Nonlinear DynDOI 10.1007/s11071-014-1459-1

ORIGINAL PAPER

Observer-based synchronization of memristive systemswith multiple networked input and output delays

Shiping Wen · Zhigang Zeng · Tingwen Huang

Received: 10 January 2014 / Accepted: 8 May 2014© Springer Science+Business Media Dordrecht 2014

Abstract This paper investigates the synchroniza-tion problem of memristive systems with multiple net-worked input and output delays via observer-basedcontrol. A memristive system is set up, and thefuzzy method has been employed to linearize thedynamical system of the memristive system; the net-worked input and output delays are considered inthe synchronization problem of this system. A trun-cated predictor feedback approach is employed todesign the observers. Under certain restrictions, a classof finite-dimensional observer-based output feedbackcontrollers is designed. A numerical example is carriedout to demonstrate the effectiveness of the proposedmethods.

Keywords Memristor · Synchronization ·Output feedback · Networked time delay

S. Wen · Z. Zeng (B)School of Automation, Huazhong University of Scienceand Technology, Wuhan 430074, Chinae-mail: zgzeng527@126.com

S. Wen · Z. ZengKey Laboratory of Image Processing and IntelligentControl of Education Ministry of China,Wuhan 430074, China

T. HuangTexas A & M University at Qatar, Doha 5825, Qatar

1 Introduction

Though the current digital computers can now pos-sess computing speed and complexity to emulate thebrain functionality of animals like a spider, mouse, andcat [1,2], the associated energy dissipation in the sys-tem grows exponentially along the hierarchy of ani-mal intelligence. However, the sequential processing offetch, decode, and execution of instructions through theclassical von Neumann bottleneck of conventional dig-ital computers has resulted in less efficient machines astheir eco-systems have grown to be increasingly com-plex [3]. For example, to perform certain cortical simu-lations at the cat scale even at 83 times slower firing rate,the IBM team [1] has to employ Blue Gene/P (BG/P), asupercomputer equipped with 1,47,456 CPUs and 144TBs of main memory. On the other hand, the humanbrain contains more than 100 billion neurons and eachneuron has more than 20,000 synapses. Efficient cir-cuit implementation of synapses, therefore, is espe-cially important to build a brain-like machine. How-ever, since shrinking the current transistor size is verydifficult, introducing a more efficient approach is essen-tial for further development of artificial intelligenceimplementations.

In 2008, the Williams group announced a successfulfabrication of a very compact and nonvolatile nanoscalememory called the memristor [4]. It was postulated byChua [5] as the fourth basic circuit element in elec-trical circuits. It is based on the nonlinear character-istics of charge and flux. By supplying a voltage or

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current to the memristor, its resistance can be altered[6]. In this way, the memristor remembers informa-tion. Several examples of successful multichip net-works of spiking neurons have been recently proposed[7–9]; however, there are still a number of practicalproblems that hinder the development of truly large-scale, distributed, and massively parallel networks ofvery large-scale integration neurons, such as how to setthe weight of individual synapses in the network. It iswell known that the changes in the synaptic connec-tions between the neurons are widely believed to con-tribute to memory storage and the activity-dependentdevelopment of artificial intelligence. These changesare thought to occur through correlation-based or Heb-bian plasticity.

On the other hand, synchronization problem of con-trol systems has attracted great attention due to itspotential applications in many fields such as securecommunications, biological systems, information sci-ence, image encryption, and pseudo-random numbergenerator [10,11]. Up to now, a wide variety of syn-chronization phenomena has been investigated suchas complete synchronization [12–14], generalized syn-chronization [15], phase synchronization [16], and lagsynchronization [17]. It is worth mentioning that inconnected electronic networks, the occurrence of timedelay is unavoidable due to finite signal transmissiontimes, switching speeds, and some other reasons. Thus,the complete synchronization of control systems is hardto implement effectively, and it is more reasonableto consider the input/output delays which have beenknown as the main sources to degrade the performanceof the control system [18]. In the last decade, significantprogress has been made on the analysis and synthesisissues for systems with various types of delays [19–33]. Meanwhile, in time-delay systems, there are manynew problems such as network-induced time delays inthe input and output terminations.

There are several categories of methods for han-dling asymptotic stability analysis and stabilization oftime-delay systems. The most efficient methods areLyapunov–Krasovskii functional-based methods. TheRazumikhin theorem-based approach also falls into thiscategory. The idea is to find a positive-definite func-tional such that its derivative along the trajectories ofthe time-delay system is negative. However, one draw-back of these methods is that in general only suffi-cient conditions can be obtained, which leads to con-servatism.

