ROTARY INVERTED PENDULUM

Transcription

1 ROARY INVERED PENDULUM AN O CHYE EO CHUN SAN SCHOOL OF ELECRICAL AND ELECRONIC ENINEERIN NANYAN ECHNOLOICAL UNIVERSIY 998/99

2 ROARY INVERED PENDULUM SUBMIED BY AN O CHYE EO CHUN SAN SCHOOL OF ELECRICAL AND ELECRONIC ENIINEERIN A final year projet report presented to Nanyang ehnologial University in partial flfillment of the reqirements for the Degree of Bahelor of Engineering April 999

3 Abstrat he Rotary Inverted Pendlm System is an eellent test-bed for teahing gital ontrol tehniqes. he system is eited by a DC motor, and is eqipped with sensors to measre the anglar splaement of the pendlm and the anglar veloity and position of the DC motor. A miro-ontroller is sed to implement a range of real-time gital ontrol tehniqes. It an be sed to demonstrate the varios pratial aspets involved in designing a ontrol system. Used as a teahing tool, it allows stdents to gain hands-on eperiene with an atal system, from olleting, filtering and analyzing data to obtain a dynami model of the system, to deing an appropriate sampling interval and designing a gital ontroller to balane the pendlm in the pright position. Deriving a mathematial model for the inverted pendlm system is the first step in the design proess. he model mst be able to desribe the response of the atal system as losely as possible. Lagrange s eqation of motion was sed in the derivation of the dynami non-linear model. he model gives the eat relationships among all the variables involved. Using the dynami non-linear model, we an derive the linear model. he nnown parameters of this model will be estimated in the System Identifiation. Data is olleted from eperiments to identify the nnown physial qantities that are reqired to omplete the model. System Identifiation sing Least Sqares Approimation is sed to estimate the parameters. One the parameters are identified, the model is sed for design of ontrollers to balane the pendlm. he ontrollers are designed sing pole plaement tehniqe. Besides balaning the pendlm in the pright position, it is desired to ontrol the position as well as the movement of the rotating arm. Controllers with proportional and/or integral ations were designed. Digital ontrollers sh as the eneralised Pretive Controller and Disrete Inremental ontroller were also implemented for better performane. Page I

4 o omplete the projet, a raphial User Interfae was developed to provide orseware to ater to sers with fferent bagrond nowledge. his will enhane the system so that it will beome a good test-bed for ontrol stdy.. Page II

5 Anowledgement Firstly, we wold lie to than or spervisor Dr. Ling e Voon for his gidane throghot the orse of the projet. His omments and sggestions have ontribted signifiantly to the sess of the projet. We wold also lie to than the Laboratory ehniians in Control Engineering Lab, Ms. Joanne ham, Ms. Song Lee Sin and Ms. Feliia Lim for their spport throghot the past year. Withot their help, many administrative matters wold not have been performed as effiiently. heir friendly sposition have reated a pleasant environment to wor in. Net, we wold lie to than or peers in Control Engineering Lab for their opinions and appraisals. heir ompanionship have made staying long hors a lesser hore. Last of all, we wold lie to than all those who have, in one way or another, ontribted to the sess of this projet. Page III

6 Contribtion FYP PROJEC Major as eo Chn Sang an o Chye Familiarisation with 5% 5% hardware Familiarisation with 5% 5% related software Researh done on 6% 4% Pendlm Mathematial Modeling 6% 4% Parameter Estimation 7% % Matlab Simlation % 8% Controller Design % 8% raphial User Interfae % 9% Final Report 6% 4% Page IV

7 Content Page Abstrat I Anowledgments Contribtion able of Contents V III IV Chapter Introdtion. Bagrond and Motivation. Objetives Chapter System Setp. Introdtion 4. Setp of System 5. Software System 6.4 Personal ompter system 6.5 Universal ontroller 7.6 Motor driver board 7.7 Power spply 8.8 Inverted pendlm apparats 8.8. Potentiometer 9.8. Enoder Chapter Mathematial Modelling. Introdtion. Co-ornate System. Non-Linear Dynami Model.4 Linearised Model.5 Continos-ime State Spae Model Page V

8 .6 Disrete-ime State Spae Model 4.7 Disrete-ime Inremental Model 4 Chapter 4 Parameter Estimation 4. Introdtion 6 4. Mathematial Model Used for Parameter Estimation 6 4. Signal Contioning Estimated Data Forward Differene Central Method Baward Differene 4.5 Unfored Osillation Response 4.6 Least Sqares Estimation Methods 4.6. Method 4.6. Method 4.7 Model Verifiation ehniqes 4 Chapter 5 Eperimental Reslts and Disssion, Parameter Estimation 5. Introdtion 5 5. est Signals and Data 5 5. Reslts of Parameter Estimation Disssion on Parameter Estimation Method Effets of Differene Approimation Method Effets of Using Unfored Osillation Response for Estimation 5.4. Reslts of Estimation of J and C Referene Model Page VI

9 5.5 Disssion on Parameter Estimation Method Referene Vales for to Overview of Estimation Reslts Effets of Differene Approimation Method Effets of Using Unfored Osillation Response for Estimation Reslts of Estimation 7 Chapter 6 Controller Design 6. Introdtion 4 6. Controllability 4 6. Aermann s Formla State Feedba Controller (Continos ime Model) State Feedba Controller with Proportional Ation (Continos ime Model) State Feedba Controller with Integral Ation (Continos ime Model) State Feedba Controller with PI Ation (Continos ime Model) State Feedba Controller (Disrete ime Model) State Feedba Controller (Disrete ime Inremental Model) 5 6. eneralized Pretive Controller 5 6. Swing Up Controller 5 6. Dead Zone Mapping 56 Page VII

10 Chapter 7 Eperimental Reslts and Disssion, Controller Design 7. Introdtion State Feedba Controller (Disrete and Continos ime Model) State Feedba Controller with Integral Ation (Continos ime) State Feedba Controller with Proportional Ation (Continos ime) State Feedba Controller with Proportional and Integral Ation (Continos ime) State Feedba Controller (Disrete ime Inremental Model) eneralized Pretive Controller Swing-Up Controller Effet of Dead Zone Mapping 75 Chapter 8 raphial User Interfae 8. Bagrond Objetives Featres raphial User Interfae Version 98 Sreen Shot Startp Sreen Introdtion Sreen System Configration Interfae Beginner Interfae Option Advane Interfae Option System Setp Interfae Mathematial Modelling Misellaneos System Identifiation 88 Page VIII

11 8.4. Controller Design Demonstration 98 Chapter 9 Conlsion 99 Chapter Reommendations. DC Motor. Swing Up Controller. Parameter Estimation.4 Simlation with Animation Referenes 4 Appen A Appen B Appen C Appen D Appen E Appen F Appen Appen H Page IX

12 INRODUCION Chapter Introdtion. Bagrond and Motivation Early stes of the inverted pendlm system was motivated by the need to design ontrollers to balane roets dring vertial tae-off. At the instane of time dring lanh, the roet is etremely nstable. Similar to the roet at lanh, the inverted pendlm reqires a ontinos orretion mehanism to stay pright, sine the system is nstable in open loop onfigration. his problem an be ompared to the roet dring lanh. Here, roet boosters have to be fired in a ontrolled manner, to maintain the roet pright. he simple linear pendlm has long proved a sefl model for more ompliated physial systems, and its behavior in the small-amplitde limit provides a realisti yet solvable eample for stdents in introdtory lasses. While the fore-free, fritionless pendlm an be solved eatly for all amplitdes in terms of ellipti integrals, the soltion is hardly illminating, rarely fond sefl, and when damping and eternal driving are inlded, the eqations of motion beome intratable (eept for small osillations). With the advent of destop ompters, however, it has beome possible to stdy in some detail the rih nonlinear dynamis of the damped, fore-driven pendlm and gain signifiant insight into its sensitivity to initial ontions for ertain vales of the system parameters. Eqally interesting, bt mh less sted in physis tets, is the state in whih the plane pendlm an be stabilized in a small-osillation mode abot the vertial by means of a sinsoidal driving fore. Althogh the eternal fore an be applied to the pivot in either the vertial or horizontal retion, or in some linear ombination, a vertially osillating pivot is the most ommon senario as in or ase. It appears that this system was first sted by Stephenson in 98 and somewhat later by van der Pol and Strtt. Page

13 INRODUCION An interesting variation on the inverting pendlm problem is the Rotary Inverted Pendlm that is being sed in this projet. (Figre ) It onsists of an arm, and a pendlm monted on a hinge. As the pendlm has a irlar trajetory, this setp overomes the spae reqirement of a onventional art type pendlm system. he rotary type system reqires less spae and is very sitable for a projet. Figre Besides lassroom theory and varios ontrol design methods, it is important to epose stdents of gital ontrol to the many pratial aspets of implementing a gital ontrol system. his reqires a good fondation in ontrol theory as well as nowledge in ompter interfaing tehniqes, system modeling, instrmentation and gital signal proessing. herefore, a ompter-based orseware, whih aims to ahieve the above objetives, has been developed. It is implemented sing MALAB, a ommon software tool for gital ontrol design and analysis. he raphial User Interfae (UI) is implemented to aid the learning of the varios elements of a gital ontrol system. Page

14 INRODUCION. Objetives he basi objetives of this projet are as follows:. o verify the mathematial model of the inverted pendlm system and the ontrol algorithms implemented by the previos FYP grop [].. o estimate the system parameters of the inverted pendlm system.. o design ontrollers that an balane the pendlm and to ompare the performane of fferent ontrollers. 4. o have an Interative User Interfae for both beginners and advaned sers of this system and to provide orseware that integrates theory with pratie. With adtion objetives as follows:. o provide a platform sitable for the teahing of gital ontrol orses.. o spplement trational lassroom lessons by sing an interative and interesting teahing approah.. o provide hands-on pratie to enhane and reinfore stdent s learning. Page

15 SYSEM SEUP Chapter System Setp. Introdtion his hapter will desribe the varios modles of the system. he system omprises si major modles, whih will be desribed in the following pages. hey are: System Software Personal Compter System Universal Controller UC96 Miroontroller Board Motor Driver Board Power Spply Inverted Pendlm Apparats ri PP- Potentiometer Enoder Page 4

