Finite-time stabilization of chaotic gyros based on a homogeneous supertwisting-like algorithm

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1 OP Conferene Series: Materials Siene Engineering PAPER OPEN ACCESS Finite-time stabilization of haoti gyros based on a homogeneous supertwisting-like algorithm To ite this artile: Pitha Khamsuwan et al 8 OP Conf. Ser.: Mater. Si. Eng View the artile online for updates enhanements. This ontent was downloaded from P address on 7/6/8 at 6:

2 8th TSME-nternational Conferene on Mehanial Engineering (TSME-CoME 7) OP Publishing OP Conf. Series: Materials Siene Engineering (8) 8 doi:.88/ x/97//8 Finite-time stabilization of haoti gyros based on a homogeneous supertwisting-like algorithm Pitha Khamsuwan *, Teerawat Sangpet, Suwat Kuntanapreeda Department of Mehanial Aerospae Engineering, King Mongkut s University of Tehnology North Bangkok, Bangkok 8, Thail * Corresponding Author: pitha7@gmail.om Abstrat. This paper presents a finite-time stabilization sheme for nonlinear haoti gyros. The sheme utilizes a supertwisting-like ontinuous ontrol algorithm for the systems of dimension more than one with a Lipshitz disturbane. The algorithm yields finite-time onvergene similar to that produes by disontinuous sliding mode ontrol algorithms. To design the ontroller, the nonlinearities in the gyro are treated as a disturbane in the system. Thanks to the dissipativeness of haoti systems, the nonlinearities also possess the Lipshitz property. Numerial results are provided to illustrate the effetiveness of the sheme.. ntrodution Gyros are important omponents in navigational, aeronautial spae systems. However, gyros an ehibit haoti behavior, whih will adversely affet the performane of the systems []. Reently, many authors have disussed the problem of ontrolling the haoti gyros [ ]. Chaoti ontrol is hallenging has been etensively studied in the past deades. The study an be lassified into haos stabilization haos synhronization. Chaos stabilization attempts to eliminate the haoti behavior of systems while haos synhronization is to ontrol a haoti system so that it follows another haoti system. Sine Ott et al. [5] introdued the idea of ontrolling haoti systems, a number of approahes have been proposed to ontrol the haoti systems [6 8]. Finite-time ontrol has reently gained the attention of the haos ontrol ommunity [9 ]. The main attrations of the finite-time ontrol inlude faster onvergene, robustness, disturbane rejetion. Sliding mode ontrol (SMC) has attrated interest in nonlinear ontrol beause of its numerous attrative properties. The properties inlude robustness against unertainties eternal disturbanes, a fast dynami response a simple design. However, hattering phenomenon is one of the major obstales for the implementation of SMC in a wide range of appliations. A number of solutions have been proposed to eliminate or redue hattering, inluding the boundary layer tehnique [] highorder SMC (HOSMC) methods [, ]. The supertwisting algorithm is one of effetive HOSMC methods. HOSMC has reently attrated interest in haos ontrol [5 7]. n this paper, we eamine the problems of stabilization of nonlinear haoti gyros based on the HOSMC. To ahieve this goal, a reently established homogeneous supertwisting-like modifiation of twisting algorithm [8] is employed. The algorithm is highly effetive for the stabilization of both system state its derivative in a finite time. Content from this work may be used under the terms of the Creative Commons Attribution. liene. Any further distribution of this work must maintain attribution to the author(s) the title of the work, journal itation DO. Published under liene by OP Publishing Ltd

3 8th TSME-nternational Conferene on Mehanial Engineering (TSME-CoME 7) OP Publishing OP Conf. Series: Materials Siene Engineering (8) 8 doi:.88/ x/97//8 The rest of the paper is organized as follows. Preliminaries are presented in Setion. Control design is desribed in Setion. n Setion, numerial results are given to illustrate the effetiveness of the ontrol sheme. Finally, onlusions are drawn in the last setion.. Preliminaries.. Chaoti gyro Consider the symmetri gyro mounted on a vibrating base as shown in figure. Here, (X, Y, Z) (, y, z) are the referene body frames, respetively. The dynamis of the gyro is desribed by z--z Euler angles: (preession), (nutation), (spin) body rates:,,. The rotational sequene of the Euler angles, starting by assuming (X, Y, Z) (, y, z) oinide, is listed as [9]:. Rotation about Z ais through angle to produe (, y, z ) frame.. Rotation about ais through angle to produe (, y, z ) frame.. Rotation about z ais through angle to produe (, y, z) frame. n the figure, is the gravity fore, l is the distane of the entre of gravity (CG) of the gyro from the origin, m M g l t is the eternal torque applied to the gyro The kineti energy of the gyro an be epressed as: y is the motion of the vibrating base. T () y z where are the polar equatorial moments of inertia of the gyro, respetively. Based on the z--z Euler angle transformation, the body rates an be epressed in terms of Euler rates,, as [9]: z Figure. A symmetri gyrosope mounted on a vibrating base.

