Synchronisation of Nonlinear Systems by Bidirectional
Coupling with Time-varying Delay
Citation for published version (APA):
Oguchi, T., Yamamoto, T., & Nijmeijer, H. (2007). Synchronisation of Nonlinear Systems by Bidirectional Coupling with Time-varying Delay. In IFAC Workshop on Time Delay Systems (pp. 12153-12158).
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SYNCHRONISATION OF NONLINEAR SYSTEMS BY BIDIRECTIONAL COUPLING
WITH TIME-VARYING DELAY
Toshiki Oguchi∗ Takashi Yamamoto∗
Henk Nijmeijer∗∗
∗Department of Mechanical Engineering,
Tokyo Metropolitan University
1-1, Minami-Osawa, Hachioji-shi, Tokyo 192-0397 Japan oguchi@ctrl.prec.metro-u.ac.jp
∗∗Department of Mechanical Engineering
Eindhoven University of Technology
P.O. Box 513, 5600 MB Eindhoven, The Netherlands h.nijmeijer@tue.nl
Abstract: This paper considers the synchronization problem for coupled nonlinear systems with time-varying delay. In previous work, we have already derived a sufficient condition for synchronisation and boundedness of two identical strictly semi-passive systems coupled using state feedback with time-delay. This method, however, requires that all states of each system are coupled to the other systems and the coupling delay is constant. In this paper we extend the conditions to identical systems coupled using output feedback with time-varying delay where a bound on the length of the delay and an upper bound of the time-derivative is known. Firstly, we show, using the small-gain theorem, that the trajectories of coupled strictly semi-dissipative systems converge to a bounded region. Then we derive a sufficient condition for synchronisation of the systems coupled with time-varying delay by using a delay range dependent on the stability criterion.
Keywords:
nonlinear systems, synchronisation, chaotic systems, time-varying delay
1. INTRODUCTION
Synchronisation phenomena are of interest of researchers in applied physics, biology, social science, engineering and in interdisciplinary re-search, and the synchronisation notion has have been investigated in order to clarify the mech-anism of synchronisation (Strogatz and Stew-art, 1993.; Strogatz, 2000; Pecora and Carroll,
1 This work was supported by the Japan Society for the Promotion of Science (JSPS) Grant-in-Aid for Science Research (No. 18560441).
1990.; Nijmeijer and Mareels, 1997) . More re-cently, applications of these phenomena to en-gineering have also been considered and anal-ysed via control theory (Nijmeijer and Rodriguez-Angeles, 2003; Pogromsky et al., 2002; Oguchi and Nijmeijer, 2005; Huijberts et al., 2007) . On the other hand, in practical situations, time-delays caused by the signal transmission affect the behaviour of coupled systems. It is therefore important to study the effect of time-delay in existing synchronisation schemes. Although the effect of time-delay in the synchronisation of
cou-pled systems has been investigated both numeri-cally and theoretinumeri-cally by a number of researchers, these works concentrate on synchronisation of sys-tems with the coupling term typically described by K(xi(t − τ ) − xj(t − τ )) or K(Cxi(t − τ ) −
Cxj(t − τ )) (Amano et al., 2006) and there are
few results for the case in which the coupling term is described by K(xi(t) − xj(t − τ )). For
the latter case, as the coupling term does not vanish if the systems synchronise, therefore even if uncoupled cell systems are bounded, coupled systems are not necessarily bounded. In our pre-vious work (Yamamoto et al., 2007) , we have de-rived a sufficient condition for synchronisation and boundedness of two identical strictly semi-passive systems coupled using state feedback with time-delay. This method, however, requires that all states of each system are coupled with the other system and the coupling delay is constant. In this paper we extend the conditions to identical cell systems coupled using output feedback with time-varying delay where a bound on the length of the delay and an upper bound of the time-derivative of the time-varying time delay is known. The paper is organized as follows. Firstly, we show, using the small-gain theorem, that the trajectories of cou-pled strictly semi-dissipative systems converge to a bounded region. Then we derive a sufficient con-dition for synchronisation of the systems coupled with time-varying delay by using a delay range dependent on the stability criterion (Fridman and Shaked, 2003).
2. PRELIMINARIES
Throughout this paper, k·k denotes the Euclidean norm. For a vector function v(t) : [0, ∞) → Rn, if kvk∞ , supt≥0kv(t)k < ∞, then we
denote v ∈ Ln
∞. Define a continuous norm by
kφkc , max−h1≤θ≤0kφ(θ)k for a vector function
φ : [−h1, 0] → Rn. Furthermore, for a vector
function v(t) defined on [−h1, ∞), we also define
kvk[−h1,∞), supt≥−h1kv(t)k.
