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itnes1. Introduction

In this paper, we consider the following class of nonlinear timedelay systems, which is created by generalizing the continuous-timemodel of FAST TCP proposed in Jin, Wei, and Low (2004), Koo,Choi, and Lee (2008) and Wang, Wei, and Low (2005):

xi(t) = xi(t)+ ii + f (xi)xi(t)+ i (1)

with the initial condition

xi() = i() 0 [ , 0] , (2)where xi := [x1(t i1(t))x2(t i2(t)) xn(t in(t))]T andi {1, 2, . . . , n}. Let R+ and Rn+ be the set of all nonnegativereal numbers and the set of all n-dimensional nonnegative realvectors, respectively, then in (1) and (2), xi(t) R+, xi Rn+, iis a positive constant, i is a positive control parameter, ij(t) is acontinuously time-varying delay such that 0 ij(t) < for all t 0 and j {1, 2, . . . , n}, i : [ , 0] 7 R+ iscontinuous, and f : Rn+ 7 R is continuously differentiable. It isnoticeable that the class of systems (1) includes a delay logisticequation considered in Gopalsamy and Weng (1993).There have been a number of published results considering the

stability of the systems in a similar form to that of (1) (Marcus &

I This paperwas not presented at any IFACmeeting. This paperwa recommendedfor publication in revised form by Associate Editor Dragan Nei under the directionof Editor (Rapid Publications) Andr L. Tits. Tel.: +82 51 510 2490; fax: +82 51 515 5190.E-mail address: jyc@pusan.ac.kr.

Westervelt, 1989; Mazenc & Niculescu, 2001; van den Driessche &Zou, 1998; Zeng, Wang, & Liao, 2003; Zhang, 2003). In Mazenc andNiculescu (2001), nonlinear time delay systems in a general formwere studied, and a sufficient condition for the delay independentglobal asymptotic stability was shown in terms of a Lyapunovfunction. However, the analysis scheme in Mazenc and Niculescu(2001) cannot be applied to (1) because only a time-invariantdelay is considered in Mazenc and Niculescu (2001) and the highnonlinearity of (1) makes it difficult to construct a Lyapunovfunction that satisfies the essential assumptions in Mazenc andNiculescu (2001) that are necessary for stability. On the other hand,nonlinear time delay systems describing neural networks are ina similar form to that of (1), and the stability properties of theneural networkswere investigated by a variety of analysis schemesin Marcus and Westervelt (1989), van den Driessche and Zou(1998), Zhang (2003) and Zeng et al. (2003). Even these analysisschemes cannot be applied to (1) because of the high nonlinearityof (1).A common approach to analyzing the stability of nonlinear

time delay systems is to use the Krasovskii stability theory orthe Razumikhin stability theory (Hale & Lunel, 2002; Jankovic,2001; Niculescu, 2001). In order to apply those theories, it is firstneeded to construct or find LyapunovKrasovskii functionals orLyapunovRazumikhin functions that are suitable for the consid-ered time delay systems. It, however, turns out that it is excessivelydifficult to obtain such functionals or functions for the analysis of(1) because of the high nonlinearity and various amounts of time-varying time delays in (1). Moreover, it is known that the analysisresults based on the Krasovskii or Razumikhin stability theory aremore or less conservative (Niculescu, 2001).In order to overcome these disadvantages, we exploit the inher-

ent properties of (1) in analysis instead of applying the KrasovskiiAutomatica 45 (2

Contents lists availa

Autom

journal homepage: www.els

Technical communique

Global stability analysis scheme for a clasJoon-Young Choi School of Electrical Engineering, Pusan National University, 30, Jangjeon-dong, Geumjeong

a r t i c l e i n f o

Article history:Received 8 December 2007Received in revised form3 June 2009Accepted 28 June 2009Available online 25 July 2009

