Research ArticleRobust Control of Underactuated SystemsHigher Order Integral Sliding Mode Approach
Sami ud Din12 Qudrat Khan34 Fazal ur Rehman1 and Rini Akmeliawati3
1Department of Electrical Engineering Capital University of Science and Technology (CUST)Kahuta Road Express Highway Islamabad 44000 Pakistan2Department of Electrical Engineering The University of Lahore (UOL) Japan Road Express Highway Islamabad 44000 Pakistan3Department of Mechatronics Engineering International Islamic University 50728 Kuala Lumpur Malaysia4Center for Advanced Studies in Telecommunications COMSATS Institute of Information Technology Islamabad 44000 Pakistan
Correspondence should be addressed to Sami ud Din engrsamiuddingmailcom
Received 25 September 2015 Revised 8 January 2016 Accepted 12 January 2016
Academic Editor Wenguang Yu
Copyright copy 2016 Sami ud Din et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper presents a robust control design for the class of underactuated uncertain nonlinear systems Either the nonlinearmodel ofthe underactuated systems is transformed into an input output form and then an integral manifold is devised for the control designpurpose or an integral manifold is defined directly for the concerned class Having defined the integral manifolds discontinuouscontrol laws are designed which are capable of maintaining slidingmode from the very beginningThe closed loop stability of thesesystems is presented in an impressive wayThe effectiveness and demand of the designed control laws are verified via the simulationand experimental results of ball and beam system
1 Introduction
The control design of underactuated systems was the mainfocus of the researchers in the current and last decadeThese systems by definition contain less number of controlinputsactuators as compared to the degree of freedom [1]This feature makes them quite different from the other non-linear plantswhere the systems operatewith the samenumberof inputs and outputs the so-called fully actuated systemsThe control design of these systems is quite demandingbecause of their vital theoretical and practical applications inthe areas of aerospace systems marine systems humanoidslocomotive systems manipulators of different kinds and soforth [2] This family also includes ball and beam system [3]TORA (translational oscillator with rotational actuator) [4]and inverted pendulum system [5]These systems are used inorder to have minimum weight cost and energy usage whilestill retaining the key features of the processes In additionanother significant feature of underactuated systems is lessdamage in case of collision with other objects which inturn provides more safety to actuators [6] Underactuation
can be raised due to the hardware failure this hardwaresolution to actuator failures can be achieved by equippingthe vehicle with redundant actuators [2] Note that in caseof fully actuated systems there exists a broad range of designtechniques in order to improve performance and robustnessThese include adaptive control optimal control feedbacklinearization and passivity However it may be difficult toapply such techniques in large class of underactuated systemsbecause sometimes these systems are not linearizable usingsmooth feedback [7] also due to the existence of unstablehidden modes in some systems Brockett [8] also provided anecessary condition for the hold of stable smooth feedbacklaw but this condition is not satisfied in the majority ofunderactuated systems Nevertheless control design expertshave employed approximate feedback linearization [9ndash11] andbackstepping control [12] Passivity-based methodology isalso used to control such systems but the main drawback inthis technique is its narrow range of applications [13] Slidingmode control is also proposed for the class of underactuatedsystems [6] but the problem with sliding mode control ispresence of chattering
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 5641478 11 pageshttpdxdoiorg10115520165641478
2 Mathematical Problems in Engineering
The aforementioned design strategies were quite suitableand resulted in satisfactory results but it is worthy to notethat the system often becomes too sensitive to disturbance inthe reaching phase of sliding mode strategy that the systemmay even become unstable Therefore in order to get rid ofthis issue the integral sliding mode strategy was proposed[14ndash16] In this paper a robust integral sliding mode control(RISMC) approach for underactuated systems is proposedThe benefit of this strategy is enhancement of robustnessfrom initial time instant It also suppresses the well-knownchattering phenomenon across the manifold Before thedesign presentation the system is suitably transformed intospecial formats An integral slidingmode strategy is proposedfor both the cases along with their comprehensive stabilityanalysis The proposed technique is practically implementedon the ball and beam system to authenticate the affectivityand efficiency of the designed algorithm Note that in thispaper our contributions are twofold The first one is thedevelopment of theRISMCand the secondone is the practicalresults of the system on the said system The rest of thepaper is organized as follows In Section 2 the problem isformulated into two special formats which further simplifythe design methodology In Section 3 the integral slidingmode strategy for both the cases is discussed in detailaccompanied by their respective stability analysis in terms ofLyapunov theory Section 4 presents the development of thecontrol laws simulation and practical results of the ball andbeam system Section 5 concludes the overall efforts beingmade in this study In the end more relevant recent articlesare enlisted
2 Problem Formulation
The dynamic equations which govern the motion of the classof underactuated system can be presented as
where 119902 and are 119899-dimensional position velocityand acceleration vectors and 119869(119902) 119862(119902 ) 119866(119902) and 119865()
represent the inertia Coriolis gravitational and fractionaltorques matrices respectively 120591 is the measured controlinput and 120575(119902 119905) represents the uncertainties in the controlinput channel whereas 119861 is the control input channel
It is assumed that rank(119869minus1(119902)119861) = 119898 and the originis considered to be the equilibrium point for the aforemen-tioned system Now the system in (1) can be rewritten inalternate form as follows
11989811
(119902) 1199021
+ 11989812
(119902) 1199022
+ ℎ1
(119902 ) = 0
11989821
(119902) 1199021
+ 11989812
(119902) 1199022
+ ℎ2
(119902 ) = 120591
(2)
where 119902 = [1199021
1199022
]119879 represents the states of the system and 119902
and point to the states In order to design a control law thesystem in (2) can be transformed into two formats which aredescribed in the subsequent study
21 System in Cascaded Form Following some algebraicmanipulations the system in (2) may be written in cascadedform as follows [17]
1199091
= 1199092
+ 1198891
1199092
= 1198911
(1199091
1199092
1199093
1199094
) + 1198892
(3)
1199093
= 1199094
1199094
= 1198912
(1199091
1199092
1199093
1199094
) + 119887 (1199091
1199092
1199093
1199094
) 120591 + 1198893
(4)
where 1199091
1199092
1199093
1199094
are measurable states of the systems suchthat 119909
1
and 1199092
are pointing to the position and velocity ofthe indirect actuated system (3) while 119909
3
and 1199094
representthe position and velocity of the directly actuated system(4) 120591 represents the controlled signal as already discussedto the system (4) input Owing to the assumption statedimmediately after (1) the inverse of 119887 exists The nonlinearfunctions 119891
1
1198912
1198774119899
rarr 119877119899 119887 119877
4119899
rarr 119877119899times119899 are smooth in
nature Now following the procedure of [6] the disturbances1198891
1198892
1198893
are deliberately introduced to get an approximatecontrollable canonical form Note that practical systems likeinverted pendulum [18] TORA [4] VTOL (vertical take-offand landing) aircraft [17] and quad rotor [19] can be putin the form presented in (3) and (4) Before proceeding tothe control design of the above cascaded form the followingassumptions are made
Assumption 1 Assume that
1198911
(0 0 0 0) = 0 (5)
This condition is necessary for the system origin to be inequilibrium point when the system is operated in closed loop
Assumption 2 1205971198911
1205971199093
is invertible or 1205971198911
1205971199094
is invertible
Assumption 3 1198911
(0 0 1199093
1199094
) = 0 is an asymptotically stablemanifold that is 119909
3
and 1199094
approaches zero
Note that Assumptions 2 and 3 lie in the category ofnonnecessary conditions These are only used when oneneeds to furnish the closed loop system with a sliding modecontroller (see for details [6])
22 Input Output Form The system in (3) and (4) canbe transformed into the following input output form whilefollowing the procedure reported in [16] Let us assume thatthe system has a nonlinear output 119910 = ℎ(119909) To this end wedenote
Assume that the system reported in (3)-(4) has a relativedegree ldquo119903rdquo with respect to the defined nonlinear outputTherefore owing to [20] one has
119910(119903)
= 119871119903
119891
ℎ (119909) + 119871119892
(119871119903minus1
119891
ℎ (119909)) 120591 + 120577 (119909 119905) (8)
subject to the following conditions
(1) 119871119892
(119871119894
119891
ℎ(119909)) = 0 forall119909 isin 119861 where 119861 indicates theneighborhood of 119909
0
for 119894 lt 119903 minus 1(2) 119871119892
(119871119903minus1
119891
ℎ(119909)) = 0 where 120577(119909 119905) represents thematched unmodeled uncertainties System (8) bydefining the transformation 119910
(119894minus1)
= 120585119894
[21] can beput in the following form
1205851
= 1205852
1205852
= 1205853
120585119899
= 120593 ( ) + 120574 () 120591 + Δ119866119898
( 119905)
(9)
where the transformed states = (1205851
1205852
120585119899
) arephase variables 120591 is the control input andΔ119866
119898
( 119905)
represents matched uncertainties It is worthy tonotice that the inverted pendulum and the ball andbeam systems can be replaced in the aforementionedform
Note that both the formats are ready to design the controllaw for these systems In the next section we outline thedesign procedure for both the forms
3 Control Law Design
The control design for the forms presented in (3)-(4) and(9) is carried out in this section which we claim as ourmain contribution in this paper The main objective inthis work is to enhance the robustness of the system fromthe very beginning of the process which is the beauty ofintegral sliding mode control In general the integral slidingmode control law appears as follows [14] In the subsequentsubsections the authors aim to present the design procedure
31 Integral Sliding Mode This variant of sliding mode pos-sesses the main features of the sliding mode like robustnessand the existence chattering across the switching manifoldOn the other hand the sliding mode occurs from the verystart which consequently provides insensitivity of distur-bance from the beginning The control law can be expressedas follows
120591 = 1205910
+ 1205911
(10)
where the first component on the right hand side of (10)governs the systems dynamics in sliding modes whereas thesecond component compensates the matched disturbancesNow the aim is to present the design of the aforesaid controlcomponents
311 Control Design for Case-1 This control design for case-1 is the main obstacle in this subsection To define both thecomponents the following terms are defined
1198901
= 1199091
1198902
= 1199092
1198903
= 1198911
(1199091
1199092
1199093
1199094
)
1198904
=1205971198911
1205971199091
1199092
+1205971198911
1205971199092
1198911
+1205971198911
1205971199093
1199094
(11)
Using these new variables the components of the controllerare designed in the following subsection For the sake ofcompleteness the design of this component is worked out viasimple pole placement Following the design procedure ofpole placement method one gets
1205910
= minus1198961
1198901
minus 1198962
1198902
minus 1198963
1198903
minus 1198964
1198904
(12)
where 119896119894
119894 = 1 2 3 4 are the gains of this control componentThis control component steers the states of the nominalsystem to their defined equilibrium Now in the subsequentstudy the design of the uncertainties compensating term ispresented An integral manifold is defined as follows
120590 = 1198881
1198901
+ 1198882
1198902
+ 1198883
1198903
+ 1198904
+ 119911 = 1205900
+ 119911 (13)
where 1205900
= 1198881
1198901
+ 1198882
1198902
+ 1198883
1198903
+ 1198904
represents the conventionalsliding manifold which is Hurwitz by definition
Now computing along (3)-(4) one has
= 1198881
(1199092
+ 1198891
) + 1198882
(1198911
(1199091
1199092
1199093
1199094
) + 1198892
)
+ 1198883
(1198891198911
(1199091
1199092
1199093
1199094
)
119889119905) +
119889
119889119905(1205971198911
1205971199091
1199092
)
+1205971198911
1205971199091
1199092
+119889
119889119905(1205971198911
1205971199092
1198911
) +1205971198911
1205971199092
1198911
+119889
119889119905(1205971198911
1205971199093
1199094
) +1205971198911
1205971199093
1198912
(1199091
1199092
1199093
1199094
)
+1205971198911
1205971199093
119887 (1199091
1199092
1199093
1199094
) 1205910
+1205971198911
1205971199093
119887 (1199091
1199092
1199093
1199094
) 1205911
+1205971198911
1205971199093
1198893
(14)
Now choose the dynamics of the integral term as follows
= minus1198881
1199092
minus 1198882
1198911
(1199091
1199092
1199093
1199094
)
minus 1198883
(1198891198911
(1199091
1199092
1199093
1199094
)
119889119905) minus
119889
119889119905(1205971198911
1205971199091
1199092
)
minus1205971198911
1205971199091
1199092
minus119889
119889119905(1205971198911
1205971199092
1198911
) minus1205971198911
1205971199092
1198911
minus119889
119889119905(1205971198911
1205971199093
1199094
) minus1205971198911
1205971199093
(1199091
1199092
1199093
1199094
) 1205910
(15)
4 Mathematical Problems in Engineering
The expression of the termwhich compensates the uncertain-ties may be written as follows
1205911
= minus(1205971198911
1205971199093
119887 (1199091
1199092
1199093
1199094
))
minus1
sdot (1205971198911
1205971199093
1198912
(1199091
1199092
1199093
1199094
) + 119870sign (120590))
(16)
The overall controller will look like
120591 = minus1198961
1198901
minus 1198962
1198902
minus 1198963
1198903
minus 1198964
1198904
minus (1205971198911
1205971199093
119887 (1199091
1199092
1199093
1199094
))
minus1
sdot (1205971198911
1205971199093
1198912
(1199091
1199092
1199093
1199094
) + 119870sign (120590))
(17)
The constants 119888119894
rsquos are control gains which are selected intel-ligently according to bounds In the forthcoming paragraphthe stability of the presented integral sliding mode is carriedout in the presence of the disturbances and uncertaintiesConsider the following Lyapunov candidate function
119881 =1
21205902
(18)
The time derivative of this function along dynamics (11)becomes
= 120590 = 120590(1198881
(1199092
+ 1198891
)
+ 1198882
(1198911
(1199091
1199092
1199093
1199094
) + 1198892
)
+ 1198883
(1198891198911
(1199091
1199092
1199093
1199094
)
119889119905) +
119889
119889119905(1205971198911
1205971199091
1199092
)
+1205971198911
1205971199091
1199092
+119889
119889119905(1205971198911
1205971199092
1198911
) +1205971198911
1205971199092
1198911
+119889
119889119905(1205971198911
1205971199093
1199094
)
+1205971198911
1205971199093
1198912
(1199091
1199092
1199093
1199094
) +1205971198911
1205971199093
(1199091
1199092
1199093
1199094
) 1205910
+1205971198911
1205971199093
119887 (1199091
1199092
1199093
1199094
) 1205911
+1205971198911
1205971199093
1198893
)
(19)
The substitution of (15)-(16) results in the following form
le minus |120590| 1205781
lt 0
or + radic21205781
radic119881 lt 0
(20)
subject to 119870 ge [(1205971198911
1205971199093
)1198893
+ 1198881
1198891
+ 1198882
1198892
+ 120578]This expression confirms the enforcement of the sliding
mode from the very beginning of the process that is 120590 rarr 0
in finite time Now we proceed to the actual systemrsquos stabilityIf one considers 119890
1
as the output of the system then 1198902
1198903
and 119890
4
become the successive derivatives of 1198901
Whenever120590 = 0 is achieved the dynamics of the transformed system
(11) will converge asymptotically to zero under the action ofthe control component (12) [22] That is in closed loop thetransformed system dynamics will be operated under (12) asfollows
[[[[[
[
1198901
1198902
1198903
1198904
]]]]]
]
=
[[[[[
[
0 1 0 0
0 0 1 0
0 0 0 1
minus1198961
minus1198962
minus1198963
minus1198964
]]]]]
]
[[[[[
[
1198901
1198902
1198903
1198904
]]]]]
]
(21)
and the disturbances will be compensated via (16)The asymptotic convergence of 119890
1
1198902
1198903
and 1198904
to zeromeans the convergence of the indirectly actuated system (3)to zero On the other hand the states of the directly actuatedsystem (4) will remain bounded that is state of (4) will havesome nonzero value in order to keep 119890
1
at zero Thus theoverall system is stabilized and the desired control objectiveis achieved
32 Control Design for Case-2 Thenominal system related to(9) can be replaced in the subsequent alternative form
1205851
= 1205852
1205852
= 1205853
120585119903
= 120594 ( 120591) + 120591
(22)
where 120594( 120591) = 120593(120585 120591) + (120574() minus 1)120591 It is assumed that120594( 120591
(119896)
) = 0 at 119905 = 0 in addition to the next suppositionthat (22) is governed by 120591
0
1205851
= 1205852
1205852
= 1205853
120585119903
= 1205910
(23)
or
= 119860120585 + 1198611205910
(24)
where
119860 = [0(119903minus1)times1
119868(119903minus1)times(119903minus1)
01times1
01times(119903minus1)
]
119861 = [0(119903minus1)times1
1]
(25)
Mathematical Problems in Engineering 5
Once again following the pole placement procedure onemayhave for the sake of simplicity the input 120591
0
which is designedvia pole placement that is
1205910
= minus119870119879
0
120585 (26)
Now to get the desired robust performance the followingsliding manifold of integral type [14] is defined
120590 (120585) = 1205900
(120585) + 119911 (27)
where 1205900
(120585) is the usual sliding surface and 119911 is the integralterm The time derivative of (27) along (9) yields
= minus(
119903minus1
sum
119894=1
119888119894
120585119894+1
+ 1205910
)
119911 (0) = minus1205900
(120585 (0))
(28)
1205911
=1
120574 ()(minus120593 ( 120591) minus (120574 () minus 1) 120591
0
minus 119870 sign120590) (29)
This control law enforces sliding mode along the slidingmanifold defined in (27) The constant 119870 can be selectedaccording to the subsequent stability analysis
Thus the final control law becomes
1205911
= minus119870119879
0
120585
+1
120574 ()(minus120593 ( 119906) minus (120574 () minus 1) 120591
0
minus 119870 sign120590)
(30)
Theorem 4 Consider that |Δ119866119898
( 119905)| le 1205731
are satisfiedthen the sliding mode against the switching manifold 120590 = 0 canbe ensured and one has
119870 ge [119870119872
1205731
+ 1205781
] (31)
where 1205781
is a positive constant
Proof Toprove that the slidingmode can be enforced in finitetime differentiating (22) along the dynamics of (3)-(4) andthen substituting (30) one has
(120585) =
119903minus1
sum
119894=1
119888119894
120585119894+1
+ 1205910
minus 119870 sign120590 + 120574 () Δ119866119898
( 119905)
+
(32)
Substituting (28) in (32) and then rearranging one obtains
] (36)The inequality in (35) presents that 120590(120585) approaches zero in afinite time 119905
119904
[23] such that
119905119904
le radic2120578minus1
1
radic119881 (120590 (0)) (37)which completes the proof
4 Illustrative Example
The control algorithms presented in Section 3 are applied tothe control design of a ball and beam systemThe assessmentof the proposed controller for the ball and beam systemis carried out on the basis of output tracking robustnessenhancement via the elimination of reaching phase andchattering-free control input in the presence of uncertainties
41 Description of the Ball and Beam System The ball andbeam system is a very sound candidate of the class ofunderactuated nonlinear system It is famous because of itsnonlinear nature and due to its wide range of applicationsin the existing era like passenger cabin balancing in luxurycars balancing of liquid fuel in vertical take-off objectsIn terms of control scenarios it is an ill-defined relativedegree system which to some extent does not support inputoutput linearization A schematic diagram with their typicalparameters of the ball and beam system is displayed in theadjacent Figure 1 and Table 1 respectively In this study theauthors use the equipment manufacture by GoogolTech Ingeneral this system is equipped with a metallic ball whichis let free to roll on a rod having a specified length havingone end fixed and the other end moved up and down via anelectric servomotorThe position of the ball can be measuredvia different techniques The measured position is used asfeedback to the system and accordingly the motor moves thebeam to balance the ball at user defined location
The motion governing equations of this system are givenbelow which are adopted from [24]
(1198981199032
+ 1198621
) + (2119898119903 119903 + 1198622
)
+ (119898119892119903 +119871
2119872119892) cos120573 = 120591
1198624
119903 minus 119903
1205732
+ 119892 sin120573 = 0
(38)
6 Mathematical Problems in Engineering
yL
R rBall
Beam
Mg
mg
Motor
d120579
z120573
mg sin 120573
Figure 1 Schematic diagram of the ball and beam system
where 120579(119905) angle is subtended to make the ball stable thelever angle is represented by 120573(119905) 119903(119905) is the position of theball on the beam and Vin(119905) is the input voltage of the motorwhereas the controlled input appears mathematically via theexpression 120591(119905) = 119862
3
Vin(119905) in the dynamic modelThe derived parameters used in the dynamic model of
this system are represented by 1198621
1198622
1198623
and 1198624
with thefollowing mathematical relations [25]
1198621
=119877119898
times 119869119898
times 119871
119862119898
times 119862119887
times 119889+ 1198691
(39)
1198622
=119871
119889(119862119898
times 119862119887
119877119898
+ 119862119887
+119877119898
times 119869119898
119862119898
times 119862119892
) (40)
1198623
= 1 +119862119898
119877119898
(41)
1198624
=7
5 (42)
The equivalent state spacemodel of this is described as followsby assuming 119909
1
= 119903 (position of ball) 1199092
= 119903 (rate of changeof position) 119909
3
= 120573 (beam angle) and 1199094
= (the rate ofchange of angle of the motor)
1199091
= 1199092
1199092
=1
1198624
(minus119892 sin (1199093
))
1199093
= 1199094
1199094
=1
11989811990921
+ 1198621
(120591 minus (21198981199091
1199092
+ 1198622
) 1199094
minus (1198981198921199091
+119871
2119872119892) cos119909
3
)
(43)
Now the output of interest is 119910 = 1199091
which representsthe position of the ball This representation is similar tothat reported in (3)-(4) In the next discussion the controllerdesign is outlined
42 Controller Design Following the procedure outlined inSection 3 the authors proceed as follows
119910 = 1199091
= 1199092
= minus119892
1198624
sin (1199093
)
119910(3)
= minus119892
1198624
1199094
cos (1199093
)
119910(4)
=1
1198624
(11989811990921
+ 1198621
)[minus120591 cos119909
3
+ (21198981199091
1199092
+ 1198622
) 1199094
cos1199093
+ (1198981198921199091
+119871
2119872119892) cos2119909
3
+ 1199092
4
(1198981199092
1
+ 1198621
) sin1199093
]
119910(4)
= 119891119904
+ ℎ119904
120591
119891119904
=119892
1198624
[(21198981199091
1199092
+ 1198622
) 1199094
+ (1198981198921199091
+ (1198712)119872119892) cos21199093
+ 1199092
4
sin1199093
11989811990921
+ 1198621
]
ℎ119904
=minus119892 cos119909
3
1198624
(11989811990921
+ 1198621
)
(44)
Mathematical Problems in Engineering 7
Table 1 Parameters and values used in equations
Parameter Description Nominal values Units
119892Gravitationalacceleration 981 ms2
119898 Mass of ball 007 kg119872 Mass of beam 015 kg119871 Length of beam 04 m
119877119898
Resistance ofarmature of the motor 9 Ω
119869119898
Moment of inertia ofmotor 735 times 10
minus4 Nmrads2
119862119898
Torque constant ofmotor 00075 NmA
119862119892
Gear ratio 428 mdash
119889
Radius of armconnected toservomotor
004 m
1198691
Moment of inertia ofbeam 0001 kgm2
119862119887
Back emf constantvalue 05625 Vrads
Now writing this in the controllable canonical form (phasevariable form) one may have
( 119905) represents the modeluncertainties Herewe discuss ISMCon ball and beam systemwith fixed step tracking as well as variable step tracking Theintegral manifold is defined as follows
120590 = 1198881
1205851
+ 1198882
1205852
+ 1198883
1205853
+ 1205854
+ 119911 (47)
The expression of the overall controller which becomes willbe as follows
1205911
= minus1198961
1205851
minus 1198962
1205852
minus 1198963
1205853
minus 1198964
1205854
+1
120574 ()(minus120593 () minus (120574 () minus 1) 120591
0
minus 119870sign120590)
= 1198881
1205851
+ 1198882
1205852
+ 1198883
1205853
+ 119891119904
+ ℎ119904
1205910
+ ℎ119904
1205911
+
= minus1198881
1199092
+1198882
119892
1198624
sin1199093
+1198883
119892
1198624
1199094
cos1199093
minus 120574 () 1205910
minus 120593 ()
(48)
As the authors are performing the reference tracking heretherefore the integral manifold and the controller will appearas follows
120590 = 1198881
(1205851
minus 119903119889
) + 1198882
1205852
+ 1198883
1205853
+ 1205854
+ 119911 (49)
1205911
= minus1198961
(1205851
minus 119903119889
) minus 1198962
1205852
minus 1198963
1205853
minus 1198964
1205854
+1
120574 ()(minus120593 ( 120591) minus (120574 () minus 1) 120591
0
minus 119870sign (120590))
(50)
where 119903119889
is the desired reference with 119903119889
119903119889
119903119889
beingbounded
43 Simulation Results The simulation study of the system iscarried out by considering the reference tracking of a squarewave signal and sinusoidal wave signal In the subsequentparagraph their respective results will be demonstrated indetail
In case the efforts are directed to track a fixed square wavesignal in the presence of disturbances the initial conditions ofthe system were set to 119909
1
(0) = 04 1199092
(0) = 1199093
(0) = 1199094
(0) = 0Furthermore the square wave was defined in the simulationcode as follows
119903119889
(119905) =
20 cm 0 le 119905 le 19
14 cm 20 le 119905 le 39
20 cm 40 le 119905 le 60
(51)
The gains of the proposed controller presented from (39) to(41) are chosen according to Table 2
The output tracking performance of the proposed controlinput when a square wave is used as desired reference outputis shown in Figure 2 It can be clearly examined that theperformance is very appealing in this caseThe correspondingsliding manifold profile is displayed in Figure 3 which clearlyindicates that the sliding mode is established from the verybeginning of the processes which in turn results in enhancedrobustnessThe controlled input signalrsquos profile is depicted inFigure 4 with its zoomed profile as shown in Figure 5 It isobvious from both the figures that the control input derivesthe system with suppressed chattering phenomenon which is
