Linear and Non-linear Feedback Control Strategies for a 4D Hyperchaotic System

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1 Pure nd Applied Mthemtics Journl 017; 6(1): doi: /j.pmj ISSN: (Print); ISSN: (Online) Liner nd Non-liner Feedbck Control Strtegies for 4D Hyperchotic System Mysoon M. Aziz, Sd Fwzi AL-Azzwi * Deprtment of Mthemtics, College of Computer Sciences nd Mthemtics, University of Mosul, Mosul, Irq Emil ddress: sd_fwzi78@yhoo.com (S. F. AL-Azzwi) * Corresponding uthor To cite this rticle: Mysoon M. Aziz, Sd Fwzi AL-Azzwi. Liner nd Non-liner Feedbck Control Strtegies for 4D Hyperchotic System. Pure nd Applied Mthemtics Journl. Vol. 6, No. 1, 017, pp doi: /j.pmj Received: Jnury 1, 017; Accepted: Jnury 13, 017; Published: Februry 16, 017 Abstrct: This pper investigtes the stbiliztion of unstble equilibrium for 4D hyperchotic system. The liner, nonliner nd speed feedbck controls re used to suppress hyperchos to this equilibrium. The Routh-Hurwitz theorem nd Lypunov's second methods re used to derive the conditions of the symptotic stbility of the controlled hyperchotic system. Theoreticl nlysis, numericl simultion nd illustrtive exmples re given to demonstrte the effectiveness of the proposed controllers. Keywords: Chos Control, Liner Feedbck Control, Non-liner Feedbck Control, Routh-Hurwitz Method, Lypunov's Second Method 1. Introduction Chos is n importnt topic in nonliner science. In mny cses, chos is sometimes undesirble, so we wish to void nd eliminte such behviors [1]. Chos control nd chos synchroniztion were once believed to be impossible until the 1990s when Ott et l. developed the OGY method to suppress chos. Recently, mny different techniques nd methods hve been proposed to chieve chos control such s impulsive control method, sliding method control, differentil geometric method, H control, dptive control method, nd so on. Among them, the feedbck control is especilly ttrctive nd hs been commonly pplied to prcticl implementtion due to its simplicity in configurtion nd implementtion []. Generlly speking, there re two min pproches for controlling chos: liner feedbck control nd non- liner feedbck control. The liner feedbck control pproch offers mny dvntges such s robustness nd computtionl complexity over the non- feedbck control method [-7]. Mny ttempts hve been mde to control hyperchos of hyperchotic systems. Recently, Yn (005)[8], Dou nd Sun et l. (009) [9], Zhu (010)[10] nd Aziz nd AL-Azzwi (015) [11] suppressed hyperchotic systems to unstble equilibrium by using feedbck control method. In 017, the Ref. [1] present some problems of these strtegies nd how tretment. Our system is generted bsed on Pn system vi stte feedbck controller which is described by the following mthemticl model: x =y x+w y =x xz z =xy z w = y where,,,, nd,,, re constnt prmeters. When prmeters =10,=8 3,=8 nd =10, system (1) is hyperchotic nd hs two Lypunov exponents, i.e. `" =0.367, `& = [13], nd hyperchotic ttrctors nd Lypunov exponents spectrum re shown in Figure 1 nd Figure respectively. The system (1) hs only one equilibrium O0,0,0,0 nd the equilibrium is n unstble under these prmeters [13]. The im of this pper is discuss three feedbck control methods for 4D hyperchotic system. The rest of the pper is orgnized s follows. section present helping results which inclined ordinry feedbck control, dislocted feedbck control, enhncing feedbck control, nd speed feedbck control. min results re present in section 3. And section 4 present the flow chts for liner nd non-liner feedbck control. Finlly, the conclusion re given in section (1)