16 SYSEM SEUP. Setp of System op view of the inverted pendlm system Figre. on the right shows the pitre of the Inverted Pendlm System. It shows the top view of the Ri PP- Inverted Pendlm apparats with the power spply nit hosed in a asing on the right beside the pendlm apparats. he UC96 Controller and Motor Driver board have been monted on the base of the Inverted Pendlm apparats. Figre. Disassembled view of the omponents Figre. on the right shows the sassembled view onsisting of the Power Spply Unit, Motor Driver Board and UC96 Miroontroller Board. From figre., we an see that the UC96 Miroontroller Board is monted on the Motor Driver Board on the top left orner of the pitre. he power spply nit onsists of the transformer and the retifier irit. Figre. Page 5

17 SYSEM SEUP. Software System he Software Interfae system (UIv98) is sed to provide an interfae to help sers stdy the system. It enhanes the effiieny of the ompter as a development platform. It also provides an integrated environment that lins all the varios sbprograms that are needed, maing the system operations transparent to the ser. Figre..4 Personal Compter System he Personal Compter system is sed as a development platform. It hosts the software interfae system and all other programs that are reqired to operate the system. Most importantly, the ompter ats as an interfae between the hardware and the ser. Fntions and proesses for analysis, omptation debgging, doshelling, downloang/ ploang of data, as well as mofiations of the internal registers of the UC96 board dring rn-time an be performed. he physial interfae sed to onnet the ompter system with the UC96 system is the serial port. Figre.4 Page 6

18 SYSEM SEUP.5 Universal Controller UC96 Miroontroller Board he Universal Controller UC96 Miroontroller Board is a single-board miroontroller system sitable for implementing a wide variety of standalone embedded systems (Figre.5). Appliations inlde instrmentation, data aqisition, ontrollers and other intelligent gital systems. he Universal Controller UC96 Miroontroller Board is powered by the Intel 8C96C high performane miroontroller, omplemented by PSD4A a fieldprogrammable peripheral IC. he ombination allows the hardware strtre to be programmed with firmware, whih maes the board etremely versatile. Figre.5.6 Motor Driver Board he Motor Driver Board is designed to drive the motor onneted to it, with a PWM signal (Figre.6). o ahieve high reliability and better onnetivity between the miroontroller and driver board, the motor driver board was designed sh that the miroontroller board sits on top of it and trn-pins are sed instead of ables. his also redes the signal path between the boards, hene reng signal propagation. Figre.6 he board also hoses reglators to onvert the 7V spply to 5Vd and Vd to drive the PWM motor. Page 7

19 SYSEM SEUP.7 Power Spply Unit A pea 7V nreglated DC power spply to the motor driver board to drive the inverted pendlm system. Besides that, the power spply nit 5VA /V transformer steps down the AC from V to Vrms. hrogh a fll-bridge retifier, DC voltage is spplied to the miroontroller. Figre.7.8 Inverted Pendlm Apparats ri PP- he Ri PP- Inverted Pendlm apparats (Figre.8) onsists of a short arm that rotates in a horizontal plane, driven by a motor monted vertially. A pendlm rod is fied at one end of the arm, whih moves in the vertial plane. At rest, the arm is stationary and the pendlm hangs downwards. he objetive is to balane the pendlm in the pright position. Figre.8 Page 8

20 SYSEM SEUP.8. Potentiometer he anglar position of the pendlm is measred by a Ω potentiometer (rotary type). A spply of 5V is applied aross the potentiometer. his signal is presented to an ADC with amplitdes of V - 5V orresponng to - 6. (Figre.9) Bt de to the non-linearity of the potentiometer, the potentiometer will only measre arately from to 4 ±. he fferene between the potentiometer reang at pright position and pendant position is not eqal to half the range of the -bit ADC (5 onts). Figre.9 herefore, to minimize the measrement error in the pendlm s anglar position, the rossover point (dead zone) was set half way between the pright and horizontal position. Hene, while performing system identifiation and balaning the pendlm, the rossover (dead zone) effet is minimal. Page 9

21 SYSEM SEUP he interfae between the potentiometer and the miroontroller board is throgh wires throgh a opling on top of the rotating drm (Figre.). his will effetively pass the signal of the potentiometer to the ontroller board. Figre..8. Enoder Position of the arm is fed-ba by the enoder in the motor shaft (Figre.). With this line enoder, it is able to give a fferential otpt signal, whih gives a total of 4 onts per revoltion. Figre. Page

22 MAHEMAICAL MODELLIN Chapter Mathematial Modelling. Introdtion o be able to analyse the system, we need to derive an arate mathematial model. In this hapter, the mathematial models sed throghot the projet will be presented. he derivations are shown in the Appenes.. Co-Ornate System m l z y Figre. he models presented in this hapter are based on the o-ornate system shown in Figre. above. he standard right-handed Cartesian o-ornate system is sed. he anglar position of the arm,, is assigned to be inreasing when the arm is rotating abot the z-ais in the right-handed sense (when the thmb points in the retion of the ais, the fingers point in the retion of inreasing angle). he referene of is taen to be the -ais. L he anglar position of the pendlm, and, are assigned to be inreasing when the pendlm is rotating abot an ais passing throgh the arm setion from the Page

23 MAHEMAICAL MODELLIN Page m gl C m l m l l m L m l C m l J l m L l m L m l m L J a t a b t R R sin sin sin sin sin os os sin m gl C m l m l l m L m l C m l J l m L l m L m l m L J a t a b t R R sin sin sin sin sin os os sin origin to the pivot point of the pendlm, in the right-handed sense. he referene of is taen from the pward vertial and the referene of is taen from the downward vertial. ransforming between fferent o-ornate systems an be easily performed as demonstrated in the derivation of the dynami model in the downward position in Appen A.. Non-Linear Dynami Model he non-linear dynami model desribes the system ompletely. It gives the eat relationships among all the variables involved. All the linear models sed for ontroller design are derived from this non-linear model. As shown in Appen A, eqation (A6), the dynami model with the pendlm in the pright position is: (.) he dynami model with the pendlm in the downward position, Appen A eqation (A), is: (.)

24 MAHEMAICAL MODELLIN Page m gl C C m l J l m L l m L m L J a t a b t R R e ef af ac ah d af C h df af B A m l J f C d m l b m gl h e l m L m L J a a b t a t R R e ef af ac ah d af C h df af af B A Linearised Model he linear model is obtained by linearising eqation (.) abot: As shown in Appen B, eqation (B), the linearised model is: (.).5 Continos-ime State Spae Model Define: (.4) Writing eqation (.) in state spae form with state variables, we obtain, as shown in Appen B eqation (B6): (.5) Agmenting the state variables with the arm position, the following model is obtained as shown in Appen B eqation (B7): (.6)

25 MAHEMAICAL MODELLIN Note that for the above models, the otpt(s) an be defined as any linear ombination(s) of the state variables speified by the following: y C where is the state variables ( or 4 )..6 Disrete-ime State Spae Model he srete-time model is reqired when sing gital ontrol tehniqes. When a system desribed by state spae eqations, y C is sretized with respet to a sampling interval s, the orresponng srete state spae model is obtained: ( ) A d ( ) Bd( ) y( ) C ( ) where is the srete time inde, d (.7) he sretization tehniqe an be applied to the ontinos time models as stated in eqations (.4) and (.5) to obtain the srete models for design or analysis prposes. A B A s s Aγ Ad e, Bd e Bdγ, Cd C.7 Disrete-ime Inremental Model he inremental model [] is an alternative form of representation and will be sed in the design and implementation of the eneralised Pretive Controller. Define the state variables to be: i ( ) ( ) ( ) Page 4

26 MAHEMAICAL MODELLIN As shown in Appen C eqation (C), the inremental model sed for this projet is: where i ( ) A i ( ) B ( ) y( ) C ( ) i A d Bd Bd A, B, C [ C ] d (.8) Page 5

27 PARAMEER ESIMAION Chapter 4 Parameter Estimation 4. Introdtion In the mathematial models derived in Chapter, the parameters of the models need to be determined for the model to be sefl. In this hapter, two tehniqes sed for parameter estimation are presented. Both the tehniqes involve the same priniples, eept that the eqations sed for estimation are slightly fferent. o be able to estimate the parameters, we eite the system with a nown test signal and ollet the data of the response of the system. With the mathematial model derived in Chapter, we then try to find the best vales of the parameters that will prode theoretial reslts that are as lose as possible to the eperimental reslts. he proess is illstrated in Figre 4. below. est Signal Physial System System Response Parameter Estimation Comparison Mathematial Model Model Response Figre Mathematial Model sed for Parameter Estimation For parameter estimation, the system will be eited with a nown test signal from its downward hanging position. he response of the system to the test signal will adhere to the dynamis desribed by the non-linear downward model. Hene it will be more onvenient to se the downward model to estimate the parameters. he Page 6

28 PARAMEER ESIMAION downward model presented in eqation (.) is reproded here for onveniene of referene: J C m L t b Ra m l m L l os sin ml sin m l sin t Ra m gl sin m L l os J m l m L l sin C m l sin 4. Signal Contioning he data olleted for parameter estimation are: Arm Veloity Pendlm Position Motor Driving Command he raw data olleted are not in Standard Units and are sally very noisy. Frthermore, sampling and gital storage of data introdes qantization and trnation errors. After onverting to Standard Units, the data are filtered before being sed for parameter estimation. he filter sed is an Infinite Implse Response (IIR) gital filter of transfer fntion:.5 H ( z).85z he freqeny response of the filter is shown in Figre 4. (a) below. (a) Figre 4. (b) Page 7

29 PARAMEER ESIMAION he magnitde response H(e jω ) of the filter is: H H jω ( e ) jω ( e ).5 jω.85e.5.85osω j.85sin ω.5 (.85osω ) (.85sin ω ).5.7osω.85 Sine the filter is of IIR type, it will introde some phase stortion to the filtered signal. o perform IIR filtering withot phase stortion, the filtered signal is reversed in seqene and passed throgh the filter again. By this ation, all the freqeny omponents of the signal pass throgh the filter twie, one in eah retion. heir invidal delays will be the same and hene there will be no phase stortion. his is implemented by the MALAB ommand filtfilt(). he eqivalent filter by this ation is doble the original filter order. he effetive freqeny magnitde response, shown in Figre 4. (b), is H(e jω ). Hene the t-off freqeny an be fond as follows: H ( e ) j ω.5.7osω.85 osω.7 ω.48 rad / sample (.85.5 ) πf.48 rad / sample f.667 yle/ sample For a sampling freqeny F s of Hz, the analoge t-off freqeny f, is F s f.667hz. Page 8

