Adaptive decentralized output feedback PI tracking control design for uncertain interconnected nonlinear systems with input quantization

Adaptive decentralized output feedback PI tracking control design for uncertain interconnected nonlinear systems with input quantization

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Adaptive decentralized output feedback PI tracking control design for uncertain interconnected nonlinear systems with input quantization Haibin Sun, Guangdeng Zong, C.L. Philip Chen PII: DOI: Reference:

S0020-0255(19)30923-5 https://doi.org/10.1016/j.ins.2019.09.072 INS 14903

To appear in:

Information Sciences

Received date: Revised date: Accepted date:

9 April 2019 24 September 2019 26 September 2019

Please cite this article as: Haibin Sun, Guangdeng Zong, C.L. Philip Chen, Adaptive decentralized output feedback PI tracking control design for uncertain interconnected nonlinear systems with input quantization, Information Sciences (2019), doi: https://doi.org/10.1016/j.ins.2019.09.072

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Highlights • Unlike the existing results in the literature, the effects of quantization, strongly interconnection and unmeasurable states are simultaneously investigated in the nonlinear system. This brings more complexity and difficulty to the analysis and synthesis. • Compared with [22] and [23], in this paper, the proposed control method is simpler in structure. • A decentralized input-driven filter is designed and an output feedback controller is developed, which is more applicable in practice. • One adaptive parameter needs only to be regulated for each subsystem, which has no connection with the number of neural network nodes. Hence, the computational overhead is greatly reduced.

1

Adaptive decentralized output feedback PI tracking control design for uncertain interconnected nonlinear systems with input quantization Haibin Suna , Guangdeng Zonga,∗, C. L. Philip Chenb a

b

School of Engineering, Qufu Normal University, Rizhao 276826, Shandong, P. R. China Department of Computer and Information Science, University of Macau, Macau 99999, China

Abstract In this paper, the problem of adaptive decentralized proportional-integral (PI) tracking control is investigated for a class of interconnected nonlinear systems with input quantization and unknown functions, where the interconnection terms are bounded by completely unknown functions. By designing an input-driven filter, the unknown states are estimated and then an adaptive decentralized output feedback PI tracking controller is constructed via the backstepping method and neural network technique. The stability of the closed-loop system is addressed based on the Lyapunov function technique plus graph theory, and all the signals in the closed-loop system are uniformly ultimately bounded. Finally, simulation results are utilized to demonstrate the effectiveness of the proposed method. Keywords: Interconnected system, PI control, output feedback, neural network, input quantization, decentralized control

1. Introduction Interconnected systems can be used to describe different kinds of systems with complex dynamical structures and a strong coupling character, for example, tuned mass damper and multimachine power systems [12, 46, 47]. Hence, the stability analysis and controller design of interconnected systems have increasingly received more attention, and many kinds of control schemes have been proposed. Firstly, based on the full states information, some centralized control methods are presented, but the computational burden of the centralized control schemes is heavy. Then, numerous decentralized controllers on the basis of local information of subsystem have been designed to solve the above problem. The existing interconnected coupling term makes it difficult to construct a satisfactory controller. Numerous control schemes have been developed according to the different properties of interconnected coupling terms. In previous studies [21, 41, 42], some control problems have been ∗

Corresponding Author. Guangdeng Zong, Tel.: +86 633 3981777, Email address: [email protected]

Preprint submitted to Information Sciences

September 27, 2019

investigated for many kinds of interconnected nonlinear systems where the interconnected terms are assumed to include subsystem inputs and outputs and deemed as weak interconnection [49]. Recently, a stronger interconnection, i.e. where the interconnected terms and their bounds are the functions of all subsystem states, is introduced to the control domain [3, 49]. Unsatisfactorily, the controllers for each subsystem use some prior information from other subsystems. In [15], a decentralized output feedback controller is proposed for a large-scale nonlinear system, where the value of the interconnected terms is bounded by the full state-dependent linear functions. Very recently, two adaptive decentralized state feedback controllers have been proposed for interconnected systems with full state-dependent nonlinear functions, where each control law depends on its subsystem information [17, 18]. Although various control methods have been developed for interconnected systems in the above references, the resulting solutions are structurally complicated and computationally expensive. As is well-known, the proportional-integral (PI) control has many merits:intuitiveness and simplicity in both structure and concept. Thus, PI control has gained many applications in practical engineering [6, 48]. Hence, many scholars have put significant effort into studying the related problem of PI control, such as the determination and adjustment of the PI gain [1, 5, 24] and [25]. In [26, 27, 28, 32], the authors have explored some control problems for complex nonlinear systems via PI control with the full states measurable. In addition, with the progress of networked control system, quantization control has become a heated issue because of its wide application in many fields [7, 23, 35, 40, 44]. Numerous control schemes have been proposed to study the quantized feedback control problem for different control systems [7, 19, 20, 31, 45]. At present, quantization control has been investigated for interconnected systems with weak interconnected terms [37], and the controller has a complex structure. There are few results discussing the quantized controller design for an interconnected system with strongly interconnected terms and unknown functions, especially with regard to to designing a simple form of output feedback controller for interconnected systems. This is an open problem and needs to be further discussed. This paper explores the output feedback PI tracking control problem for a class of interconnected nonlinear systems with input quantization and unknown functions. By means of neural network scheme and backstepping control method, an adaptive decentralized output feedback PI tracking controller is developed, where each virtual control has the same structure, i.e., PI-like control law. Using the Lyapunov function technique, the stability analysis of the closed-loop system is established. The uniformly ultimate boundedness of the closed-loop system is analyzed. Finally, the effectiveness of the developed controller is demonstrated via an application example.The main advantages of this paper include: • Unlike existing results in the literature, the effects of quantization, strong interconnection and unmeasurable states are simultaneously investigated in a nonlinear system. This brings more complexity and difficulty to the analysis and synthesis. • Compared with [17] and [18], in this paper, the proposed control method is simpler in structure. • A decentralized input-driven filter is designed, and an output feedback controller is devel3

oped, which is more applicable in practice. • One adaptive parameter needs to be regulated only for each subsystem, which has no connection to the number of neural network nodes. Hence, the computational overhead is greatly reduced. 2. Preliminaries and problem formulation 2.1. Preliminaries G = (V, E) denotes a weighted graph with nodes V = 1, 2, · · · , N and an edge set E with element (i, j). The weight matrix A = (ai j )N×N is defined for a weighted digraph G with N nodes, where ai j is associated with the arc (i, j) of the digraph. For convenience, a weighted digraph is denoted as (G, A), and it can be employed to model the topology of all links in the network. If there exists a direct path for any pair of distinct nodes, a digraph G is called strongly connected. Strong connection of a weighted digraph (G, A) is equivalent to the irreducibility of the weight matrix A. P The Laplacian matrix L = (li j )N×N of (G, A) is described as li j = −ai, j , i , j, lii = Nj=1, j,i ai j , i = 1, 2, · · · , N. 2.2. Problem formulation An interconnected nonlinear system is considered in this paper, where each node of diagraph G denotes its own internal dynamics, and directed edges in G indicates coupling terms between node dynamics. Each node’s dynamics is shown as follows (Σ0 ) : x˙i,m = xi,m+1 + fi,m ( x¯i,m ) + hi,m ( x¯1,m , · · · , x¯ N,m ), x˙i,n = q(ui ) + fi,n ( x¯i,n ) + hi,n ( x¯1,n , x¯2,n , x¯ N,n ), yi = xi,1 ,

(1)

where xi = [xi,1 , xi,2 , · · · , xi,n ]T ∈ Rn , yi ∈ R, and ui ∈ R are system states, output, and control input of subsystem i, i = 1, 2, · · · , N, and m = 1, · · · , n − 1, respectively. fi,m and hi,m denote unknown functions, x¯i,m = [xi,1 , xi,2 , · · · , xi,m ]T . q(ui ) means the quantized input with ui and the expression of quantizer is referenced in [37]. Moreover, some assumptions and lemmas are required to develop the main results. Assumption 1. The digraph G is strongly connected. Assumption 2. The nonlinear functions hi,m ( x¯1,m , x¯2,m , x¯ N,m ) satisfy |hi,m ( x¯1,m , x¯2,m , x¯ N,m )| ≤

N X

ci, j φm ( x¯ j,m )

(2)

j=1

where continuous functions φm ( x¯ j,m ), m = 1, 2, · · · , n, and constants ci, j are unknown. Remark 1. For simplicity of notation, we drop the time variable t from the states, outputs, and inputs throughout the whole paper. 4

