Chaos Control via Non-singular Terminal Sliding Mode Controller in Auto Gauge Control System Wei-wei ZHANG, Jun-tao PAN and Hong-tao SHI

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1 7 International Conference on Computer, Electronics and Communication Enineerin (CECE 7) ISBN: Chaos Control via Non-sinular Terminal Slidin Mode Controller in Auto Gaue Control System Wei-wei ZHANG, Jun-tao PAN and Hon-tao SHI School of Electrical and Information Enineerin, North Minzu University, Yinchuan 75 P.R. China Keywords: Chaos, AGC, Terminal slidin mode control, Exponential reachin law. Abstract. In this paper, new idea is proposed that roll eccentricity disturbance on aue may cause chaos under certain system parameters durin rollin throuh establishin the eccentric model. A new non-sinular terminal slidin mode controller is proposed to control the chaotic system. The new controller based on exponential reachin law improves the converence rate and reduce the reachin time. As expected from the theoretical analysis, the desined controller actualizes stability control in the chaotic system with hih converence rate. Simulation results show the effectiveness of the method. Introduction In recent years, chaos and chaos control in enineerin has increasinly ained research interest since the pioneerin work of Ott. et al []. Studies have shown that AGC system was complex nonlinear system, roll eccentricity motion obeyed the law of sine wave and the roll motion equation has the form of Duffin equation []. Considerin the disturbance there is chaos in rolled piece thickness system when the AGC system is under certain condition. Now, several control methods have been successfully applied to chaotic systems, e.. the OGY(Ott-Greboi-Yorke) method [-4], the DFC (delay feedback control) method [5-6], the slidin mode control method [7] and fuzzy control method [8] etc. In this paper, a new terminal slidin mode controller is proposed. The exponential reachin law method is used to improve the converence rate. The new terminal slidin mode controller based on exponential reachin law is non-sinular and control the chaotic system reach equilibrium in finite time. Simulation results show that the method proposed in this paper can stabilize the chaotic system in hih rate. Mill Eccentric Model In the rollin process, elastic potential enery of the roll department is 4 U kx kx 4 () Where, k and k represent the coefficient associated with the elastic deformation. Comparin with k, k is a smaller coefficient. The resilience of the rollers system is du f kx kx dx () When the system maintains a certain roll ap, the roll ap may chane with the roll eccentricity. Eccentricity of the rotatin roll will cause the rollin force chane periodically. The chane of rollin force is non-sinusoidal and its frequency is the same as the backup roll rotatin frequency. Eccentricity force can be expressed as [] F F cos( t ) i i i () 496

2 Considerin the impact of the fundamental wave to the thickness of rolled piece, the equation of roll motion can be expressed as k x k x k x F t mx cos Where, x is the roll eccentricity displacement, k is dampin coefficient, k (4) is the coefficient associated with the elastic deformation, F and represents amplitude and anular frequency of the eccentricity force separately. Transform the equation into the form k k k F x x x x t m m m m cos Let k m, k m, k m Fm, then we have (5) x x x x cost (6) The equation fits well with the form of Duffin equation and this means the chaotic phenomena may be appeared. The relationship between the chaotic phenomena and system parameters is analyzed usin the Smale-horseshoe theory and subharmonic orbits theory in the author s previous paper [9]. Chaos Control via NTSM Based on Exponential Approach Law Considerin second order non-linear system x x x f ( x) ( x) b( x) u where x [ x, x] T are system states. ( ) ( x ) is distribution and satisfies ( x ), l l slidin mode (TSM) controller is as follow (7) b x are nonlinear lubricous functions of x,, u is the controller output. The conventional terminal f x and ( ) s x x q p (8) q q p u b ( x )[ f ( x ) x x ( l )sn( s )] p (9) where, p and q( p q ) are positive odds, is a desin constant. It cannot ensure that x while x in the controller (9).the sinularity may occur even after the slidin mode x is reached. This underlines the importance of addressin the sinularity problem in traditional TSM controllers. In this paper, a nonsinular terminal slidin mode (NTSM) based on reachin law method is proposed to improve the converence rate and minish the time t s, which is also able to avoid the sinularity problem. The proposed NTSM controller based on exponential reach law (NTSM-EAL) is described in the followin. In order to avoid the sinularity problem, the slidin mode is desined as pq s x x where, p and q( p q ) are positive odds and satisfy p q. The NTSM () is equivalent to (8) when s. So the time taken to reach the equilibrium point when in the slidin mode is the same as in (9). It can be proved that [] the system dynamics does not result in terms with neative powers usin (). Sinularity problem is avoided. The derivation of () is () 497