30 PARAMEER ESIMAION Page Estimated Data From the raw data olleted, the arm aeleration, pendlm veloity and pendlm aeleration an be estimated. For sampling interval s and sample inde, there are possible ways to estimate these Forward Differene aing the forward fferene, the arm aeleration, pendlm veloity and pendlm aeleration an be estimated as: 4.4. Central Differene aing the entral fferene, the arm aeleration, pendlm veloity and pendlm aeleration an be estimated as: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) s s s s s ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) s s s s s 4 4

31 PARAMEER ESIMAION 4.4. Baward Differene aing the baward fferene, the arm aeleration, pendlm veloity and pendlm aeleration an be estimated as: ( ) ( ) ( ) ( ) ( ) s ( ) ( ) s ( ) ( ) s ( ) ( ) [ ( ) ( ) ] ( ) ( ) ( ) s s 4.5 Unfored Osillation Response A typial set of raw data is shown in Figre 4. below. (a) Unfored Osillation (b) () Figre 4. From Figre 4., observe that we an segment the data into fored response and nfored osillatory response where the nfored response is haraterised by: Page

32 PARAMEER ESIMAION he response of this setion orresponds to the dynamis of a free osillating pendlm. he dynami eqation for this response an be fond by setting the orresponng terms in eqation (A9), Appen A, to zero. he reslt is: ( ) J ml C m gl sin his will be sed as an alternative form for estimation, where appropriate. 4.6 Least Sqares Estimation Methods All the estimation methods presented here are based on the Least Sqares ehniqe [7]. Stated simply, for a system of linear eqations desribed by y A (4.) where A is an n m matri and n > m, y is a olmn vetor of n elements, is a olmn vetor of m nnowns, the Least Sqares soltion of the system of eqations is given by: (A A) - A y (4.) he raw data and the estimated data will be sed in the estimation of the nnown parameters. Method was sed by the previos FYP grop and has been a proven tehniqe. It will be sed as the referene method for omparison prposes Method Here, we ategorise the parameters into determinable and indeterminable. By determinable, we refer to those qantities that an be determined from some reliable sores or throgh performing some simple eperiments. he determinable qantities are presented in Appen F. With these vales, we se the Least Sqares ehniqe to estimate the indeterminable parameters whih are J, C, J and C. Page

33 PARAMEER ESIMAION From eqation (.), separating the nown qantities and nnown qantities, the following an be written: J C m L m l sin J C ml l os ml ml In matri form, the above an be written as: t b ml ml sin mll os R a ml sin mll sin [ ] J C J [ m L l os m l m l sin m gl sin ] [ ] m L l sin t Ra m L l os sin m gl sin t Ra t b Ra m l sin C (4.) (4.4) he left-hand side of eqations (4.) and (4.4) are nown qantities. he terms ontain the determinable parameters and data whih are obtainable from the response of the system. When the data from eah sampling instane are formed into eqations (4.) and (4.4), we get an eqation similar to (4.) whih is then sed for estimation. Mathematially, for N samples of the response of the system, we an form N eqations from (4.) as: ( ) ( ) y y y A M M y ( N ) N where is the sample inde, y y A ( ) ( ) ( ) ( ) ( ) ( N ) M J C tb ( ) m L ( ) m l ( ) sin [ ( ) ] m L l ( ) os[ ( ) ] ( ) m l ( ) ( ) sin[ ( ) ] m L l ( ) [ ] sin ( ) Ra t [ ] ( ) Ra he parameters J and C an be estimated from eqation (4.). he same tehniqe is sed for all the estimation methods and will not be stated again. Using the nfored osillation response, J and C an be estimated from: J [ m l m gl ] [ ] sin C he linear model an then be onstrted from eqation (B8) in Appen B. (4.5) Page

34 PARAMEER ESIMAION Page sin sin os sin sin os sin h C b f e b d b a [ ] ( ) ( ) [ ] [ ] [ ] sin os sin os sin sin sin f h f a e a f C f b a d a b [ ] [ ] 8 7 sin 4.6. Method Method reqires prior nowledge of the determinable parameters for estimation. he advantage of Method is that all the parameters an be estimated withot any sh nowledge. he variables defined in (.4) are reproded here: With referene to the variables above, the downward model an be written as: (4.6) (4.7) Define: (4.8) After some maniplation, the following an be obtained: (4.9) (4.) Using the nfored osillation response, 7 and 8 an be estimated by: (4.) m l J f C d m l b m gl h e l m L m L J a a b t a t R R

35 PARAMEER ESIMAION From these reslts, 5 and 6 an be estimated, sing the fll response, by: 5 [ sin ] [ sin ] 7 8 os 6 (4.) Note that the linear model an be onstrted from eqation (B9) in Appen B. 4.7 Model Verifiation ehniqes After parameter estimation, some form of verifiation is needed to he if the reslts of the estimation are arate and reasonable. One way is to ompare the response of the physial system with the simlated response. For this prpose, the downward model is onstrted sing MALAB s Simlin oolbo. Simlin is an interative tool for modelling, simlating and analysing dynami systems. he Simlin model, whih is based on eqations (4.9) and (4.) is shown in Figre 4.4 below. Figre 4.4 he reslts from parameter estimation are sed to mofy the Simlin model s parameters. he simlated response to the same test signal will give an ination of the aray of the estimated parameters. Page 4

36 EXPERIMENAL RESULS AND DISCUSSION FOR PARAMEER ESIMAION Chapter 5 Eperimental Reslts and Disssion for Parameter Estimation 5. Introdtion his hapter presents the reslts of the parameter estimation tehniqes as desribed in Chapter 4. Previos wor has shown that ertain patterns of test signals prode onsistently good reslts. Figre 4. (a) shows one sh test signal. It is a step signal that drops to zero after a finite dration. his type of test signals provides good eitation for the transient response and allows the system to ehibit the nfored osillation response. he sssion will be limited to parameter estimation sing this partilar type of test signals. As there are methods and a few options available for estimation, a omprehensive treatment is not possible. Seleted ases will be highlighted where appropriate. All sssions are based on data olleted on System II as mentioned in Appen F. 5. est Signals and Data Assming that noise affeting the signals has a mean vale of zero, it wold be ideal to se as many samples as possible for estimation sh that its effets will be minimised. However, there is a physial limitation on the memory of the miroontroller and this in trn limits the nmber of samples of data that an be olleted. Eah set of data sed is 8 samples wide. For a sampling period of ms, this orresponds to the first 8 seonds of eitation. Page 5

37 EXPERIMENAL RESULS AND DISCUSSION FOR PARAMEER ESIMAION en sets of data are olleted for analysis. he test signals for these are as desribed above in inreasing dration of -seond steps. he data olleted for analysis are presented here in Figre to Figre. hese will be referred to as data set, data set, to data set respetively. Figre 5. Figre 5. Figre 5. Figre 5.4 Figre 5.5 Figre 5.6 Page 6

38 EXPERIMENAL RESULS AND DISCUSSION FOR PARAMEER ESIMAION Figre 5.7 Figre 5.8 Figre 5.9 Figre Reslts of Parameter Estimation he reslts obtained from parameter estimation are tablated in Appen H for ease of referene. For eah method, there is an alternative to se the nfored osillation response to estimate some of the parameters. his has been desribed in Chapter. Frthermore, there are ways to estimate the arm aeleration, pendlm veloity, and pendlm aeleration. he total ombinations prode a sbstantial amont of data not sitable to be presented in this hapter. he reslts are verified sing the Simlin model. If we tae the reslts from the simlation as the referene, we an have an ination of the aray of the estimation by defining the following error onstants. Page 7

39 EXPERIMENAL RESULS AND DISCUSSION FOR PARAMEER ESIMAION Let Define: S S E E be the arm veloity obtained from simlation, be the pendlm position obtained from simlation, be the arm veloity obtained from eperiment, be the pendlm position obtained from eperiment. N [ ( ) ( )] [ ( ) ( )] S E S E Error A Error B N N he error onstants defined above are normalised with respet to the total nmber of samples N so that the errors for fferent sample size wold not ffer by too mh. hese error onstants are also tablated in Appen H. N 5.4 Disssion on Parameter Estimation Method 5.4. Effet of Differene Approimation Method his setion ssses the effets of the fferene approimation methods on the reslts of estimation sing Method. On eamination, the forward and baward fferene seqene of a signal ffers only in phase by one sample, whereas the entral fferene seqene is the average of the two seqenes. his an be epressed mathematially as: F ( ) ( ) ( ) ( ) ( ) ( ) ( ) where F B C B is the entral fferene C F ( ) B ( ) F ( ) ( ) B ( ) is the forward fferene seqene, is the baward fferene seqene, seqene. C ( ) ( ) (5.) Page 8

40 EXPERIMENAL RESULS AND DISCUSSION FOR PARAMEER ESIMAION When applied to the approimation of the arm aeleration, pendlm veloity, and pendlm aeleration (these three qantities will be olletively referred to as the derived qantities ), they prode reslts whih have very fferent impliations. Eamination of ables H, H, H5 and H6 shows that all the estimated reslts are positive vales. his is intitively orret as the moment of inertia of a body and the visos frition oeffiient of an ais annot be negative. However, eamining ables H and H4 shows that C is onsistently a negative vale as estimated by Method. Using the baward fferene approimation, the estimated vale of C is negative. For data set, sing the fll response for estimation, the reslts are shown in Figre 5., 5. and 5. for forward, baward, and entral fferene approimation respetively. Figre 5. Figre 5. Figre 5. he bold line represents the simlated reslts whereas the thin line represents the eperimental reslts. Observing the reslts visally, it an be said that the Page 9

41 EXPERIMENAL RESULS AND DISCUSSION FOR PARAMEER ESIMAION se of the entral fferene method prodes more arate reslts than sing the forward or baward fferene methods. his is tre for the data sets even thogh the nmerial error onstants defined in setion 5. may inate otherwise for ertain data sets. o eplain this phenomenon, more stes will be reqired. However, we an establish some fats that an lead s to some starting point of stdy. When performing the fferene approimation, the higher freqeny omponents are amplified while the lower freqeny omponents are attenated, similar to the ation of a high-pass filter. Averaging a seqene has an effet similar to low-pass filtering. herefore, the entral fferene tehniqe essentially ats as a band-pass filter while the forward and baward fferene ats as a high-pass filter. he transfer fntions for these approimation methods are shown below. y Y F F YF X ( ) ( z) ( z) ( z) zx ( ) ( ) S z z S ( z) X ( z) ( z ) S S y Y B B YB X ( ) ( z) ( z) ( z) ( ) ( ) X S z ( z) z X ( z) ( z ) S S S y Y C C YC X ( ) ( z) ( z) ( z) zx ( ) ( ) ( z) z X ( z) S z z S ( z z ) S S o plot the freqeny response for omparison prposes, we shift the otpt by one sample into the past and plot its response defined by: H FS ( z) z Y X z F ( z) ( z) S H BS ( z) z Y X z B ( z) ( z) z S H CS ( z) z Y X z C ( z) ( z) S Page

