decentralized adaptive stabilization for interconnected systems with dynamic input–output and...

Automatica 46 (2010) 1060–1067

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Automatica

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Brief paper

Decentralized adaptive stabilization for interconnected systems with dynamicinput–output and nonlinear interactionsI

Liang Liu, Xue-Jun Xie ∗Institute of Automation, Qufu Normal University, Qufu, Shandong Province, 273165, China

a r t i c l e i n f o

Article history:Received 2 July 2009Received in revised form24 February 2010Accepted 26 February 2010Available online 3 April 2010

Keywords:Decentralized adaptive stabilizationInterconnected systemsBacksteppingMT-filters

a b s t r a c t

This paper considers the problem of robust decentralized adaptive output feedback stabilization for aclass of interconnected systems with dynamic input and output interactions and nonlinear interactionsby using MT-filters and the backstepping design method. It is shown that the closed-loop decentralizedsystem based on MT-filters is globally uniformly bounded, all the signals except for the parameterestimates can be regulated to zero asymptotically, and the L2 and L∞ norms of the system outputs are alsobe bounded by functions of design parameters. The scheme is demonstrated by a simulation example.

© 2010 Elsevier Ltd. All rights reserved.

1. Introduction

In recent years, research on decentralized adaptive control us-ing the backstepping approach has received great attentions dueto its advantages such as improving transient performance, etc.The first result without any requirement on subsystem relative de-greewas reported byWen (1994). Since then,more general class ofsystemswith the consideration of unmodeled dynamics or nonlin-ear interactions were studied by Jain and Khorrami (1997), Jiang(2000), Wen and Soh (1997), and Zhang, Wen, and Soh (2000).More recently, decentralized adaptive stabilization for nonlinearsystems with dynamic interactions dependent upon subsystemoutputs or unmodeled dynamics was studied in Jiang and Rep-perger (2001) and Liu and Li (2002). Liu, Zhang, and Jiang (2007)gave a result for stochastic nonlinear systems with parametricuncertainties, nonlinear uncertain interactions and stochastic in-verse dynamics, and Wen and Zhou (2007) considered a class ofnonlinear systems with nonsmooth hysteresis nonlinearities and

I This work was supported by National Natural Science Foundation of China(No. 60774010, 10971256, 60974127), Natural Science Foundation of JiangsuProvince (No. BK2009083), Program for Fundamental Research of Natural Sciencesin Universities of Jiangsu Province (No. 07KJB510114), Shandong Provincial NaturalScience Foundation of China (No. ZR2009GM008, ZR2009AL014). The material inthis paper was not presented at any conference. This paper was recommended forpublication in revised form by Associate Editor Zhihua Qu under the direction ofEditor Andrew R. Teel.∗ Corresponding author. Tel.: +86 537 4455720; fax: +86 537 4455720.E-mail addresses: [email protected] (L. Liu), [email protected],

[email protected] (X.-J. Xie).

0005-1098/$ – see front matter© 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2010.03.003

higher order nonlinear interactions. In Zhou andWen (2008), a newscheme to design decentralized backstepping adaptive trackingcontrollers for a class of nonlinear interconnected systems in thepresence of external disturbances was solved by adding two newterms in the parameter update laws compared with the conven-tional backstepping method in Krstić, Kanellakopoulos, and Koko-tović (1995), in which the interactions between subsystems areunknown and allowed to satisfy a high order nonlinear bound.However, except for Jiang and Repperger (2001), Wen and Soh(1997), and Zhang et al. (2000), all the results are only applica-ble to systems with interaction effects bounded by static functionsof subsystem outputs. In Wen, Zhou, and Wang (2009), intercon-nected systems by considering both input and output dynamic in-teractions were firstly settled by using K-filters and backsteppingdesignmethod; the L2 and L∞ norms of the system outputs are alsoestablished.In the widely cited in-depth monograph on the backstepping

design method, Krstić et al. (1995) systematically studied two setsof filters, namely K-filters andMT-filters, with different merits anddemerits, and applied them respectively to the design of outputfeedback adaptive controllers. The design with MT-filters, whichwas firstly proposed by Marino and Tomei (1992) and Marinoand Tomei (1993), is motivated by the idea of using an adaptiveobserver for output feedback control.Inspired by these results, the purpose of this paper is to

further address the same problem as in Wen et al. (2009) byusing MT-filters and the backstepping design method. Our maincontributions are composed of three parts.(i) This paper considers a class of interconnected systems

with dynamic input, output and nonlinear interactions. Theseinteractions are more general than those in Wen et al. (2009).

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L. Liu, X.-J. Xie / Automatica 46 (2010) 1060–1067 1061

(ii) Due to the dynamic interactions and unmodeled dynamicsappearing in the output of the subsystem, the adaptive lawsobtained by adopting the MT-filtered transformation equation(8.156) in Krstić et al. (1995) are not available for measurement.Therefore how to construct a new filtered transformation such thatthe adaptive laws can be available for measurement constitutesone of the main difficulties in this paper.(iii) By establishing a few key lemmas which play an important

role in the analysis of stability and asymptotic convergence, weshow that the closed-loop decentralized system based on MT-filters is globally uniformly bounded, all the signals except for theparameter estimates can be regulated to zero asymptotically, andthe L2 and L∞ norms of the system outputs are also be bounded byfunctions of design parameters. The effectiveness of the proposedscheme is demonstrated by a simulation example.

