic from the Controller menu. Poition the Carriage Module a for crane mode. Pre Start and the input value will be plotted on a graph for you. You can elect V θ alone by unticking the other tick boxe on the right hand ide of the graph, leaving jut the TH (θ ) box ticked. Alo the box on the left hand ide of the creen in the controller area will log the time and extent of point of maximum diplacement. Diplace the pendulum by about 30, releae it and record the tranient repone. Record the magnitude of the point of maximum diplacement on either ide. After about 0 cycle top the logging. Etimate the period of ocillation, T. From thee value we can calculate the logarithmic decrement. Thi i calculated according to the following formula: α k+ ln( m k / m ) where m k i the magnitude of the kth overhoot. If V θ i not equal to ero when the rod i hanging motionle, then you will need to recalibrate the theta potentiometer a decribed in Appendix. Bytronic International Ltd 3.5

26 Labwork The value of the period can be calculated theoretically from the baic dynamic of a imple pendulum. The well-known formula for the period L i π. Do thi calculation uing the effective centre of ma obtained g earlier and compare the period obtained with that oberved experimentally. From the logarithmic decrement the damping ratio, ζ, may be calculated uing the formula: ζ α α. + 4π Thi ytem i very lightly damped, o in order to arrive at a figure for the damping ratio, it i bet to plot a graph of ln( m k ) veru k. The logarithmic decrement, α, i twice the lope of the graph. Suppoe for example the logarithmic decrement i 0.0, then the damping ratio work out to be You may need to modify the damping ratio figure lightly to obtain a better overall fit to the oberved repone. Auming that the damping ratio i very mall, the natural frequency, ω n, π can be deduced from the period, T uing the approximation ωn. T Suppoe T i.0 then ω n i approximately 6.6 rad/. Thu, with a natural frequency value of 6.6rad/ and a damping ratio figure of , the pendulum can be modelled approximately a a econd order ytem: G( ) ζ + + ω n ω n, i.e. G( ) Bytronic International Ltd 3.6

27 Labwork Modelling of the Carriage Servo We will now obtain a tranfer function for the carriage ervo. Thi can be ued to obtain more accurate model of the pendulum dynamic. To do thi properly, you really need a frequency repone analyer, but the method decribed here i a imple method for obtaining an approximate econd order model. Firt we are going to determine the bandwidth of the carriage ervo. Apply a H inuoidal ignal with a peak to peak amplitude of volt to the input of the ervo (ocket H). Oberve the carriage poition voltage on an ocillocope. You may ee a light ditortion becaue of hyterei that i preent, but the amplitude will be nearly the ame a the applied ignal, ie the ervo-ubytem ha unity low frequency gain which confirm the reult obtained in c above. Increae the frequency of the applied ignal until the oberved voltage ha reduced to 0.7V, ie of the low frequency value. The frequency at which thi occur i the bandwidth of the ytem, ω. b For a ytem with a damping ratio of 0.7, the natural frequency, ω n, i equal to the bandwidth. Baed on thi approximation, calculate the natural frequency, ω n. (Don t forget to work in rad/). Hence deduce the tranfer function of the ervo-ubytem auming that the damping ratio i 0.7. A an example, a ytem whoe bandwidth i 9.6H and a damping ratio of 0.7 ha the approximate tranfer function: G( ) Bytronic International Ltd 3.7

28 Labwork Labwork : Analogue Control of an Inverted Pendulum Apparatu PCS, PCS oftware, Analogue I/O card (MPIBM3). Introduction A imple pendulum wing naturally into an equilibrium poition with the centre of ma below the pivot. In order to balance the pendulum in an inverted poition the pivot mut be continuouly moved to correct the falling pendulum. Thi intereting control problem i fundamentally the ame a that involved in rocket or miile propulion. The rocket ha to balance on it engine a it i accelerated. A the rocket tend to fall over the engine thrut mut be deflected ideway to retore it coure. Meaurement of the Ma Poition The controlled variable in thi ytem i the poition of the pendulum ma, y. Unfortunately y cannot be meaured directly but it can be calculated imply from the meaurement θ and x a y x + L inθ (ee Figure 3 on page.). For mall angle the above equation approximate to with θ in radian. y x + Lθ Bytronic International Ltd 3.8