42 EXPERIMENAL RESULS AND DISCUSSION FOR PARAMEER ESIMAION he freqeny response for the shifted Forward H FS (z), Baward H BS (z) and Centre H CS (z) Differene Approimation Methods are shown in Figre 5.4, 5.5 and 5.6 respetively, for s of ms. Figre 5.4 Figre 5.5 Figre 5.6 Sine the transfer fntions are Finite Implse Response (FIR) type, all the phase responses are linear, whih implies that there will not be any phase stortion of the otpt signal. However, sine the inpt and otpt seqenes are sed in the estimation proess, the delays on the invidal freqeny omponents from the fferent seqenes old have some effets on the reslts. Page

43 EXPERIMENAL RESULS AND DISCUSSION FOR PARAMEER ESIMAION 5.4. Effet of Using Unfored Osillation Response for Estimation he nfored osillation response of the data segment an be sed to estimate J and C. ables H, H and H5 are the reslts sing the fll response for estimation. ables H, H4 and H6 are the reslts obtained sing the nfored osillation response. Eamining the invidal tables, we see that the stribtion of vales for J and C within eah table is qite small. Comparing among able H and H, H and H4, H5 and H6, we observe that the reslts of estimation sing the nfored response generally prodes slightly lower vales. Sine the stribtion of the reslts is small for these data sets, we an onlde that reslts sing the fll response and the nfored osillation response are both valid Reslts of Estimation of J and C Eamining ables H to H6 shows that the stribtion of vales of J is small. However, for C, the reslts for data sets and are somewhat higher than the rest of the data sets. Observation of the error onstants Error A and Error B within eah table, we see that for data set, the simlated reslts are losest to the eperimental data ompared to the other data sets. his is despite the fat that C estimated sing this data set is not onsistent with the rest of the data sets Referene Model From 5.4., we have shown that sing the entral fferene approimation prodes more arate reslts than sing the forward or baward fferene approimation. From 5.4., we observed that sing the fll response or the nfored osillation response wold prode qite reasonable reslts. From 5.4., we see that even thogh data sets and prode vales of C that are inonsistent with the rest of the data sets, we annot sard these reslts bease their simlated reslts are the losest to the eperimental reslts. Page

44 EXPERIMENAL RESULS AND DISCUSSION FOR PARAMEER ESIMAION Hene a logial way to determine an aeptable vale wold be to average the reslts from all the data sets. Using the fll response for parameter estimation, entral fferene approimation for estimation of the derivatives, the average of the reslts (able H5) an be fond as: Data Set J C J C Average( to ) able 5. he linear model (Appen B, eqation (B8)) formed from these reslts is: B A (5.) he orresponng open loop poles are loated at: s -.98, s 7.698, s -7.8 he simlated responses from all the data sets are shown below for the reslts shown in able 5. above. he bold line is the simlated response and the thin line is the eperimental data. Figre 5.7 Figre 5.8 Page

45 EXPERIMENAL RESULS AND DISCUSSION FOR PARAMEER ESIMAION Figre 5.9 Figre 5. Figre 5. Figre 5. Figre 5. Figre 5.4 Page 4

46 EXPERIMENAL RESULS AND DISCUSSION FOR PARAMEER ESIMAION Figre 5.5 Figre 5.6 he bode plot of the linear model an then be obtained. For the otpt being defined as the arm veloity, the bode plot is shown in Figre 5.7. Figre 5.8 shows the ase when the pendlm position is defined as the otpt. Figre 5.7 Figre Disssion on Parameter Estimation Method 5.5. Referene Vales for to 8 From the referene model obtained in Setion 5.4.4, we an establish some referene vales that are epeted of to 8 for omparison prposes. he referene vales an be allated from (4.8) and (.4) to be: Page 5

47 EXPERIMENAL RESULS AND DISCUSSION FOR PARAMEER ESIMAION 5.5. Overview of Estimation Reslts Intitively, the vales to 8 shold not be negative qantities as these are derived from all positive qantities. Looing throgh ables H7 to H, we notie that for all the reslts obtained, at least one of the parameters is a negative qantity. Frthermore, from (4.8), observe that if the estimation is error-free, the following relationship shold hold: 5 6 However, this relationship does not hold for most, if not all, of the reslts obtained. his implies that visible estimation errors eist for all the reslts. Hene in this sssion, the main objetive is to stdy if an arate linear model an be obtained despite these errors Effet of Differene Approimation Method Similar effets where sing the entral fferene approimation gives a more arate model an be demonstrated with some of the data sets obtained. For data set, sing the nfored osillation response for estimation, the reslts are shown in Figre 5.9, 5. and 5. for forward, baward, and entral fferene approimation respetively. Figre 5.9 Figre 5. Page 6

48 EXPERIMENAL RESULS AND DISCUSSION FOR PARAMEER ESIMAION Figre 5. Comparing with the referene vales against ables H8, H and H, we see that H has the average vales of, 7, 8 losest to the referene Effet of Using Unfored Osillation Response for Estimation he parameters 7 and 8 an be estimated from the nfored osillation response sing eqation (4.). Using these reslts, 5 and 6 an then be estimated from eqation (4.). ables H7, H9 and H are the reslts sing the fll response for estimation. ables H8, H and H are the reslts obtained sing the nfored osillation response. Comparing ables H7 and H8, we see that sing the fll response for estimation, the reslts for 5, 6, 7 and 8 has a relatively large stribtion and their vales are frther from the referene ompared with reslts estimated sing the nfored osillation response. he same an be said for ables H9 and H as well as ables H and H. From these, we an onlde that sing the nfored osillation response for estimation sing Method prodes more reliable reslts. Moreover, we an establish that Method is a more reliable method ompared to Method bease Method is relatively insensitive to the effets of sing the nfored osillation response for estimation Reslts of Estimation Sine it has been fond that sing the entral fferene for approimation of the derived qantities prode better reslts, and sing the nfored Page 7

49 EXPERIMENAL RESULS AND DISCUSSION FOR PARAMEER ESIMAION osillation response to estimate the parameters is more reliable, we shall limit the sssion to the reslts in able H. Compare the estimated reslts with the referene vales established in Setion First, note that the reslts of,, and 5 are nreliable. hey are negative for most of the estimation reslts. he reslts that are more agreeable with the referene vales are 6, 7 and 8. Vales of and 4 are fairly lose to the referene. When forming the linear model with eqation (B9), the vales of and 5 are not reqired, therefore we an still have an arate linear model if the errors present in are not too prominent. aing the average vales so as to minimise the overall errors, the final reslts are tablated below. Data Set 4 Average( to ) Data Set Average( to ) able 5. he linear model (Appen B, eqation (B9)) formed from these reslts is: B A he orresponng open loop poles are loated at: s -.96, s 6.97, s Comparing these with the open loop poles of the referene model, we observe that they are fairly lose. Page 8

50 EXPERIMENAL RESULS AND DISCUSSION FOR PARAMEER ESIMAION he bode plot of the linear model an then be obtained. he bode plot of this model is plotted against that of the referene model for omparison prposes. For the otpt being defined as the arm veloity, the bode plot is shown in Figre 5.. Figre 5. shows the ase when the pendlm position is defined as the otpt. Figre 5. Figre 5. From the above reslts, we see that despite the errors present in the estimation proess, Method still prodes a fairly arate linear model when ompared to Method. Page 9

51 CONROLLER DESIN Chapter 6 Controller Design 6. Introdtion In a reglatory type ontrol system, the ontroller reglates the otpt and maintains it at a desired level as speified by some referene. When the referene is time varying, the ontroller will attempt to eite the system in sh a way that the otpt will tra the referene as losely as possible. Usally, the ontrol signal applied to the plant or system is a fntion of the fferene between the referene inpt and the atal otpt, ommonly referred to as the error signal. For a real system, strbanes will at on the plant. An effetive ontrol system shold be insensitive to eternal strbanes and responsive to hanging inpts. For this projet, the primary objetive is to balane the pendlm in its pright position. he seondary objetive is to maintain the arm at a fied position. Varios ontrollers were implemented to ahieve these. o bring the pendlm from its downward stable position to its pright position, a swing-p ontroller was implemented. his hapter presents the ontrollers sed in this projet and the theoretial bagrond assoiated with them. he ontrollers sed for balaning are linear state feedba ontrollers. hey are designed based on a nown model of the system. In Chapter 5, the parameters of the linear model were estimated. For presentation of this report, the model sed is as desribed by (5.). It is reproded here for ease of referene. B A Page 4

52 CONROLLER DESIN he blo agram of the open loop system is shown below. A B y C B C y A Figre 6. he following are the thirteen ontrollers being implemented based on fferent models sed: State Feedba Controller (Continos ime Model) 4 State Feedba Controller (Continos ime Model) State Feedba Controller with Integral (I) Ation (Continos ime Model) 4 State Feedba Controller with Integral (I) Ation (Continos ime Model) State Feedba Controller with Proportional (P) Ation (Continos ime Model) 4 State Feedba Controller with Proportional (P) Ation (Continos ime Model) State Feedba Controller with PI Ation (Continos ime Model) 4 State Feedba Controller with PI Ation (Continos ime Model) State Feedba Controller (Disrete ime Model) 4 State Feedba Controller (Disrete ime Model) 4 State Feedba Controller (Inremental Model) 5 State Feedba Controller (Inremental Model) eneralised Pretive Controller Page 4

53 CONROLLER DESIN 6. Controllability o be able to se the Pole Plaement ehniqe for ontroller design, the system mst be ompletely state ontrollable. herefore, before designing any ontroller, the ontrollability of the system mst be heed first. Consider the system: A B he above system is said to be state ontrollable at t t if it is possible to onstrt an nonstrained ontrol signal that will transfer an initial state to any final state in a finite time interval t t t. If every state is ontrollable, then the system is said to be ompletely state ontrollable. he system is ompletely state ontrollable if the ran of the n n matri, S t, is n. S t n- [ B AB A B L A B] Similarly, for a system desribed by: ( ) A d ( ) B ( ) d he system is ompletely state ontrollable if the ran of the n n matri, S dt, is n. S dt n- [ B A B A B L A B ] d d d d d d d Page 4