2 6 Mysoon M. Aziz nd Sd Fwzi AL-Azzwi: Liner nd Non-liner Feedbck Control Strtegies for 4D Hyperchotic System 5. Moreover, numericl simultions nd illustrtive exmples re pplied to verify the effectiveness of chosen controllers. Figure 1. The ttrctor of system (1) in x-y-z spce. Figure. The Lypunov exponents of system (1).. Helping Results To control the hyperchotic system (1) to the unstble equilibrium O0,0,0,0, we use the feedbck control pproch to control it. Let us ssume tht the controlled hyperchotic system (1) is given by x 10y xwu " y 8x xzu & z xy 8/3zu * w 10yu ()

3 Pure nd Applied Mthemtics Journl 017; 6(1): where u ",u &,u * nd u re externl feedbck control inputs which will be suitbly derive the trjectory of the hyperchotic system, specified by x,y,z,w to the equilibrium O0,0,0,0 of uncontrolled system (i.e. u + = 0,i=1,,3,4. Remrk 1 (Ordinry feedbck control method) For the ordinry feedbck control, the system vrible is often multiplied by coefficient s the feedbck gin, nd the feedbck gin is dded to the right -hnd of the corresponding eqution [3-5, 7, 10-1, 16]. Remrk (Dislocted feedbck control method) If system vrible multiplied by coefficient is dded to the right -hnd of nother eqution, then this method is clled dislocted feedbck control [1, 3-5, 7, 10-1, 16]. Remrk 3 (Enhncing feedbck control method) It is difficult for complex system to be controlled by only one feedbck vrible, nd in such cses the feedbck gin is lwys very lrge. So we consider using multiple vribles multiplied by proper coefficient s the feedbck gin. This method is clled enhncing feedbck control [1, 3-5, 7, 10-1, 16]. Remrk 4. The bove three remrks re used in liner feedbck control, lso cn use with nonliner feedbck control, but in nonliner feedbck the system vrible multiplied by quntity which contins coefficient nd nonliner term nd dded to the right -hnd of eqution ccording to ech bove three remrks. Remrk 5 (Speed feedbck control method) For the feedbck control, the independent vrible of system function is often multiplied by coefficient s the feedbck gin, so the method is clled displcement feedbck control. Similrly, if the derivtive of n independent vrible is multiplied by coefficient s the feedbck gin, it is clled speed feedbck control [-1, 16]. Remrk 6 (Routh-Hurwitz method, [14]). All the roots of the indicted polynomil hve negtive rel prts precisely when the given conditions re met. λ & +/λ+b:a>0,b>0 λ * +/λ & +4λ+C:A>0,C>0,AB C>0 λ +Aλ * +4λ & +6λ+D:A>0,AB C> 0,AB CC A & D>0,D>0 Remrk 7 (Brblt Remrk, [15, 16]). If 89: & < nd 89: <, then lim? < 89=0. 3. Min Results Theorem 1 (Ordinry feedbck control method). Let the controlled hyperchotic system () be x =y x+w y =x xz AB z =xy z w = y where C is the feedbck coefficient. When k> EFFEGHIFFJ HFG K H&F K GHGL, system (3) will grdully &F (3) converge to the unstble equilibrium O0,0,0,0. Proof. The Jcobi mtrix of system (3) is 0 1 c k M=N 0 0 Q (4) d 0 0 The chrcteristic eqution of mtrix (4) is λ+bλ * +/λ & +4λ+C=0, (5) where /=C+,4=C,6 = According to the Routh-Hurwitz method, the Jcobin mtrix of system (4) hs four negtive rel prt eigenvlues if nd only if C+>0,>0, C+C >0 Implying /,6 >0 since, nd >0 (given) nd C is lwys positive nd /4 6 >0 it is possible under the condition k> EFFEGHIFFJ HFG K H&F K GHGL. &F Thus system (3) will grdully converge to the unstble equilibrium O0,0,0,0. The proof is completed. Illustrtive Exmple 1. the hyperchotic system (3) be x =10y x+w y =8x xz SB z =xy 8 3z w = 10y When C >8.731, system (6) will grdully converge to the unstble equilibrium O0,0,0,0 ccording to theorem 1. Sol. The chrcteristic eqution of system (6) is (6) λ+8/3λ * +10+Cλ & +10C 80λ+80=0, (7) Then the roots of eqution (7) depended of the vlue of C s following: when C=9, the eqution (7) becme λ+8/3λ * +39λ & +10λ+80=0, nd the ll roots hve negtive rel prts λ " = U *,λ & = ,λ *, = ±.6817W, therefore the system (6) is symptoticlly stble. while C =8.5, the eqution (7) becme λ+8/3λ * +38.5λ & +5λ+80=0, then not ll roots hve negtive rel prts λ " = U,λ * & = ,λ *, =0.093±.6946W therefore the system (6) is unstble. Also cn justified the sme result tht obtin in theorem 1 by using MATLAB softwre where numericl simultions re used to investigte the controlled hyperchotic system (3) using fourth-order Runge-Kutt scheme with time step The initil vlues re tken s by Y0 = 5, 0 = 5,0=10,0=15Z. The feedbck coefficients re given by C =9, C=8.5 in Figure 3-, b respectively. The behviors of the sttes 9,9,9,9 of the control

