Adaptive Variable Structure Hierarchical Fuzzy Control for a Class of High-Order Nonlinear Dynamic...

Adaptive variable structure hierarchical fuzzy control for a classof high-order nonlinear dynamic systems

Mohammad Mansouri a,n, Mohammad Teshnehlab b, Mahdi Aliyari Shoorehdeli c

a Department of Control Engineering, Faculty of Electrical Engineering, K.N. Toosi University of Technology, P.O. Box 16315-1355, Tehran, Iranb Industrial Control Center of Excellence, Faculty of Electrical Engineering, K.N. Toosi University of Technology, Tehran, Iranc Department of Mechatronics Engineering, Faculty of Electrical Engineering, K.N. Toosi University of Technology, Tehran, Iran

a r t i c l e i n f o

Article history:Received 5 April 2014Received in revised form15 November 2014Accepted 23 November 2014

Keywords:Adaptive controlFunction approximatorHierarchical fuzzy systemSwitching surfaceVariable structure control

a b s t r a c t

In this paper, a novel adaptive hierarchical fuzzy control system based on the variable structure control isdeveloped for a class of SISO canonical nonlinear systems in the presence of bounded disturbances. It isassumed that nonlinear functions of the systems be completely unknown. Switching surfaces areincorporated into the hierarchical fuzzy control scheme to ensure the system stability. A fuzzy softswitching system decides the operation area of the hierarchical fuzzy control and variable structurecontrol systems. All the nonlinearly appeared parameters of conclusion parts of fuzzy blocks located indifferent layers of the hierarchical fuzzy control system are adjusted through adaptation laws deducedfrom the defined Lyapunov function. The proposed hierarchical fuzzy control system reduces the numberof rules and consequently the number of tunable parameters with respect to the ordinary fuzzy controlsystem. Global boundedness of the overall adaptive system and the desired precision are achieved usingthe proposed adaptive control system. In this study, an adaptive hierarchical fuzzy system is used for twoobjectives; it can be as a function approximator or a control system based on an intelligent-classicapproach. Three theorems are proven to investigate the stability of the nonlinear dynamic systems. Theimportant point about the proposed theorems is that they can be applied not only to hierarchical fuzzycontrollers with different structures of hierarchical fuzzy controller, but also to ordinary fuzzycontrollers. Therefore, the proposed algorithm is more general. To show the effectiveness of theproposed method four systems (two mechanical, one mathematical and one chaotic) are consideredin simulations. Simulation results demonstrate the validity, efficiency and feasibility of the proposedapproach to control of nonlinear dynamic systems.

& 2014 ISA. Published by Elsevier Ltd. All rights reserved.

1. Introduction

The application of fuzzy systems to automation processes pro-vides insight into the structure of control systems for control of high-order nonlinear dynamic systems. Hence, it is challenging issue dueto its applications and there are significant literatures published inthis area. Generally, a typical fuzzy systemwith n input variables andm membership functions for each variable, requires mn rules. Thisphenomenon, increasing the number of rules with increasing ofnumber of inputs exponentially, is called “curse-of-dimensionality”.So, the curse of dimensionality comes to be as a serious problem tofuzzy systems' extensive application when they are applied to high-dimensional systems. To overcome this problem, Raju and Zhou [1]suggested hierarchical fuzzy systems. In hierarchical fuzzy systems

(HFSs), instead of using the standard high-dimensional fuzzy systemseveral lower dimensional fuzzy subsystems are arranged in thehierarchical structure. It makes the number of fuzzy rules growlinearly according to the number of inputs [2]. It has been proventhat HFSs are universal estimators [2–5]. Refs. [6,7] are review papersintroduced main topics of hierarchical fuzzy systems.

The rule reduction feature of HFSs make them significantly moreapplicable in the real world situations. This paper suggests twocategories for classification of researches in the HFS domain;(І)control and prediction, (ІІ) pattern recognition. There are manysimulated and implemented examples in real world such as; controlof the ball and plate system [8] control of the ball and beam system[9] control of double-inverted pendulums connected by a torsionalspring [10] control of two-link robot manipulator [10,11] control ofdouble pendulum system [10] control of rotary crane system [12]predictive control of a satellite [13] intrusion detection [14] trafficprediction [15] cancer classification [16].

