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Robust power control of multi-link single-sink optical networks with time-delays
Automatica 49 (2013) 2261–2266
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Brief paper
Robust power control of multi-link single-sink optical networks with time-delays✩ Nem Stefanovic 1 , Lacra Pavel University of Toronto, Edward S. Rogers Sr. Department of Electrical and Computer Engineering, 10 King’s College Rd., Toronto, Ontario, Canada M5S 3G4
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Article history: Received 11 September 2011 Received in revised form 2 December 2012 Accepted 2 April 2013 Available online 6 May 2013 Keywords: Time-delay Communication control applications Optical communication Robust stability Communication networks
abstract Optical networks provide high data transmission rates which require optimal power control algorithms to ensure the proper delivery of the signals. Time-delays may destabilize the control algorithms. We study the stability of game-theoretic based primal–dual control algorithms applied to multi-link singlesink optical networks. We present a perturbed optical signal-to-noise ratio (OSNR) model with timedelays. By optimizing the OSNR values, we reduce the bit error rates of the channels. We modify singular perturbation theory to include Lyapunov–Krasovskii stability theory with multiple time-delays and uncertainties. Simulations show the stabilization of perturbed systems at the expense of transient convergence time. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction We study the optical signal-to-noise ratio (OSNR) optimization problem in optical networks with time-delays. A signal’s OSNR relates to its bit error rate. Signal propagation delays in the optical fibers may destabilize the power control algorithms. A game-theoretic approach is applied in Pavel (2006, 2007) that yields a primal–dual controller. The work is generalized in Pan and Pavel (2009) for multi-link networks. In Pavel (2006, 2007) and Pan and Pavel (2009), time-delays are not considered. The single link case is analyzed in Stefanovic and Pavel (2009a) via a Lyapunov–Razumikhin approach combined with singular perturbation theory. The method in Stefanovic and Pavel (2009a) does not apply to multiple time-delays or uncertainties. Including timedelays and uncertainties in the stability analysis requires more sophisticated techniques such as Lyapunov–Krasovskii functionals with singular perturbation theory.
✩ A short version of this work appears in Stefanovic and Pavel (2009c). The material in this paper was presented at the 48th IEEE Conference on Decision and Control (CDC) and 28th Chinese Control Conference (CCC), December 16–18, 2009, Shanghai, China. This paper was recommended for publication in revised form by Associate Editor Yoshito Ohta under the direction of Editor Roberto Tempo. E-mail addresses: [email protected] (N. Stefanovic), [email protected] (L. Pavel). 1 Present address: McMaster University, Faculty of Engineering, Bachelor of
Technology - Four Year Program, 1280 Main St. West, Hamilton, Ontario, Canada, L8S 0A3. Tel.: +1 905 525 9140x20280; fax: +1 905 525 7015. 0005-1098/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.automatica.2013.04.009
Recent literature contains few results for the robust stability of optical networks with time-delays or uncertainties. The works Ghafouri-Shiraz and Karbassian (2009), Lauck, Charny, and Ramakrishnan (1996), Mokhtar, Azizoglu, and Barry (1996) and Rapet, Wang, Gangaji, and Tang (2007) explore channel use in optical CDMA networks, the effects of tuning delays for optical networks, end-to-end congestion control and communications over GEO-satellite links, respectively. In Baraniuk, Ahmed, and Khojestapour (2004), an optimal transmission rate and power control maximizes the network throughput. In Zhu and Pavel (2008), classic control techniques are applied to separately study additive uncertainty and time-delays. A converse Lyapunov Theorem is applied to time-delayed systems in Yeganefar, Pepe, and Dambrine (2007). The paper Jiang and Han (2006) studies the delay-dependent robust stability problem for linear uncertain systems using a Lyapunov–Krasovskii approach. Linear singularly perturbed systems with time-delays are studied in Dragan and Ionita (2000); Fridman (2006). The paper Glizer (2009) studies nonstandard singularly perturbed systems. In this paper, we study multi-link single-sink optical networks with time-delays. We present a time-delayed OSNR model with input multiplicative and additive uncertainty. We derive stability conditions that ensure OSNR optimization for both time-delays and uncertainties. Multi-link networks with multiple time-delays and uncertainties cause analytical complications. The main results are novel and derive via singular perturbation theory, modified to handle Lyapunov–Krasovskii functionals with time-delays and uncertainties. This paper is organized as follows. Section 2 reviews the time-delay stability results from Stefanovic and Pavel (2009a,b).
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Fig. 1. Multi-link single-sink network with time-delays.
Section 3 presents the main results of the paper. We present simulations in Section 4 and conclusions in Section 5. Fig. 2. Closed loop system with delays from (3), (4) and (1).
2. Background Stefanovic and Pavel (2009a) using a Razumikhin approach. The primal–dual algorithms from Stefanovic and Pavel (2009a) are
2.1. OSNR model with time-delays An optical network is composed of a set of links, L = {1, . . . , L}, that interconnect the optical nodes, where channels are added, dropped or rerouted. Each link, ℓ ∈ L, is composed of a series of Nℓ optical spans, containing one optical amplifier and 100 km s of fiber. The set of channels multiplexed and transmitted over the network is denoted by M = {1, . . . , n}. We denote by Ri , for i ∈ M , the set of links from the transmitter (Tx) to the receiver (Rx) in the optical path of channel i. The optical input power at Tx, the output signal at Rx, and the output noise at Rx are denoted by ui > 0, si > 0, and n¯ i > 0, respectively. The optical signal-to-noise ratio (OSNR) for channel i ∈ M , is OSNRi = si /¯ni . The optical amplifiers (OAs) within the optical spans have channel dependent gains, Gℓ,k,i . The optical fibers have wavelength independent loss coefficients, Lℓ,k . Furthermore, OAs introduce amplified spontaneous emission (ASE) noise denoted by ASEℓ,k,i . The optical span transmission, for kth span on the ℓth link, and ith channel, is given by hℓ,k,i = Gℓ,k,i Lℓ,k , ∀k = 1, . . . , Nℓ . We define
N
ℓ the ℓth link transmission as Tℓ,i = q= 1 hℓ,q,i , ∀ℓ = 1, . . . , L. We assume that all OAs operate with the same total power target P0,ℓ , below the threshold for nonlinear effects (Agrawal, 1997). Thus, Gℓ,k,i = Gℓ,i ∀k = 1, . . . , Nℓ .
f
Consider the single-sink network in Fig. 1. We denote by τj the forward propagation delay of a signal from its source, j ∈ M , to the OSNR output. We denote by τib the backward propagation delay of f
a signal from the OSNR output to the source, i. Let τi,j = τj + τib .
Lemma 1. The OSNR for the ith channel in a single-sink network is given as OSNRi (t ) =
ui (t − τi,i ) n0 , i + Γi,j uj (t − τi,j )
(1)
Γi,j =
ℓ∈Ri ∩Rj k=1
Gkℓ,i
ℓ−1 T q ,j q=1
Tq,i
ASEℓ,k,i P0,ℓ
ϵ
=η
dui (t )
and n0,i is the noise optical power at Tx for channel i.
uj,in (t ) − P0
βi − ui (t − τ ) µ ¯in (t ) 1 1 − Γi,i ui (t − τ ) − ai OSNRi,in (t )
= ρi
dt
(3)
(4)
where (3) and (4) are on ‘‘slow’’ and ‘‘fast’’ time-scales, respectively. The small gain, ϵ > 0, represents the fast dynamics of dui (t )/dt. We explicitly denote the fast time-scale by tf , and the slow time-scale by t. The two time-scales are related by t = ϵ tf . We define τ = ϵ τ˜ , where τ˜ is the time-delay τ represented in the ‘‘fast’’ time-scale tf . The parameters uj,in (t ), OSNRi,in (t ) and µ ¯ in (t ) represent the channel powers uj (t ), the OSNR measurements OSNRi (t ), and the channel price µ(t ), respectively, as inputs to the primal–dual algorithms. Our control parameters are ρi > 0. The design parameter, βi , and step size η, are both adjustable. Note that P0 is P0,ℓ for the link ℓ associated with the OSNR output. Fig. 2 depicts control algorithms (3)–(4) acting on the timedelayed single-link OSNR system (1). We substitute µ ¯ in = µ(t − τ b ), OSNRi,in (t ) = OSNRi (t ) and ui,in = u(t − τ f ). Convergence to 1 the NE is ensured if ai > j̸=i Γi,j and ai > 2 j̸=i (Γi,j + Γj,i ) and if
σ (ρ Γ˜ + Γ˜ T ρ) 2 σ¯ ((ρ Γ˜ )2 ((ρ Γ˜ )2 )T )
τ˜ <
(5)
where τ˜ is the time-delay with respect to the fast time-scale tf , σ and σ¯ are the minimum and maximum singular values, ρ = diag(ρi ), and Γ˜ is defined as 1,
Γi,j ai
(2)
j =1
Γ˜ i,j =
where Γi,j , elements of the (n × n) system matrix Γ , are
dt
i=j
j∈M
Nℓ Gkℓ,j
dµ(t )
,
i ̸= j.