Another efficient approach for dealing with time-delay systems is the predictor feedback. This approachis especially effective for input-delay systems and hasreceived renewed interest in recent years. The exponen-tial stability for linear systems with distributed actua-tor delays and under predictor-based feedback has beenestablished by the use of Lyapunov functionals [34]. Apredictor feedback controller for a general system instrict-feedback form with delayed integrators, whichis a classic example of exponentially unstable systemswith simultaneous input and state delays, is proposed[35]. Predictor-based state feedback and output feed-back controllers are designed for general nonlinearsystems with time-varying input and state delays; andan approximated predictor for nonlinear systems withdelayed input is established [36]. This paper is focusedon observer-based feedback synchronization control ofmemristive systems with multiple networked input andoutput delays.

Motivated by the above discussion, in this paper,we investigate the problem of the observer-based feed-back synchronization control for memristive systemswith multiple input and output delays. The main con-tributions of this paper can be summarized as follows:(i) a fuzzy model is set up for a memristive circuit; (ii)both multiple input and output delays are considered;and (iii) a predictor feedback sychronization controlmethod is proposed to control the synchronization errorsystems.

The rest of the paper is organized as follows.In Sect. 2, the investigated circuit and the corre-sponding dynamic model are described. In Sect. 3,infinite-dimensional observer-based output feedback isdesigned for systems with multiple output delays. Anillustrative example is discussed to demonstrate theeffectiveness of the obtained results in Sect. 4. Finally,conclusions are drawn in Sect. 5.

Notation The notation used here is fairly standard.The superscript “T ” stands for matrix transposition,R

n denotes the n-dimensional Euclidean space, Rm×n

is the set of all real matrices of dimension m×n, and Iand 0 represent the identity matrix and zero matrix,respectively. The notation P ≥ 0 means that P is realsymmetric and positive definite (semidefinite). For apositive scalar τ , let Ln,τ = L([−τ, 0],Rn) denotethe Banach space of continuous vector functions map-ping the interval [−τ, 0] into R

n with the topologyof uniform convergence, and let xt ∈ Ln,τ repre-

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Observer-based synchronization of memristive systems

Y

X

Z

W

R1

R2

R3

R4

R5

R8

R6 R7

R9

R10

R11

R12

C1

C2

C3

C4

Fig. 1 A circuit with memristor

sent the restriction of x(t) to the interval [t − τ, t]translated to [−τ, 0], that is xt (s) = x(t + s), s ∈[−τ, 0]. Finally, for a linear system characterized bythe matrix pair (A, B), it is asymptotically null con-trollable with bounded controls if (A, B) is stabi-lizable in the ordinary linear systems and all theeigenvalues of A are located on the closed left-halfplane.

2 Preliminaries

Consider the memristive circuit as in Fig. 1.We can obtain the following equations:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

−R3C1 X(t) = − R2R1

Y (t),

−R8C2Y (t) = − R7W (ϕ(t)) Z(t)+ R4 R6

R5 R7X (t)),

−R11C3 Z(t) = − R10R9�(t),

−R12C4�(t) = Z(t),

ϕ(t) = Z(t).

(1)

For the cause of convenience, denote x(t) = [x1(t),x2(t), x3(t), x4(t), x5(t)]T = [X (t), Y (t), Z(t), �(t),ϕ(t)]T , R2

R2 R3C1= α1, R5 R7

R4 R6 R8C2= α2, R10

R9 R11C3= α3,

1R12C4

= α4, and R7R8C2= 1. Therefore, system (1) can

be rewritten as

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0 50 100 150 200−40

−20

0

20

40

t

x 1

0 50 100 150 200−40

−20

0

20

t

x 2

(a)

−25 −20 −15 −10 −5 0 5 10 15 20 25−30

−25

−20

−15

−10

−5

0

5

10

15

20

x1

x 2

(b)

Fig. 2 Dynamical behaviors of memristive system (1)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

x1(t) = α1x2(t),

x2(t) = −α2x1(t)+W (x5)x3(t),

x3(t) = α3x4(t),

x4(t) = −α4x3(t),

x5(t) = x3(t),

(2)

where the piece-wise linear function W (x5(t)) is givenby

W (x5(t)) ={

a, |x5(t)| ≤ 1;b, |x5(t)| > 1.

Let α1 = 3, α2 = 3, α3 = 2, α4 = 3, a = 10, and b =−10; the dynamic behaviors of system (1) is shown inFig. 2 under the initial values of [−1, 2, 2,−1, 1].

The memristor-based circuit can be exactly repre-sented by the fuzzy model as follows:

Rule 1 IF x(t) is N1, THEN

x(t) = A1x(t), (3)

Rule 2 IF x(t) is N2, THEN

x(t) = A2x(t), (4)

where

A1 =

⎡

⎢⎢⎢⎢⎣

0 α1 0 0 0−α2 0 a 0 00 0 0 α3 00 0 −α4 0 00 0 1 0 0

⎤

⎥⎥⎥⎥⎦,

A2 =

⎡

⎢⎢⎢⎢⎣

0 α1 0 0 0−α2 0 b 0 00 0 0 α3 00 0 −α4 0 00 0 1 0 0

⎤

⎥⎥⎥⎥⎦,

N1 is |x5(t)| ≤ 1, and N2 is |x5(t)| > 1. With a cen-ter average defuzzifier, the over fuzzy system is repre-sented as

x(t) =2∑

i=1

ηi Ai x(t), (5)

where

η1 ={

1, |x5(t)| ≤ 1,0, |x5(t)| > 1,

η2 ={

0, |x5(t)| ≤ 1,1, |x5(t)| > 1.