Keywords:Nonlinear time delay systemsGlobal asymptotic stabilityInternet congestion controlFAST TCP

a b s t r a c t

We consider a class of nonlan Internet congestion contrthe global asymptotic stabilitwo sequences that represeand showing that the two sresults exemplify that thesufficient condition is a closeasymptotic stability.0005-1098/$ see front matter 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2009.06.017009) 24622466

ble at ScienceDirect

atica

evier.com/locate/automatica

s of nonlinear time delay systemsI

-gu, Busan, 609-735, South Korea

near time delay systems created by generalizing the model for FAST TCP,ol algorithm. We achieve a time delay independent sufficient condition fory of the class of systems. The sufficient condition is verified by constructingt the lower and upper bound variations of the system trajectory in time,quences converge to the equilibrium point of the system. The simulationufficient condition is valid for global asymptotic stability, and that theapproximation to the unknown necessary and sufficient condition for global

2009 Elsevier Ltd. All rights reserved.

J.-Y. Choi / Automatica

or Razumikhin stability theory. We present a time delay indepen-dent sufficient condition for the global asymptotic stability. Thesufficient condition is verified by constructing two sequences thatrepresent the lower and upper bound variations of the system tra-jectory along the current of time, and showing that the two se-quences converge to an identical point, which is the equilibriumpoint of the system.This paper is organized as follows. Section 2 states the problem.

Section 3 shows the boundedness property of the system tra-jectory. Section 4 presents a sufficient condition for the globalasymptotic stability. Section 5 provides simulation results and dis-cussions. Section 6 makes conclusions.

2. Problem statement

The following notations will be used throughout the paper forthe brevity of expression: 0 := [0 0 0]T Rn+; :=[1 2 n]T Rn+; := [11 22 nn]T Rn+; andt > 0 is an exponentially decaying to zero term. We assume thatthe function f () in (1) has the following properties, whichwill playsignificant roles in the subsequent analysis.

Assumption 1. The continuously differentiable function f : Rn+7 R in (1) satisfies the following properties:(i) There exists an equilibrium point of (1) x = [x1 x2 xn]T

Rn+ such that

x = + 1f (x)

, (3)

where f (x) > 0.(ii) f (0) 0.(iii) For all i {1, 2, . . . , n}, 0 f (x)

xifor x Rn+.

(iv) For all i {1, 2, . . . , n},there exist positive constants and isuch that i f (x)xi on {x Rn+ | < |x| 0}.

We explain detailed meanings of Assumption 1 as follows. Theproperty (i) of Assumption 1 is to guarantee the existence of theequilibrium point of (1). The property (iii) means that the functionf (x) is increasing with respect to each independent variable onits domain Rn+. The property (iv) means that once < |x| 0, f (x) is strictly increasing with respect to eachindependent variable xi, and moreover, f (x) as xi .In addition, the property (i) and (iii) also guarantee the uniquenessof the equilibrium point of (1) as shown in the following lemma.

Lemma 2. The equilibrium point of (1) represented by (3) is uniqueunder the condition(iii)of Assumption 1.

Proof. We suppose that x and y are the equilibrium points of(1). Then, from (3) and the mean value theorem, we derive thatx y =

(1f (x) 1f (y)

) = T (x y), where :=

[1 2 n]T, i := 1f (x)f (y) f (x)xix=zfor all i {1, 2, . . . , n},

z L (x, y), and L (x, y) is the line segment between x and y.Hence, it follows that

(In + T

)(x y) = 0, where In is the

n n identity matrix. On the other hand, since det (In + T) =det

(1+T

)(Chen, 1999); ii > 0 for all i {1, 2, . . . , n}

from the definition of i and i; and i 0 for all i {1, 2, . . . , n}from (i) and (iii) of Assumption 1, it follows that det

(In + T

)> 0, and the matrix

(In + T

)is invertible, which implies that

x = y. Therefore, it is proved that the equilibrium point of (1) isunique under the condition (iii) of Assumption 1. The class of systems described by (1) can cover even the systemswith the function f (x) that is highly nonlinear and expressed45 (2009) 24622466 2463

in an implicit form as long as the function f (x) satisfies theproperties in Assumption 1. The goal of this paper is to achieve asufficient condition for the global asymptotic stability of (1) withthe initial condition (2) in terms of the control parameter i underAssumption 1.

