Adaptive decentralized control for a class of interconnected nonlinear systems via backstepping approach and graph theory

Automatica 76 (2017) 87–95

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Automatica journal homepage: www.elsevier.com/locate/automatica

Brief paper

Adaptive decentralized control for a class of interconnected nonlinear systems via backstepping approach and graph theory✩ Xiao-Jian Li, Guang-Hong Yang College of Information Science and Engineering Northeastern University, Shenyang, 110004, PR China Key Laboratory of Integrated Automation of Process Industry (Ministry of Education), Northeastern University, Shenyang, 110004, PR China

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Article history: Received 14 May 2015 Received in revised form 2 July 2016 Accepted 24 September 2016

Keywords: Interconnected nonlinear systems Backstepping adaptive control Strictly decentralized control Graph theory

abstract This paper is concerned with the adaptive decentralized control problem for a class of interconnected nonlinear systems, where the interconnections are assumed to be unknown and completely nonlinear. In addition, the interconnections and their bounds are allowed to contain the states of all subsystems. The main contribution is that, a strictly decentralized control scheme with compensation mechanism is developed to achieve the desirable tracking performance. More specifically, a smooth switching function is introduced to construct adaptive control laws, where the compensation mechanism is activated only if the immediate variable involved in the backstepping design exceeds a given constant, otherwise it will be turned-off. Furthermore, by combining graph theory and Lyapunov analysis method, it is proved that all the signals of the resulting closed-loop system are globally bounded, and the tracking errors of subsystems exponentially converge to a compact set, whose radius is adjustable by choosing different controller design parameters. Finally, the effectiveness of the proposed adaptive decentralized control scheme is illustrated with a simulated example. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Interconnected systems have been used to model a wide variety of physical, natural, and artificial complex dynamical systems: such as power systems, computer and telecommunications networks, and aerospace systems, etc. Due to the complexity and the heavier computational burden of centralized control, it is often required to design decentralized controllers for interconnected systems based only on local information of subsystem. Note that one of the main obstacles in decentralized control is how to address the interconnections. Under the match conditions, that is, all interconnections lie within the range space of the control vectors, the adaptive decentralized control problems were considered in Gavel and Siljak (1989) and Shi

✩ This work was supported in part by the Funds of National Science of China (Grant no. 61273148, 61420106016, 61403070, 61621004), the Fundamental Research Funds for the Central Universities (No. N140402002, N150404025), China Postdoctoral Science Foundation Special Funded Project (No. 2015T80263), and the IAPI Fundamental Research Funds (Grant No. 2013ZCX01-01). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Changyun Wen under the direction of Editor Miroslav Krstic. E-mail addresses: [email protected] (X.-J. Li), [email protected] (G.-H. Yang).

http://dx.doi.org/10.1016/j.automatica.2016.10.019 0005-1098/© 2016 Elsevier Ltd. All rights reserved.

and Singh (1992) for large-scale systems subject to first order and high order interconnections. To remove such structural constraints on interconnections, the backstepping techniques (Krstic, Kanellakopoulos, & Kokotovic, 1995; Wen, 1994) have been widely used in the decentralized control of interconnected nonlinear systems. For example, the problems of decentralized tracking were addressed for large-scale output feedback nonlinear systems (Jiang, 2002; Krishnamurthy & Khorrami, 2003; Tong, Huo, & Li, 2014). In Chen and Li (2008) and Mehraeen, Jagannathan, and Crow (2011), adaptive neural output feedback controllers were designed for interconnected nonlinear systems in strict feedback form. The decentralized tracking control designs were further investigated in Ye (2011) for strict feedback interconnected systems with time delays. On the other hand, the corresponding decentralized control problems were also extended to stochastic nonlinear systems (Liu, Zhang, & Jiang, 2007; Xie & Xie, 2000; Zhou, Shi, Liu, & Xu, 2012). Especially, in Wen, Zhou, and Wang (2009), the backstepping-based decentralized adaptive control problem was studied for interconnected systems with interactions including subsystem inputs and outputs. In fact, a common feature of the above results is that the interconnections or their bounds for each subsystem contain its own states and the outputs of other subsystems, which are the so-called weak interconnections (Zhang & Lin, 2014).

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X.-J. Li, G.-H. Yang / Automatica 76 (2017) 87–95

Recently, some interesting works have been done for coupled systems with strongly interconnections, where both the interconnections and their bounds are allowed to be the functions of the states of all subsystems. In Chen, Liu, Liu, and Lin (2009), Hovakimyan, Lavretsky, Yang, and Calise (2005) and Yoo and Park (2009), the neural networks are introduced to approximate the bounds of interconnections, and the semi-global stability of the tracking errors is ensured. The global stability problems were considered in Narenda and Oleng (2002) and Stankovic and Siljak (2009), while the interconnections or their bounding functions are in the linear form. Considering the more general strongly nonlinear interconnections, the decentralized state feedback controller is successfully constructed in Zhang and Lin (2014) via a Lyapunov function in a product integral manner. However, just as mentioned in the paper, the control laws design for each subsystem should share some prior information from other subsystems. To the best of our knowledge, no result has yet been reported about strictly decentralized control for coupled nonlinear systems with strongly interconnections. In such control scheme, each of the controllers ui , (i = 1, 2, . . . , N ) is assumed to have access only to the information of the corresponding subsystem, but not those of the other subsystems. Motivated by the above analysis, this paper further investigates the adaptive decentralized control problem for a class of coupled nonlinear systems with strongly interconnections. The main innovation is that a strictly decentralized controller design approach, together with constructing a new compensation mechanism for the strongly interconnections, is developed to achieve the desired tracking performance. More specifically, a smooth switching function is introduced to redesign adaptive control laws, and the compensation mechanism is activated only if the immediate variable involved in the backstepping design exceeds a given constant, otherwise it will be turned-off. In this case, the singularity problem is also avoided. Furthermore, by combining a convex combination technique and Matrix Tree Theorem (Knuth, 1997), a global Lyapunov function is given such that all the signals of the resulting closed-loop system are proved to be globally bounded, and the tracking errors of each subsystem exponentially converge to an adjustable compact set, whose radius is related with the controller design parameters. The following of this paper is organized as follows: some necessary preliminaries are presented in Section 2. The backstepping based strictly decentralized adaptive controller designs are summarized in Section 3. In Section 4, an example with comparison analysis is given to illustrate the effectiveness of the proposed methods. Finally, the conclusions are given in Section 5. 2. Problem statement and preliminaries 2.1. Preliminaries A directed graph or digraph G = (V , E ) contains a set V = 1, 2, . . . , N of vertices and a set E of arcs (edges) (i, j) leading from initial vertex i to terminal vertex j. A subgraph H of G is said to be spanning if H and G have the same vertex set. A digraph G is weighted if each arc (j, i) is assigned a non-negative weight lij . The weight w(H ) of a subgraph H is the product of the weights on all its arcs. A directed path P in G is a subgraph with distinct vertices i1 , i2 , . . . , im such that its set of arcs is {(ik , ik+1 ) : k = 1, 2, . . . , m − 1}. If im = i1 , we call P a directed cycle. A connected subgraph T is a tree if it contains no cycles, directed or undirected. A tree T is rooted at vertex i, called the root, if i is not a terminal vertex of any arcs, and each of the remaining vertices is a terminal vertex of exactly one arc. A subgraph Q is unicyclic if it is a disjoint union of rooted trees whose roots form a directed cycle.

