Implementation of a Controller to Eliminate the Limit ...ResearchArticle Implementation of a Controller to Eliminate the Limit Cycle in the Inverted Pendulum on a Cart MayraAntonio-Cruz

Research ArticleImplementation of a Controller to Eliminate the Limit Cycle inthe Inverted Pendulum on a Cart

Mayra Antonio-Cruz 12 Victor Manuel Hernaacutendez-Guzmaacuten 3

Ramoacuten Silva-Ortigoza 1 and Gilberto Silva-Ortigoza4

1 Instituto Politecnico Nacional CIDETEC Area de Mecatronica y Energıa Renovable 07700 Mexico City Mexico2Instituto Politecnico Nacional UPIICSA SEPI 08400 Mexico City Mexico3Universidad Autonoma de Queretaro Facultad de Ingenierıa 76010 Queretaro QRO Mexico4Benemerita Universidad Autonoma de Puebla Facultad de Ciencias Fısico Matematicas 72570 Puebla PUE Mexico

Correspondence should be addressed to Victor Manuel Hernandez-Guzman vmhguaqmx

Received 28 July 2018 Revised 14 November 2018 Accepted 27 November 2018 Published 4 February 2019

Guest Editor Carlos-Arturo Loredo-Villalobos

Copyright copy 2019 Mayra Antonio-Cruz et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A frequency response-based linear controller is implemented to regulate the inverted pendulum on a cart at the inverted positionThe objective is to improve the performance of the control systemby eliminating the limit cycle generated by the dead-zone inducedby static friction at the actuator of the mechanism This control strategy has been recently introduced and applied by the authorsto eliminate the limit cycle in the Furuta pendulum and the pendubot systems Hence the main aim of the present paper is tostudy the applicability of the control strategy to eliminate the limit cycle in the inverted pendulum on a cart The successful resultsthat are obtained in experiments corroborate that the approach introduced by the authors to eliminate the limit cycle in the Furutapendulum and pendubot is also valid for the inverted pendulum on a cart

1 Introduction

Friction is a phenomenon that can cause nonlinear behav-ior in mechanical systems involving motion [1] Such anonlinear behavior refers in particular to a dead-zone[2] which degrades performance of the overall system bygenerating position error limit cycle and even instability[3]These problems separately have been important subjectsof study As a matter of fact compensation of friction inmechanical systems has been carried out to achieve a bettercontrol of position [4ndash8] Limit cycles generated by thestatic and Coulomb friction have been treated in [1 9ndash14]and the stability analysis of systems with friction has beenintroduced in [15ndash17] On the other hand underactuatedmechanical systems and in particular inverted pendulumshave attracted the attention of several researchers becausethey are an excellent benchmark to study position controllimit cycles and system stability [18] Motivated by thisscenario and by [19 20] this paper deals with the elimination

of limit cycles generated by a friction-induced dead-zonenonlinearity when regulating the inverted pendulum ona cart

Different authors have reported important results on limitcycles in the regulation of inverted pendulums which ingeneral can be divided into three categories (a) generationof stable limit cycles (b) reduction of limit cycles and (c)elimination of limit cycles

Regarding (a) Verduzco [21] presented a method fornonlinear systems that have 119896 zero eigenvalues when theyare linearized Such a method contemplates the existence ofa curve of Hopf bifurcation points and a change of bothcoordinates and input control The pendubot was used toillustrate the method Also for the pendubot Freidovich etal [22] proposed a feedback control strategy based onmotionplanning via virtual holonomic constraints FurthermoreFreidovich et al [23] developed a control for the Furutapendulum which was integrated by a shaping energy controla passivity-based control and an auxiliary feedback action

HindawiComplexityVolume 2019 Article ID 8271584 13 pageshttpsdoiorg10115520198271584

2 Complexity

Andary et al [24] introduced a control based on partial non-linear feedback linearization and dynamic control Aguilar etal [25] used partial feedback linearization with a two-relaycontroller which was tuned with the classic tool root locusThe two latter works are for the inertia wheel pendulum

For (b) Medrano-Cerda [26] considered a scheme basedon velocity-sign compensation in the inverted pendulum ona cart Also Vasudevan et al [27] compensated friction via apassivity-based observer for the wheeled inverted pendulumEom and Chwa [28] compensated friction system uncer-tainties and an external disturbance through a nonlinearobserver for the pendubot

With regard to (c) Hernandez-Guzman et al [19]exploited the differential flatness property of the Furutapendulum to propose a linear state feedback controller whichcan be designed to regulate the system and to eliminate limitcycles To achieve this an educational experimental andintuitive procedure based on the time response approach ieroot locus was introduced As an improvement Antonio-Cruz et al [20] presented a modified version of the controlin [19] The design of such a modified control was based onfrequency response instead of time response as in [19] whichentailed the obtention of precise formulas that facilitates thelimit cycle elimination A comparison of [19 20] showedthat [20] has better performance when dealing with the limitcycle elimination On the other hand some studies thatconsider the backlash nonlinearity in the Furuta pendulumand cart-pendulum system [29 30] have been reportedOther papers dealing with performance improvement ofinverted pendulums have been reported [31ndash34] Finallypapers related to dead-zone compensation for nonlinearsystems and suppression of limit cycles in servomechanismare [35ndash39]

Having undertaken the literature review it was found thatthe papers dealing with reduction of friction-induced limitcycles use compensation techniques that have the followingdisadvantages (i) most compensation terms are complex andrequire the numerical values of the frictional parameters [27]and (ii) undercompensation leads to steady-state error andovercompensation may induce limit cycles [9 40] Althoughan important effort has been done in [26] to reduce the limitcycle in the inverted pendulum on a cart to the authorsrsquoknowledge the elimination of the limit cycle in the invertedpendulum on a cart has not been achieved until now Onthe other hand only two previous papers [19 20] haveachieved limit cycle elimination without using compensationtechniques but just by employing a simple linear controller Inboth papers the controller was proposed by using a differen-tial flatness representation of the system and the describingfunction method was employed to study the existence oflimit cycles In [19] the controller was designed via the timeresponse approach (root locus) and in [20] via the frequencyresponse approach (Bode diagrams) Since the root locus isnot intrinsically related to the describing function methodthe corresponding design procedure presented in [19] is basedon intuitive ideas because precise formulas were not obtainedAccording to [20] this renders the repetition of the resultwhen using a different Furuta pendulumdifficult Hence [20]presented a simple and precise procedure to eliminate limit