8 Mathematical Problems in Engineering
Table 2 Parametric values used in the square wave tracking
Constants 1198621
1198622
1198623
1198701
1198702
1198703
1198704
119870
Values 12 12 011 40298 25018 60 41 5
0 10 20 30 40 50 600
00501
01502
02503
03504
Time (s)
Trac
king
per
form
ance
ActualDesired
Figure 2 Output tracking performance when a square wave is usedas referencedesired output
0 10 20 30 40 50 60
005
115
2
Time (s)
Slid
ing
surfa
ce
Sliding surface
minus05
minus1
minus15
minus2
Figure 3 Sliding manifold convergence profile in case of squarewave tracking
tolerable for the system actuators health Now from this casestudy it is concluded that integral sliding mode approach isan interesting candidate for this class
In this case study once again efforts are focused on thetracking of a sinusoidal signal which is defined as 119903
119889
(119905) =
sin(119905) in the presence of disturbances Like the previous casestudy the initial conditions of the system were set to 119909
1
(0) =
04 1199092
(0) = 1199093
(0) = 1199094
(0) = 0 In addition the gains of theproposed controller presented in (50) are chosen accordingto Table 3
The output tracking performance of the proposed controlinput when a sinusoidal signal is considered as desiredreference output is shown in Figure 6 It can be clearlyseen that the performance is excellent in this scenarioThe corresponding sliding manifold profile is displayed inFigure 7 which confirms the establishment of sliding modesfrom the starting instant and consequently enhancement ofrobustnessThe controlled input signalrsquos profile is depicted inFigure 8 It is obvious from the figure that the control input
Table 3 Parametric values used in the sinusoid wave tracking
Constants 1198621
1198622
1198623
1198701
1198702
1198703
1198704
119870
Values 12 12 011 40298 25018 230 49 5
Time (s)0 10 20 30 40 50 60
05
101520
Con
trol i
nput
Control input
minus5
minus10
minus15
minus20
Figure 4 Control input in square wave reference tracking
Time (s)
Con
trol i
nput
Control input
0 10 20 30 40 50 60
0
05
1
15
minus05
minus1
minus15
Figure 5 Zoom profile of the control input depicted in Figure 4
Time (s)
ActualDesired
0 5 10 15 20 25
Trac
king
per
form
ance
0
05
1
15
minus05
minus1
minus15
Figure 6 Output tracking performance when a sinusoidal wave isused as referencedesired output
evolves with suppressed chattering phenomenonwhich onceagain makes this design strategy a good candidate for theclass of these underactuated systems
44 Implementation Results The control technique proposedin this paper is implemented on the actual apparatus using
Mathematical Problems in Engineering 9
Time (s)
Sliding surface
0 5 10 15 20 25
0
05
1
Slid
ing
surfa
ce(v
aria
ble r
efer
ence
sign
al)
minus05
minus1
Figure 7 Sliding manifold convergence profile in case of sinusoidalwave tracking
Time (s)0 5 10 15 20 25
0
5
10
Control effort
minus5
minus10
Con
trol i
nput
ldquordquo
u
Figure 8 Control input in sinusoidal wave reference tracking
the MATLAB environment The detailed discussions arepresented below
441 Experimental Setup Description The experiment setupis equipped by GoogolTech GBB1004 with an electroniccontrol boxThe beam length is 40 cm alongwithmass of ballthat is 28 g and an intelligent IPM100 servo driver which isused formoving the ball on the beamThe experimental setupis shown in Figure 9
The input given to apparatus is the voltage Vin(119905) and theoutput is the position of themotor 120579(119905) which in otherwordsis an input for the positioning of the ball on the beam Thisapparatus uses potentiometer mounted within a slot insidethe beam to sense the position of the ball on the beam Themeasured position along the beam is fed to the AD converterof IPM100 motion drive
The power module used in GoogolTech requires 220Vand 10A input Note that the control accuracy of this manu-factured apparatus lies within the range of plusmn1mmThe typicalparameters values are listed in Table 1 The environmentused here includes Windows XP as an operating system andMATLAB 712Simulink 77 Furthermore the sampling timeused in forthcoming practical results was 2ms In the exper-imental processes the proposed controllers need velocitymeasurements which are in general not available One may
Figure 9 Experimental setup of the ball and beam equipped viaGoogolTech GBB1004
use different kind of velocity observersdifferentiator for thevelocity estimation [16] In order to make the implemen-tation easy and simple a derivative block of the Simulinkenvironment is used to provide the corresponding velocitiesmeasurements Now we are ready to discuss the results of thesystem
In this experiment the initial conditions were set to1199091
(0) = 028 1199092
(0) = 1199093
(0) = 1199094
(0) = 0 The referencesignal which is needed to be tracked is being defined in (51)In Figures 10 and 11 the tracking performance is shown Theresults reveal that the actual signal 119909
1
(119905) is pretty close tothe desired signal 119903
119889
(119905) with a steady state error which isapproximatelyplusmn0001mThe existence of this error is becauseof the apparatus
The observations of these tracking results make it clearthat the practically implemented results have very closeresemblance with the simulation result presented in Figure 2The error convergence depends on the initial conditions ofthe ball on the beam If the ball is placed very close to thedesired reference value then it will take little time to reachthe desired position On the other hand the convergence tothe desired position will take considerable time if the initialcondition is chosen far away from the desired values Thisphenomenon of convergence is according to the equipmentdesign and structure
The sliding manifold convergence and the control inputare shown in Figures 12 and 13 respectively The controlinput and the sliding manifolds show some deviations inthe first second This deviation occurs because the ball onthe beam being placed anywhere on the beam is firstmoved to one side of the beam and then ball moved to thedesired position The zoomed profile of the control inputbeing displayed in Figure 14 shows high frequency vibration(chattering) of magnitude plusmn007 This makes the proposedcontrol design algorithm an appealing candidate for this classof nonlinear systems The gains of the controller being usedin this experiment are displayed in Table 4
5 Conclusion
The control of underactuated systems because of their lessnumber of actuators than the degree of freedom is an
10 Mathematical Problems in Engineering
Table 4 Parametric values used in implementation
Constants 1198621
1198622
1198623
1198701
1198702
1198703
1198704
119870
Values 8 5 1 3 15 3 1 4
0
01
02
03
04
Trac
king
per
form
ance
0 5 10 15 20 25Time (s)
DesiredActual
16 18 20 22 240219
022022102220223
Figure 10 Output tracking performance when 119903119889
= 22 cm is set asreferencedesired output
Time (s)
DesiredActual
0
01
02
03
04
Trac
king
per
form
ance
0 10 20 30 40 50 6044 46 48 50 520196
019802
02020204
Figure 11 Output tracking performance when a square wave is usedas referencedesired output
0 5 10 15 20 25
05
101520
Time (s)
Slid
ing
surfa
ce
Sliding surface
minus5
minus10
minus15
minus20
Figure 12 Sliding surface of practical system
interesting objective among the researchers In this work anintegral sliding mode control approach due to its robustnessfrom the very beginning of the process is employed forthe control design of this class The design of the integralmanifold relied upon a transformed form The benefit of thetransformed form is that itmakes the design strategy easy and
0 5 10 15 20 25
0
2
4
6
Time (s)
Inpu
t per
form
ance
Control input
minus2
minus4
minus6
Figure 13 Control input for reference tracking
Inpu
t per
form
ance
Control input
0 5 10 15 20 25
0
05
1
Time (s)
minus05
minus1
Figure 14 Zoom profile of the control input depicted in Figure 13
simple The stability analysis and experimental results of theproposed control laws are presented which convey the goodfeatures and demand the proposed approachwhen the systemoperates under uncertainties
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Mahjoub F Mnif and N Derbel ldquoSet point stabilization ofa 2DOF underactuated manipulatorrdquo Journal of Computers vol6 no 2 pp 368ndash376 2011
[2] F Mnif ldquoVSS control for a class of underactuated Mechanicalsystemsrdquo International Journal of Computational Cognition vol3 no 2 pp 14ndash18 2005
[3] J Hauser S Sastry and P Kokotovic ldquoNonlinear control viaapproximate inputndashoutput linearization the ball and beamexamplerdquo IEEE Transactions on Automatic Control vol 37 no3 pp 392ndash398 1992
[4] M Jankovic D Fontanine and P V Kokotovic ldquoTORAexample cascade and passitivity-based control designsrdquo IEEETransactions on Control Systems Technology vol 6 pp 4347ndash4351 1996
Mathematical Problems in Engineering 11
[5] T Sugie and K Fujimoto ldquoControl of inverted pendulumsystems based on approximate linearization design and exper-imentrdquo in Proceedings of the 33rd IEEE Conference on Decisionand Control vol 2 pp 1647ndash1648 December 1994
[6] R Xu and U Ozguner ldquoSliding mode control of a class ofunderactuated systemsrdquo Automatica vol 44 no 1 pp 233ndash2412008
[7] M W Spong ldquoThe swing-up control problem for the acrobatrdquoIEEE Transactions on Systems Technology vol 15 no 1 pp 49ndash55 1995
[8] R W Brockett ldquoAsymptotic stability feedback and stabiliza-tionrdquo in Differential Geometric Control Theory pp 181ndash191Birkhauser Boston Mass USA 1983
[9] C J Tomlin and S S Sastry ldquoSwitching through singularitiesrdquoSystems and Control Letters vol 35 no 3 pp 145ndash154 1998
[10] W-H Chen and D J Ballance ldquoOn a switching control schemefor nonlinear systems with ill-defined relative degreerdquo Systemsamp Control Letters vol 47 no 2 pp 159ndash166 2002
[11] F Zhang and B Fernndez-Rodriguez ldquoFeedback linearizationcontrol of systems with singularitiesrdquo in Proceedings of the6th International Conference on Complex Systems (ICCS rsquo06)Boston Mass USA June 2006
[12] D Seto and J Baillieul ldquoControl problems in super-articulatedmechanical systemsrdquo IEEE Transactions on Automatic Controlvol 39 no 12 pp 2442ndash2453 1994
[13] M W Spong Energy Based Control of a Class of UnderactuatedMechanical Systems IFACWorld Congress 1996
[14] V I Utkin Sliding Mode Control in Electromechanical SystemsTaylor amp Francis 1999
[15] M Rubagotti A Estrada F Castanos A Ferrara and LFridman ldquoIntegral sliding mode control for nonlinear systemswith matched and unmatched perturbationsrdquo IEEE Transactionon Automatic Control vol 56 no 11 pp 2699ndash2704 2011
[16] Q Khan A I Bhatti S Iqbal and M Iqbal ldquoDynamicintegral sliding mode for MIMO uncertain nonlinear systemsrdquoInternational Journal of Control Automation and Systems vol9 no 1 pp 151ndash160 2011
[17] R Olfati-Saber ldquoNormal forms for underactuated mechanicalsystems with symmetryrdquo IEEE Transactions on Automatic Con-trol vol 47 no 2 pp 305ndash308 2002
[18] R Lozano I Fantoni and D J Block ldquoStabilization of theinverted pendulum around its homoclinic orbitrdquo Systems andControl Letters vol 40 no 3 pp 197ndash204 2000
[19] E Altug J P Ostrowski and R Mahony ldquoControl of aquadrotor helicopter using visual feedbackrdquo in Proceedings ofthe IEEE International Conference on Robotics and Automation(ICRA rsquo02) vol 1 pp 72ndash77 Washington DC USA 2002
[20] A Isidori Nonlinear Control Systems Communications andControl Engineering Series Springer Berlin Germany 3rdedition 1995
[21] H Sira-Ramirez ldquoOn the dynamical sliding mode control ofnonlinear systemsrdquo International Journal of Control vol 57 no5 pp 1039ndash1061 1993
[22] Q Khan A I Bhatti and A Ferrara ldquoDynamic sliding modecontrol design based on an integral manifold for nonlinearuncertain systemsrdquo Journal of Nonlinear Dynamics vol 2014Article ID 489364 10 pages 2014
[23] C Edwards and S K Spurgeon Sliding Modes Control Theoryand Applications Taylor amp Francis London UK 1998
[24] A M Khan A I Bhatti S U Din and Q Khan ldquoStaticamp dynamic sliding mode control of ball and beam systemrdquo
in Proceedings of the 9th International Bhurban Conferenceon Applied Sciences and Technology (IBCAST rsquo12) pp 32ndash36Islamabad Pakistan January 2012
[25] N B Almutairi and M Zribi ldquoOn the sliding mode control ofa Ball on a Beam systemrdquo Nonlinear Dynamics vol 59 no 1-2pp 222ndash239 2010
The aforementioned design strategies were quite suitableand resulted in satisfactory results but it is worthy to notethat the system often becomes too sensitive to disturbance inthe reaching phase of sliding mode strategy that the systemmay even become unstable Therefore in order to get rid ofthis issue the integral sliding mode strategy was proposed[14ndash16] In this paper a robust integral sliding mode control(RISMC) approach for underactuated systems is proposedThe benefit of this strategy is enhancement of robustnessfrom initial time instant It also suppresses the well-knownchattering phenomenon across the manifold Before thedesign presentation the system is suitably transformed intospecial formats An integral slidingmode strategy is proposedfor both the cases along with their comprehensive stabilityanalysis The proposed technique is practically implementedon the ball and beam system to authenticate the affectivityand efficiency of the designed algorithm Note that in thispaper our contributions are twofold The first one is thedevelopment of theRISMCand the secondone is the practicalresults of the system on the said system The rest of thepaper is organized as follows In Section 2 the problem isformulated into two special formats which further simplifythe design methodology In Section 3 the integral slidingmode strategy for both the cases is discussed in detailaccompanied by their respective stability analysis in terms ofLyapunov theory Section 4 presents the development of thecontrol laws simulation and practical results of the ball andbeam system Section 5 concludes the overall efforts beingmade in this study In the end more relevant recent articlesare enlisted
2 Problem Formulation
The dynamic equations which govern the motion of the classof underactuated system can be presented as
where 119902 and are 119899-dimensional position velocityand acceleration vectors and 119869(119902) 119862(119902 ) 119866(119902) and 119865()
represent the inertia Coriolis gravitational and fractionaltorques matrices respectively 120591 is the measured controlinput and 120575(119902 119905) represents the uncertainties in the controlinput channel whereas 119861 is the control input channel
It is assumed that rank(119869minus1(119902)119861) = 119898 and the originis considered to be the equilibrium point for the aforemen-tioned system Now the system in (1) can be rewritten inalternate form as follows
11989811
(119902) 1199021
+ 11989812
(119902) 1199022
+ ℎ1
(119902 ) = 0
11989821
(119902) 1199021
+ 11989812
(119902) 1199022
+ ℎ2
(119902 ) = 120591
(2)
where 119902 = [1199021
1199022
]119879 represents the states of the system and 119902
and point to the states In order to design a control law thesystem in (2) can be transformed into two formats which aredescribed in the subsequent study
21 System in Cascaded Form Following some algebraicmanipulations the system in (2) may be written in cascadedform as follows [17]
1199091
= 1199092
+ 1198891
1199092
= 1198911
(1199091
1199092
1199093
1199094
) + 1198892
(3)
1199093
= 1199094
1199094
= 1198912
(1199091
1199092
1199093
1199094
) + 119887 (1199091
1199092
1199093
1199094
) 120591 + 1198893
(4)
where 1199091
1199092
1199093
1199094
are measurable states of the systems suchthat 119909
1
and 1199092
are pointing to the position and velocity ofthe indirect actuated system (3) while 119909
3
and 1199094
representthe position and velocity of the directly actuated system(4) 120591 represents the controlled signal as already discussedto the system (4) input Owing to the assumption statedimmediately after (1) the inverse of 119887 exists The nonlinearfunctions 119891
1
1198912
1198774119899
rarr 119877119899 119887 119877
4119899
rarr 119877119899times119899 are smooth in
nature Now following the procedure of [6] the disturbances1198891
1198892
1198893
are deliberately introduced to get an approximatecontrollable canonical form Note that practical systems likeinverted pendulum [18] TORA [4] VTOL (vertical take-offand landing) aircraft [17] and quad rotor [19] can be putin the form presented in (3) and (4) Before proceeding tothe control design of the above cascaded form the followingassumptions are made
Assumption 1 Assume that
1198911
(0 0 0 0) = 0 (5)
This condition is necessary for the system origin to be inequilibrium point when the system is operated in closed loop
Assumption 2 1205971198911
1205971199093
is invertible or 1205971198911
1205971199094
is invertible
Assumption 3 1198911
(0 0 1199093
1199094
) = 0 is an asymptotically stablemanifold that is 119909
3
and 1199094
approaches zero
Note that Assumptions 2 and 3 lie in the category ofnonnecessary conditions These are only used when oneneeds to furnish the closed loop system with a sliding modecontroller (see for details [6])
22 Input Output Form The system in (3) and (4) canbe transformed into the following input output form whilefollowing the procedure reported in [16] Let us assume thatthe system has a nonlinear output 119910 = ℎ(119909) To this end wedenote
Assume that the system reported in (3)-(4) has a relativedegree ldquo119903rdquo with respect to the defined nonlinear outputTherefore owing to [20] one has
119910(119903)
= 119871119903
119891
ℎ (119909) + 119871119892
(119871119903minus1
119891
ℎ (119909)) 120591 + 120577 (119909 119905) (8)
subject to the following conditions
(1) 119871119892
(119871119894
119891
ℎ(119909)) = 0 forall119909 isin 119861 where 119861 indicates theneighborhood of 119909
0
for 119894 lt 119903 minus 1(2) 119871119892
(119871119903minus1
119891
ℎ(119909)) = 0 where 120577(119909 119905) represents thematched unmodeled uncertainties System (8) bydefining the transformation 119910
(119894minus1)
= 120585119894
[21] can beput in the following form
1205851
= 1205852
1205852
= 1205853
120585119899
= 120593 ( ) + 120574 () 120591 + Δ119866119898
( 119905)
(9)
where the transformed states = (1205851
1205852
120585119899
) arephase variables 120591 is the control input andΔ119866
119898
( 119905)
represents matched uncertainties It is worthy tonotice that the inverted pendulum and the ball andbeam systems can be replaced in the aforementionedform
Note that both the formats are ready to design the controllaw for these systems In the next section we outline thedesign procedure for both the forms
3 Control Law Design
The control design for the forms presented in (3)-(4) and(9) is carried out in this section which we claim as ourmain contribution in this paper The main objective inthis work is to enhance the robustness of the system fromthe very beginning of the process which is the beauty ofintegral sliding mode control In general the integral slidingmode control law appears as follows [14] In the subsequentsubsections the authors aim to present the design procedure
31 Integral Sliding Mode This variant of sliding mode pos-sesses the main features of the sliding mode like robustnessand the existence chattering across the switching manifoldOn the other hand the sliding mode occurs from the verystart which consequently provides insensitivity of distur-bance from the beginning The control law can be expressedas follows
120591 = 1205910
+ 1205911
(10)
where the first component on the right hand side of (10)governs the systems dynamics in sliding modes whereas thesecond component compensates the matched disturbancesNow the aim is to present the design of the aforesaid controlcomponents
311 Control Design for Case-1 This control design for case-1 is the main obstacle in this subsection To define both thecomponents the following terms are defined
1198901
= 1199091
1198902
= 1199092
1198903
= 1198911
(1199091
1199092
1199093
1199094
)
1198904
=1205971198911
1205971199091
1199092
+1205971198911
1205971199092
1198911
+1205971198911
1205971199093
1199094
(11)
Using these new variables the components of the controllerare designed in the following subsection For the sake ofcompleteness the design of this component is worked out viasimple pole placement Following the design procedure ofpole placement method one gets
1205910
= minus1198961
1198901
minus 1198962
1198902
minus 1198963
1198903
minus 1198964
1198904
(12)
where 119896119894
119894 = 1 2 3 4 are the gains of this control componentThis control component steers the states of the nominalsystem to their defined equilibrium Now in the subsequentstudy the design of the uncertainties compensating term ispresented An integral manifold is defined as follows
120590 = 1198881
1198901
+ 1198882
1198902
+ 1198883
1198903
+ 1198904
+ 119911 = 1205900
+ 119911 (13)
where 1205900
= 1198881
1198901
+ 1198882
1198902
+ 1198883
1198903
+ 1198904
represents the conventionalsliding manifold which is Hurwitz by definition
Now computing along (3)-(4) one has
= 1198881
(1199092
+ 1198891
) + 1198882
(1198911
(1199091
1199092
1199093
1199094
) + 1198892
)
+ 1198883
(1198891198911
(1199091
1199092
1199093
1199094
)
119889119905) +
119889
119889119905(1205971198911
1205971199091
1199092
)
+1205971198911
1205971199091
1199092
+119889
119889119905(1205971198911
1205971199092
1198911
) +1205971198911
1205971199092
1198911
+119889
119889119905(1205971198911
1205971199093
1199094
) +1205971198911
1205971199093
1198912
(1199091
1199092
1199093
1199094
)
+1205971198911
1205971199093
119887 (1199091
1199092
1199093
1199094
) 1205910
+1205971198911
1205971199093
119887 (1199091
1199092
1199093
1199094
) 1205911
+1205971198911
1205971199093
1198893
(14)
Now choose the dynamics of the integral term as follows
= minus1198881
1199092
minus 1198882
1198911
(1199091
1199092
1199093
1199094
)
minus 1198883
(1198891198911
(1199091
1199092
1199093
1199094
)
119889119905) minus
119889
119889119905(1205971198911
1205971199091
1199092
)
minus1205971198911
1205971199091
1199092
minus119889
119889119905(1205971198911
1205971199092
1198911
) minus1205971198911
1205971199092
1198911
minus119889
119889119905(1205971198911
1205971199093
1199094
) minus1205971198911
1205971199093
(1199091
1199092
1199093
1199094
) 1205910
(15)
4 Mathematical Problems in Engineering
The expression of the termwhich compensates the uncertain-ties may be written as follows
1205911
= minus(1205971198911
1205971199093
119887 (1199091
1199092
1199093
1199094
))
minus1
sdot (1205971198911
1205971199093
1198912
(1199091
1199092
1199093
1199094
) + 119870sign (120590))
(16)
The overall controller will look like
120591 = minus1198961
1198901
minus 1198962
1198902
minus 1198963
1198903
minus 1198964
1198904
minus (1205971198911
1205971199093
119887 (1199091
1199092
1199093
1199094
))
minus1
sdot (1205971198911
1205971199093
1198912
(1199091
1199092
1199093
1199094
) + 119870sign (120590))
(17)
The constants 119888119894
rsquos are control gains which are selected intel-ligently according to bounds In the forthcoming paragraphthe stability of the presented integral sliding mode is carriedout in the presence of the disturbances and uncertaintiesConsider the following Lyapunov candidate function
119881 =1
21205902
(18)
The time derivative of this function along dynamics (11)becomes
= 120590 = 120590(1198881
(1199092
+ 1198891
)
+ 1198882
(1198911
(1199091
1199092
1199093
1199094
) + 1198892
)
+ 1198883
(1198891198911
(1199091
1199092
1199093
1199094
)
119889119905) +
119889
119889119905(1205971198911
1205971199091
1199092
)
+1205971198911
1205971199091
1199092
+119889
119889119905(1205971198911
1205971199092
1198911
) +1205971198911
1205971199092
1198911
+119889
119889119905(1205971198911
1205971199093
1199094
)
+1205971198911
1205971199093
1198912
(1199091
1199092
1199093
1199094
) +1205971198911
1205971199093
(1199091
1199092
1199093
1199094
) 1205910
+1205971198911
1205971199093
119887 (1199091
1199092
1199093
1199094
) 1205911
+1205971198911
1205971199093
1198893
)
(19)
The substitution of (15)-(16) results in the following form
le minus |120590| 1205781
lt 0
or + radic21205781
radic119881 lt 0
(20)
subject to 119870 ge [(1205971198911
1205971199093
)1198893
+ 1198881
1198891
+ 1198882
1198892
+ 120578]This expression confirms the enforcement of the sliding
mode from the very beginning of the process that is 120590 rarr 0
in finite time Now we proceed to the actual systemrsquos stabilityIf one considers 119890
1
as the output of the system then 1198902
1198903
and 119890
4
become the successive derivatives of 1198901
Whenever120590 = 0 is achieved the dynamics of the transformed system
(11) will converge asymptotically to zero under the action ofthe control component (12) [22] That is in closed loop thetransformed system dynamics will be operated under (12) asfollows
[[[[[
[
1198901
1198902
1198903
1198904
]]]]]
]
=
[[[[[
[
0 1 0 0
0 0 1 0
0 0 0 1
minus1198961
minus1198962
minus1198963
minus1198964
]]]]]
]
[[[[[
[
1198901
1198902
1198903
1198904
]]]]]
]
(21)
and the disturbances will be compensated via (16)The asymptotic convergence of 119890
1
1198902
1198903
and 1198904
to zeromeans the convergence of the indirectly actuated system (3)to zero On the other hand the states of the directly actuatedsystem (4) will remain bounded that is state of (4) will havesome nonzero value in order to keep 119890
1
at zero Thus theoverall system is stabilized and the desired control objectiveis achieved
32 Control Design for Case-2 Thenominal system related to(9) can be replaced in the subsequent alternative form
1205851
= 1205852
1205852
= 1205853
120585119903
= 120594 ( 120591) + 120591
(22)
where 120594( 120591) = 120593(120585 120591) + (120574() minus 1)120591 It is assumed that120594( 120591
(119896)
) = 0 at 119905 = 0 in addition to the next suppositionthat (22) is governed by 120591
0
1205851
= 1205852
1205852
= 1205853
120585119903
= 1205910
(23)
or
= 119860120585 + 1198611205910
(24)
where
119860 = [0(119903minus1)times1
119868(119903minus1)times(119903minus1)
01times1
01times(119903minus1)
]
119861 = [0(119903minus1)times1
1]
(25)
Mathematical Problems in Engineering 5
Once again following the pole placement procedure onemayhave for the sake of simplicity the input 120591
0
which is designedvia pole placement that is
1205910
= minus119870119879
0
120585 (26)
Now to get the desired robust performance the followingsliding manifold of integral type [14] is defined
120590 (120585) = 1205900
(120585) + 119911 (27)
where 1205900
(120585) is the usual sliding surface and 119911 is the integralterm The time derivative of (27) along (9) yields
= minus(
119903minus1
sum
119894=1
119888119894
120585119894+1
+ 1205910
)
119911 (0) = minus1205900
(120585 (0))
(28)
1205911
=1
120574 ()(minus120593 ( 120591) minus (120574 () minus 1) 120591
0
minus 119870 sign120590) (29)
This control law enforces sliding mode along the slidingmanifold defined in (27) The constant 119870 can be selectedaccording to the subsequent stability analysis
Thus the final control law becomes
1205911
= minus119870119879
0
120585
+1
