Control of an Inverted Pendulum Johnny Lam

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1 Contro of an Inverted Penduu Johnny a Abstract he baancin of an inverted enduu by ovin a cart aon a horizonta track is a cassic robe in the area of contro his aer wi describe two ethods to swin a enduu attached to a cart fro an initia downwards osition to an uriht osition and aintain that state A noninear heuristic controer and an enery controer have been ieented in order to swin the enduu to an uriht osition After the enduu is swun u, a inear quadratic reuator state feedback otia controer has been ieented to aintain the baanced state he heuristic controer oututs a reetitive sina at the aroriate oent and is finey tuned for the secific exerienta setu he enery controer adds an aroriate aount of enery into the enduu syste in order to achieve a desired enery state he otia state feedback controer is a stabiizin controer based on a ode inearized around the uriht osition and is effective when the cartenduu syste is near the baanced state he enduu has been swun fro the downwards osition to the uriht osition usin both ethods and the exerienta resuts are reorted INRODUCION he inverted enduu syste is a standard robe in the area of contro systes hey are often usefu to deonstrate concets in inear contro such as the stabiization of unstabe systes Since the syste is inherenty noninear, it has aso been usefu in iustratin soe of the ideas in noninear contro In this syste, an inverted enduu is attached to a cart equied with a otor that drives it aon a horizonta track he user is abe to dictate the osition and veocity of the cart throuh the otor and the track restricts the cart to oveent in the horizonta direction Sensors are attached to the cart and the ivot in order to easure the cart osition and enduu joint ane, resectivey easureents are taken with a quadrature encoder connected to a utiq3 enera urose data acquisition and contro board atab/siuink is used to ieent the controer and anayze data he inverted enduu syste inherenty has two equiibria, one of which is stabe whie the other is unstabe he stabe equiibriu corresonds to a state in which the enduu is ointin downwards In the absence of any contro force, the syste wi naturay return to this state he stabe equiibriu requires no contro inut to be achieved and, thus, is uninterestin fro a contro ersective he unstabe equiibriu corresonds to a state in which the enduu oints stricty uwards and, thus, requires a contro force to aintain this osition he basic contro objective of the inverted enduu robe is to aintain the unstabe equiibriu osition when the enduu initiay starts in an uriht osition he contro objective for this roject wi focus on startin fro the stabe equiibriu osition (enduu ointin down), swinin it u to the unstabe equiibriu osition (enduu uriht), and aintainin this state ODEING A scheatic of the inverted enduu is shown in Fiure Fiure Inverted Penduu Setu A cart equied with a otor rovides horizonta otion of the cart whie cart osition,, and joint ane,, easureents are taken via a quadrature encoder By ayin the aw of dynaics on the inverted enduu syste, the equations of otion are: cos ( ) ( ) cos V cos( ) sin( ) sin( ) () cos ( ) () sin ( ) cos( ) sin( ) V, where c is the cart ass, is the enduu ass, I is the rotationa inertia, is the hafenth of the enduu, R is the otor arature resistance, r is the otor inion radius, is the otor torque constant, and is the earbox ratio Aso, for siicity, c I and note that the reationshi between force, F, and votae, V, for the otor is: () () (3) F V (4) et the state vector be defined as: x ( ) Finay, we inearize the syste about the unstabe equiibriu ( ) Note that corresonds to the (5)