(6)
Note that (5) was derived in Stefanovic and Pavel (2009a) by applying the singular perturbation theory modified to handle Razumikhin theory to (3), (4) and (1). We decomposed the closed-loop system into its reduced and boundary-layer forms, which produced a linear time-delayed system. A Razumikhin based LMI that ensured stability reduced to (5).
2.2. Game-theoretic based primal–dual control 3. Main results A game-theoretic based channel algorithm, or primal algorithm, that reaches a Nash Equilibrium (NE) and produces optimal OSNR values is presented in Pavel (2006) without time-delays. A dynamic pricing term, µ > 0, ensures that the total link power is below the threshold for nonlinear effects (Pavel, 2007). A stability analysis of the algorithms of Pavel (2006, 2007) with delays is done in
3.1. Single link with time-delay and uncertainty We first present some ancillary time-delay theory. Define
C ([−r , 0], ℜn ) as the set of all continuous functions mapping [−r , 0] to ℜn . Let C = C ([−r , 0], ℜn ).
N. Stefanovic, L. Pavel / Automatica 49 (2013) 2261–2266
Definition 1. The continuous norm is defined as ∥yt ∥c = max−r ≤θ≤0 ∥y(t + θ )∥2 where yt ∈ C , is the set, y(q) for all q ∈ [t − r , t ], and ∥ · ∥2 is the Euclidean norm. We study the linear time-invariant systems of the form x˙¯ (t ) = A0 x¯ (t ) + A1 x¯ (t − τ )
the uncertainty (1 + δi ) in u1i with its worst case (1 − q¯ i ). Thus, (14) becomes
ϵ u˙ i (t ) = ρi (7)
where A0 and A1 are (n × n) real matrices. The following restricted Lyapunov–Krasovskii lemma (Gu, Kharitonov, & Chen, 2003, Proposition 5.2, p. 150) gives conditions for the asymptotic stability of linear time-delay systems. Lemma 2. The linear time-delayed system (7) is asymptotically stable if there exists a Lyapunov–Krasovskii functional V such that for some ψ1 > 0 and ψ2 > 0
ψ1 ∥y(t )∥22 ≤ V (yt ) ≤ ψ2 ∥yt ∥2c
(8)
where the norm ∥yt ∥c is defined in Definition 1 and the derivative of V (yt ) along the trajectory V˙ (yt ) satisfies V˙ (yt ) = V ′ (yt ) where V (yt ) or ′
dyt dt dV dyt
≤ −ψ1 ∥y(t )∥22
(9)
denotes the Fréchet derivative of V .
Next, we introduce uncertainty in the OSNR model (1) for the single-link case. The system gains, Γi,j , are actually time-varying due to slow parameter drift and ASE. There is also uncertainty in the time-delays τ f and τ b due to system reconfigurations. We also have uncertainty in the transmitter noise at the sources due to imperfect measurements. Thus, we define Γ∆ ∈ ΩΓ , where
ΩΓ = {Γ + 1Γ | ∥1Γ ∥2 ≤ qˆ }
(10)
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βi n∆,0,i − µ(t − τ b ) (1 − q¯ i )ai
−
j∈Mℓ
i
Γ˜ 1i,j (1 + δj )uj (t − τ )
(16)
where Γ˜ 1i,j = Γ˜ i,j + 1Γi,j /(1 − q¯ i )ai , such that
Γ˜ i,j =
1 1 − q¯ i
,
i=j
Γi,j , (1 − q¯ i )ai
(17) i ̸= j.
Rewrite (16) and (15) using x = µ − µ∗ and zi = ui − u∗i , where µ∗ and u∗i are equilibrium points of (16) and (15), x˙ = η1row (I + diag(δj ))z (t − τ f )
ϵ z˙ = ρ β
−x(t − τ b ) − τ b ) + µ∗ )
µ∗ (x(t
(18)
− Γ˜ ∆ (I + diag(δj ))z (t − τ ) (19)
where z is a vector with elements zi , ρ = diag(ρi ), β is a column vector with elements βi , and 1row is a row vector of 1 elements. Eqs. (18) and (19) represent the perturbed, closed-loop system with time-delay shifted about the equilibrium points (µ, u∗ ). We rewrite (18) and (19) using the coordinate shift, y(t ) = z (t − τ f ) − h(x(t )), where h(x(t )) is the isolated root of the RHS of (19),
where Γ is a matrix with elements Γi,j , 1Γ is a matrix with elements 1Γi,j , and qˆ is the independent uncertainty bound. We define the uncertain input, u1j ∈ Ωj
h(x(t )) = (I + diag(δi ))−1 Γ˜ ∆−1 β
Ωj = {uj + δj uj | ∥δj ∥2 ≤ q¯ j } ∀j
and I + diag(δi ) is invertible, Γ˜ ∆ = Γ˜ + diag( ((1−¯1q )a ) )1Γ , Γ˜ is a
(11)
−x(t ) µ∗ (x(t ) + µ∗ )
(20)
i
i
where q¯ j is the independent uncertainty for input j. We define the uncertain transmitter noise, n∆,0,i ∈ Ωni ,
matrix with elements defined in (17). Later, Lemma 4 ensures Γ˜ ∆ is invertible. Next, define
Ωni = {n0,i + δni n0,i | ∥δni ∥2 ≤ qni } ∀i
f (y(t ) + h(x(t ))) = η1row I + diag(δj ) (y(t ) + h(x(t )))
(21)
g (y(t − τ )) = −ρ Γ˜ ∆ (I + diag(δi ))y(t − τ )
(22)
(12)
where qni is the uncertainty bound on the transmitter noise for channel i. The uncertain OSNR model is OSNR1i (t ) =
u1 i ( t − τ ) n∆,0,i +
Γ1i,j u1j (t − τ )
j∈Mℓ
.
(13)
ϵ u˙ i (t ) = ρi
βi + µ(t − τ b )
Γi,i ai
− 1 ui (t − τ )
1 ui ( t − τ ) − n∆,0,i + Γ1i,j u1j (t − τ ) . ai u1i (t − τ ) j∈M
to get the coordinate shifted closed-loop system
ϵ
dy(t ) dt
= g (y(t − τ )) − ϵ
(14)
ℓi
dyˆ dtf
In (3), let uin,j = (1 + δj )uj (t − τ f )
(15)
j∈M
In the foregoing analysis, we use u1j = uj + δj uj = (1 + δj )uj since uj is scalar. In the denominator of the last term of (14), we replace
(23) dh(x(t )) dx
f (y(t ) + h(x(t ))).
(24)
We next rewrite (23) and (24) using two separate time-scales. Let t = ϵ tf , where tf and t are the fast and slow time-scales, respectively. Also, let τ = ϵ τ˜ . Thus, x˙˜ (t ) = η1row Γ˜ ∆−1 β
f µ( ˙ t) = η (1 + δj )uj (t − τ ) − P0 .
x˙ = f (y(t ) + h(x(t )))
i
We substitute (13) into the primal–dual algorithms (3) and (4) to obtain the closed-loop uncertain system with time-delays. In (4), let OSNRi,in (t ) = OSNR1i (t ) and µin,i (t ) = µ(t − τ b ) to get
−˜x(t ) µ∗ (˜x(t ) + µ∗ )
= −ρ Γ˜ ∆ (I + diag(δi ))ˆy(tf − τ˜ )
(25) (26)
where yˆ (tf ) and x˜ (t ) are the distinct solutions. The following lemma presents the LMI that ensures exponential stability for the boundary-layer system (26). The proof is derived similarly to Gu et al. (2003) and is omitted here. The result is less conservative than Lemma 2 from Stefanovic and Pavel (2009a) which is based on Lyapunov–Razumikhin theory.