3 Main results

In this paper, we consider system (5) as the mastersystem, and through electronic inductors, the values ofmemristor will be presented in the corresponding slave

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Observer-based synchronization of memristive systems

system. Meanwhile, the input delays and output delaysare considered in the slave system as

y(t) =2∑

k=1

ηk Ak y(t)+l∑

i=1

Bi u(t − τi ), t ≥ 0, (6)

where y(.) = (y1(.), y2(.), . . . , y5(.))T , Bi ∈ R

n×1,and u(.) is the control input. Then we can get the syn-chronization error system as follows:

{e(t) =∑2

k=1 ηk Ake(t)+∑li=1 Bi u(t − τi ),

z(t) =∑pj=1 C j e(t − ξ j ),

(7)

where z(t) is the output of the synchronization error;τi , ξ j , i ∈ {1, 2 . . . , l}, j ∈ {1, 2, . . . , p} are theknown nonnegative constant scalars representing,respectively, the input delays and output delays. With-out loss of generality, assume that

0 ≤ τ1 < τ2 < · · · < τl = τ, (8)

0 ≤ ξ1 < ξ2 < · · · < ξp = ξ. (9)

The initial conditions for the synchronization error sys-tem (7) are assumed to be e(s),∀s ∈ [−τ, 0] andu(s),∀s ∈ [−ξ, 0], and u, z are measurable. This paperaims to design observer-based output feedback syn-chronization controllers that can be implemented easilyfor synchronization error system (7) with time delays.Considering the time-delay terms in system (7), wedefine the following matrices

⎧⎨

⎩

Bk = Bk({τi }li=1) =∑l

i=1 e−∑2

k=1 Akτi Bi ,

Ck = Ck({ξ j }pj=1) =∑p

j=1 C j e−∑2

k=1 Akξ j .(10)

In the sense of the ordinary system theory, the time-delay system (7) is stabilizable and detectable ifand only if (Ak, Bk) is stabilizable and (Ak, Ck) isdetectable. These conditions are necessary and suffi-cient for the existence of an output feedback stabiliz-able controller and, thus, are assumed to be satisfied inthis paper.

In general, this class of systems with delays can bereduced to be

� (t) =2∑

k=1

ηk

(Ak�(t)+ Bku(t)

),∀t ≥ τ, (11)

where �(t) is a new state vector defined as

�(t)= e(t)+l∑

i=1

0∫

−τi

e−∑2

k=1 ηk Ak (τi+s)Bi u(t + s)ds,

∀t ≥ τ. (12)

This method is known as prediction [41], model reduc-tion [39], or finite spectrum assignment [42]. There-fore, if the state vector e(t) is available for feedback,via designing a nominal feedback for the transformedsystem (11), then

u(t) = −2∑

k=1

ηk�k�(t) = −2∑

k=1

ηk�k

×(

e(t)+l∑

i=1

0∫

−τi

e−∑2

k=1 ηk Ak (τi+s)Bi u(t + s)ds

)

,

(13)

where t ≥ τ and �k satisfies Ak− Bk�k and is asymp-totically stable, and the synchronization error systemis equal to a finite-dimensional system and, thus, pos-sesses only a finite number of spectrum [42].

However, if the synchronization error states are notavailable for feedback, an observer should be designedto estimate the state vector using the output signal ofthe system. As the predictor- based feedback (13) doesnot utilize the state e(t) directly, but the auxiliary state�(t), a novel observer was proposed to construct theauxiliary state�(t) instead of e(t) [44]. To present thisresult, an auxiliary output is utilized:

ψ=p∑

j=1

C j e−∑2

k=1 ηk Akξ j

l∑

i=1

0∫

−(τi+ξ j )

e−∑2

k=1 ηk Ak (τi+s)

× Bi u(t + s)ds + z(t),∀t ≥ τ + ξ. (14)

As the input and output signals of synchronization errorsystem (7) are assumed to be measurable, ψ can becomputed accordingly.

Lemma 1 [44] The time-delay system (7) is reducedto the following delay-free system:

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⎧⎨

⎩

� (t) =∑2k=1 ηk

(Ak�(t)+ Bku(t)

),

ψ(t) =∑2k=1 ηkCk�(t),∀t ≥ τ + ξ.