2 enduu bein in the uriht osition he inearization of the cartenduu syste around the uriht osition is: (6) Cx y BV Ax x where (7) ; C B Finay, by substitutin the araeter vaues that corresond to the exerienta setu: (8) his syste wi aow us to desin a controer to baance the inverted enduu around the oint of inearization 3 SABIIZING CONROER DESIGN he controer desin aroach for this roject is broken u into two coonents he first art invoves the desin of an otia state feedback controer for the inearized ode that wi stabiize the enduu around the uriht osition he second art invoves the desin of a controer that swins the enduu u to the unstabe equiibriu When the enduu aroaches the inearized oint, the contro wi switch to the stabiizin controer which wi baance the enduu around the uriht osition he state feedback controer resonsibe for baancin the enduu in the uriht osition is based on a inear Quadratic Reuator (QR) desin usin the inearized syste In a QR desin, the ain atrix for a inear state feedback contro aw u x is found by iniizin a quadratic cost function of the for, (9) Ru(t)dt u(t) Qx(t) x(t) J where Q and R are weihtin araeters that enaize certain states or contro inuts he weihtin araeters chosen in the desin of the otia state feedback controer are: A R Q () Based on this desin, the controer ain atrix for the inearized syste is: ( ) () By usin this and the contro aw u x, the syste is stabiized around the inearized oint (enduu uriht) Since this contro aw is based on the inearized syste, the state feedback otia controer is ony effective when the enduu is near the uriht osition 4 SAE ESIAION For the inverted enduu exerienta setu, not a the state variabes are avaiabe for easureent In fact, ony the cart osition,, and the enduu ane,, are directy easured his eans that the cart veocity and the enduu anuar veocity are not iediatey avaiabe for use in any contro schees beyond just stabiization hus, an observer is reied uon to suy accurate estiations of the states at a cartenduu ositions A inear fu state observer can be ieented based on the inearized syste derived earier his observer is sie in desin and rovides accurate estiation of a the states around the inearized oint he observer is ieented by usin a duicate of the inearized syste dynaics and addin in a correction ter that is siy a ain on the error in the estiates he observer ain atrix is deterined by an QR desin siiar to that used to deterine the ain of the otia state feedback stabiizin controer In this case, the weihtin araeters are chosen to be: () R Q Based on this desin, the observer ain atrix is: y V

3 (3) 49 Since the inear fu state observer is based on the inearized syste, it is ony effective in estiatin the state variabes when the cartenduu syste is near the uriht osition hus, a owass fitered derivative is used to estiate the two uneasured states, cart veocity and enduu anuar veocity, when the syste is not cose to the unstabe equiibriu his ethod aroxiates the cart veocity and enduu anuar veocity by usin a finite difference and then assin it throuh a owass fiter he foowin fiter is chosen for this estiation ethod: 5s G(s) (4) s 5 he robes with such a ethod are that it introduces soe deay and has a ain that is sihty ess than one he state estiates obtained fro the fitered derivative, however, are reasonaby accurate for the swinu controers ieented in this aer 5 SWINGUP CONROER DESIGN wo different contro schees were ieented to swin the enduu fro the downwards osition to the uriht osition he first is a heuristic controer that rovides a constant votae in the aroriate direction and, thus, drives the cart back and forth aon the track reeatedy It wi reeat this action unti the enduu is cose enouh to the uriht osition such that the stabiizin controer can be triered to aintain this baanced state he second schee is an enery controer that reuates the aount of enery in the enduu his controer inuts enery into the cartenduu syste unti it attains the enery state that corresonds to the enduu in the uriht osition Siiar to the heuristic contro ethod, the enery contro ethod wi aso switch to the stabiizin controer when the enduu is cose to the uriht osition he switch that triers the stabiizin controer in both cases is activated when the enduu is within 5 of the uriht osition and the anuar veocity is sower than 5 radians er second Heuristic Controer he heuristic controer is a oicbased contro desin that deterines the direction and the oent in tie the cart shoud ove deendin on the state of the syste A secific votae ain is aied to the cart otor based on resuts fro reeated exerientation his controer wi ake the cart drive forward or back whenever the enduu crosses the downwards osition and deendin on the direction that the enduu is swinin when it reaches the downwards osition he oicbased contro desin is coetey deendent on the enduu ane, one of the avaiabe easured state variabes he contro schee wi chane the direction of the cart oveent whenever the enduu ane crosses the downwards osition Since this contro desin is based soey on the enduu ane, the downwards osition is the otia oent in tie to add enery to the enduu by ovin the cart in the aroriate direction he direction the cart oves is the oosite sin of the enduu ane iediatey after it crosses the downwards osition When the direction of the cart oveent is deterined, a constant votae ain is aied to the cart otor in that sae direction unti the enduu returns to the downwards osition his contro schee wi effectivey ove the cart back and forth aon the track reeatedy unti the enduu swins cose enouh to the uriht osition It is iortant to note that the nature of this contro schee is that the sae cart oveent is aied reardess of whether the enduu is above or beow the horizonta axis (since the sin of the enduu ane reains the sae) he nature of the cartenduu syste, however, is that the sae cart oveent that once added enery to the enduu whie it was beow the horizonta axis now actuay takes away enery fro the enduu Eventuay, the enduu wi reach a oint where it can add no ore enery to the enduu syste but it has yet to buid enouh enery to reach the uriht osition o avoid this henoenon, a switch has been ieented that chanes the votae inut to the cart otor to when the enduu is 35 fro the downwards osition As a resut, the cart wi not ove to take enery away fro the enduu syste when the enduu is hiher than 35 his wi aow the enduu to siy return to the downwards osition without osin anyore enery When the enduu crosses the downwards osition aain, the oicbased controer wi be abe to add ore enery to the enduu unti it eventuay aroaches the uriht osition he votae ain of this contro schee is deterined by reeated exerientation here is a direct correation between the tie it takes to swin the enduu to its uriht osition and the anitude of the votae ain A ain that is too hih, thouh, ay ake the enduu aroach the uriht osition with too hih a veocity and, thus, the stabiizin controer wi be unabe to baance the enduu On the other hand, a ain too ow ay not rovide enouh enery to the enduu so that it can reach the uriht osition Aso, the reiabiity of the controer in erforin the task varies deendin on the ain seected hus, reeated exerientation is required to finey tune the ain so that the enduu aroaches the uriht osition with just the riht aount of veocity and in a reasonabe aount of tie with a hih success rate Enery Controer he swinin u of a enduu fro the downwards osition can aso be accoished by controin the aount of enery in the syste he enery in the enduu syste can be driven to a desired vaue throuh the use of feedback contro By addin in enouh enery such that its vaue corresonds to the uriht osition, the enduu can be swun u to its unstabe equiibriu When the enduu is cose to the uriht osition, the stabiizin controer desined earier can be triered to catch the enduu and baance it around the unstabe equiibriu