54 CONROLLER DESIN 6. Aermann s Formla B C y B C y A A - Figre 6. Figre 6. iven an open loop ontrol system (figre 6.) A B he ontrol signal sing pole plaement tehniqe is (figre 6.): o se Aermann s formla, the system mst be ompletely state ontrollable. It is important to note that the state feedba gain matri () is not niqe for a given system, bt depends on the desired losed-loop pole loation seleted. his will also determine the speed and damping of the response. he seletion of the desired losed loop poles is a ompromise between the speed of the response of the error vetor and its sensitivity to strbanes and measrement noise. hat is, if we inrease the speed of response, then the adverse effets of strbanes and measrement noise will generally inrease. For a seond order system, the system dynamis (response harateristis) an be preisely orrelated to the loation of the desired losed loop poles and zeros of the plant. Bt for higher order systems, the loation of the losed loop poles and the system dynamis (response harateristis) are not easily orrelated. Hene, in or system, in determining the state feedba gain matri () for the system, we need to eamine the response harateristis of the system for several sets of ontroller gains () and hoose the one that gives the best overall system performane. Page 4

55 CONROLLER DESIN 6.4 and 4 State Pole Plaement (Continos ime Model) r B C y A Figre 6.4 Figre 6.4 shows the blo agram of the inverted pendlm system with state feedba. he derivation of the losed loop ontrol system with state feedba is as shown A B y C r A B( r ) Below is the final state spae model, where A an be 44 matri for 4 state or for state. ( A B) Br y C Aermann's formla is sed to allate the feedba gain matri (). Using Matlab aer (A,B, poles) the vales of the feedba gain matri () an be easily allated. Page 44

56 CONROLLER DESIN 6.5 and 4 State Pole Plaement with Proportional Controller (Continos ime Model) r p B C y A Figre 6.5 Figre 6.5 shows the blo agram of the inverted pendlm system with state feedba and proportional ontroller. he derivation of the losed loop ontrol system with state feedba and proportional ontrol is as shown A B y C p( r C) A B( pc) Bpr and below is the final state spae model. y C ( A B BpC) Bpr Where A an be a 44 matri for 4 state or for state. Matri C will be dependent on what is being defined as the otpt. If the ser hose the proportional gain to at on the pendlm position for 4 state, matri C will be [ ]. Aermann's formla is sed to allate the feedba gain matri (). Using Matlab aer(a,b, poles) the vales of the feedba gain matri () an be easily fond. he ser will design a vale for proportional gain (p). he system will then he the loations of the poles with p inlded. his an be done sing Matlab eig(a- B-pBC). Page 45

57 CONROLLER DESIN 6.6 and 4 State Pole Plaement with Integral Controller (Continos ime Model) r e i B e A C y Figre 6.6 Figre 6.6 shows the blo agram of the inverted pendlm system with state feedba and integral ontroller. he derivation of the losed loop ontrol system with state feedba and integral ontrol is as shown A B A B( e ) y C e ir ic e ( r y) i e and below is the final state spae model. e y ( A B) ic e [ C ] B r e i Where A an be a 44 matri for 4 state or for state. Matri C will be dependent on what is being defined as the otpt. If the ser hose the integral gain to at on the pendlm position for state, matri C will be [ ]. Aermann's formla is sed to allate the feedba gain matri (). Using Matlab aer(a,b, poles) the vales of the feedba gain matri () an be easily allated. he ser will design a vale for integral gain (i). he system will then he the loations of the poles with i inlded. his an be done sing Matlab ( A B) B eig. ic Page 46

58 CONROLLER DESIN 6.7 and 4 State Pole Plaement with Proportional and Integral Controller (Continos ime Model) r z e i B e w C A y p A Figre 6.7 Figre 6.7 shows the blo agram of the inverted pendlm system with state feedba and both proportional and integral ontroller. he derivation of the losed loop ontrol system with state feedba and proportional and integral ontrol is as shown A B y C z r y w w e pz e zi A B( w ) ( A B) Bw ( A B) B( e pz) ( A B) Be Bpz ( A B) Be Bp( r y) ( A B) Be Bpr BpC e zi i( r y) ir ic and below is the final state spae model. e y ( A B pbc) e [ C ] ic B pb r e i Page 47

59 CONROLLER DESIN Where A an be a 44 matri for 4 state or for state. Matri C will be dependent on what is being defined as the otpt. If the ser hose the proportional and Integral ontrol to at on the pendlm veloity for 4 state, matri C will be [ ]. Aermann's formla is sed to allate the feedba gain matri (). Using Matlab aer(a,b, poles) the vales of the feedba gain matri () an be easily allated. he ser will design a vale for proportional gain (p) and integral gain (i). he system will then he the loations of the poles with p and i inlded. his an be done sing Matlab ( A BpBC) B eig. ic Page 48

60 CONROLLER DESIN 6.8 State Feedba Controller (Disrete ime Model) he srete-time model as desribed by eqation (.7) is reproded here. ( ) A d ( ) Bd( ) y( ) C ( ) d he blo agram of the losed loop system is shown below. r() - () () z - I () y() C d B d A d Figre 6.8 Using Aermann s formla, the ontroller gains defined by an be determined for a given set of losed loop poles. he ontroller an then be fond by: For r(), ( ) r( ) ( ) ( ) ( ) When the states ontinos time model is sretized to obtain A d, the reslting ontroller is the states feedba ontroller (designed sing the srete time model). Similarly, if the 4 states ontinos time model is sretized instead, the reslting ontroller is the 4 states feedba ontroller (srete time model). Fntionally, the state feedba ontroller designed sing the ontinos or srete time model is idential if the ontroller gains are the same. Sine the system is essentially a gital system, it wold be more appropriate to se gital ontrol tehniqes for design and analysis. Page 49

61 CONROLLER DESIN 6.9 State Feedba Controller (Disrete-ime Inremental Model) he srete-time inremental model as desribed by eqation (.8) is reproded here. where i i ( ) A i ( ) B ( ) y( ) C ( ) ( ) i ( ) ( ) he blo agram of the losed loop system is shown below. r() - ( ) () B i z - I i () y() C A Figre 6.9 Using Aermann s formla, the ontroller gains defined by an be determined for a given set of losed loop poles. he ontroller an then be fond by: r For r(), ( ) r( ) i ( ) ( ) ( ) ( ) ( ) r( ) i ( ) ( ) ( ) r( ) [ ] ( ) ( ) r( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Note that the ontroller gains ats on the state variables whereas ats on the previos ontroller otpt. his tehniqe an be applied to the states srete model or the 4 states srete model to obtain the 4 states inremental model or 5 states inremental model respetively. Page 5

62 CONROLLER DESIN Page 5 6. eneralised Pretive Controller Pretive Control belongs to the lass of model based ontroller design onepts. he onept of Pretive Control [] is presented in Appen D. For this projet, the 5 states inremental model is sed for design of the eneralised Pretive Controller. he derivation of the PC ontrol law is presented in Appen E. he ontroller an be written as where For the details on the atal allation of the ontroller gains, and, refer to Appen E. his partilar ontroller is atally a speial ase of the 5 states feedba ontroller desribed in Setion 6.9 above. However, the onept behind the ontroller is more omple. For a omprehensive treatment of this sbjet, refer to State Spae Pretive Control of Inverted Pendlm [] by Yong et Leong. ( ) ( ) ( ) ( ) ( ) w w i ( ) ( ) ( ) ( ) ] [ ] [ ] [ Φ Φ I I I σ σ λ σ σ σ λ σ σ σ λ σ σ L M L M L

63 CONROLLER DESIN 6. Swing Up Controller he swing p ontroller is reqired to swing the pendlm from its downward or pendant position to its pright position. he swing p ontroller sed for this projet is an adaptation of the ontroller proposed by Frta and M Yamaita in Swingp ontrol of inverted pendlm sing psedo-state feedba []. he swing-p ontrol is based on the sbspae projeted from the state spae. he partitioned state omposed of the pendlm s anglar position and veloity, whih are seleted as psedo-state, are assigned fferent aelerations of the arm. Based on the vetor fields in the partitions of the sbspae, the trajetories onneting the pendant state to the pright state in the psedo-state spae is determined. Eah vetor field orresponds to a ontrol law giving the presribed aeleration. In the trajetory determination, the anglar veloity of the motor is negleted, and it is assmed that the aeleration is of the bang-bang type. Consider eqation (A9) as stated in Appen A. mll os ( J ml ) ml sin C m gl sin ( J m l ) C m gl sin m L l os m l sin (6.) Now, sppose that the last term of the right-hand side eqation (6.) an be negleted, that is, m l sin his assmption is valid provided that the pendlm is light enogh to be swng p with small arm veloity. Conseqently, eqation (6.) is reded to: ( J m l ) C m gl m L l os sin (6.) When the arm aeleration is onsidered as an inpt to eqation (6.), the pendlm aeleration an be determined as: J m l ( C m gl sin m L l os ) Page 5

64 CONROLLER DESIN For a given state of the pendlm s anglar position and veloity, the inremental position and veloity an be allated as: s s where s is the sampling interval. he projeted vetor fields onto the sbspae an then be fond for a given arm aeleration. he projeted vetor fields are shown in Figres 6., 6. and 6. for arm aeleration of rad/s, 6rad/s and 6rad/s respetively. Figre 6. Figre 6. Figre 6. In the above vetor fields, the downward pendant state is mared by the symbol O. he desired state for the pright position where the pendlm position is ±π and veloity is, is mared by the symbol X. he horizontal ais speifies the pendlm s anglar position and the vertial ais speifies the anglar veloity. Page 5

65 CONROLLER DESIN Observe from Figre 6. that the sbspae will onverge at the origin where the anglar position and veloity are both zero. he net step ombines these projeted vetor fields to onstrt a bang-bang state feedba. For simpliity, it is assmed here that there is a ret proportional relationship between the motor s driving ommand,, and the arm aeleration. It an be observed that there are many swithing patterns of the arm aeleration that an swing the pendlm from the pendant position to the pright position. he swithing pattern implemented in this projet is shown in Figre 6. below: Region A, - Region B, Figre 6. his swithing pattern is hosen bease it is simple, easy to adapt for fferent systems, and it is intitively lear as to how the ontrol ation ats to swing p the pendlm. he ontrol ation is applied only in regions A and B. Region A an be speified as: π <, Region B an be speified as: π, here are no ontrol ations when the pendlm is above the horizontal. In this region above the horizontal, we se the fore of gravity to retard the pendlm before it reahes its desired state. Note that the sbspae trajetories, pon leaving regions A and B, will have a higher veloity magnitde or larger absolte anglar position when ompared to the nfored system. Over a few yles of swinging, the Page 54