120574 ()(minus120593 ( 119906) minus (120574 () minus 1) 120591
0
minus 119870 sign120590)
(30)
Theorem 4 Consider that |Δ119866119898
( 119905)| le 1205731
are satisfiedthen the sliding mode against the switching manifold 120590 = 0 canbe ensured and one has
119870 ge [119870119872
1205731
+ 1205781
] (31)
where 1205781
is a positive constant
Proof Toprove that the slidingmode can be enforced in finitetime differentiating (22) along the dynamics of (3)-(4) andthen substituting (30) one has
(120585) =
119903minus1
sum
119894=1
119888119894
120585119894+1
+ 1205910
minus 119870 sign120590 + 120574 () Δ119866119898
( 119905)
+
(32)
Substituting (28) in (32) and then rearranging one obtains
] (36)The inequality in (35) presents that 120590(120585) approaches zero in afinite time 119905
119904
[23] such that
119905119904
le radic2120578minus1
1
radic119881 (120590 (0)) (37)which completes the proof
4 Illustrative Example
The control algorithms presented in Section 3 are applied tothe control design of a ball and beam systemThe assessmentof the proposed controller for the ball and beam systemis carried out on the basis of output tracking robustnessenhancement via the elimination of reaching phase andchattering-free control input in the presence of uncertainties
41 Description of the Ball and Beam System The ball andbeam system is a very sound candidate of the class ofunderactuated nonlinear system It is famous because of itsnonlinear nature and due to its wide range of applicationsin the existing era like passenger cabin balancing in luxurycars balancing of liquid fuel in vertical take-off objectsIn terms of control scenarios it is an ill-defined relativedegree system which to some extent does not support inputoutput linearization A schematic diagram with their typicalparameters of the ball and beam system is displayed in theadjacent Figure 1 and Table 1 respectively In this study theauthors use the equipment manufacture by GoogolTech Ingeneral this system is equipped with a metallic ball whichis let free to roll on a rod having a specified length havingone end fixed and the other end moved up and down via anelectric servomotorThe position of the ball can be measuredvia different techniques The measured position is used asfeedback to the system and accordingly the motor moves thebeam to balance the ball at user defined location
The motion governing equations of this system are givenbelow which are adopted from [24]
(1198981199032
+ 1198621
) + (2119898119903 119903 + 1198622
)
+ (119898119892119903 +119871
2119872119892) cos120573 = 120591
1198624
119903 minus 119903
1205732
+ 119892 sin120573 = 0
(38)
6 Mathematical Problems in Engineering
yL
R rBall
Beam
Mg
mg
Motor
d120579
z120573
mg sin 120573
Figure 1 Schematic diagram of the ball and beam system
where 120579(119905) angle is subtended to make the ball stable thelever angle is represented by 120573(119905) 119903(119905) is the position of theball on the beam and Vin(119905) is the input voltage of the motorwhereas the controlled input appears mathematically via theexpression 120591(119905) = 119862
3
Vin(119905) in the dynamic modelThe derived parameters used in the dynamic model of
this system are represented by 1198621
1198622
1198623
and 1198624
with thefollowing mathematical relations [25]
1198621
=119877119898
times 119869119898
times 119871
119862119898
times 119862119887
times 119889+ 1198691
(39)
1198622
=119871
119889(119862119898
times 119862119887
119877119898
+ 119862119887
+119877119898
times 119869119898
119862119898
times 119862119892
) (40)
1198623
= 1 +119862119898
119877119898
(41)
1198624
=7
5 (42)
The equivalent state spacemodel of this is described as followsby assuming 119909
1
= 119903 (position of ball) 1199092
= 119903 (rate of changeof position) 119909
3
= 120573 (beam angle) and 1199094
= (the rate ofchange of angle of the motor)
1199091
= 1199092
1199092
=1
1198624
(minus119892 sin (1199093
))
1199093
= 1199094
1199094
=1
11989811990921
+ 1198621
(120591 minus (21198981199091
1199092
+ 1198622
) 1199094
minus (1198981198921199091
+119871
2119872119892) cos119909
3
)
(43)
Now the output of interest is 119910 = 1199091
which representsthe position of the ball This representation is similar tothat reported in (3)-(4) In the next discussion the controllerdesign is outlined
42 Controller Design Following the procedure outlined inSection 3 the authors proceed as follows
119910 = 1199091
= 1199092
= minus119892
1198624
sin (1199093
)
119910(3)
= minus119892
1198624
1199094
cos (1199093
)
119910(4)
=1
1198624
(11989811990921
+ 1198621
)[minus120591 cos119909
3
+ (21198981199091
1199092
+ 1198622
) 1199094
cos1199093
+ (1198981198921199091
+119871
2119872119892) cos2119909
3
+ 1199092
4
(1198981199092
1
+ 1198621
) sin1199093
]
119910(4)
= 119891119904
+ ℎ119904
120591
119891119904
=119892
1198624
[(21198981199091
1199092
+ 1198622
) 1199094
+ (1198981198921199091
+ (1198712)119872119892) cos21199093
+ 1199092
4
sin1199093
11989811990921
+ 1198621
]
ℎ119904
=minus119892 cos119909
3
1198624
(11989811990921
+ 1198621
)
(44)
Mathematical Problems in Engineering 7
Table 1 Parameters and values used in equations
Parameter Description Nominal values Units
119892Gravitationalacceleration 981 ms2
119898 Mass of ball 007 kg119872 Mass of beam 015 kg119871 Length of beam 04 m
119877119898
Resistance ofarmature of the motor 9 Ω
119869119898
Moment of inertia ofmotor 735 times 10
minus4 Nmrads2
119862119898
Torque constant ofmotor 00075 NmA
119862119892
Gear ratio 428 mdash
119889
Radius of armconnected toservomotor
004 m
1198691
Moment of inertia ofbeam 0001 kgm2
119862119887
Back emf constantvalue 05625 Vrads
Now writing this in the controllable canonical form (phasevariable form) one may have
( 119905) represents the modeluncertainties Herewe discuss ISMCon ball and beam systemwith fixed step tracking as well as variable step tracking Theintegral manifold is defined as follows
120590 = 1198881
1205851
+ 1198882
1205852
+ 1198883
1205853
+ 1205854
+ 119911 (47)
The expression of the overall controller which becomes willbe as follows
1205911
= minus1198961
1205851
minus 1198962
1205852
minus 1198963
1205853
minus 1198964
1205854
+1
120574 ()(minus120593 () minus (120574 () minus 1) 120591
0
minus 119870sign120590)
= 1198881
1205851
+ 1198882
1205852
+ 1198883
1205853
+ 119891119904
+ ℎ119904
1205910
+ ℎ119904
1205911
+
= minus1198881
1199092
+1198882
119892
1198624
sin1199093
+1198883
119892
1198624
1199094
cos1199093
minus 120574 () 1205910
minus 120593 ()
(48)
As the authors are performing the reference tracking heretherefore the integral manifold and the controller will appearas follows
120590 = 1198881
(1205851
minus 119903119889
) + 1198882
1205852
+ 1198883
1205853
+ 1205854
+ 119911 (49)
1205911
= minus1198961
(1205851
minus 119903119889
) minus 1198962
1205852
minus 1198963
1205853
minus 1198964
1205854
+1
120574 ()(minus120593 ( 120591) minus (120574 () minus 1) 120591
0
minus 119870sign (120590))
(50)
where 119903119889
is the desired reference with 119903119889
119903119889
119903119889
beingbounded
43 Simulation Results The simulation study of the system iscarried out by considering the reference tracking of a squarewave signal and sinusoidal wave signal In the subsequentparagraph their respective results will be demonstrated indetail
In case the efforts are directed to track a fixed square wavesignal in the presence of disturbances the initial conditions ofthe system were set to 119909
1
(0) = 04 1199092
(0) = 1199093
(0) = 1199094
(0) = 0Furthermore the square wave was defined in the simulationcode as follows
119903119889
(119905) =
20 cm 0 le 119905 le 19
14 cm 20 le 119905 le 39
20 cm 40 le 119905 le 60
(51)
The gains of the proposed controller presented from (39) to(41) are chosen according to Table 2
The output tracking performance of the proposed controlinput when a square wave is used as desired reference outputis shown in Figure 2 It can be clearly examined that theperformance is very appealing in this caseThe correspondingsliding manifold profile is displayed in Figure 3 which clearlyindicates that the sliding mode is established from the verybeginning of the processes which in turn results in enhancedrobustnessThe controlled input signalrsquos profile is depicted inFigure 4 with its zoomed profile as shown in Figure 5 It isobvious from both the figures that the control input derivesthe system with suppressed chattering phenomenon which is
8 Mathematical Problems in Engineering
Table 2 Parametric values used in the square wave tracking
Constants 1198621
1198622
1198623
1198701
1198702
1198703
1198704
119870
Values 12 12 011 40298 25018 60 41 5
0 10 20 30 40 50 600
00501
01502
02503
03504
Time (s)
Trac
king
per
form
ance
ActualDesired
Figure 2 Output tracking performance when a square wave is usedas referencedesired output
0 10 20 30 40 50 60
005
115
2
Time (s)
Slid
ing
surfa
ce
Sliding surface
minus05
minus1
minus15
minus2
Figure 3 Sliding manifold convergence profile in case of squarewave tracking
tolerable for the system actuators health Now from this casestudy it is concluded that integral sliding mode approach isan interesting candidate for this class
In this case study once again efforts are focused on thetracking of a sinusoidal signal which is defined as 119903
119889
(119905) =
sin(119905) in the presence of disturbances Like the previous casestudy the initial conditions of the system were set to 119909
1
(0) =
04 1199092
(0) = 1199093
(0) = 1199094
(0) = 0 In addition the gains of theproposed controller presented in (50) are chosen accordingto Table 3
The output tracking performance of the proposed controlinput when a sinusoidal signal is considered as desiredreference output is shown in Figure 6 It can be clearlyseen that the performance is excellent in this scenarioThe corresponding sliding manifold profile is displayed inFigure 7 which confirms the establishment of sliding modesfrom the starting instant and consequently enhancement ofrobustnessThe controlled input signalrsquos profile is depicted inFigure 8 It is obvious from the figure that the control input
Table 3 Parametric values used in the sinusoid wave tracking
Constants 1198621
1198622
1198623
1198701
1198702
1198703
1198704
119870
Values 12 12 011 40298 25018 230 49 5
Time (s)0 10 20 30 40 50 60
05
101520
Con
trol i
nput
Control input
minus5
minus10
minus15
minus20
Figure 4 Control input in square wave reference tracking
Time (s)
Con
trol i
nput
Control input
0 10 20 30 40 50 60
0
05
1
15
minus05
minus1
minus15
Figure 5 Zoom profile of the control input depicted in Figure 4
Time (s)
ActualDesired
0 5 10 15 20 25
Trac
king
per
form
ance
0
05
1
15
minus05
minus1
minus15
Figure 6 Output tracking performance when a sinusoidal wave isused as referencedesired output
evolves with suppressed chattering phenomenonwhich onceagain makes this design strategy a good candidate for theclass of these underactuated systems
44 Implementation Results The control technique proposedin this paper is implemented on the actual apparatus using
Mathematical Problems in Engineering 9
Time (s)
Sliding surface
0 5 10 15 20 25
0
05
1
Slid
ing
surfa
ce(v
aria
ble r
efer
ence
sign
al)
minus05
minus1
Figure 7 Sliding manifold convergence profile in case of sinusoidalwave tracking
Time (s)0 5 10 15 20 25
0
5
10
Control effort
minus5
minus10
Con
trol i
nput
ldquordquo
u
Figure 8 Control input in sinusoidal wave reference tracking
the MATLAB environment The detailed discussions arepresented below
441 Experimental Setup Description The experiment setupis equipped by GoogolTech GBB1004 with an electroniccontrol boxThe beam length is 40 cm alongwithmass of ballthat is 28 g and an intelligent IPM100 servo driver which isused formoving the ball on the beamThe experimental setupis shown in Figure 9
The input given to apparatus is the voltage Vin(119905) and theoutput is the position of themotor 120579(119905) which in otherwordsis an input for the positioning of the ball on the beam Thisapparatus uses potentiometer mounted within a slot insidethe beam to sense the position of the ball on the beam Themeasured position along the beam is fed to the AD converterof IPM100 motion drive
The power module used in GoogolTech requires 220Vand 10A input Note that the control accuracy of this manu-factured apparatus lies within the range of plusmn1mmThe typicalparameters values are listed in Table 1 The environmentused here includes Windows XP as an operating system andMATLAB 712Simulink 77 Furthermore the sampling timeused in forthcoming practical results was 2ms In the exper-imental processes the proposed controllers need velocitymeasurements which are in general not available One may
Figure 9 Experimental setup of the ball and beam equipped viaGoogolTech GBB1004
use different kind of velocity observersdifferentiator for thevelocity estimation [16] In order to make the implemen-tation easy and simple a derivative block of the Simulinkenvironment is used to provide the corresponding velocitiesmeasurements Now we are ready to discuss the results of thesystem
In this experiment the initial conditions were set to1199091
(0) = 028 1199092
(0) = 1199093
(0) = 1199094
(0) = 0 The referencesignal which is needed to be tracked is being defined in (51)In Figures 10 and 11 the tracking performance is shown Theresults reveal that the actual signal 119909
1
(119905) is pretty close tothe desired signal 119903
119889
(119905) with a steady state error which isapproximatelyplusmn0001mThe existence of this error is becauseof the apparatus
The observations of these tracking results make it clearthat the practically implemented results have very closeresemblance with the simulation result presented in Figure 2The error convergence depends on the initial conditions ofthe ball on the beam If the ball is placed very close to thedesired reference value then it will take little time to reachthe desired position On the other hand the convergence tothe desired position will take considerable time if the initialcondition is chosen far away from the desired values Thisphenomenon of convergence is according to the equipmentdesign and structure
The sliding manifold convergence and the control inputare shown in Figures 12 and 13 respectively The controlinput and the sliding manifolds show some deviations inthe first second This deviation occurs because the ball onthe beam being placed anywhere on the beam is firstmoved to one side of the beam and then ball moved to thedesired position The zoomed profile of the control inputbeing displayed in Figure 14 shows high frequency vibration(chattering) of magnitude plusmn007 This makes the proposedcontrol design algorithm an appealing candidate for this classof nonlinear systems The gains of the controller being usedin this experiment are displayed in Table 4
5 Conclusion
The control of underactuated systems because of their lessnumber of actuators than the degree of freedom is an
10 Mathematical Problems in Engineering
Table 4 Parametric values used in implementation
Constants 1198621
1198622
1198623
1198701
1198702
1198703
1198704
119870
Values 8 5 1 3 15 3 1 4
0
01
02
03
04
Trac
king
per
form
ance
0 5 10 15 20 25Time (s)
DesiredActual
16 18 20 22 240219
022022102220223
Figure 10 Output tracking performance when 119903119889
= 22 cm is set asreferencedesired output
Time (s)
DesiredActual
0
01
02
03
04
Trac
king
per
form
ance
0 10 20 30 40 50 6044 46 48 50 520196
019802
02020204
Figure 11 Output tracking performance when a square wave is usedas referencedesired output
0 5 10 15 20 25
05
101520
Time (s)
Slid
ing
surfa
ce
Sliding surface
minus5
minus10
minus15
minus20
Figure 12 Sliding surface of practical system
interesting objective among the researchers In this work anintegral sliding mode control approach due to its robustnessfrom the very beginning of the process is employed forthe control design of this class The design of the integralmanifold relied upon a transformed form The benefit of thetransformed form is that itmakes the design strategy easy and
0 5 10 15 20 25
0
2
4
6
Time (s)
Inpu
t per
form
ance
Control input
minus2
minus4
minus6
Figure 13 Control input for reference tracking
Inpu
t per
form
ance
Control input
0 5 10 15 20 25
0
05
1
Time (s)
minus05
minus1
Figure 14 Zoom profile of the control input depicted in Figure 13
simple The stability analysis and experimental results of theproposed control laws are presented which convey the goodfeatures and demand the proposed approachwhen the systemoperates under uncertainties
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Mahjoub F Mnif and N Derbel ldquoSet point stabilization ofa 2DOF underactuated manipulatorrdquo Journal of Computers vol6 no 2 pp 368ndash376 2011
[2] F Mnif ldquoVSS control for a class of underactuated Mechanicalsystemsrdquo International Journal of Computational Cognition vol3 no 2 pp 14ndash18 2005
[3] J Hauser S Sastry and P Kokotovic ldquoNonlinear control viaapproximate inputndashoutput linearization the ball and beamexamplerdquo IEEE Transactions on Automatic Control vol 37 no3 pp 392ndash398 1992
[4] M Jankovic D Fontanine and P V Kokotovic ldquoTORAexample cascade and passitivity-based control designsrdquo IEEETransactions on Control Systems Technology vol 6 pp 4347ndash4351 1996
Mathematical Problems in Engineering 11
[5] T Sugie and K Fujimoto ldquoControl of inverted pendulumsystems based on approximate linearization design and exper-imentrdquo in Proceedings of the 33rd IEEE Conference on Decisionand Control vol 2 pp 1647ndash1648 December 1994
[6] R Xu and U Ozguner ldquoSliding mode control of a class ofunderactuated systemsrdquo Automatica vol 44 no 1 pp 233ndash2412008
[7] M W Spong ldquoThe swing-up control problem for the acrobatrdquoIEEE Transactions on Systems Technology vol 15 no 1 pp 49ndash55 1995
[8] R W Brockett ldquoAsymptotic stability feedback and stabiliza-tionrdquo in Differential Geometric Control Theory pp 181ndash191Birkhauser Boston Mass USA 1983
[9] C J Tomlin and S S Sastry ldquoSwitching through singularitiesrdquoSystems and Control Letters vol 35 no 3 pp 145ndash154 1998
[10] W-H Chen and D J Ballance ldquoOn a switching control schemefor nonlinear systems with ill-defined relative degreerdquo Systemsamp Control Letters vol 47 no 2 pp 159ndash166 2002
[11] F Zhang and B Fernndez-Rodriguez ldquoFeedback linearizationcontrol of systems with singularitiesrdquo in Proceedings of the6th International Conference on Complex Systems (ICCS rsquo06)Boston Mass USA June 2006
[12] D Seto and J Baillieul ldquoControl problems in super-articulatedmechanical systemsrdquo IEEE Transactions on Automatic Controlvol 39 no 12 pp 2442ndash2453 1994
[13] M W Spong Energy Based Control of a Class of UnderactuatedMechanical Systems IFACWorld Congress 1996
[14] V I Utkin Sliding Mode Control in Electromechanical SystemsTaylor amp Francis 1999
[15] M Rubagotti A Estrada F Castanos A Ferrara and LFridman ldquoIntegral sliding mode control for nonlinear systemswith matched and unmatched perturbationsrdquo IEEE Transactionon Automatic Control vol 56 no 11 pp 2699ndash2704 2011
[16] Q Khan A I Bhatti S Iqbal and M Iqbal ldquoDynamicintegral sliding mode for MIMO uncertain nonlinear systemsrdquoInternational Journal of Control Automation and Systems vol9 no 1 pp 151ndash160 2011
[17] R Olfati-Saber ldquoNormal forms for underactuated mechanicalsystems with symmetryrdquo IEEE Transactions on Automatic Con-trol vol 47 no 2 pp 305ndash308 2002
[18] R Lozano I Fantoni and D J Block ldquoStabilization of theinverted pendulum around its homoclinic orbitrdquo Systems andControl Letters vol 40 no 3 pp 197ndash204 2000
[19] E Altug J P Ostrowski and R Mahony ldquoControl of aquadrotor helicopter using visual feedbackrdquo in Proceedings ofthe IEEE International Conference on Robotics and Automation(ICRA rsquo02) vol 1 pp 72ndash77 Washington DC USA 2002
[20] A Isidori Nonlinear Control Systems Communications andControl Engineering Series Springer Berlin Germany 3rdedition 1995
[21] H Sira-Ramirez ldquoOn the dynamical sliding mode control ofnonlinear systemsrdquo International Journal of Control vol 57 no5 pp 1039ndash1061 1993
[22] Q Khan A I Bhatti and A Ferrara ldquoDynamic sliding modecontrol design based on an integral manifold for nonlinearuncertain systemsrdquo Journal of Nonlinear Dynamics vol 2014Article ID 489364 10 pages 2014
[23] C Edwards and S K Spurgeon Sliding Modes Control Theoryand Applications Taylor amp Francis London UK 1998
[24] A M Khan A I Bhatti S U Din and Q Khan ldquoStaticamp dynamic sliding mode control of ball and beam systemrdquo
in Proceedings of the 9th International Bhurban Conferenceon Applied Sciences and Technology (IBCAST rsquo12) pp 32ndash36Islamabad Pakistan January 2012
[25] N B Almutairi and M Zribi ldquoOn the sliding mode control ofa Ball on a Beam systemrdquo Nonlinear Dynamics vol 59 no 1-2pp 222ndash239 2010
Assume that the system reported in (3)-(4) has a relativedegree ldquo119903rdquo with respect to the defined nonlinear outputTherefore owing to [20] one has
119910(119903)
= 119871119903
119891
ℎ (119909) + 119871119892
(119871119903minus1
119891
ℎ (119909)) 120591 + 120577 (119909 119905) (8)
subject to the following conditions
(1) 119871119892
(119871119894
119891
ℎ(119909)) = 0 forall119909 isin 119861 where 119861 indicates theneighborhood of 119909
0
for 119894 lt 119903 minus 1(2) 119871119892
(119871119903minus1
119891
ℎ(119909)) = 0 where 120577(119909 119905) represents thematched unmodeled uncertainties System (8) bydefining the transformation 119910
(119894minus1)
= 120585119894
[21] can beput in the following form
1205851
= 1205852
1205852
= 1205853
120585119899
= 120593 ( ) + 120574 () 120591 + Δ119866119898
( 119905)
(9)
where the transformed states = (1205851
1205852
120585119899
) arephase variables 120591 is the control input andΔ119866
119898
( 119905)
represents matched uncertainties It is worthy tonotice that the inverted pendulum and the ball andbeam systems can be replaced in the aforementionedform
Note that both the formats are ready to design the controllaw for these systems In the next section we outline thedesign procedure for both the forms
3 Control Law Design
The control design for the forms presented in (3)-(4) and(9) is carried out in this section which we claim as ourmain contribution in this paper The main objective inthis work is to enhance the robustness of the system fromthe very beginning of the process which is the beauty ofintegral sliding mode control In general the integral slidingmode control law appears as follows [14] In the subsequentsubsections the authors aim to present the design procedure
31 Integral Sliding Mode This variant of sliding mode pos-sesses the main features of the sliding mode like robustnessand the existence chattering across the switching manifoldOn the other hand the sliding mode occurs from the verystart which consequently provides insensitivity of distur-bance from the beginning The control law can be expressedas follows
120591 = 1205910
+ 1205911
(10)
where the first component on the right hand side of (10)governs the systems dynamics in sliding modes whereas thesecond component compensates the matched disturbancesNow the aim is to present the design of the aforesaid controlcomponents
311 Control Design for Case-1 This control design for case-1 is the main obstacle in this subsection To define both thecomponents the following terms are defined
1198901
= 1199091
1198902
= 1199092
1198903
= 1198911
(1199091
1199092
1199093
1199094
)
1198904
=1205971198911
1205971199091
1199092
+1205971198911
1205971199092
1198911
+1205971198911
1205971199093
1199094
(11)
Using these new variables the components of the controllerare designed in the following subsection For the sake ofcompleteness the design of this component is worked out viasimple pole placement Following the design procedure ofpole placement method one gets
1205910
= minus1198961
1198901
minus 1198962
1198902
minus 1198963
1198903
minus 1198964
1198904
(12)
where 119896119894
119894 = 1 2 3 4 are the gains of this control componentThis control component steers the states of the nominalsystem to their defined equilibrium Now in the subsequentstudy the design of the uncertainties compensating term ispresented An integral manifold is defined as follows
120590 = 1198881
1198901
+ 1198882
1198902
+ 1198883
1198903
+ 1198904
+ 119911 = 1205900
+ 119911 (13)
where 1205900
= 1198881
1198901
+ 1198882
1198902
+ 1198883
1198903
+ 1198904
represents the conventionalsliding manifold which is Hurwitz by definition
Now computing along (3)-(4) one has
= 1198881
(1199092
+ 1198891
) + 1198882
(1198911
(1199091
1199092
1199093
1199094
) + 1198892
)
+ 1198883
(1198891198911
(1199091
1199092
1199093
1199094
)
119889119905) +
119889
119889119905(1205971198911
1205971199091
1199092
)
+1205971198911
1205971199091
1199092
+119889
119889119905(1205971198911
1205971199092
1198911
) +1205971198911
1205971199092
1198911
+119889
119889119905(1205971198911
1205971199093
1199094
) +1205971198911
1205971199093
1198912
(1199091
1199092
1199093
1199094
)
+1205971198911
1205971199093
119887 (1199091
1199092
1199093
1199094
) 1205910
+1205971198911
1205971199093
119887 (1199091
1199092
1199093
1199094
) 1205911
+1205971198911
1205971199093
1198893
(14)
Now choose the dynamics of the integral term as follows
= minus1198881
1199092
minus 1198882
1198911
(1199091
1199092
1199093
1199094
)
minus 1198883
(1198891198911
(1199091
1199092
1199093
1199094
)
119889119905) minus
119889
119889119905(1205971198911
1205971199091
1199092
)
minus1205971198911
1205971199091
1199092
minus119889
119889119905(1205971198911
1205971199092
1198911
) minus1205971198911
1205971199092
1198911
minus119889
119889119905(1205971198911
1205971199093
1199094
) minus1205971198911
1205971199093
(1199091
1199092
1199093
1199094
) 1205910
(15)
4 Mathematical Problems in Engineering
The expression of the termwhich compensates the uncertain-ties may be written as follows
1205911
= minus(1205971198911
1205971199093
119887 (1199091
1199092
1199093
1199094
))
minus1
sdot (1205971198911
1205971199093
1198912
(1199091
1199092
1199093
1199094
) + 119870sign (120590))
(16)
The overall controller will look like
120591 = minus1198961
1198901
minus 1198962
1198902
minus 1198963
1198903
minus 1198964
1198904
minus (1205971198911
1205971199093
119887 (1199091
1199092
1199093
1199094
))
minus1
sdot (1205971198911
1205971199093
1198912
(1199091
1199092
1199093
1199094
) + 119870sign (120590))
(17)
The constants 119888119894
rsquos are control gains which are selected intel-ligently according to bounds In the forthcoming paragraphthe stability of the presented integral sliding mode is carriedout in the presence of the disturbances and uncertaintiesConsider the following Lyapunov candidate function
119881 =1
21205902
(18)
The time derivative of this function along dynamics (11)becomes
= 120590 = 120590(1198881
(1199092
+ 1198891
)
+ 1198882
(1198911
(1199091
1199092
1199093
1199094
) + 1198892
)
+ 1198883
(1198891198911
(1199091
1199092
1199093
1199094
)
119889119905) +
119889
119889119905(1205971198911
1205971199091
1199092
)
+1205971198911
1205971199091
1199092
+119889
119889119905(1205971198911
1205971199092
1198911
) +1205971198911
1205971199092
1198911
+119889
119889119905(1205971198911