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N. Stefanovic, L. Pavel / Automatica 49 (2013) 2261–2266
Lemma 3. The boundary-layer system (26) is exponentially stable if there exist symmetric, positive definite matrices P and S1 such that τ˜ satisfies
H1 H2T
H2 H3
<0
(27)
where
H1 =
Mn2 −(AT0 )2 P
H2 =
1
PE ∗
−PA20 −S1 + µ ˜ 1 (G∗ )T G∗ (G∗ )T
0
−PA0 E ∗
τ˜ −AT0 (G∗ )T ∗ T
0
3.2. Single-sink network analysis
µ ˜ 1 (G∗ )T D∗
(D ) −I −(E ∗ )T (G∗ )T
−I ∗ H3 = D
We extend the single-link results of the previous section to the single-sink case (see Fig. 1). Let the uncertainty set for the perturbed system Γ∆ ∈ ΩΓ be defined as
0
−G∗ E ∗ −µ ˜ 1 (I − (D∗ )T D∗ )
ΩΓ = {Γ + 1Γ | ∥1Γi,row ∥2 ≤ qˆ i ∀i}
E2 = −ρ diag E∗ = ∗
G =
qˆ E2 k1
qˆ
k1
I
1
(1 − q¯ i )ai A0 diag
q¯ i k¯ i
√ q¯ i , diag k¯ i
qˆ
D = 0 0 ∗
k1
diag
q¯ i .
Lemma 4. The uncertain reduced system (25) is exponentially stable if
(Γi,j + Γj,i ) + qˆ ∀i.
(34)
i
where the parameters are defined as in Lemma 1. We formulate the closed-loop system as follows. Substitute (34) into (3) and (4), f modified such that for each channel i, τ = τi,i , τ f = τi , τ b = τib , to obtain f
x˙ = η1row (I + diag(δj ))z (t − τJ )
(35)
ϵ z˙ = ρ β¯ R(x(t − τIb )) −
Γ˜ 1i,row
1
µ∗ (r1 + µ∗ )
∥˜x∥22
(36) f
Proof. We prove asymptotic stability for (23) and (24) using a composite Lyapunov functional based on the reduced and boundary layer system functionals. For the reduced system (25), we select the Lyapunov function V (˜x) = 12 x˜ 2 . If (28) holds, then
× (I + diag(δj ))z (t − τi,J )
where x = µ−µ∗ , z is a vector with elements zi = ui −u∗i , z (t −τJ )
Theorem 1. Consider the system (18) and (19), with uncertainty sets defined in (10)–(12). There exists an ϵ ∗ > 0 such that for 0 < ϵ < ϵ ∗ the origin is asymptotically stable if (27) and (28) are satisfied.
∂V f (h(˜x)) ≤ −η1row Γ˜ ∆−1 β ∂ x˜
j∈Mℓ
(28)
We state the main theorem based on Lemmas 3 and 4.
u1i (t − τi,i ) n∆,0,i + Γ1i,j u1j (t − τi,j )
i∈M
1 2 j= ̸ i
OSNR1i (t ) =
0
The proof of the following lemma is omitted because it follows similarly to Lemma 3 in Stefanovic and Pavel (2009a). Lemma 4 incorporates uncertainties unlike Lemma 3 in Stefanovic and Pavel (2009a).
ai >
(33)
where Γ is a matrix with elements Γi,j , 1Γ is a matrix with elements 1Γi,j , 1Γi,row is an (n × n) matrix with the ith row equal to the ith row of 1Γ and the remaining terms equal to zero, and qˆ i is a direct and independent uncertainty bound. The uncertain OSNR model for multiple time-delays and uncertainties (33), (11) and (12) is
and µ ˜ 1 > 0, A0 = −ρ Γ˜ , Mn2 = τ1˜ (PA0 + AT0 P ) + S1
obtained from g (ˆy(tf − τ˜ )) as in Curtain and Zwart (1995, Theorem 2.4.6, p. 60). We define the composite Lyapunov functional χ (x(t ), yt ) = V (x(t )) + W (yt ), where V (x) = 12 x2 , W (yt ) is the converse Lyapunov functional (see Lemma 33.1 in Krasovskii, 1963 Lemma 33.1 and Yeganefar et al., 2007) of (26) with yt as the argument. By applying the composite Lyapunov function, χ (x(t ), yt ), to (23) and (24), and exploiting the Lyapunov inequalities in (29)–(32) and various Lipschitz properties, we prove asymptotic stability by Lemma 2. The rest of the proof follows as in Khalil (2002, Theorem 11.4).
(29)
f j
is a vector with elements zj (t − τ ), z (t − τi,J ) is a vector with elements zj (t − τi,j ), Γ˜ 1i,row is an (n × n) matrix of zeros except for the ith row which has the jth element Γ˜ 1i,j , β¯ = diag(βi ), and R(x(t − τIb ))
−x(t − τ1b ) = µ∗ (x(t − τ1b ) + µ∗ )
···
−x(t − τnb ) µ∗ (x(t − τnb ) + µ∗ )
T
.
We apply the following coordinate shift to (35) and (36), given by f ys (t ) = z (t − τJ ) − h¯ (x(t )), where h¯ (x(t )) is the isolated root of the RHS of (36),
where x˜ ≤ r1 for r1 > −µ∗ , for r1 arbitrarily large. By Lemma 3, the boundary-layer system (26) is exponentially stable if τ˜ satisfies (27). By Krasovskii (1963, Lemma 33.1) and Yeganefar et al. (2007), there exists a Lyapunov functional W (ˆytf ),
We apply similar steps used in (18)–(24) to derive the following reduced and boundary-layer systems
ψ∥ˆytf ∥2c ≤ W (ˆytf ) ≤ K ∥ˆytf ∥2c
(30)
x˙¯ s (t ) = η1row Γ˜ ∆−1 β¯ R(¯xs (t ))
W ′ (ˆytf ) gtf (ˆy(tf − τ˜ )) ≤ −k∥ˆytf ∥2c ′ W (ˆyt ) ≤ kF ∥ˆyt ∥c f f c
(31)
dyˆ s
˙ (ˆytf ) = where W
dW (ˆytf ) dtf
= W ′ (ˆytf )
(32) dyˆ tf dtf
= W ′ (ˆytf ) gtf , with W ′ (ˆytf )
denoting the Fréchet derivative of W and
dyˆ tf dtf
= gtf (ˆy(tf − τ˜ )) is
h¯ (x(t )) = (I + diag(δi ))−1 Γ˜ ∆−1 β¯ R(x(t )).
dtf
= −ρ
Γ˜ 1i,row (I + diag(δi ))ˆys (tf − τ˜i,i )
(37)
(38) (39)
i
where we denote by x¯ s (t ) and yˆ s (tf ) as the solutions. The following lemma ensures exponential stability for the boundary-layer system (39). The proof derived from a similar procedure in Gu et al. (2003) and is omitted here.
N. Stefanovic, L. Pavel / Automatica 49 (2013) 2261–2266
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Lemma 5. The boundary-layer system (39) is exponentially stable if there exist symmetric, positive definite matrices P and Si,j such that
Mn J1T T J2 J3T
J1 J7 J4T J5T
J2 J4 J6 J9T
J3 J5 <0 J9 J8
(40)
where
∗ −PAs,n As ), J2 = PEs = −PAs,1 Es∗ · · · −PAs,n Es∗ T 0 ··· 0 = −G∗ A · · · −G∗ A s,1 s s ,n s −I (D∗s )T = diag(µi (G∗s )T D∗s ), J6 = D∗s −I 1 = diag − Si + µi (G∗s )T G∗s τ˜i,i
J1 = (−PAs,1 As J3 J4 J5 J7
1
···
τ˜1,1
(G∗s, )T
Fig. 3. Perturbed system, no robust compensation, ρ = 0.01.
J8 = diag(−µi (I − (D∗s )T D∗s ))
J9 =
Mn =
··· ···
0 −G∗s,1 Es∗ 1
P
τ˜1,1
n
0 −G∗s,n Es∗
As,i +
n
i=1
ATs,i P
i=1
+
n n
τ˜i,i Si,j
i =1 j =1
1
, As,i = −ρ Γ˜ i,row ∀i (1 − q¯ i )ai As = (As,1 · · · As,n ), Si = diag(Si1 · · · Sin ) √ ∗ Es = diag( qi ki )Es2 As diag(k¯ i )Λ T √ qi 1 G∗s = diag In diag Λ ki k¯ i √ qi ¯ 0 diag I diag ( k ) Λ n i . D∗s = ki
Es2 = −ρ diag
0
0
Fig. 4. Perturbed system, robust compensation, ρ = 0.002.