2. Problem formulation

Consider a class of interconnected systems consisting of Nsubsystems, and the ith subsystem is modeled byyi(t) = Gi(s)(1+ νii∆ii(s))ui(t)+ Hi(s)(1+ νii∆ii(s))

×

N∑j=1

fij(t, yj)+N∑j=1

νijHij(s)uj(t)+N∑j=1

νij

×Hij(s)Fj(s)N∑k=1

fjk(t, yk)+N∑j=1

µij∆ij(s)yj(t),

i = 1, . . . ,N, (1)where ui, yi ∈ R are respectively the input and output of theith subsystem, s denotes the differential operator d

dt , Fi(s) =Di(s)Bi(s), Gi(s) =

Bi(s)Ai(s), Hi(s) =

Di(s)Ai(s), Ai(s) = sni + ai,ni−1s

ni−1 +

· · · + ai,0, Bi(s) = bi,mismi + bi,mi−1s

mi−1 + · · · + bi,0, Di(s) =(sni−1, . . . , s, 1), ∆ii(s), ∆ij(s), and Hij(s) are some rationalfunctions of s, fij(t, yj), fjk(t, yk) ∈ Rni denote the nonlinearinteractions, fij = fjk if i = j, j = k, νij andµij are positive constantscalars, i, j, k = 1, . . . ,N .

Remark 1. νijHij(s)uj and µij∆ij(s)yj denote the dynamic interac-tions from the input and output of the jth subsystem to the ith sub-system for j 6= i, or unmodeled dynamics of the ith subsystem forj = i with νij and µij indicating the strength of the interactionsor unmodeled dynamics; fij(t, yj) denotes the nonlinear interac-tions from the jth subsystem to the ith subsystem for j 6= i, or anonlinear unmodeled part of the ith subsystem for j = i; ∆ii(s)denotes unmodeled dynamics of the ith subsystem. Such interac-tions, which include dynamic input, output and nonlinear interac-tions, are more general than those in Wen and Zhou (2007), Wenet al. (2009) and Zhou and Wen (2008).

We make the following assumptions on system (1).

Assumption 1. For each subsystemGi(s) = Bi(s)Ai(s), where ai,j and bi,k

(j = 0, 1, . . . , ni − 1, k = 0, 1, . . . ,mi) are unknown constants,Bi(s) is a Hurwitz polynomial, the order ni, the relative degree%i = ni − mi, and the sign of the high frequency gain bi,mi areknown.

Assumption 2. The nonlinear interaction terms fij(t, yj) (j =1, . . . ,N) satisfy |fij(t, yj)| ≤ γij|yjψj(yj)|, where γij are knownpositive constants, and ψj(yj) are known smooth nonlinearfunctions, | · | denotes the 2-norm for any vector or matrix whichwill be used throughout the paper.

Assumption 3. For any i, j = 1, . . . ,N ,∆ij(s) and ∆ii(s) are stableand strictly proper with unity high frequency gains, and Hij(s) isstable with a unity high frequency gain and its relative degree islarger than %i.

Remark 2. Assumptions 1 and 3 are similar to those in Wen et al.(2009). Assumption 2 means that the effects of the nonlinearinteractions to a local subsystem from other subsystems or itsunmodeled part is bounded by a function of the output of thissubsystem.With this condition, it is possible for the designed localcontrollers to stabilize the interconnected systems with stronginteractions. In fact, this assumption is muchmore relaxed versionof the linear bounding conditions used in Wen (1994). It is worthnoting that ∆ij(s) satisfying Assumption 3 can often be foundin many practical situations such as power systems. Thus theassumption is reasonable in practice.

3. The design of the robust decentralized adaptive controllerbased on MT-filters

By Appendix, the ith subsystem (1) can be transformed into thefollowing state-space realization

xi = Aixi − aixi,1 + biui + fi,= Aixi + Fi(ui, yi)Tθi + aiℵi + fi,

yi = xi,1 + ℵi = cTi xi + ℵi, i = 1, . . . ,N, (2)where

ℵi = νii∆iixi,1 +N∑j=1

νijHijGjxj,1 +

N∑j=1

µij∆ijyj,

fi =N∑j=1

fij(t, yj), Ai =[0(ni−1)×1 Ini−10 01×(ni−1)

],

ai =

ai,ni−1...ai,0

, bi =

0bi,mi...bi,0

= [0bi], ci =

1...0

,Fi(ui, yi)T =

[[0(%i−1)×(mi+1)Imi+1

]ui,−Iniyi

],

θi =[bTi , a

Ti

]T. (3)

To estimate the state of each subsystem, a localMT-filter only usinginput and output is designed asξi = Aliξi, ξi ∈ Rni−1,

ΩTi = AliΩTi + BliFi(ui, yi)

T, Ωi ∈ Rςi×(ni−1), (4)where ςi = ni +mi + 1,

Ali =[−li

Ini−201×(ni−2)

], Bli =

[−li, Ini−1

],

li =[1, li,1, . . . , li,ni−1

]T=[1, lTi

]T, (5)

with li,1, . . . , li,ni−1 being the coefficients of any Hurwitz polyno-mial Li(s) = sni−1 + li,1sni−2 + · · · + li,ni−1. To reduce the orderof filters,ΩTi is decomposed intoΩ

Ti = [vi,mi , . . . , vi,0,Ξi], where

vi,j ∈ Rni−1 (j = 0, 1, . . . ,mi) is the jth vector of vi and Ξi =[δi,ni−1, . . . , δi,0] ∈ R

(ni−1)×ni , δi,k ∈ Rni−1 (k = 0, 1, . . . , ni − 1) isthe kth vector of δi. By using (8.161), (8.162) in Krstić et al. (1995)

(Ali)jeni−1,ni−1 = Blieni,ni−j, j = 0, 1, . . . , ni − 1, (6)

λi = Aliλi + eni−1,ni−1ui, λi ∈ Rni−1, (7)we know that

vi,j = (Ali)jλi, j = 0, 1, . . . ,mi. (8)From

ηi = Aliηi + eni−1,ni−1yi, ηi ∈ Rni−1, (9)

Ξi = AliΞi − Bliyi, (10)

1062 L. Liu, X.-J. Xie / Automatica 46 (2010) 1060–1067

one gets δi,j = −(Ali)jηi, j = 0, . . . , ni − 1, where ei,k denotesthe kth coordinate vector in Ri. Due to the presence of ℵi in (2), weadopt the filtered transformation