29 Labwork The voltage from the angle potentiometer, V θ, i caled by a factor a (potentiometer P5), and added, by mean of an operational amplifier circuit, to the voltage from the carriage potentiometer, V x : Vy Vx + av θ. The factor a can be adjuted to take account of different value of L. Setting up the Pendulum Diconnect all the lead from the PCS conole including the 5 way cable. Make ure that the carriage ervo gain i at a maximum and the velocity feedback potentiometer i at the value determined in Labwork. Alo enure that the ma on the pendulum rod i in the ame poition a in Labwork. Connect the et-point pot to the carriage ervo input and check that the ervo i working. Poition the carriage near the centre of the track and crew the pendulum rod into it pivot. Support the pendulum in an upright poition by uing threaded knob and upport bracket locating the end of the rod into the clearance hole machined in the pendulum weight. Tighten the wing nut onto the inide face of the upport bracket to remove any lackne. With the rod / ma aembly held teady at it centre of gravity gently move the carriage from ide to ide uing the et-point potentiometer. Monitor the voltage V y (ocket L). A thi voltage repreent the poition of the ma, it hould not vary a the carriage move. By trial and error find a value for a which minimie the variation in the calculated voltage V y. Do thi tet with great care, make a note of the etting and lock the potentiometer in thi poition. Remove the threaded knob and upport bracket. Software Linear Control of the Pendulum Connect the Control Module to the computer a decribed in Getting Started. Run the PCS oftware. Select Linear Control. Linear control operate a imple gain model: the carriage i moved by a multiple of the diplacement of the pendulum bob from the centre of the carriage (etpoint i not implemented in thi control method). Linear control demontrate the baic linear characteritic of the pendulum. Linear control will be untable. Why? Bytronic International Ltd 3.9

30 Labwork Experiment with different value of gain. Comment on the behaviour of the pendulum when the gain i et to one. Switch in the oftware phae lead controller. Try it. In thi Labwork you will implement in hardware the phae lead controller that jut failed in oftware. Why doe thi controller work in hardware and not oftware? Software Harmonic Control of the Pendulum Switch the PCS oftware to Harmonic Control. Harmonic Control i baed upon the Simple Harmonic Motion equation applied to y : d y dt yω (equation 3). n Harmonic control i defined by ubtituting x for y in the left hand ide of thi equation: d x dt yω n and implementing the reulting differential equation. Harmonic control demontrate the baic harmonic (ocillatory) behaviour of the pendulum. Experiment with different value of ω n. Harmonic control will be untable. Why? Tranfer Function for the Inverted Pendulum We will now tackle the dynamic of the pendulum in term of Laplace tranform. No decription of the theory of Laplace tranform i given here. For uch a decription, pleae refer to a fuller ource uch a one of the reference given in Appendix 6 at the end of thi manual. Initially we will neglect the dynamic of the carriage ervo, i.e. aume that it repone time i very fat in comparion with the pendulum. The inverted pendulum can be linearied and modelled by the tranfer function: Y X ( ) ω n Bytronic International Ltd 3.0

31 Labwork where ω n i the natural frequency in rad/ determined experimentally in Labwork. (Thi i equation 3 expreed in Laplace tranform). Uing the equation for the period of a pendulum thi equation can be rewritten a: Y X ( ) d where L d. g Conider the tranfer function for your pendulum. For example, if ω n i 6.6 rad/, the tranfer function i: G p ( ) Conider thi tranfer function. Examine the -plane pole/ero pattern and confirm that the open loop tranfer function predict intability. Comment on the poibility of tabiliation uing proportional control. Predict the form of cloed-loop tep repone with proportional gain of 0.5, -.0 and.0. We will now try out proportional control on the PCS unit with negative gain. Switch off the power and connect the et point potentiometer a a reference voltage to the overall inverted pendulum control ytem (link ocket A and B). Set up the compenator a a unity gain inverting amplifier. To do thi place large reitor of equal value, e.g. 00K, acro terminal D and F. Connect the output of the operational amplifier directly to the ervo reference input (link ocket G and H). Make ure the gain witch i et to negative. Support the pendulum by hand and with the gain potentiometer P, et to ero, witch on the power. You may need to adjut the et point pot to bring the carriage to the centre of the rig. Slowly increae the gain and oberve the carriage action a you move the pendulum to one ide. Do not releae the pendulum completely becaue proportional control cannot tabilie thi ytem. Examine the frequency of ocillatory behaviour at the different gain decribed above. Note particularly the behaviour with the gain et at unity. Bytronic International Ltd 3.