1205971199093
1199094
)
+1205971198911
1205971199093
1198912
(1199091
1199092
1199093
1199094
) +1205971198911
1205971199093
(1199091
1199092
1199093
1199094
) 1205910
+1205971198911
1205971199093
119887 (1199091
1199092
1199093
1199094
) 1205911
+1205971198911
1205971199093
1198893
)
(19)
The substitution of (15)-(16) results in the following form
le minus |120590| 1205781
lt 0
or + radic21205781
radic119881 lt 0
(20)
subject to 119870 ge [(1205971198911
1205971199093
)1198893
+ 1198881
1198891
+ 1198882
1198892
+ 120578]This expression confirms the enforcement of the sliding
mode from the very beginning of the process that is 120590 rarr 0
in finite time Now we proceed to the actual systemrsquos stabilityIf one considers 119890
1
as the output of the system then 1198902
1198903
and 119890
4
become the successive derivatives of 1198901
Whenever120590 = 0 is achieved the dynamics of the transformed system
(11) will converge asymptotically to zero under the action ofthe control component (12) [22] That is in closed loop thetransformed system dynamics will be operated under (12) asfollows
[[[[[
[
1198901
1198902
1198903
1198904
]]]]]
]
=
[[[[[
[
0 1 0 0
0 0 1 0
0 0 0 1
minus1198961
minus1198962
minus1198963
minus1198964
]]]]]
]
[[[[[
[
1198901
1198902
1198903
1198904
]]]]]
]
(21)
and the disturbances will be compensated via (16)The asymptotic convergence of 119890
1
1198902
1198903
and 1198904
to zeromeans the convergence of the indirectly actuated system (3)to zero On the other hand the states of the directly actuatedsystem (4) will remain bounded that is state of (4) will havesome nonzero value in order to keep 119890
1
at zero Thus theoverall system is stabilized and the desired control objectiveis achieved
32 Control Design for Case-2 Thenominal system related to(9) can be replaced in the subsequent alternative form
1205851
= 1205852
1205852
= 1205853
120585119903
= 120594 ( 120591) + 120591
(22)
where 120594( 120591) = 120593(120585 120591) + (120574() minus 1)120591 It is assumed that120594( 120591
(119896)
) = 0 at 119905 = 0 in addition to the next suppositionthat (22) is governed by 120591
0
1205851
= 1205852
1205852
= 1205853
120585119903
= 1205910
(23)
or
= 119860120585 + 1198611205910
(24)
where
119860 = [0(119903minus1)times1
119868(119903minus1)times(119903minus1)
01times1
01times(119903minus1)
]
119861 = [0(119903minus1)times1
1]
(25)
Mathematical Problems in Engineering 5
Once again following the pole placement procedure onemayhave for the sake of simplicity the input 120591
0
which is designedvia pole placement that is
1205910
= minus119870119879
0
120585 (26)
Now to get the desired robust performance the followingsliding manifold of integral type [14] is defined
120590 (120585) = 1205900
(120585) + 119911 (27)
where 1205900
(120585) is the usual sliding surface and 119911 is the integralterm The time derivative of (27) along (9) yields
= minus(
119903minus1
sum
119894=1
119888119894
120585119894+1
+ 1205910
)
119911 (0) = minus1205900
(120585 (0))
(28)
1205911
=1
120574 ()(minus120593 ( 120591) minus (120574 () minus 1) 120591
0
minus 119870 sign120590) (29)
This control law enforces sliding mode along the slidingmanifold defined in (27) The constant 119870 can be selectedaccording to the subsequent stability analysis
Thus the final control law becomes
1205911
= minus119870119879
0
120585
+1
120574 ()(minus120593 ( 119906) minus (120574 () minus 1) 120591
0
minus 119870 sign120590)
(30)
Theorem 4 Consider that |Δ119866119898
( 119905)| le 1205731
are satisfiedthen the sliding mode against the switching manifold 120590 = 0 canbe ensured and one has
119870 ge [119870119872
1205731
+ 1205781
] (31)
where 1205781
is a positive constant
Proof Toprove that the slidingmode can be enforced in finitetime differentiating (22) along the dynamics of (3)-(4) andthen substituting (30) one has
(120585) =
119903minus1
sum
119894=1
119888119894
120585119894+1
+ 1205910
minus 119870 sign120590 + 120574 () Δ119866119898
( 119905)
+
(32)
Substituting (28) in (32) and then rearranging one obtains
] (36)The inequality in (35) presents that 120590(120585) approaches zero in afinite time 119905
119904
[23] such that
119905119904
le radic2120578minus1
1
radic119881 (120590 (0)) (37)which completes the proof
4 Illustrative Example
The control algorithms presented in Section 3 are applied tothe control design of a ball and beam systemThe assessmentof the proposed controller for the ball and beam systemis carried out on the basis of output tracking robustnessenhancement via the elimination of reaching phase andchattering-free control input in the presence of uncertainties
41 Description of the Ball and Beam System The ball andbeam system is a very sound candidate of the class ofunderactuated nonlinear system It is famous because of itsnonlinear nature and due to its wide range of applicationsin the existing era like passenger cabin balancing in luxurycars balancing of liquid fuel in vertical take-off objectsIn terms of control scenarios it is an ill-defined relativedegree system which to some extent does not support inputoutput linearization A schematic diagram with their typicalparameters of the ball and beam system is displayed in theadjacent Figure 1 and Table 1 respectively In this study theauthors use the equipment manufacture by GoogolTech Ingeneral this system is equipped with a metallic ball whichis let free to roll on a rod having a specified length havingone end fixed and the other end moved up and down via anelectric servomotorThe position of the ball can be measuredvia different techniques The measured position is used asfeedback to the system and accordingly the motor moves thebeam to balance the ball at user defined location
The motion governing equations of this system are givenbelow which are adopted from [24]
(1198981199032
+ 1198621
) + (2119898119903 119903 + 1198622
)
+ (119898119892119903 +119871
2119872119892) cos120573 = 120591
1198624
119903 minus 119903
1205732
+ 119892 sin120573 = 0
(38)
6 Mathematical Problems in Engineering
yL
R rBall
Beam
Mg
mg
Motor
d120579
z120573
mg sin 120573
Figure 1 Schematic diagram of the ball and beam system
where 120579(119905) angle is subtended to make the ball stable thelever angle is represented by 120573(119905) 119903(119905) is the position of theball on the beam and Vin(119905) is the input voltage of the motorwhereas the controlled input appears mathematically via theexpression 120591(119905) = 119862
3
Vin(119905) in the dynamic modelThe derived parameters used in the dynamic model of
this system are represented by 1198621
1198622
1198623
and 1198624
with thefollowing mathematical relations [25]
1198621
=119877119898
times 119869119898
times 119871
119862119898
times 119862119887
times 119889+ 1198691
(39)
1198622
=119871
119889(119862119898
times 119862119887
119877119898
+ 119862119887
+119877119898
times 119869119898
119862119898
times 119862119892
) (40)
1198623
= 1 +119862119898
119877119898
(41)
1198624
=7
5 (42)
The equivalent state spacemodel of this is described as followsby assuming 119909
1
= 119903 (position of ball) 1199092
= 119903 (rate of changeof position) 119909
3
= 120573 (beam angle) and 1199094
= (the rate ofchange of angle of the motor)
1199091
= 1199092
1199092
=1
1198624
(minus119892 sin (1199093
))
1199093
= 1199094
1199094
=1
11989811990921
+ 1198621
(120591 minus (21198981199091
1199092
+ 1198622
) 1199094
minus (1198981198921199091
+119871
2119872119892) cos119909
3
)
(43)
Now the output of interest is 119910 = 1199091
which representsthe position of the ball This representation is similar tothat reported in (3)-(4) In the next discussion the controllerdesign is outlined
42 Controller Design Following the procedure outlined inSection 3 the authors proceed as follows
119910 = 1199091
= 1199092
= minus119892
1198624
sin (1199093
)
119910(3)
= minus119892
1198624
1199094
cos (1199093
)
119910(4)
=1
1198624
(11989811990921
+ 1198621
)[minus120591 cos119909
3
+ (21198981199091
1199092
+ 1198622
) 1199094
cos1199093
+ (1198981198921199091
+119871
2119872119892) cos2119909
3
+ 1199092
4
(1198981199092
1
+ 1198621
) sin1199093
]
119910(4)
= 119891119904
+ ℎ119904
120591
119891119904
=119892
1198624
[(21198981199091
1199092
+ 1198622
) 1199094
+ (1198981198921199091
+ (1198712)119872119892) cos21199093
+ 1199092
4
sin1199093
11989811990921
+ 1198621
]
ℎ119904
=minus119892 cos119909
3
1198624
(11989811990921
+ 1198621
)
(44)
Mathematical Problems in Engineering 7
Table 1 Parameters and values used in equations
Parameter Description Nominal values Units
119892Gravitationalacceleration 981 ms2
119898 Mass of ball 007 kg119872 Mass of beam 015 kg119871 Length of beam 04 m
119877119898
Resistance ofarmature of the motor 9 Ω
119869119898
Moment of inertia ofmotor 735 times 10
minus4 Nmrads2
119862119898
Torque constant ofmotor 00075 NmA
119862119892
Gear ratio 428 mdash
119889
Radius of armconnected toservomotor
004 m
1198691
Moment of inertia ofbeam 0001 kgm2
119862119887
Back emf constantvalue 05625 Vrads
Now writing this in the controllable canonical form (phasevariable form) one may have
( 119905) represents the modeluncertainties Herewe discuss ISMCon ball and beam systemwith fixed step tracking as well as variable step tracking Theintegral manifold is defined as follows
120590 = 1198881
1205851
+ 1198882
1205852
+ 1198883
1205853
+ 1205854
+ 119911 (47)
The expression of the overall controller which becomes willbe as follows
1205911
= minus1198961
1205851
minus 1198962
1205852
minus 1198963
1205853
minus 1198964
1205854
+1
120574 ()(minus120593 () minus (120574 () minus 1) 120591
0
minus 119870sign120590)
= 1198881
1205851
+ 1198882
1205852
+ 1198883
1205853
+ 119891119904
+ ℎ119904
1205910
+ ℎ119904
1205911
+
= minus1198881
1199092
+1198882
119892
1198624
sin1199093
+1198883
119892
1198624
1199094
cos1199093
minus 120574 () 1205910
minus 120593 ()
(48)
As the authors are performing the reference tracking heretherefore the integral manifold and the controller will appearas follows
120590 = 1198881
(1205851
minus 119903119889
) + 1198882
1205852
+ 1198883
1205853
+ 1205854
+ 119911 (49)
1205911
= minus1198961
(1205851
minus 119903119889
) minus 1198962
1205852
minus 1198963
1205853
minus 1198964
1205854
+1
120574 ()(minus120593 ( 120591) minus (120574 () minus 1) 120591
0
minus 119870sign (120590))
(50)
where 119903119889
is the desired reference with 119903119889
119903119889
119903119889
beingbounded
43 Simulation Results The simulation study of the system iscarried out by considering the reference tracking of a squarewave signal and sinusoidal wave signal In the subsequentparagraph their respective results will be demonstrated indetail
In case the efforts are directed to track a fixed square wavesignal in the presence of disturbances the initial conditions ofthe system were set to 119909
1
(0) = 04 1199092
(0) = 1199093
(0) = 1199094
(0) = 0Furthermore the square wave was defined in the simulationcode as follows
119903119889
(119905) =
20 cm 0 le 119905 le 19
14 cm 20 le 119905 le 39
20 cm 40 le 119905 le 60
(51)
The gains of the proposed controller presented from (39) to(41) are chosen according to Table 2
The output tracking performance of the proposed controlinput when a square wave is used as desired reference outputis shown in Figure 2 It can be clearly examined that theperformance is very appealing in this caseThe correspondingsliding manifold profile is displayed in Figure 3 which clearlyindicates that the sliding mode is established from the verybeginning of the processes which in turn results in enhancedrobustnessThe controlled input signalrsquos profile is depicted inFigure 4 with its zoomed profile as shown in Figure 5 It isobvious from both the figures that the control input derivesthe system with suppressed chattering phenomenon which is
8 Mathematical Problems in Engineering
Table 2 Parametric values used in the square wave tracking
Constants 1198621
1198622
1198623
1198701
1198702
1198703
1198704
119870
Values 12 12 011 40298 25018 60 41 5
0 10 20 30 40 50 600
00501
01502
02503
03504
Time (s)
Trac
king
per
form
ance
ActualDesired
Figure 2 Output tracking performance when a square wave is usedas referencedesired output
0 10 20 30 40 50 60
005
115
2
Time (s)
Slid
ing
surfa
ce
Sliding surface
minus05
minus1
minus15
minus2
Figure 3 Sliding manifold convergence profile in case of squarewave tracking
tolerable for the system actuators health Now from this casestudy it is concluded that integral sliding mode approach isan interesting candidate for this class
In this case study once again efforts are focused on thetracking of a sinusoidal signal which is defined as 119903
119889
(119905) =
sin(119905) in the presence of disturbances Like the previous casestudy the initial conditions of the system were set to 119909
1
(0) =
04 1199092
(0) = 1199093
(0) = 1199094
(0) = 0 In addition the gains of theproposed controller presented in (50) are chosen accordingto Table 3
The output tracking performance of the proposed controlinput when a sinusoidal signal is considered as desiredreference output is shown in Figure 6 It can be clearlyseen that the performance is excellent in this scenarioThe corresponding sliding manifold profile is displayed inFigure 7 which confirms the establishment of sliding modesfrom the starting instant and consequently enhancement ofrobustnessThe controlled input signalrsquos profile is depicted inFigure 8 It is obvious from the figure that the control input
Table 3 Parametric values used in the sinusoid wave tracking
Constants 1198621
1198622
1198623
1198701
1198702
1198703
1198704
119870
Values 12 12 011 40298 25018 230 49 5
Time (s)0 10 20 30 40 50 60
05
101520
Con
trol i
nput
Control input
minus5
minus10
minus15
minus20
Figure 4 Control input in square wave reference tracking
Time (s)
Con
trol i
nput
Control input
0 10 20 30 40 50 60
0
05
1
15
minus05
minus1
minus15
Figure 5 Zoom profile of the control input depicted in Figure 4
Time (s)
ActualDesired
0 5 10 15 20 25
Trac
king
per
form
ance
0
05
1
15
minus05
minus1
minus15
Figure 6 Output tracking performance when a sinusoidal wave isused as referencedesired output
evolves with suppressed chattering phenomenonwhich onceagain makes this design strategy a good candidate for theclass of these underactuated systems
44 Implementation Results The control technique proposedin this paper is implemented on the actual apparatus using
Mathematical Problems in Engineering 9
Time (s)
Sliding surface
0 5 10 15 20 25
0
05
1
Slid
ing
surfa
ce(v
aria
ble r
efer
ence
sign
al)
minus05
minus1
Figure 7 Sliding manifold convergence profile in case of sinusoidalwave tracking
Time (s)0 5 10 15 20 25
0
5
10
Control effort
minus5
minus10
Con
trol i
nput
ldquordquo
u
Figure 8 Control input in sinusoidal wave reference tracking
the MATLAB environment The detailed discussions arepresented below
441 Experimental Setup Description The experiment setupis equipped by GoogolTech GBB1004 with an electroniccontrol boxThe beam length is 40 cm alongwithmass of ballthat is 28 g and an intelligent IPM100 servo driver which isused formoving the ball on the beamThe experimental setupis shown in Figure 9
The input given to apparatus is the voltage Vin(119905) and theoutput is the position of themotor 120579(119905) which in otherwordsis an input for the positioning of the ball on the beam Thisapparatus uses potentiometer mounted within a slot insidethe beam to sense the position of the ball on the beam Themeasured position along the beam is fed to the AD converterof IPM100 motion drive
The power module used in GoogolTech requires 220Vand 10A input Note that the control accuracy of this manu-factured apparatus lies within the range of plusmn1mmThe typicalparameters values are listed in Table 1 The environmentused here includes Windows XP as an operating system andMATLAB 712Simulink 77 Furthermore the sampling timeused in forthcoming practical results was 2ms In the exper-imental processes the proposed controllers need velocitymeasurements which are in general not available One may
Figure 9 Experimental setup of the ball and beam equipped viaGoogolTech GBB1004
use different kind of velocity observersdifferentiator for thevelocity estimation [16] In order to make the implemen-tation easy and simple a derivative block of the Simulinkenvironment is used to provide the corresponding velocitiesmeasurements Now we are ready to discuss the results of thesystem
In this experiment the initial conditions were set to1199091
(0) = 028 1199092
(0) = 1199093
(0) = 1199094
(0) = 0 The referencesignal which is needed to be tracked is being defined in (51)In Figures 10 and 11 the tracking performance is shown Theresults reveal that the actual signal 119909
1
(119905) is pretty close tothe desired signal 119903
119889
(119905) with a steady state error which isapproximatelyplusmn0001mThe existence of this error is becauseof the apparatus
The observations of these tracking results make it clearthat the practically implemented results have very closeresemblance with the simulation result presented in Figure 2The error convergence depends on the initial conditions ofthe ball on the beam If the ball is placed very close to thedesired reference value then it will take little time to reachthe desired position On the other hand the convergence tothe desired position will take considerable time if the initialcondition is chosen far away from the desired values Thisphenomenon of convergence is according to the equipmentdesign and structure
The sliding manifold convergence and the control inputare shown in Figures 12 and 13 respectively The controlinput and the sliding manifolds show some deviations inthe first second This deviation occurs because the ball onthe beam being placed anywhere on the beam is firstmoved to one side of the beam and then ball moved to thedesired position The zoomed profile of the control inputbeing displayed in Figure 14 shows high frequency vibration(chattering) of magnitude plusmn007 This makes the proposedcontrol design algorithm an appealing candidate for this classof nonlinear systems The gains of the controller being usedin this experiment are displayed in Table 4
5 Conclusion
The control of underactuated systems because of their lessnumber of actuators than the degree of freedom is an
10 Mathematical Problems in Engineering
Table 4 Parametric values used in implementation
Constants 1198621
1198622
1198623
1198701
1198702
1198703
1198704
119870
Values 8 5 1 3 15 3 1 4
0
01
02
03
04
Trac
king
per
form
ance
0 5 10 15 20 25Time (s)
DesiredActual
16 18 20 22 240219
022022102220223
Figure 10 Output tracking performance when 119903119889
= 22 cm is set asreferencedesired output
Time (s)
DesiredActual
0
01
02
03
04
Trac
king
per
form
ance
0 10 20 30 40 50 6044 46 48 50 520196
019802
02020204
Figure 11 Output tracking performance when a square wave is usedas referencedesired output
0 5 10 15 20 25
05
101520
Time (s)
Slid
ing
surfa
ce
Sliding surface
minus5
minus10
minus15
minus20
Figure 12 Sliding surface of practical system
interesting objective among the researchers In this work anintegral sliding mode control approach due to its robustnessfrom the very beginning of the process is employed forthe control design of this class The design of the integralmanifold relied upon a transformed form The benefit of thetransformed form is that itmakes the design strategy easy and
0 5 10 15 20 25
0
2
4
6
Time (s)
Inpu
t per
form
ance
Control input
minus2
minus4
minus6
Figure 13 Control input for reference tracking
Inpu
t per
form
ance
Control input
0 5 10 15 20 25
0
05
1
Time (s)
minus05
minus1
Figure 14 Zoom profile of the control input depicted in Figure 13
simple The stability analysis and experimental results of theproposed control laws are presented which convey the goodfeatures and demand the proposed approachwhen the systemoperates under uncertainties
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Mahjoub F Mnif and N Derbel ldquoSet point stabilization ofa 2DOF underactuated manipulatorrdquo Journal of Computers vol6 no 2 pp 368ndash376 2011
[2] F Mnif ldquoVSS control for a class of underactuated Mechanicalsystemsrdquo International Journal of Computational Cognition vol3 no 2 pp 14ndash18 2005
[3] J Hauser S Sastry and P Kokotovic ldquoNonlinear control viaapproximate inputndashoutput linearization the ball and beamexamplerdquo IEEE Transactions on Automatic Control vol 37 no3 pp 392ndash398 1992
[4] M Jankovic D Fontanine and P V Kokotovic ldquoTORAexample cascade and passitivity-based control designsrdquo IEEETransactions on Control Systems Technology vol 6 pp 4347ndash4351 1996
Mathematical Problems in Engineering 11
[5] T Sugie and K Fujimoto ldquoControl of inverted pendulumsystems based on approximate linearization design and exper-imentrdquo in Proceedings of the 33rd IEEE Conference on Decisionand Control vol 2 pp 1647ndash1648 December 1994
[6] R Xu and U Ozguner ldquoSliding mode control of a class ofunderactuated systemsrdquo Automatica vol 44 no 1 pp 233ndash2412008
[7] M W Spong ldquoThe swing-up control problem for the acrobatrdquoIEEE Transactions on Systems Technology vol 15 no 1 pp 49ndash55 1995
[8] R W Brockett ldquoAsymptotic stability feedback and stabiliza-tionrdquo in Differential Geometric Control Theory pp 181ndash191Birkhauser Boston Mass USA 1983
[9] C J Tomlin and S S Sastry ldquoSwitching through singularitiesrdquoSystems and Control Letters vol 35 no 3 pp 145ndash154 1998
[10] W-H Chen and D J Ballance ldquoOn a switching control schemefor nonlinear systems with ill-defined relative degreerdquo Systemsamp Control Letters vol 47 no 2 pp 159ndash166 2002
[11] F Zhang and B Fernndez-Rodriguez ldquoFeedback linearizationcontrol of systems with singularitiesrdquo in Proceedings of the6th International Conference on Complex Systems (ICCS rsquo06)Boston Mass USA June 2006
[12] D Seto and J Baillieul ldquoControl problems in super-articulatedmechanical systemsrdquo IEEE Transactions on Automatic Controlvol 39 no 12 pp 2442ndash2453 1994
[13] M W Spong Energy Based Control of a Class of UnderactuatedMechanical Systems IFACWorld Congress 1996
[14] V I Utkin Sliding Mode Control in Electromechanical SystemsTaylor amp Francis 1999
[15] M Rubagotti A Estrada F Castanos A Ferrara and LFridman ldquoIntegral sliding mode control for nonlinear systemswith matched and unmatched perturbationsrdquo IEEE Transactionon Automatic Control vol 56 no 11 pp 2699ndash2704 2011
[16] Q Khan A I Bhatti S Iqbal and M Iqbal ldquoDynamicintegral sliding mode for MIMO uncertain nonlinear systemsrdquoInternational Journal of Control Automation and Systems vol9 no 1 pp 151ndash160 2011
[17] R Olfati-Saber ldquoNormal forms for underactuated mechanicalsystems with symmetryrdquo IEEE Transactions on Automatic Con-trol vol 47 no 2 pp 305ndash308 2002
[18] R Lozano I Fantoni and D J Block ldquoStabilization of theinverted pendulum around its homoclinic orbitrdquo Systems andControl Letters vol 40 no 3 pp 197ndash204 2000
[19] E Altug J P Ostrowski and R Mahony ldquoControl of aquadrotor helicopter using visual feedbackrdquo in Proceedings ofthe IEEE International Conference on Robotics and Automation(ICRA rsquo02) vol 1 pp 72ndash77 Washington DC USA 2002
[20] A Isidori Nonlinear Control Systems Communications andControl Engineering Series Springer Berlin Germany 3rdedition 1995
[21] H Sira-Ramirez ldquoOn the dynamical sliding mode control ofnonlinear systemsrdquo International Journal of Control vol 57 no5 pp 1039ndash1061 1993
[22] Q Khan A I Bhatti and A Ferrara ldquoDynamic sliding modecontrol design based on an integral manifold for nonlinearuncertain systemsrdquo Journal of Nonlinear Dynamics vol 2014Article ID 489364 10 pages 2014
[23] C Edwards and S K Spurgeon Sliding Modes Control Theoryand Applications Taylor amp Francis London UK 1998
[24] A M Khan A I Bhatti S U Din and Q Khan ldquoStaticamp dynamic sliding mode control of ball and beam systemrdquo
in Proceedings of the 9th International Bhurban Conferenceon Applied Sciences and Technology (IBCAST rsquo12) pp 32ndash36Islamabad Pakistan January 2012
[25] N B Almutairi and M Zribi ldquoOn the sliding mode control ofa Ball on a Beam systemrdquo Nonlinear Dynamics vol 59 no 1-2pp 222ndash239 2010
The expression of the termwhich compensates the uncertain-ties may be written as follows
1205911
= minus(1205971198911
1205971199093
119887 (1199091
1199092
1199093
1199094
))
minus1
sdot (1205971198911
1205971199093
1198912
(1199091
1199092
1199093
1199094
) + 119870sign (120590))
(16)
The overall controller will look like
120591 = minus1198961
1198901
minus 1198962
1198902
minus 1198963
1198903
minus 1198964
1198904
minus (1205971198911
1205971199093
119887 (1199091
1199092
1199093
1199094
))
minus1
sdot (1205971198911
1205971199093
1198912
(1199091
1199092
1199093
1199094
) + 119870sign (120590))
(17)
The constants 119888119894
rsquos are control gains which are selected intel-ligently according to bounds In the forthcoming paragraphthe stability of the presented integral sliding mode is carriedout in the presence of the disturbances and uncertaintiesConsider the following Lyapunov candidate function
119881 =1
21205902
(18)
The time derivative of this function along dynamics (11)becomes
= 120590 = 120590(1198881
(1199092
+ 1198891
)
+ 1198882
(1198911
(1199091
1199092
1199093
1199094
) + 1198892
)
+ 1198883
(1198891198911
(1199091
1199092
1199093
1199094
)
119889119905) +
119889
119889119905(1205971198911
1205971199091
1199092
)
+1205971198911
1205971199091
1199092
+119889
119889119905(1205971198911
1205971199092
1198911
) +1205971198911
1205971199092
1198911
+119889
119889119905(1205971198911
1205971199093
1199094
)
+1205971198911
1205971199093
1198912
(1199091
1199092
1199093
1199094
) +1205971198911
1205971199093
(1199091
1199092
1199093
1199094
) 1205910
+1205971198911
1205971199093
119887 (1199091
1199092
1199093
1199094
) 1205911
+1205971198911
1205971199093
1198893
)
(19)
The substitution of (15)-(16) results in the following form
le minus |120590| 1205781
lt 0
or + radic21205781
radic119881 lt 0
(20)
subject to 119870 ge [(1205971198911
1205971199093
)1198893
+ 1198881
1198891
+ 1198882
1198892
+ 120578]This expression confirms the enforcement of the sliding
mode from the very beginning of the process that is 120590 rarr 0
in finite time Now we proceed to the actual systemrsquos stabilityIf one considers 119890
1
as the output of the system then 1198902
1198903
and 119890
4
become the successive derivatives of 1198901
Whenever120590 = 0 is achieved the dynamics of the transformed system
(11) will converge asymptotically to zero under the action ofthe control component (12) [22] That is in closed loop thetransformed system dynamics will be operated under (12) asfollows
[[[[[
[
1198901
1198902
1198903
1198904
]]]]]
]
=
[[[[[
[
0 1 0 0
0 0 1 0
0 0 0 1
minus1198961
minus1198962
minus1198963
minus1198964
]]]]]
]
[[[[[
[
1198901
1198902
1198903
1198904
]]]]]
]
(21)
and the disturbances will be compensated via (16)The asymptotic convergence of 119890
1
1198902
1198903
and 1198904
to zeromeans the convergence of the indirectly actuated system (3)to zero On the other hand the states of the directly actuatedsystem (4) will remain bounded that is state of (4) will havesome nonzero value in order to keep 119890