Based on the universal approximation property and in order toreach on-line learning ability in fuzzy systems, a number of stable

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ISA Transactions

http://dx.doi.org/10.1016/j.isatra.2014.11.0140019-0578/& 2014 ISA. Published by Elsevier Ltd. All rights reserved.

n Corresponding author.E-mail addresses: [email protected] (M. Mansouri),

[email protected] (M. Teshnehlab),[email protected] (M. Aliyari Shoorehdeli).

Please cite this article as: Mansouri M, et al. Adaptive variable structure hierarchical fuzzy control for a class of high-order nonlineardynamic systems. ISA Transactions (2014), http://dx.doi.org/10.1016/j.isatra.2014.11.014i

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adaptive fuzzy control schemes have been developed. The stabilitystudy in such schemes is carried out by using the Lyapunov designapproach. There are two distinct methods in designing of a fuzzyadaptive control system; direct and indirect approaches. In thedirect scheme, the fuzzy system is used for approximating anunknown ideal controller [17–19]. On the other hand, the indirectscheme uses fuzzy systems to estimate the plant dynamics andthen produces a control law based on these estimations [20–22].Till now, there is no investigation on the subject of Lyapunov basedadaptive hierarchical fuzzy control due to difficulty in deriving theadaptation laws for the parameters appearing in the nonlinearlyparameterized form. One notable paper is [8] where hierarchicalfuzzy CMAC is used as Lyapunov based indirect adaptive controller.In the mentioned paper the HFCMAC is used for estimating theunknown nonlinear function (f ðXÞ) and control gain (bðXÞ). How-ever in our work we approximate two unknown nonlinear func-tions; (f ðXÞ=bðXÞ) and (1=bðXÞ). The main advantage of our workwith respect to [8] is that we consider the error of all tunableparameters of HFS as terms of the proposed Lyapunov functionwhich guarantees convergence of them to their optimal valueswhile in [8] just the error of the last layer's parameter ofhierarchical fuzzy CMAC is considered in the Lyapunov function.Notice that in [29] the sliding surface is produced in hierarchicalmanner and adaptive fuzzy system is used for approximatingunknown nonlinear function, but in this study adaptive hierarch-ical fuzzy system is investigated and the sliding surface is used forconvergence of errors and it is not in the hierarchical structure. So,this work is completely different from [29].

In this study, a new adaptive variable structure hierarchicalfuzzy control(VSHFC) system is presented. The proposed hierarch-ical fuzzy control system is able to reduce the number of ruleswith respect to the ordinary fuzzy control system. Some tunableparameters in the hierarchical fuzzy control (HFC) systems appearnonlinearly. So, the determination of the adaptive laws for suchparameters is a nontrivial task. In this study, first a new descrip-tion of the HFS for being a function approximator is presented.Then, this study offers a novel adaptive control method to tune allparameters of conclusion parts of fuzzy blocks located in differentlayers of the HFC. Taylor series expansion is applied for parameterswhich appear nonlinearly. The fuzzy switching system decides theoperation area of the HFC and the variable structure control.Global boundedness of the overall adaptive system and the desiredprecision are established with the new adaptive control systemusing the Lyapunov-based stability theorem. In order to prove theproposed theorems, nonlinear systems are categorized into twocategories; in the first category, the control gain is considered asunity and in the other category it is considered as non-unity. Thetheorems are proved for both categories. The proposed adaptivecontrol system decreases the number of rules and consequentlythe number of tunable parameters with respect to the ordinaryfuzzy control system. Finally, results of proposed theorems aresimulated on four systems. The systems are categorized asrobotics, mechanical, mathematical and physical systems.

It is important to notice that the proposed VSHFC systemdescribed in this paper differs from those described in theliterature from these aspects: (i) For the first time, the HFC systemis combined with the variable structure control. Also, its non-linearly appearing adjustable parameters are tuned using adapta-tion laws achieved by the Lyapunov approach. Then the drivenadaptation laws are made robust using σ-modification. (ii) There isan extra term in the Lyapunov approach to overcome the boundeddisturbances of nonlinear systems. (iii) In order to gain softswitching system, it is considered as a fuzzy system.

