Balancing an Inverted Pendulum Project Description Consider the problem of controlling an inverted pendulum on a moving base, as shown in Figure 1. The pendulum is assumed to be ideal, i.e., the pivot at the moving base is frictionless and its rod is massless. The transfer function of the plant is G(s)= -1 M b L s 2- (M b+M s )g M b L . Page 1 of 4 Figure 1: An inverted pendulum on a moving base Input ? g M b M s EE 141 Course Project Winter 2012 The design objective is to balance the pendulum in the presence of disturbance inputs. The system is illustrated in Figure 2 as a block diagram. For this project, use a lead/lag compensator of the form D(s)= -K(s+a) s+b The input/output of the system are summarized in Table 1 and the system parameters are given in Table 2. Input Description Value/Unit ? d (t) Reference input 0°, t>0 w(t) Disturbance (unit step) 1°, t>0 Output Description Value/Unit ? (t) Angle away from the balance center See Table 3. Table 1: System input/output description Page 2 of 4 Figure 2: A block diagram representation D(s) G(s) + - ? d (t) ? (t) + + w(t) EE 141 Course Project Winter 2012 Parameter ID Description Value/Unit M s Mass of the sphere 10 kg M b Mass of the moving base 100 kg L Length of the pendulum 1 m g Standard gravity 9.81 m/s2 a Lead/lag compensator parameter 1 5 b Lead/lag compensator parameter 2 10 Table 2: System parameters The output signal specifications, based on the system I/O description, are listed in Table 2. Performance Parameter Specification Settling time (1% criterion) < 10="" seconds="" percent="" overshoot="">< 40%="" steady-state="" tracking="" error="">< 0.1° table 3: output specification the project task is to develop a set of matlab scripts and/or functions to aid in the control system design. the guidelines in developing the scripts/functions are as follows. the matlab scripts/functions should: a) compute the closed-loop transfer function from the disturbance to the output, given a value for k , b) draw the bode plots of the closed-loop system, c) estimate and display the resonant peak and resonant frequency, d) estimate and display the second-order system parameters ? and ? n , e) estimate and display the settling time and percent overshoot using the estimated ? and ? n , f) plot the response ? (t) to the unit step disturbance with appropriate labels. use an iterative process to determine k. estimate the initial value of k by applying concepts introduced during the course. if the performance specifications are not satisfied, change k and rerun your matlab scripts/functions. the iterations may be implemented as conditional loops in your scripts/functions to automate the process or may be manually executed. note: if the iteration is manually executed, include the list the values of k that are evaluated in the documentation. page 3 of 4 ee 141 course project winter 2012 project requirements i. group size requirement the number of the project group members shall not exceed 3. ii. delivery items requirement the following items shall be delivered to the course instructor: a. documentation (project report) generate a report that includes: i. project description summary: in your own words, summarize briefly what the problem is and provide the group member list (name and student id) ii. analysis: describe briefly any analytical steps performed in the project and highlight all intermediate results iii. design verification and conclusion: provide the proof(s) that the controller has met all the performance specifications b. matlab files all matlab files generated and used during the project. include short in-line comments as coding documentation for key lines of codes. iii. delivery format and method requirement a. the report shall be in electronic form. if the report is hand-written, provide its scanned image pdf file. b. the report and matlab scripts/functions shall be zip-archived. c. the archived file shall be delivered to the instructor via an e-mail to taeroh[at]ymail.com with the subject line “ee 141 project report.” d. no project that is received by the instructor after 11:59 pm, march 11, 2012 shall be graded for any partial credit. grading criteria i. completeness (~75%) ii. demonstration of course material mastery (~15%) iii. presentation (~10%) page 4 of 4 0.1°="" table="" 3:="" output="" specification="" the="" project="" task="" is="" to="" develop="" a="" set="" of="" matlab="" scripts="" and/or="" functions="" to="" aid="" in="" the="" control="" system="" design.