Remark 2. Based on Assumption 2, we know the interconnections rely on the states of all subsystems. Then the system (1) can be employed to describe many state-space models of interconnected nonlinear systems, e.g. [29, 34]. Lemma 1 [50]. The hysteretic quantizer q(ui ) is rewritten as q(ui ) = ui + %i ,

(3)

where %i = q(ui ) − ui ∈ R satisfies the following inequalities %2i ≤ τ2i u2i , ∀ |ui | ≥ ui,min ,

%2i

≤

u2i,min , ∀

|ui | ≤ ui,min ,

(4) (5)

i where τi = 1− , 0 < i < 1 is the measure of quantization density. 1+i Lemma 2 [38, 22]. For the following system

˙ˆ = −γβ(t) ˆ + oρ(t) β(t)

(6)

ˆ 0 ) ≥ 0, we have β(t) ˆ ≥ 0. with o > 0, γ > 0, function ρ(t) > 0 and initial condition β(t Lemma 3 [9]. For a given constant µ > 0 and variable s ∈ R, the following condition holds lim tanh2 (s/µ)/s = 0. s→0

Lemma 4 [9]. Consider the set Ω := {X | | X |< 0.8814υ}. Then, for any X < Ω, the following inequality holds 1 − 2 tanh2

X υ

≤ 0.

The control objective is to develop an adaptive decentralized output feedback PI tracking controller for interconnected nonlinear systems (1), such that each subsystem output yi can follow a reference signal yi,r and all signals in the closed-loop system are bounded in the presence of input quantization and unknown functions. 3. Adaptive decentralized output feedback PI tracking controller design and stability analysis 3.1. Adaptive decentralized output feedback PI tracking controller design A decentralized input-driven filter is designed as x˙ˆi,m = xˆi,m+1 − li,m xˆi,1 , x˙ˆi,n = q(ui ) − li,n xˆi,1 , 5

(7)

where xˆi = [ xˆi,1 , xˆi,2 , · · · , xˆi,n ]T is an observer state, and li,m is an observer gain such that the following matrix   −li,1  Ai,c =  ... In−1  −li,n 0 · · ·

0

    

is a strict Hurwitz matrix. Thus, there exists a positive definite matrix Pi such that ATi,c Pi + Pi Ai,c = −Qi , Qi > 0.

(8)

Define the observer errors as ei,m = xi,m − xˆi,m and ei,n = xi,n − xˆi,n . Then the observer error system is described as e˙ i,m = ei,m+1 − li,m ei,1 + fi,m ( x¯i,m ) +hi,m ( x¯1,m , · · · , x¯ N,m ) + li,m yi , e˙ i,n = −li,n ei,1 + fi,n ( x¯i,n ) +hi,n ( x¯1,n , · · · , x¯ N,n ) + li,n yi ,

(9)

which can be expressed as the following form e˙ i = Ai,c ei + Fi ( x¯i ) + Hi (x1 , · · · , xN ),

(10)

where ei = [ei,1 , ei,2 , · · · , ei,n ]T ,

Fi ( x¯i ) = [Fi,1 ( x¯i,1 ), · · · , Fi,n ( x¯i,n )]T

= [ fi,1 ( x¯i,1 ) + li,1 yi , · · · , fi,n ( x¯i,n ) + li,n yi ]T , Hi (x1 , · · · , xN ) = [hi,1 ( x¯1,1 , · · · , x¯ N,1 ), · · · , hi,n ( x¯1,n , · · · , x¯ N,n )]T .

Combining (1) and (7) with (10), we have y˙ i = x˙ˆi,m = x˙ˆi,n = e˙ i =

xˆi,2 + ei,2 + fi,1 ( x¯i,1 ) + hi,1 ( x¯1,1 , · · · , x¯ N,1 ), xˆi,m+1 − li,m xˆi,1 , q(ui ) − li,n xˆi,1 , Ai,c ei + Fi ( x¯i ) + Hi (x1 , · · · , xN ).

(11)

The PI tracking controller will be designed for the system (11). Step 1: Consider the first equation in (11), i.e., y˙ i = xˆi,2 + ei,2 + fi,1 ( x¯i,1 ) + hi,1 ( x¯1,1 , · · · , x¯ N,1 ). 6

(12)

The tracking R t error is defined as si,1 = yi − yi,r and the general tracking error is presented as zi,1 = si,1 + εi,1 0 si,1 ds, εi,1 > 0. Then the derivative of zi,1 is expressed as z˙i,1 = xˆi,2 + ei,2 + fi,1 ( x¯i,1 ) +hi,1 ( x¯1,1 , · · · , x¯ N,1 ) − y˙ i,r + εi,1 si,1 .

(13)

The quadratic Lyapunov function is selected as 1 1 ˜2 Vi,1 = eTi Pi ei + z2i,1 + θ, 2 2γi i

(14)

where Pi > 0, and γi > 0, θ˜i = θi − θˆi , θˆi is the estimation of θi specified by θi = max1≤i≤N ||θi,∗ j ||2 ; 1 ≤ i ≤ N, 1 ≤ j ≤ o j , and θi,∗ j is the weight vector of neural network being defined at step j, o j denotes the dimension of weight vector. Remark 3. To reduce the computational load of neural approximation, the minimal-learningparameter (MLP) scheme [2] is employed. It is implied that we estimate θi instead of θi,∗ j . In this case, for each subsystem only one parameter needs to regulated, so the required adaptive parameters will greatly decrease. Computing the derivative of (14) along systems (10) and (12) leads to ˙ i,1 = eTi (Pi Ai,c + ATi,c Pi )ei + 2eTi Pi Fi ( x¯i ) V +2eTi Pi Hi (x1 , · · · , xN ) +zi,1 ( xˆi,2 + ei,2 + fi,1 ( x¯i,1 ) − y˙ i,r + εi,1 si,1 ) 1 +hi,1 ( x¯1,1 , · · · , x¯ N,1 ) − θ˜i θ˙ˆi . γi Here, some inequalities are first presented as follows: 2ei Pi Fi ( x¯i ) ≤ ||Pi ||2 eTi ei + FiT Fi n X 2 T 2 ≤ ||Pi || ei ei + Fi,1 , j=1

2ei Pi Hi (x1 , · · · , xN ) ≤ ||Pi ||2 eTi ei + HiT Hi n X h2i, j ≤ ||Pi ||2 eTi ei + j=1

≤

||Pi ||2 eTi ei

+

h2i,1

+2

n X

h2i, j ,

j=2

1 zi,1 hi,1 ( x¯1,1 , · · · , x¯ N,1 ) ≤ z2i,1 + h2i,1 4 7

(15)

N X 1 2 ≤ z + ( ci, j φ1 ( x¯ j,1 ))2 4 i,1 j=1

N X 1 2 Nc2i, j φ21 ( x¯ j,1 )) zi,1 + ( 4 j=1

≤

N

1 2 X ≤ λi, j φ21 ( x¯i,1 ) zi,1 + 4 j=1 +

n X j=1

  λi, j φ21 ( x¯ j,1 ) − φ21 ( x¯i,1 ) ,

(16)

where λi, j = Nc2i, j . Combining (8), (15), with (16) gives rise to

1 ˜ ˙ˆ 2 2 ˙ i,1 ≤ −(λ V min (Qi ) − 2||Pi || )||ei || − γ θi θi i +zi,1 ( xˆi,2 + ei,2 + fi,1 ( x¯i,1 ) − y˙ i,r + εi,1 si,1 ) N X 1 2 + z2i,1 + Fi,1 +2 λi, j φ21 ( x¯i,1 ) 4 j=1 +

n X j=1

+

n X

 λi, j (φ21 ( x¯ j,1 ) − φ21 ( x¯i,1 )) Fi,2 j + 2

j=2

n X

h2i, j .