(15)

Based on this lemma, the conventional observer-based output controller for the original synchronizationerror system (7) can be constructed as

⎧⎪⎨

⎪⎩

(t) =∑2k=1 ηk(

Ak(t)+ Bku(t)+ Lk(ψ(t)− Ck(t))),

u(t) = −∑2k=1 ηk�k(t), (0) = 0,∀t ≥ 0,

(16)

where �k and Lk satisfy the conditions that Ak− Bk�k

and Ak − LkCk are both asymptotically stable, and

ψ(ε)=p∑

j=1

C j e−∑2

k=1 ηk Akξ j

l∑

i=1

0∫

−(τi+ξ j )

e−∑2

k=1 ηk Ak (τi+s)

× Bi u(ε + s)ds+z(ε),∀ε ∈ [0, τ+ξ ]. (17)

It is notable that both the state feedback in (11) andthe observer-based output feedback in (16) are infi-nite dimensional and are generally very hard to beimplemented since the control u(t) and the auxiliaryoutput ψ(t) involve the past information of u(t), asu(t + s), s ∈ [−τ − ξ, 0]. As a result, such controllerscan only be implemented by approximating the integralterms involving u(t+s), s ∈ [−τ−ξ, 0]with a numer-ical quadrature rule. However, for the state feedbackcase, it has been shown that the control law approxi-mated by numerical quadrature methods such as Simp-son rule may lose its ability to stabilize synchronizationerror system (7), no matter how precise the approxima-tion is [43]. A specific example has been carried out byMirkin [38].

To obtain the main results, this paper gives severalassumptions as follows.H1 The matrix pair (Ak, Bk) ∈ (Rn×n × R

n×m) iscontrollable with all the eigenvalues of Ak being onthe imaginary axis.

Under this assumption, if the feedback gain �k =�k(δ) : (0, 1] → R

m×n is properly designed as

limδ−→0+

�k(δ) = 0, limδ→0+

1

δ||�k(δ)|| <∞, (18)

the distributed terms in (13) are dominated by theterm −�ke(t) and thus can be neglected safely inu(t) when δ is small enough. Thus, the predictor-based feedback law (13) can be truncated as u(t) =−∑2

k=1 ηk�k(δ)e(t). Regarding the stability of thesynchronization error system under the truncated pre-dictor feedback, we introduce the following lemma.

Lemma 2 [45] Assume that (Ak, Bk) satisfies H1.Then, there exists a scalar δ∗ = δ∗({τi }li=1) such that

u(t) = −2∑

k=1

ηk�k(δ)e(t) = −BTk Pk(δ)e(t), (19)

which asymptotically stabilizes the synchronizationerror system (7) for all δ ∈ (0, δ∗], where Pk(δ) isthe unique positive definite solution to the followingalgebraic Riccatic equation (ARE)

ATk Pk + Pk Ak − Pk Bk BT

k Pk = −βPk . (20)

The main advantage of the truncated predictor feed-back over the predictor feedback (13) is that the numer-ical problem met in the implementation of the integral(distributed) terms in (13) is entirely avoided. In thispaper, under H1, and via the predictor based observer(16) and the following truncated predictor feedbackapproach, we will present the observer-based outputfeedback controllers which are easy to be implemented.

In consequence, system (26) is stable if and only ifall the zeros of ,bk(sג δ) = 0 are located in the openleft-half plane. For a given δ ≥ 0. Set the real parts ofthe right-most zeros of ,bk(sג δ) = 0 as

λbmax(δ) = max{Re{s} : ,bk(sג δ) = 0}. (21)

Then, system (26) is asymptotically stable if and onlyif λb

max(δ) < 0. The convergence rate of system (26)is completely determined by λb

max(δ); in other words,the smaller the value of λb

max(δ), the faster is the stateconvergence to the origin.

There exists a scalar δsup > 0 such that λbmax(δ) <

0,∀δ ∈ (0, δsup), (0, δsup) is the maximal intervalaround 0 such that ,b(sג δ) = 0 has no unstable zeros,and δsup can be calculated by the following algorithm:

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Observer-based synchronization of memristive systems

Algorithm 1 Calculate δsup

Initialization:Set δa → 0; ε→ 10−5;δb large enough to satisfy λb

max(δb) > 0 in Eq. (21)Iteration:1: while |δa − δb| > ε do2: δt ← 0.5 ∗ (δa + δb);3: if λb

max(δt ) < 0 then4: δa → 0.5 ∗ (δa + δb)

5: else6: if λb

max(δt ) >= 0 then7: δb → 0.5 ∗ (δa + δb)

8: end if9: end if

10: end whileSet δsup → δa .