√
Here, Λ = diag(diag( q¯ i )), and Γ˜ i,row is an (n × n) matrix with the ith row equal to the ith row of Γ˜ and all other entries set to zero. The matrix G∗s,i is the ith (2n2 × n) submatrix of G∗s . We define G∗s, =
G∗s,i . The parameters ki , k¯ i and µi are design variables such
that k¯ n×(i−1)+j < ki for all i, j = 1, . . . , n. The matrix In is an (n × n) identity matrix. The control parameters to satisfy (40) are ρi and ai . We rewrite Lemma 4 to account for multiple uncertainties qˆ i . The proof is similar to the proof in Lemma 3 in Stefanovic and Pavel (2009a). Lemma 6. The uncertain reduced system (38) is exponentially stable if ai >
1 2 j= ̸ i
(Γi,j + Γj,i ) + qˆ i ∀i.
(41)
We now state the main stability theorem based on Lemmas 5 and 6. The proof follows as in Theorem 1. Theorem 2. Consider the system (35) and (36), with uncertainty sets defined in (33), (11) and (12). There exists an ϵ ∗ > 0 such that for 0 < ϵ < ϵ ∗ the origin is asymptotically stable if (40) and (41) hold.
4. Simulations We simulate the single-sink network in Fig. 1 using realistic network data. Single-sinks realistically appear as substructures within larger optical network systems. There are four channels on five links with one common output. The total round trip delays for channels 1–4 are 20 ms, 30 ms, 40 ms and 50 ms, respectively. The amplifier gain is parabolic in shape with G = −4 × 1016 × (λ − 1555 × 10−9 )2 + 15 decibels, where λ represents the channel wavelength. The βi values are selected by the same pricing strategy as in Stefanovic and Pavel (2009a). Light propagates at approximately 200,000 km/s within the fiber for a round-trip time-delay of 10 ms on the link. The channel and link algorithms are updated every 5 ms, and 0.5 s, respectively, i.e., ϵ = 0.01. We fix ki = 1.1 and k¯ i = 1. We introduce a uniform uncertainty into the system, i.e., 1Γi,j = w , for w = q/n × diag(ai /ρi ), where n is the size of the system matrix, which implies ∥F ∥2 ≤ q. We select the uncertainty parameters q¯ i = qni = 0.3 for 30% uncertainty in the input signals and noise. We select qˆ i = 1.07 × 10−2 , which is a large uniform uncertainty. Fig. 3 shows the system response without uncertainty compensation. The LMI condition (40) is not satisfied, and we see instability in the channels. We compensate by decreasing ρi to 0.002 and increasing ai by a factor of 4 to satisfy (40). Fig. 4 shows
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this case. The LMI (40) is very complicated, possessing multiple time-delays. We do not have explicit equations to obtain theoretical upper bounds for each of the time-delays in the system. We rely on tuning ρi and ai to ensure (40) is satisfied. 5. Conclusions We studied a primal–dual control approach for the single-sink optical networks with time-delays. We derived stability conditions using a combined Lyapunov–Krasovskii and singular perturbation theory approach. Simulations showed robust compensators stabilized uncertain optical systems at the expense of convergence time. Acknowledgment We gratefully acknowledge the support of the Natural Science and Engineering Research Council of Canada.
Mokhtar, A., Azizoglu, M., & Barry, R. A. (1996). Impact of tuning delay on the performance of bandwidth-limited optical broadcast networks with uniform traffic. IEEE Journal on Selected Areas in Communications, 14(5), 935–944. Pan, Yan, & Pavel, Lacra (2009). Games with coupled propagated constraints in optical networks. In GameNets. Turkey, May. Pavel, L. (2006). A noncooperative game approach to OSNR optimization in optical networks. IEEE Transactions on Automatic Control, 51(5), 848–852. Pavel, L. (2007). An extension of duality to a game-theoretic framework. Automatica, 43(2), 226–237. Rapet, P., Wang, R., Gangaji, S., & Tang, L. (2007). Experimental investigation of rate-based and delay-tolerant transmission mechanisms over geo-satellite communication links. In The proceedings of WCNC 2007 (pp. 3717–3721). Stefanovic, N., & Pavel, L. (2009a). A stability analysis with time-delay of primal–dual power control in optical links. Automatica, 45(5), 1319–1325. Stefanovic, N., & Pavel, L. (2009b). An analysis of stability with time-delay of link level power control in optical networks. Automatica, 45(1), 149–154. Stefanovic, N., & Pavel, L. (2009c). Robust power control of single sink optical networks with time-delays. In 48th IEEE conference on decision and control (pp. 2034–2039). Shanghai, China, December. Yeganefar, N., Pepe, P., & Dambrine, M. (2007). Input-to-state stability and exponential stability for time-delay systems: further results. In IEEE conference on decision and control (pp. 2059–2064). December. Zhu, Quanyan, & Pavel, Lacra (2008). State-space approach to pricing design in OSNR Nahs game. In 17th IFAC world congress (pp. 12001–12006). July.
References Agrawal, G. P. (1997). Fiber-optic communication systems. New York: John Wiley and Sons. Baraniuk, R.G., Ahmed, N., & Khojestapour, M.A. (2004). Delay-limited throughput maximization for fading channels using rate and power control. In Globecomm 2004 (pp. 3459–3463). Curtain, R. F., & Zwart, H. (1995). An introduction to infinite dimensional linear systems theory. New York: Springer-Verilag. Dragan, V., & Ionita, A. (2000). Exponential stability for singularly perturbed systems with state delays. In Proc. of the 6th colloquium on the qualitative theory of differential equations (pp. 1–8). Szeged, Hungary, August. Fridman, E. (2006). Robust sampled-data h∞ control of linear singularly perturbed systems. IEEE Transactions on Automatic Control, 51(3), 470–475. Ghafouri-Shiraz, H., & Karbassian, M. M. (2009). IP routing and traffic analysis in coherent optical CDMA networks. Journal of Lightwave Technology, 27(10), 1262–1268. Glizer, V. Y. (2009). l2 -stabilizability conditions for a class of nonstandard singularly perturbed functional-differential systems. Dynamics of Continuous, Discrete and Impulsive Systems, Series B: Applications and Algorithms, 16, 181–213. Gu, K., Kharitonov, V. L., & Chen, J. (2003). Stability of time-delay systems. Boston: Birkhäuser. Jiang, Xiefu, & Han, Qing-Long (2006). Delay-dependent robust stability for uncertain linear systems with interval time-varying delay. Automatica, 42, 1059–1065. Khalil, H. K. (2002). Nonlinear systems. New Jersey: Prentice-Hall. Krasovskii, N. (1963). Stability of motion. Stanford University Press. Lauck, A., Charny, A., & Ramakrishnan, K. K. (1996). Time scale analysis and scalability issues for explicit rate allocation in ATM networks. IEEE Transactions on Networking, 4(4), 569–581.
Nem Stefanovic obtained his Ph.D. in electrical and computer engineering from the University of Toronto in 2010. He spent one year developing and analyzing control systems for the ACR-1000 and EC6 nuclear reactors while at Atomic Energy of Canada. Since September 2010, Dr. Stefanovic has been an Assistant Professor at McMaster University in the Faculty of Engineering, Bachelor of Technology program. His area of expertise is the control of optical networks, linear and nonlinear systems, time-delay systems, and process automation.
Lacra Pavel (M’92–SM’04) received the Ph.D. degree in electrical engineering from Queen’s University, Canada, in 1996. She spent a year at the Institute for Aerospace research (NRC) in Ottawa as a NSERC Postdoctoral Fellow. From 1998 to 2002, she worked in the optical communications industry. In August 2002 she joined the Electrical and Computer Engineering Department at University of Toronto, where she is currently an Associate Professor. Her research interests include system control and optimization in optical networks, game theory, robust and H-infinity optical control. Dr. Pavel served as Publications Chair of Conference on Decision and Control 2006, Technical Program Committees of INFOCOM 2007, IEEE Control Applications Conference 2005; Associate Chair (Control) on the Program Committee of IEEE Canadian Conference of Electrical and Computer Engineering 2004.