(17)

j=2

N P

λi, j φ21 ( x¯i,1 ) are included in (17), which are difficult to be directly z  compensated via controller design. To solve this difficulty, a hyperbolic tangent function tanh ηi,1i,1 first given in [8] is employed to cope with the above two terms. 2 Remark 4. Two terms Fi,1 and

j=1

By using this technique, we have zi,1 zi,1 2 2 ))Fi,1 + 2 tanh2 ( )Fi,1 , ηi,1 ηi,1 N X  ! 2 zi,1 2 2 λi, j φ1 ( x¯i,1 ) = 2 1 − 2tanh ηi,1 j=1 2 Fi,1 = (1 − 2 tanh2 (

×

N X

λi, j φ21 ( x¯i,1 )

2

+ 4tanh

j=1

The unknown function is defined as

N  zi,1  X

ηi,1

λi, j φ21 ( x¯i,1 ).

j=1

Φi,1 (Zi,1 ) = fi,1 ( x¯i,1 ) − y˙ i,r + εi,1 si,1 − εi,2 8

Z

t 0

si,2 ds

(18)

zi,1 2 1  2 tanh2 ( )Fi,1 zi,1 ηi,1 N   X 2 zi,1 +4tanh λi, j φ21 ( x¯i,1 ) , ηi,1 j=1 +

where Zi,1 = [xi,1 , xˆi,2 , yi,r , y˙ i,r , θˆi ]T , εi,2 , si,2 shall be defined later. z

z

Remark 5. The function tanh2 ( ηi,1i,1 ) is introduced into (18) due to its good property, i.e., z1i,1 tanh2 ( ηi,1i,1 ) is well-defined as zi,1 = 0 based on Lemma 2, and thus can be estimated ! via neural network tech  N P z z 2 and 1 − 2tanh2 ηi,1i,1 λi, j φ21 ( x¯i,1 ) will be dealt nique. In addition, the terms (1 − 2 tanh2 ( ηi,1i,1 ))Fi,1 j=1

with in the next section. In this case, the design difficulty from the terms Fi,1 and overcome.

N P

j=1

λi, j φ1 ( x¯i,1 ) is

To reconstruct the unknown nonlinear function, the neural network technique [4, 10, 13, 30, 33, 39, 43] is used and then the unknown function Φi,1 (Zi,1 ) can be expressed as follows: ∗T Φi,1 (Zi,1 ) = θi,1 ξi,1 (Zi,1 ) + δ∗i,1 ,

(19)

where ξi,1 (Zi,1 ) ∈ Ro1 is a radial basis function vector with o1 > 1 as the neural network node ∗ number. θi,1 denotes the ideal weight and δ∗i,1 means the approximation error with |δ∗i,1 | ≤ δi,1 . Combining (17)-(19) produces 1 ˜ ˙ˆ 2 2 ˙ i,1 ≤ −(λ V min (Qi ) − 2||Pi || )||ei || − γ θi θi i ∗T + zi,1 ( xˆi,2 + ei,2 + θi,1 ξi,1 (Zi,1 ) + δ∗i,1 Z t zi,1 1 2 + εi,2 si,2 ds) + z2i,1 + (1 − 2 tanh2 ( ))Fi,1 4 η i,1 0 N  ! X 2 zi,1 + 2 1 − 2tanh λi, j φ21 ( x¯i,1 ) ηi,1 j=1 +2

N X j=1

n n X   X Fi,2 j + 2 h2i, j . λi, j φ21 ( x¯ j,1 ) − φ21 ( x¯i,1 ) + j=2

j=2

Some inequalities are presented as follows 1 2 1 zi,1 + e2i,1 ≤ z2i,1 + eTi ei , 4 4 1 2 ∗T ∗ 2 || + ai zi,1 θi,1 ξi,1 (Zi,1 ) ≤ zi,1 ||ξi,1 (Zi,1 )||2 ||θi,1 4ai 1 2 ≤ z ||ξi,1 (Zi,1 )||2 θi + ai , 4ai i,1 zi,1 ei,1 ≤

9

(20)

1 2 z + δ2i,1 . 4 i,1

zi,1 δ∗i,1 ≤

(21)

Rt Define the error as si,2 = xˆi,2 − αi,1 and the generalized error as zi,2 = si,2 + εi,2 0 si,2 ds, where εi,2 > 0 and αi,1 is a virtual control law, which can be designed as the following PI-like form Z t αi,1 = −(ki,P1 + ∆ki,P1 )si,1 − (ki,I1 + ∆ki,I1 ) si,1 ds, (22) 0

where ki,P1 = λi,1 , ki,I1 = εi,1 λi,1 with εi,1 > 0 and λi,1 > 0, ∆ki,P1 and ∆ki,I1 are the time-varying terms, which can be given as follows: ∆ki,P1 =

1 3 + ||ξi,1 (Zi,1 )||2 θˆi , ∆ki,I1 = εi,1 ∆ki,P1 . 4 4ai

(23)

Then we can obtain zi,1 ( xˆi,2 + εi,2

Z

t

0

si,2 ds) = zi,1 αi,1 + zi,1 zi,2

3 1 = −λi,1 z2i,1 − ( + ||ξi,1 ||2 θˆi )z2i,1 + zi,1 zi,2 . 4 4ai

(24)

Substituting (21)-(24) into (20) gives rise to 2 2 ˙ i,1 ≤ −(λ V min (Qi ) − 2||Pi || − 1)||ei || + zi,1 zi,2 1 −λi,1 z2i,1 + θ˜i (Πθi,1i − θ˙ˆi ) γi zi,1 2 +(1 − 2 tanh2 ( ))Fi,1 ηi,1 N  ! X 2 zi,1 +2 1 − 2tanh λi, j φ21 ( x¯i,1 ) ηi,1 j=1

+2

N X j=1

+

n X

  λi, j φ21 ( x¯ j,1 ) − φ21 ( x¯i,1 )

Fi,2 j + 2

j=2

n X

h2i, j + ai + δ2i,1 ,

(25)

j=2

where Πθi,1i = γi

1 ||ξi,1 ||2 z2i,1 . 4ai

(26)

Step 2: The derivative zi,2 is presented as z˙i,2 = xˆi,3 − li,2 xˆi,1 − α˙ i,1 + εi,2 si,2 . 10

(27)

The Lyapunov function is chosen as follows: 1 Vi,2 = Vi,1 + z2i,2 . 2

(28)

Computing the derivative of (28) yields 2 2 ˙ i,2 ≤ −(λ V min (Qi ) − 2||Pi || − 1)||ei || + zi,1 zi,2 1 −λi,1 z2i,1 + θ˜i (Πθi,1i − θ˙ˆi ) γi zi,1 2 +(1 − 2 tanh2 ( ))Fi,1 ηi,1 N  ! X 2 zi,1 +2 1 − 2tanh λi, j φ21 ( x¯i,1 ) ηi,1 j=1

+2 +

N X

j=1 n X

  λi, j φ21 ( x¯ j,1 ) − φ21 ( x¯i,1 )

Fi,2 j

+2

j=2

n X

h2i, j + ai + δ2i,1

j=2

+zi,2 ( xˆi,3 − li,2 xˆi,1 − α˙ i,1 + εi,2 si,2 ).

(29)

Note that zi,2 2 zi,2 2 ))Fi,2 + 2 tanh2 ( )Fi,2 , ηi,2 ηi,2 N  ! X 2 zi,2 ≤ 2 1 − 2tanh λi, j φ22 ( x¯i,2 ) ηi,2 j=1

2 Fi,2 = (1 − 2 tanh2 (

2h2i,2

+4tanh

+2

N X j=1

2

N  zi,2  X

ηi,2

λi, j φ22 ( x¯i,2 )

j=1

  λi, j φ22 ( x¯ j,2 ) − φ22 ( x¯i,2 ) .

The unknown function can be defined as Φi,2 (Zi,2 ) = zi,1 − li,2 xˆi,1 − α˙ i,1 + εi,2 si,2 Z t zi,2 2 1  2 tanh2 ( )Fi,2 −εi,3 si,3 ds + zi,2 ηi,2 0 N  zi,2  X  +4tanh2 λi, j φ22 ( x¯i,2 ) , ηi,2 j=1 11

(30)

where Zi,2 = [xi,1 , xˆi,2 , xˆi,3 , yi,r , y˙ i,r , θˆi ]T , εi,3 , si,3 are defined later. To reconstruct the unknown function, the neural network technique is used and then the unknown function Φi,2 (Zi,2 ) can be expressed as follows ∗T Φi,2 (Zi,2 ) = θi,2 ξi,2 (Zi,2 ) + δ∗i,2 ,

(31)

where ξi,2 (Zi,2 ) ∈ Ro2 is a radial basis function vector with o2 > 1 being the neural network node ∗ number. θi,2 denotes the ideal weight and δ∗i,2 means the approximation error with |δ∗i,2 | ≤ δi,2 . Substituting (30) and (31) into (29) leads to 2 2 ˙ i,2 ≤ −(λ V min (Qi ) − 2||Pi || − 1)||ei || 2 X zi,l 1 ˜ θi ˙ˆ 2 2 (1 − 2 tanh2 ( ))Fi,l − λi,1 zi,1 + θi (Πi,1 − θi ) + γi η i,l l=1 ! N 2 X  zi,l  X λi, j φ21 ( x¯i,l ) +2 1 − 2tanh2 η i,l j=1 l=1

+2

2 X N X l=1 j=1

+

n X

Fi,2 j

  λi, j φ2l ( x¯ j,l ) − φ21 ( x¯i,l )

+2

j=3

n X

h2i, j + 2ai + δ2i

j=3

+ zi,2 ( xˆi,3 +

(32)

∗T θi,2 ξi,2 (Zi,2 )

+ εi,3

Some inequalities are presented as follows:

Z

0

t

si,3 ds + δ∗i,2 ).