3. Boundedness

In this section, we investigate the boundedness properties ofthe solution of (1) xi(t). The following lemma shows that xi(t) andf (xi) in (1) are bounded below.

Lemma 3. xi(t) and f (xi) described by (1) are bounded below asxi(t) > 0 and f (xi) 0 for all t > 0 and i {1, 2, . . . , n}.Proof. Suppose that xi(0) > 0 and xi(t) = 0 for some value of t .Since xi(0) > 0, by continuity of solutions, such a value of t mustbe strictly greater than zero. Let t1 = inf{t : t > 0, xi(t) = 0}, thenxi(t1) should be non-positive. From (1), however, we obtain thederivative of xi(t) at t1 as xi(t1) = xi(t1)+ ii+f (xi)xi(t1)+ i =i > 0, which is a positive value, and this gives us a contradiction.Therefore, no such t1 exists, and it follows that xi(t) > 0 for allt > 0.Next, we consider the case when xi(0) = 0. In this case, we

obtain xi(0) from (1) as xi(0) = xi(0)+ ii+f (xi)xi(0)+i = i >0, which implies that xi(t) is increasing at t = 0, and by continuityof solutions, there must exist a time t2 > 0 such that xi(t) > 0for 0 < t < t2. Then, in the same way as in the case of xi(0) > 0,we can prove that xi(t) > 0 for all t > 0. Accordingly, it is provedthat xi(t) > 0 for all t > 0 with the initial condition xi(0) 0.In addition, from the initial condition (2), we have xi(t) 0 forall t , which implies along with (ii) and (iii) of Assumption 1that f (xi) 0 for all t > 0. Note that since it holds from Lemma 3 and the initial condition (2)that xi(t) 0 for all t and for all i {1, 2, . . . , n}, it wasadequate to select the domain of the function f (xi) in (1) as Rn+.The following lemma shows that xi(t) in (1) is bounded above.

Lemma 4. xi(t) described by (1) is bounded above for all i {1,2, . . . , n}.Proof. By Lemma 3 and (iii) of Assumption 1, it holds thatf (0, . . . , 0, xi(t ii), 0, . . . , 0) f (xi) for all xi Rn+,from which we obtain in view of (1) that xi(t) xi(t) +

ii+f (0,...,0,xi(tii),0,...,0)xi(t)+i.Hence, considering the comparisonlemma (Khalil, 1996), the upper boundedness of xi(t) is impliedby the upper boundedness of yi(t) described by the following firstorder system:

yi(t) = yi(t)+ ii + f (0, . . . , 0, yi(t ii), 0, . . . , 0)yi(t)+ i, (4)

where we define the equilibrium point as

yi := i(1+ i

f (0, . . . , 0, yi , 0, . . . , 0)

).

Now, we show the upper boundedness of yi(t) in (4) byconsidering two possible cases: yi(t) is non-oscillatory about yi oroscillatory about yi .If yi(t) is not oscillatory about yi , then there exists a T0 such

that either yi(t) yi for t T0 or yi(t) yi for t T0. If thesecond alternative holds, then we already have an upper boundof yi(t). If the first alternative holds, then we have yi(t) 0 fort T0 + by (4), which implies that limt yi(t) = y0 for some

yi y0

2464 J.-Y. Choi / Automatica

yi , and let yi(t) yi denote an arbitrary local maximum of yi(t),then 0 = yi(t) = yi(t) + ii+f (0,...,0,yi(tii),0,...,0)yi(t) + i,which implies along with Assumption 1 that

0 f (0, . . . , 0, yi(t ii), 0, . . . , 0) f (0, . . . , 0, yi , 0, . . . , 0) , (5)that is, yi(t ii) yi . A consequence of (5) is that thereexists [t , t] such that yi() = yi . Integrating (4) from tot, we have yi(t) = yi +

t

(yi(s)+ ii+f (0,...,0,yi(sii),0,...,0)yi(s)

+ i)ds yi + i

tds yi + i

tt ds = yi + i < ,

which completes the proof.

