Observer-based adaptive fuzzy control of a class of MIMO non-strict feedback nonlinear systems

Accepted Manuscript

Observer-Based Adaptive Fuzzy Control of A Class of MIMO Non-Strict Feedback Nonlinear Systems Na Wang, Shaocheng Tong PII: DOI: Reference:

S0016-0032(18)30325-9 10.1016/j.jfranklin.2018.05.013 FI 3447

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Journal of the Franklin Institute

Received date: Revised date: Accepted date:

23 September 2017 15 March 2018 4 May 2018

Please cite this article as: Na Wang, Shaocheng Tong, Observer-Based Adaptive Fuzzy Control of A Class of MIMO Non-Strict Feedback Nonlinear Systems, Journal of the Franklin Institute (2018), doi: 10.1016/j.jfranklin.2018.05.013

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ACCEPTED MANUSCRIPT

Observer-Based Adaptive Fuzzy Control of A

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Class of MIMO Non-Strict Feedback Nonlinear Systems∗

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Na Wang, Shaocheng Tong†

College of Science, Liaoning University of Technology, Jinzhou, Liaoning, 121001, P. R. China

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Abstract: This paper investigates the adaptive fuzzy control design problem of multi-input and multi-output (MIMO) non-strict feedback nonlinear systems. The considered control

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systems contain unknown control directions and dead zones. Fuzzy logic systems (FLSs)

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are utilized to approximate the unknown nonlinear functions, and the state observers are designed to estimate immeasurable states. By constructing a dead zone compensator and

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introducing a Nussbaum gain function into the backstepping technique, an adaptive fuzzy output feedback control method is developed. The proposed adaptive fuzzy controller is

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proved to guarantee the semi-globally uniformly ultimately bounded (SGUUB) of the closedloop system, and can solve the control design problems of unmeasured states, unknown control directions and dead zones. The simulation results are given to demonstrate the effectiveness of the proposed control method. ∗

This work was supported by the National Natural Science Foundation of China (Nos.

61773188,

61573175). †

Corresponding author. Tel.: +86-416-4198002; Fax: +86-416-4199415. E-mail: [email protected]

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Keywords: Adaptive fuzzy control, non-strict feedback systems, MIMO nonlinear systems, immeasurable states, unknown control directions, dead zones

1

Introduction

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The adaptive fuzzy and neural control design methods of strict feedback nonlinear system via backstepping technique have been attracted a great deal of attention in recent years, the main idea is that FLSs or neural networks (NNs) are applied to the unknown functions

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in the nonlinear system, and by combining adaptive backstepping control design with other robust control design theories, many typical adaptive fuzzy and neural control design methods are developed. The early researches all focused on the single-input and single-output (SISO) nonlinear systems [1–12]. Afterwards, many authors extended the results on SISO

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nonlinear systems to MIMO nonlinear systems [13–20]. Among them, the authors in [14–16]

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developed fuzzy or NN control methods for MIMO nonlinear systems with the unknown control directions by using the Nussbaum-type function. The literatures in [17,18] presented

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fuzzy control schemes for MIMO nonlinear systems with dead zones. In addition, [19, 20]

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investigated the problems of both unknown control directions and dead zones to guarantee the stability of MIMO nonlinear systems. However, the aforementioned works all require

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that the controlled systems have a strict feedback form and can not be applied to those nonlinear systems in the non-strict feedback form. It should be pointed out that unlike a strict feedback nonlinear system, the function fi (·) in a non-strict feedback nonlinear system contains the whole state variables, not a partial state vector. Therefore, the previous adaptive fuzzy or NN backstepping control methods can not be applied to solve the control problem of a non-strict feedback nonlinear 2

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system. In order to handle this problem, the authors [21] first developed an adaptive fuzzy state feedback control design for the non-strict feedback nonlinear systems by assuming the unknown functions must satisfy the condition |fi,j (x)| ≤ ϕi,j (|xi |). In the adaptive control design framework of [21], the literatures [22,23] proposed adaptive NN state feedback control

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design schemes for non-strict feedback systems with backlash-like hysteresis or dead-zone, respectively. [24] investigated the output feedback control design problem of unknown control direction in the control design. More recently, the authors in [25] proposed new adaptive fuzzy state feedback and output feedback control algorithms for the non-strict feedback

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systems in the unified framework of adaptive backstepping control, where the restrictive assumption in [21–24] is removed. In the adaptive control design framework of [25], the literatures [26–28] studied adaptive fuzzy output feedback control schemes for SISO non-

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strict feedback systems with/without dead-zone. Although many scholars have paid more attention to investigating the control problem of non-strict feedback system, there are few

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works on the observer-based fuzzy control for MIMO non-strict feedback nonlinear systems

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with both unknown control directions and dead zones. Based on the above works, in this paper, the adaptive fuzzy control design method for

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MIMO non-strict feedback nonlinear system with both unknown control directions and dead

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zones is studied. The semi-globally uniformly ultimately bounded (SGUUB) of the closedloop system is proved by using Lyapunov function theory. The proposed adaptive fuzzy control method has two advantages. One is that the adaptive backstepping control design does not assume that the unknown nonlinear functions satisfy monotonically increasing property [21–24], it thus has extended the application scopes of nonlinear systems. The other is that the proposed adaptive control method has solved control design problems of MIMO non-strict feedback nonlinear system with unmeasured states, unknown control directions 3

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and dead zones. The remainder of the paper is organized as follows. The control problem is formulated in Section 2. A fuzzy state observer is developed in Section 3. In Section 4, the control design and stability analysis are given. In Section 5, the simulation studies illustrating the

Problem Formulations and Preliminaries

2.1

Problem Formulation

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2

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effectiveness of the method are given. Finally, we conclude this paper in Section 6.

i,j

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Consider a class of MIMO non-strict feedback nonlinear system:     χ˙ i,1 = χi,2 + fi,1 (χ)         χ˙ = χ + f (χ) 1 ≤ j ≤ n − 1 i,j+1

i,j

(1)

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   χ˙ i,ni = dmi Di (ui ) + fi,ni (χ)         yi = χi,1 1 ≤ i ≤ N

i

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where χi = [χi,1 , . . . , χi,ni ]T ∈ Rni (χ = [χT1 , . . . , χTN ]T ) denotes the state vector of the system

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and it is assumed that the system variable χi,j (i = 1, . . . , N , j = 2, . . . , ni ) are unmeasured directly; fi,j (·) (i = 1, . . . , N , j = 2, . . . , ni ) are unknown smooth functions. ui ∈ R is

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the control input to the dead zone in the subsystem, yi ∈ R is the control output of the ith nonlinear system, dmi is referred to as a control coefficient and represents an unknown nonzero constant control gain. Remark 1. If the system (1) does not contain the terms of unknown control directions and dead zones, especially, the functions fi,j (x) satisfy that |fi,j (x)| ≤ ϕi,j (|xi |), where ϕi,j (·) is a monotonically increasing and bounded function, it will become the controlled system in 4

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[15] and [16]. Control objective: By utilizing the fuzzy logic systems to approximating the unknown functions, this study will develop an observer-based adaptive fuzzy control design method for system (1) such that the all variables in closed-loop system to be SGUUB. Moreover, the

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tracking errors can be kept smaller. According to [29], Di (ui ) ∈ R is defined as the output of the dead zone, and it can be expressed as

0 if − bi,l < ui < bi,r        σi,l (ui + bi,l ) if ui ≤ bi,l

(2)

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Di (ui ) =

    σi,r (ui − bi,r ) if ui ≥ bi,r    

where σi,r , σi,l represent the slopes of the dead zone and σi,r > 0, σi,l < 0. bi,r and bi,l are

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the width parameters of the dead zone. In this paper, σi,r , σi,l , bi,r ≥ 0 and bi,l ≤ 0 are unknown.