Given a weighted digraph G with N vertices, define the weight matrix Λ = (lij )N ×N whose entry lij equals the weight of arc (j, i). For our purpose, we denote a weighted digraph as (G, Λ), and it is used in this paper to model the topology of all links in the networks. A digraph G is strongly connected if, for any pair of distinct vertices, there exists a directed path from one to the other. A weighted digraph (G, Λ) is strongly connected if and only if the weight matrix Λ is irreducible. The Laplacian matrix of (G, Λ) is defined as



l1k

 k̸=1   −l21  L=  .  .  .  −lN1

−l12 

l2k

··· ···

k̸=2

.. .

−lN2

..

. ···

−l1N



  −l2N   . ..   .  

(1)

lNk

k̸=N

2.2. Problem statement In this paper, we consider an interconnected nonlinear systems built on the digraph G by assigning each vertex its own internal dynamics and then coupling these vertex dynamics based on directed arcs in G. Assume that each vertex dynamics is described by the following strict-feedback system: x˙ i,m (t ) = fi,m (¯xi,m ) + gi,m (¯xi,m )xi,m+1 + hi,m (¯x1,m , x¯ 2,m , . . . x¯ N ,m ), x˙ i,n (t ) = fi,n (¯xi,ni ) + gi,ni (¯xi,ni )ui

+ hi,n (¯x1,n , x¯ 2,n , . . . x¯ N ,n ), yi (t ) =xi,1 , m = 1, 2, . . . , n − 1,

(2)

where xi = [xi,1 , xi,2 , . . . xi,n ] ∈ R , ui ∈ R and yi ∈ R are the system state, control input, and output of the subsystem i, i = 1, 2, . . . , N, respectively. fi,m , gi,m , are known smooth functions, and hi,m , m = 1, 2, . . . , n are unknown interconnection terms. Throughout this paper, we denote x¯ i,m = [xi,1 , xi,2 , . . . xi,m ]T . T

n

Remark 1. Since the interconnections depend on the states of all subsystems, the system (2) can be used to describe many state space models of interconnected nonlinear systems, and the details can be found in Niculescu (2001), Spooner and Passino (1999) and Tang, Tomizuka, Guerrero, and Montemayor (2000), etc. Moreover, the controller design also requires the following assumptions: Assumption 1. The digraph G is strongly connected. Assumption 2. There exist positive constants Y0 , Y1 , . . . , Yn−1 such that the desired trajectory yi,d (t ) and its time derivatives (n−1)

satisfy |yi,d (t )| ≤ Y0 , |˙yi,d (t )| ≤ Y1 , |¨yi,d (t )| ≤ Y2 , . . . |yi,d Yn−1 .

(t )| ≤

Assumption 3. The functions gi,m , i = 1, 2 · · · N , m = 1, 2, . . . , n, are known, and there exist positive constants g¯i,m and g such that g¯i,m ≥ |gi,m (¯xi,m )| ≥ g

i ,m

> 0.

i,m

Not that if the interconnected functions between any adjacent subsystems are not zero, then Assumption 1 holds according to the definition given in Section 2.1. In addition, Assumptions 2–3 have been widely used in the existing backstepping control design, such as Chen et al. (2009), Krstic et al. (1995) and Tee, Ge, and Tay (2009). Now, the considered problem is formulated as follows.

X.-J. Li, G.-H. Yang / Automatica 76 (2017) 87–95

Problem. The main objective is to design strictly adaptive decentralized control scheme for the system (2), such that the output of each subsystem can track the desired trajectory yi,d (t ), and all the signals of the resulting closed-loop system are globally bounded.

In this section, a strictly adaptive decentralized tracking controller is constructed by using new backstepping design technique and graph theory result. To this end, we start by introducing the following switching functions

m(z ) =

cosn



  

π 2

1,

sinn

|z | ≤ a

π |z |2 − a2 2 b 2 − a2 0,



where ki,l > 0 is constant and α˙ i,l−1 is given by

α˙ i,l−1 =

l −1  ∂αi,l−1



,

a < |z | < b

(3)

r =1

1 2

x˙ i,m (t ) = fi,m (¯xi,m ) + gi,m (¯xi,m )xi,m+1 ,

.

2

zi2,l ,

l = 2, 3, . . . , n − 2

(8)

then one can obtain the following equation according to Krstic et al. (1995) V˙ i,n−2 = −

n−2 

ki,l zi2,l + gi,n−2 zi,n−2 zi,n−1 .

(9)

Step n − 1: To address the nonlinear interconnected term, the last two steps of the considered backstepping design are different from the standard ones Krstic et al. (1995). Now, define the stabilizing function αi,n−1 as 1 gi,n−1

(−fi,n−1 − ki,n−1 (gi2,n−1 + 1)zi,n−1

+ α˙ i,n−2 − gi,n−2 zi,n−2 ),

(10)

and choose the following quadratic Lyapunov function candidate

i = 1, 2, . . . , N .