4 8 Mysoon M. Aziz nd Sd Fwzi AL-Azzwi: Liner nd Non-liner Feedbck Control Strtegies for 4D Hyperchotic System hyperchotic system (6) show converging to O0,0,0,0 when C9 (Figure 3,.) nd divergence to O0,0,0,0 when C8.5 (Figure 3,b.) Figure 3. The difference of the stte of the controlled system (3) with the control gin chnge. Theorem (Dislocted feedbck control method). Let u " C, u & u * u 0 nd the controlled hyperchotic system () be x y xw AB y x xz z xy z w y where C is the feedbck coefficient. When C3 [K H\ [, system (8) will grdully converge to the unstble equilibrium O0,0,0,0. Proof. The Jcobi mtrix of system (8) is (8) C 0 1 M] c ^ (9) d 0 0 The chrcteristic eqution of mtrix (9) is λbλ * /λ & 4λC0, (10) where /,4C,6 By using Routh-Hurwitz method, the Jcobin mtrix of system (9) hs four negtive rel prt eigenvlues if nd only if 30,30 C 30, Implying /,6 30 since, c nd d re positive (given) nd k is lwys positive nd /4 6 30, it is possible under the condition C3 [K H\. Thus system (8) will grdully converge [ to the unstble equilibrium O0,0,0,0. The proof is completed. Illustrtive Exmple : Let the hyperchotic system (8) be x 10y xw AB y 8x xz z xy 8 3z w 10y (11) When C311, system (11) will grdully converge to the unstble equilibrium O0,0,0,0 ccording to theorem. Sol. The chrcteristic eqution of system (11) is λ8/3λ * 10λ & 8C 80λ800 (1) Then the roots of eqution (1) depended of the vlue of k s following: when C11.1, the eqution (1) becme λ8/3λ * 10λ & 30.8λ800, nd the ll roots hve negtive rel prts λ " U *,λ & ,λ *, V5.3499W, therefore the system (11) is symptoticlly stble. While C10.9, the eqution (1) becme λ8/3λ * 10λ & 5.λ800, then not ll roots hve negtive rel prts λ " U *,λ & ,λ *, V5.341W therefore the system (11) is unstble.