In the following subsections, we review some re-sults derived in our previous work (Yamamoto et al., 2007).
2.1 Strict semi-passivity and strict semi-dissipativity
Consider the nonlinear system
˙x(t) = f (x, u) , y(t) = h(x) (t ≥ 0) (1) with state x ∈ Rn, input u ∈ Rm, output y ∈ Rm,
f : Rn× Rm→ Rn and h : Rn→ Rm.
Then we introduce strict semi-passivity and strict semi-dissipativity as follows.
Definition 1.(strict semi-passivity). System (1) is said to be strictly semi-passive, if there exist a C1
-class function V : Rn → R, class-K
∞ functions
α(·), α(·) and α(·) satisfying
α(kxk) ≤ V (x) ≤ α(kxk) (2) ˙
V (x) ≤ −α(kxk) − H(x) + yTu (3) for all x ∈ Rn, u ∈ Rm, y ∈ Rm, where the
function H(x) satisfies the following condition: kxk ≥ η ⇒ H(x) ≥ 0 (4) for a positive real number η.
Definition 2.(strict semi-dissipativity). System (1) is said to be strictly semi-dissipative with respect to the supply rate q(u, y), if there exist a C1-class
function V : Rn → R, class-K
∞ functions α(·),
α(·) and α(·) satisfying (2) and ˙
V (x) ≤ −α(kxk) − H(x) + q(u, y) (5) for all x ∈ Rn, u ∈ Rm, y ∈ Rm, where the
function H(x) satisfies (4).
Remark 3. The system is strictly semi-passive if the supply rate q(u, y) = yTu in (5) for all u ∈
Rm.
For a strictly semi-dissipative system, the follow-ing lemma can be proved in a similar way as the argument of the input-to-state stability(ISS) in (Isidori, 1999).
Lemma 4. Suppose that system (1) is strictly semi-dissipative with respect to the supply rate β(kuk) for a class-K function β(·). Then there exist class-K functions ρ(·), γ(·) and a real number η > 0 such that the response x(t) of (1) with the initial state x(0) = x0 satisfies
kxk∞≤ max{ρ(kx0k), γ(kuk∞), ρ(η)}, lim sup t→∞ kx(t)k ≤ max{γ(lim sup t→∞ ku(t)k), ρ(η)} (6) for any input u ∈ Lm
∞ and any x0 ∈ Rn. Here
functions ρ(·) and γ(·) are given by ρ(r) = α−1◦ α(r)
γ(r) = α−1◦ α ◦ α−1◦ κβ(r) (r ≥ 0) (7) where κ > 1, functions α(·), α(·) and α(·) satisfy (2) and (5) and η is some positive real number satisfying the condition(4).
2.2 Small-gain theorem
Consider the following two systems
˙xi(t) = fi(xi, ui) , yi(t) = Cixi (t ≥ 0)
(8) where xi ∈ Rn, ui ∈ Rm, yi ∈ Rm, Ci ∈ Rm×n
and fi : Rn × Rm → Rn with initial conditions
Suppose that each system (8) is strictly semi-dissipative. Then from Lemma 4, each system has the properties (6). Now, we consider the case in which these two systems are coupled by the following inputs containing time-varying delay,
u1(t) =y2(t − τ (t)) = C2x2(t − τ1(t))
u2(t) =y1(t − τ (t)) = C1x1(t − τ2(t)).
(9)
where τ (t) is a time-varying delay and the initial conditions of xi for i = 1, 2 are respectively given
by
xi(θ) = φi(θ) (−h1≤ θ ≤ 0)
xi(0) = φi(0) = x0i
(10)
where φi: [−τ, 0] → Rn.
Define class-K functions as π1(r) = γ1(σmax(C2) · r)
π2(r) = γ2(σmax(C1) · r) (r ≥ 0). (11)
where γi(·) are defined as equation (7) and σmax(·)
denotes the maximum singular value of a matrix. Then we obtain the following lemma.
Lemma 5. For coupled system (8) with the cou-pling term (9), if the functions π1(·) and π2(·) in
(11) satisfy
π1◦ π2(r) < r for all r > 0, (12)
then the trajectories x1(t) and x2(t) satisfy
lim sup t→∞ kx1(t)k ≤ max{π1◦ ρ2(η2), ρ1(η1)} lim sup t→∞ kx2(t)k ≤ max{π2◦ ρ1(η1), ρ2(η2)} (13) where ρi(·) are defined by equation (7) and ηi
satisfies condition(4).