3 p p s x x x x x x q q p q p q p p q x x ( f ( x) ( x) b( x) u) q Usin exponential reach law, ivin s ks sn( s), k,. p p q Let ( x) x, ivin q desined as followin u b x x x f ( )[ ( ) ( x) ( x )( kssn( s)) ( x)] x ( x )( f ( ) ( ) b( ) u) ks sn( s) x x x The control sinal is Considerin ( x ) is uncertain, l is used to restricted the uncertainty ( ) () x in the controller. At the same time, constant is accede to the control sinal to avoid zero. The new controller is desined as u b ( x)[( ( x ) ) x f( x) ( ( x ) ) ( ks sn( s) l sn( s))] where, is a positive constant. Theorem. For system (7) with the NTSM (), if control is desined as () u b ( x)[( ( x ) ) x f( x) x ks s l s ( ( ) ) ( sn( ) sn( ))] p p q where ( x) x,, p and q are positive odds and satisfy p q,, k,, q ( x ) l, l, then the NTSM surface () will be reached in finite time. Furthermore, the states will convere to zero in finite time. Proof. For the NTSM (), its derivative is s x ( x )[( ( x ) ) x ( ( x ) ) ( ks sn( s) l sn( s)) ( x)] is small enouh, ( x )( ( x ) ) holds, then s ks sn( s) ( x )[ l sn( s) ( x)] When x,since, p and q ( p q ) are positive odds and p q,there is ( x ),then s kssn( s). Inore the affect of uncertainty ( x ), then it has When x, then ( x ),ivin s ks sn( s) x s ks sn( s) x For both x and x, it is obtained s ks sn( s). The converence rate of s is in exponential form. The system can reach the slidin mode s within finite time. Once the slidin surface is reached, it can be easily seen that NTSM-EAL controller () is equivalent to the TSM controller (9), so the time taken to reach the equilibrium point x from x in the slidin mode is the same. Therefore, the NTSM surface can be reached in finite time. The states will reach equilibrium point in finite time. This completes the proof. Simulation results is iven to control the chaotic system in part II. The initial states of the system (7) is [.,]. The parameters of NTSM-EAL is q, p 5, l.,.,., k,.,the simulation results is showed in Fiure () 498

4 . A. State response of ntsm-eal system B. Control sinal of ntsm-eal system C. Slidin Surface of NTSM-EAL System D. Phase Plot of NTSM-EAL System Fiure. Non-sinular terminal slidin model control based on exponential approach law. As can be seen from Fiure, the time system taken from initial value convere to the equilibrium point is less than s, owin this to the slidin mode surface reached the zero in a very short time. The amplitude of control sinal is in a low level PDC FL NTSM-EAL NTSM x time(s) Fiure. System output via 4 control methods. To compare the control effectiveness, the NTSM, parallel distributed compensation method (PDC) based on TS fuzzy model and feedback linearization method (FL) are applied to control the chaotic system. The parameters of the PDC are selected as literature []. The parameters of the NTSM are as follows. q, p 5, l.,.,. The initial states of the system is [.5,]. The results of 4 kinds control methods are shown in Fiure. The control effectiveness of the NTSM- EAL is better than NTSM, PDC and PL method both in the response rapidity and steady-state behavior. It takes 5s to radually make the system stable usin PDC method. Althouh the converence speed is fast usin the FL method, in the case of small error, the effect of reulation is weaker and it takes lon time to eliminate the steady-state error. 499

5 Conclusion New idea is proposed that roll eccentricity disturbance on aue is chaotic durin rollin throuh establishin the eccentric model. Then terminal slidin mode controller is used to control the chaos system. By analyzin the sinularity problem and the reachin time, a new non-sinular terminal slidin mode controller based on exponential reachin law is proposed to improve the converence rate and minish the reachin time. As expected from the theoretical analysis and verified by the simulation results, the desined controller achieves chaos stabilization with hih converence rate. Acknowledment This work is supported by National Natural Science Foundation (NNSF) of China under Grant 646. References [] Ott E., Greboi C., Yorke J.A., Controllin chaos, Phys. Rev. Lett., vol. 64, pp. 96-, 99. [] Zhan Libin, Pen Kaixian, Ton Chaonan, Research on chaotic phenomenon in hot strip continuous rollin aue control system, Control Enineerin, vol., pp. 7-75,. [] Shinbort T., Ott E., Greboi C., Yorke J., Usin chaos to direct trajectories to tarets, Phys. Rev. Lett., vol. 65, pp. 5-8, 99. [4] Ott E., Chaos in dynamical systems. nd ed., Cambride University Press,, pp [5] Mehta N.J., Henderson R.M., Controllin chaos to enerate aperiodic orbits, Phys. Rev. A., vol. 44, pp , 99. [6] Arecchi F.T., Boccaletti S., Ciofini M., Meucci R., The control of chaos: Theoretical schemes and experimental realizations, Int. J. Bifurcat. Chaos, vol. 8, pp , Au., 998. [7] Yu X., Variable structure control approach for controllin chaos, Chaos, Solitons & Fractals, vol. 8, pp , Sep., 997. [8] Radu-Emil Precup, Lorenz System Stabilization Usin Fuzzy Controllers, International Journal of Computers, Communications and Control, vol., pp , Mar., 7. [9] Hui Wan, Weiwei Zhan, Restraint Stratey for Chaos in Auto Gaue Control System, The Open Automation and Control Systems Journal, vol. 6, pp. 5-, 4. [] Fen Y., Yu X., Man Z, Non-sinular terminal slidin mode control of riid manipulators, Automatic, vol. 8, pp ,. [] Qian Junlei, Approach of chaotic control based on T-S fuzzy model, Journal of system simulation, vol. 7, pp , Dec. 5. 5