3.1 Infinite-dimensional observer-based outputfeedback

3.1.1 Multiple output delays

Based on the discussion above, we propose the follow-ing observer-based output feedback controller:

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(t) =∑2k=1 ηk

{

Ak(t)+∑li=1 Bi u(t − τi )

+ Lk

(z(t)−∑p

j=1 C j(t − ξ j ))}

,

u(t) = −∑2k=1 ηk�k(t), ∀t ≥ 0,

(22)

where �k and Lk are the gains to be specified. The ini-tial conditions are z(s),∀s ∈ [−ξ, 0], and u(s),∀s ∈[−τ, 0). Although this observer is also infinite dimen-sional, it is easier to implement than the infinite-dimensional observer (16) since only the past informa-tion of u(t) at the time points t − τi , i ∈ {1, 2, . . . , l}and the past information of (t) at the time pointst − ξ j , j ∈ {1, 2, . . . , p} are required.

Theorem 1 Assume that the matrix pair (Ak, Bk) sat-isfies assumption H1 and (Ak, Ck) is observable. Let�k = �k(δ) be defined as in (19) and Lk = Lk(υ) =�CT

k with �k being the unique positive definite solutionto the following ARE:

Ak�k + �k ATk − �kCT

k Ck�k = −υ�k, (23)

where υ > 0. Then for any given arbitrarily largeyet bounded input delays {τi }li=1 and output delays{ξ j }pj=1, there exist two scalars

δ∗ = δ∗({τi }li=1), υ∗ = υ∗({ξ j }pj=1), (24)

such that the observer-based output feedback con-troller (22) asymptotically stabilizes the synchroniza-tion error system (7) ∀δ ∈ (0, δ∗] and υ ∈ (0, υ∗].Proof Denote ρ = e(t) − (t). Then it follows from(7) and (22) that

ρ(t) =2∑

k=1

ηk

{

Akρ(t)− Lk

( p∑

j=1

C jρ(t − ξ j ))}

.

(25)

Furthermore, under the controller u(t) in (22), the syn-chronization error system (7) can be rewritten as

e(t) =2∑

k=1

ηk

{

Ake(t)−l∑

i=1

�k(t − τi )

}

=2∑

k=1

{

ηk Ake(t)−l∑

i=1

Bi �ke(t − τi )

−l∑

i=1

Bi �ke(t − τi )

}

. (26)

The characteristic polynomial of the controlled systemconsisting of (25) and (26) is

k(s)ג = ,bk(sג δ)גck(s, υ), (27)

where ,bk(sג δ) and ,ck(sג υ) are, respectively, definedby

,bk(sג δ) � det(

s In −2∑

k=1

ηk(

Ak −l∑

i=1

Bi e−τi s

�k(δ))),

(28)

,ck(sג υ) � det(

s In −2∑

k=1

ηk(

ATk −

p∑

j=1

CTj e−ξ j s LT (υ)

)).

(29)

From Lemma 2, there exists a scalar δ∗ = δ∗({τi }li=1)

such that all the zeros of the characteristic quasi-polynomial ,bk(sג δ) are on the open left-half plane for

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all δ ∈ (0, δ∗]. In the same way, there exists a υ∗ =υ∗({ξ j }pj=1) such that all the zeros of ,ck(sג υ) = 0are on the open left-half plane for all υ ∈ (0, υ∗]. Theproof is completed.

In the process to design the observer-based feed-back controller (22), a separation principle is employedto design the feedback gains �k and Lk separately.Furthermore, �k and L fall into the category oflow gain feedback; in other words, larger values of{τi }li=1({ξ j }pj=1) allow only smaller values of δ(υ) and,consequently, smaller values of ||�k ||(||Lk ||).

As the delayed observer states (t − ξ j ), j ∈{1, 2, . . . , p} are fed back in (22), which is not desirablesince the delay effect will degrade the performances ofthe observer, on the other hand, it makes implementingthis observer expensive.

3.1.2 A single output delay

If there exists only a single delay in the output of syn-chronization error system (7), an alterative observer canbe designed, and the error system (7) can be rewrittenas

{e(t) =∑2

k=1 ηk Ake(t)+∑li=1 Bi u(t − τi ),

z(t) = C1e(t − ξ1),(30)

where C1 ∈ Rr×n is a constant matrix and ξ1 > 0 is

a known nonnegative scalar as the output delay. Theobserver for this system is designed as

⎧⎪⎪⎨

⎪⎪⎩

(t) =∑2k=1 ηk

{Ak(t)+∑l

i=1 Bi u(t − τi − ξ1)

+ Lk(z(t)− C1(t))},

u(t) = −∑2k=1 ηk�keAξ1(t),

(31)

where L is a given constant matrix such that Ak−LkC1

is Hurwitz, �k = �k(δ) is defined as in (19), and theinitial conditions are assumed to be (0) and u(s), s ∈[−(τ+ξ1), 0]. Then, we can get the following theorem.