66 CONROLLER DESIN anglar position will reah ±π with a low veloity. o allow adjstments for the speed of swing-p ation and the nmber of swings, let: S M where s is a ser spplied onstant between and, M is the maimm allowable motor driving ommand. Page 55

67 CONROLLER DESIN 6. Dead Zone Mapping When the motor s torqe annot overome the effetive moment of inertia of the load and the frition on the ais of rotation, the arm will be stationary. his is alled the dead zone effet of the motor. he ideal and atal torqe-inpt relationship is illstrated in Figres 6.4 and 6.5 respetively. Here, a simplisti linear model is assmed. τ M τ τ M τ - M M - M -dz dz M -τ M -τ M Figre 6.4 Figre 6.5 Dead zone mapping may be sed to ompensate for this effet. It is a simple mapping fntion that maps all driving signals that are within the dead zone to the bondary of the dead zone. he reslting torqe-inpt relationship is shown in Figre 6.6 below. τ M τ - M -dz dz M -τ M Figre 6.6 Note that the dead zone bondary for positive and negative ommands need not be the same. It is illstrated as sh for simpliity of presentation only. Page 56

68 EXPERIMENAL RESULS AND DISCUSSION FOR CONROLLER DESIN Chapter 7 Eperimental Reslts and Disssion for Controller Design 7. Introdtion his hapter presents the reslts obtained from the fferent ontrollers. It has to be stated that bease of the dead zone of the motor, the eperimental reslts have deviated too far from the simlated reslts that there annot be any meaningfl omparisons. As sh, in this report, no attempts have been made to ompare the reslts with the simlated reslts. Frthermore, there are no apparats available to be able to introde a onsistent strbane to the fferent ontrollers. However, the performane of the fferent ontrollers an be ompared with respet to other ontrollers. All the reslts are obtained from System II as stated in Appen F. he ontrollers were designed based on models derived from (5.) of Chapter 5. Eept for the PC ontroller, all the other ontrollers were designed sing Pole Plaement tehniqe. he measrable qantities are the pendlm position and the inremental arm position. As no state observers were implemented, the nmeasrable state variables are approimated by: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) s where () is the inremental arm position, is the sample inde. s Page 57

69 EXPERIMENAL RESULS AND DISCUSSION FOR CONROLLER DESIN he design proess is mostly throgh trial and error as there are no partilar speifiations to meet. he best responses are then adopted for presentation in this hapter. he reslts for the ontrollers are obtained with dead zone mapping trned off. he effets of dead zone mapping will be sssed separately. Page 58

70 EXPERIMENAL RESULS AND DISCUSSION FOR CONROLLER DESIN 7. State Feedba Controller (Disrete and Continos ime Model) he state and 4 state feedba ontrollers are designed sing pole plaement tehniqe. he srete and ontinos time models are shown below, based on the referene model obtained in Chapter 5. ( ) A d ( ) Bd( ) ( ) ( ).6 ( ) A B B 4 ( ) A 4d 4( ) B4d( ) ( ).99 ( ).9864 ( ) ( ).9 A ( ) ( ) ( ).57.9 ( ) ( ) ( ) ( ) ( ) ( ) he ontrollability matries for the state models are: S S d Ran Ran ( S ) ( S ) d Page 59

71 EXPERIMENAL RESULS AND DISCUSSION FOR CONROLLER DESIN he ontrollability matries for the 4 state models are: S S 4d Ran(S Ran(S 4 4d ) 4 ) 4 Sine all the ontrollability matries have fll ran, the pole plaement tehniqe an then be sed to design the state feedba ontrollers. he losed loop poles for the state feedba system are hosen to be at: i i.957 ( State Disrete ime Model) and: i i ( State Continos ime Model) he feedba ontroller s gains ( ) that orrespond to the above poles are shown below. his set of gains will be sed throghot this hapter for omparison and sssion. [-8 5 5] Page 6

72 EXPERIMENAL RESULS AND DISCUSSION FOR CONROLLER DESIN he ontroller response is shown below: Figre 7. From Figre 7., we an see that the system was able to balane itself pright. Note that the arm position information is wrapped arond sh that it will always be within the range of ±π. here is no ontrol over the arm position. Frthermore, the arm veloity has a non-zero steady state error and the arm an be seen to be rotating in the lowise retion at a fairly onstant speed. he arm position is not a state variable in the model and hene the ontroller otpt will be independent of the arm position. he arm position an be better ontrolled sing the 4 states feedba ontroller as the arm position is a state variable of the 4 states linear model. he losed loop poles for the 4 state feedba system are hosen to be at: i i (4 state Disrete ime Model) i i (4 State Continos ime Model) Page 6

73 EXPERIMENAL RESULS AND DISCUSSION FOR CONROLLER DESIN he feedba ontroller s gains ( 4 ) that orrespond to the above poles are shown below. 4 [ ] he ontroller response is shown below: Figre 7. From Figre 7., we an see that the arm position is now restrited within some finite bonds. here is an improvement on the ontrol of the arm position sing the 4 states feedba ontroller as ompared to the states feedba ontroller as epeted. For the state feedba ontroller, the otpt an be defined as the arm veloity, pendlm position, and pendlm veloity. For the 4 state feedba ontroller, the otpt an also be defined as the arm position. he integral ontroller (I), or proportional ontroller (P), and the proportional and integral ontroller (PI) an be onfigred to at on eah of these otpts. As sh, the nmber of possible ombinations is very large and a treatment of every ontroller wold be very intensive. he sssion on the I ontroller, P ontroller and PI ontroller will be limited to its effet on the system with state feedba, taen as a plant to be ontrolled. Page 6

74 EXPERIMENAL RESULS AND DISCUSSION FOR CONROLLER DESIN 7. State Feedba Controller with Integral Ation (Continos ime) When an integrator is added to a system and the loop losed, the steady state error de to step inpts will rede to zero. his inreases the system type by. he integral ontroller is sed primarily to improve the steady state response of ontrol systems. his setion eplores the effets of sing the integral ontroller to improve the steady state response of the state feedba system taen as a whole. aing the system with state feedba as the plant to be ontrolled, the poles are reproded below: i i With an integral gain of 8 whih ats on the arm veloity, the losed loop poles are: i i i i he system s response is shown below: Figre 7. By having the integral ation on the arm veloity, the arm veloity shold rede to zero theoretially. From the reslts above, we see that the arm position is fltating Page 6

75 EXPERIMENAL RESULS AND DISCUSSION FOR CONROLLER DESIN abot zero. Even thogh the arm veloity annot rede to zero, as it shold, the mean of the arm veloity over a period of time an be said to be near zero. he reason for this sparity is de to the prominent effet of the dead zone. When the orreting error signal falls within the dead zone of the motor, it will not be able to drive the motor and hene the error will inrease ntil a level that is otside the dead zone before the ontroller an tae effetive ation. his partilar ontroller ombination is effetively a 4 state feedba ontroller with ontroller gains at: 4 [ ] Comparing this ontroller s response to that of the 4 state feedba ontroller, we an see that the responses are qite similar in terms of the magnitde of fltation of the arm and pendlm position. Page 64

76 EXPERIMENAL RESULS AND DISCUSSION FOR CONROLLER DESIN 7.4 State Feedba with Proportional Control (Continos ime) A proportional ontroller is essentially an amplifier with an adjstable gain. A proportional ontroller inreases the speed of the response and improves the transient harateristis. he gain vales ( matri) that orrespond to the poles are as below whih is similar to the one se in the state ontroller With a proportional gain of (p ) ating on the pendlm position. he losed loop poles shifted from I I to i i Page 65

77 EXPERIMENAL RESULS AND DISCUSSION FOR CONROLLER DESIN Figre 7.4 From figre 7.4 above, we an see that the driving inpt had inreased. his verified the proportional ontroller ation, whih ats lie an amplifier to the inpt error signal. From the figre, we an see that the system response is faster. he veloity is abot three or for times that of the state feedba ontroller. Page 66

78 EXPERIMENAL RESULS AND DISCUSSION FOR CONROLLER DESIN 7.5 State and 4 State Feedba Controller with Proportional and Integral Control (Continos ime) he Proportional and Integral is basially a lag ompensator. his ontroller an improve the steady state harateristis. By properly designing the PI ontroller, it is possible to mae the transient response relatively small, bt may ase the speed of response to slow down. his is bease the PI ontroller, being a low pass filter, will attenate the high freqeny omponents of the signal. With p 5 and i8 set he losed loop poles shifted from I I for the state feedba to i i e-8 With PI ontrol ation on pendlm position. Figre 7.5 With PI ation, the system performane ompared to the state feedba ontroller is jst slightly better. his is de to the adtional parameters (P gain and I gain) that made it ffilt to fine tne. Page 67

79 EXPERIMENAL RESULS AND DISCUSSION FOR CONROLLER DESIN Page State Feedba Controller (Inremental Model) (Disrete ime) he inremental models sed have inlded an etra state variable (-). Eamination of this state variable reveals that it is effetively performing an integral ation over all the other state variables. Assming that all the initial states of the state variables are all zero, then the following an be written: Observe that the ontroller otpt is performing an integral ation, that is, smming p all the previos state variables over a period of time. As sh, we wold epet good steady state response from the ontrollers designed sing this inremental model. he 4 state and 5 state inremental model are shown below: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) i i i i i i i i i i i i i i ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) i i i i B A B A

80 EXPERIMENAL RESULS AND DISCUSSION FOR CONROLLER DESIN For the 4 state inremental model, the poles are loated at: z.995, z.987, z, z 4.7 For the 5 state inremental model, the poles are loated at: z.995, z.987, z, z 4, z 5.7 he ontrollability matries are: S 5i S4i Ran Ran ( S ) 4i 4 ( S ) 5 5i Sine all the ontrollability matries have fll ran, the pole plaement tehniqe an then be sed to design the state feedba ontrollers. he losed loop poles for the 4 state (inremental model) feedba system are hosen to be at:.97.59i i.9.76 he feedba ontroller s gains ( 4i ) that orrespond to the above poles are shown below. 4i [-8.4] Page 69

81 EXPERIMENAL RESULS AND DISCUSSION FOR CONROLLER DESIN he ontroller response is shown below: Figre 7.6 From Figre 7.6, we see that its response is mh more sperior ompared to the states feedba ontroller. he arm veloity, in this ase, is very low. he pendlm position is fltating very losely abot zero. However, the motor driving ommand is qite large. he losed loop poles for the 5 state (inremental model) feedba system are hosen to be at: i i he feedba ontroller s gains ( 4i ) that orrespond to the above poles are shown below. 5i [-5 -.4] Page 7