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Automatica journal homepage: www.elsevier.com/locate/automatica
Brief paper
Robust power control of multi-link single-sink optical networks with time-delays✩ Nem Stefanovic 1 , Lacra Pavel University of Toronto, Edward S. Rogers Sr. Department of Electrical and Computer Engineering, 10 King’s College Rd., Toronto, Ontario, Canada M5S 3G4
article
info
Article history: Received 11 September 2011 Received in revised form 2 December 2012 Accepted 2 April 2013 Available online 6 May 2013 Keywords: Time-delay Communication control applications Optical communication Robust stability Communication networks
abstract Optical networks provide high data transmission rates which require optimal power control algorithms to ensure the proper delivery of the signals. Time-delays may destabilize the control algorithms. We study the stability of game-theoretic based primal–dual control algorithms applied to multi-link singlesink optical networks. We present a perturbed optical signal-to-noise ratio (OSNR) model with timedelays. By optimizing the OSNR values, we reduce the bit error rates of the channels. We modify singular perturbation theory to include Lyapunov–Krasovskii stability theory with multiple time-delays and uncertainties. Simulations show the stabilization of perturbed systems at the expense of transient convergence time. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction We study the optical signal-to-noise ratio (OSNR) optimization problem in optical networks with time-delays. A signal’s OSNR relates to its bit error rate. Signal propagation delays in the optical fibers may destabilize the power control algorithms. A game-theoretic approach is applied in Pavel (2006, 2007) that yields a primal–dual controller. The work is generalized in Pan and Pavel (2009) for multi-link networks. In Pavel (2006, 2007) and Pan and Pavel (2009), time-delays are not considered. The single link case is analyzed in Stefanovic and Pavel (2009a) via a Lyapunov–Razumikhin approach combined with singular perturbation theory. The method in Stefanovic and Pavel (2009a) does not apply to multiple time-delays or uncertainties. Including timedelays and uncertainties in the stability analysis requires more sophisticated techniques such as Lyapunov–Krasovskii functionals with singular perturbation theory.
✩ A short version of this work appears in Stefanovic and Pavel (2009c). The material in this paper was presented at the 48th IEEE Conference on Decision and Control (CDC) and 28th Chinese Control Conference (CCC), December 16–18, 2009, Shanghai, China. This paper was recommended for publication in revised form by Associate Editor Yoshito Ohta under the direction of Editor Roberto Tempo. E-mail addresses: [email protected] (N. Stefanovic), [email protected] (L. Pavel). 1 Present address: McMaster University, Faculty of Engineering, Bachelor of
Technology - Four Year Program, 1280 Main St. West, Hamilton, Ontario, Canada, L8S 0A3. Tel.: +1 905 525 9140x20280; fax: +1 905 525 7015. 0005-1098/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.automatica.2013.04.009
Recent literature contains few results for the robust stability of optical networks with time-delays or uncertainties. The works Ghafouri-Shiraz and Karbassian (2009), Lauck, Charny, and Ramakrishnan (1996), Mokhtar, Azizoglu, and Barry (1996) and Rapet, Wang, Gangaji, and Tang (2007) explore channel use in optical CDMA networks, the effects of tuning delays for optical networks, end-to-end congestion control and communications over GEO-satellite links, respectively. In Baraniuk, Ahmed, and Khojestapour (2004), an optimal transmission rate and power control maximizes the network throughput. In Zhu and Pavel (2008), classic control techniques are applied to separately study additive uncertainty and time-delays. A converse Lyapunov Theorem is applied to time-delayed systems in Yeganefar, Pepe, and Dambrine (2007). The paper Jiang and Han (2006) studies the delay-dependent robust stability problem for linear uncertain systems using a Lyapunov–Krasovskii approach. Linear singularly perturbed systems with time-delays are studied in Dragan and Ionita (2000); Fridman (2006). The paper Glizer (2009) studies nonstandard singularly perturbed systems. In this paper, we study multi-link single-sink optical networks with time-delays. We present a time-delayed OSNR model with input multiplicative and additive uncertainty. We derive stability conditions that ensure OSNR optimization for both time-delays and uncertainties. Multi-link networks with multiple time-delays and uncertainties cause analytical complications. The main results are novel and derive via singular perturbation theory, modified to handle Lyapunov–Krasovskii functionals with time-delays and uncertainties. This paper is organized as follows. Section 2 reviews the time-delay stability results from Stefanovic and Pavel (2009a,b).
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Fig. 1. Multi-link single-sink network with time-delays.
Section 3 presents the main results of the paper. We present simulations in Section 4 and conclusions in Section 5. Fig. 2. Closed loop system with delays from (3), (4) and (1).
2. Background Stefanovic and Pavel (2009a) using a Razumikhin approach. The primal–dual algorithms from Stefanovic and Pavel (2009a) are
2.1. OSNR model with time-delays An optical network is composed of a set of links, L = {1, . . . , L}, that interconnect the optical nodes, where channels are added, dropped or rerouted. Each link, ℓ ∈ L, is composed of a series of Nℓ optical spans, containing one optical amplifier and 100 km s of fiber. The set of channels multiplexed and transmitted over the network is denoted by M = {1, . . . , n}. We denote by Ri , for i ∈ M , the set of links from the transmitter (Tx) to the receiver (Rx) in the optical path of channel i. The optical input power at Tx, the output signal at Rx, and the output noise at Rx are denoted by ui > 0, si > 0, and n¯ i > 0, respectively. The optical signal-to-noise ratio (OSNR) for channel i ∈ M , is OSNRi = si /¯ni . The optical amplifiers (OAs) within the optical spans have channel dependent gains, Gℓ,k,i . The optical fibers have wavelength independent loss coefficients, Lℓ,k . Furthermore, OAs introduce amplified spontaneous emission (ASE) noise denoted by ASEℓ,k,i . The optical span transmission, for kth span on the ℓth link, and ith channel, is given by hℓ,k,i = Gℓ,k,i Lℓ,k , ∀k = 1, . . . , Nℓ . We define
N
ℓ the ℓth link transmission as Tℓ,i = q= 1 hℓ,q,i , ∀ℓ = 1, . . . , L. We assume that all OAs operate with the same total power target P0,ℓ , below the threshold for nonlinear effects (Agrawal, 1997). Thus, Gℓ,k,i = Gℓ,i ∀k = 1, . . . , Nℓ .
f
Consider the single-sink network in Fig. 1. We denote by τj the forward propagation delay of a signal from its source, j ∈ M , to the OSNR output. We denote by τib the backward propagation delay of f
a signal from the OSNR output to the source, i. Let τi,j = τj + τib .
Lemma 1. The OSNR for the ith channel in a single-sink network is given as OSNRi (t ) =
ui (t − τi,i ) n0 , i + Γi,j uj (t − τi,j )
(1)
Γi,j =
ℓ∈Ri ∩Rj k=1
Gkℓ,i
ℓ−1 T q ,j q=1
Tq,i
ASEℓ,k,i P0,ℓ
ϵ
=η
dui (t )
and n0,i is the noise optical power at Tx for channel i.
uj,in (t ) − P0
βi − ui (t − τ ) µ ¯in (t ) 1 1 − Γi,i ui (t − τ ) − ai OSNRi,in (t )
= ρi
dt
(3)
(4)
where (3) and (4) are on ‘‘slow’’ and ‘‘fast’’ time-scales, respectively. The small gain, ϵ > 0, represents the fast dynamics of dui (t )/dt. We explicitly denote the fast time-scale by tf , and the slow time-scale by t. The two time-scales are related by t = ϵ tf . We define τ = ϵ τ˜ , where τ˜ is the time-delay τ represented in the ‘‘fast’’ time-scale tf . The parameters uj,in (t ), OSNRi,in (t ) and µ ¯ in (t ) represent the channel powers uj (t ), the OSNR measurements OSNRi (t ), and the channel price µ(t ), respectively, as inputs to the primal–dual algorithms. Our control parameters are ρi > 0. The design parameter, βi , and step size η, are both adjustable. Note that P0 is P0,ℓ for the link ℓ associated with the OSNR output. Fig. 2 depicts control algorithms (3)–(4) acting on the timedelayed single-link OSNR system (1). We substitute µ ¯ in = µ(t − τ b ), OSNRi,in (t ) = OSNRi (t ) and ui,in = u(t − τ f ). Convergence to 1 the NE is ensured if ai > j̸=i Γi,j and ai > 2 j̸=i (Γi,j + Γj,i ) and if
σ (ρ Γ˜ + Γ˜ T ρ) 2 σ¯ ((ρ Γ˜ )2 ((ρ Γ˜ )2 )T )
τ˜ <
(5)
where τ˜ is the time-delay with respect to the fast time-scale tf , σ and σ¯ are the minimum and maximum singular values, ρ = diag(ρi ), and Γ˜ is defined as 1,
Γi,j ai
(2)
j =1
Γ˜ i,j =
where Γi,j , elements of the (n × n) system matrix Γ , are
dt
i=j
j∈M
Nℓ Gkℓ,j
dµ(t )
,
i ̸= j.