Theorem 2 Assume that the matrix pair (Ak, Bk) sat-isfies assumption H1 and (Ak,C1) is observable. Thenfor any given arbitrarily large yet bounded input delays{τi }li=1 and output delay ξ1, there exists a scalarδ∗ = δ∗({τi }li=1, ξ1) such that the observer-based out-put feedback controller (30) asymptotically stabilizessystem (30) ∀δ ∈ (0, δ∗].

Proof Set a new state vector

ω(t) = e(t − ξ1),∀t ≥ ξ1. (32)

The system (30) can be rewritten as

{ω(t) =∑2

k=1 ηk Akω(t)+∑li=1 Bi u(t − τi − ξ1),

z(t) = C1ω(t),∀t ≥ ξ1,

(33)

which has only input delays. Let ρ(t) = ω(t)− y(t) =e(t − ξ1)− y(t),∀t ≥ ξ1. Then from (31) and (33), wecan get

ρ(t) =2∑

k=1

ηk(

Ak − LkC1)ρ(t),∀t ≥ ξ1. (34)

In the same way, let u(t) = −2∑

k=1ηk�k(t), then

e(t) =2∑

k=1

ηk

{Ake(t)−

l∑

i=1

Bi �k(t − τi )}

=2∑

k=1

ηk

{Ake(t)−

l∑

i=1

Bi �ke(t − τi − ξ1)

+l∑

i=1

Bi �kρ(t − τi )},∀t ≥ ξ1. (35)

As Ak − LkC1 is Hurwitz, from Eq. (34) the system(35) is asymptotically stable if and only if there exists

˙e(t) =2∑

k=1

ηk

{Ake(t)−

l∑

i=1

Bi �k e(t − τi − ξ1)},

∀t ≥ ξ1. (36)

According to Lemma 2, if �k = BTk Pk , where

Bk �l∑

i=1

e−

2∑

k=1ηk Ak (τi+ξ1)

Bi = e−

2∑

k=1ηk Akξ1

Bk, (37)

and Pk is the unique positive definite solution to thefollowing ARE:

ATk Pk + Pk Ak − Pk Bk BT Pk = −δ Pk, (38)

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Observer-based synchronization of memristive systems

consequently, there is a scalar δ∗ = δ∗({τi }li=1, ξ1)

such that system (36) is asymptotically stable ∀δ ∈

(0, δ∗]. Let Pk = e−AT ξ1 Pke−

2∑

k=1ηk Akξ1

, Eq. (38)equals to Eq. (20), then

�k = BTk Pk

= BTk e−

2∑

k=1ηk AT

k ξ1e

2∑

k=1ηk AT

k ξ1Pke

2∑

k=1ηk Akξ1

= BTk Pke

2∑

k=1ηk Akξ1

= �ke

2∑

k=1ηk Akξ1

. (39)

This completes the proof.

3.1.3 Single input and single output time-varyingdelays

In this part, it is useful to utilize the state transformationby (32) to study observer-based output feedback controlof the synchronization error system with time-varyingdelays

{e(t) =∑2

k=1 ηk Ake(t)+ B1u(μ(t)),

z(t) = C1e(ν(t)),(40)

where Ak ∈ Rn×n , B1 ∈ R

n×m , and C1 ∈ Rr×n are

constant matrices; and μ(t), ν(t) : R+ → R are con-tinuously differentiable functions that incorporate theinput and output delays, respectively. μ(t) and ν(t) aredefined as follows:

μ(t) = t − τ(t), ν(t) = t − ξ(t), (41)

where τ(t), ξ(t) : R+ → R

+ are the time-varyingdelays, and satisfy the following conditions.H2. The function μ, ν : R+ → R

+ are continuouslydifferentiable, invertible, and exactly known functions,such that ∀t ∈ R

+

0 < μ ≤ μ(t) ≤ μ <∞, (42)

0 < ν ≤ ν(t) ≤ ν <∞, (43)

and the delays τ(t) and ξ(t) are bounded—namely,there exist two finite yet arbitrarily large numbers τ

and ξ such that

0 ≤ τ(t) ≤ τ , 0 ≤ ξ(t) ≤ ξ,∀t ∈ R. (44)

Then, the inverse functions of μ(t) and ν(t) exist suchthat μ−1(t) − t and ν−1 − t are bounded; thus, thereexist two finite positive constants ςμ and ςν , such that

0 ≤ μ−1(t)− t ≤ ςμ, 0 ≤ ν−1(t)− t ≤ ςν,∀t ∈ R.