1
at zero Thus theoverall system is stabilized and the desired control objectiveis achieved
32 Control Design for Case-2 Thenominal system related to(9) can be replaced in the subsequent alternative form
1205851
= 1205852
1205852
= 1205853
120585119903
= 120594 ( 120591) + 120591
(22)
where 120594( 120591) = 120593(120585 120591) + (120574() minus 1)120591 It is assumed that120594( 120591
(119896)
) = 0 at 119905 = 0 in addition to the next suppositionthat (22) is governed by 120591
0
1205851
= 1205852
1205852
= 1205853
120585119903
= 1205910
(23)
or
= 119860120585 + 1198611205910
(24)
where
119860 = [0(119903minus1)times1
119868(119903minus1)times(119903minus1)
01times1
01times(119903minus1)
]
119861 = [0(119903minus1)times1
1]
(25)
Mathematical Problems in Engineering 5
Once again following the pole placement procedure onemayhave for the sake of simplicity the input 120591
0
which is designedvia pole placement that is
1205910
= minus119870119879
0
120585 (26)
Now to get the desired robust performance the followingsliding manifold of integral type [14] is defined
120590 (120585) = 1205900
(120585) + 119911 (27)
where 1205900
(120585) is the usual sliding surface and 119911 is the integralterm The time derivative of (27) along (9) yields
= minus(
119903minus1
sum
119894=1
119888119894
120585119894+1
+ 1205910
)
119911 (0) = minus1205900
(120585 (0))
(28)
1205911
=1
120574 ()(minus120593 ( 120591) minus (120574 () minus 1) 120591
0
minus 119870 sign120590) (29)
This control law enforces sliding mode along the slidingmanifold defined in (27) The constant 119870 can be selectedaccording to the subsequent stability analysis
Thus the final control law becomes
1205911
= minus119870119879
0
120585
+1
120574 ()(minus120593 ( 119906) minus (120574 () minus 1) 120591
0
minus 119870 sign120590)
(30)
Theorem 4 Consider that |Δ119866119898
( 119905)| le 1205731
are satisfiedthen the sliding mode against the switching manifold 120590 = 0 canbe ensured and one has
119870 ge [119870119872
1205731
+ 1205781
] (31)
where 1205781
is a positive constant
Proof Toprove that the slidingmode can be enforced in finitetime differentiating (22) along the dynamics of (3)-(4) andthen substituting (30) one has
(120585) =
119903minus1
sum
119894=1
119888119894
120585119894+1
+ 1205910
minus 119870 sign120590 + 120574 () Δ119866119898
( 119905)
+
(32)
Substituting (28) in (32) and then rearranging one obtains
] (36)The inequality in (35) presents that 120590(120585) approaches zero in afinite time 119905
119904
[23] such that
119905119904
le radic2120578minus1
1
radic119881 (120590 (0)) (37)which completes the proof
4 Illustrative Example
The control algorithms presented in Section 3 are applied tothe control design of a ball and beam systemThe assessmentof the proposed controller for the ball and beam systemis carried out on the basis of output tracking robustnessenhancement via the elimination of reaching phase andchattering-free control input in the presence of uncertainties
41 Description of the Ball and Beam System The ball andbeam system is a very sound candidate of the class ofunderactuated nonlinear system It is famous because of itsnonlinear nature and due to its wide range of applicationsin the existing era like passenger cabin balancing in luxurycars balancing of liquid fuel in vertical take-off objectsIn terms of control scenarios it is an ill-defined relativedegree system which to some extent does not support inputoutput linearization A schematic diagram with their typicalparameters of the ball and beam system is displayed in theadjacent Figure 1 and Table 1 respectively In this study theauthors use the equipment manufacture by GoogolTech Ingeneral this system is equipped with a metallic ball whichis let free to roll on a rod having a specified length havingone end fixed and the other end moved up and down via anelectric servomotorThe position of the ball can be measuredvia different techniques The measured position is used asfeedback to the system and accordingly the motor moves thebeam to balance the ball at user defined location
The motion governing equations of this system are givenbelow which are adopted from [24]
(1198981199032
+ 1198621
) + (2119898119903 119903 + 1198622
)
+ (119898119892119903 +119871
2119872119892) cos120573 = 120591
1198624
119903 minus 119903
1205732
+ 119892 sin120573 = 0
(38)
6 Mathematical Problems in Engineering
yL
R rBall
Beam
Mg
mg
Motor
d120579
z120573
mg sin 120573
Figure 1 Schematic diagram of the ball and beam system
where 120579(119905) angle is subtended to make the ball stable thelever angle is represented by 120573(119905) 119903(119905) is the position of theball on the beam and Vin(119905) is the input voltage of the motorwhereas the controlled input appears mathematically via theexpression 120591(119905) = 119862
3
Vin(119905) in the dynamic modelThe derived parameters used in the dynamic model of
this system are represented by 1198621
1198622
1198623
and 1198624
with thefollowing mathematical relations [25]
1198621
=119877119898
times 119869119898
times 119871
119862119898
times 119862119887
times 119889+ 1198691
(39)
1198622
=119871
119889(119862119898
times 119862119887
119877119898
+ 119862119887
+119877119898
times 119869119898
119862119898
times 119862119892
) (40)
1198623
= 1 +119862119898
119877119898
(41)
1198624
=7
5 (42)
The equivalent state spacemodel of this is described as followsby assuming 119909
1
= 119903 (position of ball) 1199092
= 119903 (rate of changeof position) 119909
3
= 120573 (beam angle) and 1199094
= (the rate ofchange of angle of the motor)
1199091
= 1199092
1199092
=1
1198624
(minus119892 sin (1199093
))
1199093
= 1199094
1199094
=1
11989811990921
+ 1198621
(120591 minus (21198981199091
1199092
+ 1198622
) 1199094
minus (1198981198921199091
+119871
2119872119892) cos119909
3
)
(43)
Now the output of interest is 119910 = 1199091
which representsthe position of the ball This representation is similar tothat reported in (3)-(4) In the next discussion the controllerdesign is outlined
42 Controller Design Following the procedure outlined inSection 3 the authors proceed as follows
119910 = 1199091
= 1199092
= minus119892
1198624
sin (1199093
)
119910(3)
= minus119892
1198624
1199094
cos (1199093
)
119910(4)
=1
1198624
(11989811990921
+ 1198621
)[minus120591 cos119909
3
+ (21198981199091
1199092
+ 1198622
) 1199094
cos1199093
+ (1198981198921199091
+119871
2119872119892) cos2119909
3
+ 1199092
4
(1198981199092
1
+ 1198621
) sin1199093
]
119910(4)
= 119891119904
+ ℎ119904
120591
119891119904
=119892
1198624
[(21198981199091
1199092
+ 1198622
) 1199094
+ (1198981198921199091
+ (1198712)119872119892) cos21199093
+ 1199092
4
sin1199093
11989811990921
+ 1198621
]
ℎ119904
=minus119892 cos119909
3
1198624
(11989811990921
+ 1198621
)
(44)
Mathematical Problems in Engineering 7
Table 1 Parameters and values used in equations
Parameter Description Nominal values Units
119892Gravitationalacceleration 981 ms2
119898 Mass of ball 007 kg119872 Mass of beam 015 kg119871 Length of beam 04 m
119877119898
Resistance ofarmature of the motor 9 Ω
119869119898
Moment of inertia ofmotor 735 times 10
minus4 Nmrads2
119862119898
Torque constant ofmotor 00075 NmA
119862119892
Gear ratio 428 mdash
119889
Radius of armconnected toservomotor
004 m
1198691
Moment of inertia ofbeam 0001 kgm2
119862119887
Back emf constantvalue 05625 Vrads
Now writing this in the controllable canonical form (phasevariable form) one may have
( 119905) represents the modeluncertainties Herewe discuss ISMCon ball and beam systemwith fixed step tracking as well as variable step tracking Theintegral manifold is defined as follows
120590 = 1198881
1205851
+ 1198882
1205852
+ 1198883
1205853
+ 1205854
+ 119911 (47)
The expression of the overall controller which becomes willbe as follows
1205911
= minus1198961
1205851
minus 1198962
1205852
minus 1198963
1205853
minus 1198964
1205854
+1
120574 ()(minus120593 () minus (120574 () minus 1) 120591
0
minus 119870sign120590)
= 1198881
1205851
+ 1198882
1205852
+ 1198883
1205853
+ 119891119904
+ ℎ119904
1205910
+ ℎ119904
1205911
+
= minus1198881
1199092
+1198882
119892
1198624
sin1199093
+1198883
119892
1198624
1199094
cos1199093
minus 120574 () 1205910
minus 120593 ()
(48)
As the authors are performing the reference tracking heretherefore the integral manifold and the controller will appearas follows
120590 = 1198881
(1205851
minus 119903119889
) + 1198882
1205852
+ 1198883
1205853
+ 1205854
+ 119911 (49)
1205911
= minus1198961
(1205851
minus 119903119889
) minus 1198962
1205852
minus 1198963
1205853
minus 1198964
1205854
+1
120574 ()(minus120593 ( 120591) minus (120574 () minus 1) 120591
0
minus 119870sign (120590))
(50)
where 119903119889
is the desired reference with 119903119889
119903119889
119903119889
beingbounded
43 Simulation Results The simulation study of the system iscarried out by considering the reference tracking of a squarewave signal and sinusoidal wave signal In the subsequentparagraph their respective results will be demonstrated indetail
In case the efforts are directed to track a fixed square wavesignal in the presence of disturbances the initial conditions ofthe system were set to 119909
1
(0) = 04 1199092
(0) = 1199093
(0) = 1199094
(0) = 0Furthermore the square wave was defined in the simulationcode as follows
119903119889
(119905) =
20 cm 0 le 119905 le 19
14 cm 20 le 119905 le 39
20 cm 40 le 119905 le 60
(51)
The gains of the proposed controller presented from (39) to(41) are chosen according to Table 2
The output tracking performance of the proposed controlinput when a square wave is used as desired reference outputis shown in Figure 2 It can be clearly examined that theperformance is very appealing in this caseThe correspondingsliding manifold profile is displayed in Figure 3 which clearlyindicates that the sliding mode is established from the verybeginning of the processes which in turn results in enhancedrobustnessThe controlled input signalrsquos profile is depicted inFigure 4 with its zoomed profile as shown in Figure 5 It isobvious from both the figures that the control input derivesthe system with suppressed chattering phenomenon which is
8 Mathematical Problems in Engineering
Table 2 Parametric values used in the square wave tracking
Constants 1198621
1198622
1198623
1198701
1198702
1198703
1198704
119870
Values 12 12 011 40298 25018 60 41 5
0 10 20 30 40 50 600
00501
01502
02503
03504
Time (s)
Trac
king
per
form
ance
ActualDesired
Figure 2 Output tracking performance when a square wave is usedas referencedesired output
0 10 20 30 40 50 60
005
115
2
Time (s)
Slid
ing
surfa
ce
Sliding surface
minus05
minus1
minus15
minus2
Figure 3 Sliding manifold convergence profile in case of squarewave tracking
tolerable for the system actuators health Now from this casestudy it is concluded that integral sliding mode approach isan interesting candidate for this class
In this case study once again efforts are focused on thetracking of a sinusoidal signal which is defined as 119903
119889
(119905) =
sin(119905) in the presence of disturbances Like the previous casestudy the initial conditions of the system were set to 119909
1
(0) =
04 1199092
(0) = 1199093
(0) = 1199094
(0) = 0 In addition the gains of theproposed controller presented in (50) are chosen accordingto Table 3
The output tracking performance of the proposed controlinput when a sinusoidal signal is considered as desiredreference output is shown in Figure 6 It can be clearlyseen that the performance is excellent in this scenarioThe corresponding sliding manifold profile is displayed inFigure 7 which confirms the establishment of sliding modesfrom the starting instant and consequently enhancement ofrobustnessThe controlled input signalrsquos profile is depicted inFigure 8 It is obvious from the figure that the control input
Table 3 Parametric values used in the sinusoid wave tracking
Constants 1198621
1198622
1198623
1198701
1198702
1198703
1198704
119870
Values 12 12 011 40298 25018 230 49 5
Time (s)0 10 20 30 40 50 60
05
101520
Con
trol i
nput
Control input
minus5
minus10
minus15
minus20
Figure 4 Control input in square wave reference tracking
Time (s)
Con
trol i
nput
Control input
0 10 20 30 40 50 60
0
05
1
15
minus05
minus1
minus15
Figure 5 Zoom profile of the control input depicted in Figure 4
Time (s)
ActualDesired
0 5 10 15 20 25
Trac
king
per
form
ance
0
05
1
15
minus05
minus1
minus15
Figure 6 Output tracking performance when a sinusoidal wave isused as referencedesired output
evolves with suppressed chattering phenomenonwhich onceagain makes this design strategy a good candidate for theclass of these underactuated systems
44 Implementation Results The control technique proposedin this paper is implemented on the actual apparatus using
Mathematical Problems in Engineering 9
Time (s)
Sliding surface
0 5 10 15 20 25
0
05
1
Slid
ing
surfa
ce(v
aria
ble r
efer
ence
sign
al)
minus05
minus1
Figure 7 Sliding manifold convergence profile in case of sinusoidalwave tracking
Time (s)0 5 10 15 20 25
0
5
10
Control effort
minus5
minus10
Con
trol i
nput
ldquordquo
u
Figure 8 Control input in sinusoidal wave reference tracking
the MATLAB environment The detailed discussions arepresented below
441 Experimental Setup Description The experiment setupis equipped by GoogolTech GBB1004 with an electroniccontrol boxThe beam length is 40 cm alongwithmass of ballthat is 28 g and an intelligent IPM100 servo driver which isused formoving the ball on the beamThe experimental setupis shown in Figure 9
The input given to apparatus is the voltage Vin(119905) and theoutput is the position of themotor 120579(119905) which in otherwordsis an input for the positioning of the ball on the beam Thisapparatus uses potentiometer mounted within a slot insidethe beam to sense the position of the ball on the beam Themeasured position along the beam is fed to the AD converterof IPM100 motion drive
The power module used in GoogolTech requires 220Vand 10A input Note that the control accuracy of this manu-factured apparatus lies within the range of plusmn1mmThe typicalparameters values are listed in Table 1 The environmentused here includes Windows XP as an operating system andMATLAB 712Simulink 77 Furthermore the sampling timeused in forthcoming practical results was 2ms In the exper-imental processes the proposed controllers need velocitymeasurements which are in general not available One may
Figure 9 Experimental setup of the ball and beam equipped viaGoogolTech GBB1004
use different kind of velocity observersdifferentiator for thevelocity estimation [16] In order to make the implemen-tation easy and simple a derivative block of the Simulinkenvironment is used to provide the corresponding velocitiesmeasurements Now we are ready to discuss the results of thesystem
In this experiment the initial conditions were set to1199091
(0) = 028 1199092
(0) = 1199093
(0) = 1199094
(0) = 0 The referencesignal which is needed to be tracked is being defined in (51)In Figures 10 and 11 the tracking performance is shown Theresults reveal that the actual signal 119909
1
(119905) is pretty close tothe desired signal 119903
119889
(119905) with a steady state error which isapproximatelyplusmn0001mThe existence of this error is becauseof the apparatus
The observations of these tracking results make it clearthat the practically implemented results have very closeresemblance with the simulation result presented in Figure 2The error convergence depends on the initial conditions ofthe ball on the beam If the ball is placed very close to thedesired reference value then it will take little time to reachthe desired position On the other hand the convergence tothe desired position will take considerable time if the initialcondition is chosen far away from the desired values Thisphenomenon of convergence is according to the equipmentdesign and structure
The sliding manifold convergence and the control inputare shown in Figures 12 and 13 respectively The controlinput and the sliding manifolds show some deviations inthe first second This deviation occurs because the ball onthe beam being placed anywhere on the beam is firstmoved to one side of the beam and then ball moved to thedesired position The zoomed profile of the control inputbeing displayed in Figure 14 shows high frequency vibration(chattering) of magnitude plusmn007 This makes the proposedcontrol design algorithm an appealing candidate for this classof nonlinear systems The gains of the controller being usedin this experiment are displayed in Table 4
5 Conclusion
The control of underactuated systems because of their lessnumber of actuators than the degree of freedom is an
10 Mathematical Problems in Engineering
Table 4 Parametric values used in implementation
Constants 1198621
1198622
1198623
1198701
1198702
1198703
1198704
119870
Values 8 5 1 3 15 3 1 4
0
01
02
03
04
Trac
king
per
form
ance
0 5 10 15 20 25Time (s)
DesiredActual
16 18 20 22 240219
022022102220223
Figure 10 Output tracking performance when 119903119889
= 22 cm is set asreferencedesired output
Time (s)
DesiredActual
0
01
02
03
04
Trac
king
per
form
ance
0 10 20 30 40 50 6044 46 48 50 520196
019802
02020204
Figure 11 Output tracking performance when a square wave is usedas referencedesired output
0 5 10 15 20 25
05
101520
Time (s)
Slid
ing
surfa
ce
Sliding surface
minus5
minus10
minus15
minus20
Figure 12 Sliding surface of practical system
interesting objective among the researchers In this work anintegral sliding mode control approach due to its robustnessfrom the very beginning of the process is employed forthe control design of this class The design of the integralmanifold relied upon a transformed form The benefit of thetransformed form is that itmakes the design strategy easy and
0 5 10 15 20 25
0
2
4
6
Time (s)
Inpu
t per
form
ance
Control input
minus2
minus4
minus6
Figure 13 Control input for reference tracking
Inpu
t per
form
ance
Control input
0 5 10 15 20 25
0
05
1
Time (s)
minus05
minus1
Figure 14 Zoom profile of the control input depicted in Figure 13
simple The stability analysis and experimental results of theproposed control laws are presented which convey the goodfeatures and demand the proposed approachwhen the systemoperates under uncertainties
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Mahjoub F Mnif and N Derbel ldquoSet point stabilization ofa 2DOF underactuated manipulatorrdquo Journal of Computers vol6 no 2 pp 368ndash376 2011
[2] F Mnif ldquoVSS control for a class of underactuated Mechanicalsystemsrdquo International Journal of Computational Cognition vol3 no 2 pp 14ndash18 2005
[3] J Hauser S Sastry and P Kokotovic ldquoNonlinear control viaapproximate inputndashoutput linearization the ball and beamexamplerdquo IEEE Transactions on Automatic Control vol 37 no3 pp 392ndash398 1992
[4] M Jankovic D Fontanine and P V Kokotovic ldquoTORAexample cascade and passitivity-based control designsrdquo IEEETransactions on Control Systems Technology vol 6 pp 4347ndash4351 1996
Mathematical Problems in Engineering 11
[5] T Sugie and K Fujimoto ldquoControl of inverted pendulumsystems based on approximate linearization design and exper-imentrdquo in Proceedings of the 33rd IEEE Conference on Decisionand Control vol 2 pp 1647ndash1648 December 1994
[6] R Xu and U Ozguner ldquoSliding mode control of a class ofunderactuated systemsrdquo Automatica vol 44 no 1 pp 233ndash2412008
[7] M W Spong ldquoThe swing-up control problem for the acrobatrdquoIEEE Transactions on Systems Technology vol 15 no 1 pp 49ndash55 1995
[8] R W Brockett ldquoAsymptotic stability feedback and stabiliza-tionrdquo in Differential Geometric Control Theory pp 181ndash191Birkhauser Boston Mass USA 1983
[9] C J Tomlin and S S Sastry ldquoSwitching through singularitiesrdquoSystems and Control Letters vol 35 no 3 pp 145ndash154 1998
[10] W-H Chen and D J Ballance ldquoOn a switching control schemefor nonlinear systems with ill-defined relative degreerdquo Systemsamp Control Letters vol 47 no 2 pp 159ndash166 2002
[11] F Zhang and B Fernndez-Rodriguez ldquoFeedback linearizationcontrol of systems with singularitiesrdquo in Proceedings of the6th International Conference on Complex Systems (ICCS rsquo06)Boston Mass USA June 2006
[12] D Seto and J Baillieul ldquoControl problems in super-articulatedmechanical systemsrdquo IEEE Transactions on Automatic Controlvol 39 no 12 pp 2442ndash2453 1994
[13] M W Spong Energy Based Control of a Class of UnderactuatedMechanical Systems IFACWorld Congress 1996
[14] V I Utkin Sliding Mode Control in Electromechanical SystemsTaylor amp Francis 1999
[15] M Rubagotti A Estrada F Castanos A Ferrara and LFridman ldquoIntegral sliding mode control for nonlinear systemswith matched and unmatched perturbationsrdquo IEEE Transactionon Automatic Control vol 56 no 11 pp 2699ndash2704 2011
[16] Q Khan A I Bhatti S Iqbal and M Iqbal ldquoDynamicintegral sliding mode for MIMO uncertain nonlinear systemsrdquoInternational Journal of Control Automation and Systems vol9 no 1 pp 151ndash160 2011
[17] R Olfati-Saber ldquoNormal forms for underactuated mechanicalsystems with symmetryrdquo IEEE Transactions on Automatic Con-trol vol 47 no 2 pp 305ndash308 2002
[18] R Lozano I Fantoni and D J Block ldquoStabilization of theinverted pendulum around its homoclinic orbitrdquo Systems andControl Letters vol 40 no 3 pp 197ndash204 2000
[19] E Altug J P Ostrowski and R Mahony ldquoControl of aquadrotor helicopter using visual feedbackrdquo in Proceedings ofthe IEEE International Conference on Robotics and Automation(ICRA rsquo02) vol 1 pp 72ndash77 Washington DC USA 2002
[20] A Isidori Nonlinear Control Systems Communications andControl Engineering Series Springer Berlin Germany 3rdedition 1995
[21] H Sira-Ramirez ldquoOn the dynamical sliding mode control ofnonlinear systemsrdquo International Journal of Control vol 57 no5 pp 1039ndash1061 1993
[22] Q Khan A I Bhatti and A Ferrara ldquoDynamic sliding modecontrol design based on an integral manifold for nonlinearuncertain systemsrdquo Journal of Nonlinear Dynamics vol 2014Article ID 489364 10 pages 2014
[23] C Edwards and S K Spurgeon Sliding Modes Control Theoryand Applications Taylor amp Francis London UK 1998
[24] A M Khan A I Bhatti S U Din and Q Khan ldquoStaticamp dynamic sliding mode control of ball and beam systemrdquo
in Proceedings of the 9th International Bhurban Conferenceon Applied Sciences and Technology (IBCAST rsquo12) pp 32ndash36Islamabad Pakistan January 2012
[25] N B Almutairi and M Zribi ldquoOn the sliding mode control ofa Ball on a Beam systemrdquo Nonlinear Dynamics vol 59 no 1-2pp 222ndash239 2010
Once again following the pole placement procedure onemayhave for the sake of simplicity the input 120591
0
which is designedvia pole placement that is
1205910
= minus119870119879
0
120585 (26)
Now to get the desired robust performance the followingsliding manifold of integral type [14] is defined
120590 (120585) = 1205900
(120585) + 119911 (27)
where 1205900
(120585) is the usual sliding surface and 119911 is the integralterm The time derivative of (27) along (9) yields
= minus(
119903minus1
sum
119894=1
119888119894
120585119894+1
+ 1205910
)
119911 (0) = minus1205900
(120585 (0))
(28)
1205911
=1
120574 ()(minus120593 ( 120591) minus (120574 () minus 1) 120591
0
minus 119870 sign120590) (29)
This control law enforces sliding mode along the slidingmanifold defined in (27) The constant 119870 can be selectedaccording to the subsequent stability analysis
Thus the final control law becomes
1205911
= minus119870119879
0
120585
+1
120574 ()(minus120593 ( 119906) minus (120574 () minus 1) 120591
0
minus 119870 sign120590)
(30)
Theorem 4 Consider that |Δ119866119898
( 119905)| le 1205731
are satisfiedthen the sliding mode against the switching manifold 120590 = 0 canbe ensured and one has
119870 ge [119870119872
1205731
+ 1205781
] (31)
where 1205781
is a positive constant
Proof Toprove that the slidingmode can be enforced in finitetime differentiating (22) along the dynamics of (3)-(4) andthen substituting (30) one has
(120585) =
119903minus1
sum
119894=1
119888119894
120585119894+1
+ 1205910
minus 119870 sign120590 + 120574 () Δ119866119898
( 119905)
+
(32)
Substituting (28) in (32) and then rearranging one obtains
] (36)The inequality in (35) presents that 120590(120585) approaches zero in afinite time 119905
119904
[23] such that
119905119904
le radic2120578minus1
1
radic119881 (120590 (0)) (37)which completes the proof
4 Illustrative Example
The control algorithms presented in Section 3 are applied tothe control design of a ball and beam systemThe assessmentof the proposed controller for the ball and beam systemis carried out on the basis of output tracking robustnessenhancement via the elimination of reaching phase andchattering-free control input in the presence of uncertainties
41 Description of the Ball and Beam System The ball andbeam system is a very sound candidate of the class ofunderactuated nonlinear system It is famous because of itsnonlinear nature and due to its wide range of applicationsin the existing era like passenger cabin balancing in luxurycars balancing of liquid fuel in vertical take-off objectsIn terms of control scenarios it is an ill-defined relativedegree system which to some extent does not support inputoutput linearization A schematic diagram with their typicalparameters of the ball and beam system is displayed in theadjacent Figure 1 and Table 1 respectively In this study theauthors use the equipment manufacture by GoogolTech Ingeneral this system is equipped with a metallic ball whichis let free to roll on a rod having a specified length havingone end fixed and the other end moved up and down via anelectric servomotorThe position of the ball can be measuredvia different techniques The measured position is used asfeedback to the system and accordingly the motor moves thebeam to balance the ball at user defined location
The motion governing equations of this system are givenbelow which are adopted from [24]
(1198981199032
+ 1198621
) + (2119898119903 119903 + 1198622
)
+ (119898119892119903 +119871
2119872119892) cos120573 = 120591
1198624
119903 minus 119903
1205732
+ 119892 sin120573 = 0
(38)
6 Mathematical Problems in Engineering
yL
R rBall
Beam
Mg
mg
Motor
d120579
z120573
mg sin 120573
Figure 1 Schematic diagram of the ball and beam system
where 120579(119905) angle is subtended to make the ball stable thelever angle is represented by 120573(119905) 119903(119905) is the position of theball on the beam and Vin(119905) is the input voltage of the motorwhereas the controlled input appears mathematically via theexpression 120591(119905) = 119862
3
Vin(119905) in the dynamic modelThe derived parameters used in the dynamic model of
this system are represented by 1198621
1198622
1198623
and 1198624
with thefollowing mathematical relations [25]
1198621
=119877119898
times 119869119898
times 119871
119862119898
times 119862119887
times 119889+ 1198691
(39)
1198622
=119871
119889(119862119898
times 119862119887
119877119898
+ 119862119887
+119877119898
times 119869119898
119862119898
times 119862119892
) (40)
1198623
= 1 +119862119898
119877119898
(41)
1198624
=7
5 (42)
The equivalent state spacemodel of this is described as followsby assuming 119909