χi = xi −[−ℵi

ξi +ΩTi θi

], (11)

from which, (2), (4), (5), and a tedious but straightforwardcalculation, (2) can be represented as

χi = Aiχi + li(ω0i + ωTi θi)+ (ai + seni,1)ℵi + fi,yi = χi,1, (12)

where ω0i = ξi,1, ωTi = FTi,1 +Ω

Ti,1, χi,1, ξi,1, F

Ti,1, andΩ

Ti,1 represent

the first rows of χi, ξi, F Ti andΩTi , respectively. Since θi is unknown,

the adaptive observer for χi can be chosen as

˙χ i = Aiχi + K0i(yi − χi,1)+ li(ω0i + ωTi θi), (13)

where θi is the estimate of θi and K0i = (Ai + c0iIni)li, c0i > 0 is aconstant. Defining the observer error

εi = χi − χi, (14)

and noting that yi− χi,1 = χi,1− χi,1 = eTni,1(χi− χi), by (12)–(14),after some simple calculations, one obtains

εi = A0iεi + liωTi θi + (ai + seni,1)ℵi + fi, (15)

where θi = θi − θi, A0i = Ai − K0ieTni,1. Obviously, eTni,1(sIni −

A0i)−1li = 1/(s+c0i). By (11), (14),wehave xi = χi+[−ℵi

ξi +ΩTi θi

]+εi.

By (3), (5), (12), one has

yi = χi,2 + ω0i + ωTi θi + (s+ ai,ni−1)ℵi + fi,1, (16)

where fi,1 represents the first element of fi. From (3), the definitionsofΩTi and ω

Ti , it follows that

ωTi =[vi,(mi,1), . . . , vi,(0,1),Ξi,1 − yie

Tni,1

], (17)

where vi,(j,1) (j = 0, 1, . . . ,mi) denotes the first entry of vi,j andΞi,1 denotes the first row of Ξi, respectively. Combining (3), (14),(16) with (17) leads to

yi = χi,2 + ω0i + ωTi θi + εi,2= bi,mivi,(mi,1) + χi,2 + ω0i + ω

Ti θi + εi,2,

where εi,2 = εi,2 + (s + ai,ni−1)ℵi + fi,1, ωTi = [0, vi,(mi−1,1), . . . ,

vi,(0,1),Ξi,1 − yieTni,1]. Now we replace (2) with a new system,whose states depend on filters (4), (7), (9) and thus are availablefor control design

yi = bi,mivi,(mi,1) + χi,2 + ω0i + ωTi θi + εi,2,

vi,(mi,j) = vi,(mi,j+1) − li,jvi,(mi,1), j = 1, . . . , %i − 2,

vi,(mi,%i−1) = ui + vi,(mi,%i) − li,%i−1vi,(mi,1), (18)

where vi,(mi,j) (j = 1, . . . , %i) is the jth element of vi,mi . Define achange of coordinates

zi,1 = yi, zi,j = vi,(mi,j−1) − αi,j−1, j = 2, . . . , %i. (19)

Let us choose the control law as

ui = αi,%i − vi,(mi,%i), αi,1 = ρiαi,1,

αi,1 = −ci,1zi,1 − χi,2 − ω0i − ωTi θi − c∗

i zi,1ψ2i ,

αi,2 = −bi,mizi,1 − ci,2zi,2 +∂αi,1

∂ρi

˙ρ i +∂αi,1

∂θiΓiτi,2 + βi,2,

αi,j = −zi,j−1 − ci,jzi,j +∂αi,j−1

∂ρi

˙ρ i +∂αi,j−1

∂θiΓiτi,j

−

j−1∑k=2

σik,jzi,k + βi,j,

βi,j =∂αi,j−1

∂yi(χi,2 + ω0i + ω

Ti θi)+

∂αi,j−1

∂ξiAliξi

+

mi+j−2∑k=1

∂αi,j−1

∂λi,k(−li,kλi,1 + λi,k+1)+

∂αi,j−1

∂ηi

× (Aliηi + eni−1,ni−1yi)+ li,j−1vi,(mi,1) +∂αi,j−1

∂χi×[Aiχi + K0i(yi − χi,1)+ li(ω0i + ωTi θi)],

σik,j =∂αi,k−1

∂θiΓi∂αi,j−1

∂yiωi, (20)

where ci,k = ci,k + di,k(∂αi,k−1∂yi

)2, αi,0 = yi, k = 1, . . . , %i, αi,j, σik,j,j = 3, . . . , %i, βi,j, j = 2, . . . , %i, ψi is defined as in Assumption 2,c∗i is a designed parameter.

Remark 3. Let us compare these two kinds of filters. The reduced-order MT-filters are more simpler than the full-order K-filters,while the anti-disturbance ability of the reduced-order MT-filtersis weaker than K-filters.Choose the following adaptive laws

τi,0 = r1,iωiεi,1,τi,1 = (ωi − ρiαi,1eni+mi+1,1)zi,1 + τi,0,

τi,j = τi,j−1 −∂αi,j−1

∂yiωizi,j, j = 2, . . . , %i,

˙θ i = Γiτi,%i = Γi

[Wθ i(zi, t)zi + r1,iωiεi,1

],

˙ρ i = −γisgn(bi,mi)αi,1zi,1, (21)where ρi is the estimate of ρi = 1/bi,mi , Γi, r1,i, γi > 0 areparameters. The system is compactly written as

zi = Azi(zi, t)zi +Wθ i(zi, t)Tθi − bi,mi αi,1ρie%i,1+Wεi(zi, t)εi,2, (22)

where

Azi =

−ci,1 − c∗i ψ

2i bi,mi 0 · · · 0

−bi,mi −ci,2 1+ σi2,3 · · · σi2,%i0 −1− σi2,3 −ci,3 · · · σi3,%i...

......

. . ....