The rest of this paper is organized as follows. Problem for-mulation is given in Section 2. Section 3 presents the formulationof the hierarchical fuzzy system as function approximators. Also, a

theorem for describing the function approximation errors isdeveloped and proven. In Section 4, the scheme of VSHFC systemis proposed to control of SISO nonlinear systems. In addition,according to what is achieved from the theorem (presented inSection 3), two theorems and their conditions are presented andproven to ensure the asymptotically stability of nonlinear systems.Simulation results are provided in Section 5 to demonstrate theperformance of the proposed method. The paper is discussed andconcluded in Section 6.

2. Problem formulation

This study focuses on the design of adaptive control algorithmsfor a class of dynamic SISO affine nonlinear systems which can beexpressed in the canonical form as follows:

x nð Þ tð Þþ f x tð Þ; _x tð Þ;…; x n�1ð ÞðtÞ� �¼ b x tð Þ; _x tð Þ;…; x n�1ð Þ� �u tð ÞþdðtÞ

yðtÞ ¼ xðtÞ ð1Þwhere uðtÞ is a control input, f ðXÞ is an unknown function, bðXÞ isthe control gain and dðtÞ is the bounded disturbance such thatdðtÞ�� oD. The control objective is to force the states of the system,X ¼ x; _x;…; x n�1ð Þ� �T , to follow a specified desired trajectory,Xd ¼ xd; _xd;…; x n�1ð Þ

d

h iT. Considering e¼ X�Xd as the tracking

error vector, our goal is to design a suitable control law, uðtÞ,which ensures that e-0 as t-1.

In this paper, the hierarchical fuzzy system (HFS) is utilized toprovide the control effort. The majority of solutions using adaptivecontrol approaches has focused on the situation where an explicitlinear parameterization of the unknown function is possible.Although, it is unfeasible for HFSs. Thus, the challenge addressedin this paper is development of adaptive HFCs where the explicitlinear parameterization of the adaptive control system isimpossible.

In the next Section, the hierarchical fuzzy system is introducedto approximate functions. In addition, the novel description of thefunction approximation errors in HFS is proposed.

3. Hierarchical fuzzy system as a function approximator

In this section, the HFS is presented as a function approximator.First, the structure of HFS is given. Then, Theorem 1 is proposed todescribe the function approximator errors in HFS.

3.1. Hierarchical fuzzy system

In this section, the hierarchical fuzzy system considered in thisstudy is introduced. Fig. 1 shows a typical HFS with n�1 layerswhere FIS is used for representation of fuzzy inference system. Asshown in Fig. 1, every layer consists of one block which is anordinary fuzzy system with two inputs. It is proven that the

FIS1x1

x2

x3

y 1

FIS2FIS3

x4

y2

yn-2

FISn-1xn

yn-1

Fig. 1. Structure of the typical hierarchical fuzzy system.

M. Mansouri et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎2





hierarchical fuzzy systemwith this structure can be minimized thenumber of rules [3].

Generally, the output of thep-th block with employing thezero-order Takagi-Sugeno fuzzy system with Gaussian member-ship functions for inputs, singleton fuzzifier, sum-product infer-ence engine, and center average defuzzifier is as follow:

yp ¼∑rp

ip ¼ 1μipAip

xpþ1� �

:μipBip

yp�1

� �θipp

∑rpip ¼ 1μ

ipAip

xpþ1� �

:μipBip

yp�1

� � ð2Þ

where xpþ1 is the ðpþ1Þ-th input, yp�1 is the output of theðp�1Þ-th layer, θip

p is the center ofthe consequent part membershipfunctions, μip

Aipð:Þ and μip

Bipð:Þ are the membership functions of

xpþ1 and yp�1 respectively and rp is the number of rules for p-thblock. It is necessary to indicate that for the first layer, y0 isconsidered as x2. It is possible to rewrite (2) in vector form asshown in (3).

yp ¼ θTpΓp xpþ1; yp�1

� �; p¼ 1;…;n�1 ð3Þ

where θpϵRrp�1 is a vector of centers of consequent parts andΓpϵR1�rp is a vector of the normalized part due to the antecedentpart of rules. Please notice that centers of consequent parts ofprevious layers appear nonlinearly in the Γpð:Þ.