1
= 119903 (position of ball) 1199092
= 119903 (rate of changeof position) 119909
3
= 120573 (beam angle) and 1199094
= (the rate ofchange of angle of the motor)
1199091
= 1199092
1199092
=1
1198624
(minus119892 sin (1199093
))
1199093
= 1199094
1199094
=1
11989811990921
+ 1198621
(120591 minus (21198981199091
1199092
+ 1198622
) 1199094
minus (1198981198921199091
+119871
2119872119892) cos119909
3
)
(43)
Now the output of interest is 119910 = 1199091
which representsthe position of the ball This representation is similar tothat reported in (3)-(4) In the next discussion the controllerdesign is outlined
42 Controller Design Following the procedure outlined inSection 3 the authors proceed as follows
119910 = 1199091
= 1199092
= minus119892
1198624
sin (1199093
)
119910(3)
= minus119892
1198624
1199094
cos (1199093
)
119910(4)
=1
1198624
(11989811990921
+ 1198621
)[minus120591 cos119909
3
+ (21198981199091
1199092
+ 1198622
) 1199094
cos1199093
+ (1198981198921199091
+119871
2119872119892) cos2119909
3
+ 1199092
4
(1198981199092
1
+ 1198621
) sin1199093
]
119910(4)
= 119891119904
+ ℎ119904
120591
119891119904
=119892
1198624
[(21198981199091
1199092
+ 1198622
) 1199094
+ (1198981198921199091
+ (1198712)119872119892) cos21199093
+ 1199092
4
sin1199093
11989811990921
+ 1198621
]
ℎ119904
=minus119892 cos119909
3
1198624
(11989811990921
+ 1198621
)
(44)
Mathematical Problems in Engineering 7
Table 1 Parameters and values used in equations
Parameter Description Nominal values Units
119892Gravitationalacceleration 981 ms2
119898 Mass of ball 007 kg119872 Mass of beam 015 kg119871 Length of beam 04 m
119877119898
Resistance ofarmature of the motor 9 Ω
119869119898
Moment of inertia ofmotor 735 times 10
minus4 Nmrads2
119862119898
Torque constant ofmotor 00075 NmA
119862119892
Gear ratio 428 mdash
119889
Radius of armconnected toservomotor
004 m
1198691
Moment of inertia ofbeam 0001 kgm2
119862119887
Back emf constantvalue 05625 Vrads
Now writing this in the controllable canonical form (phasevariable form) one may have
( 119905) represents the modeluncertainties Herewe discuss ISMCon ball and beam systemwith fixed step tracking as well as variable step tracking Theintegral manifold is defined as follows
120590 = 1198881
1205851
+ 1198882
1205852
+ 1198883
1205853
+ 1205854
+ 119911 (47)
The expression of the overall controller which becomes willbe as follows
1205911
= minus1198961
1205851
minus 1198962
1205852
minus 1198963
1205853
minus 1198964
1205854
+1
120574 ()(minus120593 () minus (120574 () minus 1) 120591
0
minus 119870sign120590)
= 1198881
1205851
+ 1198882
1205852
+ 1198883
1205853
+ 119891119904
+ ℎ119904
1205910
+ ℎ119904
1205911
+
= minus1198881
1199092
+1198882
119892
1198624
sin1199093
+1198883
119892
1198624
1199094
cos1199093
minus 120574 () 1205910
minus 120593 ()
(48)
As the authors are performing the reference tracking heretherefore the integral manifold and the controller will appearas follows
120590 = 1198881
(1205851
minus 119903119889
) + 1198882
1205852
+ 1198883
1205853
+ 1205854
+ 119911 (49)
1205911
= minus1198961
(1205851
minus 119903119889
) minus 1198962
1205852
minus 1198963
1205853
minus 1198964
1205854
+1
120574 ()(minus120593 ( 120591) minus (120574 () minus 1) 120591
0
minus 119870sign (120590))
(50)
where 119903119889
is the desired reference with 119903119889
119903119889
119903119889
beingbounded
43 Simulation Results The simulation study of the system iscarried out by considering the reference tracking of a squarewave signal and sinusoidal wave signal In the subsequentparagraph their respective results will be demonstrated indetail
In case the efforts are directed to track a fixed square wavesignal in the presence of disturbances the initial conditions ofthe system were set to 119909
1
(0) = 04 1199092
(0) = 1199093
(0) = 1199094
(0) = 0Furthermore the square wave was defined in the simulationcode as follows
119903119889
(119905) =
20 cm 0 le 119905 le 19
14 cm 20 le 119905 le 39
20 cm 40 le 119905 le 60
(51)
The gains of the proposed controller presented from (39) to(41) are chosen according to Table 2
The output tracking performance of the proposed controlinput when a square wave is used as desired reference outputis shown in Figure 2 It can be clearly examined that theperformance is very appealing in this caseThe correspondingsliding manifold profile is displayed in Figure 3 which clearlyindicates that the sliding mode is established from the verybeginning of the processes which in turn results in enhancedrobustnessThe controlled input signalrsquos profile is depicted inFigure 4 with its zoomed profile as shown in Figure 5 It isobvious from both the figures that the control input derivesthe system with suppressed chattering phenomenon which is
8 Mathematical Problems in Engineering
Table 2 Parametric values used in the square wave tracking
Constants 1198621
1198622
1198623
1198701
1198702
1198703
1198704
119870
Values 12 12 011 40298 25018 60 41 5
0 10 20 30 40 50 600
00501
01502
02503
03504
Time (s)
Trac
king
per
form
ance
ActualDesired
Figure 2 Output tracking performance when a square wave is usedas referencedesired output
0 10 20 30 40 50 60
005
115
2
Time (s)
Slid
ing
surfa
ce
Sliding surface
minus05
minus1
minus15
minus2
Figure 3 Sliding manifold convergence profile in case of squarewave tracking
tolerable for the system actuators health Now from this casestudy it is concluded that integral sliding mode approach isan interesting candidate for this class
In this case study once again efforts are focused on thetracking of a sinusoidal signal which is defined as 119903
119889
(119905) =
sin(119905) in the presence of disturbances Like the previous casestudy the initial conditions of the system were set to 119909
1
(0) =
04 1199092
(0) = 1199093
(0) = 1199094
(0) = 0 In addition the gains of theproposed controller presented in (50) are chosen accordingto Table 3
The output tracking performance of the proposed controlinput when a sinusoidal signal is considered as desiredreference output is shown in Figure 6 It can be clearlyseen that the performance is excellent in this scenarioThe corresponding sliding manifold profile is displayed inFigure 7 which confirms the establishment of sliding modesfrom the starting instant and consequently enhancement ofrobustnessThe controlled input signalrsquos profile is depicted inFigure 8 It is obvious from the figure that the control input
Table 3 Parametric values used in the sinusoid wave tracking
Constants 1198621
1198622
1198623
1198701
1198702
1198703
1198704
119870
Values 12 12 011 40298 25018 230 49 5
Time (s)0 10 20 30 40 50 60
05
101520
Con
trol i
nput
Control input
minus5
minus10
minus15
minus20
Figure 4 Control input in square wave reference tracking
Time (s)
Con
trol i
nput
Control input
0 10 20 30 40 50 60
0
05
1
15
minus05
minus1
minus15
Figure 5 Zoom profile of the control input depicted in Figure 4
Time (s)
ActualDesired
0 5 10 15 20 25
Trac
king
per
form
ance
0
05
1
15
minus05
minus1
minus15
Figure 6 Output tracking performance when a sinusoidal wave isused as referencedesired output
evolves with suppressed chattering phenomenonwhich onceagain makes this design strategy a good candidate for theclass of these underactuated systems
44 Implementation Results The control technique proposedin this paper is implemented on the actual apparatus using
Mathematical Problems in Engineering 9
Time (s)
Sliding surface
0 5 10 15 20 25
0
05
1
Slid
ing
surfa
ce(v
aria
ble r
efer
ence
sign
al)
minus05
minus1
Figure 7 Sliding manifold convergence profile in case of sinusoidalwave tracking
Time (s)0 5 10 15 20 25
0
5
10
Control effort
minus5
minus10
Con
trol i
nput
ldquordquo
u
Figure 8 Control input in sinusoidal wave reference tracking
the MATLAB environment The detailed discussions arepresented below
441 Experimental Setup Description The experiment setupis equipped by GoogolTech GBB1004 with an electroniccontrol boxThe beam length is 40 cm alongwithmass of ballthat is 28 g and an intelligent IPM100 servo driver which isused formoving the ball on the beamThe experimental setupis shown in Figure 9
The input given to apparatus is the voltage Vin(119905) and theoutput is the position of themotor 120579(119905) which in otherwordsis an input for the positioning of the ball on the beam Thisapparatus uses potentiometer mounted within a slot insidethe beam to sense the position of the ball on the beam Themeasured position along the beam is fed to the AD converterof IPM100 motion drive
The power module used in GoogolTech requires 220Vand 10A input Note that the control accuracy of this manu-factured apparatus lies within the range of plusmn1mmThe typicalparameters values are listed in Table 1 The environmentused here includes Windows XP as an operating system andMATLAB 712Simulink 77 Furthermore the sampling timeused in forthcoming practical results was 2ms In the exper-imental processes the proposed controllers need velocitymeasurements which are in general not available One may
Figure 9 Experimental setup of the ball and beam equipped viaGoogolTech GBB1004
use different kind of velocity observersdifferentiator for thevelocity estimation [16] In order to make the implemen-tation easy and simple a derivative block of the Simulinkenvironment is used to provide the corresponding velocitiesmeasurements Now we are ready to discuss the results of thesystem
In this experiment the initial conditions were set to1199091
(0) = 028 1199092
(0) = 1199093
(0) = 1199094
(0) = 0 The referencesignal which is needed to be tracked is being defined in (51)In Figures 10 and 11 the tracking performance is shown Theresults reveal that the actual signal 119909
1
(119905) is pretty close tothe desired signal 119903
119889
(119905) with a steady state error which isapproximatelyplusmn0001mThe existence of this error is becauseof the apparatus
The observations of these tracking results make it clearthat the practically implemented results have very closeresemblance with the simulation result presented in Figure 2The error convergence depends on the initial conditions ofthe ball on the beam If the ball is placed very close to thedesired reference value then it will take little time to reachthe desired position On the other hand the convergence tothe desired position will take considerable time if the initialcondition is chosen far away from the desired values Thisphenomenon of convergence is according to the equipmentdesign and structure
The sliding manifold convergence and the control inputare shown in Figures 12 and 13 respectively The controlinput and the sliding manifolds show some deviations inthe first second This deviation occurs because the ball onthe beam being placed anywhere on the beam is firstmoved to one side of the beam and then ball moved to thedesired position The zoomed profile of the control inputbeing displayed in Figure 14 shows high frequency vibration(chattering) of magnitude plusmn007 This makes the proposedcontrol design algorithm an appealing candidate for this classof nonlinear systems The gains of the controller being usedin this experiment are displayed in Table 4
5 Conclusion
The control of underactuated systems because of their lessnumber of actuators than the degree of freedom is an
10 Mathematical Problems in Engineering
Table 4 Parametric values used in implementation
Constants 1198621
1198622
1198623
1198701
1198702
1198703
1198704
119870
Values 8 5 1 3 15 3 1 4
0
01
02
03
04
Trac
king
per
form
ance
0 5 10 15 20 25Time (s)
DesiredActual
16 18 20 22 240219
022022102220223
Figure 10 Output tracking performance when 119903119889
= 22 cm is set asreferencedesired output
Time (s)
DesiredActual
0
01
02
03
04
Trac
king
per
form
ance
0 10 20 30 40 50 6044 46 48 50 520196
019802
02020204
Figure 11 Output tracking performance when a square wave is usedas referencedesired output
0 5 10 15 20 25
05
101520
Time (s)
Slid
ing
surfa
ce
Sliding surface
minus5
minus10
minus15
minus20
Figure 12 Sliding surface of practical system
interesting objective among the researchers In this work anintegral sliding mode control approach due to its robustnessfrom the very beginning of the process is employed forthe control design of this class The design of the integralmanifold relied upon a transformed form The benefit of thetransformed form is that itmakes the design strategy easy and
0 5 10 15 20 25
0
2
4
6
Time (s)
Inpu
t per
form
ance
Control input
minus2
minus4
minus6
Figure 13 Control input for reference tracking
Inpu
t per
form
ance
Control input
0 5 10 15 20 25
0
05
1
Time (s)
minus05
minus1
Figure 14 Zoom profile of the control input depicted in Figure 13
simple The stability analysis and experimental results of theproposed control laws are presented which convey the goodfeatures and demand the proposed approachwhen the systemoperates under uncertainties
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Mahjoub F Mnif and N Derbel ldquoSet point stabilization ofa 2DOF underactuated manipulatorrdquo Journal of Computers vol6 no 2 pp 368ndash376 2011
[2] F Mnif ldquoVSS control for a class of underactuated Mechanicalsystemsrdquo International Journal of Computational Cognition vol3 no 2 pp 14ndash18 2005
[3] J Hauser S Sastry and P Kokotovic ldquoNonlinear control viaapproximate inputndashoutput linearization the ball and beamexamplerdquo IEEE Transactions on Automatic Control vol 37 no3 pp 392ndash398 1992
[4] M Jankovic D Fontanine and P V Kokotovic ldquoTORAexample cascade and passitivity-based control designsrdquo IEEETransactions on Control Systems Technology vol 6 pp 4347ndash4351 1996
Mathematical Problems in Engineering 11
[5] T Sugie and K Fujimoto ldquoControl of inverted pendulumsystems based on approximate linearization design and exper-imentrdquo in Proceedings of the 33rd IEEE Conference on Decisionand Control vol 2 pp 1647ndash1648 December 1994
[6] R Xu and U Ozguner ldquoSliding mode control of a class ofunderactuated systemsrdquo Automatica vol 44 no 1 pp 233ndash2412008
[7] M W Spong ldquoThe swing-up control problem for the acrobatrdquoIEEE Transactions on Systems Technology vol 15 no 1 pp 49ndash55 1995
[8] R W Brockett ldquoAsymptotic stability feedback and stabiliza-tionrdquo in Differential Geometric Control Theory pp 181ndash191Birkhauser Boston Mass USA 1983
[9] C J Tomlin and S S Sastry ldquoSwitching through singularitiesrdquoSystems and Control Letters vol 35 no 3 pp 145ndash154 1998
[10] W-H Chen and D J Ballance ldquoOn a switching control schemefor nonlinear systems with ill-defined relative degreerdquo Systemsamp Control Letters vol 47 no 2 pp 159ndash166 2002
[11] F Zhang and B Fernndez-Rodriguez ldquoFeedback linearizationcontrol of systems with singularitiesrdquo in Proceedings of the6th International Conference on Complex Systems (ICCS rsquo06)Boston Mass USA June 2006
[12] D Seto and J Baillieul ldquoControl problems in super-articulatedmechanical systemsrdquo IEEE Transactions on Automatic Controlvol 39 no 12 pp 2442ndash2453 1994
[13] M W Spong Energy Based Control of a Class of UnderactuatedMechanical Systems IFACWorld Congress 1996
[14] V I Utkin Sliding Mode Control in Electromechanical SystemsTaylor amp Francis 1999
[15] M Rubagotti A Estrada F Castanos A Ferrara and LFridman ldquoIntegral sliding mode control for nonlinear systemswith matched and unmatched perturbationsrdquo IEEE Transactionon Automatic Control vol 56 no 11 pp 2699ndash2704 2011
[16] Q Khan A I Bhatti S Iqbal and M Iqbal ldquoDynamicintegral sliding mode for MIMO uncertain nonlinear systemsrdquoInternational Journal of Control Automation and Systems vol9 no 1 pp 151ndash160 2011
[17] R Olfati-Saber ldquoNormal forms for underactuated mechanicalsystems with symmetryrdquo IEEE Transactions on Automatic Con-trol vol 47 no 2 pp 305ndash308 2002
[18] R Lozano I Fantoni and D J Block ldquoStabilization of theinverted pendulum around its homoclinic orbitrdquo Systems andControl Letters vol 40 no 3 pp 197ndash204 2000
[19] E Altug J P Ostrowski and R Mahony ldquoControl of aquadrotor helicopter using visual feedbackrdquo in Proceedings ofthe IEEE International Conference on Robotics and Automation(ICRA rsquo02) vol 1 pp 72ndash77 Washington DC USA 2002
[20] A Isidori Nonlinear Control Systems Communications andControl Engineering Series Springer Berlin Germany 3rdedition 1995
[21] H Sira-Ramirez ldquoOn the dynamical sliding mode control ofnonlinear systemsrdquo International Journal of Control vol 57 no5 pp 1039ndash1061 1993
[22] Q Khan A I Bhatti and A Ferrara ldquoDynamic sliding modecontrol design based on an integral manifold for nonlinearuncertain systemsrdquo Journal of Nonlinear Dynamics vol 2014Article ID 489364 10 pages 2014
[23] C Edwards and S K Spurgeon Sliding Modes Control Theoryand Applications Taylor amp Francis London UK 1998
[24] A M Khan A I Bhatti S U Din and Q Khan ldquoStaticamp dynamic sliding mode control of ball and beam systemrdquo
in Proceedings of the 9th International Bhurban Conferenceon Applied Sciences and Technology (IBCAST rsquo12) pp 32ndash36Islamabad Pakistan January 2012
[25] N B Almutairi and M Zribi ldquoOn the sliding mode control ofa Ball on a Beam systemrdquo Nonlinear Dynamics vol 59 no 1-2pp 222ndash239 2010
Figure 1 Schematic diagram of the ball and beam system
where 120579(119905) angle is subtended to make the ball stable thelever angle is represented by 120573(119905) 119903(119905) is the position of theball on the beam and Vin(119905) is the input voltage of the motorwhereas the controlled input appears mathematically via theexpression 120591(119905) = 119862
3
Vin(119905) in the dynamic modelThe derived parameters used in the dynamic model of
this system are represented by 1198621
1198622
1198623
and 1198624
with thefollowing mathematical relations [25]
1198621
=119877119898
times 119869119898
times 119871
119862119898
times 119862119887
times 119889+ 1198691
(39)
1198622
=119871
119889(119862119898
times 119862119887
119877119898
+ 119862119887
+119877119898
times 119869119898
119862119898
times 119862119892
) (40)
1198623
= 1 +119862119898
119877119898
(41)
1198624
=7
5 (42)
The equivalent state spacemodel of this is described as followsby assuming 119909
1
= 119903 (position of ball) 1199092
= 119903 (rate of changeof position) 119909
3
= 120573 (beam angle) and 1199094
= (the rate ofchange of angle of the motor)
1199091
= 1199092
1199092
=1
1198624
(minus119892 sin (1199093
))
1199093
= 1199094
1199094
=1
11989811990921
+ 1198621
(120591 minus (21198981199091
1199092
+ 1198622
) 1199094
minus (1198981198921199091
+119871
2119872119892) cos119909
3
)
(43)
Now the output of interest is 119910 = 1199091
which representsthe position of the ball This representation is similar tothat reported in (3)-(4) In the next discussion the controllerdesign is outlined
42 Controller Design Following the procedure outlined inSection 3 the authors proceed as follows
119910 = 1199091
= 1199092
= minus119892
1198624
sin (1199093
)
119910(3)
= minus119892
1198624
1199094
cos (1199093
)
119910(4)
=1
1198624
(11989811990921
+ 1198621
)[minus120591 cos119909
3
+ (21198981199091
1199092
+ 1198622
) 1199094
cos1199093
+ (1198981198921199091
+119871
2119872119892) cos2119909
3
+ 1199092
4
(1198981199092
1
+ 1198621
) sin1199093
]
119910(4)
= 119891119904
+ ℎ119904
120591
119891119904
=119892
1198624
[(21198981199091
1199092
+ 1198622
) 1199094
+ (1198981198921199091
+ (1198712)119872119892) cos21199093
+ 1199092
4
sin1199093
11989811990921
+ 1198621
]
ℎ119904
=minus119892 cos119909
3
1198624
(11989811990921
+ 1198621
)
(44)
Mathematical Problems in Engineering 7
Table 1 Parameters and values used in equations
Parameter Description Nominal values Units
119892Gravitationalacceleration 981 ms2
119898 Mass of ball 007 kg119872 Mass of beam 015 kg119871 Length of beam 04 m
119877119898
Resistance ofarmature of the motor 9 Ω
119869119898
Moment of inertia ofmotor 735 times 10
minus4 Nmrads2
119862119898
Torque constant ofmotor 00075 NmA
119862119892
Gear ratio 428 mdash
119889
Radius of armconnected toservomotor
004 m
1198691
Moment of inertia ofbeam 0001 kgm2
119862119887
Back emf constantvalue 05625 Vrads
Now writing this in the controllable canonical form (phasevariable form) one may have
( 119905) represents the modeluncertainties Herewe discuss ISMCon ball and beam systemwith fixed step tracking as well as variable step tracking Theintegral manifold is defined as follows
120590 = 1198881
1205851
+ 1198882
1205852
+ 1198883
1205853
+ 1205854
+ 119911 (47)
The expression of the overall controller which becomes willbe as follows
1205911
= minus1198961
1205851
minus 1198962
1205852
minus 1198963
1205853
minus 1198964
1205854
+1
120574 ()(minus120593 () minus (120574 () minus 1) 120591
0
minus 119870sign120590)
= 1198881
1205851
+ 1198882
1205852
+ 1198883
1205853
+ 119891119904
+ ℎ119904
1205910
+ ℎ119904
1205911
+
= minus1198881
1199092
+1198882
119892
1198624
sin1199093
+1198883
119892
1198624
1199094
cos1199093
minus 120574 () 1205910
minus 120593 ()
(48)
As the authors are performing the reference tracking heretherefore the integral manifold and the controller will appearas follows
120590 = 1198881
(1205851
minus 119903119889
) + 1198882
1205852
+ 1198883
1205853
+ 1205854
+ 119911 (49)
1205911
= minus1198961
(1205851
minus 119903119889
) minus 1198962
1205852
minus 1198963
1205853
minus 1198964
1205854
+1
120574 ()(minus120593 ( 120591) minus (120574 () minus 1) 120591
0
minus 119870sign (120590))
(50)
where 119903119889
is the desired reference with 119903119889
119903119889
119903119889
beingbounded
43 Simulation Results The simulation study of the system iscarried out by considering the reference tracking of a squarewave signal and sinusoidal wave signal In the subsequentparagraph their respective results will be demonstrated indetail
In case the efforts are directed to track a fixed square wavesignal in the presence of disturbances the initial conditions ofthe system were set to 119909
1
(0) = 04 1199092
(0) = 1199093
(0) = 1199094
(0) = 0Furthermore the square wave was defined in the simulationcode as follows
119903119889
(119905) =
20 cm 0 le 119905 le 19
14 cm 20 le 119905 le 39
20 cm 40 le 119905 le 60
(51)
The gains of the proposed controller presented from (39) to(41) are chosen according to Table 2
The output tracking performance of the proposed controlinput when a square wave is used as desired reference outputis shown in Figure 2 It can be clearly examined that theperformance is very appealing in this caseThe correspondingsliding manifold profile is displayed in Figure 3 which clearlyindicates that the sliding mode is established from the verybeginning of the processes which in turn results in enhancedrobustnessThe controlled input signalrsquos profile is depicted inFigure 4 with its zoomed profile as shown in Figure 5 It isobvious from both the figures that the control input derivesthe system with suppressed chattering phenomenon which is
8 Mathematical Problems in Engineering
Table 2 Parametric values used in the square wave tracking
Constants 1198621
1198622
1198623
1198701
1198702
1198703
1198704
119870
Values 12 12 011 40298 25018 60 41 5
0 10 20 30 40 50 600
00501
01502
02503
03504
Time (s)
Trac
king
per
form
ance
ActualDesired
Figure 2 Output tracking performance when a square wave is usedas referencedesired output
0 10 20 30 40 50 60
005
115
2
Time (s)
Slid
ing
surfa
ce
Sliding surface
minus05
minus1
minus15
minus2
Figure 3 Sliding manifold convergence profile in case of squarewave tracking
tolerable for the system actuators health Now from this casestudy it is concluded that integral sliding mode approach isan interesting candidate for this class
In this case study once again efforts are focused on thetracking of a sinusoidal signal which is defined as 119903
119889
(119905) =
sin(119905) in the presence of disturbances Like the previous casestudy the initial conditions of the system were set to 119909
1
(0) =
04 1199092
(0) = 1199093
(0) = 1199094
(0) = 0 In addition the gains of theproposed controller presented in (50) are chosen accordingto Table 3
The output tracking performance of the proposed controlinput when a sinusoidal signal is considered as desiredreference output is shown in Figure 6 It can be clearlyseen that the performance is excellent in this scenarioThe corresponding sliding manifold profile is displayed inFigure 7 which confirms the establishment of sliding modesfrom the starting instant and consequently enhancement ofrobustnessThe controlled input signalrsquos profile is depicted inFigure 8 It is obvious from the figure that the control input
Table 3 Parametric values used in the sinusoid wave tracking
Constants 1198621
1198622
1198623
1198701
1198702
1198703
1198704
119870
Values 12 12 011 40298 25018 230 49 5
Time (s)0 10 20 30 40 50 60
05
101520
Con
trol i
nput
Control input
minus5
minus10
minus15
minus20
Figure 4 Control input in square wave reference tracking
Time (s)
Con
trol i
nput
Control input
0 10 20 30 40 50 60
0
05
1
15
minus05
minus1
minus15
Figure 5 Zoom profile of the control input depicted in Figure 4
Time (s)
ActualDesired
0 5 10 15 20 25
Trac
king
per
form
ance
0
05
1
15
minus05
minus1
minus15
Figure 6 Output tracking performance when a sinusoidal wave isused as referencedesired output
evolves with suppressed chattering phenomenonwhich onceagain makes this design strategy a good candidate for theclass of these underactuated systems
44 Implementation Results The control technique proposedin this paper is implemented on the actual apparatus using
Mathematical Problems in Engineering 9
Time (s)
Sliding surface
0 5 10 15 20 25
0
05
1
Slid
ing
surfa
ce(v
aria
ble r
efer
ence
sign
al)
minus05
minus1
Figure 7 Sliding manifold convergence profile in case of sinusoidalwave tracking
Time (s)0 5 10 15 20 25
0
5
10
Control effort
minus5
minus10
Con
trol i
nput
ldquordquo
u
Figure 8 Control input in sinusoidal wave reference tracking
the MATLAB environment The detailed discussions arepresented below
441 Experimental Setup Description The experiment setupis equipped by GoogolTech GBB1004 with an electroniccontrol boxThe beam length is 40 cm alongwithmass of ballthat is 28 g and an intelligent IPM100 servo driver which isused formoving the ball on the beamThe experimental setupis shown in Figure 9
The input given to apparatus is the voltage Vin(119905) and theoutput is the position of themotor 120579(119905) which in otherwordsis an input for the positioning of the ball on the beam Thisapparatus uses potentiometer mounted within a slot insidethe beam to sense the position of the ball on the beam Themeasured position along the beam is fed to the AD converterof IPM100 motion drive
The power module used in GoogolTech requires 220Vand 10A input Note that the control accuracy of this manu-factured apparatus lies within the range of plusmn1mmThe typicalparameters values are listed in Table 1 The environmentused here includes Windows XP as an operating system andMATLAB 712Simulink 77 Furthermore the sampling timeused in forthcoming practical results was 2ms In the exper-imental processes the proposed controllers need velocitymeasurements which are in general not available One may
Figure 9 Experimental setup of the ball and beam equipped viaGoogolTech GBB1004