1 2 z ||ξi,2 (Zi,2 )||2 θi + ai , 4ai i,2 1 2 ≤ z + δ2i,2 . 4 i,2

∗T zi,2 θi,2 ξi,2 (Zi,2 ) ≤

zi,2 δ∗i,2

(33)

Rt Define the error as si,3 = xˆi,3 − αi,2 and the generalized error as zi,3 = si,3 + εi,3 0 si,3 ds, where εi,3 > 0 and αi,2 is a virtual control law, which can be designed as the following PI-like form Z t αi,2 = −(ki,P2 + ∆ki,P2 )si,2 − (ki,I2 + ∆ki,I2 ) si,2 ds, (34) 0

where ki,P2 = λi,2 , ki,I2 = εi,2 λi,2 with εi,2 > 0 and λi,2 > 0, ∆ki,P2 and ∆ki,I2 are the time-varying terms, which can be given as follows: ∆ki,P2 =

1 3 + ||ξi,2 (Zi,2 )||2 θˆi , ∆ki,I2 = εi,2 ∆ki,P2 . 4 4ai

Then we can obtain zi,2 ( xˆi,3 + εi,3

Z

0

t

si,3 ds) = zi,2 αi,2 + zi,2 zi,3 12

(35)

3 1 = −λi,2 z2i,2 − ( + ||ξi,2 ||2 θˆi )z2i,2 + zi,2 zi,3 . 4 4ai

(36)

Substituting (33)-(36) into (32) gives rise to 2 2 ˙ i,2 ≤ −(λ V min (Qi ) − 2||Pi || − 1)||ei || 2 X 1 λi,l z2i,l + θ˜i (Πθi,2i − θ˙ˆi ) + zi,2 zi,3 − γi l=1

2 X zi,l 2 (1 − 2 tanh2 ( ))Fi,l η i,l l=1 2 N X  ! X 2 zi,l +2 1 − 2tanh λi, j φ2l ( x¯i,l ) η i,l j=1 l=1

+

+2

2 X N X l=1 j=1

+2

n X j=3

n   X 2 2 λi, j φl ( x¯ j,l ) − φl ( x¯i,l ) + Fi,2 j j=3

h2i, j ( x¯l, j , · · · , x¯ N, j ) + ai +

2 X

δ2i, j ,

(37)

j=1

where Πθi,2i

=

2 X

γi

l=1

1 ||ξi,l ||2 z2i,l . 4ai

(38)

Step m (3 ≤ m ≤ n − 1): Suppose that the virtual control laws αi,1 , · · · , αi,m−1 and tuning function i Πθi,m−1 are designed. Define the error as si,m = xˆi,m − αi,m−1 and the generalized error as zi,m = Rt si,m + 0 si,m ds. Computing the derivative of zi,m yields z˙i,m = xˆi,m+1 − li,m xˆi,1 − α˙ i,m−1 + εi,m si,m .

(39)

The Lyapunov function is chosen as follows: 1 Vi,m = Vi,m−1 + z2i,m . 2 Calculating the derivative of (40) gives rise to 2 2 ˙ i,m ≤ −(λ V min (Qi ) − 2||Pi || − 1)||ei || m−1 X 1 i + zi,m−1 zi,m − λi,l z2i,l + θ˜i (Πθi,m−1 − θ˙ˆi ) γi l=1

+

m−1 X l=1

(1 − 2 tanh2 (

zi,l 2 ))Fi,l ηi,l 13

(40)

m−1  N X X 2 zi,l λi, j φ2l ( x¯i,l ) +2 1 − 2tanh ( ) η i,l j=1 l=1

+2

N m−1 X X l=1 j=1

+

n X

Fi,2 j

  λi, j φ2l ( x¯ j,l ) − φ2l ( x¯i,l )

+2

n X

h2i, j

j=m

j=m

+ (m − 1)ai +

(41) m−1 X

δi, j

j=1

+ zi,m ( xˆi,m+1 − li,m xˆi,1 − α˙ i,m−1 + εi,m si,m ). 2 2 ˙ i,m ≤ −(λ V min (Qi ) − 2||Pi || − 1)||ei || m−1 X 1 i λi,l z2i,l + θ˜i (Πθi,m−1 + zi,m−1 zi,m − − θ˙ˆi ) γi l=1

+

m−1 X

(1 − 2 tanh2 (

l=1 m−1  X

+2

l=1

+2

N zi,l  X 1 − 2tanh ( ) λi, j φ2l ( x¯i,l ) ηi,l j=1 2

m−1 X N X l=1 j=1

+

n X

zi,l 2 ))Fi,l ηi,l

  λi, j φ2l ( x¯ j,l ) − φ2l ( x¯i,l )

Fi,2 j + 2

j=m

n X j=m

h2i, j + (m − 1)ai +

(42) m−1 X

δi, j

j=1

+ zi,m ( xˆi,m+1 − li,m xˆi,1 − α˙ i,m−1 + εi,m si,m ). Note that zi,m 2 zi,m 2 ))Fi,m )F , + 2 tanh2 ( ηi,m ηi,m i,m N  X 2 zi,m ) ≤ 2 1 − 2tanh ( λi, j φ2m ( x¯i,m ) ηi,m j=1

2 Fi,m = (1 − 2 tanh2 (

2h2i,m

+ 4tanh2 +2

N X j=1

N  zi,m  X

ηi,m

λi, j φ2m ( x¯i,m )

j=1

  λi, j φ2m ( x¯ j,m ) − φ2m ( x¯i,m ) .

Define the unknown function as Φi,m (Zi,m ) = zi,m−1 − li,m xˆi,1 − α˙ i,m−1 14

(43)

+εi,m si,m − εi,m+1

Z

t

si,m+1 ds

0

zi,m 2 1  2 tanh2 ( )F zi,m ηi,m i,m N   X 2 zi,m +4tanh λi, j φ2m ( x¯i,m ) , ηi,m j=1 +

where Zi,m = [xi,1 , xˆi,2 , · · · , xˆi,m+1 , yi,r , y˙ i,r , θˆi ]T , εi,m+1 , si,m+1 shall be defined later. In order to reconstruct the unknown function, the neural network technique is used and then the unknown function Φi,m (Zi,m ) is expressed as ∗T Φi,m (Zi,m ) = θi,m ξi,m (Zi,m ) + δ∗i,m ,

(44)

where ξi,m (Zi,m ) ∈ Rom is radial basis function vector with om > 1 being the neural network node ∗ denotes the ideal weight and δ∗i,m means the approximation error with |δ∗i,m | ≤ δi,m . number, θi,m Substituting (43) and (44) into (42) leads to 2 2 ˙ i,m ≤ −(λ V min (Qi ) − 2||Pi || − 1)||ei || m−1 X 1 i − λi,l z2i,l + θ˜i (Πθi,m−1 − θ˙ˆi ) γi l=1 m X zi,l 2 + (1 − 2 tanh2 ( ))Fi,l η i,l l=1

m  N X X 2 zi,l 1 − 2tanh ( ) +2 λi, j φ2l ( x¯i,l ) η i,l j=1 l=1

+2

m X N X l=1 j=1

+

n X

Fi,2 j

  λi, j φ2l ( x¯ j,l ) − φ2l ( x¯i,l ) +2

j=m+1

+ zi,m ( xˆi,m+1 +

n X

h2i, j

j=m+1

(45)

+ (m − 1)ai +

∗T θi,m ξi,m (Zi,m )

+

δ∗i,m

m−1 X j=1

+ εi,m+1

Some inequalities are presented as follows:

δ2i, j

Z

0

t

si,m+1 ds).