32 Labwork Stabiliation uing Phae Lead Compenation A compenator of the form G + d ( ) c + ( d /0 ) i uggeted for the pendulum. For detail on the choice and deign of thi compenator ee Ref., pp 98-0 (Appendix 6). 0. For the Bytronic pendulum d The actual compenator recommended imply ue round number. It i not uggeted that thi compenator i in any way optimum. Better compenator may be deigned taking into account the dynamic of the ervo carriage (ee Ref., pp. 0-04, Appendix 6). The operational amplifier on the conole can be ued to implement a phae lead compenator. We will ue the compenator: G + 0. ( ) c Figure : Inverted Pendulum Compenator Bytronic International Ltd 3.

33 Labwork Combine thi compenator with the G p tranfer function obtained above and plot the new root locu. Can the ytem now be tabilied? What value of gain will reult in a table cloed-loop ytem? Chooe a gain which give the cloed-loop ytem dominant pole in a reaonable damping ratio. To implement thi tranfer function a a phyical compenator, conider the following: In general for an op amp circuit: V V output input Z Z f input where Z f i the feedback impedance round the op amp and Z input i the impedance between the input voltage and the umming junction. For a parallel circuit coniting of a reitance, R, and a capacitance, C Z + + C R RC R Thu for R R R and C input.0µ F and C f 0.µ F, we obtain: V V output input + RCinput RC f Add the compenator component a hown in the diagram above. Once more upport the pendulum by hand and with the gain P et to ero witch on the power. Slowly increae the gain until the pendulum balance itelf. Make a note of thi critical value of gain. Increae the gain to the value calculated earlier and oberve the repone. Try gently altering the et-point value to make the pendulum move along the track. Try gently tapping the pendulum to knock it over. The inverted pendulum under the control of a phae lead compenator i reaonably robut. If any problem are experienced try etting a to.7, gain to.5, compenator component a above, ervo gain to maximum and velocity feedback to approximately three-quarter. Bytronic International Ltd 3.3

34 Labwork The Effect of Hyterei in the Servo Sub-Sytem The tranfer function derived above i predicted to be table. You will however note the ervo hunting continuouly from ide to ide. Thi behaviour i due to hyterei in the carriage ervo. For more detail refer to Ref. (Appendix 6). Bytronic International Ltd 3.4

35 Labwork Labwork 3: Fuy Control of the Inverted Pendulum Apparatu PCS, PCS oftware, Analogue I/O card (MPIBM3). Introduction Fuy Control i a form of control of variable which exhibit multiple mode of behaviour. Providing the behaviour can be broken down into dicrete area, the input variable are broken down in term of the different mode of behaviour of the ytem and the control repone adjuted accordingly. The fuy controller provided in the PCS oftware i a fixed controller (though the method of accumulation and defuification can be changed). The controller wa developed uing the Fuy Control Package available from Bytronic International Ltd. Thi package i deigned to teach the baic principle of fuy control and although an outline i given here, it may be neceary to refer to the Fuy Control Package for better undertanding. Decription of the Fuy Controller In the cae of the Pendulum, the variable to be controlled i the poition of the pendulum bob, y. A we know, y exhibit both linear and ocillatory behaviour and thee mode are hard to eparate. The pendulum fuy controller work on both the error and the change in the error. The fuy controller attempt to handle both variable by aigning dicrete range of behaviour to each. According to which variable i more ignificant, the repone of the fuy controller i altered. Bytronic International Ltd 3.5