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Assumption 1 [29] [30]. Assume that there exist known constants σi,r min , σi,r max , σi,l min

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and σi,l max such that 0 < σi,r min ≤ σi,r ≤ σi,r max , 0 < σi,l min ≤ σi,l ≤ σi,l max , respectively. The dead zone inverse technique is useful for compensating the dead zone effects [29]. In

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order to achieve the control objective, we set ui,b as the control signal from the controller.

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According to (2), the control signal ui can be described as follows: ui = Di−1 (ui,b ) =

ui,b + ˆbi,r,σ ui,b + ˆbi,l,σ ωi + (1 − ωi ) σ ˆi,r σ ˆi,l

(3)

where ˆbi,r,σ , ˆbi,l,σ , σ ˆi,r and σ ˆi,l are the estimations of σi,r bi,r , σi,l bi,l , σi,r and σi,l , respectively. ωi is defined as ωi =

    1 if ui,b ≥ 0    0 others 5

(4)

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By substituting (3) and (4) into (2), we can obtain that ui,b + ˆbi,r,σ ui,b + ˆbi,l,σ Di (ui ) − ui,b = (˜bi,r,σ − σ ˜i,r )ωi + (˜bi,l,σ − σ ˜i,l )(1 − ωi ) + δi,b σ ˆi,r σ ˆi,l

(5)

where σ ˜i,r = σ ˆi,r − σi,r , σ ˜i,l = σ ˆi,l − σi,l , ˜bi,r,σ = ˆbi,r,σ − bi,r,σ and ˜bi,l,σ = ˆbi,l,σ − bi,l,σ are

ςi,l =

2.2

    1 if bi,l < ui < 0

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and

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parameters errors. δi,b = −σi,r ςi,r (ui − bi,r ) − σi,l ςi,l (ui − bi,l ) is bounded, where     1 if 0 ≤ ui < bi,r ςi,r =    0 others    0 others

Nussbaum Function Propertie

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To deal with the unknown control directions problem in the system (1), the Nussbaum gain

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technique is developed in this paper. A Nussbaum-type function N (ξ) is given as follows: lim sup 1s

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s→∞

lim inf

s→∞

1 s

Rs 0

Rs 0

N (ξ)dξ = ∞

(6)

N (ξ)dξ = −∞

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The Nussbaum functions are commonly employed are ξ 2 sin(ξ 2 ), ξ 2 cos(ξ 2 ) and exp(ξ 2 ) cos(ξ 2 ). In this study, a Nussbaum function is chosen as exp(ξ 2 ) cos(ξ 2 ).

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Lemma 1 [14] [16]. ξj (t) is defined as a smooth function on [0, tf ], N (ξj ) is a Nussbaum

gain function. Consider the system (1), for any t ∈ [0, tf ], if there exists a function V (t, x) ≥ 0 and satisfies V˙ (t, x) ≤ −ψV (t, x) + then, V (t, x), ξj (t) and

Pn

j=1 τj [βj N

0

n X

τj [βj N 0 (ξj ) + 1]ξ˙j + φ

(7)

j=1

(ξj ) + 1]ξ˙j are bounded on [0, tf ], where ψ > 0, φ > 0

and τj are suitable constants. 6

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2.3

Fuzzy Logic Systems

In this paper, the unknown nonlinear functions included in the control system are approximated by employing FLS. The FLS consists of four parts: the knowledge base, the fuzzifier, the fuzzy inference engine and the defuzzifier. The fuzzy rule base for FLS comprises the

Rl : If x1 is F1l and x2 is F2l and and xn is Fnl , Then, y is Gl , l = 1, 2, . . . , N

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following form:

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where x = [x1 , · · · , xn ]T and y are the input and output of the fuzzy logic system, respectively. Rl and Gl are fuzzy sets in R, respectively. l = 1, 2, . . . , N , i = 1, 2, . . . , n. N is the rules

(8)

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number. According to [17], the fuzzy logic system can be described as follows: Q PN ¯l ni=1 µFil (xi ) l=1 y y(x) = PN Qn l=1 [ i=1 µFil (xi )]

where µFil (xi ) is fuzzy function of Fil and y¯l = maxy∈R µGl (y).

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The fuzzy basis functions can be defined as follows: Πni=1 µFil (xi ) ϕl = PN n l=1 [Πi=1 µFil (xi )]

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Define the desired weight vector θ = [¯ y1 , y¯2 , . . . , y¯N ]T = [θ1 , θ2 , . . . , θN ]T , and fuzzy basis function vector ϕT (x) = [ϕ1 (x), ϕ2 (x), . . . , , ϕN (x)], then FLS (2) is expressed by the

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following form

y(x) = θT ϕ(x)

(9)

Lemma 2 [15] [21]. Let f (x) be a continuous function defined on a compact set Ω. Then, for giving a positive constant ε arbitrarily, there exists a FLS (9) such that

sup f (x) − θT ϕ(x) ≤ ε x∈Ω

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3

Fuzzy State Observer Design

Let xi = χi /dmi = [χi,1 /dmi , . . . , χi,ni /dmi ]T = [xi,1 , . . . , xi,ni ]T , x = [xT1 , . . . , xTN ]T , Fi,j (x) =

   x˙ i,ni = Di (ui ) + Fi,ni (x)         yi = dmi xi,1 1 ≤ i ≤ N

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fi,j (χ)/dmi , then the system (1) can be described as the following form:     x˙ i,1 = xi,2 + Fi,1 (x)         x˙ i,j = xi,j+1 + Fi,j (x) 1 ≤ j ≤ ni − 1

(10)

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According to Lemma 2, the nonlinear function Fi,j (x) (1 ≤ j ≤ ni ) in (10) can be approximated by the following FLS

T Fˆi,j (ˆ x|θˆi,j ) = θˆi,j ϕi,j (ˆ x)

(11)

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where xˆ = [ˆ xT1 , . . . , xˆTN ]T is the estimation of x = [xT1 , . . . , xTN ]T . ∗ as Define the optimal parameter vector θi,j

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ˆ ˆ = arg min [sup Fi,j (ˆ x|θi,j ) − Fi,j (x) ] θˆi,j ∈Ωi,j x∈U ˆ x ˆ∈U

(12)

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∗ θi,j

where Ωi,j , U and Uˆ are compact regions for θˆi,j , x and xˆ , respectively. Also define the

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fuzzy minimum approximation error εi,j (ˆ x) as ∗ εi,j = Fi,j (x) − Fˆi,j (ˆ x|θi,j )

(13)

There exist known positive constants ε∗i,j such that |εi,j | ≤ ε∗i,j . The system (1) can be expressed equivalently in the following form: x˙ i = Ai xi + Ki yi +

ni X j=1

∗ Bi,j Fˆi,j (ˆ x|θi,j ) + εi + Bi ui,b

8

(14)

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   where Ai =    



−K i,1 .. .