(4)

1 2

zi2,n−1

(11)

V˙ i,n−1 = V˙ i,n−2 + zi,n−1 (fi,n−1 + gi,n−1 (zi,n + αi,n−1 ) − α˙ i,n−2 ). (12) Substituting (9) and (10) into (12), the derivative of Vi,n−1 becomes

Assumption 4. The interconnection terms satisfy cij ϕn (¯xj,n )

(5)

V˙ i,n−1 = −

n−1 

In the following, a new adaptive control method including switching mechanism is developed to address the matching interconnections. Since the backstepping design is very mature, the details of the first n − 2 steps are omitted here, and we only introduce the following necessary notations l = 2, 3, . . . , n.

(6)

gi,1 αi,1 = −fi,1 − ki,1 zi,1 + y˙ i,d gi,l αi,l = −fi,l − ki,l zi,l + α˙ i,l−1 − gi,l−1 zi,l−1 (7)



ki,n−1 gi,n−1 zi,n−1

+

− 

2 ki,n−1

zi2,n 4ki,n−1

.

(13)

Step n: The following Lyapunov function is given Vi,n = Vi,n−1 +

1 2

zi2,n

(14)

and the control input ui is defined as ui = (m(zi,n ))u1i + (1 − m(zi,n ))u2i

(15)

where

     1 1 1  u = − f − k + z + α ˙  i , n i , n i , n i , n − 1 i  gi,n 4ki,n−1         2 1 1 −fi,n − ki,n + zi,n + α˙ i,n−1 ui = gi,n 4ki,n−1      N    1   − zi2,n + rij ϕn2 (¯xi,n )  2zi,n

Design the stabilizing functions as

−

2

zi,n

where ϕn (¯xj,n ) are known continuous nonlinear functions, and cij are known constants. Remark 2. Assumption 4 is more relaxed than the existing one in Chen and Li (2008), Liu and Huang (2001), Liu et al. (2007), Panagi and Polycarpou (2011), Shi and Singh (1992), Xie and Xie (2000) and Zhou et al. (2012), where the interconnected terms are outputdependent or bounded by the state-dependent linear functions. Under Assumption 4, a challenging issue is how to compensate the effects of the interconnections.

 ki,l zi2,l

l =1

j =1

l = 2, . . . , n − 2,

( r +1 )

yi,d

then it follows that

At this step, the following assumption on the interconnected terms is made for system (4).

zi,l = xi,l − αi,l−1 ,

1

Vi,n−1 = Vi,n−2 +

x˙ i,n (t ) = fi,n (¯xi,ni ) + gi,ni (¯xi,ni )ui

zi,1 = xi,1 − yi,d ,

r =0

∂ y(i,rd)

zi2,1

Vi,l = Vi,l−1 +

αi,n−1 =

In this paper, we firstly investigate the system (2) with matching interconnected terms. In other words, the following systems are considered

|hi,n (¯x1,n , x¯ 2,n , . . . x¯ N ,n )| ≤

l−1  ∂αi,l−1

l =1

3.1. The interconnections satisfy matching condition

+ hi,n (¯x1,n , x¯ 2,n , . . . x¯ N ,n ) yi (t ) = xi,1 , m = 1, 2, . . . , n − 1,

Vi,1 =

|z | ≥ b

where a < b are two given positive constants. According to the Lemma 3 of Wu, Chen, and Li (2015), the function m(z ) has continuous n-order derivatives.

N 

∂ xi,r

(fi,r + gi,r xi,r +1 ) +

Considering the following quadratic Lyapunov function candidate

3. Backstepping control design

   

89

(16)

j =1

and rij = Ncij2 . In addition, m(zi,n ) is defined in (3) with a = εi,n1 , b = εi,n2 , where εi,n1 , εi,n2 are two given positive constants. To analyze the convergence of the tracking errors and other signals of the closed-loop system, the following three cases are considered.

90

X.-J. Li, G.-H. Yang / Automatica 76 (2017) 87–95

Case I: we firstly consider the case |zi,n | ≤ εi,n1 . According to (3), (15) and (16), we have ui = u1i . Then Vi,n becomes V˙ i,n = −

n−1 

ki,l zi2,l

−

 

l =1

Case III: Finally, we consider the case εi,n1 < |zi,n | < εi,n2 . From (3), (15) and (16), we have

2

zi,n

ui =

ki,n−1 gi,n−1 zi,n−1 −  2 ki,n−1

− ki,n zi2,n + zi,n hi,n (¯x1,n , x¯ 2,n , . . . x¯ N ,n ).

(17)



1

− fi,n − ki,n +

gi,n

+

m(zi,n ) − 1

1

2

V˙ i,n = −

(18)

On the other hand, if the constants ki,l are chosen such that ki,l ≥ then together with |zi,n | ≤ εi,n1 and (13) we have

 rij ϕ (¯xi,n ) 2 n

(24)

1 , 2

4ki,n−1



−

ki,n−1 gi,n−1 zi,n−1

2 +

zi2,n

+ zi,n (fi,n + gi,n ui

4ki,n−1

+ hi,n (¯x1,n , x¯ 2,n , . . . x¯ N ,n ) − α˙ i,n−1 )  n   2 ≤− ki,l zi,l − ki,n−1 gi,n−1 zi,n−1 l =1

− 

. ≤ e Vi,n−1 (0) +

εi2,n1

−t

4ki,n−1

+

m(zi,n )

N 1

+

2 j=1

, which



2

2 ki,n−1

(19)

From (19), we know Vi,n−1

2

zi,n

4ki,n−1

εi2,n1

 ki,l zi2,l

2 ki,n−1

zi2,n

l =1

≤ −Vi,n−1 +

n−1 

− 

− ϕ (¯xi,n )).

ki,l zi2,l +

N 

zi,n + α˙ i,n−1

j =1

zi,n

2 n

n −1 

+

4ki,n−1

l =1

2 j =1

j =1

V˙ i,n−1 ≤ −

zi2,n



1

which yields

(zi2,n + h2i,n (¯x1,n , x¯ 2,n , . . . x¯ N ,n )) 2   N N  1 1 2 2 ≤ zi,n + rij ϕn (¯xi,n ) + rij (ϕn2 (¯xj,n ) ≤



2zi,n

Using Assumption 4 gives zi,n hi,n (¯x1,n , x¯ 2,n , . . . x¯ N ,n )



zi2,n

+

N 

 rij ϕ (¯xi,n ) 2 n

j =1

rij (ϕn2 (¯xj,n ) − ϕn2 (¯xi,n )).