3. SYNCHRONISATION
3.1 Problem formulation
We consider two identical cell systems
Σi: ( ˙xi(t) = Axi+ f (xi) + Bui yi(t) = Cxi (t ≥ 0) (i = 1, 2) (14) where xi ∈ Rn, ui ∈ Rm, yi ∈ Rm, A ∈ Rn×n, B ∈ Rn×m, C ∈ Rm×n, f : Rn × Rm → Rn is
Lipschitz continuous and φi : [−h1, 0] → Rn with
h1 > 0. The initial condition for each system is
given by xi(θ) = φi(θ) (−h1≤ θ ≤ 0).
Now we assume that each system is strictly semi-passive, CB is nonsingular and each system for i = 1, 2 coupled with each other by the following controller (Figure 1).
ui(t) = Kij(yi(t) − yj(t − τi(t))) (i, j = 1, 2, i 6= j)
(15)
where τ1(t) = τ2(t), τ(t) is a time-varying delay
satisfying
0 ≤ τ (t) ≤ h1, k ˙τ (t)k ≤ d (16)
and Ki are gain matrices such as
Kij = Kji, K < 0. (17)
Here we formulate synchronisation of coupled
− + K + − Delay ˙x2= Ax2+ f (x2) + Bu2 y2= Cx2 K ˙x1= Ax1+ f (x1) + Bu1 y1= Cx1 Delay
Fig. 1. Coupled systems
systems as shown below.
Definition 6. If there exists a positive real number r such that the trajectories xi(t) of the systems
(14), (15) with initial conditions φi such that
kφ1− φ2k ≤ r satisfies
kx1(t) − x2(t)k → 0 as t → ∞,
then the coupled systems (14) with coupling in-puts (15) are asymptotically synchronised.
Therefore our goal in this paper is to derive synchronisation conditions for systems (14) con-nected with (15).
3.2 Boundedness of the trajectories
Firstly we show under suitable assumptions the boundedness of the coupled systems (14) with coupling inputs (15).
Since the matrix CB is nonsingular, the system (14) can be transformed to the following normal form (Lozano et al., 2000).
˙yi(t) = a(yi, zi) + CBui (18)
˙zi(t) = q(yi, zi) (19)
for i = 1, 2, where zi∈ Rn−m and
yi
zi
= Φxi
for a nonsingular matrix Φ , CT NTT with N ∈ R(n−m)×n such that N B = 0 and functions
a : Rm× Rn−m → Rm and q : Rm× Rn−m →
Rn−m are globally Lipschitz continuous.
˙zi(t) = q(yi, zi)
˙yi(t) = a(yi, zi) + CBui
yi
ui
zi
Fig. 2. Decomposition of a cell system
• The system (18) is strictly semi-dissipative with βy(kzik) + yiTui, where βy∈ K.
• The system (19) is strictly semi-dissipative with βz(kyik), where βz∈ K.
These assumptions mean that there exist positive definite C1-class functions V
y and Vz for the
systems (18) and (19) such that the following inequalities hold. ˙ Vy(yi) ≤ −ǫy(kyik) − Hy(yi) + βy(kzik) + yiTui ≤ −αi(kyik) − Hy(yi) + βy(kzik) + βij(kyj,τk) ˙ Vz(zi) ≤ −αz(kzik) − Hz(zi) + βz(kyik) where αi(r) = ǫy(r) − 1 2λmax(K)r 2 βij(r) = − 1 2λmin(K)r 2.
Therefore, from Lemma 4, lim sup
t→∞ kyi(t)k ≤ max{γy(lim supt→∞ kzik),
γij(lim sup
t→∞ kyjk), ρy(ηy)}
lim sup
t→∞
kzi(t)k ≤ max{γz(lim sup t→∞ kyik), ρz(ηz)} hold, where ρy(·) = α−1y ◦ αy(·) , γy(·) = ρy◦ α−1y ◦ κβy(·) ρz(·) = α−1z ◦ αz(·) , γz(·) = ρz◦ α−1z ◦ κβz(·) γij(·) = ρy◦ α−1ij ◦ κβij(·). (20) with αy(r) + αij(r) = αi(r) and αij(r) = ǫy(r) −
αy(r) −12λmax(K)r2 such that αij∈ K∞.
As a result, applying Lemma 5, we obtain the following theorem.
Theorem 7. For all r > 0, if (
γy◦ γz(r) < r
γij(r) < r
(21)
hold, then the trajectories of the system (14) converge to a set
Ω, {xi∈ Rn| kyik ≤ sy and kzik ≤ sz} (22)
where sy , max{ρy(ηy), γy ◦ ρz(ηz)}, sz ,
max{ρz(ηz), γz◦ ρy(ηy)}.