3.2. Novel description of the function approximation error in HFS

In this section, it is assumed that the HFS is used for functionapproximation and a novel description of estimation error ispresented. Assume a function approximation error is defined asfollows:

ε Xð Þ9 f Xð Þ� θT

n�1Γn�1ðθ1; θ2;…; θn�2Þ ð4Þ

where θT

n�1Γn�1ð:Þ is the estimated output of an HFS approxima-tion of f Xð Þ as described in (3) and θ

T

i with for i¼ 1;…;n�1 is thevector of centers of consequent parts of i-th block. The parametersin the estimation should then be tuned to provide the effectiveapproximation architecture. The error between the ideal HFS andthe function f Xð Þ is defined as:

εf Xð Þ9 f Xð Þ� f n Xð Þ ¼ f Xð Þ�θnTn�1Γ

n

n�1ðθn

1;θn

2;…;θn

n�2Þ ð5Þ

where θnTn�1Γ

n

n�1ð:Þ is the output of an ideal HFS such that allparameters of HFS, θnT

i with i¼ 1;…;n�1, are in their optimalvalues. Since the optimal values of HFS are unknown, using theestimation function f Xð Þ ¼ θ

T

n�1Γn�1ð:Þ the approximation errorbetween f ðXÞ and f Xð Þ needs to be established. Theorem 1 explainshow this is accomplished. Please note that Γn�1ð:Þ is a function ofthe center of consequent part membership functions and theinputs of fuzzy blocks in the previous layers, but the only tunableparameters of it are just consequent part's parameters. In thispaper, after this the argument of Γn�1ð:Þ is neglected for simplicityand it is shown just as Γn�1.

The following theorem is proposed for HFS function approx-imation errors.

Theorem 1. Let us define the estimation errors of parameter vectorsas:~θ i ¼ θn

i � θi i¼ 1;…;n�1 ð6Þ

the function approximation error ε Xð Þ ¼ f Xð Þ� f ðXÞ, when the HFSwith ðn�1Þ layers, n is the dimension of the state vector X,are usedfor function estimation and all membership functions are

considered as Gaussian functions, can be expressed as:

ε Xð Þ ¼ ~θTn�1 Γn�1� ∑

n�2

i ¼ 1

∂Γn�1

∂θn

i

jθ ¼ θ θi

!þ θ

T

n�1 ∑n�2

i ¼ 1

∂Γn�1

∂θn

i

jθ ¼ θ~θ i

!þR

ð7Þwhere R is a residual term which satisfies:

Rj jrθnTf :Yf ð8Þ

where θn

f ARn is an unknown constant vector composed of optimalweight matrices and some bounded constants and Yf ¼ 1;½jjθn�1jj; jjθn�2jj;…; jjθ1jj�T is a known function vector.

Proof 1. Using (5), the function approximation error in (4) can bewritten as:

ε Xð Þ ¼ εf Xð Þþ f n Xð Þ� θT

n�1Γn�1 ¼ εf Xð ÞþθnTn�1Γ

n

n�1� θT

n�1Γn�1

ð9ÞSo, the function approximation error can be written as follows:

ε Xð Þ ¼ εf Xð Þþ ~θTn�1

~Γn�1þ θT

n�1~Γn�1þ ~θ

Tn�1Γn�1 ð10Þ

In order to obtain a mathematical expression for ~Γn�1, the Taylor'sseries expansion of Γn

n�1 is performed about θn

i ¼ θi f or i¼ 1;…;

n�2. This leads to

Γn

n�1 x;θnT1 ;…;θnT

n�2

� �¼ Γn�1þ ∑

n�2

i ¼ 1

∂Γn�1

∂θn

i

jθ ¼ θ θn

i � θi

� �þH:O:T

ð11Þwhere H:O:T denotes the sum of high-order terms in Taylor's seriesexpansion and ∂Γn� 1