The following lemma shows an ultimate lower bound of xi(t) in (1).

Lemma 5. xi(t) described by (1) is ultimately bounded below asxi(t) > i for all sufficiently large t and i {1, 2, . . . , n}.Proof. From Lemma 3, we have i

i+f (xi )xi(t) > 0 for all t > 0.Applying this inequality to (1) yields xi(t) > xi(t) + i, fromwhich we obtain by the comparison lemma that

xi(t) > i t i {1, 2, . . . , n}. (6)Now, from Lemma 3, Lemma 4, and (6), theremust exist a constant > 0 such that < i

i+f (xi )xi(t) < for all sufficiently large t ,from which we have xi(t) > xi(t) + i + for all sufficientlylarge t . Applying the comparison lemma again, we obtain thatxi(t) > i + t for all sufficiently large t , which implies thatxi(t) > i for all sufficiently large t and for all i {1, 2, . . . , n}.

The following lemmas show the relationship between an ultimateupper bound and an ultimate lower bound of xi(t) in (1).

Lemma 6. Let := [1 2 n]T Rn+ be a constant vectorsuch that i i xi for all i {1, 2, . . . , n}. If f () > 0, andxi(t) > (i t) for all sufficiently large t and i {1, 2, . . . , n} in(1), then xi(t) < i

(1+ if ()

)+ t for all sufficiently large t and

i {1, 2, . . . , n}, where i(1+ if ()

) xi for all i {1, 2, . . . , n}.

Proof. Since f () in (1) is continuously differentiable, f () > 0implies that f ( Et) > 0 for all sufficiently large t , whereEt := [t t t ] Rn+. On the other hand, since xi(t) >(i t) (i t) for all sufficiently large t and ij(t) < , we have xj(t ij) > (j t) (j t) for allsufficiently large t and j {1, 2, . . . , n}, which implies in viewof (iv) of Assumption 1 and the condition f () > 0 that f (xi) >f ( Et) f ( Et) > 0 for all sufficiently large t and i {1, 2, . . . , n}. Applying this inequality to (1), and using Lemma 4,(iv) of Assumption 1, and the mean value theorem, we obtain thefollowing derivation: xi(t) < xi(t) + ii+f (Et )xi(t) + i 0, which implies thati

(1+ if ()

) i

(1+ if (x)

)= xi .

Lemma 7. Let := [1 2 n]T Rn+ be a constant vectorsuch that i xi for all i {1, 2, . . . , n}. If f () > 0, andxi(t) < (i + t) for all sufficiently large t and i {1, 2, . . . , n} in(1), then xi(t) > i

(1+ if ()

) t for all sufficiently large t and i {1, 2, . . . , n}, where i(1+ if ()

) xi for all i {1, 2, . . . , n}.45 (2009) 24622466

Proof. In the sameway as in the proof of Lemma 6, this lemma canbe proved.

4. Global asymptotic stability

In this section, we establish a sufficient condition for the globalasymptotic stability of (1). For this purpose,we exploit the inherentproperty of (1) instead of applying the Krasovskii or Razumikhinstability theory. We construct two kinds of sequence {ki } and{ki } for i {1, 2, . . . , n}, where k denotes the sequence index,that represent the lower and upper bound variations of xi(t)respectively along the progress of time under the condition thatf () > 0.We start with Lemma 5, which shows that xi(t) > i for all

sufficiently large t and i {1, 2, . . . , n}, and choose 1i as 1i := ifor all i {1, 2, . . . , n}. Then, we have xi(t) > (1i t) forall sufficiently large t and i {1, 2, . . . , n}, and we obtain fromLemma 6 that xi(t) < i

(1+ i

f (1)

)+ t := 1i + t for all

sufficiently large t , where 1i xi and k :=[k1

k2 kn

]Tfor k = 1, 2, 3, . . .. Then, applying (1i + t) to Lemma 7 yieldsxi(t) > i

(1+ i

f (1)

) t := 2i t for all sufficiently large t ,

where 2i xi and k :=[k1

k2...