0 −σi2,%i −σi3,%i · · · −ci,%i

,

Wεi =

[1,−

∂αi,1

∂yi, . . . ,−

∂αi,%i−1

∂yi

]T,

W Tθ i = WεiωTi − ρiαi,1e%i,1e

Tςi,1 ∈ R

%i×ςi . (23)

Remark 4. Let us discuss the implementation problem of theadaptive laws of these two design methods. If we adopt the samedesign procedure and (44), (45) by using the K-filters as in Wenet al. (2009), one obtains

˙θ i = Γiτi,%i = Γi

(τi,%i−1 −

∂αi,%i−1

∂yiωizi,%i

)= · · ·

= Γi

((ωi − ρiαi,1eςi,1

)zi,1 −

%i∑j=2

∂αi,j−1

∂yiωizi,j

);

obviously, ˙θ i can be implemented. While for the controller designbased onMT-filters, ifwe still adopt theMT-filtered transformationχi = xi −

[0

ξi +ΩTi θi

]used in Krstić et al. (1995), then from (2) it


follows that

χi = Aiχi + li(ω0i + ωTi θi)+ aiℵi + fi,

yi = xi,1 + ℵi = χi,1 + ℵi. (24)

By (21), one gets τi,%i = τi,%i−1 −∂αi,%i−1∂yi

ωizi,%i = · · · = r1,iωiεi,1 +

(ωi − ρiαi,1eςi,1)zi,1 −∑%ij=2

∂αi,j−1∂yi

ωizi,j. From (14) and (24), εi,1 =χi,1−χi,1 = yi−ℵi−χi,1. Sinceℵi is not available formeasurement,it concludes that εi,1 and τi,%i are not available for measurement;

hence, ˙θ i = Γiτi,%i is unable to be implemented. This is the maindifference with the design using K-filters, and this problem is easyto be neglected.

In this paper, by introducing a new filtered transformation (11),it follows that εi,1

(14)= χi,1 − χi,1

(12)= yi − χi,1; thus,

˙θ i = Γiτi,%i can

be implemented.

4. Main results

We first introduce the similarity transformations[εi,1πi

],

[εi,1Tiεi

]=

[eTni,1Ti

]εi, (25)[

χi,1ϕi

],

[χi,1Tiχi

]=

[eTni,1Ti

]χi, (26)

for i = 1, 2, . . . ,N , where Ti = [Alieni−1,1, Ini−1] = [Ali, eni−1,ni−1].From (5), the definitions of K0i, A0i and Ti, it is easy to verify thefollowing properties

Tili = 0, TiK0i = Ali li,

TiA0i = AliTi, K0i = c0ili +[li0

]. (27)

Combining (15), (25) with (27), one obtains

πi = Aliπi + Ti[(ai + seni,1)ℵi + fi]

= Aliπi + Ti[aiℵi + (s+ ai,ni−1)eni,1ℵi + fi], (28)

where ai = (0, ai,ni−2, . . . , ai,0)T. From the definition of Ti and (25),

one has

εi,2 − li,1εi,1 = πi,1. (29)

Thus, from (15), (29), the definitions of K0i and A0i, it follows that

εi,1 = −(c0i + li,1)εi,1 + εi,2 + ωTi θi

= −c0iεi,1 + πi,1 + ωTi θi + (s+ ai,ni−1)ℵi + fi,1. (30)

By the definition of A0i, (13) can be expressed as

˙χ i = A0iχi + K0iyi + li(ω0i + ωTi θi). (31)

From (26), (27) and (31), it follows that

ϕi = Ti ˙χ i = Aliϕi + Aliliyi. (32)

Let ∆ii(s)xi,1,∆ij(s)yj, Hij(s)G−1j (s)xj,1 be achieved as

˙f i,i = Af i,i fi,i + bf i,ixi,1,

∆ii(s)xi,1 = (1, 0, . . . , 0)fi,i, (33)gi,j = Agi,jgi,j + bgi,jyj,

∆ij(s)yj = (1, 0, . . . , 0)gi,j, (34)

hi,j = Ahi,jhi,j + bhi,jxj,1,

Hij(s)G−1j (s)xj,1 = (1, 0, . . . , 0)hi,j. (35)

By Assumption 3, Af i,i, Agi,j and Ahi,j(i, j = 1, . . . ,N) are Hurwitz.By (33)–(35), one has

|∆ii(s)xi,1|2 = |(1, 0, . . . , 0)fi,i|2 ≤ |Φi|2, (36)

|∆ij(s)yj|2 = |(1, 0, . . . , 0)gi,j|2 ≤ |Φi|2, (37)

|Hij(s)G−1j (s)xj,1|2= |(1, 0, . . . , 0)hi,j|2 ≤ |Φi|2, (38)

where Φi = [zTi , εi,1, πTi , f

Ti,i, g

Ti,1, . . . , g

Ti,N , h

Ti,1, . . . , h

Ti,N ]

T, Φ =[ΦT1 , . . . ,Φ

TN ]T.

Lemma 1. The effects of the interactions and unmodeled dynamicsare bounded by

|∆ii(s)xi,1|2 ≤ |Φ|2, (39)∣∣∣∣∣ N∑j=1

∆ij(s)yj

∣∣∣∣∣2

≤ ki0|Φ|2, (40)

∣∣∣∣∣ N∑j=1

Hij(s)G−1j (s)xj,1

∣∣∣∣∣2

≤ ki0|Φ|2, (41)

∣∣∣∣∣(s+ ai,ni−1) N∑j=1

∆ij(s)yj

∣∣∣∣∣2

≤ ki3|Φ|2, (42)

|(s+ ai,ni−1)∆ii(s)xi,1|2+

∣∣∣∣∣(s+ ai,ni−1) N∑j=1

Hij(s)G−1j (s)xj,1

∣∣∣∣∣2

≤ [ki1(1+ µ2 + 2ki0µ2)+ ki2]|Φ|2, (43)

where µ = max1≤i≤Nµi, µi = max1≤j≤Nνij, µij, ki0, ki1, ki2 andki3 are positive constants which are independent of µij and νij.