="" the="" guidelines="" in="" developing="" the="" scripts/functions="" are="" as="" follows.="" the="" matlab="" scripts/functions="" should:="" a)="" compute="" the="" closed-loop="" transfer="" function="" from="" the="" disturbance="" to="" the="" output,="" given="" a="" value="" for="" k="" ,="" b)="" draw="" the="" bode="" plots="" of="" the="" closed-loop="" system,="" c)="" estimate="" and="" display="" the="" resonant="" peak="" and="" resonant="" frequency,="" d)="" estimate="" and="" display="" the="" second-order="" system="" parameters="" and="" n="" ,="" e)="" estimate="" and="" display="" the="" settling="" time="" and="" percent="" overshoot="" using="" the="" estimated="" and="" n="" ,="" f)="" plot="" the="" response="" (t)="" to="" the="" unit="" step="" disturbance="" with="" appropriate="" labels.="" use="" an="" iterative="" process="" to="" determine="" k.="" estimate="" the="" initial="" value="" of="" k="" by="" applying="" concepts="" introduced="" during="" the="" course.="" if="" the="" performance="" specifications="" are="" not="" satisfied,="" change="" k="" and="" rerun="" your="" matlab="" scripts/functions.="" the="" iterations="" may="" be="" implemented="" as="" conditional="" loops="" in="" your="" scripts/functions="" to="" automate="" the="" process="" or="" may="" be="" manually="" executed.="" note:="" if="" the="" iteration="" is="" manually="" executed,="" include="" the="" list="" the="" values="" of="" k="" that="" are="" evaluated="" in="" the="" documentation.="" page="" 3="" of="" 4="" ee="" 141="" course="" project="" winter="" 2012="" project="" requirements="" i.="" group="" size="" requirement="" the="" number="" of="" the="" project="" group="" members="" shall="" not="" exceed="" 3.="" ii.="" delivery="" items="" requirement="" the="" following="" items="" shall="" be="" delivered="" to="" the="" course="" instructor:="" a.="" documentation="" (project="" report)="" generate="" a="" report="" that="" includes:="" i.="" project="" description="" summary:="" in="" your="" own="" words,="" summarize="" briefly="" what="" the="" problem="" is="" and="" provide="" the="" group="" member="" list="" (name="" and="" student="" id)="" ii.="" analysis:="" describe="" briefly="" any="" analytical="" steps="" performed="" in="" the="" project="" and="" highlight="" all="" intermediate="" results="" iii.="" design="" verification="" and="" conclusion:="" provide="" the="" proof(s)="" that="" the="" controller="" has="" met="" all="" the="" performance="" specifications="" b.="" matlab="" files="" all="" matlab="" files="" generated="" and="" used="" during="" the="" project.="" include="" short="" in-line="" comments="" as="" coding="" documentation="" for="" key="" lines="" of="" codes.="" iii.="" delivery="" format="" and="" method="" requirement="" a.="" the="" report="" shall="" be="" in="" electronic="" form.="" if="" the="" report="" is="" hand-written,="" provide="" its="" scanned="" image="" pdf="" file.="" b.="" the="" report="" and="" matlab="" scripts/functions="" shall="" be="" zip-archived.="" c.="" the="" archived="" file="" shall="" be="" delivered="" to="" the="" instructor="" via="" an="" e-mail="" to="" taeroh[at]ymail.com="" with="" the="" subject="" line="" “ee="" 141="" project="" report.”="" d.="" no="" project="" that="" is="" received="" by="" the="" instructor="" after="" 11:59="" pm,="" march="" 11,="" 2012="" shall="" be="" graded="" for="" any="" partial="" credit.="" grading="" criteria="" i.="" completeness="" (~75%)="" ii.="" demonstration="" of="" course="" material="" mastery="" (~15%)="" iii.="" presentation="" (~10%)="" page="" 4="" of="">