Ini −1

−Ki,ni 0 · · · 0

   , Ki = [Ki,1 , . . . , Ki,n ]T , Bi,j = [0 · · · 1 · · · 0]Tn ×1 , Bi = i i  | {z }  j 

[0, . . . , 1]T and εi = [εi,1 , . . . , εi,ni ]T .

xˆ˙ i = Ai xˆi + Ki yi +

ni X

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Construct the fuzzy observer as T φi,j (ˆ x) + Bi ui,b Bi,j θˆi,j

j=1

(15)

Choose the vector Ki such that matrix Ai is a strict Hurwitz matrix. Therefore, for given

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a positive definite matrix Qi = QTi > 0, the following matrix equation holds: ATi Pi + Pi Ai = −2Qi where Pi is a positive definite matrix.

(16)

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Define ei = [ei,1 , . . . , ei,ni ]T = xi − xˆi to be an observer error, and then from (14) and (15), ni X

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we have

e˙ i = Ai ei +

T Bi,j [θ˜i,j ϕi,j (ˆ x)] + εi

(17)

j=1

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∗ ∗ where θ˜i,j = θi,j − θˆi,j is the adaptive parameter error vector, θˆi,j is the estimation of θi,j ,

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i = 1, . . . , N , j = 1, . . . , ni .

Consider the Lyapunov function candidate V0 =

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is

V˙ 0 = −

N X

eTi Qi ei +

i=1

N X i=1

PN

T i=1 ei Pi ei

 2, the time derivative of V0

ni X T eTi Pi ( Bi,j θ˜i,j ϕi,j (ˆ x) + εi )

(18)

j=1

By using the Youngs inequality ab ≤ a2 /2 + b2 /2 and the inequality 0 < ϕTi,j (·)ϕi,j (·) ≤ 1, we have eTi Pi (

ni P

j=1

≤

T Bi,j θ˜i,j ϕi,j (ˆ x) + εi )

ni +1 kei k2 2

+

1 T ˜ kPi k2 θ˜i,j θi,j 2

9

(19) +

1 kPi k2 ε∗2 i 2

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where ε∗i = [ε∗i,1 , . . . , ε∗i,ni ]T . Substituting (19) into (18) yields N X i=1

2

−λ0 kei k +

ni N X X 1 i=1 j=1

PN

where λ0 = λmin (Q) − (ni + 1)/2 and M0 =

4

i=1

2

T ˜ kPi k2 θ˜i,j θi,j + M0

 kPi k2 ε∗2 2. i

(20)

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V˙ 0 ≤

Adaptive Fuzzy Control Design and Stability Anal-

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ysis

In this part, the adaptive fuzzy controller will be developed by utilizing the backstepping technique, and the Lyapunov function stability theory is utilized to demonstrate the systems

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stability.

Define the tracking error of the system as

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zi,1 = yi − yi,r

(21) j = 2, . . . , ni

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zi,j = xˆi,j − αi,j−1

where αi,j−1 is a virtual control function and will be given in the following steps.

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Step 1: According to (21) and xi,2 = xˆi,2 + ei,2 , one has z˙i,1 = y˙ i − y˙ i,r

(22)

= dmi (αi,1 + zi,2 + ei,2 +

∗T θi,1 ϕi,1 (ˆ x)

+ εi,1 ) − y˙ i,r

Consider the following Lyapunov function candidate: V1 =

N X 1 2 1 ˜T ˜ 1 ˜T ˜ + θi,1 θi,1 + Θi,1 Θi,1 ] + V0 [ zi,1 2 2r 2¯ r i,1 i,1 i=1

(23)

˜ i,l = Θ∗ − Θ ˆ i,l , Θ∗ = θ∗ 2 , and Θ ˆ i,l where ri,1 > 0 and r¯i,1 > 0 are design parameters, Θ i,l i,l i,l is the estimation of Θ∗i,l , l = 1, . . . , ni − 1.

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From (22) and (23), the time derivative of V1 is V1 = =

N P

i=1 N P

[zi,1 z˙i,1 +

1 ˜T ˜ θ θ˙ ri,1 i,1 i,1

+

˙ ] 1 ˜T ˜ Θ Θ r¯i,1 i,1 i,1

+ V˙ 0

∗T {zi,1 [dmi (αi,1 + zi,2 + ei,2 + θi,1 ϕi,1 (ˆ x) + εi,1

i=1

(24)

T T ∗T ϕi,1 (ˆ x1 )) − y˙ i,r ]} ϕi,1 (ˆ x1 ) + θ˜i,1 ϕi,1 (ˆ x1 ) + θˆi,1 − θi,1 N P

i=1

P ˙ θ˜T θˆ − ri,1 i,1 i,1 N

1

i=1

˙ 1 ˜T ˆ Θ Θ r¯i,1 i,1 i,1

+ V˙ 0

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−

Since 0 < ϕTi,j (x)ϕi,j (x) ≤ 1, by using Youngs inequality, we have ∗T ∗T dmi zi,1 (θi,1 ϕi,1 (ˆ x) − θi,1 ϕi,1 (ˆ x1 )) ≤

µd¯mi 2 ∗ 2 zi,1 Θi,1 + 2 µ

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1 1 2 zi,1 dmi (ei,2 + εi,1 ) ≤ d¯mi zi,1 + kei k2 + ε∗2 2 2 i,1

(25) (26)

where µ > 0 is a design parameter, d¯mi is a known constant and dmi ≤ d¯mi . Substituting (25)-(26) into (24) results in

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N P T ϕi,1 (ˆ x1 ) + V˙ 1 ≤ [zi,1 (dmi αi,1 + d¯mi zi,1 + d¯mi θˆi,1 i=1

[λ0 kei k2 − 21 kei k2 ] +

i=1

N P ˙ 1 ˆ T [− ri,1 θ˜i,1 x1 )] + θi,1 zi,1 + d¯mi ϕi,1 (ˆ

i=1

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+

N P

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+ µ2 + 12 ε∗2 i,1 ] −

µd¯mi ˆ i,1 zi,1 Θ 2

ni N P P

− y˙ i,r ) + d¯mi zi,1 zi,2

1 T ˜ θi,j kPi k2 θ˜i,j 2

i=1 j=1 N P ˜ T [− 1 Θ ˆ˙ Θ i,1 r¯i,1 i,1 i=1

+

+ M0

(27)