(45)

Let

ϕ(t) = ν(μ(t)). (46)

Then, it is obvious that ϕ(t) has all the properties ofμ(t)—especially, ϕ−1(t)− t is bounded for all t and

|ϕ(t)− t | = |t − τ(t)− ξ(t − τ(t))− t |≤ τ + ξ,∀t ∈ R. (47)

Thus, the observer-based feedback control for the syn-chronization error system (40) can be designed as

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(t) =∑2k=1 ηk ν(t)

(Ak(t)+ B1u(ϕ(t))+ Lk(z(t)− C1(t))

),

u(t) = −∑2k=1 ηk�k(t, δ)(t)

= −∑2k=1 ηk BT

1 Pke∑2

k=1 ηk Ak (ϕ−1(t)−t)(t),

(48)

where the initial condition is assumed to (s),∀s ∈[−ξ, 0], u(s), s ∈ [−(τ+ξ), 0), Lk is a constant matrixsuch that Ak− LkC1 is Hurwitz, and Pk = Pk(δ) is theunique positive definite solution to the following ARE:

ATk Pk + Pk Ak − Pk B1 BT

1 Pk = −δPk . (49)

As ∀t, ϕ−1(t) − t is bounded, the time-varying gainmatrix ∀t,�(t, δ) is also bounded.

To prove that controller (48) can stabilize the syn-chronization error system (40), the following resultabout state feedback stabilization of (40) is obtainedvia truncated predictor feedback

Lemma 3 Let μ(t) satisfy H2, and assume that(Ak, B1) is controllable and all the eigenvalues of Ak

are on the imaginary axis. Then, there exists a number

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S. Wen et al.

δ∗ = δ∗(τ ) such that the following truncated predictorfeedback

u(t) = −2∑

k=1

ηk BT1 Pk(δ)e

2∑

k=1ηk Ak (μ

−1(t)−t)e(t),

∀δ ∈ (0, δ∗] (50)

asymptotically stabilizes synchronization error system(40), where Pk is the unique positive definite solutionto ARE (49).

Based on this lemma, the following result can beproved.

Theorem 3 Let assumption H2 be satisfied. Assumethat (Ak, B1) is controllable, (Ak,C1) is observable,and all the eigenvalues of Ak are on the imaginaryaxis. Then there exists a scalar δ∗ = δ∗(τ + ξ) suchthat the observer-based output feedback (49) asymp-totically stabilizes synchronization error system (40).

Proof Set a new state vector

ω(t) = e(ν(t)). (51)

By system (40), we can get

ω(t) = ν(t)2∑

k=1

ηk

(Akω + B1u(ϕ(t))

)(52)

and z(t) = C1ω(t). Let ρ(t) = ω − (t). From Eq.(49) and Eq. (52), we can get

ρ(t) = ν(t)2∑

k=1

ηk(Ak − LkC1)ρ(t). (53)

Let V (t) = ∑2k=1 ηkρ

T (t)�kρ(t), where �k > 0 sat-isfies

(Ak − LkC1)T�k + �k(Ak − LkC1) = −In . (54)

Then

V (t) = −ν(t)||e(t)||2 ≤ −ν||e(t)||2, (55)

which means system (53) is asymptotically stable.Meanwhile

e(t) =2∑

k=1

ηk

{Ake(t)− B1�k(t, δ)e(ϕ(t))

+ B1�k(t, δ)ρ(μ(t))}. (56)

As the dynamics of ρ(t) is exponentially stable and∀t,�k(t, δ) is bounded, system (56) is asymptoticallystable if and only if there exists

˙e(t) =2∑

k=1

ηk

(Ake(t)− B1�k(t, δ)e(ϕ(t))

). (57)

The stability of system (57) is guaranteed by Lemma 3.

It is obvious that the observer gain Lk and the feed-back gain �k(t, δ) are designed independent of eachother. Furthermore, from (57), the determination of thevalue of the parameter δ in �k(t, δ) is also indepen-dent of Lk . These indicate that the separation principleholds for the observer-based controller (48).

Remark 1 This paper aims to generalize the idea of thetruncated predictor feedback in [45,46] to the designof observer-based output feedback controllers for syn-chronized error systems with both input and outputdelays. As a result, the implementation difficulty inher-ent in the predictor-based output feedback controllers.In particular, two kinds of observer-based output feed-back controllers are proposed using the truncated pre-dictor feedback approach.

4 A numerical example

Consider the memristive system (1), its correspond-ing synchronization error system (7), and the input andoutput delays of this system as in the form of (7) with

B1 = [0 0 0 0 3]T , B2 = [0 0 0 0 2]T ,C1 = [0 0 0 0 1],C2 = [0 0 0 0 3],

and τ1 = 0, τ2 = 2, ξ1 = 0, and ξ2 = 1. For the mem-ristive system (5), λ(Ai ) = {0,±2.7386i,±2.4495i},i = 1, 2.