(6)
Note that (5) was derived in Stefanovic and Pavel (2009a) by applying the singular perturbation theory modified to handle Razumikhin theory to (3), (4) and (1). We decomposed the closed-loop system into its reduced and boundary-layer forms, which produced a linear time-delayed system. A Razumikhin based LMI that ensured stability reduced to (5).
2.2. Game-theoretic based primal–dual control 3. Main results A game-theoretic based channel algorithm, or primal algorithm, that reaches a Nash Equilibrium (NE) and produces optimal OSNR values is presented in Pavel (2006) without time-delays. A dynamic pricing term, µ > 0, ensures that the total link power is below the threshold for nonlinear effects (Pavel, 2007). A stability analysis of the algorithms of Pavel (2006, 2007) with delays is done in
3.1. Single link with time-delay and uncertainty We first present some ancillary time-delay theory. Define
C ([−r , 0], ℜn ) as the set of all continuous functions mapping [−r , 0] to ℜn . Let C = C ([−r , 0], ℜn ).
N. Stefanovic, L. Pavel / Automatica 49 (2013) 2261–2266
Definition 1. The continuous norm is defined as ∥yt ∥c = max−r ≤θ≤0 ∥y(t + θ )∥2 where yt ∈ C , is the set, y(q) for all q ∈ [t − r , t ], and ∥ · ∥2 is the Euclidean norm. We study the linear time-invariant systems of the form x˙¯ (t ) = A0 x¯ (t ) + A1 x¯ (t − τ )
the uncertainty (1 + δi ) in u1i with its worst case (1 − q¯ i ). Thus, (14) becomes
ϵ u˙ i (t ) = ρi (7)
where A0 and A1 are (n × n) real matrices. The following restricted Lyapunov–Krasovskii lemma (Gu, Kharitonov, & Chen, 2003, Proposition 5.2, p. 150) gives conditions for the asymptotic stability of linear time-delay systems. Lemma 2. The linear time-delayed system (7) is asymptotically stable if there exists a Lyapunov–Krasovskii functional V such that for some ψ1 > 0 and ψ2 > 0
ψ1 ∥y(t )∥22 ≤ V (yt ) ≤ ψ2 ∥yt ∥2c
(8)
where the norm ∥yt ∥c is defined in Definition 1 and the derivative of V (yt ) along the trajectory V˙ (yt ) satisfies V˙ (yt ) = V ′ (yt ) where V (yt ) or ′
dyt dt dV dyt
≤ −ψ1 ∥y(t )∥22
(9)
denotes the Fréchet derivative of V .
Next, we introduce uncertainty in the OSNR model (1) for the single-link case. The system gains, Γi,j , are actually time-varying due to slow parameter drift and ASE. There is also uncertainty in the time-delays τ f and τ b due to system reconfigurations. We also have uncertainty in the transmitter noise at the sources due to imperfect measurements. Thus, we define Γ∆ ∈ ΩΓ , where
ΩΓ = {Γ + 1Γ | ∥1Γ ∥2 ≤ qˆ }
(10)
2263
βi n∆,0,i − µ(t − τ b ) (1 − q¯ i )ai
−
j∈Mℓ
i
Γ˜ 1i,j (1 + δj )uj (t − τ )
(16)
where Γ˜ 1i,j = Γ˜ i,j + 1Γi,j /(1 − q¯ i )ai , such that
Γ˜ i,j =
1 1 − q¯ i
,
i=j
Γi,j , (1 − q¯ i )ai
(17) i ̸= j.
Rewrite (16) and (15) using x = µ − µ∗ and zi = ui − u∗i , where µ∗ and u∗i are equilibrium points of (16) and (15), x˙ = η1row (I + diag(δj ))z (t − τ f )
ϵ z˙ = ρ β
−x(t − τ b ) − τ b ) + µ∗ )
µ∗ (x(t
(18)
− Γ˜ ∆ (I + diag(δj ))z (t − τ ) (19)
where z is a vector with elements zi , ρ = diag(ρi ), β is a column vector with elements βi , and 1row is a row vector of 1 elements. Eqs. (18) and (19) represent the perturbed, closed-loop system with time-delay shifted about the equilibrium points (µ, u∗ ). We rewrite (18) and (19) using the coordinate shift, y(t ) = z (t − τ f ) − h(x(t )), where h(x(t )) is the isolated root of the RHS of (19),
where Γ is a matrix with elements Γi,j , 1Γ is a matrix with elements 1Γi,j , and qˆ is the independent uncertainty bound. We define the uncertain input, u1j ∈ Ωj
h(x(t )) = (I + diag(δi ))−1 Γ˜ ∆−1 β
Ωj = {uj + δj uj | ∥δj ∥2 ≤ q¯ j } ∀j
and I + diag(δi ) is invertible, Γ˜ ∆ = Γ˜ + diag( ((1−¯1q )a ) )1Γ , Γ˜ is a
(11)
−x(t ) µ∗ (x(t ) + µ∗ )
(20)
i
i
where q¯ j is the independent uncertainty for input j. We define the uncertain transmitter noise, n∆,0,i ∈ Ωni ,
matrix with elements defined in (17). Later, Lemma 4 ensures Γ˜ ∆ is invertible. Next, define
Ωni = {n0,i + δni n0,i | ∥δni ∥2 ≤ qni } ∀i
f (y(t ) + h(x(t ))) = η1row I + diag(δj ) (y(t ) + h(x(t )))
(21)
g (y(t − τ )) = −ρ Γ˜ ∆ (I + diag(δi ))y(t − τ )
(22)
(12)
where qni is the uncertainty bound on the transmitter noise for channel i. The uncertain OSNR model is OSNR1i (t ) =
u1 i ( t − τ ) n∆,0,i +
Γ1i,j u1j (t − τ )
j∈Mℓ
.
(13)
ϵ u˙ i (t ) = ρi
βi + µ(t − τ b )
Γi,i ai
− 1 ui (t − τ )
1 ui ( t − τ ) − n∆,0,i + Γ1i,j u1j (t − τ ) . ai u1i (t − τ ) j∈M
to get the coordinate shifted closed-loop system
ϵ
dy(t ) dt
= g (y(t − τ )) − ϵ
(14)
ℓi
dyˆ dtf
In (3), let uin,j = (1 + δj )uj (t − τ f )
(15)
j∈M
In the foregoing analysis, we use u1j = uj + δj uj = (1 + δj )uj since uj is scalar. In the denominator of the last term of (14), we replace
(23) dh(x(t )) dx
f (y(t ) + h(x(t ))).
(24)
We next rewrite (23) and (24) using two separate time-scales. Let t = ϵ tf , where tf and t are the fast and slow time-scales, respectively. Also, let τ = ϵ τ˜ . Thus, x˙˜ (t ) = η1row Γ˜ ∆−1 β
f µ( ˙ t) = η (1 + δj )uj (t − τ ) − P0 .
x˙ = f (y(t ) + h(x(t )))
i
We substitute (13) into the primal–dual algorithms (3) and (4) to obtain the closed-loop uncertain system with time-delays. In (4), let OSNRi,in (t ) = OSNR1i (t ) and µin,i (t ) = µ(t − τ b ) to get
−˜x(t ) µ∗ (˜x(t ) + µ∗ )
= −ρ Γ˜ ∆ (I + diag(δi ))ˆy(tf − τ˜ )
(25) (26)
where yˆ (tf ) and x˜ (t ) are the distinct solutions. The following lemma presents the LMI that ensures exponential stability for the boundary-layer system (26). The proof is derived similarly to Gu et al. (2003) and is omitted here. The result is less conservative than Lemma 2 from Stefanovic and Pavel (2009a) which is based on Lyapunov–Razumikhin theory.