82 EXPERIMENAL RESULS AND DISCUSSION FOR CONROLLER DESIN he ontroller response is shown below: Figre 7.7 From Figre 7.7, we an observe that among all the ontrollers, the 5 state feedba ontroller gives the best response. he arm position an be onstrained within a very small region of approimately ± degrees while the pendlm is ept pright. Page 7

83 EXPERIMENAL RESULS AND DISCUSSION FOR CONROLLER DESIN 7.7 eneralized Pretive Controller he model sed for implementation of the PC ontroller is the 5 state inremental model as stated in Setion 7.6 above. In State Spae Pretive Control of Inverted Pendlm [] by Yong et Leong, the athor gave a omprehensive treatment of the sbjet and a systemati way of tning the ontroller. he ontroller designed for this report has the following parameters: Ny, Ny, N, σ, σ, λ. he orresponng ontroller gains are: [ ] he losed loop poles an then be fond to be:.97.57i i i i he ontroller response is shown below: Figre 7.8 It an be seen that the PC response is very similar to the 5 state feedba ontroller response designed sing pole plaement tehniqe. he PC ontroller and the 5 state feedba ontroller are both based on the inremental model whih has the Page 7

84 EXPERIMENAL RESULS AND DISCUSSION FOR CONROLLER DESIN effet of integrating all the state variables. his effetively redes the steady state errors even thogh the dead zone of the motor had shown to overwhelm other forms of steady state ompensation. herefore we an onlde that sing the 5 state inremental model for design of ontrollers will give the best response among all the ontrollers presented here. Page 7

85 EXPERIMENAL RESULS AND DISCUSSION FOR CONROLLER DESIN 7.8 Swing-Up Controller he swing-p ontroller an be fine tned sh that the pendlm is swng p to its pright position in motions, one swing in either retion to gain momentm, and the seond swing in the opposite retion to reah the pright position. his finetning is done by adjsting the amplitde of the driving signal. he amplitde where the swing-p an be ahieved in motions will be referred to as the ritial amplitde. For a swing-p gain of.54, orresponng to a driving amplitde of onts, the following response is ahieved. Figre 7.9 We see that the pendlm is swng p with lean motions, after whih the balaning ontroller too over ontrol. Note that the glith appearing at arond the th to 4 th sample of the pendlm position is ased by an instrmentation error. he ritial amplitde an be easily fond for a partilar set by the trial and error approah. It shold be noted that it will give onsistently good reslts only if the same initial ontions are met every time, that is, the pendlm s starting position is at the downward position and the veloity is zero. However, this initial ontion will not be met if the balaning ontroller was to fail and the swing-p ontroller was to tae over ontrol. It wold start off from some non-zero pendlm position and veloity and sbjet to that, the swing-p may fail bease the final veloity may be too high for the balaning ontroller to ath on. Page 74

86 EXPERIMENAL RESULS AND DISCUSSION FOR CONROLLER DESIN 7.9 Effet of Dead Zone Mapping he effets of dead zone mapping an only be observed if the ontroller driving otpt signals are small. o demonstrate the effets, a 5 state feedba ontroller was designed sing the pole plaement tehniqe whih will prode small driving signals. he ontroller gains are: 5i [ ] he ontroller response withot dead zone mapping is shown in Figre 7. below. Figre 7. he ontroller response with dead zone mapping is shown in Figre 7. below. Figre 7. Comparing the above responses, it an be seen that the arm position, as well as the pendlm position, is reglated better with the dead zone mapping trned on. his an only be said for ontrollers that prode small ontrol signals whih lie within the dead zone. Otherwise, there are no notieable effets Page 75

87 RAPHICAL USER INERFACE Chapter 8 raphial User Interfae 8. Bagrond he operations for varios proesses are often interonneted and rather ompliated and teos to perform. Even the operation of getting the pendlm rnning, varios shelling proesses that to be done. his an prove to be repelling to the ser, ase not only does he need to waste preios time. He also needs to nderstand and perform the messy, teos and nfriendly operations. In proess, areless mistaen will also arise. his is de to the fat that most of the pendlm ontrol software e.g. ECM, LIB96 et are Dos environment programs. While all the omptation and allation are done in Matlab environment. Constant shelling between Dos, ECM, Windows and Matlab environment has to be performed. Frstration and onfsion always hinder the ser in many instanes, as we have eperiened. aing to retrieve data from the pendlm as an eample, ser will have to shell ot of Matlab, ot to Windows and then into ECM (Dos) environment. hen he will have to wait for the data to be downloaded, after that he will have to toggle ba to Windows and rn Rawmat (Dos) to onvert the data into Matlab data file. After this he will have to shellba to Matlab environment whih is anhor in Window environment. o do this, withot the help of an integrated UI, an prove to be very repelling and error prone. Page 76

88 RAPHICAL USER INERFACE 8. Objetives Most engineering orses ontain abstrat onepts that are important for the nderstanng of material espeially in ontrol stdy. he main objetive for implementing a UI is mae ompliated and teos operations transparent to the ser. he idea is to let the ser have all the time and onentration to design and implement their design. In that aspet, the transparent operations of the UI have been designed in sh a thoght in mind. It mst be ser-friendly, aptivating, easy to operated and aid ser nderstanng. Yet it mst not limit the freedom of design and mst also be stimlating to the ser. In this 98 version of the UI, we have maimized Matlab ver 4. graphial ability to it s fllest to reate a beatifl front-end interfae for the ser. With an integrated UI, the problem of data storage will also be solved at the same time. As the nmber ontrollers and design inreases, files and programs handling will be more ompliated. With a standard database system that omes with the UI, ser will have no problems figring ot where are the files and program. It will also allow fferent sers to qily arry on researh and stdy on any ompter system that have been installed with this UI interfae. Setion 8.4 will serve as gide and an instrtion set to operate the UI. Page 77

89 RAPHICAL USER INERFACE 8. Featres he UI is bilt flly in Matlab environment. User with fferent nowledge in ontrol engineering has been onsidered while designing the UI. he UI is designed to provide a ommon platform for both advane ser and beginner ser. he whole interfae will integrate all the operations and mae it transparent to the ser. his enables the ser to be flly onentrate in the learning proess. he program also vides the proess into varios modles, whih allow ser to perform the desire proess at any time he wants. he Interfae is ser-friendly, aptivating, easy to operated and aid ser nderstanng. It have ser with variety and does not limit the freedom of design and mst also be stimlating to the ser. A ommon database system will inter-lin eah part of the proess. his solve messy file storage problem. hs allowing data to be ommon in all part of the program. A Pentim system with 64MB of ram is reommended to rn this program. his is bease the UI Interfae program is bilt will intensive graphial enhanement and error heing. Page 78

90 RAPHICAL USER INERFACE 8.4 raphial User Interfae Version 98 (Sreen Shots and Introdtion) 8.4. Startp Sreen o begin, start Matlab and hange retory to Rn retory. On Matlab prompt, type <o>. Figre 8. on the right will appear after the go ommand. he welome sreen will notify the ser the starting of the program. his at as a welome men, and show the rrent version of the UI. o proeed with the program, the ser will have to li <Contine > to enter to the UI. Figre Introdtion Sreen he introdtion sreen (Figre 8.). It shows the objetives of this UI interfae program. his sreen allows the ser to aess the Beginner and Advane. Depenng on the ser ontrol nowledge, he has the freedom to go into either of them by liing on <Beginner> or <Advane> btton. Figre 8. System Configration an be aess by liing on the <System Config> btton. Page 79

91 RAPHICAL USER INERFACE 8.4. System Configration Interfae he System Configration Interfae option (Figre 8.) splay the system parameters sed by the software in all its allation. Eample lie armatre rotation onts whih refers to the enoder resoltion in ont/rev. In ase the enoder has to be hange, we only need et the parameter here. he whole program will still fntion with the new enoder. Figre 8. he ser an also et the system parameters sed for Method of System Identifiation Beginner Interfae Option he Beginner Interfae option (Figre 8.4d) will at as a teahing aid. It will eplain for the whole system. It is design to offer step by step gide to the new ser to this system. System modeling, identifiation, design, implementation and the testing of ontroller are being gided step by step. With this 98 version of the UI, we have maimized Matlab graphial ability to it s fllest to reate a beatifl front-end interfae for the ser. Figre 8.4 System Setp and Mathematial Model Interfae an only be invoed in this interfae. Page 8

92 RAPHICAL USER INERFACE Advane Interfae Option As for the Advane Interfae Option (Figre 8.5), the ser will have less worly desription and notes. Advane option interfae, the ser will retly go into the varios proesses he desire to perform. He will not enonter detail gide and eplanation sreens as it is assmed the ser is familiar with the system. his fferene is to let the advane ser do what he need, faster withot all those detail eplanation. Figre 8.5 Etra featres lie mltiple System Identifiation an only be aess from this option. Page 8

93 RAPHICAL USER INERFACE System Setp Interfae System Setp Main Interfae sreen shot (Figre 8.6) an be invoes only from the Beginner Interfae Option. he interfae onsists of 5 sreen. hese 5 sreen will desribe the basially the system setp, the apparats and eqipment sed. his interfae will also offer a detail eplanation of eah of the fntions of eah of the omponents. (See Chapter for fll details) Figre 8.6 here are a total of 5 sreen shots that mae p the System Setp Interfae. he objetive for this interfae is to let the ser now the apparats and omponents of the Inverted Pendlm System. Software desription sreen shot (Figre 8.7) desribes the fntion of this UI. How this UI will integrate the entire environment and software. Proving a transparent, easy to se, friendly, fast, interesting, aptivating et. learning platform for the ser. Figre 8.7 Page 8

94 RAPHICAL USER INERFACE Personal ompter system sreen shot (Figre 8.8) state the fntion and roles of the personal ompter in this setp. How it being se as a sed as a development platform. Figre 8.8 RI PP- Inverted Pendlm Apparats sreen shot (Figre 8.9) provide a detail desription of the devies and eqipment of the pendlm system and the enoder and potentiometer is sed. Figre 8.9 Universal Controller UC96 Miroontroller sreen shot (Figre 8.) desribe the singleboard miroontroller system sed for the pendlm system. How this ontroller an be sitable for implementing a wide variety of standalone embedded systems. Figre 8. Page 8

95 RAPHICAL USER INERFACE Motor Driver Board Power Spply Unit sreen shot (Figre 8.) present the design of the motor drive board. How to give rise to high reliability and better onnetivity between the miroontroller and driver board and to drive the motor retly from the board. Figre 8. his sreen also desribed the power spply system Page 84