36 Labwork Thu when the rate of change of y i large when the pendulum might be in danger of falling over the controller act to low the movement. When the pendulum i a large ditance away from the et point, the controller attempt to move toward it. The mot important apect i the eparation of thee variable: when acting to balance the pendulum, movement to the et point i et aide. Moving to the et point will only be done when the pendulum i balanced. In fact thee action are in oppoition and the pendulum will often balance ome ditance from the et point. The firt tep, the breaking down of the variable down into dicrete area of behaviour i known a fuification. Fuification Thu error i broken down into five dicrete region -ve Big, -ve Small, Zero, +ve Small, +ve Big. Each of thee range i indicated above either by a ingle diagonal line (edge, -ve Big and +ve Big) or by a triangle (in the middle, the other). The carriage poition ignal range from -0 to +0 and thee five region cover the range. Change in error i imilarly broken down into five range: -ve Far, -ve Near, Zero, +ve Near, +ve Far. The height of the triangle at each point indicate the degree of memberhip of that range. Thu a pendulum change in error ignal of -4 would be aid to be 0.6 a member of -ve Far. Range overlap o -4 would alo be aid to be 0.4 a member of -ve Near. A pendulum change in error ignal of -0.5 would be aid to be 0.9 a member of Zero and 0. a member of -ve Near. Bytronic International Ltd 3.6

37 Labwork The degree of memberhip indicate the weight that i given from the ignal to that range. The pendulum change in error ignal actually range from -0 to +0 but ignal greater than plu or minu 5 indicate that the pendulum i irrevocably in the proce of falling over. Rule The ignal from the two variable Error and Change in Error are now allowed to interact according to the fuy rule bae. The rule bae ued here i: Error -ve Big -ve Small Zero +ve Small +ve Big -ve Far -ve High -ve High -ve Low -ve Low Zero -ve Near -ve High -ve Low -ve Low Zero +ve Low d/dt (-y) Zero -ve Low -ve Low Zero +ve Low +ve Low +ve Near -ve Low Zero +ve Low +ve Low +ve High +ve Far Zero +ve Low +ve Low +ve High +ve High Thu an error range of ve Big combine with a change of error range of Zero to give and output of -ve Low. +ve Small combine with -ve Near to give an output of Zero. The advantage of the fuy controller i that different requirement of the pendulum and carriage in different circumtance can be accommodated. Thu in the rule bae above, when the change in error i -ve Far or +ve Far, the repone i alway -ve or +ve, regardle of the value of Error. Thi correpond to the ituation that the pendulum i in danger of toppling over and need to be tabilied: the poitional behaviour can be dicarded. For maller value of change in error, the change in error behaviour i le ignificant and the error value can be given more emphai. Thi allow the fuy controller to control the two different input variable. Fuy controller are relevant when there are two or more input variable which interact, or a ingle input variable which demontrate different mode of behaviour. When a ingle, table, variable i to be controlled, PID control i better. The control action i calculated from the overall table (ee below). When conidering each individual output action, we need to calculate the weight given to each of thee output range. Thu a ignal of 0. ve Big and 0.8 ve Far give an output ignal of +ve High which need a weight. There are two way of calculating thi weight: multiplication of the two value, or taking the minimum of the two value. Multiplication give a weight to +ve High of 0.6. Minimum give a weight of 0.. Thee two Bytronic International Ltd 3.7

38 Labwork method are known a Product and Minimum. You can chooe between thee in the fuy controller. Defuification The output ignal now need to be tranformed back into a phyical ignal. The phyical output range i 5 to +5 in order to allow ome ignal to be given extra weight over ±0. Zero, 0, obviouly correpond to not moving the carriage. The phyical output correponding to each range i: -ve High -5 -ve Low -7.5 Zero 0 +ve Low ve High +5. At thi point we may have, ay, an error vector of 0.3 +ve Small, 0.4 Zero, and a change in error vector of 0.7 Zero, 0.3 +ve Near. How do we convert thi into a ignal? Firtly, we ubtitute the output trength for the output range in the table above: Error -ve Big -ve Small Zero +ve Small +ve Big ve Far ve Near d/dt (-y) Zero ve Near ve Far Then we convert the relative trength of each ignal in the table into an overall ignal. There are two method. The firt method weight the ignal according to their relative trength. Uing product accumulation: (0.4*0.7* *0.7* *0.3* *0.3*7.5) / (0.4* * * *0.3) 4.5. Thi i known a Centre of Gravity defuification. The other take the larget ignal. (Product and minimum accumulation are not relevant here): 0.4>0.3>0,0,0 and 0.7>0.3>0,0,0 o output ignal (0.4 column, 0.7 row): 0. Thi method i known a Middle of Maxima. Selection between thoe two method i available in the fuy controller. Bytronic International Ltd 3.8