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N. Stefanovic, L. Pavel / Automatica 49 (2013) 2261–2266
Lemma 3. The boundary-layer system (26) is exponentially stable if there exist symmetric, positive definite matrices P and S1 such that τ˜ satisfies
H1 H2T
H2 H3
<0
(27)
where
H1 =
Mn2 −(AT0 )2 P
H2 =
1
PE ∗
−PA20 −S1 + µ ˜ 1 (G∗ )T G∗ (G∗ )T
0
−PA0 E ∗
τ˜ −AT0 (G∗ )T ∗ T
0
3.2. Single-sink network analysis
µ ˜ 1 (G∗ )T D∗
(D ) −I −(E ∗ )T (G∗ )T
−I ∗ H3 = D
We extend the single-link results of the previous section to the single-sink case (see Fig. 1). Let the uncertainty set for the perturbed system Γ∆ ∈ ΩΓ be defined as
0
−G∗ E ∗ −µ ˜ 1 (I − (D∗ )T D∗ )
ΩΓ = {Γ + 1Γ | ∥1Γi,row ∥2 ≤ qˆ i ∀i}
E2 = −ρ diag E∗ = ∗
G =
qˆ E2 k1
qˆ
k1
I
1
(1 − q¯ i )ai A0 diag
q¯ i k¯ i
√ q¯ i , diag k¯ i
qˆ
D = 0 0 ∗
k1
diag
q¯ i .
Lemma 4. The uncertain reduced system (25) is exponentially stable if
(Γi,j + Γj,i ) + qˆ ∀i.
(34)
i
where the parameters are defined as in Lemma 1. We formulate the closed-loop system as follows. Substitute (34) into (3) and (4), f modified such that for each channel i, τ = τi,i , τ f = τi , τ b = τib , to obtain f
x˙ = η1row (I + diag(δj ))z (t − τJ )
(35)
ϵ z˙ = ρ β¯ R(x(t − τIb )) −
Γ˜ 1i,row
1
µ∗ (r1 + µ∗ )
∥˜x∥22
(36) f
Proof. We prove asymptotic stability for (23) and (24) using a composite Lyapunov functional based on the reduced and boundary layer system functionals. For the reduced system (25), we select the Lyapunov function V (˜x) = 12 x˜ 2 . If (28) holds, then
× (I + diag(δj ))z (t − τi,J )
where x = µ−µ∗ , z is a vector with elements zi = ui −u∗i , z (t −τJ )
Theorem 1. Consider the system (18) and (19), with uncertainty sets defined in (10)–(12). There exists an ϵ ∗ > 0 such that for 0 < ϵ < ϵ ∗ the origin is asymptotically stable if (27) and (28) are satisfied.
∂V f (h(˜x)) ≤ −η1row Γ˜ ∆−1 β ∂ x˜
j∈Mℓ
(28)
We state the main theorem based on Lemmas 3 and 4.
u1i (t − τi,i ) n∆,0,i + Γ1i,j u1j (t − τi,j )
i∈M
1 2 j= ̸ i
OSNR1i (t ) =
0
The proof of the following lemma is omitted because it follows similarly to Lemma 3 in Stefanovic and Pavel (2009a). Lemma 4 incorporates uncertainties unlike Lemma 3 in Stefanovic and Pavel (2009a).
ai >
(33)
where Γ is a matrix with elements Γi,j , 1Γ is a matrix with elements 1Γi,j , 1Γi,row is an (n × n) matrix with the ith row equal to the ith row of 1Γ and the remaining terms equal to zero, and qˆ i is a direct and independent uncertainty bound. The uncertain OSNR model for multiple time-delays and uncertainties (33), (11) and (12) is
and µ ˜ 1 > 0, A0 = −ρ Γ˜ , Mn2 = τ1˜ (PA0 + AT0 P ) + S1
obtained from g (ˆy(tf − τ˜ )) as in Curtain and Zwart (1995, Theorem 2.4.6, p. 60). We define the composite Lyapunov functional χ (x(t ), yt ) = V (x(t )) + W (yt ), where V (x) = 12 x2 , W (yt ) is the converse Lyapunov functional (see Lemma 33.1 in Krasovskii, 1963 Lemma 33.1 and Yeganefar et al., 2007) of (26) with yt as the argument. By applying the composite Lyapunov function, χ (x(t ), yt ), to (23) and (24), and exploiting the Lyapunov inequalities in (29)–(32) and various Lipschitz properties, we prove asymptotic stability by Lemma 2. The rest of the proof follows as in Khalil (2002, Theorem 11.4).
(29)
f j
is a vector with elements zj (t − τ ), z (t − τi,J ) is a vector with elements zj (t − τi,j ), Γ˜ 1i,row is an (n × n) matrix of zeros except for the ith row which has the jth element Γ˜ 1i,j , β¯ = diag(βi ), and R(x(t − τIb ))
−x(t − τ1b ) = µ∗ (x(t − τ1b ) + µ∗ )
···
−x(t − τnb ) µ∗ (x(t − τnb ) + µ∗ )
T
.
We apply the following coordinate shift to (35) and (36), given by f ys (t ) = z (t − τJ ) − h¯ (x(t )), where h¯ (x(t )) is the isolated root of the RHS of (36),
where x˜ ≤ r1 for r1 > −µ∗ , for r1 arbitrarily large. By Lemma 3, the boundary-layer system (26) is exponentially stable if τ˜ satisfies (27). By Krasovskii (1963, Lemma 33.1) and Yeganefar et al. (2007), there exists a Lyapunov functional W (ˆytf ),
We apply similar steps used in (18)–(24) to derive the following reduced and boundary-layer systems
ψ∥ˆytf ∥2c ≤ W (ˆytf ) ≤ K ∥ˆytf ∥2c
(30)
x˙¯ s (t ) = η1row Γ˜ ∆−1 β¯ R(¯xs (t ))
W ′ (ˆytf ) gtf (ˆy(tf − τ˜ )) ≤ −k∥ˆytf ∥2c ′ W (ˆyt ) ≤ kF ∥ˆyt ∥c f f c
(31)
dyˆ s
˙ (ˆytf ) = where W
dW (ˆytf ) dtf
= W ′ (ˆytf )
(32) dyˆ tf dtf
= W ′ (ˆytf ) gtf , with W ′ (ˆytf )
denoting the Fréchet derivative of W and
dyˆ tf dtf
= gtf (ˆy(tf − τ˜ )) is
h¯ (x(t )) = (I + diag(δi ))−1 Γ˜ ∆−1 β¯ R(x(t )).
dtf
= −ρ
Γ˜ 1i,row (I + diag(δi ))ˆys (tf − τ˜i,i )
(37)
(38) (39)
i
where we denote by x¯ s (t ) and yˆ s (tf ) as the solutions. The following lemma ensures exponential stability for the boundary-layer system (39). The proof derived from a similar procedure in Gu et al. (2003) and is omitted here.
N. Stefanovic, L. Pavel / Automatica 49 (2013) 2261–2266
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Lemma 5. The boundary-layer system (39) is exponentially stable if there exist symmetric, positive definite matrices P and Si,j such that
Mn J1T T J2 J3T
J1 J7 J4T J5T
J2 J4 J6 J9T
J3 J5 <0 J9 J8
(40)
where
∗ −PAs,n As ), J2 = PEs = −PAs,1 Es∗ · · · −PAs,n Es∗ T 0 ··· 0 = −G∗ A · · · −G∗ A s,1 s s ,n s −I (D∗s )T = diag(µi (G∗s )T D∗s ), J6 = D∗s −I 1 = diag − Si + µi (G∗s )T G∗s τ˜i,i
J1 = (−PAs,1 As J3 J4 J5 J7
1
···
τ˜1,1
(G∗s, )T
Fig. 3. Perturbed system, no robust compensation, ρ = 0.01.
J8 = diag(−µi (I − (D∗s )T D∗s ))
J9 =
Mn =
··· ···
0 −G∗s,1 Es∗ 1
P
τ˜1,1
n
0 −G∗s,n Es∗
As,i +
n
i=1
ATs,i P
i=1
+
n n
τ˜i,i Si,j
i =1 j =1
1
, As,i = −ρ Γ˜ i,row ∀i (1 − q¯ i )ai As = (As,1 · · · As,n ), Si = diag(Si1 · · · Sin ) √ ∗ Es = diag( qi ki )Es2 As diag(k¯ i )Λ T √ qi 1 G∗s = diag In diag Λ ki k¯ i √ qi ¯ 0 diag I diag ( k ) Λ n i . D∗s = ki
Es2 = −ρ diag
0
0
Fig. 4. Perturbed system, robust compensation, ρ = 0.002.