(25)

means that zi,l exponentially converge to the following compact ε set {zi,l | |zi,l | ≤ √ i,n1 , l = 1, 2 · · · n − 1}. Combining (6), (7),

Similar to the inequalities (19) and (20), it is known that, for Case III, there must exist a positive constant Mi (εi,n2 ) such that

(10) and Assumption 3, we get that both xi,l , (l = 1, 2 · · · n) and αi,l , (l = 1, 2 · · · n − 1) also exponentially converge to a compact set. Furthermore, from Assumption 4, it is known

zi2,n +

2ki,n−1

zi2,n

+

N 

rij ϕ (¯xi,n ) ≤ Mi (εi,n1 )

V˙ i,n ≤ −

n 



1 2

zi,n



ki,l zi2,l −

1 2 j =1

Case II: Here, we consider the case |zi,n | ≥ εi,n2 . From (3) and (15) we have ui = u2i , and it follows that

 ki,l zi2,l −

ki,n−1 gi,n−1 zi,n−1

2

zi,n

− 

+

2 ki,n−1

zi2,n 4ki,n−1

2 j=1

+ hi,n (¯x1,n , x¯ 2,n , . . . x¯ N ,n ) − α˙ i,n−1 )  n   zi,n ≤− ki,l zi2,l − ki,n−1 gi,n−1 zi,n−1 −  N 1

2 j =1

(22)

V˙ i,n ≤ −

n 

(27)

 ki,l zi2,l

−



l =1

zi,n

2

ki,n−1 gi,n−1 zi,n−1 −  2 ki,n−1

+ Mi (εi,n1 , εi,n2 ) +

N 1

2 j=1

rij (ϕn2 (¯xj,n ) − ϕn2 (¯xi,n ))

(28)



1 2

Mi (εi,n1 ),

1 2

Mi (εi,n2 )

 (29)

and Mi (εi,n1 ), Mi (εi,n2 ) are given in (20) and (26), respectively. Now, we construct a global Lyapunov function V (t ):

2 V (t ) =

2 ki,n−1

rij (ϕn2 (¯xj,n ) − ϕn2 (¯xi,n )).

rij (ϕn2 (¯xj,n ) − ϕn2 (¯xi,n )).

Mi (εi,n1 , εi,n2 ) = max

+ zi,n (fi,n + gi,n u2i

l =1

2

where

l =1

+

N 1

1

Now, combining the Cases I–III, we have (21)



ki,n−1 gi,n−1 zi,n−1

+ Mi (εi,n2 )

−  +

rij (ϕn2 (¯xj,n )



−

2

zi,n

2

− ϕn2 (¯xi,n )).

n−1 

 ki,l zi2,l

l =1

N

Mi (εi,n1 ) +

n 

2 ki,n−1

ki,n−1 gi,n−1 zi,n−1 −  2 ki,n−1

l =1

V˙ i,n = −

(26)

j =1

(20)

where Mi (εi,n1 ) denotes a positive constant related with the given constant εi,n1 . Substituting (18) and (20) into (17) yields

+

rij ϕn2 (¯xi,n ) ≤ Mi (εi,n2 ).

Substituting (26) into (25), we have 2 n

j =1

V˙ i,n ≤ −

N 

N 

βi Vi,n (t )

(30)

i =1

(23)

where βi denote the cofactor of the ith diagonal element of L in (1) with lij replaced by rij . According to Assumption 1 and Kirchhoff’s Matrix Tree Theorem (Knuth, 1997), it is known that βi > 0.

X.-J. Li, G.-H. Yang / Automatica 76 (2017) 87–95

From (28) and (30), we have N

V˙ (t ) ≤ −

n

N



βi ki,l zi2,l +



i=1 l=1 N N 1 

+

βi Mi (εi,n1 , εi,n2 )

i=1

2 i=1 j=1

βi rij (ϕn2 (¯xj,n ) − ϕn2 (¯xi,n )).

(31)

Next, based on the Theorem 2.2 of Li and Shuai (2010), an important property on the interconnected terms is derived. In fact, according to the Theorem 2.2 of Li and Shuai (2010), we have

91

and Li (2008), Krstic et al. (1995) and Tee et al. (2009). According to Assumption 3, yi,d (t ) is bounded, then xi,1 is also bounded. Together with the boundedness of y˙ i,d (t ) and the continuousness of fi,1 , it is known that αi,1 is also bounded, which implies the boundedness of xi,2 from (7). In this case, the boundedness of αi,2 can be ensured since it is a continuous function of x¯ i,2 , zi,1 , zi,2 , yi,d , y˙ i,d and y¨ i,d . This leads to the boundedness of xi,3 . Repeat the above lines, we know xi,l , l = 1, 2, . . . , n and αi,l , l = 1, 2, . . . , n − 1 are all bounded. Combining (3), (15), and (16), it can be shown that the control inputs ui (i = 1, 2, . . . , N ) are also bounded, which completes the proof.

Without loss of generality, for any directed cycle CQ , the set E (CQ ) can be described as

Remark 3. By using Theorem 2.2 in Li and Shuai (2010), a new result related with the interconnections is derived in (35). Combining (18) and (35), it is known that only the first part of (18) needs to be compensated. Therefore, the smooth switching functions-based control laws are designed in (15), which ensure the global convergence of the tracking errors and the signals of the closed-loop system. In addition, since u2i in (15) is zero for zi,n ≤ εi,n1 , the singularity problem in controller design is also avoided.

E (CQ ) = {(ik , ik+1 ) | k = 1, . . . , m − 1, m ≤ N , im = i1 }.

3.2. The interconnections satisfy mismatching condition

N 

βi rij (ϕ (¯xj,n ) − ϕ (¯xi,n )) 2 n

2 n

i,j=1

=





w(Q)

(ϕn2 (¯xs,n ) − ϕn2 (¯xr ,n )).

(32)

(s,r )∈E (CQ )

Q∈Q

(33)

Via (32), we have



In this subsection, we consider the system (2) with mismatching nonlinear interconnections, which satisfy the following assumption.