Pendulum

Cart

m J

2l

M f

0

Figure 1 Inverted pendulum on a cart

cycles caused by the friction-induced dead-zone nonlinearityIn that direction the controller and the design procedurefor limit cycle elimination introduced in [20] for the Furutapendulum and pendubot systems are applied to the invertedpendulum on a cart in the present paper with the aim ofinvestigating the possibility to eliminate the limit cycle in thisunderactuated mechanism

It is recalled that the advantage of the controller proposedin [20] with regard to compensation techniques is thatthe model or the characterization of the dead-zone is notrequired whereas the advantage with regard to [19] is thatthe combination of the describing function method andfrequency response allows providing precise formulas thatrender the design of the linear controller and in consequencethe experimental elimination of limit cycles easier

The rest of the paper is organized as follows Section 2presents the differential flatness model of the linear approx-imation of the inverted pendulum on a cart as well as adescription of the real prototype used in experiments Sec-tion 3 presents the linear controller to regulate the invertedpendulum on a cart and the procedure to design it Theprocedure to eliminate the dead-zone nonlinearity-inducedlimit cycles in the system is briefly described in Section 4Lastly Section 5 gives the conclusions

2 Inverted Pendulum on a Cart

This section presents the differential flatness model of theinverted pendulum on a cart as well as the description of theprototype used in the experimental procedure to eliminatelimit cycle

21 Flatness Model The inverted pendulum on a cart shownin Figure 1 consists of a cart that has linear motion on alimited rail in the horizontal plane and in one dimensionThismotion is due to a force applied by a transmission systemactuated by a motor On the cart a pendulum is attachedwhich can move angularly in the vertical plane which isparallel to the cart movement The parameters and variablesof this system appear in Figure 1 and are denoted as follows119872 and 120585 are the mass and the translational position of thecart respectively whereas 119898 2119897 and 120601 are the mass lengthand angular position of the pendulum respectively Lastly 119891is the force applied to the cart and 119892 is acceleration of gravity

Complexity 3

The approximate linear model of the inverted pendulumon a cart

119909120575 = A119909120575 +B119906120575 (1)

with

119909120575 =[[[[[[

1199091205751119909120575211990912057531199091205754

]]]]]]

=[[[[[[[[

120585 minus 120585120585 minus 120585

120601 minus 120601120601 minus 120601

]]]]]]]]

119906120575 = 119891 minus 119891 (2)

A =[[[[[[[[

0 1 0 00 0 minus119898119892119872 00 0 0 10 0 (119872 + 119898)119892119897119872 0

]]]]]]]]

B =[[[[[[[[[

011198720

minus 1119897119872

]]]]]]]]]

(3)

around the operation point

[120585 120585 120601 120601]119879 = [0 0 0 0]119879 119891 = 0 (4)

is controllable [41] and in consequence is differentially flat[42] Ch 2Therefore the flat output 119865 associated with (1) isdefined by [42] Ch 2

119865 = 120582 [0 0 0 1] 119862minus10 119909120575 (5)

where 120582 is an arbitrary nonzero constant convenientlychosen as 120582 = minus119892(119897119872) and 1198620 = [B AB A2B A3B]is the controllability matrix of system (1) After calculations 119865and its first four time derivatives are obtained

119865 = 1199091205751 + 1198971199091205753 (6)

= 1199091205752 + 1198971199091205754 (7)

= 1198921199091205753 (8)

119865(3) = 1198921199091205754 (9)

119865(4) = (119872 + 119898)119892119897119872 minus 119892119897119872119906120575 (10)

This last expression represents the differential flat model thatdescribes the dynamics (1) and since 119891 = 0 it can be writtenas

119891 = (119872 + 119898) minus 119897119872119892 119865(4) (11)

22 Description of the Prototype The prototype of theinverted pendulum on a cart used in the experimentalprocedure to eliminate limit cycle is shown in Figure 2 Ingeneral this prototype has four subsystems (i) Mechanicalstructure (ii) actuator and sensors (iii) power stage and (iv)data acquisition and processing which are described below

Cart

Pendulum

Belt Pulley

DC motor

Encoder

Servo-drivePower supply

DS1104 board

Figure 2 Prototype of the inverted pendulum on a cart

(i) Mechanical structure refers to the mechanical ele-ments that compose the mechanism that is a cartmounted on a limited rail the transmission system(toothed belt and two pulleys) and the pendulum

(ii) Actuator and sensors consist in a Pittman 14204S006DC motor and two incremental encoders used tomeasure the angular positions of the pulleys andpendulum Since the DC motor is connected to onepulley the angular position of this pulley is used tocompute the linear position of the cart The encoderused tomeasure the angular position of the pulley has500 PPR and is included in the chassis of the motorThe encoder used to measure the angular positionof the pendulum has 1024 PPR and is fabricated byBaumer in the model ITD01B14

(iii) Power stage is integrated by an HF100W-SF-24switched power supply and an AZ12A8DDC servo-drive manufactured by Advanced Motion ControlsThis latter possesses an inner current-loop driven bya PI controller which ensures that the current of theDC motor 119894119898 reaches the current imposed by thecontrol signal 119894 that is 119894119898 997888rarr 119894 This means that adesired torque signal can be implemented through thedynamic relation of torque-current 120591 = 119896119898119894 asymp 119896119898119894119898where 120591 and 119896119898 are the torque and torque constantof the DC motor respectively Since the input of theinverted pendulumon a cart is the force119891 the desiredtorque is converted to the desired force by using 120591 =119891119903 with 119903 being the radius of the pulley connected tothe DC motor

