Bond Graph Modelling and Nonlinear Control of an Inverted Pendulum

⎡ 0 ⎤⎢ ⎥⎢0⎥⎢Cg( x)= − ⎥ , (7)⎢ E( x)⎥⎢ B cos x ⎥2⎢ ⎥⎣ E( x)⎦where we denoted:E2( x) B x − AC=22A = mr + J2 cos ,, B = mrl ,2C = ml , D = mgl .As the output variable, the angular position α ( t)isconsidered:= h( x)= α() t = x () t(8)y24. THE FEEDBACK LINEARIZING METHODWe consider the following form of the state spaceequations in the case of nonlinear system:x&= f ( x)+ ∑ gi( x)uyj= h ( x)jmi=1j = 1... miwhere f ( x),g1(x),g2( x),...,gm( x)are smooth vectorfields (see Isidori (1995), Fossard and Normand-Cyrot(1993)).The problem of exact linearization via feedback anddiffeomorphism consists in transforming a nonlinearsystem (9) into a linear one using a state feedback and acoordinate transformation of the systems state.Let’s introduce now the Lie derivative of the functionh( x) : Rn → R along the vector fieldf ( x)= [ f1(x),...,f ( x)]:= ∂in∂= ∑n h(x)Lfh( x)fi( x)(10)i 1 xDefinition. A multivariable nonlinear system of the form0(9) has a relative degree r ,..., r } at a point x if:{ 1 mkLg jLfhi( x)= 0(11)for all 1 ≤ j ≤ m , for all 1 ≤ i ≤ m , for all k ≤ r i−1, and0for x in a neighbourhood of x , the m × m matrix:r1−1⎡ LgL h (x)1 f 1⎢ r2−1=⎢ LgL h (x)1 f 2A(x)⎢ .⎢r −1⎢⎣LgLfhm(x)1is nonsingular at........LLLr1−1g m fr2−1g mLfm 10x = x .g mL h ⎤1(x)⎥h2(x)⎥. ⎥⎥rm−Lfhm(x)⎥⎦(9)(12)Remark: Let be a SISO nonlinear system of the form (9),0which has the relative degree r at a point x .The state feedback:r[ − L h(x + v]1u =f)r− 1L L h(x)gftransforms the nonlinear system into a system, whoseinput-output behaviour is the same with a linear systemhaving the transfer function:1H () s = .rsTheorem. Let be the nonlinear system of the form (9).Suppose the matrix g ( x0 ) has rank m. Then the statespace exact linearization problem is solvable if and onlyif: for each 0 ≤ i ≤ n −1, the distribution G i has constant0dimension near x ; the distribution Gn−1has dimensionn ; for each 0 ≤ i ≤ n − 2 , the distribution Giisinvolutive.For system (5) we have:LL0f2fhh0( x) = 0; L L h( x)( x)B=xg2 24fcos x2= 0; LE1 B cos x2L h =g fE( x)A( Bx3x4+ D)sin x2( x) −(13)Thus, we see that the system has relative degree r = 2 . Inthis situation, the state feedback:1 2u = ( −Lh(x)v)1 f+(14)L L h(x)gftransforms the system (5) into a system whose inputoutputbehavior is identical to that a linear system having1a transfer function: H ( s)= .2sImposing on the linear system an additional feedback ofthe form:v = c (ref− c x&(15)0α − x2)12then, the obtained system has a linear input-outputbehavior, described by the following transfer functionH ( s)c0= (16)2s + c1s+ c0In relation (15)deflection.αrefis the imposed reference of the armThe coefficients c 0, c1in (16) are determined using apole placement procedure. The values of thesecoefficients are chosen such that the behaviour of theentire closed loop system has a desired shape (Bobasu etal. (1998a)).

5. SIMULATION RESULTSIn order to test the behaviour and the performance of theproposed nonlinear control strategy, extensive simulationswere performed using the Quanser Inverted Pendulumexperiment.The values of the inverted pendulum parameters are:m = 0.128kg;r = 0.158m;l = 0.35 m;22J = 0.0015kgm ; g = 9.81m/sThe design parameters are set to:c0= 100 , c1= 14Fig. 5 presents the time evolution of the pendulum armdeflection α (in radians) for α ref= 0 .1.210.80.60.40.20[rad]Time [s]-0.20 0.5 1 1.5 2 2.5 3Fig. 5. Time evolution of pendulum arm deflectionIt can be seen that the behaviour of the controlled systemis quite good, and the inverted pendulum is stabilized.In Fig. 6 the time profile of the angular velocity α& isshown.1[rad/s]0Time [s]-1-2-3-4-50 0.5 1 1.5 2 2.5 3Fig. 6. Time evolution of pendulum angular velocity α&6. CONCLUSIONSIn this paper our interest was pointed towards the BondGraph modelling and the design of feedback linearizingtechnique for an Inverted Pendulum system.The Bond Graph model of the system was built up writingfirst the word Bond Graph containing words instead ofstandard symbols for the main components and bonds forpower and signal exchange, and then replacing words bystandards elements which contain precise mathematical orfunctional relations. The system was decomposed intothree subsystems that were modelled separately. Byjoining together these three models, we obtained thecomplete Bond Graph model of the Quanser RotaryInverted Pendulum system. The model was created andsimulated using 20sim modelling and simulationenvironment.The nonlinear control method based on the feedbacklinearizing technique provides an alternative solution toexisting classical linear methods. The implementation ofthe method requires a complete knowledge of the statevariables (or the use of a state observer).REFERENCESKarnopp, D., and Rosenberg, R. (1974). SystemDynamics: A Unified Approach. John Wiley, NewYork.Thoma, J. (1975). Introduction to Bond Graphs and TheirApplications. Perg. Press, Oxford.Dauphin-Tanguy, G. (2000). Les Bond Graphs. HermesSci., Paris.Păstrăvanu, O., Ibănescu, R. (2001). Bond-graphlanguage in modeling and simulation of physicaltechnicalsystems. Gh. Asachi, Iasi.Damic, V., Montgomery, J. (2002). Mechatronics byBond Graphs. Springer, Germany.Gawthrop, P.J., and Bevan, G.P. (2007). A TutorialIntroduction for Control Engineers. IEEE ControlSystems, 27, 24-45.Isidori, A. (1995). Nonlinear control systems. Springer-Verlag, Berlin.Bobasu, E., Petre, E., Popescu, D. (1998). On nonlinearcontrol for electric induction motors. In ProcessControl'98, (1), 44-47. Pardubice, Czech Republic.Bobasu, E., Ionete, C., Selisteanu, D. (1998). Onnonlinear control for electric d.c. motors. In ProcessControl'98, (1), 42-46. Pardubice, Slovak Republic.Bodson, M., Chiasson, J., Novotnac R. (1994). High-Performance Induction Motor Control Via Input-Output Linearization. IEEE Contr. Systems, 14 (4),25-33.Fossard, A.J., Normand-Cyrot, D. (1993). Systemesnonlineaires. Masson, Paris.Raumer, T., Dion, J. M., Dugard, L., Thomas, J.L. (1994).Applied Nonlinear Control of an Induction MotorUsing Digital Signal Processing. IEEE Trans. Contr.Systems Technology, 2 (4), 327- 335.