Proof. The proof is given in Appendix.

By Lemma 1, we can conclude the following corollary.

Corollary 1. ℵi and (s+ ai,ni−1)ℵi satisfy

|ℵi|2≤ 3µ2i (2ki0 + 1)|Φ|

2,

|(s+ ai,ni−1)ℵi|2≤ 3µ2i [ki1(1+ µ

2+ 2ki0µ2)

+ ki2 + ki3]|Φ|2, (44)

where µi, ki0, ki1, ki2, ki3 are defined as in Lemma 1.


Next, we give a key lemma which will be used in the stabilityanalysis.

Lemma 2. If ∆(s) is stable and strictly proper, then there alwaysexists a constant λ > 0 such that ∆∗(s) = 1/(1 + λ∆(s)) is properand stable for any λ ∈ [0, λ).


We give main results in this paper.

Theorem 1. Consider the decentralized adaptive control systemsconsisting of the system (1), the local adaptive controllers (20), (21),and the filters (4), (7), (10). Under Assumptions 1–3, there alwaysexist two positive constants µ∗ and ci such that for any µ ∈ [0, µ∗),c∗i ≥ ci, and any initial value xi(0), where µ, c∗i are definedin Lemma 1 and (20), i = 1, . . . ,N, all the signals in the closed-loopsystem are globally uniformly bounded, and all the signals except forθi, ρi converge asymptotically to zero.


Proof. A Lyapunov function for the ith subsystem is defined as

Vi =12

(|zi|2 + θTi Γ

−1i θi + |bi,mi |γ

−1i ρ2i + r1,iε

2i,1

)+ r2,iπTi Pliπi

+ lf i,i fTi,iPf i,i fi,i +

N∑j=1

lgi,jgTi,jPgi,jgi,j +N∑j=1

lhi,jhTi,jPhi,jhi,j, (45)

where r1,i, r2,i, lf i,i, lgi,j, lhi,j are some constant parameters to bechosen later, θi = θi − θi, ρi = ρi − ρi, and Pli, Pf i,i, Pgi,j and Phi,jsatisfy ATliPli + PliAli = −I , A

Tf i,iPf i,i + Pf i,iAf i,i = −I , A

Tgi,jPgi,j +

Pgi,jAgi,j = −I , AThi,jPhi,j + Phi,jAhi,j = −I . By (21)–(23), (28), (30)and (33)–(35), one has

Vi = −di,1z2i,1 + εi,2zi,1 −%i∑j=2

di,j

(∂αi,j−1

∂yi

)2z2i,j

− εi,2

%i∑j=2

(∂αi,j−1

∂yi

)zi,j −

12c0ir1,iε2i,1

+ r1,iεi,1[πi,1 + (s+ ai,ni−1)ℵi + fi,1] −12r2,i|πi|2

+ 2r2,iπTi PliTi[aiℵi + (s+ ai,ni−1)ℵieni,1 + fi]

−14lf i,i|fi,i|

2+ 2lf i,i f

Ti,iPf i,ibf i,izi,1 −

18ci,1z2i,1 −

14lf i,i|fi,i|

2

− 2lf i,i fTi,iPf i,ibf i,iℵi −

12

N∑j=1

lgi,j|gi,j|2

+ 2N∑j=1

lgi,jgTi,jPgi,jbgi,jzj,1 −14ci,1z2i,1 −

14

N∑j=1

lhi,j|hi,j|2

+ 2N∑j=1

lhi,jhTi,jPhi,j · bhi,jzj,1 −18ci,1z2i,1 −

14

N∑j=1

lhi,j|hi,j|2

− 2N∑j=1

lhi,jhTi,jPhi,jbhi,jℵj −Λi, (46)

where Λi =∑%ij=2 ci,jz

2i,j +

12 c0ir1,iε

2i,1 +

12 r2,i|πi|

2+12 lf i,i|fi,i|

2+

12

∑Nj=1 lgi,j|gi,j|

2+12

∑Nj=1 lhi,j|hi,j|

2+12 ci,1z

2i,1 + c

∗

i z2i,1ψ

2i . Taking

1d0i=

∑%ij=1

1di,j, lf i,i ≤

ci,132|Pf i,ibf i,i|

2 , lgi,j ≤cj,1

8N|Pgi,jbgi,j|2, lhi,j ≤

cj,132N|Phi,jbhi,j|2

, and using the complete square inequality, one obtains

Vi ≤14d0i

ε2i,2 +r1,i2c0i[πi,1 + (s+ ai,ni−1)ℵi + fi,1]

2

+ 2r2,i|PliTi|2[aiℵi + (s+ ai,ni−1)ℵieni,1 + fi]2

−38ci,1z2i,1 +

N∑j=1

38Ncj,1z2j,1 + 4

N∑j=1

lhi,j|Phi,jbhi,j|2

× |ℵj|2+ 4lf i,i|Pf i,ibf i,i|

2|ℵi|

2−Λi

≤3ε2i,24d0i+ kai|(s+ ai,ni−1)ℵi|

2+ kai|fi|2 + kbi|ℵi|2

+3r1,i2c0i

π2i,1 −38ci,1z2i,1 +

N∑j=1

38Ncj,1z2j,1 −Λi, (47)

where kai = 3/(4d0i) + 3r1,i/(2c0i) + 6r2,i|PliTi|2, kbi =(6r2,i|PliTi|2|ai|2 + 4

∑Nj=1 lhj,i|Phj,ibhj,i|

2+ 4lf i,i|Pf i,ibf i,i|

2). By

Assumption 2, one can show that

kai|fi|2 ≤N∑j=1

γijz2j,1ψ2j , (48)

where γij = O(γ 2ij ), which depends on d0i, c0i, r1,i, r2,i, |Pli| and |Ti|,indicates the coupling strength from the jth subsystem to the ithsubsystem. Clearly, there exist constants γ ∗ji (j = 1, . . . ,N) andci =