5 Pure nd Applied Mthemtics Journl 017; 6(1): The feedbck coefficients re given by C11.1, C10.9 in Figure 4-, b respectively. The behviors of the sttes 9,9,9,9 of the control hyperchotic system (11) show converging to O0,0,0,0. when C11.1 (Figure 4,.) nd divergence to O0,0,0,0 when C10.9 in (Figure 4,b.). Figure 4. The difference of the stte of the controlled system (11) with the control gin chnge. Theorem 3 (Enhncing feedbck control method): Let u " C, u & C,u * u 0 nd the controlled hyperchotic system () be x y xw A_ y x xz AB z xy z w y (13) where C is the feedbck coefficient. When C * 3C & & C & 30, system (13) will grdully converge to the unstble equilibrium O0,0,0,0. Proof. Anlogously s in proof of Theorem 1 nd Theorem. Illustrtive Exmple 3: Let the hyperchotic system (13) be c x 10y xw A_ y 8x xz AB b ` z xy U * z w 10y (14) When C , system (14) will grdully converge to the unstble equilibrium O0,0,0,0 ccording to theorem 3. Sol. The chrcteristic eqution of system (14) is λ8/3λ * C10λ & C & 10k 80λ800, (15) Then the roots of eqution (15) depended of the vlue of k s following: when C13, the eqution (15) becme λ8/3λ * 36λ & 19λ800, nd the ll roots hve negtive rel prts λ " U *,λ & ,λ *, V.7967W therefore the system (14) is symptoticlly stble. while C1, the eqution (15) becme λ8/3λ * 34λ & 16λ800, then not ll roots hve negtive rel prts λ " U *,λ & ,λ *, V.8196W therefore the system (14) is unstble. The feedbck coefficients re given by C13,C1 in Figure 5-, b respectively. The behviors of the sttes 9,9,9,9 of the control hyperchotic system (14) show converging to O0,0,0,0 when C13 (Figure 5,.) nd divergence to O0,0,0,0 when C1 (Figure 5,b.).

6 10 Mysoon M. Aziz nd Sd Fwzi AL-Azzwi: Liner nd Non-liner Feedbck Control Strtegies for 4D Hyperchotic System Figure 5. The difference of the stte of the controlled system (14) with the control gin chnge. Theorem 4 (Speed feedbck control method). Let u C, u " u & u * 0 nd the controlled hyperchotic system be x y xw y x xz z xy z w y ky (16) where C is the feedbck coefficient. When C3 [K H\ [, system (16) will grdully converge to the unstble equilibrium O0,0,0,0. Proof. Anlogously s in proof of Theorem 1 nd Theorem. Illustrtive Exmple 4. Let the hyperchotic system (16) be c x 10y xw y 8x xz b ` z xy U * z w 10y ky (17) When C311, system (17) will grdully converge to the unstble equilibrium O0,0,0,0 ccording to theorem 4. Sol. The chrcteristic eqution of system (17) is λ8/3λ * 10λ & 8C 80λ800 (18) Then the roots of eqution (18) depended of the vlue of C s following: when C11.5, the eqution (18) becme λ8/3λ * 10λ & 4λ800, nd the ll roots hve negtive rel prts λ " 8 3,λ & ,λ *, 0.583V5.5997W therefore the system (17) is symptoticlly stble. while C10.5, the eqution (18) becme λ8/3λ * 10λ & 14λ800, then not ll roots hve negtive rel prts λ " U *,λ & ,λ *, V5.0115W therefore the system (17) is unstble. The behviors of the sttes xt,yt,zt,wt of the control hyperchotic system (17) show converging to O0,0,0,0 when C11.5 (Figure 6,) nd divergence to O0,0,0,0 when C10.5 (Figure 6,b).

7 Pure nd Applied Mthemtics Journl 017; 6(1): Figure 6. The difference of the stte of the controlled system (17) with the control gin chnge. Theorem 5 (Nonliner feedbck control method). Consider the controlled hyperchotic system () s follows: cx y xw y x xz Ce " b z xy z ` w y Ce & (19) Where Ce " f " &,Ce & f & &,f ",f & re positive feedbck gin, then system (19) will converge to the unstble equilibrium point O0,0,0,0. Proof. Construct the Lypunov function g 1 & & & & 1 f " Ce " C & 1 f & Ce & C & where C is prmeter to be determined. The derivtion of g is: & C & & C & j j Y,,,Zj j j i k 0 C d 0 o p qo 1 n m d m mn Q m 0 C m l To ensure the system (19) symptoticlly stble, the symmetric mtrix P should be positive definite, nd when P stisfy the following conditions: c 1.30 & 30.C [Hr b 3.C [Hr & 30 `4.C & " 1& C " & 30 we cn choose the constnt C s g Ce " C & Ce & C & C 3st & 4, 1& I1 & & 16 & u 8 so tht g is positive definite nd g is negtive semi-definite. However, cnnot get immeditely the controlled system (19) is symptoticlly stble t equilibrium O0,0,0,0 i.e.,x,y,z,w < nd Ce " C,Ce & C <, for this reson, x,y,z,w < then, we hve??? vλ wxy z & 9 vo p qov} g~9g0 9 g0