(iv) Data acquisition and processing corresponds to aDS1104 board from dSPACE Matlab-Simulink andControlDesk Through this hardware and softwarethe variables of the system are read which allowsimplementation of the controller It is important tosay that in all experiments the velocities 120585 and 120601wereestimated via a derivative block of Simulink and thesampling period was set to 1 ms

The mechanical parameters of the inverted pendulum on acart are presented below which were found bymeasuring thelength of the pendulum andweighing the cart and pendulum

4 Complexity

2119897 = 0200 times 10minus3m

119898 = 0034 kg119872 = 0385 kg

3 Linear Controller

This section presents a linear state feedback controller forposition regulation in the inverted pendulumon a cart whichis derived by considering flat model (11) Also the analysisof the existence of limit cycles is given and the procedure todesign the controller is described

31 Controller Proposal On the one hand after applying theLaplace transform to (11) the following transfer function isobtained

119865 (119904)119891 (119904) = minus119892 (119897119872)1199042 [1199042 minus 119892 (119872 + 119898) (119897119872)] (12)

where 119865(119904) and 119891(119904) stand for Laplace transforms of the flatoutput and the applied force respectively

On the other hand the dead-zone nonlinearity has thecharacteristic function shown in Figure 3 whose parametersare described in [19] An approximate frequency responsedescription of a dead-zone nonlinearity is the followingdescribing function [43] Ch 5 where it is assumed that thenonlinearity input 119890 is a sinusoidal function of time withamplitude 119860 and frequency 120596

119873(119860) = 2119896120587 [[1205872 minus arcsin ( 120575119860) minus 120575119860radic1 minus ( 120575119860)2]

] (13)

This ldquotransfer functionrdquo119873(119860) is real positive and frequencyindependent but dependent on the input amplitude 119860 Itsmaximal value is 119873(119860) = 119896 gt 0 which is reached as 119860 997888rarrinfin and its minimal value tends to zero if 119860 997888rarr 120575

Since transfer function (12) has similar characteristics tothat of the Furuta pendulum obtained in [19 20] hencethe control scheme proposed in [20] to control the Furutapendulum and the pendubot that is

119891 (119904) = 119896V119865 (119904) 1199043 + 120573119865 (119904) 1199042 + 119896119889119865 (119904) 119904 + 119896119901119865 (119904) (14)

where 119896V 120573 119896119889 and 119896119901 are the control gains is also used tocontrol (12) The block diagram of the plant (12) in closedloop with controller (14) and considering the dead-zonenonlinearity is shown in Figure 4 There positive feedbackis used due to minus119892(119897119872) lt 0

Lastly when considering the linear state feedback con-troller

119906120575 = 119891 = minus11989611199091205751 minus 11989621199091205752 minus 11989631199091205753 minus 11989641199091205754 (15)

k

k

e

c

minus

Figure 3 Representation of a dead-zone nonlinearity

f(s)

ks

kds + kp

minusg

lM

s2minus(M+m) g

lM

1

s2

F(s)

Figure 4 Closed-loop system considering a dead-zone nonlinear-ity

+ ceR(s) = 0 minus

minus

eG(s)

Figure 5 Equivalent representation of block diagram in Figure 4

where 1198961 1198962 1198963 and 1198964 are the gains of the controller 119906120575 and119909120575 stand as defined previously in (2) and using (6)-(10) thefollowing equivalence between (14) and (15) is found

1198961 = minus1198961199011198962 = minus1198961198891198963 = minus (120573119892 + 119897119896119901) 1198964 = minus (119896V119892 + 119897119896119889)

(16)

32 Analysis of Limit Cycle Existence The analysis of limitcycle existence in the inverted pendulum on a cart is carriedout as in [19 20] for the Furuta pendulum and pendubot thatis by using the describing function method This is because theapproximation of the dead-zone nonlinearity is based on thedescribing function

The describing function method [43] Ch 5 suggestsrepresenting the system in the standard form shown inFigure 5 Also such a method requires that the linear timeinvariant system 119866(119904) behaves as a low-pass filter Thus thestandard form is obtained applying block algebra on Figure 4In this case the nonlinearity input is 119890 = 119891(119904)while the lineartime invariant system 119866(119904) is

119866 (119904) = 1198661 (119904) 1198662 (119904) (17)

where

1198661 (119904) = 119892 (119897119872)1199042 [1199042 minus 119892 (119872 + 119898) (119897119872)] (18)

1198662 (119904) = 119896V1199043 + 1205731199042 + 119896119889119904 + 119896119901 (19)

Complexity 5

Im

G(j)

minus1

N(A) rarr infin

rarr 0

minus1

N(A)rarr minusinfin

|G (j)|

A rarr A rarr infin

minus1

N(A)rarr minus

1

k

Re

Figure 6 Polar plot of 119866(119895120596) and minus1119873(119860)

are the plant and controller respectively and 119892(119897119872) gt 0Furthermore the magnitude of (17) behaves as a low-passfilter since 119866(119904) has four poles and only three zeros Thena limit cycle may exist if [43] Ch 5

119866(119895120596) = minus 1119873 (119860) (20)

which implies that the polar plot of 119866(119895120596) intersects thenegative real axis in the open interval (minusinfinminus1119896) This isbecause minus1119873(119860) is real and negative Hence the oscillationfrequency 120596120590 and the amplitude of the oscillation 119860 arefound as the values of 120596 in 119866(119895120596) and 119860 in minus1119873(119860) at thepoint 120590 where their plots intersect [43] Ch 5 The graphicrepresentation of this is shown in Figure 6