(ϕn2 (¯xs,n ) − ϕn2 (¯xr ,n ))

(s,r )∈E (CQ )

= (ϕn2 (¯xi1 ,n ) − ϕn2 (¯xi2 ,n )) + (ϕn2 (¯xi2 ,n ) − ϕn2 (¯xi3 ,n )) + · · · (ϕ (¯xim ,n ) − ϕ (¯xi1 ,n )) = 0 2 n

2 n

Assumption 5. The interconnection terms satisfy (34)

|hi,m (¯x1,m , x¯ 2,m , . . . x¯ N ,m )| ≤

which implies that

N 

cij ϕm (¯xj,m )

(37)

j =1 N N 1 

2 i=1 j=1

βi rij (ϕn2 (¯xj,n ) − ϕn2 (¯xi,n )) = 0.

(35)

Subsequently, combining (31) and (35) results in V˙ (t ) ≤ −

N  n 

βi ki,l zi2,l +

i=1 l=1

N 

βi Mi (εi,n1 , εi,n2 ).

Similar to the previous subsection and (3), define (36)

i=1

The aforementioned design and analysis procedures are summarized in the following theorem. Theorem 1. Consider the interconnected nonlinear system (4), and suppose that Assumptions 1–4 hold. If the constants ki,l are chosen such that ki,l ≥ 21 , then the controller (15) with the virtual control functions (7) and (10) ensure that all the signals in the closedloop system are globally bounded, and the tracking errors of each subsystem converge to the following compact set D

  N 2  D = zi,1 | |zi,1 | ≤  βi Mi (εi,n1 , εi,n2 ),  βi i=1   i = 1, 2, . . . , N   

with exponential decay rate

+

 N

i=1

zi,1 = xi,1 − yi,d , zi,l = xi,l − αi,l−1 ,

βi Mi (εi,n1 , εi,n2 )), i = 1, 2, . . . , N , l =

1, 2, . . . , n. Therefore, the tracking errors zi,1 converge to the compact set D with exponential decay rate 21 . In addition, the following discussion on the boundedness of the signals of the closed-loop system is similar to the one of Chen

 1,   

m(zi,l ) =



cosn

  

π 2

 sinn

0,

l = 2 , 3 , . . . , n,

π |zi,l |2 − εi2,l1 2 εi2,l2 − εi2,l1

|z | ≤ εi,l1

 ,

εi,l1 < |z | < εi,l2 |z | ≥ εi,l2 (38)

where εi,l1 < εi,l2 , (l = 1, 2, . . . , n) are two given positive constants. In addition, we design the stabilizing functions as

αi,l = m(zi,l )αia,l + (1 − m(zi,l ))αib,l ,

l = 1, 2, . . . , n

ui = αi,n

(39)

where

αia,1 = αib,1 =

1 . 2

Proof. From (36), the following inequality holds V (t ) ≤ e−t V (0)+  N 1 , which implies that |zi,l | ≤ i=1 βi Mi (εi,n1 , εi,n2 ) if ki,l ≥ 2 √ t √ √ 2 (e− 2 Vn (0) βi

where ϕm (¯xj,m ), m = 1, 2, . . . , n − 1 are known smooth and bounded functions, that is, there exist positive constants ϕ¯ i,m such that |ϕm (¯xj,m (t ))| ≤ ϕ¯ j,m . In addition, ϕn (¯xj,n (t )) are known continuous functions, and cij are known constants.

αia,l = αib,l =

1 gi,1 1

(−fi,1 − ki,1 zi,1 + y˙ i,d )  −fi,1 − ki,1 zi,1 + y˙ i,d −

gi,1 1 gi,l 1

2zi,1

 zi2,1 +

N 

 rij ϕ12 (¯xi,1 )

j =1

(−fi,l − ki,l zi,l + α˙ i,l−1 − gi,l−1 zi,l−1 ),  − fi,l − ki,l zi,l + α˙ i,l−1 − gi,l−1 zi,l−1

gi,l

−

1

1 2zi,l

 zi2,l

+

N  j =1

 rij ϕ (¯xi,l ) 2 l

,

l = 2, 3, . . . , n.

(40)

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X.-J. Li, G.-H. Yang / Automatica 76 (2017) 87–95

In (40), ki,l > 0 are given constants and α˙ i,l−1 are given by

which yields

∂αi,l−1  ∂αi,l−1 = (fi,r + gi,r xi,r +1 ) + ∂t ∂ x i ,r r =1 l −1

α˙ i,l−1

ki,1 zi2,1

V˙ i,1 ≤ −

l −1  ∂αi,l−1

+

r =0

(r +1) yi,d . ∂ y(i,rd)

N 

+

rij ϕ (¯xi,1 )

ki,1 zi2,1

≤−

as Vi,1 =

2

zi2,1

(41)

+

then it follows that V˙ i,1 = zi,1 (fi,1 + gi,1 (zi,2 + αi,1 ) − y˙ i,d + hi,1 (¯x1,1 , x¯ 2,1 , . . . x¯ N ,1 )). (42)

V˙ i,1 ≤ zi,1 (fi,1 + gi,1 (zi,2 + αi,1 ) − y˙ i,d )   N N  1 1 2 2 + zi,1 + rij ϕ1 (¯xi,1 ) + rij (ϕ12 (¯xj,1 ) − ϕ12 (¯xi,1 )). 2

2 j=1

j =1

(43) Similar to Section 3.1, the following three cases are considered according to the values of |zi,1 |. Case I: Here, we firstly consider the case |zi,1 | ≤ εi,11 . According to (38)–(40), we have αi,1 = αia,1 , and V˙ i,1 ≤ −ki,1 zi2,1 + gi,1 zi,1 zi,2 +



1

zi2,1 +

2



N



+

2 j =1

zi2,1 +

rij (ϕ (¯xj,1 ) − ϕ (¯xi,1 )).

rij ϕ (¯xi,1 ) ≤ Mi (εi,11 , ϕ¯ i,1 )

(45)

where Mi (εi,11 , ϕ¯ i,1 ) denotes a positive constant related with the given constants εi,11 and ϕ¯ i,1 . Then, substituting (45) into (44), the derivative of Vi,1 becomes V˙ i,1 ≤ −ki,1 zi2,1 + gi,1 zi,1 zi,2 +

+

N 1

2 j=1

1 2

Mi (εi,11 , ϕ¯ i,1 )

rij (ϕ12 (¯xj,1 ) − ϕ12 (¯xi,1 )).