1 2 z ||ξi,m (Zi,m )||2 θi + ai , 4ai i,m 1 2 zi,m δ∗i,m ≤ z + δ2i,m . 4 i,m = xˆi,m+1 − αi,m and the generalized error as Z t zi,m+1 = si,m+1 + εi,m+1 si,m+1 ds, ∗T ξi,m (Zi,m ) ≤ zi,m θi,m

Define the error as si,m+1

15

0

(46)

where εi,m+1 > 0 and αi,m is a virtual control law, which can be designed as the following PI-like form αi,m = −(ki,Pm + ∆ki,Pm )si,m Z t −(ki,Im + ∆ki,Im ) si,m ds,

(47)

0

where ki,Pm = λi,m , ki,Im = εi,m λi,m with εi,m > 0 and λi,m > 0, ∆ki,Pm and ∆ki,Im are the time-varying terms, which can be given as follows: 3 1 + ||ξi,m (Zi,m )||2 θˆi , 4 4ai = εi,m ∆ki,Pm .

∆ki,Pm = ∆ki,Im

(48)

Then we can obtain Z

zi,m ( xˆi,m+1 + εi,m+1

t

si,m+1 ds)

0

= zi,m αi,m + zi,m zi,m+1 3 1 = −λi,m z2i,m − ( + ||ξi,m ||2 θˆi )z2i,m + zi,m zi,m+1 . 4 4ai

(49)

Substituting (46)-(49) into (45) gives rise to 2 2 ˙ i,m ≤ −(λ V min (Qi ) − 2||Pi || − 1)||ei || m X 1 i − θ˙ˆi ) + zi,m zi,m+1 − λi,l z2i,l + θ˜i (Πθi,m γ i l=1 m X zi,l 2 + (1 − 2 tanh2 ( ))Fi,l ηi,l l=1 m  N X X 2 zi,l +2 1 − 2tanh ( ) λi, j φ21 ( x¯i,l ) ηi,l j=1 l=1

+2 +2

m X N X

n  X  2 2 λi, j φl ( x¯ j,l ) − φ1 ( x¯i,l ) + Fi,2 j

l=1 j=1 n X h2i, j ( x¯1, j , · · · j=m+1

, x¯ N, j ) + mai +

j=m+1 m X

δ2i, j ,

(50)

j=1

where i Πθi,m =

m X

γi

l=1

16

1 ||ξi,l ||2 z2i,l . 4ai

(51)

i Step n: Suppose that the virtual control laws αi,1 , · · · , αi,n−1 and tuning function Πθi,n−1 are designed. Rt Define the error as si,n = xˆi,n − αi,n−1 and the generalized error as zi,n = si,n + εi,n 0 si,n ds. By calculating the first order derivative of zi,n , we have

z˙i,n = q(ui ) − li,n xˆi,1 − α˙ i,n−1 + εi,n si,n .

(52)

The Lyapunov function is chosen as follows: 1 Vi,n = Vi,n−1 + z2i,n . 2

(53)

Computing the derivative of (53) leads to 2 2 ˙ i,n ≤ −(λ V min (Qi ) − 2||Pi || − 1)||ei || + zi,n−1 zi,n n−1 X 1 i λi,l z2i,l + θ˜i (Πθi,n−1 − − θ˙ˆi ) γ i l=1 n−1 X

zi,l 2 ))Fi,l η i,l l=1 n−1 N X  ! X 2 zi,l +2 1 − 2tanh λi, j φ2l ( x¯i,l ) η i,l j=1 l=1

+

+2

(1 − 2 tanh2 (

n−1 X N X l=1 j=1

+

2 Fi,n

+

  λi, j φ2l ( x¯ j,l ) − φ2l ( x¯i,l )

2h2i,n

+ (n − 1)ai +

n−1 X

δ2i, j

j=1

+ zi,n (q(ui ) − li,n xˆi,1 − α˙ i,n−1 + εi,n si,n ).

(54)

Note that zi,n zi,n 2 2 ))Fi,n + 2 tanh2 ( )Fi,n , ηi,n ηi,n N  zi,n  X ≤ 2 1 − 2tanh2 ( ) λi, j φ2n ( x¯i,n ) ηi,n j=1

2 Fi,n = (1 − 2 tanh2 (

2h2i,n

2

+ 4tanh +2

N X j=1

N  zi,n  X

ηi,n

λi, j φ2n ( x¯i,n )

j=1

  λi, j φ2n ( x¯ j,n ) − φ2n ( x¯i,n ) .

Define the unknown function as follows:

Φi,n (Zi,n ) = zi,n−1 − li,n xˆi,1 − α˙ i,n−1 + εi,n si,n 17

(55)

N   X 1  2 zi,n 2 zi,n 2 + λi, j φ2n ( x¯i,n ) , 2 tanh ( )Fi,n + 4tanh zi,n ηi,n ηi,n j=1

where Zi,n = [xi,1 , xˆi,2 , · · · , xˆi,n , yi,r , y˙ i,r , θˆi ]T . In order to reconstruct the unknown function, the neural network technique is used and then the unknown function Φi,n (Zi,n ) can be expressed ∗T Φi,n (Zi,n ) = θi,n ξi,n (Zi,n ) + δ∗i,n ,

(56)

where ξi,n (Zi,n ) ∈ Ron is radial basis function vector with on > 1 being the neural network node ∗ denotes the ideal weight and δ∗i,n means the approximation error with |δ∗i,n | ≤ δi,n . number, θi,n Substituting (55)(56) into (54) and combining with (3) yields 2 2 ˙ i,n ≤ −(λ V min (Qi ) − 2||Pi || − 1)||ei || n−1 n X X zi,l 1 ˜ θi ˙ 2 2 ˆ λi,l zi,l + θi (Πi,n−1 − θi ) + − (1 − 2 tanh2 ( ))Fi,l γ η i i,l l=1 l=1 n  N X X 2 zi,l 1 − 2tanh ( ) +2 λi, j φ2l ( x¯i,l ) η i,l j=1 l=1

+2

n X N X l=1 j=1

  λi, j φ2l ( x¯ j,l ) − φ2l ( x¯i,l )

+ zi,n (ui + %i +

∗T θi,n ξi,2 (Zi,n )

+

δ∗i,n )

(57) + (n − 1)ai +

n−1 X

δ2i, j .

j=1

By using Lemma 1, the following inequality holds zi,n %i (t) ≤ τi |zi,n ui | + ui,min |zi,n |.

(58)

Some inequalities are presented as 1 2 z ||ξi,n (Zi,n )||2 θi + ai , 4ai i,n 1 2 zi,n δ∗i,n ≤ zi,n + δ2i,n , 4 1 2 ui,min |zi,n | ≤ zi,n + u2i,min . 4

∗T zi,n θi,n ξi,n (Zi,n ) ≤

Substituting (58) and (59) into (57) leads to 2 2 ˙ i,n ≤ −(λ V min (Qi ) − 2||Pi || − 1)||ei || n−1 X 1 i − λi,l z2i,l + θ˜i (Πθi,n−1 − θ˙ˆi ) γ i l=1

18

(59)

+

n X l=1

(1 − 2 tanh2 (

zi,l 2 ))Fi,l ηi,l

n  N X X 2 zi,l +2 1 − 2tanh ( ) λi, j φ2l ( x¯i,l ) ηi,l j=1 l=1

+2

n X N X l=1 j=1

  λi, j φ2l ( x¯ j,l ) − φ2l ( x¯i,l )

1 2 z ||ξi,n (Zi,n )||2 θi 4ai i,n n X 1 + zi,n ui + τi |zi,n ui | + z2i,n + nai + δ2i, j + u2i,min . 2 j=1

+

(60)

3.2. Stability analysis In the following theorem, the main result is stated. Theorem 1. Consider the uncertain interconnected systems (1) with hysteretic quantizer and virtual control laws (22), (34), and (47). The controller and adaptive laws are constructed as 1  (ki,Pn + ∆ki,Pn )si,n 1 − τi Z t  +(ki,In + ∆ki,In ) si,n ds ,

ui = −

θ˙ˆi = Πθi,ni − γi ωi θˆi ,

(61)

0

(62)

where ki,Pn and ki,In are constants, which can be expressed as ki,Pn = λi,n , ki,In = εi,n λi,n with εi,n > 0 and λi,n > 0. ∆ki,Pn and ∆ki,In are the time-varying terms and given as follows: 1 3 + ||ξi,n (Zi,n )||2 θˆi , 4 4ai = εi,n ∆ki,Pn , n X 1 = γi ||ξi,l ||2 z2i,l . 4ai l=1

∆ki,Pn = ∆ki,In Πθi,ni

(63) (64)