3.3 A Synchronisation Condition
From the above discussion, the behaviour of each system coupled with each other converges to the
set (22) under the assumptions. By using this property, we consider a synchronisation condition for coupled systems with time-varying delay. The dynamics of the error e(t), x1(t) − x2(t) is
given by
˙e(t) = (A+ BKC)e(t)+ BKCe(t− τ (t))+ φ(e, x2)
(23) where φ(e, x2) = f (e + x2) − f (x2). If the error
system (23) has e = 0 as an asymptotically stable equilibrium, the behaviours of the two systems (14) synchronise. Therefore the synchronisation problem can be reduced to the stability problem of the error dynamics (23).
From the Lyapunov indirect method, if the lin-earised system around the origin:
˙e(t) = (A + BKC + D(x2)) e(t) + BKCe(t − τ (t))
, A0(x2)e(t) + A1e(t − τ (t)) (24) where D(x2) , ∂φ(e,x2) ∂e e=0 , A0(x2) , A + BKC + D(x2) and A1, BKC, is asymptotically
stable, the origin of the original error dynamics (23) is locally asymptotic stable. In addition, note that x2 converges to Ω from the discussion in
Section 3.2.
Theorem 8. For all x2 ∈ Ω, if there exist n × n
matrices P1 = P1T > 0, P2, P3, W1, W2,R =
RT ≥ 0,S = ST ≥ 0 satisfying the following linear
matrix inequality: Ψ1 Ψ2 h1Φ1 −W1TA1 ΨT 2 Ψ3 h1Φ2 −W2TA1 h1ΦT1 h1ΦT2 −h1R 0 −AT1W1 −AT1W2 0 −(1 − d)S < 0 (25) where Ψ1=(A0(x2) + A1)TP2+ P2T(A0(x2) + A1) + W1TA1+ AT1W1+ S Ψ2=P1− P2T+ (A0(x2) + A1)TP3+ AT1W1 Ψ3= − P3− P3T+ h1AT1RA1 Φ1=W1T + P2T, Φ2= W2T + P3T
Remark 9. This theorem is a special case of the result derived by Fridman et al. (Fridman and Shaked, 2003) and can be proved in a similar way as Theorem 1.
As the LMI (25) is affine with respect to the system matrices A0(x2) and A1, this result can be
extended to a stability criterion for the polytopic systems.
Since x2 is bounded, each element of D(x2) is
also bounded. As a result, the approximated error dynamics (24) can be rewritten by the following polytopic system:
˙e(t) = m X i=1 piAi0e(t) + A1e(t − τ (t)) where Ai
0 = A0+ Di are constant matrices and
pi(x2) ∈ [0, 1] are polytopic coordinates satisfying
the convex sum propertyPm
i=1pi(x2) = 1. Using
vertex systems, we can obtain the following poly-topic linear differential inclusion (PLDI)
˙e(t) ∈ ConA10e(t)+A1e(t − τ (t)), · · · ,
Am
0e(t) + A1e(t − τ (t))
o (26) where Co denotes a convex envelope. Therefore we can obtain the following stability criterion. Theorem 10. Consider the PLDI (26). This sys-tem is asymptotically stable if there exist n × n matrices Pi
1 = P1i T
> 0, P2, P3, W1i, W2i,R =
RT ≥ 0 and S = ST ≥ 0 for i = 1, 2, . . . , m
satisfying the following linear matrix inequality: Ψi1 Ψi2 h1Φ1 −W1i T A1 ΨiT2 Ψ3 h1Φ2 −W2i T A1 h1ΦT1 h1ΦT2 −h1R 0 −AT 1W1i −AT1W2i 0 −(1 − d)S < 0 (27) where Ψi1=(Ai0+ A1)TP2+ P2T(Ai0+ A1) + Wi 1 T A1+ AT1W1i+ Si Ψi 2=P1i− P2T+ (Ai0+ A1)TP3+ AT1W1i Ψ3= − P3− P3T+ h1AT1RA1 Φ1=W1i T + P2T, Φ2= W2i T + P3T. 4. EXAMPLE Consider 2 coupled Lorenz systems.
˙xi(t) = σ(xi2− xi1) rxi1− xi2− xi1xi3 −bxi3+ xi1xi2 + Bui , yi= Cxi (28) where σ = 10, r = 28, b = 8/3 and BT = C =0 1 0 0 0 1 . The inputs are defined by
ui(t) = K(yi(t) − yj(t − τ (t))), (i = j = 1, 2).