(46)

V˙ i,1 ≤ −ki,1 zi2,1 + gi,1 zi,1 zi,2 2 j =1

rij (ϕ12 (¯xj,1 ) − ϕ12 (¯xi,1 )).

(47)

N 1

2 j=1

1 gi,1

+

(50)

1 2

Mi (εi,12 , ϕ¯ i,1 ) (51)

rij (ϕ12 (¯xj,1 ) − ϕ12 (¯xi,1 ))

= max

1 2

Mi (εi,11 , ϕ¯ i,1 ),

(52)



2zi,1

 zi2,1

+

N  j=1

 rij ϕ (¯xi,1 ) 2 1

2

Mi (εi,12 , ϕ¯ i,1 )

 (53)

1 2

zi2,m

(54)

then it follows that V˙ i,m = V˙ i,m−1 + zi,m (fi,m + gi,m (zi,m+1 + αi,m ) − α˙ i,m−1 + hi,m (¯x1,m , x¯ 2,m , . . . x¯ N ,m )).

(55)

Subsequently, referring to Step 1, one can obtain the following inequality by using (38)–(40), V˙ i,m ≤ −

m 

ki,l zi2,l + gi,m zi,m zi,m+1 +

m 

Mi (εi,l1 ,

l=1

εi,l2 , ϕ¯ i,l ) +

m N 1 

2 l=1 j=1

rij (ϕl2 (¯xj,l ) − ϕl2 (¯xi,l ))

(56)

where Mi (εi,l1 , εi,l2 , ϕ¯ i,1 ) are positive constants associated with the given constants εi,l1 , εi,l2 and ϕ¯ i,l . Step n: Finally, the following Lyapunov function is given

− fi,1 − ki,1 zi,1 + y˙ i,d

m(zi,1 ) − 1

1

and Mi (εi,11 , ϕ¯ i,1 ), Mi (εi,12 , ϕ¯ i,1 ) are given in (45) and (50), respectively. Step m, (m = 2, . . . , n − 1): At this step, we choose the following Lyapunov function

l =1

Case III: Finally, we consider the case εi,11 < |zi,1 | < εi,12 . From (38)–(40), we have

αi,1 =

(49)

rij (ϕ12 (¯xj,1 ) − ϕ12 (¯xi,1 )).

Vi,m = Vi,m−1 +

Case II: Next, we consider the case |zi,1 | ≥ εi,12 . From (38)–(40), we have αi,1 = αib,1 , and it follows that

N 1

rij ϕ (¯xi,1 )

j=1

Now, considering the Cases I–III, we have

N

+

 2 1

rij ϕ12 (¯xi,1 ) ≤ Mi (εi,12 , ϕ¯ i,1 ).



2 j =1

+

N 

Mi (εi,11 , εi,12 , ϕ¯ i,1 )

2 1

1

zi2,1

where

j=1

+

2



rij (ϕ12 (¯xj,1 ) − ϕ12 (¯xi,1 )).

V˙ i,1 ≤ −ki,1 zi2,1 + gi,1 zi,1 zi,2 +

N

+

+ gi,1 zi,1 zi,2 +

1

rij (ϕ12 (¯xj,1 ) − ϕ12 (¯xi,1 ))

Substituting (50) into (49), we have

(44)

Using Assumption 5 gives



2 j=1

j =1

j =1

2 1

2 j=1

N 

+ 2 1

N 1

N 1

V˙ i,1 ≤ −ki,1 zi2,1 + gi,1 zi,1 zi,2 + Mi (εi,11 , εi,12 , ϕ¯ i,1 )

rij ϕ12 (¯xi,1 )

N

1

+

zi2,1

2

Again, by using Assumption 5, it is known that, for III, there must exist a positive constant Mi (εi,12 , ϕ¯ i,1 ) such that

According to (18), we have

zi2,1

 2 1

j =1

Step 1: We consider the quadratic Lyapunov function candidate 1

+ gi,1 zi,1 zi,2 +



m(zi,1 )

(48)

Vi,n = Vi,n−1 +

1 2

zi2,n

(57)

X.-J. Li, G.-H. Yang / Automatica 76 (2017) 87–95

93

which follows V˙ i,n = V˙ i,n−1 + zi,n (fi,n + gi,n ui − α˙ i,n−1 + hi,n (¯x1,n , x¯ 2,n , . . . x¯ N ,n )). (58) Substituting (38)–(40) into (58), we have V˙ i,n ≤ −

n 

ki,l zi2,l +

l =1

Mi (εi,l1 , εi,l2 , ϕ¯ i,1 )

l =1

n N 1 

+

n 

2 l=1 j=1

rij (ϕl2 (¯xj,l ) − ϕl2 (¯xi,l ))

(59)

where Mi (εi,l1 , εi,l2 , ϕ¯ i,1 ) are given in (56) with l = 1, 2, . . . , n. Now, we construct a global Lyapunov function V (t ): V (t ) =

N 

βi Vi,n (t )

(60)

i=1

where βi > 0 is given in (30). Combining (59) and (60), we have V˙ (t ) ≤ −

N  n 

βi ki,l zi2,l +

i=1 l=1

+

N  n 

From the systematic design procedure given in Section 3.2, it is known that all the signals are globally bounded if the constants ki,1 ≥ 1, and the details are omitted here due to the space limitation.

βi Mi (εi,l1 , εi,l2 , ϕ¯ i,1 )

i=1 l=1

n N N 1 

2 l =1 i =1 j =1

βi rij (ϕl2 (¯xj,l ) − ϕl2 (¯xi,l )).

(61)

N N

βi rij (ϕl2 (¯xj,l ) −

From (32)–(35), it is known that

i=1

j =1

ϕ (¯xi,l )) = 0 holds for ∀l ∈ {1, 2, . . . , n}. Therefore, if ki,l are chosen large enough such that ki,l ≥ 12 , we have 2 l

V˙ (t ) ≤ − V (t ) +

N  n 

βi Mi (εi,l1 , εi,l2 , ϕ¯ i,1 ).