∂θn

ijθ ¼ θϵR

ln� 1�li is derivative of Γn

n�1 with respectto θn

i at ðθ1; θ2;…; θn�2Þ in which ln�1 and li are dimensions ofΓn�1 and θ

n

i , respectively.Considering ~Γn�1 ¼Γn

n�1�Γn�1, (11) can be expressed as:

~Γn�1 ¼ ∑n�2

i ¼ 1

∂Γn�1

∂θn

i

jθ ¼ θ θn

i � θi

� �þH:O:T ð12Þ

Using (12), Eq. (10) can be written as:

ε xð Þ ¼ εf xð Þþ ~θTn�1 ∑

n�2

i ¼ 1

∂Γn�1

∂θn

i

jθ ¼ θ θn

i � θi

� �þH:O:T

!

þ θT

n�1 ∑n�2

i ¼ 1

∂Γn�1

∂θn

i

jθ ¼ θ θn

i � θi

� �þH:O:T

!þ ~θ

Tn�1Γn�1 ð13Þ

Finally, it is concluded that

ε xð Þ ¼ ~θTn�1 Γn�1� ∑

n�2

i ¼ 1

∂Γn�1

∂θn

i

jθ ¼ θ θi

!þ θ

T

n�1 ∑n�2

i ¼ 1

∂Γn�1

∂θn

i

jθ ¼ θ~θ i

!þR

ð14Þwhere

R¼ εf xð Þþ ~θTn�1 ∑

n�2

i ¼ 1

∂Γn�1

∂θn

i

jθ ¼ θθn

i

!þθnT

n�1: H:O:Tð Þ ð15Þ

Applying the triangular inequality to (12) leads to

JH:O:Tjj ¼ ~Γn�1� ∑n�2

i ¼ 1

∂Γn�1

∂θn

i

jθ ¼ θ~θ ijjr jj ~Γn�1jjþ ∑

n�2

i ¼ 1

∂Γn�1

∂θn

i

��:J ~θ i J

��

��

ð16ÞSince Gaussian membership functions and its derivative arealways bounded by constants, it is inferred that

JH:O:T JrCn�1þ ∑n�2

i ¼ 1Ci:J ~θ i J ð17Þ

where Ci f or i¼ 1;…;n�1 are some constants due to the fact thatwas mentioned before about the Gaussian membership function's

M. Mansouri et al. / ISA Transactions ∎ (∎∎∎∎) ∎∎∎–∎∎∎ 3





boundedness. It is obvious that

Jθn

i Jrθi i¼ 1;…;n�1 ð18ÞThe term R in (15) can be bounded as follow:

Rj j ¼ εf xð Þþ ~θTn�1 ∑

n�2

i ¼ 1

∂Γn�1

∂θn

i

jθ ¼ θθn

i

!þθnT

n�1: H:O:Tð Þ��

�� ð19Þ

Considering the fact that εf ðxÞ�� rεn, where εn is a constant and

using (17), (18) and (19) can be written as:

Rj jrεnþ θn�1þ J θn�1 J� �

: ∑n�2

i ¼ 1Ci:θi

þθn�1: Cn�1þ ∑n�2

i ¼ 1Ci: Jθiþ θi J� � !

¼ εnþ2θn�1: ∑n�2

i ¼ 1Ci:θiþCn�1:θn�1

þ J θn�1 J : ∑n�2

i ¼ 1Ci:θiþθn�1: ∑

n�2

i ¼ 1Ci: jjθijj� �

ð20Þ


Rj jr θf n ;θf n� 1;…;θf 1

� �: 1; J θn�1 J ; J θn�2 J ;…; J θ1 Jh iT

¼ θnTf :Yf

ð21Þwhere

θf n ¼ 2θn�1: ∑n�2

i ¼ 1Ci:θiþεn

θf n� 1¼ ∑

n�2

i ¼ 1Ci:θi

θf i ¼ Ciθi i¼ 1;…;n�2

Remark Theorem 1 expresses the high order terms of Taylorseries expansion as a linearly parameterizable structure withrespect to the residual term. In addition, the residual term isbounded by a linear expression with a known vector of functions.On the other hand, there is no explicit expressions for optimalparameters. Therefore, the linearly parameterized form of theresidual term makes it possible to develop the stable learningmethod using adaptive control techniques.