µd¯mi 2 zi,1 ] 2

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Design the virtual control function αi,1 , the smooth function ξi , the adaptive laws of θˆi,1 ˆ i,1 as and Θ

(28)

˙ θˆi,1 = d¯mi ri,1 ϕi,1 (ˆ x1 )zi,1 − γi,1 θˆi,1

(29)

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µd¯mi T ˆ i,1 − y˙ i,r + ci,1 zi,1 ] αi,1 = Ni0 (ξi )[d¯mi zi,1 + d¯mi θˆi,1 ϕi,1 (ˆ x1 ) + zi,1 Θ 2

¯ ˆ˙ i,1 = r¯i,1 µdmi z 2 − γ¯i,1 Θ ˆ i,1 Θ 2 i,1 zi,1 ¯ µd¯mi T ˆ i,1 − y˙ i,r + ci,1 zi,1 ] ξ˙i = ϕi,1 (ˆ x1 ) + [dmi zi,1 + d¯mi θˆi,1 zi,1 Θ τi 2 where ci,1 , γi,1 and γ¯i,1 are positive design parameters. 11

(30) (31)

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Substituting (28)-(31) into (27) yields P PP1 T ˜ θi,j + M1 V˙ 1 ≤ − λ1 kei k2 + kPi k2 θ˜i,j 2 N

+

i=1 N P

i=1 N P

i=1

i=1 j=1

2 [d¯mi zi,1 zi,2 − ci,1 zi,1 + τi (d¯mi N 0 i (ξi ) + 1)ξ˙i ] γi,1 ˜T ˆ ( ri,1 θi,1 θi,1 +

where λ1 = λ0 − 1/2 and M1 = M0 +

PN

∗2 i=1 εi,1

(32)

γ ¯i,1 ˜ T ˆ Θ Θ ) r¯i,1 i,1 i,1

 2 + µ/2.

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+

ni

N

Step 2 : According to (22), define zi,2 as zi,2 = xˆi,2 − αi,1 , and the time of derivative of zi,2 is

T T ∗T ∗T ϕi,2 (ˆ x2 ) − θ˜i,2 ϕi,2 (ˆ x) z˙i,2 = zi,3 + αi,2 + θi,2 ϕi,2 (ˆ x) − θi,2 ϕi,2 (ˆ x2 ) + θ˜i,2 i,1 ∗T ∗T T − ∂xi,1 [ei,2 + θi,1 ϕi,1 (ˆ x) − θi,1 ϕi,1 (ˆ x1 ) + θ˜i,1 ϕi,1 (ˆ x1 ) + εi,1 ]

−

ni P

∂αi,1 ∗T ϕi,j (ˆ x) (θi,j ∂x ˆi,j

j=1

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∂α

T ¯ i,2 ϕi,j (ˆ x)) + H − θ˜i,j

where xˆ2 = [ˆ x1 , xˆ2 ]T and

−

∂αi,1 y˙ ∂yi,r i,r

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˙ ∂αi,1 ˆ ˆ i,1 Θi,1 ∂Θ

∂αi,1 [ˆ x ∂xi,1 i,2

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¯ i,2 = Ki,2 ei,1 + θˆT ϕi,2 (ˆ x2 ) − H i,2 −

(33)

−

ni P

j=1

T ϕi,1 (ˆ x1 )] − + θˆi,1

∂αi,1 (ˆ xi,j+1 ∂x ˆi,j

˙ ∂αi,1 ˆ θ ∂ θˆi,1 i,1

+ Ki e1 )

Consider the following Lyapunov function candidate: (34)

CE

PT

N X 1 2 1 ˜T ˜ 1 ˜T ˜ V2 = [ zi,2 θi,2 θi,2 + Θ Θi,2 ] + V1 + 2 2ri,2 2¯ ri,2 i,2 i=1

where ri,2 > 0 and r¯i,2 > 0 are design parameters. From (32), (33) and (34), the time

AC

derivative of V2 satisfies

N P V˙ 2 = [zi,2 z˙i,2 +

=

1 ˜T ˜ θ θ˙ ri,2 i,2 i,2

+

˙ ] 1 ˜T ˜ Θ Θ r¯i,2 i,2 i,2

+ V˙ 1

i=1 N P

T T ∗T ∗T {zi,2 [zi,3 + αi,2 + θi,2 ϕi,2 (ˆ x2 ) − θ˜i,2 ϕi,2 (ˆ x) ϕi,2 (ˆ x) − θi,2 ϕi,2 (ˆ x2 ) + θ˜i,2

−

ni P

i=1

i,1 ∗T ∗T T − ∂xi,1 (ei,2 + θi,1 ϕi,1 (ˆ x) − θi,1 ϕi,1 (ˆ x1 ) + θ˜i,1 ϕi,1 (ˆ x1 ) + εi,1 )

∂α

−

j=1 N P i=1

∂αi,1 ∗T (θi,j ϕi,j (ˆ x) ∂x ˆi,j

T ¯ i,2 ]} − θ˜i,j ϕi,j (ˆ x)) + H

P ˙ θ˜T θˆ − ri,2 i,2 i,2

˙ 1 ˜T ˆ Θ Θ r¯i,2 i,2 i,2

1

N

i=1

12

+ V˙ 1

(35)

ACCEPTED MANUSCRIPT

By using Youngs inequality, we can obtain that ∗T ∗T zi,2 [θi,2 ϕi,2 (ˆ x) − θi,2 ϕi,2 (ˆ x2 )] ≤

1 2 T ϕi,2 (ˆ x) ≤ zi,2 + −zi,2 θ˜i,2 2

(37) (38)

∂αi,1 1 2 ∂αi,1 2 1 ∗2 εi,1 ≤ zi,2 ( ) + εi,1 ∂xi,1 2 ∂xi,1 2

(39)

∂αi,1 ∗T µ 2 ∂αi,1 2 2 ∗ ∗T [(θi,1 ϕi,1 (ˆ x) − θi,1 ϕi,1 (ˆ x1 ))] ≤ zi,2 ( ) + Θi,1 ∂xi,1 2 ∂xi,1 µ

(40)

1 2 ∂αi,1 2 ∂αi,1 ˜T [θi,1 ϕi,1 (ˆ x1 )] ≤ zi,2 ( ) + ∂xi,1 2 ∂xi,1

1 ˜T ˜ θ θi,1 2 i,1

AN US

−zi,2 −zi,2

1 ˜T ˜ θ θi,2 2 i,2

∂αi,1 1 2 ∂αi,1 2 1 e2 ≤ zi,2 ( ) + kei k2 ∂xi,1 2 ∂xi,1 2

−zi,2 −zi,2

(36)

CR IP T

−zi,2

µ 2 ∗ 2 zi,2 Θi,2 + 2 µ

∂αi,1 ∗T 1 1 T ˜ T 2 ∂αi,1 2 θi,j (θi,j ϕi,j (ˆ ) + Θ∗i,j + θ˜i,j x) − θ˜i,j ϕi,j (ˆ x)) ≤ zi,2 ( ∂ xˆi,j ∂ xˆi,j 2 2