∑Nj=1 γ

∗

ji , and for any constant γji satisfying γji ≤ γ∗

ji ,

c∗i ≥N∑j=1

γji if c∗i ≥ ci. (49)

By the definition ofΛi, (29), (48), and choosing r1,i ≥ 6l2i,1/(c0id0i),r2,i ≥ 6r1,i/c0i + 6/d0i, one has

Vi ≤ kai|(s+ ai,ni−1)ℵi|2+ kbi|ℵi|2 −

r1,ic0i4

ε2i,1 −3ci,18z2i,1

+

N∑j=1

3cj,18Nz2j,1 −

r2,i4π2i,1 −

(c∗i z

2i,1ψ

2i −

N∑j=1

γijz2j,1ψ2j

)

−

%i∑j=2

ci,jz2i,j −12lf i,i|fi,i|

2−12

N∑j=1

lgi,j|gi,j|2

−12

N∑j=1

lhi,j|hi,j|2 −ci,12z2i,1. (50)

Taking qi = minci,1/4, ci,2, . . . , ci,%i , r1,ic0i/4, r2,i/4, lf i,i/2,min1≤j≤Nlgi,j/2, lhi,j/2, by Corollary 1,

Vi ≤ −qi|Φi|2 −3ci,18z2i,1 +

N∑j=1

3cj,18Nz2j,1 −

ci,14z2i,1

−

(c∗i z

2i,1ψ

2i −

N∑j=1

γijz2j,1ψ2j

)+ 3kbiµ2i (2ki0 + 1)|Φ|

2

+ 3kaiµ2i [ki1(1+ µ2+ 2ki0µ2)+ ki2 + ki3]|Φ|2. (51)

Choosing the Lyapunov function for the total interconnectedsystem as V =

∑Ni=1 Vi and using (49), (51), one has

V ≤ −N∑i=1

(q− κi2µ2 − κi1µ4)|Φ|2 −14

N∑i=1

ci,1z2i,1, (52)

where q = min1≤i≤Nqi/N, κi1 = 3kaiki1(2ki0 + 1), κi2 =3kbi(2ki0 + 1) + 3kai(ki1 + ki2 + ki3). Since q, κi1 and κi2 aresome constants independent of µ, there exists a constant µ =

min1≤i≤N

√12κi1

√κ2i2 + 2κi1q−

κi22κi1

, such that for any µ ∈

[0, µ),

V ≤ −qN2|Φ|2 −

14

N∑i=1

ci,1z2i,1, (53)

from which and (45), we conclude that zi, yi, εi,1, πi, fi,i, gi,j, hi,j,θi, ρi, θi, ρi are globally uniformly bounded. Furthermore, by theLasalle–Yoshizama theorem in Krstić et al. (1995), zi, εi,1, πi, fi,i,gi,j, hi,j(i, j = 1, . . . ,N) converge to zero asymptotically,which together with Assumption 2 imply that fij(t, yj)(i, j =1, . . . ,N) converge asymptotically to zero. From (4), (9), (10)and (32), it is clear that ξi, ηi,Ξi, ϕi are asymptotically stable.

Noting that[eTni,1Ti

]is nonsingular and (25), one concludes that


εi is asymptotically stable. From (44), one gets the asymptoticconvergence of ℵi, which together with (2) imply that xi,1 isasymptotically stable. By εi,1 = yi − χi,1, χi,1 is asymptoticallystable. From (14) and (26), it follows that χi and χi areasymptotically stable. By (1), (7), we have

λij(t) =sj−1 + li,1sj−2 + · · · + li,j−1

Li(s)ui(t)

=sj−1 + li,1sj−2 + · · · + li,j−1

Li(s)Ai(s)

Bi(s)(1+ νii∆ii(s))

×

[yi(t)−

N∑j=1

νijHij(s)Gj(s)

xj,1 −N∑j=1

µij∆ij(s)yj

−Di(s)Ai(s)

(1+ νii∆ii(s))N∑j=1

fij(t, yj)

]. (54)

For ∆ii(s), by applying Lemma 2, we know that there exists λi >0 such that 1/(1 + νii∆ii(s)) is stable and proper for νii ∈[0, λi). Defining µ∗ = min1≤i≤Nλi, µ, and noting that Li(s) andBi(s) are Hurwitz polynomials, it is easy to show that λij(i =1, . . . ,N, j = 1, . . . ,mi) converge asymptotically to zero for anyµ ∈ [0, µ∗). Thus, by (8), (19), (20) and the boundedness of θi andρi, vi,(mi,1) = zi,2 + αi,1 is also asymptotically stable. From (8), it

follows that vi,(j,k) = [j+k−1︷︸︸︷∗, . . . , ∗, 1][λi1, . . . , λi,j+k]T, which implies

that λi,mi+1 = vi,(mi,1) − [

mi︷︸︸︷∗, . . . , ∗][λi1, . . . , λi,mi ]

T convergesasymptotically to zero, where vi,(j,k) denotes the kth element ofvi,j. Applying the same arguments as above, it can be shown thatλi converges asymptotically to zero for any i = 1, . . . ,N . xi areasymptotically stable by the definition of ΩTi and (11). From (20),it follows that all the inputs ui are globally uniformly bounded andconverge to zero asymptotically.

We now derive bounds for system output yi(t) on both L2 andL∞ norms. As shown in (53), the derivative of V is given by V ≤−14

∑Ni=1 ci,1z

2i,1. Obviously ‖yi(t)‖

22 =

∫∞

0 |zi,1|2dt ≤ 4

ci,1V (0),

‖yi(t)‖∞ ≤√2V (0). Consider the zero initial values fi,i(0) = 0,

gi,j(0) = 0, hi,j(0) = 0, and since the initial values zi,q(0) dependon ci,1, γi,Γi, we can set zi,q(0), q = 2, . . . , %i to zero as followsvi,(mi,q) = αi,q(yi(0), ξi(0), ηi(0), λi,m+q(0), χi(0), ρi(0), θi(0)),from which, one has V (0) =

∑Ni=1

12 (|yi(0)|

2+ θTi (0)Γ

−1i θi(0) +

|bi,mi |γ−1i ρ2i (0)+r1,iε

2i,1(0)+2r2,iπ

Ti (0)·Pliπi(0)). Thus, the bounds

for yi(t) are established and stated in the following theorem.