33 Controller Design The design of the controller gains119896V 120573 119896119889 and 119896119901 is achieved as described in [20] for theFuruta pendulum case In this section particularities of thecontroller design for the inverted pendulum on a cart caseare introduced Since the following transfer function of thetwo internal loops is obtained

minus119892 (119897119872)1199042 + 119892 (119897119872) 119896V119904 + [119892 (119897119872)120573 minus 119892 (119872 + 119898) (119897119872)] (21)

when the dead-zone is omitted from Figure 4 then 119896V and 120573must satisfy the following conditions

119896V gt 0 and 119892119897119872120573 gt (119872 + 119898)119892119897119872 (22)

to ensure that all coefficients of the characteristic polynomialof the transfer function in (21) are positive

Now note that when replacing 119904 by 119895120596 in (18) the phaseof 1198661(119895120596) is minus360∘ for all 120596 ge 0 because 119892(119897119872) gt 0 and eachone of the factors 1minus1205962 lt 0 and 1(minus1205962minus119892(119872+119898)(119897119872)) lt0 introduces a phase of minus180∘ Hence with the intention offorcing the polar plot of 119866(119895120596) to intersect the negative realaxis ie to render phase of 119866(119895120596) equal to minus180∘ at some120596 gt 0 the frequency analysis performed for the controller1198662(119895120596) in [20] is applied Such an analysis is described belowto facilitate the reference

The phase of 1198662(119895120596)must be as follows

ang1198662 (119895120596) = arctan(119896119889120596 minus 119896V1205963119896119901 minus 1205731205962 ) = +180∘ (23)

This implies that the following conditions have to be satisfied

119896119889120596 minus 119896V1205963 = 0 (24)

119896119901 minus 1205731205962 lt 0 (25)

From (24) the following relation to find 119896119889 is obtained119896119889 = 119896V1205962 (26)

Lastly in order to compute 119896119901 119904 is replaced by 119895120596 in controller(19) to obtain the following

1198662 (119895120596) = 119896V (119895120596)3 + 120573 (119895120596)2 + 119896119889 (119895120596) + 119896119901= 119895 (119896119889120596 minus 119896V1205963) + (119896119901 minus 1205731205962) (27)

whose magnitude is determined by

10038161003816100381610038161198662 (119895120596)1003816100381610038161003816 = radic(119896119889120596 minus 119896V1205963)2 + (119896119901 minus 1205731205962)2 (28)

Hence when solving (28) for 119896119901 the formula below isobtained

119896119901 = plusmnradic10038161003816100381610038161198662 (119895120596)10038161003816100381610038162 minus (119896119889120596 minus 119896V1205963)2 + 1205731205962 (29)

Therefore the sign in this latter expression has to be chosenso that (25) is accomplished

Finally to propose the frequency 120596 = 120596120590 at which it isdesired that the polar plot of 119866(119895120596) intersects the negativereal axis it is necessary to compute 119896119901 and 119896119889 Likewisethe magnitude |1198662(119895120596120590)| that must be introduced by thecontroller has to be known for which a desired magnitudeof 119866(119895120596) when 120596 = 120596120590 has to be proposed Then from

1003816100381610038161003816119866 (119895120596120590)1003816100381610038161003816 = 10038161003816100381610038161198661 (119895120596120590)1003816100381610038161003816 sdot 10038161003816100381610038161198662 (119895120596120590)1003816100381610038161003816 (30)

the magnitude |1198662(119895120596120590)| can be computed by

10038161003816100381610038161198662 (119895120596120590)1003816100381610038161003816 =1003816100381610038161003816119866 (119895120596120590)100381610038161003816100381610038161003816100381610038161198661 (119895120596120590)1003816100381610038161003816 (31)

Since |1198661(119895120596120590)| can be obtained from the Bode diagrams of1198661(119895120596) then Bode diagrams are a suitable tool to design thecontroller gains 119896V 120573 119896119889 and 119896119901

Until here the procedure and formulas to compute thecontroller gains have been described The procedure tochoose such gains in order to eliminate limit cycle due todead-zone nonlinearity is presented in the next section

4 Experimental Procedure for LimitCycle Elimination

In this section the experimental procedure introduced in[20] is applied to eliminate limit cycles in the inverted

6 Complexity

pendulum on a cart In [20] the procedure to eliminate limitcycle was executed departing from knowing the numericalvalue120575 of the dead-zone nonlinearity of the Furuta pendulumand pendubot Also in that paper it was mentioned that theprocedure can be applied without requiring the knowledgeof such a parameter Thus the procedure in [20] is appliedhere for the inverted pendulum on a cart without requiringthe knowledge of 120575 Additional steps that help to betteraddress the procedure which do not modify the generalityof the procedure introduced in [20] are indicated Alsoparticularities of the application of the procedure in theinverted pendulum on a cart are indicated in each step

Before starting the application of the procedure for limitcycle elimination in the inverted pendulum on a cart theconjecture established in [20] is recalled below

Conjecture According to the dead-zone nonlinearity charac-teristic function depicted in Figure 3 if |119890| le 120575 then a zerovalue appears at the plant input 119888 = 0 ie the force appliedby the motor to the inverted pendulum on a cart is zeroand the mechanism might rest at the operation point definedin (4) Since the threshold 120575 is uncertain because friction isuncertain it is natural to wonder whether it is possible torender 119860 lt 120575 in experiments despite (13) being only validfor119860 ge 120575 Recall that 119860 ge |119890| because 119860 is the amplitude of 119890Then the mechanism might rest at the operation point if 119860 ischosen to be small enough ie the limit cycle might vanishunder these conditions

It is also recalled that according to Figure 6 with thepurpose of reducing the amplitude of the limit cycle the polarplot of119866(119895120596)must intersect the negative real axis at a point 120590located farther to the left of the point minus1119896 = minus1This latter iscomputed by considering 119896 = 1 which is a value usually setfor a conventional DCmotorThis suggests that |119866(119895120596120590)| ≫ 1and this must occur at an oscillation frequency 120596 = 120596120590