use different kind of velocity observersdifferentiator for thevelocity estimation [16] In order to make the implemen-tation easy and simple a derivative block of the Simulinkenvironment is used to provide the corresponding velocitiesmeasurements Now we are ready to discuss the results of thesystem
In this experiment the initial conditions were set to1199091
(0) = 028 1199092
(0) = 1199093
(0) = 1199094
(0) = 0 The referencesignal which is needed to be tracked is being defined in (51)In Figures 10 and 11 the tracking performance is shown Theresults reveal that the actual signal 119909
1
(119905) is pretty close tothe desired signal 119903
119889
(119905) with a steady state error which isapproximatelyplusmn0001mThe existence of this error is becauseof the apparatus
The observations of these tracking results make it clearthat the practically implemented results have very closeresemblance with the simulation result presented in Figure 2The error convergence depends on the initial conditions ofthe ball on the beam If the ball is placed very close to thedesired reference value then it will take little time to reachthe desired position On the other hand the convergence tothe desired position will take considerable time if the initialcondition is chosen far away from the desired values Thisphenomenon of convergence is according to the equipmentdesign and structure
The sliding manifold convergence and the control inputare shown in Figures 12 and 13 respectively The controlinput and the sliding manifolds show some deviations inthe first second This deviation occurs because the ball onthe beam being placed anywhere on the beam is firstmoved to one side of the beam and then ball moved to thedesired position The zoomed profile of the control inputbeing displayed in Figure 14 shows high frequency vibration(chattering) of magnitude plusmn007 This makes the proposedcontrol design algorithm an appealing candidate for this classof nonlinear systems The gains of the controller being usedin this experiment are displayed in Table 4
5 Conclusion
The control of underactuated systems because of their lessnumber of actuators than the degree of freedom is an
10 Mathematical Problems in Engineering
Table 4 Parametric values used in implementation
Constants 1198621
1198622
1198623
1198701
1198702
1198703
1198704
119870
Values 8 5 1 3 15 3 1 4
0
01
02
03
04
Trac
king
per
form
ance
0 5 10 15 20 25Time (s)
DesiredActual
16 18 20 22 240219
022022102220223
Figure 10 Output tracking performance when 119903119889
= 22 cm is set asreferencedesired output
Time (s)
DesiredActual
0
01
02
03
04
Trac
king
per
form
ance
0 10 20 30 40 50 6044 46 48 50 520196
019802
02020204
Figure 11 Output tracking performance when a square wave is usedas referencedesired output
0 5 10 15 20 25
05
101520
Time (s)
Slid
ing
surfa
ce
Sliding surface
minus5
minus10
minus15
minus20
Figure 12 Sliding surface of practical system
interesting objective among the researchers In this work anintegral sliding mode control approach due to its robustnessfrom the very beginning of the process is employed forthe control design of this class The design of the integralmanifold relied upon a transformed form The benefit of thetransformed form is that itmakes the design strategy easy and
0 5 10 15 20 25
0
2
4
6
Time (s)
Inpu
t per
form
ance
Control input
minus2
minus4
minus6
Figure 13 Control input for reference tracking
Inpu
t per
form
ance
Control input
0 5 10 15 20 25
0
05
1
Time (s)
minus05
minus1
Figure 14 Zoom profile of the control input depicted in Figure 13
simple The stability analysis and experimental results of theproposed control laws are presented which convey the goodfeatures and demand the proposed approachwhen the systemoperates under uncertainties
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Mahjoub F Mnif and N Derbel ldquoSet point stabilization ofa 2DOF underactuated manipulatorrdquo Journal of Computers vol6 no 2 pp 368ndash376 2011
[2] F Mnif ldquoVSS control for a class of underactuated Mechanicalsystemsrdquo International Journal of Computational Cognition vol3 no 2 pp 14ndash18 2005
[3] J Hauser S Sastry and P Kokotovic ldquoNonlinear control viaapproximate inputndashoutput linearization the ball and beamexamplerdquo IEEE Transactions on Automatic Control vol 37 no3 pp 392ndash398 1992
[4] M Jankovic D Fontanine and P V Kokotovic ldquoTORAexample cascade and passitivity-based control designsrdquo IEEETransactions on Control Systems Technology vol 6 pp 4347ndash4351 1996
Mathematical Problems in Engineering 11
[5] T Sugie and K Fujimoto ldquoControl of inverted pendulumsystems based on approximate linearization design and exper-imentrdquo in Proceedings of the 33rd IEEE Conference on Decisionand Control vol 2 pp 1647ndash1648 December 1994
[6] R Xu and U Ozguner ldquoSliding mode control of a class ofunderactuated systemsrdquo Automatica vol 44 no 1 pp 233ndash2412008
[7] M W Spong ldquoThe swing-up control problem for the acrobatrdquoIEEE Transactions on Systems Technology vol 15 no 1 pp 49ndash55 1995
[8] R W Brockett ldquoAsymptotic stability feedback and stabiliza-tionrdquo in Differential Geometric Control Theory pp 181ndash191Birkhauser Boston Mass USA 1983
[9] C J Tomlin and S S Sastry ldquoSwitching through singularitiesrdquoSystems and Control Letters vol 35 no 3 pp 145ndash154 1998
[10] W-H Chen and D J Ballance ldquoOn a switching control schemefor nonlinear systems with ill-defined relative degreerdquo Systemsamp Control Letters vol 47 no 2 pp 159ndash166 2002
[11] F Zhang and B Fernndez-Rodriguez ldquoFeedback linearizationcontrol of systems with singularitiesrdquo in Proceedings of the6th International Conference on Complex Systems (ICCS rsquo06)Boston Mass USA June 2006
[12] D Seto and J Baillieul ldquoControl problems in super-articulatedmechanical systemsrdquo IEEE Transactions on Automatic Controlvol 39 no 12 pp 2442ndash2453 1994
[13] M W Spong Energy Based Control of a Class of UnderactuatedMechanical Systems IFACWorld Congress 1996
[14] V I Utkin Sliding Mode Control in Electromechanical SystemsTaylor amp Francis 1999
[15] M Rubagotti A Estrada F Castanos A Ferrara and LFridman ldquoIntegral sliding mode control for nonlinear systemswith matched and unmatched perturbationsrdquo IEEE Transactionon Automatic Control vol 56 no 11 pp 2699ndash2704 2011
[16] Q Khan A I Bhatti S Iqbal and M Iqbal ldquoDynamicintegral sliding mode for MIMO uncertain nonlinear systemsrdquoInternational Journal of Control Automation and Systems vol9 no 1 pp 151ndash160 2011
[17] R Olfati-Saber ldquoNormal forms for underactuated mechanicalsystems with symmetryrdquo IEEE Transactions on Automatic Con-trol vol 47 no 2 pp 305ndash308 2002
[18] R Lozano I Fantoni and D J Block ldquoStabilization of theinverted pendulum around its homoclinic orbitrdquo Systems andControl Letters vol 40 no 3 pp 197ndash204 2000
[19] E Altug J P Ostrowski and R Mahony ldquoControl of aquadrotor helicopter using visual feedbackrdquo in Proceedings ofthe IEEE International Conference on Robotics and Automation(ICRA rsquo02) vol 1 pp 72ndash77 Washington DC USA 2002
[20] A Isidori Nonlinear Control Systems Communications andControl Engineering Series Springer Berlin Germany 3rdedition 1995
[21] H Sira-Ramirez ldquoOn the dynamical sliding mode control ofnonlinear systemsrdquo International Journal of Control vol 57 no5 pp 1039ndash1061 1993
[22] Q Khan A I Bhatti and A Ferrara ldquoDynamic sliding modecontrol design based on an integral manifold for nonlinearuncertain systemsrdquo Journal of Nonlinear Dynamics vol 2014Article ID 489364 10 pages 2014
[23] C Edwards and S K Spurgeon Sliding Modes Control Theoryand Applications Taylor amp Francis London UK 1998
[24] A M Khan A I Bhatti S U Din and Q Khan ldquoStaticamp dynamic sliding mode control of ball and beam systemrdquo
in Proceedings of the 9th International Bhurban Conferenceon Applied Sciences and Technology (IBCAST rsquo12) pp 32ndash36Islamabad Pakistan January 2012
[25] N B Almutairi and M Zribi ldquoOn the sliding mode control ofa Ball on a Beam systemrdquo Nonlinear Dynamics vol 59 no 1-2pp 222ndash239 2010
( 119905) represents the modeluncertainties Herewe discuss ISMCon ball and beam systemwith fixed step tracking as well as variable step tracking Theintegral manifold is defined as follows
120590 = 1198881
1205851
+ 1198882
1205852
+ 1198883
1205853
+ 1205854
+ 119911 (47)
The expression of the overall controller which becomes willbe as follows
1205911
= minus1198961
1205851
minus 1198962
1205852
minus 1198963
1205853
minus 1198964
1205854
+1
120574 ()(minus120593 () minus (120574 () minus 1) 120591
0
minus 119870sign120590)
= 1198881
1205851
+ 1198882
1205852
+ 1198883
1205853
+ 119891119904
+ ℎ119904
1205910
+ ℎ119904
1205911
+
= minus1198881
1199092
+1198882
119892
1198624
sin1199093
+1198883
119892
1198624
1199094
cos1199093
minus 120574 () 1205910
minus 120593 ()
(48)
As the authors are performing the reference tracking heretherefore the integral manifold and the controller will appearas follows
120590 = 1198881
(1205851
minus 119903119889
) + 1198882
1205852
+ 1198883
1205853
+ 1205854
+ 119911 (49)
1205911
= minus1198961
(1205851
minus 119903119889
) minus 1198962
1205852
minus 1198963
1205853
minus 1198964
1205854
+1
120574 ()(minus120593 ( 120591) minus (120574 () minus 1) 120591
0
minus 119870sign (120590))
(50)
where 119903119889
is the desired reference with 119903119889
119903119889
119903119889
beingbounded
43 Simulation Results The simulation study of the system iscarried out by considering the reference tracking of a squarewave signal and sinusoidal wave signal In the subsequentparagraph their respective results will be demonstrated indetail
In case the efforts are directed to track a fixed square wavesignal in the presence of disturbances the initial conditions ofthe system were set to 119909
1
(0) = 04 1199092
(0) = 1199093
(0) = 1199094
(0) = 0Furthermore the square wave was defined in the simulationcode as follows
119903119889
(119905) =
20 cm 0 le 119905 le 19
14 cm 20 le 119905 le 39
20 cm 40 le 119905 le 60
(51)
The gains of the proposed controller presented from (39) to(41) are chosen according to Table 2
The output tracking performance of the proposed controlinput when a square wave is used as desired reference outputis shown in Figure 2 It can be clearly examined that theperformance is very appealing in this caseThe correspondingsliding manifold profile is displayed in Figure 3 which clearlyindicates that the sliding mode is established from the verybeginning of the processes which in turn results in enhancedrobustnessThe controlled input signalrsquos profile is depicted inFigure 4 with its zoomed profile as shown in Figure 5 It isobvious from both the figures that the control input derivesthe system with suppressed chattering phenomenon which is
8 Mathematical Problems in Engineering
Table 2 Parametric values used in the square wave tracking
Constants 1198621
1198622
1198623
1198701
1198702
1198703
1198704
119870
Values 12 12 011 40298 25018 60 41 5
0 10 20 30 40 50 600
00501
01502
02503
03504
Time (s)
Trac
king
per
form
ance
ActualDesired
Figure 2 Output tracking performance when a square wave is usedas referencedesired output
0 10 20 30 40 50 60
005
115
2
Time (s)
Slid
ing
surfa
ce
Sliding surface
minus05
minus1
minus15
minus2
Figure 3 Sliding manifold convergence profile in case of squarewave tracking
tolerable for the system actuators health Now from this casestudy it is concluded that integral sliding mode approach isan interesting candidate for this class
In this case study once again efforts are focused on thetracking of a sinusoidal signal which is defined as 119903
119889
(119905) =
sin(119905) in the presence of disturbances Like the previous casestudy the initial conditions of the system were set to 119909
1
(0) =
04 1199092
(0) = 1199093
(0) = 1199094
(0) = 0 In addition the gains of theproposed controller presented in (50) are chosen accordingto Table 3
The output tracking performance of the proposed controlinput when a sinusoidal signal is considered as desiredreference output is shown in Figure 6 It can be clearlyseen that the performance is excellent in this scenarioThe corresponding sliding manifold profile is displayed inFigure 7 which confirms the establishment of sliding modesfrom the starting instant and consequently enhancement ofrobustnessThe controlled input signalrsquos profile is depicted inFigure 8 It is obvious from the figure that the control input
Table 3 Parametric values used in the sinusoid wave tracking
Constants 1198621
1198622
1198623
1198701
1198702
1198703
1198704
119870
Values 12 12 011 40298 25018 230 49 5
Time (s)0 10 20 30 40 50 60
05
101520
Con
trol i
nput
Control input
minus5
minus10
minus15
minus20
Figure 4 Control input in square wave reference tracking
Time (s)
Con
trol i
nput
Control input
0 10 20 30 40 50 60
0
05
1
15
minus05
minus1
minus15
Figure 5 Zoom profile of the control input depicted in Figure 4
Time (s)
ActualDesired
0 5 10 15 20 25
Trac
king
per
form
ance
0
05
1
15
minus05
minus1
minus15
Figure 6 Output tracking performance when a sinusoidal wave isused as referencedesired output
evolves with suppressed chattering phenomenonwhich onceagain makes this design strategy a good candidate for theclass of these underactuated systems
44 Implementation Results The control technique proposedin this paper is implemented on the actual apparatus using
Mathematical Problems in Engineering 9
Time (s)
Sliding surface
0 5 10 15 20 25
0
05
1
Slid
ing
surfa
ce(v
aria
ble r
efer
ence
sign
al)
minus05
minus1
Figure 7 Sliding manifold convergence profile in case of sinusoidalwave tracking
Time (s)0 5 10 15 20 25
0
5
10
Control effort
minus5
minus10
Con
trol i
nput
ldquordquo
u
Figure 8 Control input in sinusoidal wave reference tracking
the MATLAB environment The detailed discussions arepresented below
441 Experimental Setup Description The experiment setupis equipped by GoogolTech GBB1004 with an electroniccontrol boxThe beam length is 40 cm alongwithmass of ballthat is 28 g and an intelligent IPM100 servo driver which isused formoving the ball on the beamThe experimental setupis shown in Figure 9
The input given to apparatus is the voltage Vin(119905) and theoutput is the position of themotor 120579(119905) which in otherwordsis an input for the positioning of the ball on the beam Thisapparatus uses potentiometer mounted within a slot insidethe beam to sense the position of the ball on the beam Themeasured position along the beam is fed to the AD converterof IPM100 motion drive
The power module used in GoogolTech requires 220Vand 10A input Note that the control accuracy of this manu-factured apparatus lies within the range of plusmn1mmThe typicalparameters values are listed in Table 1 The environmentused here includes Windows XP as an operating system andMATLAB 712Simulink 77 Furthermore the sampling timeused in forthcoming practical results was 2ms In the exper-imental processes the proposed controllers need velocitymeasurements which are in general not available One may
Figure 9 Experimental setup of the ball and beam equipped viaGoogolTech GBB1004
use different kind of velocity observersdifferentiator for thevelocity estimation [16] In order to make the implemen-tation easy and simple a derivative block of the Simulinkenvironment is used to provide the corresponding velocitiesmeasurements Now we are ready to discuss the results of thesystem
In this experiment the initial conditions were set to1199091
(0) = 028 1199092
(0) = 1199093
(0) = 1199094
(0) = 0 The referencesignal which is needed to be tracked is being defined in (51)In Figures 10 and 11 the tracking performance is shown Theresults reveal that the actual signal 119909
1
(119905) is pretty close tothe desired signal 119903
119889
(119905) with a steady state error which isapproximatelyplusmn0001mThe existence of this error is becauseof the apparatus
The observations of these tracking results make it clearthat the practically implemented results have very closeresemblance with the simulation result presented in Figure 2The error convergence depends on the initial conditions ofthe ball on the beam If the ball is placed very close to thedesired reference value then it will take little time to reachthe desired position On the other hand the convergence tothe desired position will take considerable time if the initialcondition is chosen far away from the desired values Thisphenomenon of convergence is according to the equipmentdesign and structure
The sliding manifold convergence and the control inputare shown in Figures 12 and 13 respectively The controlinput and the sliding manifolds show some deviations inthe first second This deviation occurs because the ball onthe beam being placed anywhere on the beam is firstmoved to one side of the beam and then ball moved to thedesired position The zoomed profile of the control inputbeing displayed in Figure 14 shows high frequency vibration(chattering) of magnitude plusmn007 This makes the proposedcontrol design algorithm an appealing candidate for this classof nonlinear systems The gains of the controller being usedin this experiment are displayed in Table 4
5 Conclusion
The control of underactuated systems because of their lessnumber of actuators than the degree of freedom is an
10 Mathematical Problems in Engineering
Table 4 Parametric values used in implementation
Constants 1198621
1198622
1198623
1198701
1198702
1198703
1198704
119870
Values 8 5 1 3 15 3 1 4
0
01
02
03
04
Trac
king
per
form
ance
0 5 10 15 20 25Time (s)
DesiredActual
16 18 20 22 240219
022022102220223
Figure 10 Output tracking performance when 119903119889
= 22 cm is set asreferencedesired output
Time (s)
DesiredActual
0
01
02
03
04
Trac
king
per
form
ance
0 10 20 30 40 50 6044 46 48 50 520196
019802
02020204
Figure 11 Output tracking performance when a square wave is usedas referencedesired output
0 5 10 15 20 25
05
101520
Time (s)
Slid
ing
surfa
ce
Sliding surface
minus5
minus10
minus15
minus20
Figure 12 Sliding surface of practical system
interesting objective among the researchers In this work anintegral sliding mode control approach due to its robustnessfrom the very beginning of the process is employed forthe control design of this class The design of the integralmanifold relied upon a transformed form The benefit of thetransformed form is that itmakes the design strategy easy and
0 5 10 15 20 25
0
2
4
6
Time (s)
Inpu
t per
form
ance
Control input
minus2
minus4
minus6
Figure 13 Control input for reference tracking
Inpu
t per
form
ance
Control input
0 5 10 15 20 25
0
05
1
Time (s)
minus05
minus1
Figure 14 Zoom profile of the control input depicted in Figure 13
simple The stability analysis and experimental results of theproposed control laws are presented which convey the goodfeatures and demand the proposed approachwhen the systemoperates under uncertainties
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Mahjoub F Mnif and N Derbel ldquoSet point stabilization ofa 2DOF underactuated manipulatorrdquo Journal of Computers vol6 no 2 pp 368ndash376 2011
[2] F Mnif ldquoVSS control for a class of underactuated Mechanicalsystemsrdquo International Journal of Computational Cognition vol3 no 2 pp 14ndash18 2005
[3] J Hauser S Sastry and P Kokotovic ldquoNonlinear control viaapproximate inputndashoutput linearization the ball and beamexamplerdquo IEEE Transactions on Automatic Control vol 37 no3 pp 392ndash398 1992
[4] M Jankovic D Fontanine and P V Kokotovic ldquoTORAexample cascade and passitivity-based control designsrdquo IEEETransactions on Control Systems Technology vol 6 pp 4347ndash4351 1996
Mathematical Problems in Engineering 11
[5] T Sugie and K Fujimoto ldquoControl of inverted pendulumsystems based on approximate linearization design and exper-imentrdquo in Proceedings of the 33rd IEEE Conference on Decisionand Control vol 2 pp 1647ndash1648 December 1994
[6] R Xu and U Ozguner ldquoSliding mode control of a class ofunderactuated systemsrdquo Automatica vol 44 no 1 pp 233ndash2412008
[7] M W Spong ldquoThe swing-up control problem for the acrobatrdquoIEEE Transactions on Systems Technology vol 15 no 1 pp 49ndash55 1995
[8] R W Brockett ldquoAsymptotic stability feedback and stabiliza-tionrdquo in Differential Geometric Control Theory pp 181ndash191Birkhauser Boston Mass USA 1983
[9] C J Tomlin and S S Sastry ldquoSwitching through singularitiesrdquoSystems and Control Letters vol 35 no 3 pp 145ndash154 1998
[10] W-H Chen and D J Ballance ldquoOn a switching control schemefor nonlinear systems with ill-defined relative degreerdquo Systemsamp Control Letters vol 47 no 2 pp 159ndash166 2002
[11] F Zhang and B Fernndez-Rodriguez ldquoFeedback linearizationcontrol of systems with singularitiesrdquo in Proceedings of the6th International Conference on Complex Systems (ICCS rsquo06)Boston Mass USA June 2006
[12] D Seto and J Baillieul ldquoControl problems in super-articulatedmechanical systemsrdquo IEEE Transactions on Automatic Controlvol 39 no 12 pp 2442ndash2453 1994
[13] M W Spong Energy Based Control of a Class of UnderactuatedMechanical Systems IFACWorld Congress 1996
[14] V I Utkin Sliding Mode Control in Electromechanical SystemsTaylor amp Francis 1999
[15] M Rubagotti A Estrada F Castanos A Ferrara and LFridman ldquoIntegral sliding mode control for nonlinear systemswith matched and unmatched perturbationsrdquo IEEE Transactionon Automatic Control vol 56 no 11 pp 2699ndash2704 2011
[16] Q Khan A I Bhatti S Iqbal and M Iqbal ldquoDynamicintegral sliding mode for MIMO uncertain nonlinear systemsrdquoInternational Journal of Control Automation and Systems vol9 no 1 pp 151ndash160 2011
[17] R Olfati-Saber ldquoNormal forms for underactuated mechanicalsystems with symmetryrdquo IEEE Transactions on Automatic Con-trol vol 47 no 2 pp 305ndash308 2002
[18] R Lozano I Fantoni and D J Block ldquoStabilization of theinverted pendulum around its homoclinic orbitrdquo Systems andControl Letters vol 40 no 3 pp 197ndash204 2000
[19] E Altug J P Ostrowski and R Mahony ldquoControl of aquadrotor helicopter using visual feedbackrdquo in Proceedings ofthe IEEE International Conference on Robotics and Automation(ICRA rsquo02) vol 1 pp 72ndash77 Washington DC USA 2002
[20] A Isidori Nonlinear Control Systems Communications andControl Engineering Series Springer Berlin Germany 3rdedition 1995
[21] H Sira-Ramirez ldquoOn the dynamical sliding mode control ofnonlinear systemsrdquo International Journal of Control vol 57 no5 pp 1039ndash1061 1993
[22] Q Khan A I Bhatti and A Ferrara ldquoDynamic sliding modecontrol design based on an integral manifold for nonlinearuncertain systemsrdquo Journal of Nonlinear Dynamics vol 2014Article ID 489364 10 pages 2014
[23] C Edwards and S K Spurgeon Sliding Modes Control Theoryand Applications Taylor amp Francis London UK 1998
[24] A M Khan A I Bhatti S U Din and Q Khan ldquoStaticamp dynamic sliding mode control of ball and beam systemrdquo
in Proceedings of the 9th International Bhurban Conferenceon Applied Sciences and Technology (IBCAST rsquo12) pp 32ndash36Islamabad Pakistan January 2012
[25] N B Almutairi and M Zribi ldquoOn the sliding mode control ofa Ball on a Beam systemrdquo Nonlinear Dynamics vol 59 no 1-2pp 222ndash239 2010
Table 2 Parametric values used in the square wave tracking
Constants 1198621
1198622
1198623
1198701
1198702
1198703
1198704
119870
Values 12 12 011 40298 25018 60 41 5
0 10 20 30 40 50 600
00501
01502
02503
03504
Time (s)
Trac
king
per
form
ance
ActualDesired
Figure 2 Output tracking performance when a square wave is usedas referencedesired output
0 10 20 30 40 50 60
005
115
2
Time (s)
Slid
ing
surfa
ce
Sliding surface
minus05
minus1
minus15
minus2
Figure 3 Sliding manifold convergence profile in case of squarewave tracking
tolerable for the system actuators health Now from this casestudy it is concluded that integral sliding mode approach isan interesting candidate for this class
In this case study once again efforts are focused on thetracking of a sinusoidal signal which is defined as 119903
119889
(119905) =
sin(119905) in the presence of disturbances Like the previous casestudy the initial conditions of the system were set to 119909
1
(0) =
04 1199092
(0) = 1199093
(0) = 1199094
(0) = 0 In addition the gains of theproposed controller presented in (50) are chosen accordingto Table 3
The output tracking performance of the proposed controlinput when a sinusoidal signal is considered as desiredreference output is shown in Figure 6 It can be clearlyseen that the performance is excellent in this scenarioThe corresponding sliding manifold profile is displayed inFigure 7 which confirms the establishment of sliding modesfrom the starting instant and consequently enhancement ofrobustnessThe controlled input signalrsquos profile is depicted inFigure 8 It is obvious from the figure that the control input
Table 3 Parametric values used in the sinusoid wave tracking
Constants 1198621
1198622
1198623
1198701
1198702
1198703
1198704
119870
Values 12 12 011 40298 25018 230 49 5
Time (s)0 10 20 30 40 50 60
05
101520
Con
trol i
nput
Control input
minus5
minus10
minus15
minus20
Figure 4 Control input in square wave reference tracking
Time (s)
Con
trol i
nput
Control input
0 10 20 30 40 50 60
0
05
1
15
minus05
minus1
minus15
Figure 5 Zoom profile of the control input depicted in Figure 4
Time (s)
ActualDesired
0 5 10 15 20 25
Trac
king
per
form
ance
0
05
1
15
minus05
minus1
minus15
Figure 6 Output tracking performance when a sinusoidal wave isused as referencedesired output
evolves with suppressed chattering phenomenonwhich onceagain makes this design strategy a good candidate for theclass of these underactuated systems
44 Implementation Results The control technique proposedin this paper is implemented on the actual apparatus using
Mathematical Problems in Engineering 9
Time (s)
Sliding surface
0 5 10 15 20 25
0
05
1
Slid
ing
surfa
ce(v
aria
ble r
efer
ence
sign
al)
minus05
minus1
Figure 7 Sliding manifold convergence profile in case of sinusoidalwave tracking
Time (s)0 5 10 15 20 25
0
5
10
Control effort
minus5
minus10
Con
trol i
nput
ldquordquo
u
Figure 8 Control input in sinusoidal wave reference tracking
the MATLAB environment The detailed discussions arepresented below
441 Experimental Setup Description The experiment setupis equipped by GoogolTech GBB1004 with an electroniccontrol boxThe beam length is 40 cm alongwithmass of ballthat is 28 g and an intelligent IPM100 servo driver which isused formoving the ball on the beamThe experimental setupis shown in Figure 9
The input given to apparatus is the voltage Vin(119905) and theoutput is the position of themotor 120579(119905) which in otherwordsis an input for the positioning of the ball on the beam Thisapparatus uses potentiometer mounted within a slot insidethe beam to sense the position of the ball on the beam Themeasured position along the beam is fed to the AD converterof IPM100 motion drive
The power module used in GoogolTech requires 220Vand 10A input Note that the control accuracy of this manu-factured apparatus lies within the range of plusmn1mmThe typicalparameters values are listed in Table 1 The environmentused here includes Windows XP as an operating system andMATLAB 712Simulink 77 Furthermore the sampling timeused in forthcoming practical results was 2ms In the exper-imental processes the proposed controllers need velocitymeasurements which are in general not available One may