Substituting (36)-(42) into (35) results in

(41) (42)

i=1

+ Hi,2 ] + zi,2 zi,3 } +

M

N P ˆ i,2 + 1 zi,2 + 5 zi,2 ( ∂αi,1 )2 + µ zi,2 ( ∂αi,1 )2 V˙ 2 ≤ {zi,2 [αi,2 + 12 zi,2 Θ 2 2 ∂xi,1 2 ∂xi,1

ni N N P P P ˙ 1 ˆ 1 ∗ T [− ri,2 θ˜i,2 θi,2 zi,2 + ϕi,2 (ˆ x2 )] + Θ 2 i,j

i=1

i=1 j=1

i=1 N P

¯

ED

N N P ˜ T [− 1 Θ ˆ˙ + µ z 2 ] + P [ 1 θ˜T θ˜i,2 + 1 θ˜T θ˜i,1 ] + Θ i,2 r¯i,2 i,2 2 i,2 2 i,2 2 i,1 i=1

¯

2 ci,1 zi,1

0

1)ξ˙i ]

CE

PT

+ τi (dmi N i (ξi ) + [dmi zi,1 zi,2 − i=1 ni N P N P P 2 ˜T ˜ 1 + θ − λ1 kei k2 + M1 θ kP k i i,j i,j 2 i=1 j=1 i=1 N P + [ µ2 + µ2 Θ∗i,1 + 12 kei k2 + 12 ε∗2 i,1 ] i=1 +

(43)

AC

¯ i,2 + zi,2 Pni (∂αi,1 /ˆ where Hi,2 = H xi,j )2 . j=1

ˆ i,2 as Design the virtual control function αi,2 , the adaptive laws of θˆi,2 and Θ 1 ˆ i,2 − 1 zi,2 + Hi,2 − d¯m zi,1 − µ + 5 zi,2 ( ∂αi,1 )2 − ci,2 zi,2 αi,2 = − zi,2 Θ i 2 2 2 ∂xi,1

(44)

˙ θˆi,2 = ri,2 ϕi,2 (ˆ x2 )zi,2 − γi,2 θˆi,2

(45)

ˆ˙ i,2 = r¯i,2 µ z 2 − γ¯i,2 Θ ˆ i,2 Θ 2 i,2

(46)

13

ACCEPTED MANUSCRIPT

where ci,2 , γi,2 and γ¯i,2 are positive design parameters. Substituting (44)-(46) into (43) results in N P 2 P 2 [−ci,k zi,k + V˙ 2 ≤

γi,k ˜T ˆ θ θ ri,k i,k i,k

+

γ ¯i,k ˜ T ˆ Θ Θ ] r¯i,k i,k i,k

where λ2 = λ1 − 1/2 and M2 = M1 +

PN

i=1

+ zi,2 zi,3

1)ξ˙i ]

(47)

CR IP T

i=1 k=1 N P T ˜ T ˜ θi,2 + 21 θ˜i,1 θi,1 + τi (d¯mi N 0 i (ξi ) + + [ 21 θ˜i,2 i=1 ni N P N P P 2 ˜T ˜ 1 + θ θ − λ2 kei k2 + M2 kP k i,j i i,j 2 i=1 i=1 j=1

   Pni ∗ [2Θ∗i,1 µ + ε∗2 i,1 2 + 2/µ + j=1 Θi,j 2].

Step j (3 ≤ j ≤ ni − 1) : Since zi,j = xˆi,j − αi,j−1 , the time derivative of zi,j is

∂αi,j−1 {ei,2 ∂xi,1

−

ni P

j=1

∗T ∗T T + θi,1 ϕi,1 (ˆ x) − θi,1 ϕi,1 (ˆ x1 ) + θ˜i,1 ϕi,1 (ˆ x1 ) + εi,1 }

∂αi,j−1 ∗T ϕi,j (ˆ x) (θi,j ∂x ˆi,j

where xˆj = [ˆ x1 , . . . , xˆj ]T and

k=1

∂αi,j−1 [ˆ xi,2 ∂xi,1

˙ ∂αi,j−1 ˆ ˆ i,k Θi,k ∂Θ

−

j−1 P

k=1

∂αi,j−1

(k−1)

∂yi,r

(k)

yi,r −

j−1 P ∂αi,j−1 ˆ˙ T ϕi,1 (ˆ x1 )] − + θˆi,1 θi,k ∂ θˆ

PT

−

j−1 P

T ¯ i,j ϕi,j (ˆ x)) + H − θ˜i,j

ED

¯ i,j = Ki,j ei,1 + θˆT ϕi,j (ˆ xj ) − H i,j

(48)

M

−

AN US

∗T ∗T T T z˙i,j = zi,j+1 + αi,j + θi,j ϕi,j (ˆ x) − θi,j ϕi,j (ˆ xj ) + θ˜i,j ϕi,j (ˆ xj ) − θ˜i,j ϕi,j (ˆ x)

ni P

k=1

∂αi,j−1 ˙ xˆi,k ∂x ˆi,k

−

k=1 ni P

j=1

i,k

∂αi,j−1 (ˆ xi,j+1 ∂x ˆi,j

+ Ki e1 )

Choose the following Lyapunov function candidate:

AC

CE

N X 1 2 1 ˜T ˜ 1 ˜T ˜ Vj = [ zi,j + θi,j θi,j + Θ Θi,j ] + Vj−1 2 2ri,j 2¯ ri,j i,j i=1

14

(49)

ACCEPTED MANUSCRIPT

where ri,j > 0 and r¯i,j > 0 are design parameters. Then we have N P V˙ j = [zi,j z˙i,j +

=

i=1 N P

1 ˜T ˜ θ θ˙ ri,j i,j i,j

+

˙ ] 1 ˜T ˜ Θ Θ r¯i,j i,j i,j

+ V˙ j−1

ˆ i,j + 1 zi,j + ( µ+5 )zi,j ( {zi,j [αi,j + 12 zi,j Θ 2 2

i=1

+ Hi,j ] + zi,j zi,j+1 } − λj−1 kei k2 + Mj−1 +

∂αi,j−1 2 ) ∂xi,1 N P

i=1

τi (d¯mi N 0 i (ξi ) + 1)ξ˙i

+

i=1 k=1 j N P P

i=1

T ˜ θi,k + [ 21 θ˜i,k

i=1 k=2 N P + [ µ2 + µ2 Θ∗i,1 i=1

¯ i,j + zi,j where Hi,j = H

γi,k−1 ˜T θˆ θ ri,k−1 i,k−1 i,k−1

2 + [−ci,k−1 zi,k−1

Pni

j=1

j−1 ˜T ˜ θ θ ] 2 i,1 i,1

+

ni N P P

+

1 T ˜ θi,j kPi k2 θ˜i,j 2

i=1 j=1 ni N P P

+ 12 kei k2 + 12 ε∗2 i,1 ] +

(∂αi,j−1 /ˆ xi,j )2 .