The procedure to eliminate the limit cycle induced bythe dead-zone nonlinearity when regulating position in theinverted pendulum on a cart was experimentally applied asfollows

(1) Bode diagrams of the plant 1198661(119895120596) were plotted asshown in Figure 7 For this (18) was used

(2) The frequency 120596120590 = 6 rads and the magnitude|119866(119895120596120590)| = 22 were initially proposed The value of120596120590 was proposed since this renders 119891120590 = 120596120590(2120587) asymp09549 Hz which is a reasonable frequency in Hertzfor the experimental prototype that was built Usingthe value of120596120590 and Bode diagrams plotted in Figure 7the following magnitude in dB was measured10038161003816100381610038161198661 (119895120596120590)1003816100381610038161003816dB = minus261dB (32)

which was converted into10038161003816100381610038161198661 (119895120596120590)1003816100381610038161003816 = 10|1198661(119895120596120590)|dB20 = 00495 (33)

The latter numerical valuewas used in (31) to compute|1198662(119895120596120590)| finding the following10038161003816100381610038161198662 (119895120596120590)1003816100381610038161003816 = 4441570 = 4440406 (34)

minus150

minus100

minus50

0

50

Mag

nitu

de (d

B)

100 101 102minus361

minus3605

minus360

minus3595

minus359

Phas

e (de

g)

Frequency (rads)

System GpFrequency (rads) 6Magnitude (dB) -261

Figure 7 Bode diagrams of 1198661(119904)

The numerical values of 120596120590 and |1198662(119895120596120590)| shall beused to compute 119896119901

(3) 119896V = 094 and 120573 = 125 were selected satisfying (22)that is rendering all coefficients of the characteristicpolynomial of (21) positive Also the proposed 119896V and120573 achieve that the sign of the square root in (29) isnegative which is implied from (25) and that 119896119901 gt0 According to Figure 4 this latter is necessary toensure closed-loop stability Note that in order toavoid negative values for 119896119901 it is clear from (29) and(25) that larger values of either 120573 or 120596120590 are requiredFrom the second degree characteristic polynomial in(21) it is concluded that a larger 120573 is possible if rootsof this characteristic polynomial are farther from theorigin This is accomplished since 119896V = 094 and120573 = 125 assign real poles of (21) at minus2258894 andminus136275 In the case that the selection of 120573 does notachieve 119896119901 gt 0 and the designer prefers to increase120596120590 instead of increasing 120573 the designer must go backto step (2)

(4) With the numerical values in steps (2) and (3) (29)and (26) 119896119889 and 119896119901 were computed as follows

119896119889 = 3384119896119901 = 59594 (35)

For (29) a ldquominusrdquo sign was chosen because this renders119896119901 minus 1205731205962120590 = minus4440406 lt 0 Notice that this ensuresthat 119896119901 is real and positive and hence closed-loopstability is ensured If this were not the case thedesigner would have to go back to step (3)

(5) Through the Bode diagrams of the compensatedsystem 119866(119904) shown in Figure 8 it was corroboratedthat the open-loop system had the desired phaseminus180∘ at the desired frequency and magnitude 120596120590 =6 rads and |119866(119895120596120590)| = 22 asymp 269 dB respectivelyThe corresponding polar plot of 119866(119895120596) is depicted inFigure 9

Complexity 7

minus50

0

50

100

150

Mag

nitu

de (d

B)

10minus2 10minus1 100 101 102 103minus360

minus270

minus180

minus90

Phas

e (de

g)

Frequency (rads)

System GFrequency (rads) 6Phase (deg) -180

System GFrequency (rads) 6Magnitude (dB) 269

Figure 8 Bode diagrams of 119866(119904)

Real Axis

System GReal -22Imag -00136Frequency (rads) 601

minus60 minus40 minus20 0 20 40

minus250

minus200

minus150

minus100

minus50

0

50

100

150

200

Imag

inar

y A

xis

Figure 9 Polar plot of 119866(119895120596)

(6) Once 119896V 120573 119896119889 and 119896119901 were known the relations in(16) were used to find the following numerical valuesfor the gains of linear state feedback controller (15)

1198961 = minus595941198962 = minus33841198963 = minus12322091198964 = minus126054

(36)

Using these gains 1198961 1198962 1198963 and 1198964 linear state feed-back controller (15) was experimentally implementedto regulate the prototype of the inverted pendulum ona cart depicted in Figure 2 Since (15) only stabilizesthe prototype at 119909120575 = 0 when operating close to(4) the pendulum was manually taken to near such

an operation point Hence the following switchingcondition was used

119891 = (15) for radic(120601 minus 120601)2 + 1206012 le 030 for radic(120601 minus 120601)2 + 1206012 gt 03 (37)

The experimental results obtained when using (37)with (36) are shown in Figure 10 where a limit cycleis observed Since there is noise in the control signal119891 the amplitude and frequency of the limit cycle aredifficult to measure there But as 119890 = 119891(119904) is linearlyrelated to 119865(119904) through (14) the analysis in SectionIII about limit cycle is also valid for 119865 Hence theamplitude and the frequency of 119865 were measured toobserve the behavior of the limit cycle The measuredamplitude of the limit cycle is denoted as 119860119865 andwas computed by summing the maximal and theminimum absolute values of 119865 whereas the measuredfrequency of the limit cycle is denoted as 120596120590119865 and wascomputed using the following

120596120590119865 = 2120587119899119905119891 minus 119905119894 (38)

where 119899 is the number of oscillations that occurredin the time interval between 119905119894 and 119905119891 Thus 119860119865 =04323 m and 120596120590119865 = 02922 rads were obtained

(7) As a limit cycle appeared in the previous step |119866(119895120596120590)|was increased and we went back to step (3) When|119866(119895120596120590)| = 38 was reached 119896V = 166 and 120573 =215 were selected Then 119896119889 = 597600 and 119896119901 =70208 were computed Thus the following gains forcontroller (15) were computed