4 he syste is defined such that the enery, E, is zero in the uriht osition he enery of the enduu can be written as E cos (5) ω where ω (6) 4I and is the ass of the enduu, is the hafenth of the enduu, is the acceeration of ravity, and I is the rotationa inertia hus, the enery in the enduu is a function of the enduu ane and the enduu anuar veocity Note aso that the enery corresondin to the enduu in the downwards osition is he oa of the contro schee is to add enery into the syste unti the vaue corresonds to the enduu in the uriht osition he contro aw ieented to achieve the desired enery is ( k( E E )) sin( cos, a sat ) V (7) where k is a desin araeter and E is the desired enery eve he contro outut, a, is the acceeration of the ivot which can be transated to a votae inut to the cart otor by usin equation (4) and the fact that: F a (8) for the syste In this contro schee, the sat V function is defined as the vaue for which the votae suied to the cart saturates his controer essentiay uses enduu ane and enduu anuar veocity to deterine the direction the cart shoud ove at any oint in tie A roortiona controer that scaes with the aount of enery sti required to achieve the desired enery state dictates the aount of votae aied to the cart otor he vaue of the araeter V in sat V dictates the axiu aount of contro sina avaiabe and thus the axiu aount of enery increase to the enduu syste he vaue of k deterines how uch the contro favors usin the axiu contro inut to achieve the desired enery state his contro is effective in increasin the enery of the enduu to a desired vaue When used as a swinu contro ethod, the desired vaue corresonds to the enery of the enduu in its uriht osition his wi aow the switch to be triered so that the stabiizin controer can be used to catch the enduu and baance it around the unstabe equiibriu oint uriht osition and baances the enduu around the unstabe equiibriu oint he heuristic controer was finey tuned to swin the enduu by ayin a constant votae of 36 V Reeated exerientation with this votae ain showed that this controer was successfu in swinin the enduu to an uriht osition for the stabiizin controer to aintain the baanced state about 75% of the tie A ot of the controer outut durin an exerienta run for the heuristic controer is shown in Fiure Fiure Pot of Contro Outut for the Heuristic Controer It is iortant to note that the swinu controer takes aroxiatey 5 seconds to reach the uriht osition he oint at which the stabiizin controer catches the enduu in the uriht osition is ceary disayed in the ot Aso, the contro outut to the cart otor aternates between 36 V and 36 V as deterined by enduu ane At a itte under 7 seconds, the contro outut aso beins to outut V at sa stretches of tie since the enduu ane is beyond 35 fro the downwards osition hus, it takes about another 55 seconds for the enduu to et fro beyond 35 fro the downwards osition to within 5 of the uriht osition he corresondin ot of the enduu ane is shown in Fiure 3 Each swin increases the enduu ane sihty unti the enduu is cose to its unstabe equiibriu he controer takes about 3 swins before the enduu is cose enouh to the uriht osition for the stabiizin controer to catch it he oint in which the stabiizin controer is activated is discernibe fro the ot Aso, once activated, the enduu ane reains fairy constant around the baanced osition 6 EXPERIENA RESUS Resuts were athered fro the ieentation of both swinu contro ethods Data were coected fro exerienta runs where each contro schee swins u the enduu fro an initiay downwards osition to an Fiure 3 Pot of the Penduu Ane for the Heuristic Controer