Then all the signals of the closed-loop system are uniformly ultimately bounded. Proof. See Appendix. The design procedure of the proposed control scheme is shown in Fig. 1. Remark 6. The developed control strategies (22), (34), (47), and (61) have the same control structure, i.e., PI-like controller, which contains both a the proportional and integral part. It should be pointed out that the controller gains are made up of two components: constant terms and time varying terms. 19

Remark 7. With the demand for transient performance of real systems, the prescribed performance control has become an important research topic and a large number of results have been reported [16, 11]. To improve the system performance, it is necessary to further consider the prescribed performance control for interconnected nonlinear systems with strongly interconnected terms. x i , m

x i ,m 1  f i ,m ( x i ,m )  h i , m ( x1,m , " , x N ,m ) ,

x i , n

yi 

q ( u i )  f i ,n ( x i ,n )  hi ,n ( x1,n , x 2 ,n , x N ,n ) ,

yi

yi ,r 

x i ,1 ,

xˆi ,m xˆ

xˆi ,2

i ,n

q (ui )

xˆi , m 1  li , m xˆi ,1 , q (ui )  li ,n xˆi ,1 ,

xˆi ,n t

si ,1  H i ,1 ³ si ,1ds, H i ,1 ! 0, si ,1

zi ,1

yi  yi , r

0

t

(ki , P1  'ki , P1 ) si ,1  (ki , I1  'ki , I1 ) ³ si ,1ds

D i ,1

0

t

si ,2  H i ,2 ³ si ,2 ds

xˆi ,2  D i ,1 , zi ,2

si ,2

0

t

(ki , P2  'ki , P2 ) si ,2  (ki , I2  'ki , I2 ) ³ si ,2 ds

D i ,2

0

# xˆi ,m

t

si ,m  ³ si ,m ds

xˆi ,m  D i , m 1 , zi ,m

si ,m

0

zi ,1

D i ,m

t

zi ,2

0

# zi , n

(ki , Pm  'ki , Pm ) si ,m  (ki , I m  'ki , I m ) ³ si ,m ds

# xˆi ,n  D i ,n 1 , zi ,n

si , n



ui



Tˆi

t

si , n  H i ,n ³ si ,n ds 0

t 1 ((ki,Pn  'ki, Pn )si,n  (ki,In  'ki,In ) ³0 si,n ds) 1W i

3Ti ,in  J iZiTˆi , 3Ti ,in

n

¦J l 1

i

1 || [i ,l ||2 zi2,l 4ai

Fig. 1. Block diagram of control system

4. An application example An inverted pendulum system is described as follows: x˙1,1 = x1,2 , x˙2,1 = x2,2 , 1 m1 gr kr2 x˙1,2 = u1 + ( − ) sin(x1,1 ) J1 J1 4J1 20

kr kr2 (l − b) + sin(x2,1 ), 2J1 4J1 m2 gr kr2 1 u2 + ( − ) sin(x2,1 ) = J2 J2 4J2 kr kr2 + (l − b) + sin(x1,1 ), 2J2 4J2 = x1,1 , y2,1 = x2,1 , +

x˙2,2

y1,1

where x1,1 and x1,2 denote the angular displacements of the pendulums from vertical. m1 and m2 are the pendulum end masses, J1 and J2 are the inertia moments, k denotes the spring constant, r is the pendulum height, l is the spring natural length, b is distance between the pendulum hinges, and g means the gravitational acceleration. According to [36], the system parameters are listed as follows. m1 = 2kg, m2 = 2.5kg, J1 = 5kg, J2 = 6.25kg, b = 0.5m, k = 100N/m, r = 0.5m, l = 0.5m, g = 9.81m/s2 . The parameters of quantizer are listed as follows 1 = 0.3, 2 = 0.4, u1,min = 0.2, u2,min = 0.4. The initial condition is chosen as x1,1 = 0.1, x1,2 = −0.3, x2,1 = 0.1, x2,2 = −0.2. The reference signals are given as y1,r = 0.2 + 0.5 sin(0.5t), y2,r = −0.1 + 0.3 cos(2t). The controller parameters are chosen as k11 l11 ε1,1 σ1,1

= = = =

10, k12 = 10, k21 = 10, k22 = 20, γ1 = 20, 18, l12 = 81, l21 = 28, l22 = 196, γ2 = 10, ε1,2 = ε2,1 = ε2,2 = 2, a1 = a2 = 0.2, 0.7, σ1,2 = 6, σ2,1 = 0.6, σ2,2 = 7.

The centres of neural networks cover the compact set: [−5, 5] × [−10, 10]. In Fig. 2-5, the simulation results are shown. Curves of reference signals, outputs, and tracking errors are illustrated in Fig. 2. From Fig. 2, we can see that the outputs can effectively track the reference signals. Fig. 3 presents the state variables x1,2 and x2,2 , which shows that these variables are bounded. In Fig. 4, the quantized control inputs are depicted. Curves of the adaptive parameters θˆi are presented in Fig. 5.

21

1 y1

0.8

y1r

y1−y1r

0.6 0.4 0.2 0 −0.2 −0.4

0

20

40

60

Time,s

80

100

120

0.4 y2

y2r

y2−y2r

0.2 0 −0.2 −0.4 −0.6

0

20

40

60

Time,s

80

100

120

Fig. 2. Curves of outputs yi , reference signals yi,r and tracking error signals

22

2 1.5

x1,2

1 0.5 0 −0.5 −1

0

20

40

0

20

40

60

80

100

120

60

80

100

120

Time,s

1

x2,2

0.5

0

−0.5

−1

Time,s

Fig. 3. Curves of state variables x1,2 and x2,2

23

100

q(u1)

50

0

−50

−100

0

20

40

0

20

40

60

80

100

120

60

80

100

120

Time,s

150

q(u2)

100 50 0 −50 −100

Time,s

Fig. 4. Curves of quantized control inputs

24

25 20

ˆ1 θ

15 10 5 0

0

20

40

0

20

40

60

80

100

120

60

80

100

120

Time,s

1.4 1.2

ˆ2 θ

1 0.8 0.6 0.4 0.2 0

Time,s

Fig. 5. Curves of adaptive parameters

25

5. Conclusions In this paper, the problem of adaptive decentralized output feedback PI tracking control for interconnected nonlinear system with input quantization has been discussed. The proposed adaptive decentralized controller is simple in structure and can guarantee that the closed-loop system is uniformly ultimately bounded. Finally, the effectiveness of the developed method has been demonstrated via an example.

APPENDIX PROOF OF THEOREM 1 From Lemma 4 and adaptive laws (62), we have θˆi ≥ 0 and τi |zi,n ui | ≤ −τi zi,n ui .

(65)

Using (65) and substituting (61), (62) into (60) leads to 2 2 ˙ i,n ≤ −(λ V min (Qi ) − 2||Pi || − 1)||ei || n X − λi,l z2i,l + ωi θ˜ θˆi l=1

+

n X l=1

+2 +2

(1 − 2 tanh2 (

zi,l 2 ))Fi,l ηi,l

N n  X zi,l  X λi, j φ21 ( x¯i,l ) 1 − 2tanh2 ( ) η i,l j=1 l=1 n X N X

l=1 j=1 n X

+ nai +

j=1

  λi, j φ2l ( x¯ j,l ) − φ2l ( x¯i,l ) δ2i, j + u2i,min .

(66)

In light of the following inequality 1 1 ωi θ˜i θˆi ≤ − ωi θ˜i2 + ωi θi2 , 2 2 we have 2 2 ˙ i,n ≤ −(λ V min (Qi ) − 2||Pi || − 1)||ei || n n X zi,l 1 ˜2 X 2 2 − λi,l zi,l − ωi θi + (1 − 2 tanh2 ( ))Fi,l 2 η i,l l=1 l=1

26

(67)

n  N X X 2 zi,l λi, j φ2l ( x¯i,l ) +2 1 − 2tanh ( ) η i,l j=1 l=1

+2

n X N X

  λi, j φ2l ( x¯ j,l ) − φ21 ( x¯i,l ) + χi ,

l=1 j=1

where χi = nai +

n P

j=1

(68)

+ 12 ωi θi2 . Now, a Lyapunov function is constructed as

δ2i, j + u2

i,min

V(t) =

N X i=1

βi Vi,n (t)

(69)

where βi denotes the cofactor of the ith diagonal element ai j replaced by λi j . Choose κi = λmin (Qi )− 2||Pi ||2 − 1. Combining (68) with (69) leads to ˙ ≤ V +

N n X X   1 βi − κi ||ei ||2 − λi,l z2i,l − ωi θ˜i2 2 i=1 l=1

N X

n X

βi

(1 − 2 tanh2 (

i=1 l=1 N n  X X

+2

+2

βi

1 − 2tanh2 (

i=1

l=1

N X

n X N X

βi

i=1

zi,l ))F 2 ηi,l i,l

l=1 j=1

(70)

N zi,l  X ) λi, j φ2l ( x¯i,l ) ηi,l j=1

N   X 2 2 λi, j φl ( x¯ j,l ) − φl ( x¯i,l ) + βi χi . i=1

By Theorem 2.2 of [14] and [17], we obtain N X N X i=1 j=1

βi λi, j (φ2l ( x¯ j,l ) − φ2l ( x¯i,l )) = 0.