It can be verified that (Ak, Bi ), i, k = 1, 2 is notcontrollable and (Ak,C j ), j, k = 1, 2 is not observ-able. However, according to (10), we can get

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Observer-based synchronization of memristive systems

0 50 100 150 200 250 300 350 400−40

−30

−20

−10

0

10

20

30

40

t

e 1(t)

(a)

0 50 100 150 200 250 300 350 400−30

−20

−10

0

10

20

30

t

e 2(t)

(b)

0 50 100 150 200 250 300 350 400−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

t

e 3(t)

(c)

0 50 100 150 200 250 300 350 400−1

−0.5

0

0.5

1

1.5

2

2.5

t

e 4(t)

(d)

0 50 100 150 200 250 300 350 400−60

−40

−20

0

20

40

60

t

e 5(t)

(e)

Fig. 3 Dynamical behaviors of the synchronization error system (7)

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S. Wen et al.

0 50 100 150 200 250 300 350 400−50

−40

−30

−20

−10

0

10

20

30

40

50

t

ρ 1(t)

(a)

0 50 100 150 200 250 300 350 400−60

−40

−20

0

20

40

60

t

ρ 2(t)

(b)

0 50 100 150 200 250 300 350 400−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

t

ρ 3(t)

(c)

0 50 100 150 200 250 300 350 400−2

−1.5

−1

−0.5

0

0.5

1

t

ρ 4(t)

(d)

0 50 100 150 200 250 300 350 400−100

−80

−60

−40

−20

0

20

40

60

80

t

ρ 5(t)

(e)

Fig. 4 Dynamical behaviors of the observer error ρ(t) = e(t)− (t)

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Observer-based synchronization of memristive systems

Bk = [2 2 2 2 5]T , Ck = [3 3 1.1036 3 4],k = 1, 2,

(Ak, Bk), k = 1, 2, is controllable, and (Ak, Ck), k =1, 2, is observable. Therefore, assumption H1 is satis-fied.

Let δ = 0.1; from (23), we can get

P1 =

⎡

⎢⎢⎢⎢⎣

0.0028 −0.0000 −0.0093 −0.0105 0.0004−0.0000 0.0027 0.0162 −0.0041 0.0003−0.0093 0.0162 0.2381 0.0094 0.0031−0.0105 −0.0041 0.0094 0.1098 −0.00250.0004 0.0003 0.0031 −0.0025 0.0058

⎤

⎥⎥⎥⎥⎦,

P2 =

⎡

⎢⎢⎢⎢⎣

0.0026 −0.0001 0.0067 0.0101 −0.0003−0.0001 0.0025 −0.0160 0.0013 −0.00040.0067 −0.0160 0.2250 0.0158 0.00400.0101 0.0013 0.0158 0.1042 −0.0019−0.0003 −0.0004 0.0040 −0.0019 0.0058

⎤

⎥⎥⎥⎥⎦;

then, the feedback gain can be obtained as

�1 =[

0.0320 −0.0313 −0.5245 −0.1969 −0.0316],

�2 =[

0.0370 −0.0263 −0.4831 −0.2534 −0.0316].

By the same way, let υ = 0.1; from (20), we can get

�1 =

⎡

⎢⎢⎢⎢⎣

0.1252 0.0166 −0.1289 −0.1748 0.05410.0166 0.1079 0.2678 −0.0840 0.0026−0.1289 0.2678 1.6663 −0.0117 0.0364−0.1748 −0.0840 −0.0117 0.5872 −0.18550.0541 0.0026 0.0364 −0.1855 0.1656

⎤

⎥⎥⎥⎥⎦,

�2 =

⎡

⎢⎢⎢⎢⎣

0.0813 0.0061 −0.0033 0.0585 −0.04570.0061 0.0981 −0.2659 −0.0916 −0.0265−0.0033 −0.2659 1.2722 0.2594 0.16860.0585 −0.0916 0.2594 0.3985 −0.0790−0.0457 −0.0265 0.1686 −0.0790 0.1460

⎤

⎥⎥⎥⎥⎦;

then, the feedback gain can be obtained as

L1 =[−0.0250 0.4276 2.3663 0.2304 0.3162

]T,

L2 =[

0.2514 −0.3615 2.0491 1.0665 0.3162]T.

We can design the infinite-dimensional observer as(22). The state e(t) and observer error ρ(t) = e(t) −(t) with this observer are shown in Figs. 3–4.

5 Conclusion

This paper considered the synchronization problem ofmemristive systems with multiple networked input and

output delays via observer-based control. A truncatedpredictor feedback approach is employed to designthe observers. Under certain restrictions, two classesof observer-based output feedback controllers weredesigned. A numerical example is carried out to demon-strate the effectiveness of the proposed methods.

Acknowledgments This work was supported by the Nat-ural Science Foundation of China under Grant 61125303, theNational Basic Research Program of China (973 Program) underGrant 2011CB710606, the Program for Science and Technol-ogy in Wuhan of China under Grant 2014010101010004, andthe Program for Changjiang Scholars and Innovative ResearchTeam in University of China under Grant IRT1245. This publica-tion was made possible by NPRP grant � 4-1162-1-181 from theQatar National Research Fund (a member of Qatar Foundation).The statements made herein are solely the responsibility of theauthor[s].

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