96 RAPHICAL USER INERFACE Mathematial Modeling he Mathematial Modeling Interfae an only be aess nder the Beginner Interfae Option. his set of interfae onsists of sreen. Jst lie the System Setp Interfae, it is design to provide new ser to this system to provide them with some basi bagrond nowledge of ontrol theory sing those that sed here. his set of interfaing sreen wills desribes the derivations of the mathematial model se by the system. It will over the whole mathematial derivation proess. How we se the Lagrange s Eqation of Motion of formlate the dynami model. How we finally get the inremental model for 5 state inremental ontroller and eneral Pretive ontroller. he Mathematial Modelling option main interfae sreen shot (figre 8.), welome the ser to the Mathematial Modelling series of interfae. Figre 8. Figre 8., shows the oornate system whih is sed in the derivation proess. Figre 8. Page 85

97 RAPHICAL USER INERFACE Figre 8.4, shows how the Lagrange s Eqation of Motion and the motor eqation is sed. Figre 8.4 Figre 8.5, show how the linearized model is obtained. Figre 8.5 Figre 8.6, show the srete model and it s derivation. Figre 8.6 Page 86

98 RAPHICAL USER INERFACE Misellaneos his interfae (figre 8.7) an be aess both nder the Beginner or Advane Interfae Option. his interfae is design to find ot (measre) two sets of parameters of the system. he dead zone and swing-p referene parameters are important, bease they are need in balaning the system. (Refer to hapter 6.5 ) he ser shold aess this interfae from time to time or when the pendlm set is hange. his is bease of the fferent pendlm have fferent harateristi de to wear and tear. Figre 8.7, show the Misellaneos sreen. he instrtions to follow are flly domented in the interfae. As the two proesses tae some time to perform and the system might seem not to be woring. Users are advie to follow the instrtions areflly. Figre 8.7 Page 87

99 RAPHICAL USER INERFACE System Identifiation his set of interfaes an be aess both nder the Beginner or Advane Interfae Option. Bt ser may eperiene some fferenes when aess this interfae from the fferent two option. his is de to the design of the system atering for fferent ser nowledge level in the system. his interfae is design to ollet the data from the system and perform the estimation proess to omplete the proess of system modelling. he objetives of this interfae are to provide a platform for ser to olleted data in an easy and hassle free matter. he interfae is also eqipped with verifiation faility so that the ser an he if the data olleted are orret. Beside that this interfae will serve as a gide to the ser to gide him along step by step in the proess of System Identifiation. System Identifiation onsists of two main modle, whih are Parameter Estimation and est Signal Design. est Signal Design Interfae is the same for both Beginner and Advane Interfae Option. Bt for Parameter Estimation, Beginner Interfae Option an only aess the Single Parameter Estimation, where else ser from the Advane Interfae Option an aess both single parameter estimation and mlti-parameter estimation interfae. his set of interfae is one of the most graphial intense of the whole UI. It pshes Matlab Ver4. graphial apability to it s fllest espeially in the single parameter estimation Interfae. Where the data have to be proess, splay and graphially enhane while eeping tra of the ser ontrol hanges. Page 88

100 RAPHICAL USER INERFACE Figre 8.6 will appear only aessible from the Beginner Interfae Option. For this option the ser is restrited only to single parameter estimation method interfae. After this he will be bring to a few more sreens of eplanation and gide. Figre 8.8 he sreen shot eample on yor left (figre 8.9), is an eample of an eplanation sreen that will only appear if the ser aess System Identifiation from the Beginner Interfae Option. Figre 8.9 Figre 8. show interfae will only be shown when aess from the Advane Interfae Option. For this option the ser is not restrited only to single parameter estimation method, he also allow into the mlti-parameter estimation interfae. Figre 8.. Page 89

101 RAPHICAL USER INERFACE he single parameter estimation interfae (figre 8.). his option allows the ser to perform eperiment to ollet data for parameter estimation to omplete System Identifiation. As single parameter estimation interfae is design espeially for new ser to the system. he interfae is design to gide the ser along in the progress. Figre 8. he sreen shots on the left (figre 8. and figre 8.) shows proess as it is being performed. he interfae will show the ser how the data (waveforms) are being filtered and proessed step by step. his allows the ser to now what is atally happening and give him the ontrol to learn rather than jst following instrtion. his set of interfae also at as an verifiation tool to allow the ser to he whether the data or system is being model orretly. Figre 8. Figre 8. Page 9

102 RAPHICAL USER INERFACE Mlti-parameter estimation interfae (figre 8.4). User an only aess this interfae form the Advane Interfae Option. User an allow mae the system to ollet mltiple set of data form the system and have it analyzed and proessed. his will average the reslt and rede the error. herefore have a better estimation. Figre 8.4 Physial Parameter sreen (figre 8.5) will appear when the ser li on <Physial Parameter>. his sreen is to let ser save some of the vales for method of Parameter Estimation in System Identifiation. Figre 8.5 Page 9

103 RAPHICAL USER INERFACE he reslt sreens (figre 8.6). he otpt of System Identifiation is to get a model that an model the system. Both the mlti parameter estimation interfae and single parameter estimation interfae of the System Identifiation will prode this reslt sreen at the end of the estimation proess. his sreen wills self invoe in the mlti parameter estimation interfae. Where in the single parameter estimation interfae the ser have to li to the <Reslt> btton to see this sreen. Figre 8.6 Finally the sreen (figre 8.7) will allow the ser to save the model for Control Design. he save file will be in the format of : filename.lm in the datam sbretory. Figre 8.7 he reslt sreen an also invoe Matlab Simlin (Figre.8) where we have design an animation to simlate the pendlm system with the estimated parameters. From the simlation the ser an verify and he the reslt enjoy the graphials. Note that the ser while in Simlin, have to pll down Simlation toolbar and li on <Start> to begin. Figre.8 Page 9

104 RAPHICAL USER INERFACE Design of a test signal for System Identifiation is not only to let the ser have some variety and fn. he main objetive is to mae System Identifiation of the Modelling Proess more arate and versatile. Letting the ser design a test signal, he an ompare the system response and see the fferent in the data obtain. De to wear and tear de to prolong se, shafts and joints will may ehibit fferent frition onstant. Other fator sh as dead zone of the motor et an be better map and ths data obtain will be more arate. he welome sreen (figre 8.9) will eplain what the ser hoie in designing the test signal. By defalt a test signal with a ont of 6 and se period is defalted into the system. he ser an revert to the defalt test signal anytime by pressing the <Defalt> btton form this sreen. Figre 8.9 he ser an also hoose to load a previosly saved test signal by liing on the load btton. he save file will be in the format of: filename.ts in the datam sbretory. (figre 8.) Figre 8. Page 9

105 RAPHICAL USER INERFACE Form the main est Signal Interfae sreen, the system also allow the ser to mofy the standard waveform period or magnitde by pressing on the <Standard> btton (figre 8.). Figre 8. By pressing on the <Design> btton on the main page, ser an aess the waveform design interfae. his interfae allows the ser design the time for eah of period and designs the magnitde of eah period. After the satisfatory design is obtain, he an save the test signal for ftre se. (figre 8.) Figre 8. his set of interfaes also provides the ser to view the test signal anytime he need (figre 8.). He jst has to li on <View est Signal> and the signal waveform will be plotted for the ser to see. his featre is available on every one of the test signal interfae sreens. Figre 8. Page 94

106 RAPHICAL USER INERFACE 8.4. Controller Design he most interesting part of the program. his is the finale in the whole orseware. he ser will design a ontroller to balane the Pendlm System here. he ontroller design will mae se of the linear model estimated form the System Identifiation Interfae. Using the model estimated in System Identifiation. he ontroller design will base on that model to design the ontroller sing Pole Plaement Method (Refer to hapter ). here are a total of ontrollers implemented for the ser to hoose from. If the ser end the Controller Design Interfae withot performing or going throgh System Identifiation. A sreen prompt will as him to perform System Identifiation or to load a previosly saved model. (Figre 8.4) Figre 8.4 Loang a saved model will ase the system to let the ser go into the main page of Controller Design Interfae. Another way is to perform System Identifiation.(figre 8.5) Figre 8.5 Page 95

107 RAPHICAL USER INERFACE Controller Design Main Interfae will let the ser have a hane to selet from twelve ind of ontroller design. (figre 8.6) One the type is seleted, the gain and variables will atomatially loaded into the system. Figre 8.6 he following sreen shots, shows samples of sreen shots ontroller design (Figre 8.7, 8.8, 8.9, 8.4). Depenng on the type on ontroller the ser selets, the design proedre varies. All ontroller are design are sing pole plaement method. he basi designs proedres are as follow. Figre 8.7 First enter the vales of the poles, and li on the find gain btton. Depenng on the Continos ime or Disrete ime option the ser have hoosen. he system will warn the ser whether his design is stable or not. he ser an also vary the ompted gains to see the hanges in the pole positions to design the system. Figre 8.8 Page 96

108 RAPHICAL USER INERFACE Another featre is a <Root Los> btton. his btton will plot the system show the ser the root los plots. From there, the ser an see and verify his design. If the ser hoose design option with either integral or proportional ontrol or both, he an se the plot to verify his design and this is very sefl. He will see what will happen to the system when he varies the PI gain. Figre 8.9 Another implementation is we allow the ser to hoose the state variable that the ontroller (P, I or PI ) at on. A stable set of gain design whih is stable and ating on arm veloity may not be stable if it at on the pendlm position. Finally, one he satisfied he old invoe the system to rn the pendlm to verify his designed ontroller. He an also save the designed ontroller for demonstration in the Demonstration Interfae Option. Figre 8.4 he system will also prompt the ser if his design is not stable. Page 97

109 RAPHICAL USER INERFACE 8.4. Demonstration User an aess this interfae (figre 8.9) either from the Advane or Beginner Option. his interfae is design to let the ser show off the ontrollers he have designed. User need jst to selet from the list of twelve ontroller similarly to the one fond in Controller Design Main Interfae. By li on <Demo> the system will rn the save design by itself. Figre 8.9 his fntion reates versatile for the ser. Not only allow the ser to save his best design for demonstration. It also provide a plae for the ser to eep a referene set perfet ontroller design for referene. Note : Maltlab only aept raw format graphi file type. Yo have to visit to onvert yor graphi file into 8 bit HIS raw format. Matlab ommand lie imread will prode great image ompare to reang in retly as bmp or jpg format. Page 98