(71)

Then (70) can be expressed as ˙ ≤− V +2

N X i=1

N X i=1

+

N X i=1

βi

βi µi Vi,n +

βi χi ≤ −µV +

+χ+2

N X i=1

βi

i=1

n X l=1

(1 − 2 tanh2 (

zi,l ))F 2 ηi,l i,l

N zi,l  X 1 − 2tanh ( ) λi, j φ2l ( x¯i,l ) ηi,l j=1

n  X l=1

N X

2

N X

βi

i=1

n X l=1

(1 − 2 tanh2 (

zi,l ))F 2 ηi,l i,l

n  N X zi,l  X βi 1 − 2tanh2 ( ) λi, j φ2l ( x¯i,l ), η i,l j=1 l=1

27

(72)

n o N P where µi = min κi , 2λi,1 , · · · , 2λi,n , ωi γi , µ = mini {µi }, i = 1, · · · , N, χ = βi χi . i=1

Note that the second and third terms in (72) may be positive or negative, lying on the value of zi,l . Thus we present the following discussion. 1). zi,l ∈ Ωi,l satisfying |zi,l | < 0.8814ηi,l , i = 1, · · · , N, l = 1, · · · , n. From the adaptive laws (62), we can obtain that δˆ i and θˆi are bounded because of |zi,l | < 0.8814ηi,l . Note that θi and δi are constants, the estimation errors θ˜i and δ˜ i are bounded. Combining this with Assumption 3, we have xi,1 is bounded, then the boundedness of αi,1 is obtained. From this chain of reasoning, the boundedness of xi,l , αi,l , and ui can be derived. Then we can obtain the following inequality N X i=1

+2

βi

n X l=1

N X i=1

βi

(1 − 2 tanh2 (

zi,l ))F 2 ηi,l i,l

(73)

n  N X zi,l  X 1 − 2tanh2 ( ) λi, j φ2l ( x¯i,l ) ≤ E, η i,l j=1 l=1

where E is a positive constant. Therefore, (72) is rewritten as ˙ ≤ −µV + E + χ. V

(74)

From (74), we deduce V ≤ e−µt V(0) +

E+χ (1 − e−µt ). µ

(75)

Therefore, all the signals in the closed-loop system converge to a compact set Θ1 as below n E + χo Θ1 = (zi,l , θ˜i , δ˜ i )|V(t) ≤ . µ

(76)

2). zi,l < Ωi,l satisfying |zi,l | ≥ 0.8814ηi,l . 2 In this case, by using Lemma 4 and the fact that λi, j φ2l ( x¯i,l ) ≥ 0 and Fi,l ≥ 0, the inequalities N  zi,l  X βi 1 − 2tanh2 ( ) λi, j φ2l ( x¯i,l ) ≤ 0, ηi,l j=1 zi,l 2 and βi (1 − 2 tanh2 ( ))Fi,l ≤0 ηi,l

(77) (78)

˙ ≤ −µV + χ. Therefore, all the signals in the closed-loop hold. Then, (72) can be rewritten as V system converge to a compact set Θ2 given below n χo Θ2 = (zi,l , θ˜i , δ˜ i )|V ≤ . (79) µ

3). zi,l ∈ Ωi,l and zm,n < Ωm,n for (i, l) , (m, n).

28

Under this case, (72) can be expressed as ˙ ≤ −µV + V

X

(i,l)∈

P

I

N X  2 zi,l λi, j φ2l ( x¯i,l ) 2βi 1 − 2tanh ( ) ηi,l j=1

N  X 2 zi,l λi, j φ2l ( x¯i,l ) + 2βi 1 − 2tanh ( ) P η i,l j=1 (m,n)∈ M X   z i,l 2 + βi 1 − 2tanh2 ( ) Fi,l P η i,l (i,l)∈ I X  zi,l  2 + + χ, βi 1 − 2tanh2 ( ) Fi,l P ηi,l (m,n)∈

X

(80)

M

where

P

(i,l)∈

P

I

is the subsystem considering of zi,l ∈ Ωi,l ,

P

(m,n)∈

P

means the subsystem considering

M

  P N P P 2 zi,l 2 ( x ¯ ) ≤ 0 and of zm,n < Ωm,n . By using Lemma 4, β (1 − 2tanh λ φ βi (1 − ) i,l i i, j l P P ηi,l j=1 (m,n)∈ M (m,n)∈ M z  2 2tanh2 ηi,li,l )Fi,l ≤ 0. (80) is rewritten as N  X 2 zi,l βi 1 − 2tanh ( ) λi, j φ2l ( x¯i,l ) P ηi,l j=1 (i,l)∈ I X   zi,l 2 + βi 1 − 2tanh2 ( ) Fi,l + χ. P η i,l (i,l)∈

˙ ≤ −µV + 2 V

X

(81)

I

Note that 2

P

(i,l)∈

P

I

 P   N P z 2 zi,l 2 βi 1 − 2tanh2 ( ηi,li,l ) β 1 − 2tanh ( ) Fi,l ≤ E. Similarly, we λi, j φ2l ( x¯i,l ) + P i ηi,l j=1

(i,l)∈

I

can deduce that all the signals in the closed-loop system can converge to a set Θ1 .

Remark 8. From (72), to obtain good control performance, we can add the value of χ and reduce the value of µ with suitable selection of the design parameters in χ and µ. Thus, a specific control performance and control action is achieved via regulation parameters in practice. Acknowledgments This work was supported in part by the National Natural Science Foundation of China (61773236, 61773235, 61873331, 61803225), part by the Taishan Scholar Project of Shandong Province (TSQN20161033, ts201712040), and the Postdoctoral Science Foundation of China (2017M612236, 2019T120574) and partially by Interdisciplinary Scientific Research Projects of Qufu Normal University under Grant (xkjjc201905). Declaration of Competing Interest The authors declare that they have no conflict of interests. 29