In the next Section, what is achieved from Theorem 1 is appliedto develop the VSHF control system for a class of high-ordernonlinear dynamic systems.

4. Adaptive variable structure hierarchical fuzzy control

In this Section, VSHFC system is proposed to control of SISOnonlinear systems. First, the structure of VSHFC is presented.Second, it is utilized to control of the nonlinear systems. Twotheorems are proven to ensure the asymptotically stability of

nonlinear systems described in (1) with considering b Xð Þ ¼ 1 orb Xð Þa1.

4.1. Structure of VSHF control system

The schematic diagram of the proposed VSHFC system is shownin Fig. 2. This structure differs from those described in [23] wherethe RBF neural network is used instead of the HFC presented here.As it is shown in the figure, the control signal composed of threeterms; (1) linear feedback, (2) sliding mode, (3) HFS. The modulateblock plays the role of weighting function which produces theappropriate value between zero and one based on the error for thevariable structure control and hierarchy fuzzy control systems.

The switching surface considered as an error metric is definedas:

s tð Þ ¼ ddt

þλ n�1

e tð Þ with λ40 ð22Þ

which can be rewritten as s tð Þ ¼ΛTeðtÞ with ΛT ¼ ½λn�1;

n�1ð Þλn�2;…;1�. The equation s tð Þ ¼ 0 demonstrates a time-

varying hyperplane in Rn such that the tracking error vectordecreases exponentially to zero on it, so that perfect tracking canbe asymptotically achieved by keeping up this condition [25].Therefore, the control objective becomes the design of a controllerto forces tð Þ ¼ 0. The time derivative of switching surface can bewritten as:

_s tð Þ ¼ �x nð Þd tð ÞþΛT

Ve tð Þþbu� f Xð ÞþdðtÞ ð23Þ

where ΛTV ¼ ½0; λn�1

; n�1ð Þλn�2;…; ðn�1Þλ�. The objective of this

paper is determination of adaption laws for θi i¼ 1;…;n�1 usingthe error Eq. (23) such that all signals remain bounded and s tð Þ ¼ 0.

Main results are presented in two steps. In the first stage, thecontrol gain considered as unity i.e. b Xð Þ ¼ 1 and the globalboundedness of all system signal is established. This lets usconcentrate on the structure of the adaptive controller, whichstably tunes all parameters in the HFC which appear in thenonlinearly parameterized form. In the second stage, results areextended to the case of the non-unity control gain i.e. bðXÞa1.This gives a complete solution for the control problem.

4.2. Asymptotically stability of nonlinear systems with consideringb Xð Þ ¼ 1

The stability of the closed-loop system when b Xð Þ ¼ 1 isestablished in the following theorem.

Theorem 2. If adaptive control laws, presented in by defining theswitching surface presented in (29), and considering the parameteradjust rules given in (30) to (33) are applied to the nonlinear dynamicsystem introduced in (1) with b Xð Þ ¼ 1 (according to Fig. 2), then allstates in the adaptive system will remain bounded, and the trackingerrors will be asymptotically bounded by e ið ÞðtÞ

�� r2iλi�nþ1ϕ;f or i¼ 1;…;n�1.

u tð Þ ¼ �kdsϕ tð Þþufd tð Þþ 1�m tð Þð Þufu tð Þþm tð Þusl tð ÞþurðtÞ ð24Þ

ufd tð Þ ¼ x nð Þd tð Þ�ΛT

Ve tð Þ ð25Þ

ufu tð Þ ¼ θT

n�1Γn�1� θT

f Yf satðsðtÞ=ϕÞ ð26Þ

usl tð Þ ¼ �ksl tð ÞsatðsðtÞ=ϕÞ ð27Þ

ur tð Þ ¼ �ρ:signðsϕÞ ð28Þ

sϕ tð Þ ¼ s tð Þ�ϕsatðsðtÞ=ϕÞ ð29Þ

X(t)

Xd(t)

LinearFeedback

Variablestructure

Modulator

Π

ΠHFS

∑

ufd(t)

Usl(t)

Ufu(t)

X(t)System

-KdSΦ

Fig. 2. Schematic diagram of the proposed VSHFC.