For a real number k > 0, set coupling gain K as K = −kI2×2 and time-delays τ (t) satisfies
0 ≤ τ (t) < 0.05 and ˙τ (t) ≤ 0.05.
Now we decompose each system into two subsys-tems by defining ˜yi and zi as follows.
˜
yi=xi2, xi3− r T
, zi= xi1
In addition, we define storage functions as Vy(˜yi), 1
2y˜ T
i y˜i and Vz(zi) , 12zi2 for the corresponding
subsystems. Then the time derivative of Vy(˜yi)
along the trajectory of (28) satisfies ˙ Vy(˜yi) ≤ −αi(k˜yik) − Hy(˜yi) + βij(k˜yj,τk) (29) where ǫy(r) = εr2 αi(r) = ε −λmax(K) 2 r2= ε +k 2 r2 βij(r) = − λmin(K) 2 r 2= k 2r 2 Hy(˜yi) = (1 − ε)˜yi12 + (b − ε)˜y2i2+ br˜yi2
with ε = 0.01. Here, for ηy= 28.9, Hy(˜yi) satisfies
k˜yik ≥ ηy ⇒ Hy(˜yi) ≥ 0.
Similarly, for the function Vz(zi),
˙
Vz(zi) = −σzi2+ σ ˜yi1zi
≤ −αz(kzik) + βz(k˜yik) (30)
holds, where αz(r) = σ2r2 and βz(r) = σ2r2. This
means that ηz= 0.
In addition, from the definitions of the storage functions Vy(˜yi) and Vz(zi), we can choose ρy and
ρz as ρy(r) = ρz(r) = r for any r ≥ 0. Therefore
γij(r) = ρy◦ α−1ij ◦ κβij(r).
As a result, γij are given by
γij(r) = s κk 2 k 2+ ε r (31)
satisfying γij < r with κ sufficiently close to
1. Furthermore, γz(r) = r,∀r ≥ 0 and since
βy(kzk) = 0, γy(r) = 0 which means (21) holds.
Finally, we obtain sy = sz= ηy= 28.9.
Therefore, from Theorem 7, the trajectories xi(t)
converge to the set Ω = { ˜yT i zi T ∈ R3 k˜yi(t)k ≤ sy and kzi(t)k ≤ sz} = {xi∈ R3 (x2i2+ (xi3− r)2) 1 2 ≤ 28.9 and kxi1k ≤ 28.9}.
Figure 3 shows the behaviour of x1(t). In this
fig-ure, the cylinder illustrates the estimated bound-ary of the set Ω. From this figure, we know that the trajectories converge to the set Ω. In this simulation, the time-varying delay is given by
τ (t) = 0.01 + 0.002 sin 3t sin 7t. Setting k = 20, there exist 3 × 3 matrices Pi
1 =
Pi 1 T
> 0, P2, P3, W1i, W2i,R = RT ≥ 0 and
S = ST ≥ 0 for i = 1, 2, . . . , 8 satisfying the LMI
condition (27) for all vertex systems.
Figure 4 and 5 show the behaviour of each com-ponent of x1(t) and e(t), respectively. We know
that the error e converges to zero, which means the synchronisation of these systems is perfectly accomplished.
−40 −20 0 20 40 −50 0 50 −20 0 20 40 60 x 11 x 12 x 13
Fig. 3. Phase portrait of system 1
−10 0 10 20 30 40 50 −20 0 20 x11 −10 0 10 20 30 40 50 −20 0 20 x12 −100 0 10 20 30 40 50 50 Time x13
Fig. 4. Behaviours of the state x1(t)
−10 0 10 20 30 40 50 −50 0 50 e1 −10 0 10 20 30 40 50 −50 0 50 e2 −10 0 10 20 30 40 50 −50 0 50 Time e3
Fig. 5. Behaviours of the error e(t)
5. CONCLUSION
In this paper, we have considered synchronisation of nonlinear systems coupled with time-varying transmittal delay. By extending our previous re-sult, we obtained a sufficient condition for syn-chronisation of two systems coupled using out-put feedback with time-varying delay. Throughout this paper, we assume that the structure of the coupled system is symmetric. If the network struc-ture is symmetric, it is expected that the analysis introduced in this paper can be applied to larger number of coupled systems. However, if τ1(t) is
not equal to τ2(t), the symmetric structure
col-lapses and the error dynamics does not have a
triv-ial solution. For such a case, we can consider the coupled system as a system with uncertainties and will be able to apply our previous work (Oguchi and Nijmeijer, 2006) in order to estimate an upper bound of the synchronisation error.
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