PDF《Rssler systems with uncertain》

2011 Chinese Physical Society and IOP Publishing Ltd http://www.iop.org/journals/cpb http://cpb.iphy.ac.cn 080507-1 Chin. Phys. B Vol. 20, No.

8 (2011)

080507 Fig. 1. Chaotic attractor of a double-delayed R¨ ossler system. 3. Synchronization of double- delayed R¨ ossler systems with uncertain parameters Assume that we have two double-delayed R¨ ossler systems where the master (or drive) system is denoted as x and the slave (or response) system denoted as y. For system (1), the master (or drive) and the slave (or response) systems can be described as ? ? ? ? ? ? ? B x1 = ?x2 ? x3 + ax1(t ? τ1) + bx1(t ? τ2), B x2 = x1 + ax2, B x3 = 0.2 + x1x3 ? cx3, (2) and ? ? ? ? ? ? ? B y1 = ?y2 ? y3 + ? ay1(t ? τ1) + ? by1(t ? τ2) + u1, B y2 = y1 + ? ay2 + u2, B y3 = 0.2 + y1y3 ? ? cy3 + u3, (3) where ? a, ? b, and ? c are the parameters of the slave sys- tem which need to be estimated, and u1, u2, and u3 are the nonlinear controller such that the two double- delayed R¨ ossler systems can be synchronized, i.e. ? ? ? ? ? ? ? limt→∞ y1 ? x1 = 0, limt→∞ y2 ? x2 = 0, limt→∞ y3 ? x3 = 0. (4) The error states are ? ? ? ? ? ? ? B e1 = B y1 ? B x1, B e2 = B y2 ? B x2, B e3 = B y3 ? B x3. (5) The error dynamical system can be written as ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? B e1 = ?y2 ? y3 + ? ay1(t ? τ1) + ? by1(t ? τ2) + u1 ? (?x2 ? x3 + ax1(t ? τ1) + bx1(t ? τ2)), B e2 = y1 + ? ay2 + u2 ? (x1 + ax2), B e3 = 0.2 + y1y3 ? ? cy3 + u3 ? (0.2 + x1x3 ? cx3). (6) Our aim is to ?nd control law u1, u2, and u3 for stabilizing the error variables of the systems at the ori- gin. At the end of this part we propose the following control law and update the rule for the system ? ? ? ? ? ? ? u1 = k1(y1 ? x1), u2 = k2(y2 ? x2), u3 = x1x3 ? y1y3. (7) The update rule for the three unknown parameters ? a, ? b, and ? c are ? ? ? ? ? ? ? B ? a = ?e1y1(t ? τ1) ? e2y2, B ? b = ?e1y1(t ? τ2), B ? c = e3y3. (8) 080507-2 Chin. Phys. B Vol. 20, No.

8 (2011)

080507 Theorem

1 For the given k1 ≤ ?5/2 ? (a2 + b2 )/4, k2 ≤ ?2?a, the synchronization between drive system (2) and response system (3) will occur by control law (7) and update rule (8). Proof Choose the following function V(e1,e2,e3) =

1 2 (e2

1 + e2

2 + e2 3) +

1 2 (e2 a + e2 b + e2 c) + ∫

0 ?τ1 (e2 1(t + θ) + e2 2(t + θ) + e2 3(t + θ))dθ + ∫

0 ?τ2 (e2 1(t + β) + e2 2(t + β) + e2 2(t + β))dβ, (9) where ea = ? a ? a, eb = ? b ? b, ec = ? c ? c. The time derivative of V along the trajectory of error system (5) is B V = (e1 B e1 + e2 B e2 + e3 B e3 + ea B ea + eb B eb + ec B ec) + (e2

1 ? e2 1(t ? τ1)) + (e2

2 ? e2 2(t ? τ1)) + (e2

3 ? e2 3(t ? τ1)) + (e2

1 ? e2 1(t ? τ2)) + (e2

2 ? e2 2(t ? τ2)) + (e2

3 ? e2 3(t ? τ2)), B V = e1(?e2 ? e3 + ae1(t ? τ1) + be1(t ? τ2) + eay1(t ? τ1) + eby1(t ? τ2) + k1e1) + e2(e1 + ae2 + eay2 + k2e2) + e3(?ce3 ? ecy3) + ea B ea + eb B eb + ec B ec + (e2

1 ? e2 1(t ? τ1)) + (e2

2 ? e2 2(t ? τ1)) + (e2

3 ? e2 3(t ? τ1)) + (e2

1 ? e2 1(t ? τ2)) + (e2

2 ? e2 2(t ? τ2)) + (e2

3 ? e2 3(t ? τ2)) = ?e1e3 + ae1e1(t ? τ1) + be1e1(t ? τ2) + e1(eay1(t ? τ1) + eby1(t ? τ2)) + k1e2

1 + ae2

2 + e2eay2 + k2e2

2 ? ce2

3 ? e3ecy3 + ea B ea + eb B eb + ec B ec + (e2

1 ? e2 1(t ? τ1)) + (e2

2 ? e2 2(t ? τ1)) + (e2

3 ? e2 3(t ? τ1)) + (e2

1 ? e2 1(t ? τ2)) + (e2

2 ? e2 2(t ? τ2)) + (e2

3 ? e2 3(t ? τ2)) ≤ (

5 2 + a2

4 + b2

4 + k1 ) e2

1 + (a +

2 + k2)e2

2 + (

5 2 ? c ) e2

3 + ea(B ea + e1y1(t ? τ1) + e2y2) + eb(B eb + e1y1(t ? τ2)) + ec(B ec ? e3y3). Inserting Eq. (8) into the above equation yields the following expression B V = ? eT Ke, where eT = (e1, e2, e3)T , and K = ? ? ? ? ? ? ? ( ?