1198961 = minus702081198962 = minus5976001198963 = minus21161711198964 = minus222606

(39)

When implementing (37) with (39) the resultsdepicted in Figure 11 were obtained There it can beobserved that the limit cycle was partially eliminatedand that 119860119865 = 01371 m when it appearsSince in the previous experiment the limit cycle waspartially eliminated |119866(119895120596120590)|was incremented so that|119866(119895120596120590)| = 40 In this case 119896V = 175 and 120573 =225 were chosen 119896119889 = 63 and 119896119901 = 26535 werecomputed and the following gains of (15) were found

1198961 = minus265351198962 = minus631198963 = minus22099031198964 = minus234675

(40)

8 Complexity

(a)

(b)

F

(c)

f

(d)

Figure 10 Experimental results when using (36)

(a)

(b)

F

(c)

f

(d)

Figure 11 Experimental results when |119866(119895120596120590)| = 38 and 120596120590 = 6 rads

Complexity 9

Although it may be thought that this time the limitcycle would disappear after executing the experimentof controller (37) with (40) considerable vibrationin the prototype was observed and limit cycle reap-peared instead of being eliminated See experimentalresults in Figure 12 where 119860119865 = 02901 m and120596120590119865 = 02365 rads were measured It is important tohighlight that although limit cycle was not eliminatedso far it was actually reduced since 119860119865 = 02901 m lt119860119865 = 04323 m This is in accordance with theconjecture Also note that noise in the control signal119891 is more noticeable because |119866(119895120596120590)| was increased(see Figures 10(d) 11(d) and 12(d))

(8) Since in the previous step limit cycle was not elim-inated and considerable vibration was observed inthe prototype (see noise in Figure 12(d)) 120596120590 wasincreased to 8 rads |119866(119895120596120590)| = 22was set again andwe went back to step (3) As limit cycle still remainsbut with a reduced amplitude of oscillation |119866(119895120596120590)|was increased again As an example of reduction oflimit cycle with regard to the experimental results inFigure 12 the experimental results when |119866(119895120596120590)| =34 are depicted in Figure 13 There it is remarkablethat limit cycle was partially eliminated and littleoscillations appeared with 119860119865 = 00786 m which isless than the amplitude of limit cycle associated withFigures 11 and 12To obtain the results in Figure 13 the following gainsof controller (15) were used

1198961 = minus483511198962 = minus97281198963 = minus22611351198964 = minus246392

(41)

Such gains were found departing from selecting 119896V =152 and 120573 = 23 and computing 119896119889 = 9728 and 119896119901 =48351(9) Finally limit cycle disappeared when |119866(119895120596120590)| = 36

and 120596120590 = 8 rads For that 119896V = 161 and 120573 = 245were chosen 119896119889 = 10304 and 119896119901 = 145313 werecomputed and the following gains of controller (15)were found

1198961 = minus1453131198962 = minus103041198963 = minus24179811198964 = minus260981

(42)

The obtained experimental results are shown in Fig-ure 14

From the experimental results it was observed that foreach 120596120590 there is a maximum value of |119866(119895120596120590)| allowed

by the prototype of the inverted pendulum on a cart toperform experiments This is because noise in the controlsignal was increased as |119866(119895120596120590)| was increased The effectof this noise was reflected in the prototype as noticeablevibration when |119866(119895120596120590)| reached some high value Thuslower frequencies allow larger magnitudes of 119866(119895120596) and atlarger frequencies the magnitude of119866(119895120596)must be decreasedto avoid noticeable vibration in the closed-loop system and toapproach to the limit cycle elimination Another observationis that the experimental results corroborate the conjectureie that limit cycle is eliminated as selecting controller gainssuch that the polar plot of 119866(119895120596) crosses the negative realaxis at a point located farther to the left Furthermore anadditional observation from the experiments is that limitcycle elimination is accomplished as frequency 120596120590 where thepolar plot of 119866(119895120596) crosses the negative real axis is chosenlarger Note that these same observations were made for theFuruta pendulum in [20]

On the other hand some differences were found whencomparing frequency120596120590119865 of the experiments with the desiredone These differences are mainly due to the following

(i) According to [43] Ch 5 since the describing functionmethod has an approximate nature some inaccura-cies are found in results (a) the predicted amplitudeand frequency might not be accurate (b) a predictedlimit cycle might actually not exist or (c) an existinglimit cycle is not predicted the first kind of inaccu-racy ie (a) being quite common

(ii) Dead-zone ldquotransfer functionrdquo (13) is an idealiza-tion of the nonlinear phenomenon that is actuallypresented in the practical plant Hence not all thedynamics of the dead-zone nonlinearity is concen-trated in (13)

Until here it has been shown that the controller andthe applied procedure allow elimination of the limit cyclein the inverted pendulum on a cart but it was previouslycommented that 120575 is uncertain because friction is uncertainThis latter implies that knowing the exact value of 120575 isdifficult which acts as a disturbance For this reason com-pensation techniques had to be used to face limit cycle issueonline Thus it becomes interesting to know the behaviorof the linear controller here implemented for the invertedpendulum on a cart when the limit cycle in the systemchanges due to the conditions of operation Figure 15 presentsthe results when gains (42) are implemented for the systemunder study without previously performing an experimentthat is without ldquowarming uprdquo the actuator Note that theseconditions of operation are different from those when theresults of Figure 14 were obtained because then several exper-iments were consecutively performed before eliminating thelimit cycle that is the actuator of the system was ldquowarmeduprdquo In Figure 15 it can be observed that in different occasionsa limit cycle reappears which is natural since static friction isgreater when there is no previous movement (120575 is different)But it is important to remark from Figure 15 is that limit cycleis eliminated after reappearing Thus it can be concluded thatthe simple linear controller here implemented is feasible androbust enough to eliminate limit cycle

10 Complexity

(a)

(b)

F

(c)

f

(d)