he enery controer is ieented with the desin araeter, k, chosen to be 65 Aso, as a resut of the friction in the cartenduu syste and the aroxiation ade in equation (8), the desired enery was offset to

5 he enery controer is ieented with the desin araeter, k, chosen to be 65 Aso, as a resut of the friction in the cartenduu syste and the aroxiation ade in equation (8), the desired enery was offset to a vaue sihty hiher than he aroriate offset can be deterined throuh exerientation In these exerients, the offset is raised to E 7 Reeated exerientation on the enery controer showed that this controer was reiabe at east 9% of the tie A ot of the controer outut durin an exerienta run usin the enery controer is shown in Fiure 4 Fiure 4 Pot of the Contro Outut for the Enery Controer It is iortant to note that the enery contro takes aroxiatey seconds to reach the uriht osition he contro outut initiay aternates between 55 V and 55 V since it attets to increase the enery of the syste as quicky as it ossiby can by usin its axiu contro outut (in this case, the saturation is defined to be at 55 V) When the enduu is cose to the uriht osition, the contro outut starts to decrease in anitude since the contro outut is based on the difference between the enery of the syste and the desired vaue As with the heuristic controer, the oint at which the stabiizin controer is activated is ceary discernibe on the ot he corresondin ot of the enduu ane for the enery controer is shown in Fiure 5 Note that with each swin the enduu ane is increased sihty his controer takes about swins before the enduu is cose to the uriht osition It is easy to see that the stabiizin controer is abe to catch the enduu and baance it once the enery controer successfuy swins the enduu to the uriht osition 7 CONCUSIONS wo swinu contro schees have been ieented that wi switch to a stabiizin controer when the enduu is near the uriht osition in order to baance the enduu Both controers are caabe of successfuy swinin a enduu fro an initiay downwards osition to the uriht osition and baancin the enduu around that oint he enery contro haens to be ore robust and reiabe than the heuristic controer in successfuy swinin the enduu to the uriht osition As the data indicates, the enery controer is aso sihty faster than the heuristic controer ieented Another advantae in the enery controer is that it is caabe of reachin the uriht osition even if it runs out of track enth and beins to run into the was at the end of the track he heuristic controer ieented in this aer, on the other hand, wi iediatey fai once the cart hits the end of the track Both swinu ethods sti require utie swins to reach the uriht osition and aso require a stabiizin controer to catch the enduu in the uriht osition Overa, it is seen that the enery controer is ore convenient to swin u a enduu to its unstabe equiibriu than the heuristic controer It has been shown, however, that both controers can be effective in swinin a enduu to the uriht osition fro the downwards osition 8 REFERNCES Astro, J and Furuta, Swinin u a Penduu by Enery Contro, Autoatica, Vo 36, Sith, R S, ECE 47b/ECE 38 Course Webaes, htt://wwwccececeucsbedu/eoe/sith/ Eker, J, and J Astro, A Noninear Observer for the Inverted Penduu, 8th IEEE Conference on Contro Aication, 996 Chun, CC and J Hauser, Noninear Contro of a Swinin Penduu, Autoatica, Vo 3, 995 Fiure 5 Pot of the Penduu Ane for the Enery Controer

Journal of World s Electrical Engineering and Technology J. World. Elect. Eng. Tech. 3(2): 83-88, 2014

Journal of World s Electrical Engineering and Technology J. World. Elect. Eng. Tech. 3(2): 83-88, 2014 ORGNA ARCE Received a. Acceted Au., Scienceine Pubication www.science-ine.co -5 Journa of Word s Eectrica Enineerin and echnoo J. Word. Eect. En. ech. : 8-88, JWEE Coarison of QR and Poe Paceent Desin