(50)

γ ¯i,k ˜ T ˆ Θ Θ ] r¯i,k i,k−1 i,k−1

i=1 j=1

1 ∗ Θ 2 i,j

AN US

+

i=1 j N P P

CR IP T

N N P P ˙ 1 ˆ T ˜ T [− 1 Θ ˆ˙ + µ z 2 ] [− ri,j θi,j zi,j + ϕi,j (ˆ xj )] + + θ˜i,j Θ i,j r¯i,j i,j 2 i,j

ˆ i,j as Design the virtual control function αi,j , the adaptive laws of θˆi,j and Θ (51)

˙ θˆi,j = ri,j ϕi,j (ˆ xj )zi,j − γi,j θˆi,j

(52)

2 ˆ˙ i,j = r¯i,j µ zi,j ˆ i,j Θ − γ¯i,j Θ 2

(53)

PT

ED

M

1 ˆ 1 µ+5 ∂αi,j−1 2 αi,j = − zi,j Θ zi,j ( ) − ci,j zi,j i,j − zi,j + Hi,j − zi,j−1 − 2 2 2 ∂xi,1

CE

where ci,j , γi,j and γ¯i,j are positive design parameters.

AC

Substituting (51)-(53) into (50) yields j N P P 2 V˙ j ≤ [−ci,k zi,k + i=1 k=1 j N P P

+

+

i=1 k=2 ni N P P

i=1 j=1

1 ˜T ˜ θ θ 2 i,k i,k

γi,k ˜T ˆ θ θ ri,k i,k i,k

+

N P

i=1

1 T ˜ kPi k2 θ˜i,j θi,j 2

where λj = λj−1 − 1/2 and Mj = Mj−1 + Step ni :

+

γ ¯i,k ˜ T ˆ Θ Θ ] r¯i,k i,k i,k

+ zi,k zi,k+1

τi (d¯mi N 0 i (ξi ) + 1)ξ˙i − +

N P

i=1

j−1 ˜T ˜ θ θ 2 i,1 i,1

+ Mj

N P

i=1

λj kei k2

   Pni ∗ ∗2 ∗ [2Θ µ + ε 2 + 2/µ + Θ 2]. i,1 i,1 j i=1 j=1

PN

(54)

The actual system input will be given in the final step. According to (5) and

15

ACCEPTED MANUSCRIPT

(21), the time of derivative of zi,ni is z˙i,ni = xˆ˙ i,ni − α˙ i,ni −1

+ δi,b +

ui,b +ˆbi,r,σ σ ˜i,r )ωi σ ˆi,r

∂αi,ni −1 [ei,2 ∂xi,1

+

+ (˜bi,l,σ −

T θ˜i,1 ϕi,1 (ˆ x1 )

where ∂αi,ni −1 [ˆ xi,2 ∂xi,1

nP i −1 j=1

∂αi,ni −1 ˆ ˙ ˆ i,j Θi,j ∂Θ

−

nP i −1 j=1

∂αi,ni −1 (j) (j−1) yi,r ∂y i,r

(55)

nP i −1 ∂αi,ni −1 ˆ ˙ T + θˆi,1 ϕi,1 (ˆ x1 )] − θi,j ∂ θˆ

−

j=1

ni P

j=1

i,j

∂αi,ni −1 ˙ xˆi,j ∂x ˆi,j

AN US

−

− ωi )

+ εi,1 ]

T T + Hi,ni + θ˜i,n ϕ (ˆ x) − θ˜i,n ϕ (ˆ x) i i,ni i i,ni

T Hi,ni = Ki,ni ei,1 + θˆi,n ϕ (ˆ x) − i i,ni

ui,b +ˆbi,l,σ σ ˜i,l )(1 σ ˆi,l

CR IP T

= ui,b + (˜bi,r,σ +

Consider the following Lyapunov function candidate V as:

N X 1 2 1 ˜T ˜ 1 2 1 ˜2 1 ˜2 1 2 + θri,n θri,ni + σ ˜i,r + σ ˜i,l + bi,r,σ + b ] + Vni −1 (56) V = [ zi,n i i 2 2ri,ni 2ρi,1 2ρi,2 2ρi,3 2ρi,4 i,l,σ i=1

ED

From (55) and (56), we have N P V˙ = {zi,ni [ui,b + δi,b −

+

i

1 ˙ [˜ σi,l (− ρi,2 σ ˆ i,l +

i=1 N P

i=1

ui,b +ˆbi,l,σ (1 σ ˆi,l

ui,b +ˆbi,r,σ ωi zi,ni )] σ ˆi,r

− ωi )zi,ni )] + V˙ ni −1

(57)

˙ 1 ˆ [˜bi,r,σ (− ρi,3 bi,r,σ + zi,ni ωi )]

˙ 1 ˆ [˜bi,l,σ (− ρi,4 bi,l,σ + zi,ni (1 − ωi ))]

AC

+

i=1 N P

T T ϕ (ˆ x)]} ϕi,1 (ˆ x1 ) + εi,1 ) + Hi,ni + θ˜i,n + θ˜i,1 i i,ni

N P ˙ 1 ˆ 1 ˙ T θ˜i,n [− θ + z ϕ (ˆ x )] + [˜ σi,r (− ρi,1 σ ˆ i,r + i,ni i,ni ri,n i,ni i

CE

+

i=1 N P

∂αi,ni −1 (ei,2 ∂xi,1

PT

i=1 N P

M

where ri,ni , ρi,1 , ρi,2 , ρi,3 and ρi,4 are positive design parameters.

+

i=1

By utilizing Youngs inequality, we can obtain 1 2 1 ∗2 zi,ni δi,b ≤ zi,n + δi,b i 2 2

(58)

1 2 1 T ˜ T −zi,ni θ˜i,n ϕ (ˆ x) ≤ zi,n + θ˜i,n θi,n i i,ni i 2 2 i i

(59)

∗ ∗ where δi,b is a known constant and |δi,b | ≤ δi,b .