Theorem 2. Consider the initial values zi,q(0) = 0, q = 2, . . . , %i,fi,i(0) = 0, gi,j(0) = 0 and hi,j(0) = 0, the L2 and L∞ norms of outputyi(t) are given by

‖yi(t)‖2 ≤2√ci,1

[N∑i=1

12

(|yi(0)|2 + θTi (0)Γ

−1i θi(0)

+ |bi,mi |γ−1i ρ2i (0)+ r1,iε

2i,1(0)+ 2r2,iπ

Ti (0)Pliπi(0)

)]1/2, (55)

‖yi(t)‖∞ ≤√2

[N∑i=1

12

(|yi(0)|2 + θTi (0)Γ

−1i θi(0)+ |bi,mi |γ

−1i

× ρ2i (0)+ r1,iε2i,1(0)+ 2r2,iπ

Ti (0)Pliπi(0)

)]1/2. (56)

Remark 5. By establishing Lemmas 1 and 2, Corollary 1, we givethe rigorous analysis of stability and asymptotic convergence inthe closed-loop decentralized system based on MT-filters, whichconstitutes another contribution in this paper.

Remark 6. Similar to (93) and (94) in Wen et al. (2009), oneconsiders more general systems

xi = Aixi + aiΨi(yi)+ biσi(yi)ui + fi,

yi = (1+ νii∆ii)xi,1 +N∑j=1

νijeT1hij(xj,1)+N∑j=1

µijeT1gij(yj),

i = 1, . . . ,N,

where Ψi(yi) ∈ Rni×ni are nonlinear function matrices, σi(yi) ∈ Ris nonlinear function, fi is defined as in (3), hij and gij are dynamicinteractions or unmodeled dynamics, e1 = [1, 0, . . . , 0]T. For sucha class of systems, Assumptions 4.1–4.3 which are used in Wenet al. (2009) are also needed. By these assumptions, following theproof procedure of Theorem 1, some similar conclusions can beobtained. Due to the limited space, the details are deleted.

5. A simulation example

Considered the following interconnected systems

yi(t) =1

s(s+ ai,1)

(1+

νii

s+ 2

)ui(t)+

[s, 1]s(s+ ai,1)

(1+

νii

s+ 2

)×

[y1 + sin y2y2 sin y2

]+

2∑j=1

νij

(s+ 1)3uj(t)+

2∑j=1

νij[s, 1](s+ 1)3

×


]+

2∑j=1

µij

s+ 1yj(t), i = 1, 2, (57)

(57) can be transformed into the state-space realization

xi = Aixi −[ai,1ai,0

]xi,1 +

[0bi

]ui + fi

yi = xi,1 + ℵi, i = 1, 2, (58)

where xi =[xi,1xi,2

], Ai =

[0 10 0

], fi =


], ℵi =

νiis+2xi,1 +∑2

j=1 νijs(s+aj,1)(s+1)3

xj,1 +∑2j=1

µijs+1yj.

MT-filters are chosen as

ξi = −liξi, ηi = −liηi + yi, λi = −liλi + ui. (59)

The change of coordinates are zi,1 = yi, zi,2 = λi − αi,1, i = 1, 2.The observer is given by

˙χ i =

[0 10 0

]χi +

[c0i + lic0ili

](yi − χi,1)+

[1li

](ξi + ω

Ti θi), (60)

where ωTi = [λi, liηi − yi,−ηi]. The control law and adaptive laware chosen as

αi,1 = ρiαi,1,

αi,1 = −(ci,1 + di,1)zi,1 − χi,2 − ξi − ωTi θi − c∗

i z2i,1ψ

2i ,

ui = −θi1zi,1 −

[ci,2 + di,2

(∂αi,1

∂yi

)2]zi,2 +

∂αi,1

∂ρi

˙ρ i

+∂αi,1

∂θi

˙θ i +

∂αi,1

∂yi(χi,2 + ξi + ω

Ti θi)+

∂αi,1

∂ηiηi

+∂αi,1

∂ξiξi +

∂αi,1

∂χi2

˙χ i,2 + liλi, (61)


Fig. 1. The responses of the closed-loop system.

˙θ i = Γi

(zi,1 + r1,izi,1 − r1,iχi,1 −

∂αi,1

∂yizi,2

)ωi

−Γi[ρiαi,1zi,1, 0, 0]T,

˙ρ i = −γi sgn(bi)αi,1zi,1, (62)

where θi and ρi are the estimates of θi = [bi, ai,1, ai,0]T and ρi =1/bi. In simulation, we choose the system parameters a1,1 = −1,a2,1 = 2, a1,0 = a2,0 = 0, b1 = b2 = 1, the design parametersc01 = 0.8, c02 = 1, li = 0.3, ci,j = 1, di,j = 0.1, νij = µij =0.1, r1,i = 0.3, γi = 0.5,Γi = 0.2, c∗i = 4, and the initialvalues x1,1(0) = 1, x1,2(0) = x2,2(0) = 0.5, x2,1(0) = 0.4,χi,j(0) = 1 , λi(0) = ηi(0) = 0, ξi(0) = 1, ρi(0) = 0.5,θi(0) = [0.2, 0.5, 0.6]T, i, j = 1, 2. Fig. 1 gives the responses ofthe closed-loop system with MT-filters.

6. A concluding remark

Our futurework is to extend the proposedmethodology tomoregeneral systems, such as stochastic interconnected systems withSiISS inverse dynamics in Yu and Xie (2010), and Ai, ci in (2) withmore general forms. Another issue is to apply the scheme to apractical example.