Figure 12 Experimental results when |119866(119895120596120590)| = 40 and 120596120590 = 6 rads

(a)

(b)

F

(c)

f

(d)

Figure 13 Experimental results when |119866(119895120596120590)| = 34 and 120596120590 = 8 rads

Complexity 11

(a)

(b)

F

(c)

f

(d)

Figure 14 Experimental results when |119866(119895120596120590)| = 36 and 120596120590 = 8 rads

(a)

(b)

F

(c)

f

(d)

Figure 15 Experimental results when gains (42) are implemented without previously performing an experiment in the prototype

12 Complexity

5 Conclusion

A linear controller based on the frequency response approachand an experimental procedure introduced recently by theauthors for the Furuta pendulum and the pendubot has beensuccessfully applied to eliminate the limit cycle in the invertedpendulumon a cartTherefore from the experimental resultsthe following can be concluded (i) The inverted pendulumon a cart has similar behavior to that of the Furuta pendulumunder the effect of linear state feedback controller (15) whenit is designed through frequency response-based controller(14) (ii)The applicability of the approach introduced in [20]to eliminate limit cycle is confirmed for another invertedpendulum corroborating that the approach can eliminatelimit cycles in different inverted pendulums (iii) Robustnessof the controller is verified when conditions of operationchange

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

The authors declare that the research was conducted inthe absence of any commercial financial or personal rela-tionships that could be construed as a potential conflict ofinterest

Acknowledgments

Thisworkwas supported by Secretarıa de Investigacion y Pos-grado del Instituto Politecnico Nacional Mexico The workof M Antonio-Cruz has been supported by the CONACYT-Mexico and BEIFI-IPN scholarships V M Hernandez-Guzman and G Silva-Ortigoza thank the support givenby the SNI-Mexico Lastly R Silva-Ortigoza acknowledgesfinancial support from IPNprograms EDI and SIBE and fromSNI-Mexico

References

[1] B Armstrong-Helouvry P Dupont and C C deWit ldquoA surveyof models analysis tools and compensation methods for thecontrol of machines with frictionrdquo Automatica vol 30 no 7pp 1083ndash1138 1994

[2] G Tao and P V Kokotovic ldquoAdaptive control of plants withunknown dead-zonesrdquo IEEE Transactions on Automatic Con-trol vol 39 no 1 pp 59ndash68 1994

[3] G Tao and F L LewisAdaptive Control of Nonsmooth DynamicSystems Springer New York NY USA 2001

[4] J Sandoval R Kelly and V Santibanez ldquoInterconnectionand damping assignment passivity-based control of a classof underactuated mechanical systems with dynamic frictionrdquoInternational Journal of Robust and Nonlinear Control vol 21no 7 pp 738ndash751 2011

[5] A Nejadfard M J Yazdanpanah and I Hassanzadeh ldquoFrictioncompensation of double inverted pendulum on a cart using

locally linear neuro-fuzzymodelrdquoNeural Computing andAppli-cations vol 22 no 2 pp 337ndash347 2013

[6] D Xia L Wang and T Chai ldquoNeural-network-friction com-pensation-based energy swing-up control of pendubotrdquo IEEETransactions on Industrial Electronics vol 61 no 3 pp 1411ndash1423 2014

[7] C Aguilar-Avelar R Rodrıguez-Calderon S Puga-Guzmanand J Moreno-Valenzuela ldquoEffects of nonlinear friction com-pensation in the inertia wheel pendulumrdquo Journal ofMechanicalScience and Technology vol 31 no 9 pp 4425ndash4433 2017

[8] A Keck J Zimmermann andO Sawodny ldquoFriction parameteridentification and compensation using the elastoplastic frictionmodelrdquoMechatronics vol 47 pp 168ndash182 2017

[9] H Olsson and K J Astrom ldquoFriction generated limit cyclesrdquoIEEE Transactions on Control Systems Technology vol 9 no 4pp 629ndash636 2001

[10] R H Hensen M J van de Molengraft and M SteinbuchldquoFriction induced hunting limit cycles a comparison betweenthe LuGre and switch friction modelrdquo Automatica vol 39 no12 pp 2131ndash2137 2003

[11] L Marton ldquoOn analysis of limit cycles in positioning systemsnear Striebeck velocitiesrdquoMechatronics vol 18 no 1 pp 46ndash522008

[12] S-L Chen K K Tan and S Huang ldquoFriction modelingand compensation of servomechanical systems with dual-relayfeedback approachrdquo IEEE Transactions on Control SystemsTechnology vol 17 no 6 pp 1295ndash1305 2009

[13] S-L Chen K K Tan and S Huang ldquoLimit cycles inducedin type-1 linear systems with PID-type of relay feedbackrdquoInternational Journal of Systems Science vol 40 no 12 pp 1229ndash1239 2009

[14] M M Z Shahadat T Mizuno Y Ishino and M TakasakildquoEffect of nonlinearity caused by friction on a negative stiffnesscontrol systemrdquo IEEE Transactions on Control Systems Technol-ogy vol 22 no 4 pp 1385ndash1395 2014

[15] S Jeon and M Tomizuka ldquoStability of controlled mechanicalsystems with ideal Coulomb frictionrdquo Journal of DynamicSystems Measurement and Control vol 130 no 1 pp 011013-1ndash011013-9 2008

[16] R Rascon D Rosas and D Hernandez-Balbuena ldquoRegulationcontrol of an underactuated mechanical system with discon-tinuous friction and backlashrdquo International Journal of AppliedMathematics and Computer Science vol 27 no 4 pp 785ndash7972017

[17] A Bisoffi M Da Lio A R Teel and L Zaccarian ldquoGlobalasymptotic stability of a PID control system with Coulombfrictionrdquo Institute of Electrical and Electronics Engineers Trans-actions on Automatic Control vol 63 no 8 pp 2654ndash2661 2018