√
Here, Λ = diag(diag( q¯ i )), and Γ˜ i,row is an (n × n) matrix with the ith row equal to the ith row of Γ˜ and all other entries set to zero. The matrix G∗s,i is the ith (2n2 × n) submatrix of G∗s . We define G∗s, =
G∗s,i . The parameters ki , k¯ i and µi are design variables such
that k¯ n×(i−1)+j < ki for all i, j = 1, . . . , n. The matrix In is an (n × n) identity matrix. The control parameters to satisfy (40) are ρi and ai . We rewrite Lemma 4 to account for multiple uncertainties qˆ i . The proof is similar to the proof in Lemma 3 in Stefanovic and Pavel (2009a). Lemma 6. The uncertain reduced system (38) is exponentially stable if ai >
1 2 j= ̸ i
(Γi,j + Γj,i ) + qˆ i ∀i.
(41)
We now state the main stability theorem based on Lemmas 5 and 6. The proof follows as in Theorem 1. Theorem 2. Consider the system (35) and (36), with uncertainty sets defined in (33), (11) and (12). There exists an ϵ ∗ > 0 such that for 0 < ϵ < ϵ ∗ the origin is asymptotically stable if (40) and (41) hold.
4. Simulations We simulate the single-sink network in Fig. 1 using realistic network data. Single-sinks realistically appear as substructures within larger optical network systems. There are four channels on five links with one common output. The total round trip delays for channels 1–4 are 20 ms, 30 ms, 40 ms and 50 ms, respectively. The amplifier gain is parabolic in shape with G = −4 × 1016 × (λ − 1555 × 10−9 )2 + 15 decibels, where λ represents the channel wavelength. The βi values are selected by the same pricing strategy as in Stefanovic and Pavel (2009a). Light propagates at approximately 200,000 km/s within the fiber for a round-trip time-delay of 10 ms on the link. The channel and link algorithms are updated every 5 ms, and 0.5 s, respectively, i.e., ϵ = 0.01. We fix ki = 1.1 and k¯ i = 1. We introduce a uniform uncertainty into the system, i.e., 1Γi,j = w , for w = q/n × diag(ai /ρi ), where n is the size of the system matrix, which implies ∥F ∥2 ≤ q. We select the uncertainty parameters q¯ i = qni = 0.3 for 30% uncertainty in the input signals and noise. We select qˆ i = 1.07 × 10−2 , which is a large uniform uncertainty. Fig. 3 shows the system response without uncertainty compensation. The LMI condition (40) is not satisfied, and we see instability in the channels. We compensate by decreasing ρi to 0.002 and increasing ai by a factor of 4 to satisfy (40). Fig. 4 shows
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this case. The LMI (40) is very complicated, possessing multiple time-delays. We do not have explicit equations to obtain theoretical upper bounds for each of the time-delays in the system. We rely on tuning ρi and ai to ensure (40) is satisfied. 5. Conclusions We studied a primal–dual control approach for the single-sink optical networks with time-delays. We derived stability conditions using a combined Lyapunov–Krasovskii and singular perturbation theory approach. Simulations showed robust compensators stabilized uncertain optical systems at the expense of convergence time. Acknowledgment We gratefully acknowledge the support of the Natural Science and Engineering Research Council of Canada.
Mokhtar, A., Azizoglu, M., & Barry, R. A. (1996). Impact of tuning delay on the performance of bandwidth-limited optical broadcast networks with uniform traffic. IEEE Journal on Selected Areas in Communications, 14(5), 935–944. Pan, Yan, & Pavel, Lacra (2009). Games with coupled propagated constraints in optical networks. In GameNets. Turkey, May. Pavel, L. (2006). A noncooperative game approach to OSNR optimization in optical networks. IEEE Transactions on Automatic Control, 51(5), 848–852. Pavel, L. (2007). An extension of duality to a game-theoretic framework. Automatica, 43(2), 226–237. Rapet, P., Wang, R., Gangaji, S., & Tang, L. (2007). Experimental investigation of rate-based and delay-tolerant transmission mechanisms over geo-satellite communication links. In The proceedings of WCNC 2007 (pp. 3717–3721). Stefanovic, N., & Pavel, L. (2009a). A stability analysis with time-delay of primal–dual power control in optical links. Automatica, 45(5), 1319–1325. Stefanovic, N., & Pavel, L. (2009b). An analysis of stability with time-delay of link level power control in optical networks. Automatica, 45(1), 149–154. Stefanovic, N., & Pavel, L. (2009c). Robust power control of single sink optical networks with time-delays. In 48th IEEE conference on decision and control (pp. 2034–2039). Shanghai, China, December. Yeganefar, N., Pepe, P., & Dambrine, M. (2007). Input-to-state stability and exponential stability for time-delay systems: further results. In IEEE conference on decision and control (pp. 2059–2064). December. Zhu, Quanyan, & Pavel, Lacra (2008). State-space approach to pricing design in OSNR Nahs game. In 17th IFAC world congress (pp. 12001–12006). July.
References Agrawal, G. P. (1997). Fiber-optic communication systems. New York: John Wiley and Sons. Baraniuk, R.G., Ahmed, N., & Khojestapour, M.A. (2004). Delay-limited throughput maximization for fading channels using rate and power control. In Globecomm 2004 (pp. 3459–3463). Curtain, R. F., & Zwart, H. (1995). An introduction to infinite dimensional linear systems theory. New York: Springer-Verilag. Dragan, V., & Ionita, A. (2000). Exponential stability for singularly perturbed systems with state delays. In Proc. of the 6th colloquium on the qualitative theory of differential equations (pp. 1–8). Szeged, Hungary, August. Fridman, E. (2006). Robust sampled-data h∞ control of linear singularly perturbed systems. IEEE Transactions on Automatic Control, 51(3), 470–475. Ghafouri-Shiraz, H., & Karbassian, M. M. (2009). IP routing and traffic analysis in coherent optical CDMA networks. Journal of Lightwave Technology, 27(10), 1262–1268. Glizer, V. Y. (2009). l2 -stabilizability conditions for a class of nonstandard singularly perturbed functional-differential systems. Dynamics of Continuous, Discrete and Impulsive Systems, Series B: Applications and Algorithms, 16, 181–213. Gu, K., Kharitonov, V. L., & Chen, J. (2003). Stability of time-delay systems. Boston: Birkhäuser. Jiang, Xiefu, & Han, Qing-Long (2006). Delay-dependent robust stability for uncertain linear systems with interval time-varying delay. Automatica, 42, 1059–1065. Khalil, H. K. (2002). Nonlinear systems. New Jersey: Prentice-Hall. Krasovskii, N. (1963). Stability of motion. Stanford University Press. Lauck, A., Charny, A., & Ramakrishnan, K. K. (1996). Time scale analysis and scalability issues for explicit rate allocation in ATM networks. IEEE Transactions on Networking, 4(4), 569–581.
Nem Stefanovic obtained his Ph.D. in electrical and computer engineering from the University of Toronto in 2010. He spent one year developing and analyzing control systems for the ACR-1000 and EC6 nuclear reactors while at Atomic Energy of Canada. Since September 2010, Dr. Stefanovic has been an Assistant Professor at McMaster University in the Faculty of Engineering, Bachelor of Technology program. His area of expertise is the control of optical networks, linear and nonlinear systems, time-delay systems, and process automation.
Lacra Pavel (M’92–SM’04) received the Ph.D. degree in electrical engineering from Queen’s University, Canada, in 1996. She spent a year at the Institute for Aerospace research (NRC) in Ottawa as a NSERC Postdoctoral Fellow. From 1998 to 2002, she worked in the optical communications industry. In August 2002 she joined the Electrical and Computer Engineering Department at University of Toronto, where she is currently an Associate Professor. Her research interests include system control and optimization in optical networks, game theory, robust and H-infinity optical control. Dr. Pavel served as Publications Chair of Conference on Decision and Control 2006, Technical Program Committees of INFOCOM 2007, IEEE Control Applications Conference 2005; Associate Chair (Control) on the Program Committee of IEEE Canadian Conference of Electrical and Computer Engineering 2004.