_θn�1 ¼ 1�m tð Þð Þ:sϕ tð Þ:P�1

n�1: ∑n�2

i ¼ 1

∂Γn�1

∂θn

i

jθ ¼ θ θi�Γn�1

!ð30Þ

_θj ¼ m tð Þ�1ð Þ:sϕ tð Þ:P�1

j : θT

n�1∂Γn�1

∂θn

j

jθ ¼ θ

! !T

; j¼ 1;…;n�2

ð31Þ

_θf ¼ 1�m tð Þð Þ: sϕðtÞ

�� :P�1n :Yf ð32Þ

_ρ¼ γ: sϕ�� ð33Þ

where mðtÞ is the output of the zero-order Takagi-Sugeno fuzzysystemwith singleton fuzzifier, sum-product inference engine, andcenter average defuzzifier fuzzy systemwhose input is just sðtÞ andit has two Gaussian membership functions for the input variableand its value is 0om tð Þo1. Also, P�1

i ϵRri�ri ; i¼ 1;…;n�1whereri is the dimension of θi and P�1

n ϵRn�n. The satð:Þ function isdefined as follows

sat yð Þ ¼�1 if yo�1y if �1ryr11 if y41

8><>: ð34Þ

Proof 2. With substituting (24) in (23), _s tð Þ can be written as:_s tð Þ ¼ �kdsϕ tð Þþ 1�m tð Þð Þ:ufu tð Þþm tð Þ:usl tð Þ� f Xð Þþur tð Þþdðx; tÞ

¼ �kdsϕ tð Þþ 1�m tð Þð Þ: ufu tð Þ� f Xð Þ� �þm tð Þ:ðusl tð Þ� f Xð ÞÞþur tð ÞþdðtÞð35Þ

Using (26) and (7), ufu tð Þ can be written as:

ufu tð Þ ¼ f Xð Þ�ε Xð Þ� θT

f Yf sats tð Þϕ

� �¼ f Xð Þþ ~θ

Tn�1 ∑

n�2

i ¼ 1

∂Γn�1

∂θn

i


!

� θT

n�1 ∑n�2

i ¼ 1

∂Γn�1

∂θn

i

jθ ¼ θ~θ i

!�R� θ

T

f Yf sats tð Þϕ

ð36Þ

Eq. (35) can be expressed as:

_s tð Þ ¼ �kdsϕ tð Þþ 1�m tð Þð Þ: ~θTn�1 ∑

n�2

i ¼ 1

∂Γn�1

∂θn

i


!(

� θT

n�1 ∑n�2

i ¼ 1

∂Γn�1

∂θn

i

jθ ¼ θ~θ i

!)� 1�m tð Þð Þ: Rþ θ

T

f Yf sats tð Þϕ

þm tð Þ: usl tð Þ� f Xð Þð Þþur tð ÞþdðtÞ ð37ÞConsider the Lyapunov function candidate as:

V tð Þ ¼ 12

s2ϕ tð Þþ ~θTn�1Pn�1

~θn�1þ ∑n�2

i ¼ 1

~θTi Pi

~θ iþ ~θTf Pn

~θ f þ1γρ�ρn� �2 !

ð38ÞIt is necessary to indicate that the last term in the Lyapunovfunction is just for designing urwhich is added to control signal inorder to compensate the disturbance. Taking the derivative of (38)gives

_V tð Þ ¼ sϕ tð Þ_sϕ tð Þ� ~θTn�1Pn�1

_θn�1� ∑

n�2

i ¼ 1

~θTi Pi

_θi� ~θ

Tf Pn

_θf þ

1γ_ρ ρ�ρn� �

ð39ÞApplying (24) to (32) (in 39) leads to

_V tð Þ ¼ �kds2ϕ tð Þþ 1�m tð Þð Þ:sϕ tð Þ: ~θTn�1 ∑

n�2

i ¼ 1

∂Γn�1

∂θn

i


!