16

ACCEPTED MANUSCRIPT

Design the controller ui,b , the adaptive laws of θˆi,ni , σ ˆi,r , σ ˆi,l , ˆbi,r,σ and ˆbi,l,σ as (60)

˙ θˆi,ni = ri,ni ϕi,ni (ˆ x)zi,ni − γi,ni θˆi,ni

(61)

ui,b + ˆbi,r,σ σ ˆ˙ i,r = ρi,1 ωi zi,ni − ρ¯i,1 σ ˆi,r σ ˆi,r

(62)

CR IP T

1 ∂αi,ni −1 2 ui,b = −Hi,ni − zi,ni − zi,ni −1 − zi,ni ( ) − ci,ni zi,ni 2 ∂xi,1

ui,b + ˆbi,l,σ σ ˆ˙ i,l = ρi,2 (1 − ωi )zi,ni − ρ¯i,2 σ ˆi,l σ ˆi,l

(63)

ˆb˙ i,r,σ = ρi,3 zi,n ωi − ρ¯i,3ˆbi,r,σ i

(64)

AN US

ˆb˙ i,l,σ = ρi,4 zi,n (1 − ωi ) − ρ¯i,4ˆbi,l,σ i

(65)

where ci,ni , γi,ni , ρ¯i,1 , ρ¯i,2 , ρ¯i,3 and ρ¯i,4 are positive design parameters. Substituting (58)-(65) into (57) yields ni N P P 2 [−ci,j zi,j + V˙ ≤

γi,j ˜T ˆ θ θ ] ri,j i,j i,j

+

N nP N i −1 P P γ ¯i,j ˜ T ˆ Θ Θ + r¯i,j i,j i,j

ni −1 ˜T ˜ θi,1 θi,1 2

(66)

PT

ED

M

i=1 j=1 i=1 j=1 i=1 N P ρ¯i,4 ˜ ρ¯ ρ¯ ρ¯ ˜ bi,r,σˆbi,r,σ + ρi,4 bi,l,σˆbi,l,σ ) + ( ρi,1 σ ˜i,r σ ˆi,r + ρi,2 σ ˜i,l σ ˆi,l + ρi,3 i,1 i,2 i,3 i=1 N N P P + τi (d¯mi N 0 i (ξi ) + 1)ξ˙i − λni kei k2 + Mni i=1 i=1 ni ni N P N P P P 1 1 ˜T ˜ T ˜ θi,j θ θ + kPi k2 θ˜i,j + 2 i,j i,j 2 i=1 j=2 i=1 j=1

CE

where λni = λni −1 − 1/2 and Mni = Mni −1 +

PN

i=1

  ∗2 [ε∗2 i,1 2 + δi,b 2].

AC

Based on Youngs inequality, we can get the following inequalities 1 ∗T ∗ 1 T ˜ T ˆ θ˜i,j θi,j ≤ θi,j θi,j − θ˜i,j θi,j 2 2

(67)

1 ∗T ∗ 1 ˜T ˜ ˜T Θ ˆ Θ i,j i,j ≤ Θi,j Θi,j − Θi,j Θi,j 2 2

(68)

1 2 1 2 σ ˜i,r σ ˆi,r ≤ σi,r − σ ˜ 2 2 i,r

(69)

1 2 1 2 σ ˜i,l σ ˆi,l ≤ σi,l − σ ˜ 2 2 i,l

(70)

˜bi,r,σˆbi,r,σ ≤ 1 b2 − 2 i,r,σ 17

1 ˜2 b 2 i,r,σ

(71)

ACCEPTED MANUSCRIPT

1 ˜2 b 2 i,l,σ

˜bi,l,σˆbi,l,σ ≤ 1 b2 − 2 i,l,σ

(72)

Substituting (67)-(72) into (66) results in ni ni N P P P γ 2 − ( 2ri,j V˙ ≤ {−λni kei k2 − ci,j zi,j − i,j

−

+

j=1

ρ¯i,1 2 σ ˜ 2ρi,1 i,r

N P

i=1

where

γ ¯i,j ˜ T ˜ Θ Θ 2¯ ri,j i,j i,j

−

γ

ni −1 2

− − ( 2ri,1 i,1

ρ¯i,2 2 σ ˜ 2ρi,2 i,l

−

ρ¯i,3 ˜2 b ρi,3 i,r,σ

−

−

kPi k2 ˜T ˜ )θi,1 θi,1 2

ρ¯i,4 ˜2 b } ρi,4 i,l,σ

τi (d¯mi N 0 i (ξi ) + 1)ξ˙i + φ

φ=

N P

ρ¯

σ2 + ( 2ρi,1 i,1 i,r

i=1 N P

+

(

ni P

i=1 j=1

ρ¯i,2 2 σ 2ρi,2 i,l

+

kPi k2 ˜T ˜ )θi,j θi,j 2

ρ¯i,3 2 b 2ρi,3 i,r,σ

+

(73)

ρ¯i,4 2 b ) 2ρi,4 i,l,σ

AN US

−

j=2

j=1

−

CR IP T

i=1 nP i −1

ni 2

∗T θ ∗ γi,j θi,j i,j 2ri,j

+

nP i −1 j=1

∗ γ ¯i,j Θ∗T i,j Θi,j ) 2¯ ri,j

+ Mni

Let ψ = min{λni /λmax (Pi ), 2ci,k , 2¯ ρi,1 , 2¯ ρi,2 , 2¯ ρi,3 , 2¯ ρi,4 , γi,1 − (ni − 1/2)ri,1 − kPi k2 ri,1 , γi,j −

M

ni ri,j − kPi k2 ri,j , γ¯i,l }, j = 2, . . . , ni , l = 1, . . . , ni − 1, k = 1, . . . , ni . Then, (73) can be rewritten as

N X

ED

V˙ ≤ −ψV +

τi (d¯mi N 0 i (ξi ) + 1)ξ˙i + φ

PT

Thus, according to Lemma 1, the term

CE

Then define φmax = maxt∈[0,tf ]

PN

(74)

i=1

PN

0 ¯ i=1 τi (dmi N i (ξi )

0 ¯ i=1 τi (dmi N i (ξi )

+ 1)ξ˙i is bounded on [0, tf ].

+ 1)ξ˙i and rewrite (74) as

V˙ ≤ −ψV + φ¯

(75)

AC

where φ¯ = φmax + φ.

Integrating (75) over [0, t] yields V ≤ e−ψt (V (0) −

φ¯ φ¯ )+ ψ ψ

(76)

˜ i,k , i = 1, . . . , N , j = 1, . . . , ni , k = From (76), we can conclude that ei , zi,j , θ˜i,j and Θ ∗ ˆ i,k are 1, . . . , ni − 1 are bounded. Since θi,j and Θ∗i,k are constants, the signals θˆi,j and Θ

18

ACCEPTED MANUSCRIPT

bounded. Moreover, it is easy to follow that χi and χˆi are also bounded. Since t → ∞, the term exp(−ψt) tends to 0. Meanwhile, for any η > 0, there exist T ≥ 0 such that

CR IP T

 for all t > T , exp(−ψt)(V (0) − φ¯ ψ) < η. Then, for t > T , the observer error and the q   tracking error converge to the compact sets Ωei = {ei (t)| kei k ≤ (2(φ¯ ψ + η) λmin (P )} q  and Ωzi = {zi | |zi | ≤ 2(φ¯ ψ + η), respectively. Moreover, according to [31], by choosing the appropriate design parameters, we can make the compact sets be as small as possible.

The properties of the proposed adaptive control method can be summarized as the following theorem.

AN US

Theorem 1: For the nonlinear system (1), under Assumption 1 and Lemmas 1-2, the virtual control function (28), (44) and (51), the parameter adaptive laws (29), (30), (45), (46), (52), (53) and (61), and the actual controller (60) can ensure that all the signals in the

ED

be kept small compact sets.