[18] D J Block K J Astrom andMW SpongThe Reaction WheelPendulum MW Spong Ed Morgan amp Claypool ChampaignIL USA 2007

[19] V M Hernandez-Guzman M Antonio-Cruz and R Silva-Ortigoza ldquoLinear state feedback regulation of a Furuta pendu-lum design based on differential flatness and root locusrdquo IEEEAccess vol 4 pp 8721ndash8736 2016

[20] M Antonio-Cruz V M Hernandez-Guzman and R Silva-Ortigoza ldquoLimit cycle elimination in inverted pendulumsFuruta pendulum and pendubotrdquo IEEEAccess vol 6 pp 30317ndash30332 2018

[21] F Verduzco ldquoControl of oscillations from the 119896-zero bifurca-tionrdquoChaos SolitonsampFractals vol 33 no 2 pp 492ndash504 2007

Complexity 13

[22] L Freidovich A Robertsson A Shiriaev and R JohanssonldquoPeriodic motions of the Pendubot via virtual holonomicconstraints theory and experimentsrdquo Automatica vol 44 no3 pp 785ndash791 2008

[23] L Freidovich A Shiriaev F Gordillo F Gomez-Estern and JAracil ldquoPartial-energy-shaping control for orbital stabilizationof high-frequency oscillations of the Furuta pendulumrdquo IEEETransactions on Control Systems Technology vol 17 no 4 pp853ndash858 2009

[24] S Andary A Chemori and S Krut ldquoControl of the underac-tuated inertia wheel inverted pendulum for stable limit cyclegenerationrdquo Advanced Robotics vol 23 no 15 pp 1999ndash20142009

[25] L T Aguilar I M Boiko L M Fridman and L B FreidovichldquoGenerating oscillations in inertia wheel pendulum via two-relay controllerrdquo International Journal of Robust and NonlinearControl vol 22 no 3 pp 318ndash330 2012

[26] G AMedrano-Cerda ldquoRobust computer control of an invertedpendulumrdquo IEEE Control Systems Magazine vol 19 no 3 pp58ndash67 1999

[27] H Vasudevan A M Dollar and J B Morrell ldquoDesign forcontrol of wheeled inverted pendulum platformsrdquo Journal ofMechanisms and Robotics vol 7 no 4 pp 1ndash12 2015

[28] M Eom and D Chwa ldquoRobust swing-up and balancing controlusing a nonlinear disturbance observer for the pendubot systemwith dynamic frictionrdquo IEEE Transactions on Robotics vol 31no 2 pp 331ndash343 2015

[29] G Pujol and L Acho ldquoStabilization of the Furuta pendulumwith backlash using Hinfin-LMI technique experimental valida-tionrdquo Asian Journal of Control vol 12 no 4 pp 460ndash467 2010

[30] A T Azar and F E Serrano ldquoStabilization of mechanicalsystems with backlash by PI loop shapingrdquo International Journalof System Dynamics Applications vol 5 no 3 pp 21ndash46 2016

[31] J Moreno-Valenzuela C Aguilar-Avelar S A Puga-Guzmanand V Santibanez ldquoAdaptive neural network control for thetrajectory tracking of the Furuta pendulumrdquo IEEE Transactionson Cybernetics vol 46 no 12 pp 3439ndash3452 2016

[32] M Antonio Cruz R Silva Ortigoza CMarquez Sanchez V MHernandez Guzman J Sandoval Gutierrez and J C HerreraLozada ldquoParallel computing as a tool for tuning the gains ofautomatic control lawsrdquo IEEE Latin America Transactions vol15 no 6 pp 1189ndash1196 2017

[33] A Zhang X Lai M Wu and J She ldquoNonlinear stabilizingcontrol for a class of underactuated mechanical systems withmulti degree of freedomsrdquo Nonlinear Dynamics vol 89 no 3pp 2241ndash2253 2017

[34] T Ortega-Montiel R Villafuerte-Segura C Vazquez-Aguileraand L Freidovich ldquoProportional retarded controller to stabilizeunderactuated systems with measurement delays Furuta pen-dulum case studyrdquo Mathematical Problems in Engineering vol2017 Article ID 2505086 12 pages 2017

[35] X-S Wang C-Y Su and H Hong ldquoRobust adaptive controlof a class of nonlinear systems with unknown dead-zonerdquoAutomatica vol 40 no 3 pp 407ndash413 2004

[36] J Zhou C Wen and Y Zhang ldquoAdaptive output controlof nonlinear systems with uncertain dead-zone nonlinearityrdquoIEEE Transactions on Automatic Control vol 51 no 3 pp 504ndash511 2006

[37] S Ibrir W F Xie and C-Y Su ldquoAdaptive tracking of nonlinearsystems with non-symmetric dead-zone inputrdquo Automaticavol 43 no 3 pp 522ndash530 2007

[38] C-H Liao F-C Chou P-C Tung and Y-D Chen ldquoSuppres-sion of limit cycles in servo systems using gain limit com-pensatorrdquo IEICE Transactions on Fundamentals of ElectronicsCommunications and Computer Sciences vol E91-A no 11 pp3293ndash3296 2008

[39] S Jeon ldquoIntegrator leakage for limit cycle suppression inservo mechanisms with stictionrdquo Journal of Dynamic SystemsMeasurement andControl vol 134 no 3 pp 034502-1ndash034502-8 2012

[40] D Putra H Nijmeijer and N van de Wouw ldquoAnalysis ofundercompensation and overcompensation of friction in 1DOFmechanical systemsrdquo Automatica vol 43 no 8 pp 1387ndash13942007

[41] I Fantoni and R Lozano Non-linear Control for UnderactuatedMechanichal Systems Springer London UK 2002

[42] H Sira-Ramırez and S K Agrawal Differentially flat systemsMarcel Dekker Inc New York NY USA 2004

[43] J J Slotine and W Li Applied nonlinear control Prentice-HallNew Jersey NJ USA 1989

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