References [1] K. H. Ang, G. Chong, and Y. Li. PID control system analysis, design, and technology. IEEE Transactions on Control Systems Technology, 13(4):559–576, 2005. [2] B. Chen, X. P. Liu, K. F. Liu, and C. Lin. Direct adaptive fuzzy control of nonlinear strict-feedback systems. Automatica, 45(6):1530–1535, 2009. [3] B. Chen, X. P. Liu, K. F. Liu, and C. Lin. Novel adaptive neural control design for nonlinear mimo time-delay systems. Automatica, 45(6):1554–1560, 2009. [4] C. L. P. Chen, G. X. Wen, Y. J. Liu, and Z. Liu. Observer-based adaptive backstepping consensus tracking control for high-order nonlinear semi-strict-feedback multiagent systems. IEEE Transactions on Cybernetics, 46(7):1591–1601, 2016. [5] M. Cheng, Q. Sun, and E. Zhou. New self-tuning fuzzy PI control of a novel doubly salient permanent-magnet motor drive. IEEE Transactions on Industrial Electronics, 53(3):814–821, 2006. [6] J. W. Choi and S. C. Lee. Anti-windup strategy for PI-type speed controller. IEEE Transactions on Industrial Electronics, 56(6):2039–2046, 2009. [7] H. J. Gao and T. W. Chen. A new approach to quantized feedback control systems. Automatica, 44(2):534–542, 2008. [8] S. S. Ge and K. P. Tee. Approximation-based control of nonlinear MIMO time-delay systems. Automatica, 43(1):31–43, 2007. [9] S. Z. S. Ge and K. P. Tee. Approximation-based control of nonlinear MIMO time-delay systems. Automatica, 43(1):31–43, 2007. [10] W. He, Y. Chen, and Z. Yin. Adaptive neural network control of an uncertain robot with full-state constraints. IEEE Transactions on Cybernetics, 46(3):620–629, 2016. [11] C. Hua, L. Zhang, and X. Guan. Decentralized output feedback adaptive NN tracking control for time-delay stochastic nonlinear systems with prescribed performance. IEEE Transactions on Neural Networks and Learning Systems, 26(11):2749–2759, Nov 2015. [12] L. Jiang, Q. H. Wu, and J. Y. Wen. Decentralized nonlinear adaptive control for multimachine power systems via high-gain perturbation observer. IEEE Transactions on Circuits and Systems I: Regular Papers, 51(10):2052– 2059, 2004. [13] D. Li, Y. Liu, S. Tong, C. L. P. Chen, and D. Li. Neural networks-based adaptive control for nonlinear state constrained systems with input delay. IEEE Transactions on Cybernetics, 49(4):1249–1258, 2019. [14] M. Y. Li and Z. S. Shuai. Global-stability problem for coupled systems of differential equations on networks. Journal of Differential Equations, 248(1):1–20, 2010. [15] S. Li, C. K. Ahn, and Z. Xiang. Decentralized stabilization for switched large-scale nonlinear systems via sampled-data output feedback. IEEE Systems Journal, page doi:10.1109/JSYST.2019.2903297, 2019. [16] S. Li, C. K. Ahn, and Z. R. Xiang. Adaptive fuzzy control of switched nonlinear time-varying delay systems with prescribed performance and unmodeled dynamics. Fuzzy Sets and Systems, 371:40 – 60, 2019. [17] X. J. Li and G. H. Yang. Adaptive decentralized control for a class of interconnected nonlinear systems via backstepping approach and graph theory. Automatica, 76:87–95, 2017. [18] X. J. Li and G. H. Yang. Neural-network-based adaptive decentralized fault-tolerant control for a class of interconnected nonlinear systems. IEEE Transactions on Neural Networks and Learning Systems, 29(1):144– 155, 2018. [19] D. Liberzon and J. Hespanha. Stabilization of nonlinear systems with limited information feedback. IEEE Transactions on Automatic Control, 50(6):910–915, 2005. [20] J. L. Liu and N. Elia. Quantized feedback stabilization of non-linear affine systems. International Journal of Control, 77(3):239–249, 2004. [21] S. J. Liu, J. F. Zhang, and Z. P. Jiang. Decentralized adaptive output-feedback stabilization for large-scale stochastic nonlinear systems. Automatica, 43(2):238–251, 2007. [22] Z. Liu, F. Wang, Y. Zhang, and C. L. P. Chen. Fuzzy adaptive quantized control for a class of stochastic nonlinear uncertain systems. IEEE Transactions on Cybernetics, 46(2):524–534, 2016. [23] R. K. Miller, A. N. Michel, and J. A. Farrell. Quantizer effects on steady-state error specifications of digital feedback control systems. IEEE Transactions on Automatic Control, 34(6):651–654, 1989.

30

[24] A. O’Dwyer. Handbook of PI and PID controller tuning rules. World Scientific, 2009. [25] R. Shahnazi, R. Mohammad, and T. Akbarzadeh-T. PI adaptive fuzzy control with large and fast disturbance rejection for a class of uncertain nonlinear systems. IEEE Transactions on Fuzzy Systems, 16(1):187–197, 2008. [26] Y. D. Song, J. X. Guo, and X. C. Huang. Smooth neuroadaptive PI tracking control of nonlinear systems with unknown and nonsmooth actuation characteristics. IEEE Transactions on Neural Networks and Learning Systems, 28(9):2183–2195, 2017. [27] Y. D. Song, Z. Y. Shen, L. He, and X. C. Huang. Neuroadaptive control of strict feedback systems with full-state constraints and unknown actuation characteristics: An inexpensive solution. IEEE Transactions on Cybernetics, 48(11):3126–3134, 2018. [28] Y. D. Song, Y. J. Wang, and C. Y. Wen. Adaptive fault-tolerant PI tracking control with guaranteed transient and steady-state performance. IEEE Transactions on Automatic Control, 62(1):481–487, 2017. [29] J. T. Spooner and K. M. Passino. Decentralized adaptive control of nonlinear systems using radial basis neural networks. IEEE Transactions on Automatic control, 44(11):2050–2057, 1999. [30] H. Sun, L. Hou, G. Zong, and X. Yu. Adaptive decentralized neural network tracking control for uncertain interconnected nonlinear systems with input quantization and time delay. IEEE Transactions on Neural Networks and Learning Systems, page 10.1109/TNNLS.2019.2919697, 2019. [31] H. Sun, L. Hou, G. Zong, and X. Yu. Fixed-time attitude tracking control for spacecraft with input quantization. IEEE Transactions on Aerospace and Electronic Systems, 55(1):124–134, 2019. [32] H. Sun, G. Zong, and C. K. Ahn. Quantized decentralized adaptive neural network pi tracking control for uncertain interconnected nonlinear systems with dynamic uncertainties. IEEE Transactions on Systems, Man, and Cybernetics: Systems, pages 1–14, 2019. [33] H. B. Sun and L. Guo. Neural network-based DOBC for a class of nonlinear systems with unmatched disturbances. IEEE Transactions on Neural Networks and Learning Systems, 28(2):482–489, 2017. [34] Y. Tang, M. Tomizuka, G. Guerrero, and G. Montemayor. Decentralized robust control of mechanical systems. IEEE Transactions on Automatic Control, 45(4):771–776, 2000. [35] S. Tatikonda and S. Mitter. Control under communication constraints. IEEE Transactions on Automatic Control, 49(7):1056–1068, 2004. [36] S. C. Tong, B. Y. Huo, and Y. M. Li. 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, 2014. [37] C. L. Wang, C. Y. Wen, Y. Lin, and W. Wang. Decentralized adaptive tracking control for a class of interconnected nonlinear systems with input quantization. Automatica, 81:359–368, 2017. [38] M. Wang, S. Y. Zhang, B. Chen, and F. Luo. Direct adaptive neural control for stabilization of nonlinear timedelay systems. Science China Information Sciences, 53(4):800–812, 2010. [39] R. Wang and S. Fei. Output tracking for nonlinear discrete-time systems via fuzzy control approach. Journal of the Franklin Institute, 352(10):4147–4162, 2015. [40] R. Wang, L. Hou, G. Zong, S. Fei, and D. Yang. Stability and stabilization of continuous-time switched systems: A multiple discontinuous convex Lyapunov function approach. International Journal of Robust and Nonlinear Control, 29(5):1499–1514, 2019. [41] C. Y. Wen. Decentralized adaptive regulation. IEEE Transactions on Automatic Control, 39(10):2163–2166, 1994. [42] C. Y. Wen, J. Zhou, and W. Wang. Decentralized adaptive backstepping stabilization of interconnected systems with dynamic input and output interactions. Automatica, 45(1):55–67, 2009. [43] G. Wen, S. S. Ge, C. L. P. Chen, F. Tu, and S. Wang. Adaptive tracking control of surface vessel using optimized backstepping technique. IEEE Transactions on Cybernetics, 49(9):3420–3431, 2019. [44] W. S. Wong and R. W. Brockett. Systems with finite communication bandwidth constraints. ii. stabilization with limited information feedback. IEEE Transactions on Automatic Control, 44(5):1049–1053, 1999. [45] H. Wu, Z. Liu, Y. Zhang, and C. L. P. Chen. Adaptive fuzzy quantized control for nonlinear systems with hysteretic actuator using a new filter-connected quantizer. IEEE Transactions on Cybernetics, 2018, doi:10.1109/TCYB.2018.2864166. [46] D. Z. Xu, J. X. Liu, X. G. Yan, and W. X. Yan. A novel adaptive neural network constrained control for a multiarea interconnected power system with hybrid energy storage. IEEE Transactions on Industrial Electronics,

31

65(8):6625–6634, 2018. [47] D. Yang, G. Zong, and H. R. Karimi. H∞ refined anti-disturbance control of switched lpv systems with application to aero-engine. IEEE Transactions on Industrial Electronics, page 10.1109/TIE.2019.2912780, 2019. [48] Y. Yi, L. Guo, and H. Wang. Constrained PI tracking control for output probability distributions based on two-step neural networks. IEEE Transactions on Circuits and Systems I: Regular Papers, 56(7):1416–1426, 2009. [49] X. Zhang and Y. Lin. Nonlinear decentralized control of large-scale systems with strong interconnections. Automatica, 50(9):2419–2423, 2014. [50] J. Zhou, C. Y. Wen, and G. H. Yang. Adaptive backstepping stabilization of nonlinear uncertain systems with quantized input signal. IEEE Transactions on Automatic Control, 59(2):460–464, 2014.

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