8 1 Mysoon M. Aziz nd Sd Fwzi AL-Azzwi: Liner nd Non-liner Feedbck Control Strtegies for 4D Hyperchotic System where λ wxy z is the smllest eigenvlue of the positive mtrix P. nd so s x,y,z,w &, ccording to the Brblt Lemm, implies x,y,z,w 0 s t 0, therefore the controlled hyperchotic system (19) is symptoticlly stble to equilibrium point, This completes the proof. 4. Flow Chrt of Liner nd Non-liner Feedbck Control Strtegies The following flow chrts explins briefly the Liner nd non-liner feedbck control strtegies respectively Figure 7. Flow chrt of Liner feedbck control strtegies. Figure 8. Flow chrt of non-liner feedbck control strtegies.

9 Pure nd Applied Mthemtics Journl 017; 6(1): Conclusions In this pper, the control problem of 4D hyperchotic system is investigted: ordinry feedbck control, dislocted feedbck control, enhncing feedbck control, speed feedbck control re used to suppress hyperchos to unstble equilibrium. The Routh-Hurwitz nd Lypunov's second methods re pplied to derive the conditions of the symptotic stbility of the controlled hyperchotic system. Theoreticl nlysis,numericl simultion nd illustrtive exmples re given to demonstrte the effectiveness of the proposed controllers. References [1] X. Chngjin, L. Peilun, Chos control for 4D hyperchotic system, Applied Mechnics nd Mterils. 418 (013) [] H. Wng, G. Ci, Controlling hyperchos in novel hyperchotic system, Journl of Informtion nd Computing Science, 4 (009) [3] C. Yng, C. H. To, P. Wng, Comprison of feedbck control methods for hyperchotic Lorenz system, Physics Letters A 374 (010) [4] C. Zhu, Feedbck control method for stbilizing unstble equilibrium points in new chotic system, Nonliner nlysis, 71 (009) [5] C. Zhu, Z. Chen, Feedbck control strtegies for the Liu chotic system, Phys. Lett. A 37 (008) [6] C. To, C. Yng, Y. Luo, H. Xiong, F. Hu, Speed feedbck control of chotic system, Chos Solitons Frctls 3 (005) [7] S. Png, Y. Liu, A new hyperchotic system from the L system nd its control. Journl of Computtionl nd Applied Mthemtics, 35 (011) [8] Z. Yn, Controlling hyperchos in the new hyperchotic chen system, Appl. Mth. Comput. 168 (005) [9] Q. Dou, J. A. Sun, W. S. Dun, K. P. Lu, Controlling hyperchos in the new hyperchotic system, Commun. Nonliner Sci. Numer., 14 (009) [10] C. Zhu, Controlling hyperchos in hyperchotic Lorenz system using feedbck controllers, Appl. Mth. Comput. 16 (010) [11] M. M. Aziz, S. F. AL-Azzwi.. Using Feedbck Control Methods to Suppress Modified Hyperchotic Pn System, Computtionl nd Applied Mthemtics Journl. 1 (3), (015) [1] M. M. Aziz, S. F. AL-Azzwi, Some Problems of feedbck control strtegies nd its tretment, Journl of Mthemtics Reserch; 9 (1), (017) [13] S. F. AL-Azzwi, Study of dynmicl properties nd effective of stte u for hyperchotic pn systems, Al-Rfiden J. Comput. Sci. Mth. 10 (013) [14] S. F. AL-Azzwi, Stbility nd bifurction of pn chotic system by using Routh-Hurwitz nd Grdn method, Appl. Mth. Comput. 19 (01) [15] W. Xing, C. Ling, M. Sheng, T. Xin, Nonliner feedbck control of novel hyperchotic system nd its circuit implementtion, 19 (010) [16] C. To, C. Yng, Three control strtegies for the Lorenz chotic system, Chos Solitons Frctls 35 (008)