4 8th TSME-nternational Conferene on Mehanial Engineering (TSME-CoME 7) OP Publishing OP Conf. Series: Materials Siene Engineering (8) 8 doi:.88/ x/97//8 os, () os y, () z os. () By substituting equations (), () () in equation (), the kineti energy beomes The potential energy of the gyro an be epresses as: T os. (5) V M g l l t os, Thus, the Lagrangian of the system an be epressed as []: os M l l os L T V g t. (7) Sine do not appear epliitly in the Lagrangian, they are yli oordinates. Thus, the momentum integrals: L P os os, (8) L P os, (9) are onserved. Then, where written as: os os z, () os, () are onstants. From equations () (), the epressions for The Routhian of the system is epressed as []: Substitute equations (7), () () in equation () yields R (6) an be os, () os os. () R L. () os M l l tos. (5) g By assuming the torque m omprises the linear-plus-ubi dissipative torque d D D the ontrol torque, the equation of motion an be epressed in the oordinate of as: d dt R R D D where D D are onstants. Substituting (5) in (6) yields (6)

5 8th TSME-nternational Conferene on Mehanial Engineering (TSME-CoME 7) OP Publishing OP Conf. Series: Materials Siene Engineering (8) 8 doi:.88/ x/97//8 os os D D (7) M g ( l l t)( ) Note that, from equations () (), = is true for all. By ug os = = D when =. Sine, equation (7) beomes D By epresg equation (8) in a state equation form as in [], we obtain: os are onstants, the equality M gl M gl ( t)( ). (8) ( f t)( ) u where, are the state variables, is the ontrol input, / z,, D, M g l, f M gl are positive onstants. When f 6,, the gyro with zero input beomes haoti []. An eample of the haoti behaviour for the initial onditions ( (), ()) (, ) is shown in figure. Note that all trajetories are bounded e haoti systems are dissipative. D.5,,,.5 f Homogeneous supertwisting-like algorithm Consider the dynamial system: where that t u t u are the state variables, u satisfies the Lipshitz ondition with onstant L. is the ontrol input t (9), () is the disturbane. t is assumed (a) (b) Figure. Chaoti behaviour: (a) Phase trajetory (b) Time history

6 8th TSME-nternational Conferene on Mehanial Engineering (TSME-CoME 7) OP Publishing OP Conf. Series: Materials Siene Engineering (8) 8 doi:.88/ x/97//8 A homogeneous ontinuous supertwisting-like algorithm for systems of dimension more than one has reently been introdued in [8]. The algorithm generalizes the well-known supertwisting algorithm to higher dimensional systems. t yields finite-time onvergene similar to that produed by disontinuous sliding mode ontrol algorithms. Based on the homogeneous ontinuous supertwisting-like algorithm [8], the ontrol law for systems of dimension two with a Lipshitz disturbane is given as: t t t u sds sds t t t sgn sgn sgn t sgn sds () t t t where,, are ontrol parameters. The ontrol law () yields finite-time onvergene of both state variables to the origin if the following onditions hold for the ontrol parameters [8]: L L L. (). Control design The ontrol objetive is to suppress the haos governed by equation (9) by driving the state variables to the origin in finite time. To ahieve the objetive, we define the ontrol law as: where u st adopts the supertwisting-like ontrol law (). Substituting the ontrol law () into equation (9) yields u st u. () os The system () an be written in the form () by defining: t ( f t)( ) u st. () os f t. (5) n order to adopt the ontrol law (), must possess the Lipshitz property. A suffiient ondition for a funtion to be Lipshitz is its derivative being bounded. Thus, we eamine the derivative of instead. By taking the derivative of with respet to time, it yields t t t os f f t os os tos os os os tos os os f f os t os t (6) 5