Fig. 1. Tracking errors zi,1 (t ) with ε1,11 = ε1,21 = 2.4 and ε1,12 = ε1,22 = 2.6.

(62)

i =1 l =1

Remark 4. From (7), (10), (15), (39) and (40), it is known that the proposed decentralized control scheme is a strict one. In other words, the ith controller design only uses the local information of the ith subsystem, but not those of the other subsystems. Therefore, from the viewpoint of strictly decentralized control, the paper improves the result of Zhang and Lin (2014). 4. Example In this section, we consider the following interconnected nonlinear systems:

Now, the following theorem can be derived.

x˙ 1,1 (t ) = x1,2 + 0.2 sin(x2,1 )

Theorem 2. Consider the interconnected nonlinear system (2), and suppose that Assumptions 1–3 and 5 hold. If the constants ki,l are chosen such that ki,l ≥ 21 , then the control laws (39) and (40) guarantee that all the signals in the closed-loop system are globally bounded, and the tracking errors of each subsystem converge to the following compact set

x˙ 1,2 (t ) = (1 − x21,1 )x1,2 − x1,1 + u1 + x2,2 sin(x2,2 ) + d(t )

  N  n 2  E = zi,1 | |zi,1 | ≤  βi Mi (εi,l1 , εi,l2 , ϕ¯ i,1 ),  βi i=1 l=1   i = 1, 2, . . . , N   

Proof. Following the same lines of Theorem 1, one can complete the proof. Finally, the robust performance of the proposed decentralized control scheme is discussed. Suppose that the external disturbances di,m (t ), m = 1, 2, . . . , n, with known bounds d¯ i,m enter the system (2). Then, one can obtain the corresponding adaptive robust control law by modifying the stabilizing functions αib,1 and

αib,l , l = 2, 3, . . . , n, in (39) and (40):   N  1 b a 2 2 2 αi,1 = αi,1 − zi,1 + rij ϕ1 (¯xi,1 ) + d¯ i,1 α =α − b i ,l

a i ,l

1 2zi,l gi,l

j =1

 zi2,l

+

N  j =1

 2 ¯ rij ϕ (¯xi,l ) + di,l . 2 l

x˙ 2,2 (t ) = (1 + x2,1 x22,2 ) + y2 (t ) = x2,1

with exponential decay rate 12 .

2zi,1 gi,1

y1 (t ) = x1,1 ˙x2,1 (t ) = x2,2 + 0.2 sin(x1,1 )

(63)

1 1 + x22,2

u2 + x1,2 cos(x1,2 ) (64)

where d(t ) = 0.5 ∗ sin(t ) in the subsystem 1 denotes the external disturbance. It is assumed that the tracking trajectories are y1,d = 0.2 + 0.3 sin(t ), y2,d = 1 + 0.3 cos(2t ), and the initial conditions are x1 (0) = [1, 3]T , x2 (0) = [−1, 2]T . Moreover, from (63) and (64), the bounded functions of the interconnections are selected as ϕ1 (¯xi,1 ) = | sin(xi,1 )|, ϕ2 (¯xi,2 ) = |xi,2 | with r1,1 = 0, r1,2 = 1, r2,1 = 1, r2,2 = 0. In addition, we choose d¯ 1,1 = 0.5, k1,1 = 2, k1,2 = 2, k2,1 = 2, k2,2 = 2, ε1,11 = ε1,21 = 2.4 and ε1,12 = ε1,22 = 2.6. Then by using the control laws given in (39) with αia,1 and αia,2 given in (40), αib,1 and αib,2 given in (63), the tracking errors can be derived, which are plotted in Fig. 1. Also, the control inputs are shown in Fig. 2. From these two figures, we observer that the tracking errors and the control inputs of the closed-loop system are all bounded. Furthermore, if we choose ε1,11 = ε1,21 = 0.4 and ε1,12 = ε1,22 = 0.6, the tracking errors and control inputs are displayed in Figs. 3–4. The comparisons with Figs. 1–2 show that one can improve the tracking performance by paying more control efforts. In fact, this conclusion has been proven in Theorems 1–2, where the tracking error bounds are related with the controller design parameters involved in the smooth switching functions (15) and (38). On the other hand, the comparison results also

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strictly adaptive decentralized control approach to achieve the desired tracking performance. To this end, a backstepping technique based controller design scheme is given, together with introducing a new compensation mechanism. Moreover, it is proved that the tracking errors of each subsystem converge to an adjustable compact set with exponential decay rate. Finally, the effectiveness of the proposed adaptive decentralized control approach has been illustrated with a simulated example. References

Fig. 2. Control inputs zi,1 (t ) with ε1,11 = ε1,21 = 2.4 and ε1,12 = ε1,22 = 2.6.

Fig. 3. Tracking errors zi,1 (t ) with ε1,11 = ε1,21 = 0.4 and ε1,12 = ε1,22 = 0.6.

Fig. 4. Control inputs zi,1 (t ) with ε1,11 = ε1,21 = 0.4 and ε1,12 = ε1,22 = 0.6.

illustrate that there should exist a tradeoff between the tracking performance and control efforts. In conclusion, the simulation results illustrate that the proposed decentralized control approaches are valid for the nonlinear systems with strongly interconnections under the Assumptions 1–5. Moreover, just as mentioned in Remark 4, no extra information exchange between different subsystems is required for the developed strictly decentralized controller design. 5. Conclusion In this paper, the problem of adaptive decentralized tracking control of coupled nonlinear systems with strongly interconnections has been investigated. The main contribution is to develop a