Figure 9 Experimental setup of the ball and beam equipped viaGoogolTech GBB1004
use different kind of velocity observersdifferentiator for thevelocity estimation [16] In order to make the implemen-tation easy and simple a derivative block of the Simulinkenvironment is used to provide the corresponding velocitiesmeasurements Now we are ready to discuss the results of thesystem
In this experiment the initial conditions were set to1199091
(0) = 028 1199092
(0) = 1199093
(0) = 1199094
(0) = 0 The referencesignal which is needed to be tracked is being defined in (51)In Figures 10 and 11 the tracking performance is shown Theresults reveal that the actual signal 119909
1
(119905) is pretty close tothe desired signal 119903
119889
(119905) with a steady state error which isapproximatelyplusmn0001mThe existence of this error is becauseof the apparatus
The observations of these tracking results make it clearthat the practically implemented results have very closeresemblance with the simulation result presented in Figure 2The error convergence depends on the initial conditions ofthe ball on the beam If the ball is placed very close to thedesired reference value then it will take little time to reachthe desired position On the other hand the convergence tothe desired position will take considerable time if the initialcondition is chosen far away from the desired values Thisphenomenon of convergence is according to the equipmentdesign and structure
The sliding manifold convergence and the control inputare shown in Figures 12 and 13 respectively The controlinput and the sliding manifolds show some deviations inthe first second This deviation occurs because the ball onthe beam being placed anywhere on the beam is firstmoved to one side of the beam and then ball moved to thedesired position The zoomed profile of the control inputbeing displayed in Figure 14 shows high frequency vibration(chattering) of magnitude plusmn007 This makes the proposedcontrol design algorithm an appealing candidate for this classof nonlinear systems The gains of the controller being usedin this experiment are displayed in Table 4
5 Conclusion
The control of underactuated systems because of their lessnumber of actuators than the degree of freedom is an
10 Mathematical Problems in Engineering
Table 4 Parametric values used in implementation
Constants 1198621
1198622
1198623
1198701
1198702
1198703
1198704
119870
Values 8 5 1 3 15 3 1 4
0
01
02
03
04
Trac
king
per
form
ance
0 5 10 15 20 25Time (s)
DesiredActual
16 18 20 22 240219
022022102220223
Figure 10 Output tracking performance when 119903119889
= 22 cm is set asreferencedesired output
Time (s)
DesiredActual
0
01
02
03
04
Trac
king
per
form
ance
0 10 20 30 40 50 6044 46 48 50 520196
019802
02020204
Figure 11 Output tracking performance when a square wave is usedas referencedesired output
0 5 10 15 20 25
05
101520
Time (s)
Slid
ing
surfa
ce
Sliding surface
minus5
minus10
minus15
minus20
Figure 12 Sliding surface of practical system
interesting objective among the researchers In this work anintegral sliding mode control approach due to its robustnessfrom the very beginning of the process is employed forthe control design of this class The design of the integralmanifold relied upon a transformed form The benefit of thetransformed form is that itmakes the design strategy easy and
0 5 10 15 20 25
0
2
4
6
Time (s)
Inpu
t per
form
ance
Control input
minus2
minus4
minus6
Figure 13 Control input for reference tracking
Inpu
t per
form
ance
Control input
0 5 10 15 20 25
0
05
1
Time (s)
minus05
minus1
Figure 14 Zoom profile of the control input depicted in Figure 13
simple The stability analysis and experimental results of theproposed control laws are presented which convey the goodfeatures and demand the proposed approachwhen the systemoperates under uncertainties
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Mahjoub F Mnif and N Derbel ldquoSet point stabilization ofa 2DOF underactuated manipulatorrdquo Journal of Computers vol6 no 2 pp 368ndash376 2011
[2] F Mnif ldquoVSS control for a class of underactuated Mechanicalsystemsrdquo International Journal of Computational Cognition vol3 no 2 pp 14ndash18 2005
[3] J Hauser S Sastry and P Kokotovic ldquoNonlinear control viaapproximate inputndashoutput linearization the ball and beamexamplerdquo IEEE Transactions on Automatic Control vol 37 no3 pp 392ndash398 1992
[4] M Jankovic D Fontanine and P V Kokotovic ldquoTORAexample cascade and passitivity-based control designsrdquo IEEETransactions on Control Systems Technology vol 6 pp 4347ndash4351 1996
Mathematical Problems in Engineering 11
[5] T Sugie and K Fujimoto ldquoControl of inverted pendulumsystems based on approximate linearization design and exper-imentrdquo in Proceedings of the 33rd IEEE Conference on Decisionand Control vol 2 pp 1647ndash1648 December 1994
[6] R Xu and U Ozguner ldquoSliding mode control of a class ofunderactuated systemsrdquo Automatica vol 44 no 1 pp 233ndash2412008
[7] M W Spong ldquoThe swing-up control problem for the acrobatrdquoIEEE Transactions on Systems Technology vol 15 no 1 pp 49ndash55 1995
[8] R W Brockett ldquoAsymptotic stability feedback and stabiliza-tionrdquo in Differential Geometric Control Theory pp 181ndash191Birkhauser Boston Mass USA 1983
[9] C J Tomlin and S S Sastry ldquoSwitching through singularitiesrdquoSystems and Control Letters vol 35 no 3 pp 145ndash154 1998
[10] W-H Chen and D J Ballance ldquoOn a switching control schemefor nonlinear systems with ill-defined relative degreerdquo Systemsamp Control Letters vol 47 no 2 pp 159ndash166 2002
[11] F Zhang and B Fernndez-Rodriguez ldquoFeedback linearizationcontrol of systems with singularitiesrdquo in Proceedings of the6th International Conference on Complex Systems (ICCS rsquo06)Boston Mass USA June 2006
[12] D Seto and J Baillieul ldquoControl problems in super-articulatedmechanical systemsrdquo IEEE Transactions on Automatic Controlvol 39 no 12 pp 2442ndash2453 1994
[13] M W Spong Energy Based Control of a Class of UnderactuatedMechanical Systems IFACWorld Congress 1996
[14] V I Utkin Sliding Mode Control in Electromechanical SystemsTaylor amp Francis 1999
[15] M Rubagotti A Estrada F Castanos A Ferrara and LFridman ldquoIntegral sliding mode control for nonlinear systemswith matched and unmatched perturbationsrdquo IEEE Transactionon Automatic Control vol 56 no 11 pp 2699ndash2704 2011
[16] Q Khan A I Bhatti S Iqbal and M Iqbal ldquoDynamicintegral sliding mode for MIMO uncertain nonlinear systemsrdquoInternational Journal of Control Automation and Systems vol9 no 1 pp 151ndash160 2011
[17] R Olfati-Saber ldquoNormal forms for underactuated mechanicalsystems with symmetryrdquo IEEE Transactions on Automatic Con-trol vol 47 no 2 pp 305ndash308 2002
[18] R Lozano I Fantoni and D J Block ldquoStabilization of theinverted pendulum around its homoclinic orbitrdquo Systems andControl Letters vol 40 no 3 pp 197ndash204 2000
[19] E Altug J P Ostrowski and R Mahony ldquoControl of aquadrotor helicopter using visual feedbackrdquo in Proceedings ofthe IEEE International Conference on Robotics and Automation(ICRA rsquo02) vol 1 pp 72ndash77 Washington DC USA 2002
[20] A Isidori Nonlinear Control Systems Communications andControl Engineering Series Springer Berlin Germany 3rdedition 1995
[21] H Sira-Ramirez ldquoOn the dynamical sliding mode control ofnonlinear systemsrdquo International Journal of Control vol 57 no5 pp 1039ndash1061 1993
[22] Q Khan A I Bhatti and A Ferrara ldquoDynamic sliding modecontrol design based on an integral manifold for nonlinearuncertain systemsrdquo Journal of Nonlinear Dynamics vol 2014Article ID 489364 10 pages 2014
[23] C Edwards and S K Spurgeon Sliding Modes Control Theoryand Applications Taylor amp Francis London UK 1998
[24] A M Khan A I Bhatti S U Din and Q Khan ldquoStaticamp dynamic sliding mode control of ball and beam systemrdquo
in Proceedings of the 9th International Bhurban Conferenceon Applied Sciences and Technology (IBCAST rsquo12) pp 32ndash36Islamabad Pakistan January 2012
[25] N B Almutairi and M Zribi ldquoOn the sliding mode control ofa Ball on a Beam systemrdquo Nonlinear Dynamics vol 59 no 1-2pp 222ndash239 2010
Figure 7 Sliding manifold convergence profile in case of sinusoidalwave tracking
Time (s)0 5 10 15 20 25
0
5
10
Control effort
minus5
minus10
Con
trol i
nput
ldquordquo
u
Figure 8 Control input in sinusoidal wave reference tracking
the MATLAB environment The detailed discussions arepresented below
441 Experimental Setup Description The experiment setupis equipped by GoogolTech GBB1004 with an electroniccontrol boxThe beam length is 40 cm alongwithmass of ballthat is 28 g and an intelligent IPM100 servo driver which isused formoving the ball on the beamThe experimental setupis shown in Figure 9
The input given to apparatus is the voltage Vin(119905) and theoutput is the position of themotor 120579(119905) which in otherwordsis an input for the positioning of the ball on the beam Thisapparatus uses potentiometer mounted within a slot insidethe beam to sense the position of the ball on the beam Themeasured position along the beam is fed to the AD converterof IPM100 motion drive
The power module used in GoogolTech requires 220Vand 10A input Note that the control accuracy of this manu-factured apparatus lies within the range of plusmn1mmThe typicalparameters values are listed in Table 1 The environmentused here includes Windows XP as an operating system andMATLAB 712Simulink 77 Furthermore the sampling timeused in forthcoming practical results was 2ms In the exper-imental processes the proposed controllers need velocitymeasurements which are in general not available One may
Figure 9 Experimental setup of the ball and beam equipped viaGoogolTech GBB1004
use different kind of velocity observersdifferentiator for thevelocity estimation [16] In order to make the implemen-tation easy and simple a derivative block of the Simulinkenvironment is used to provide the corresponding velocitiesmeasurements Now we are ready to discuss the results of thesystem
In this experiment the initial conditions were set to1199091
(0) = 028 1199092
(0) = 1199093
(0) = 1199094
(0) = 0 The referencesignal which is needed to be tracked is being defined in (51)In Figures 10 and 11 the tracking performance is shown Theresults reveal that the actual signal 119909
1
(119905) is pretty close tothe desired signal 119903
119889
(119905) with a steady state error which isapproximatelyplusmn0001mThe existence of this error is becauseof the apparatus
The observations of these tracking results make it clearthat the practically implemented results have very closeresemblance with the simulation result presented in Figure 2The error convergence depends on the initial conditions ofthe ball on the beam If the ball is placed very close to thedesired reference value then it will take little time to reachthe desired position On the other hand the convergence tothe desired position will take considerable time if the initialcondition is chosen far away from the desired values Thisphenomenon of convergence is according to the equipmentdesign and structure
The sliding manifold convergence and the control inputare shown in Figures 12 and 13 respectively The controlinput and the sliding manifolds show some deviations inthe first second This deviation occurs because the ball onthe beam being placed anywhere on the beam is firstmoved to one side of the beam and then ball moved to thedesired position The zoomed profile of the control inputbeing displayed in Figure 14 shows high frequency vibration(chattering) of magnitude plusmn007 This makes the proposedcontrol design algorithm an appealing candidate for this classof nonlinear systems The gains of the controller being usedin this experiment are displayed in Table 4
5 Conclusion
The control of underactuated systems because of their lessnumber of actuators than the degree of freedom is an
10 Mathematical Problems in Engineering
Table 4 Parametric values used in implementation
Constants 1198621
1198622
1198623
1198701
1198702
1198703
1198704
119870
Values 8 5 1 3 15 3 1 4
0
01
02
03
04
Trac
king
per
form
ance
0 5 10 15 20 25Time (s)
DesiredActual
16 18 20 22 240219
022022102220223
Figure 10 Output tracking performance when 119903119889
= 22 cm is set asreferencedesired output
Time (s)
DesiredActual
0
01
02
03
04
Trac
king
per
form
ance
0 10 20 30 40 50 6044 46 48 50 520196
019802
02020204
Figure 11 Output tracking performance when a square wave is usedas referencedesired output
0 5 10 15 20 25
05
101520
Time (s)
Slid
ing
surfa
ce
Sliding surface
minus5
minus10
minus15
minus20
Figure 12 Sliding surface of practical system
interesting objective among the researchers In this work anintegral sliding mode control approach due to its robustnessfrom the very beginning of the process is employed forthe control design of this class The design of the integralmanifold relied upon a transformed form The benefit of thetransformed form is that itmakes the design strategy easy and
0 5 10 15 20 25
0
2
4
6
Time (s)
Inpu
t per
form
ance
Control input
minus2
minus4
minus6
Figure 13 Control input for reference tracking
Inpu
t per
form
ance
Control input
0 5 10 15 20 25
0
05
1
Time (s)
minus05
minus1
Figure 14 Zoom profile of the control input depicted in Figure 13
simple The stability analysis and experimental results of theproposed control laws are presented which convey the goodfeatures and demand the proposed approachwhen the systemoperates under uncertainties
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Mahjoub F Mnif and N Derbel ldquoSet point stabilization ofa 2DOF underactuated manipulatorrdquo Journal of Computers vol6 no 2 pp 368ndash376 2011
[2] F Mnif ldquoVSS control for a class of underactuated Mechanicalsystemsrdquo International Journal of Computational Cognition vol3 no 2 pp 14ndash18 2005
[3] J Hauser S Sastry and P Kokotovic ldquoNonlinear control viaapproximate inputndashoutput linearization the ball and beamexamplerdquo IEEE Transactions on Automatic Control vol 37 no3 pp 392ndash398 1992
[4] M Jankovic D Fontanine and P V Kokotovic ldquoTORAexample cascade and passitivity-based control designsrdquo IEEETransactions on Control Systems Technology vol 6 pp 4347ndash4351 1996
Mathematical Problems in Engineering 11
[5] T Sugie and K Fujimoto ldquoControl of inverted pendulumsystems based on approximate linearization design and exper-imentrdquo in Proceedings of the 33rd IEEE Conference on Decisionand Control vol 2 pp 1647ndash1648 December 1994
[6] R Xu and U Ozguner ldquoSliding mode control of a class ofunderactuated systemsrdquo Automatica vol 44 no 1 pp 233ndash2412008
[7] M W Spong ldquoThe swing-up control problem for the acrobatrdquoIEEE Transactions on Systems Technology vol 15 no 1 pp 49ndash55 1995
[8] R W Brockett ldquoAsymptotic stability feedback and stabiliza-tionrdquo in Differential Geometric Control Theory pp 181ndash191Birkhauser Boston Mass USA 1983
[9] C J Tomlin and S S Sastry ldquoSwitching through singularitiesrdquoSystems and Control Letters vol 35 no 3 pp 145ndash154 1998
[10] W-H Chen and D J Ballance ldquoOn a switching control schemefor nonlinear systems with ill-defined relative degreerdquo Systemsamp Control Letters vol 47 no 2 pp 159ndash166 2002
[11] F Zhang and B Fernndez-Rodriguez ldquoFeedback linearizationcontrol of systems with singularitiesrdquo in Proceedings of the6th International Conference on Complex Systems (ICCS rsquo06)Boston Mass USA June 2006
[12] D Seto and J Baillieul ldquoControl problems in super-articulatedmechanical systemsrdquo IEEE Transactions on Automatic Controlvol 39 no 12 pp 2442ndash2453 1994
[13] M W Spong Energy Based Control of a Class of UnderactuatedMechanical Systems IFACWorld Congress 1996
[14] V I Utkin Sliding Mode Control in Electromechanical SystemsTaylor amp Francis 1999
[15] M Rubagotti A Estrada F Castanos A Ferrara and LFridman ldquoIntegral sliding mode control for nonlinear systemswith matched and unmatched perturbationsrdquo IEEE Transactionon Automatic Control vol 56 no 11 pp 2699ndash2704 2011
[16] Q Khan A I Bhatti S Iqbal and M Iqbal ldquoDynamicintegral sliding mode for MIMO uncertain nonlinear systemsrdquoInternational Journal of Control Automation and Systems vol9 no 1 pp 151ndash160 2011
[17] R Olfati-Saber ldquoNormal forms for underactuated mechanicalsystems with symmetryrdquo IEEE Transactions on Automatic Con-trol vol 47 no 2 pp 305ndash308 2002
[18] R Lozano I Fantoni and D J Block ldquoStabilization of theinverted pendulum around its homoclinic orbitrdquo Systems andControl Letters vol 40 no 3 pp 197ndash204 2000
[19] E Altug J P Ostrowski and R Mahony ldquoControl of aquadrotor helicopter using visual feedbackrdquo in Proceedings ofthe IEEE International Conference on Robotics and Automation(ICRA rsquo02) vol 1 pp 72ndash77 Washington DC USA 2002
[20] A Isidori Nonlinear Control Systems Communications andControl Engineering Series Springer Berlin Germany 3rdedition 1995
[21] H Sira-Ramirez ldquoOn the dynamical sliding mode control ofnonlinear systemsrdquo International Journal of Control vol 57 no5 pp 1039ndash1061 1993
[22] Q Khan A I Bhatti and A Ferrara ldquoDynamic sliding modecontrol design based on an integral manifold for nonlinearuncertain systemsrdquo Journal of Nonlinear Dynamics vol 2014Article ID 489364 10 pages 2014
[23] C Edwards and S K Spurgeon Sliding Modes Control Theoryand Applications Taylor amp Francis London UK 1998
[24] A M Khan A I Bhatti S U Din and Q Khan ldquoStaticamp dynamic sliding mode control of ball and beam systemrdquo
in Proceedings of the 9th International Bhurban Conferenceon Applied Sciences and Technology (IBCAST rsquo12) pp 32ndash36Islamabad Pakistan January 2012
[25] N B Almutairi and M Zribi ldquoOn the sliding mode control ofa Ball on a Beam systemrdquo Nonlinear Dynamics vol 59 no 1-2pp 222ndash239 2010
Figure 10 Output tracking performance when 119903119889
= 22 cm is set asreferencedesired output
Time (s)
DesiredActual
0
01
02
03
04
Trac
king
per
form
ance
0 10 20 30 40 50 6044 46 48 50 520196
019802
02020204
Figure 11 Output tracking performance when a square wave is usedas referencedesired output
0 5 10 15 20 25
05
101520
Time (s)
Slid
ing
surfa
ce
Sliding surface
minus5
minus10
minus15
minus20
Figure 12 Sliding surface of practical system
interesting objective among the researchers In this work anintegral sliding mode control approach due to its robustnessfrom the very beginning of the process is employed forthe control design of this class The design of the integralmanifold relied upon a transformed form The benefit of thetransformed form is that itmakes the design strategy easy and
0 5 10 15 20 25
0
2
4
6
Time (s)
Inpu
t per
form
ance
Control input
minus2
minus4
minus6
Figure 13 Control input for reference tracking
Inpu
t per
form
ance
Control input
0 5 10 15 20 25
0
05
1
Time (s)
minus05
minus1
Figure 14 Zoom profile of the control input depicted in Figure 13
simple The stability analysis and experimental results of theproposed control laws are presented which convey the goodfeatures and demand the proposed approachwhen the systemoperates under uncertainties
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Mahjoub F Mnif and N Derbel ldquoSet point stabilization ofa 2DOF underactuated manipulatorrdquo Journal of Computers vol6 no 2 pp 368ndash376 2011
[2] F Mnif ldquoVSS control for a class of underactuated Mechanicalsystemsrdquo International Journal of Computational Cognition vol3 no 2 pp 14ndash18 2005
[3] J Hauser S Sastry and P Kokotovic ldquoNonlinear control viaapproximate inputndashoutput linearization the ball and beamexamplerdquo IEEE Transactions on Automatic Control vol 37 no3 pp 392ndash398 1992
[4] M Jankovic D Fontanine and P V Kokotovic ldquoTORAexample cascade and passitivity-based control designsrdquo IEEETransactions on Control Systems Technology vol 6 pp 4347ndash4351 1996
Mathematical Problems in Engineering 11
[5] T Sugie and K Fujimoto ldquoControl of inverted pendulumsystems based on approximate linearization design and exper-imentrdquo in Proceedings of the 33rd IEEE Conference on Decisionand Control vol 2 pp 1647ndash1648 December 1994
[6] R Xu and U Ozguner ldquoSliding mode control of a class ofunderactuated systemsrdquo Automatica vol 44 no 1 pp 233ndash2412008
[7] M W Spong ldquoThe swing-up control problem for the acrobatrdquoIEEE Transactions on Systems Technology vol 15 no 1 pp 49ndash55 1995
[8] R W Brockett ldquoAsymptotic stability feedback and stabiliza-tionrdquo in Differential Geometric Control Theory pp 181ndash191Birkhauser Boston Mass USA 1983
[9] C J Tomlin and S S Sastry ldquoSwitching through singularitiesrdquoSystems and Control Letters vol 35 no 3 pp 145ndash154 1998
[10] W-H Chen and D J Ballance ldquoOn a switching control schemefor nonlinear systems with ill-defined relative degreerdquo Systemsamp Control Letters vol 47 no 2 pp 159ndash166 2002
[11] F Zhang and B Fernndez-Rodriguez ldquoFeedback linearizationcontrol of systems with singularitiesrdquo in Proceedings of the6th International Conference on Complex Systems (ICCS rsquo06)Boston Mass USA June 2006
[12] D Seto and J Baillieul ldquoControl problems in super-articulatedmechanical systemsrdquo IEEE Transactions on Automatic Controlvol 39 no 12 pp 2442ndash2453 1994
[13] M W Spong Energy Based Control of a Class of UnderactuatedMechanical Systems IFACWorld Congress 1996
[14] V I Utkin Sliding Mode Control in Electromechanical SystemsTaylor amp Francis 1999
[15] M Rubagotti A Estrada F Castanos A Ferrara and LFridman ldquoIntegral sliding mode control for nonlinear systemswith matched and unmatched perturbationsrdquo IEEE Transactionon Automatic Control vol 56 no 11 pp 2699ndash2704 2011
[16] Q Khan A I Bhatti S Iqbal and M Iqbal ldquoDynamicintegral sliding mode for MIMO uncertain nonlinear systemsrdquoInternational Journal of Control Automation and Systems vol9 no 1 pp 151ndash160 2011
[17] R Olfati-Saber ldquoNormal forms for underactuated mechanicalsystems with symmetryrdquo IEEE Transactions on Automatic Con-trol vol 47 no 2 pp 305ndash308 2002
[18] R Lozano I Fantoni and D J Block ldquoStabilization of theinverted pendulum around its homoclinic orbitrdquo Systems andControl Letters vol 40 no 3 pp 197ndash204 2000
[19] E Altug J P Ostrowski and R Mahony ldquoControl of aquadrotor helicopter using visual feedbackrdquo in Proceedings ofthe IEEE International Conference on Robotics and Automation(ICRA rsquo02) vol 1 pp 72ndash77 Washington DC USA 2002
[20] A Isidori Nonlinear Control Systems Communications andControl Engineering Series Springer Berlin Germany 3rdedition 1995
[21] H Sira-Ramirez ldquoOn the dynamical sliding mode control ofnonlinear systemsrdquo International Journal of Control vol 57 no5 pp 1039ndash1061 1993
[22] Q Khan A I Bhatti and A Ferrara ldquoDynamic sliding modecontrol design based on an integral manifold for nonlinearuncertain systemsrdquo Journal of Nonlinear Dynamics vol 2014Article ID 489364 10 pages 2014
[23] C Edwards and S K Spurgeon Sliding Modes Control Theoryand Applications Taylor amp Francis London UK 1998
[24] A M Khan A I Bhatti S U Din and Q Khan ldquoStaticamp dynamic sliding mode control of ball and beam systemrdquo
in Proceedings of the 9th International Bhurban Conferenceon Applied Sciences and Technology (IBCAST rsquo12) pp 32ndash36Islamabad Pakistan January 2012
[25] N B Almutairi and M Zribi ldquoOn the sliding mode control ofa Ball on a Beam systemrdquo Nonlinear Dynamics vol 59 no 1-2pp 222ndash239 2010
[5] T Sugie and K Fujimoto ldquoControl of inverted pendulumsystems based on approximate linearization design and exper-imentrdquo in Proceedings of the 33rd IEEE Conference on Decisionand Control vol 2 pp 1647ndash1648 December 1994
[6] R Xu and U Ozguner ldquoSliding mode control of a class ofunderactuated systemsrdquo Automatica vol 44 no 1 pp 233ndash2412008
[7] M W Spong ldquoThe swing-up control problem for the acrobatrdquoIEEE Transactions on Systems Technology vol 15 no 1 pp 49ndash55 1995
[8] R W Brockett ldquoAsymptotic stability feedback and stabiliza-tionrdquo in Differential Geometric Control Theory pp 181ndash191Birkhauser Boston Mass USA 1983
[9] C J Tomlin and S S Sastry ldquoSwitching through singularitiesrdquoSystems and Control Letters vol 35 no 3 pp 145ndash154 1998
[10] W-H Chen and D J Ballance ldquoOn a switching control schemefor nonlinear systems with ill-defined relative degreerdquo Systemsamp Control Letters vol 47 no 2 pp 159ndash166 2002
[11] F Zhang and B Fernndez-Rodriguez ldquoFeedback linearizationcontrol of systems with singularitiesrdquo in Proceedings of the6th International Conference on Complex Systems (ICCS rsquo06)Boston Mass USA June 2006
[12] D Seto and J Baillieul ldquoControl problems in super-articulatedmechanical systemsrdquo IEEE Transactions on Automatic Controlvol 39 no 12 pp 2442ndash2453 1994
[13] M W Spong Energy Based Control of a Class of UnderactuatedMechanical Systems IFACWorld Congress 1996
[14] V I Utkin Sliding Mode Control in Electromechanical SystemsTaylor amp Francis 1999
[15] M Rubagotti A Estrada F Castanos A Ferrara and LFridman ldquoIntegral sliding mode control for nonlinear systemswith matched and unmatched perturbationsrdquo IEEE Transactionon Automatic Control vol 56 no 11 pp 2699ndash2704 2011
[16] Q Khan A I Bhatti S Iqbal and M Iqbal ldquoDynamicintegral sliding mode for MIMO uncertain nonlinear systemsrdquoInternational Journal of Control Automation and Systems vol9 no 1 pp 151ndash160 2011
[17] R Olfati-Saber ldquoNormal forms for underactuated mechanicalsystems with symmetryrdquo IEEE Transactions on Automatic Con-trol vol 47 no 2 pp 305ndash308 2002
[18] R Lozano I Fantoni and D J Block ldquoStabilization of theinverted pendulum around its homoclinic orbitrdquo Systems andControl Letters vol 40 no 3 pp 197ndash204 2000
[19] E Altug J P Ostrowski and R Mahony ldquoControl of aquadrotor helicopter using visual feedbackrdquo in Proceedings ofthe IEEE International Conference on Robotics and Automation(ICRA rsquo02) vol 1 pp 72ndash77 Washington DC USA 2002
[20] A Isidori Nonlinear Control Systems Communications andControl Engineering Series Springer Berlin Germany 3rdedition 1995
[21] H Sira-Ramirez ldquoOn the dynamical sliding mode control ofnonlinear systemsrdquo International Journal of Control vol 57 no5 pp 1039ndash1061 1993
[22] Q Khan A I Bhatti and A Ferrara ldquoDynamic sliding modecontrol design based on an integral manifold for nonlinearuncertain systemsrdquo Journal of Nonlinear Dynamics vol 2014Article ID 489364 10 pages 2014
[23] C Edwards and S K Spurgeon Sliding Modes Control Theoryand Applications Taylor amp Francis London UK 1998
[24] A M Khan A I Bhatti S U Din and Q Khan ldquoStaticamp dynamic sliding mode control of ball and beam systemrdquo
in Proceedings of the 9th International Bhurban Conferenceon Applied Sciences and Technology (IBCAST rsquo12) pp 32ndash36Islamabad Pakistan January 2012
[25] N B Almutairi and M Zribi ldquoOn the sliding mode control ofa Ball on a Beam systemrdquo Nonlinear Dynamics vol 59 no 1-2pp 222ndash239 2010