7 8th TSME-nternational Conferene on Mehanial Engineering (TSME-CoME 7) OP Publishing OP Conf. Series: Materials Siene Engineering (8) 8 doi:.88/ x/97//8 Sine haoti systems are dissipative, unbounded only if os os os our only at t Therefore,. However, e os lim os os lim has no gularity. Thus, we an onlude that t is Lipshitz with the onstant L. are bounded. t is obvious that t an beome have a gularity. n this ase, the gularity an t.75, is bounded (i.e., t L, L ma ).. Numerial simulations n this setion, numerial simulations are utilized to illustrate the effetiveness of the supertwisting-like algorithm-based ontrol sheme to suppress the haos of haoti gyros. The parameters of the gyros are set as follows:. The fourth-order Runge-Kutta method with a time step of. seond was used in all simulations. First, the Lipshitz onstant L is estimated. From equation (6), we obtain ma where t X,, os ma f ma ma ma X R, os ma R ma ma.5,.5 os os os os tos os os os ma os ma os os f X f are upper bounds of f 5.5 ma os f f os f ma os t f tos f ma ma f f os os os R( t) os t (7), respetively. From figure (b), it an be found that X. By eamining the value of R (t ) over, ) (see figure ), it results in. ma onstant L is hosen. ma t R. From equation (7), it yields 57 (. Here, the Lipshitz 6

8 8th TSME-nternational Conferene on Mehanial Engineering (TSME-CoME 7) OP Publishing OP Conf. Series: Materials Siene Engineering (8) 8 doi:.88/ x/97// R Net, the parameters,, Figure. Plot of R versus of u st are seleted. Here, taken satisfying the ondition () were hosen suh that the desired transient response of the ontrol system is ahieved. Simulation results are depited in figure. The initial ondition of the system was set as ( (), ()) (, ). The ontroller was ativated at time equal to 5 se. The result shows that the state variables onverged to the origin in less than.5 se after the ontroller was ativated were u Figure. State variables ontrol input of the ontrolled haoti gyro. 7

9 8th TSME-nternational Conferene on Mehanial Engineering (TSME-CoME 7) OP Publishing OP Conf. Series: Materials Siene Engineering (8) 8 doi:.88/ x/97//8 n addition, a omparison with a sliding mode ontroller based on the twisting algorithm (TA) [, 5] was made. The twisting algorithm ontrol law was given as: t k sign t u( t) ksign (8) where were ontrol parameters. The parameters, were hosen suh that the ontrol system had a similar response to the supertwisting-like algorithm-based ontrol system (i.e., onverged to the origin within about.5 se). A simulation result is depited in figure. 5. From figures 5, both ontrollers () (8) yield similar responses of the state variables, whereas the ontroller () provides markedly smaller hattering ontrol signal. This illustrates the benefit of the supertwisting-like algorithm. k, k, k k u Figure 5. State variables ontrol input of the ontrolled haoti gyro ug TA. 5. Conlusions n this paper, a finite time ontrol sheme based on a reently developed homogeneous supertwistinglike ontinuous ontrol algorithm is proposed for stabilization of haoti gyros. Numerial simulations have illustrated that the proposed ontrol sheme an be effetively applied to a haos stabilization problem. By omparing with the twisting algorithm based ontrol sheme, both ontrol shemes provide similar ontrol responses, whereas the proposed sheme results in smaller hattering ontrol signal. Referenes [] Chen H K J. of Sound Vibration 55() pp 79 7 [] Yan J-J, Hung M-L Liao T-L 6 J. of Sound Vibration 98(-) p 98 6 [] Wang C-C Yau H-T Nonlinear Dynamis 66() pp 5 65 [] Yang C-C, Ou C-J Communiations in Nonlinear Siene Numerial Simulation 8() pp [5] Ott E, Grebogi C Yorke J A 99 Physial Review Letters 6(-) pp [6] Chen G Applied Mathematis Computation 7(7) pp [7] Chadl M Zelinka Computers Mathematis with Appliations 68() pp 7 [8] Khamsuwan P Kuntanapreeda S 6 J. of Computational Nonlinear Dynamis (5) p 5 8

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