Chen, W. S., & Li, J. M. (2008). Decentralized output-feedback neural control for systems with unknown interconnections. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 38(1), 258–266. Chen, B., Liu, X. P., Liu, K. F., & Lin, C. (2009). Novel adaptive neural control design for nonlinear MIMO time-delay systems. Automatica, 45(6), 1554–1560. Gavel, D. T., & Siljak, D. D. (1989). Decentralized adaptive control: Structural conditions for stability. IEEE Transactions on Automatic Control, 34(4), 413–426. Hovakimyan, N., Lavretsky, E., Yang, B. J., & Calise, A. J. (2005). Coordinated decentralized adaptive output feedback control of interconnected systems. IEEE Transactions on Neural Nerworks, 16(1), 185–194. Jiang, Z. P. (2002). Decentralized disturbance attenuating output-feedback trackers for large-scale nonlinear systems. Automatica, 38(8), 1407–1415. Knuth, D. E. (1997). The art of computer programming. Vol. 1 (3rd ed.). Reading: Addison-Wesley. Krishnamurthy, P., & Khorrami, F. (2003). Decentralized control and disturbance attenuation for large-scale nonlinear systems in generalized output-feedback canonical form. Automatica, 39(11), 1923–1933. Krstic, M., Kanellakopoulos, I., & Kokotovic, P. V. (1995). Nonlinear and adaptive control design. New York: Wiley and Sons. Li, M. Y., & Shuai, Z. S. (2010). Global-stability problem for coupled systems of differential equations on networks. Journal of Differential Equations, 248(1), 1–20. Liu, X. P., & Huang, G. S. (2001). Global decentralized robust stabilization for interconnected uncertain nonlinear systems with multiple inputs. Automatica, 37(9), 1435–1442. Liu, S. J., Zhang, J. F., & Jiang, Z. P. (2007). Decentralized adaptive output-feedback stabilization for large-scale stochastic nonlinear systems. Automatica, 43(2), 238–251. Mehraeen, S., Jagannathan, S., & Crow, M. L. (2011). Decentralized dynamic surface control of large-scale interconnected systems in strict-feedback form using neural networks with asymptotic stabilization. IEEE Transactions on Neural Nerworks, 22(11), 1709–1722. Narenda, K. S., & Oleng, N. (2002). Exact output tracking in decentralized adaptive control systems. IEEE Transactions on Automatic Control, 47(2), 390–395. Niculescu, S. L. (2001). Delay effects on stability: A robust control approach. New York: Springer-Verlag. Panagi, P., & Polycarpou, M. M. (2011). Distributed fault accommodation for a class of interconnected nonlinear systems with partial communication. IEEE Transactions on Automatic Control, 56(12), 2962–2967. Shi, L., & Singh, S. K. (1992). Decentralized adaptive controller design for large-scale systems with higher order interconnections. IEEE Transactions on Automatic Control, 37(8), 1106–1118. Spooner, J. T., & Passino, K. M. (1999). Decentralized adaptive control of nonlinear systems using radial basis neural networks. IEEE Transactions on Automatic Control, 44(11), 2050–2057. Stankovic, S. S., & Siljak, D. D. (2009). Robust stabilization of nonlinear interconnected systems by decentralized dynamic output feedback. Systems & Control Letters, 58(4), 271–275. Tang, Y., Tomizuka, M., Guerrero, G., & Montemayor, G. (2000). Decentralized robust control of mechanical systems. IEEE Transactions on Automatic Control, 45(4), 771–776. Tee, K. P., Ge, S. S., & Tay, E. H. (2009). Barrier Lyapunov functions for the control of output-constrained nonlinear systems. Automatica, 45(4), 918–927. Tong, S. C., Huo, B. Y., & Li, Y. M. (2014). Observer-based adaptive decentralized fuzzy fault-tolerant control of nonlinear large-scale systems with actuator failures. IEEE Transactions on Fuzzy Systems, 22(1), 1–15. Wen, C. Y. (1994). Decentralized adaptive regulation. IEEE Transactions on Automatic Control, 39(10), 2163–2166. Wen, C. Y., Zhou, J., & Wang, W. (2009). Decentralized adaptive backstepping stabilization of interconnected systems with dynamic input and output interactions. Automatica, 45(1), 55–67. Wu, J., Chen, W. S., & Li, J. (2015). Fuzzy-approximation-based global adaptive control for uncertain strict-feedback systems with apriori known tracking accuracy. Fuzzy Sets and Systems, 273, 1–25. Xie, S. L., & Xie, L. H. (2000). Decentralized stabilization of a class of interconnected stochastic nonlinear systems. IEEE Transactions on Automatic Control, 45(1), 132–137. Ye, X. D. (2011). Decentralized adaptive stabilization of large-scale nonlinear timedelay systems with unknown high-frequency-gain signs. IEEE Transactions on Automatic Control, 56(6), 1473–1478. Yoo, S. J., & Park, J. B. (2009). Neural-network-based decentralized adaptive control for a class of large-scale nonlinear systems with unknown time-varying delays. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 39(5), 1316–1323. Zhang, X., & Lin, Y. (2014). Nonlinear decentralized control of large-scale systems with strong interconnections. Automatica, 50(9), 2419–2423.

X.-J. Li, G.-H. Yang / Automatica 76 (2017) 87–95 Zhou, Q., Shi, P., Liu, H. H., & Xu, S. Y. (2012). Neural-Network-based decentralized adaptive output-feedback control for large-scale stochastic nonlinear systems. IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, 42(6), 1608–1619.

Xiao-Jian Li received the B.S. and M.S. degrees in mathematics from Northeast Normal University, China, in 2003 and 2006, respectively, and the Ph.D. degree in Control Theory and Engineering from Northeastern University, China, in 2011. He is currently an associate professor at the College of Information Science and Engineering, Northeastern University. His research interests include fault diagnosis, fault-tolerant control, fuzzy control, and decentralized control with applications in Cyber–Physical Systems.

95

Guang-Hong Yang received the B.S. and M.S. degrees in mathematics from Northeast University of Technology, China, in 1983 and 1986, respectively, and the Ph.D. degree in control engineering from Northeastern University, China (formerly, Northeast University of Technology), in 1994. He was a Lecturer/Associate Professor with Northeastern University from 1986 to 1995. He joined the Nanyang Technological University in 1996 as a Postdoctoral Fellow. From 2001 to 2005, he was a Research Scientist/Senior Research Scientist with the National University of Singapore. He is currently a Professor at the College of Information Science and Engineering, Northeastern University. His current research interests include fault-tolerant control, fault detection and isolation, nonfragile control systems design, and robust control. Dr. Yang is an Associate Editor for the International Journal of Control, Automation, and Systems (IJCAS), the International Journal of Systems Science (IJSS), the IET Control Theory & Applications, and the IEEE Transactions on Fuzzy Systems.