39 Labwork Ue of the Fuy Controller Connect up the PCS, witch on, run the PCS oftware, elect Fuy control and pre Start. Fuy control i a reaonably good controller for balancing the pendulum. Try uing Product and Minimum accumulation, Centre of Gravity and Middle of Maxima defuification. Set point poition are amplified: if et point i et to 0., the pendulum will move approximately an inch down the track, to roughly the poition where or ought to be. Thi i clearly een when uing a inuoidal input. Bytronic International Ltd 3.9

40 Labwork Labwork 4: Direct Digital Controller Deign and Implementation: Inverted Pendulum Apparatu PCS, PCS oftware and Analogue I/O card (MPIBM3). Introduction Balancing the inverted pendulum uing a direct digital controller i by no mean a trivial exercie, either from the point of view of deign or implementation. You can do thi labwork imply uing the deign given. In order to deign controller you will need acce to ome Computer Aided Control Sytem Deign (CACSD) oftware. To implement the Z Controller you can ue the PCS oftware. Alternatively a program that implement a control trategy in term of difference equation at a particular ample rate can be written relatively eaily. The procedure for implementing thi i given in the PCS oftware help file. The inverted pendulum controller deign i particularly difficult becaue the model obtained experimentally decribe the large ignal behaviour of the ytem. However, near the point of balance it i the mall ignal behaviour that i ignificant. In thi behaviour factor uch a friction and tiction become pronounced. Thu the model ued for deign i not the one that decribe the behaviour well near the normal operating point. Thi i rather different from proce control ituation where the mall ignal behaviour i le affected by the type of non-linearitie generally preent in uch ytem. Bytronic International Ltd 3.0

41 Labwork Thu in deigning the controller a very conervative approach mut be adopted and the rule derived for well behaved linear ytem mut be interpreted with a healthy cepticim and invariably ome trial and error i neceary to arrive at a workable robut olution. Deign of a DD Controller uing a Simplified Model Deign Criteria and Sample Rate Selection In thi exercie we hall deign a -domain compenator uing a implified model of the inverted pendulum in order to how the approach. The ervo dynamic and any damping in the pendulum pivot are neglected, ie the plant dynamic can be adequately decribed a G ( ) p ω. n For the purpoe of thi exercie a natural frequency of 6.6 rad/ will be aumed, i.e. a period of jut over one econd. The important deciion are the deired cloed-loop behaviour and the ample time. Thee two factor are linked in the analyi. An expoition follow. A the open loop ytem i econd order, it i not unreaonable to demand that the underlying dynamic of the cloed-loop ytem hould alo be econd order. However, the controller that reult produce an inadequate control action. In order to achieve a more robut control ytem, the target cloed-loop dynamic will be pecified a third order. For implicity we will chooe a characteritic equation with repeated root. Now a third order ytem with three identical lag of 0.5 ha a 5% ettling time of.5 which i a reaonable pecification for the ytem conidering the underlying dynamic of the pendulum. Bytronic International Ltd 3.

42 Labwork A well a the dynamic deign criterion, a target mut be et for the teady-tate performance of the control ytem. In the analogue deign, a Type 0 phae advance compenator wa employed. With thi compenator a forward gain of wa jut achievable. Thi correpond to teady-tate gain of K + K For the DD controller, however, we will pecify teady-tate et-point following, i.e. unity cloed-loop gain. Thi require F ( ). The guideline for ample time election for DD controller ugget that the ampling frequency hould exceed about 0 time the cloed loop bandwidth. (See Ref. ). Now a ytem with three repeated lag of time contant 0.5 ha a (-3dB) bandwidth of about 0.08/ H. Hence the ampling frequency hould exceed 6.4H (50m ample time). Too high a ampling rate caue real -plane pole to be bunched very cloe to in the -domain. The ame guideline recommend that the ampling frequency hould be le than 50 / τ 0 where τ 0 i the dominant open-loop time contant of the ytem. Now the ytem we are dealing with ha a pair of real pole at ±6.6. Thu the ampling frequency hould be le than 50*6.6 30H (3.m ample time). A certain amount of trial and error i required, but a ampling time of (8.H) doe produce a working controller and thi figure i in agreement with the theoretical limit decribed above. 5 In the -domain a time contant of 0.5 map to the point e on the real axi. For T thi point on the -domain i approximately Thu the deired cloed-loop characteritic equation i: ( 0.8) 3 0. T 0. Bytronic International Ltd 3.