M

closed-loop systems are SGUUB. And also, the tracking errors and the observer errors can

The configuration of the aforementioned adaptive fuzzy output feedback control scheme is

Simulation Example

CE

5

PT

shown in Fig. 1.

AC

In this section, a simulation example is given to demonstrate the effectiveness of the proposed adaptive control method.

19

ACCEPTED MANUSCRIPT

Consider the following MIMO nonlinear systems:     x˙ 1,1 = f1,1 (x) + x1,2    

(77)

CR IP T

x˙ 1,2 = f1,2 (x) + dm1 D1 (u1 )        y1 = x1,1     x˙ 2,1 = f2,1 (x) + x2,2    

(78) x˙ 2,2 = f2,2 (x) + dm2 D2 (u2 )        y2 = x2,1   where x = [x1,1 , x1,2 , x2,1 , x2,2 ]T , f1,1 (x) = x1,2 (1 + x41,1 ), f1,2 (x) = x1,2 exp(x21,1 ), f2,1 (x) =

AN US

  x2,2 (1 + x22,1 + x42,1 ), f2,2 (x) = −x2,2 sin(x1,1 x1,2 ) (1 + x22,1 ), dm1 = 1.6, dm2 = 1, d¯m1 = 4 and d¯m2 = 3. The dead zone model (2) parameters are selected as m1,r = 1, m1,l = 2, b1,r = −0.1, b1,l = 0.7, m2,r = 1, m2,l = 2, b2,r = 0.1 and b2,l = 0.4.

M

The fuzzy membership functions are chosen as follows: 2

2

2

2

ED

2 (ˆ 1 (ˆ xi,j ) = e−0.5(ˆxi,j −1) , xi,j ) = e−0.5(ˆxi,j −2) , µFi,j µFi,j

3 (ˆ 4 (ˆ xi,j ) = e−0.5(ˆxi,j −0) , µFi,j xi,j ) = e−0.5(ˆxi,j +1) , µFi,j 2

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5 (ˆ xi,j ) = e−0.5(ˆxi,j +2) , i = 1, 2, j = 1, 2 µFi,j

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Then, the FLSs can be expressed in the form T fˆi,j (ˆ x|θˆi,j ) = θˆi,j ϕi,j (ˆ x), i = 1, 2, j = 1, 2

Setting the parameters K1,1 = 0.49, K1,2 = 0.79, K2,1 = 0.45 and K2,2 = 0.4 , the state

observer (15) is xˆ˙ i,1 = xˆi,2 + fˆi,1 (ˆ x|θˆi,1 ) + Ki,1 (xi,1 − xˆi,1 ) xˆ˙ i,2 = ui,b + fˆi,2 (ˆ x|θˆi,2 ) + Ki,2 (xi,1 − xˆi,1 ) yˆi,1 = xˆi,1 20

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The parameters in intermediate control functions, the control input, parameters adaptive functions are chosen as c1,1 = 30, c1,2 = 35, c2,1 = 15, c2,2 = 25, r1,1 = r1,2 = 1, r2,1 = r2,2 = 1, r¯1,1 = r¯2,1 = 10, γ1,1 = γ1,2 = 10, γ2,1 = γ2,2 = 10, γ¯1,1 = γ¯2,1 = 10, µ = 10, τ1 = 50 and τ2 = 50.

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Design the intermediate function αi,1 ,(i = 1, 2), the control input ui,b and the parameters adaptation functions as

T ˆ 1,1 − cos(t) + 30z1,1 ] φ1,1 (ˆ x1 ) + 20z1,1 Θ α1,1 = N10 (ξ1 )[4z1,1 + 4θˆ1,1

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T ˆ 2,1 − cos(t) + 15z2,1 ] φ2,1 (ˆ x1 ) + 15z2,1 Θ α2,1 = N20 (ξ2 )[3z2,1 + 3θˆ2,1

˙ θˆ1,i = 4φ1,i (ˆ x1 )z1,i − 10θˆ1,i ˙ θˆ2,i = φ2,i (ˆ x1 )z2,i − 10θˆ2,i

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ˆ˙ 1,1 = 20z 2 − 10Θ ˆ i,1 Θ 1,1

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2 ˆ˙ 2,1 = 25z1,1 ˆ i,1 Θ − 10Θ T u1,b = −36z1,2 − z1,1 − θˆ1,2 φ1,2 (ˆ x2 ) − 0.79e1,1

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T u2,b = −26z1,2 − z1,1 − θˆ2,2 φ2,2 (ˆ x2 ) − 0.4e1,1

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Besides, initial conditions are chosen as [x1,1 (0) x1,2 (0) x2,1 (0) x2,2 (0)] = [0.01 0 0.01 0], [ˆ x1,1 (0) xˆ1,2 (0) xˆ2,1 (0) xˆ2,2 (0)] = [0.01 0.1 0 0] and the others initial values are chosen as

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zero.

The simulation results are given in Figs. 2-9. Among them, Figs. 2-3 express the trajec-

tories of the system output yi and the reference tracking signal yi,r (i = 1, 2); Figs. 4-5 show the trajectories of x1,j and the estimations xˆ1,j (j = 1, 2), respectively; Figs. 6-7 indicate the trajectories of x2,j and the estimations xˆ2,j (j = 1, 2), respectively; Figs. 8-9 stand for the trajectories of ui,b (i = 1, 2). 21

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From the simulation results, we can see that the proposed adaptive fuzzy output feedback control method can ensure that all the system variables are bounded. And also, the tracking errors can be kept in a small neighborhood of the desired trajectory.

Conclusion

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This study has presented an adaptive fuzzy control method for a class of MIMO non-strict feedback nonlinear systems. The proposed control method can not only get over the algebraic

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loop problem without assuming that the unknown nonlinear functions satisfy monotonically increasing property in the control design, but also eliminate the limitation of unmeasured states. And also, the problems of both unknown dead zones and control directions have been solved. The stability of the closed-loop system and the tracking performance of the

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control system have been demonstrated by utilizing the Lyapunov function theory. Finally,

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the simulation results have further checked the effectiveness of the proposed control design method and theory. In the future, the problems of adaptive fuzzy control for large-scale

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non-strict feedback nonlinear systems would be considered.

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Trans. Cybern. 47 (9) (2017) 2400-2412. [27] S.C. Tong, Y.M. Li, S. Sui, Adaptive Fuzzy Output Feedback Control for Switched Nonstrict-Feedback Nonlinear Systems with Input Nonlinearities, IEEE Trans. Fuzzy

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Fig. 1 Block diagram of the control design method.

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2 1.5 1 0.5 0

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Fig. 3 The trajectories of y2 (black line) and y2,r (red line).

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Fig. 4 The trajectories of x1,1 (black line) and x ˆ1,1 (red line). 4 3 2

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Fig. 5 The trajectories of x1,2 (black line) and x ˆ1,2 (red line).

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Fig. 6 The trajectories of x2,1 (black line) and x ˆ2,1 (red line). 4 3 2

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Fig. 7 The trajectories of x2,2 (black line) and x ˆ2,2 (red line).

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