Appendix. Proof of (2)–(3)

Define xi,1 =Bi(s)Ai(s)ui +

Di(s)Ai(s)fi, ℵi = νii∆iixi,1 +

∑Nj=1 νij

HijGjxj,1 +∑N

j=1 µij∆ijyj, fi =∑Nj=1 fij(t, yj). By (1), we have

yi = (1+ νii∆ii(s))xi,1 +N∑j=1

νijHij(s)Gj(s)

xj,1 +N∑j=1

µij∆ij(s)yj

= xi,1 + ℵi. (63)With Ai(s), Bi(s) andDi(s) in (1), sinceHi(s) andGi(s) have the samedenominator rational polynomial, there exist minimal realizationmatrices Ai =

[−ai

Ini−101×(ni−1)

], bi =

[0bi

], cTi = [1, 0, . . . , 0],

ai = [ai,ni−1, . . . , ai,0]T, bi = [bi,mi , . . . , bi,0]

T such that cTi (sIni −Ai)−1bi = Bi(s)/Ai(s), cTi (sIni − Ai)

−1= Di(s)/Ai(s). Obviously, by

Chen (1999), xi,1 can be achieved by xi = Aixi+ biui+ fi, xi,1 = cTi xi.By (3), it follows thatxi = Aixi + biui + fi= Aixi + [−ai, 0ni×(ni−1)]xi + biui + fi= Aixi − aixi,1 + biui + fi.

Proof of Lemma 1. By (36)–(38), and Lemma 1 in Wen et al.(2009), (39)–(42) can be easily proved, where ki0, ki3 are positiveconstants independent of µij and νij. By (36), one obtains

|(s+ ai,ni−1)∆ii(s)xi,1|2= |(1, 0, . . . , 0)(Af i,i fi,i + bf i,i · xi,1)

+ ai,ni−1∆ii(s)xi,1|2

≤ ki1|xi,1|2 + ki2|Φ|2. (64)

Following the same procedure, one also obtains |(s + ai,ni−1)∑Nj=1 Hij(s)G

−1j (s)xj,1|

2≤ ki1

∑Nj=1 |xj,1|

2+ ki2|Φ|2. From (2)

and (3), one gets xi,1 = zi,1 − νii∆iixi,1 −∑Nj=1 νijHijG

−1j xj,1 −∑N

j=1 µij∆ijyj. Then by (39)–(41), one obtains |xi,1|2≤ 4(1+ ν2ii +

ki0max1≤j≤Nν2ij + ki0max1≤j≤Nµ2ij)|Φ|

2, so

|(s+ ai,ni−1)∆iixi,1|2+

∣∣∣∣∣(s+ ai,ni−1) N∑j=1

HijGjxj,1

∣∣∣∣∣2

≤ ki1N∑i=1

|xi,1|2 + ki2|Φ|2

≤ 4ki1N∑i=1

(1+ µ2i + 2ki0µ2i )|Φ|

2+ ki2|Φ|2

≤ [ki1(1+ µ2 + 2ki0µ2)+ ki2]|Φ|2, (65)

where ki1 = ki1 + ki1, ki2 = ki2 + ki2, ki1 = 4Nki1, µ =max1≤i≤Nµi, µi = max1≤j≤Nνij, µij, ki1, ki2 are independent ofµij and νij.

Proof of Corollary 1. By (3) and Lemma 1, one has

|ℵi|2≤ 3

|νii∆ii(s)xi,1|2 +∣∣∣∣∣ N∑j=1

νijHij(s)Gj(s)

xj,1

∣∣∣∣∣2

+

∣∣∣∣∣ N∑j=1

µij∆ij(s)yj

∣∣∣∣∣2

≤ 3(ν2ii + ki0 max1≤j≤N

ν2ij + ki0 max1≤j≤Nµ2ij

)|Φ|2

≤ 3µ2i (1+ 2ki0)|Φ|2, (66)

and

|(s+ ai,ni−1)ℵi|2≤ 3

[|(s+ ai,ni−1)νii∆ii(s)xi,1|

2

+

∣∣∣∣∣(s+ ai,ni−1) N∑j=1

νijHij(s)Gj(s)

xj,1

∣∣∣∣∣2

+

∣∣∣∣∣(s+ ai,ni−1) N∑j=1

µij∆ij(s)yj

∣∣∣∣∣2]

≤ 3µ2i [ki1(1+ µ2+ 2ki0µ2)

+ ki2 + ki3]|Φ|2, (67)

where µi is defined as in Lemma 1.

Proof of Lemma 2. Defining ∆(s) = a(s)/b(s), one has ∆∗(s) =b(s)/(b(s) + λa(s)). Since ∆(s) is strictly proper, ∆∗(s) is proper.Assume that λ1, . . . , λk are eigenvalues of b(s), λ1(λ), . . . , λk(λ)are eigenvalues of b(s) + λa(s). Since b(s) is Hurwitz, λ1, . . . , λkare stable.


When λ→ 0, it is obvious that b(s) + λa(s)→ b(s), and thenλi(λ) → λi, i = 1, . . . , k. By the continuity of λ1(λ), . . . , λk(λ)on λ, then for any λi(λ), there always exists a positive constantλ∗i such that λi(λ) is always stable in λ ∈ [0, λ

∗

i ). Let us chooseλ = min1≤i≤kλ∗i , for any λ ∈ [0, λ), λ1(λ), . . . , λk(λ) are allstable; thus, b(s)+ λa(s) is Hurwitz.

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Liang Liu Master student at Qufu Normal University. Hiscurrent research interests include decentralized adaptivecontrol and stochastic nonlinear control.

Xue-Jun Xie received the Ph.D. degree from the Instituteof Systems Science, Chinese Academy of Sciences, in 1999.He is currently Professor in QufuNormal University, China.His current research interests include stochastic nonlinearcontrol systems and adaptive control.

decentralized adaptive stabilization for interconnected systems with dynamic input–output and...

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