� 1�m tð Þð Þ:sϕ tð Þ:θT

n�1 ∑n�2

i ¼ 1

∂Γn�1

∂θn

i

jθ ¼ θ~θ i

!� 1�m tð Þð Þ:sϕ tð Þ:

� Rþ θT

f Yf sats tð Þϕ

þm tð Þ:sϕ tð Þ: usl tð Þ� f Xð Þð Þ� ~θ

Tn�1Pn�1:

� 1�m tð Þð Þ:sϕ tð Þ:P�1n�1: ∑

n�2

i ¼ 1

∂Γn�1

∂θn

i


!( )

þ ∑n�2

j ¼ 1

~θTj Pj 1�m tð Þð Þ:sϕ tð Þ:P�1

j :θT

n�1∂Γn�1

∂θn

j

jθ ¼ θ

!

� ~θTf :Pn: 1�m tð Þð Þ: sϕ tð Þ

�� :P�1n :Yf þs∅:ur tð Þþs∅:d tð Þþ1

γ_ρ ρ�ρn� �

ð40ÞWith some mathematical simplification and using upper

bounds in applying the triangular inequality and also using thefact that Rj joθnT

f :Yf ; _sϕ tð Þ ¼ _s tð Þ; sϕ tð Þsat s tð Þ=ϕ� �¼ sϕ tð Þ�� result in

the following equation. It is necessary to mention that if sj jo∅then s∅ ¼ _s∅ ¼ 0, while if sj j4∅ then s∅ ¼ _s.

_V tð Þ ¼ �kds2ϕ tð Þþm tð Þ:sϕ tð Þ: usl tð Þ� f Xð Þð Þ� 1�m tð Þð Þ: sϕ tð Þ�� :θT

f :Yf

� 1�m tð Þð Þ:sϕ tð Þ:R� 1�m tð Þð Þ: sϕ tð Þ�� : θn

f � θf

� �T:Yf

�ρ: sϕ�� þsϕ:d tð Þþ1


r�kds2ϕ tð Þþm tð Þ: �ksl s∅ðtÞ

��

þ s∅ðtÞ�� : f ðXÞ�� 1�m tð Þð Þ:sϕ tð Þ:R� 1�m tð Þð Þ: sϕ tð Þ

�� :θnTf :Yf

�ρ: sϕ�� þsϕ:d x; tð Þþ1


r�kds2ϕ tð Þ� 1�m tð Þð Þ:sϕ tð Þ:R

� 1�m tð Þð Þ: sϕ tð Þ�� :θnT

f :Yf �ρ: sϕ�� þ s∅

�� :Dþ1γ_ρ ρ�ρn� �

r�kds2ϕ tð Þ�ρ: sϕ

�� þ s∅�� :Dþ1

γ_ρ ρ�ρn� �¼ �kds

2ϕ tð Þ�ρn: sϕ

�� þ ρn�ρ� �

: sϕ�� þ s∅

�� :Dþ1γ_ρ ρ�ρn� �¼ �kds

2ϕ tð Þ�ρn: sϕ

�� þD sϕ�� ð41Þ

Since ρn is a designing parameter, considering it as Drρn gives

_Vr�kds2∅ ð42Þ

Therefore, all signals in the system presented in (1) are bounded.Since s∅ðtÞ is bounded, it can be shown that, if eð0Þ is bounded,then eðtÞ is also bounded for all t. On the other hand, since XdðtÞ isbounded by designing it, XðtÞ is as well. In order to investigate theasymptotic convergence of the tracking error, it is necessary toprove that s∅ tð Þ-0 as t-1. This can be achieved by applyingBarbalat's Lemma to the continuous, non-negative function asfollow:

V1 tð Þ ¼ V tð Þ�Z t

0ð _V tð Þþkds

2∅ tð Þdt ) _V1 tð Þ ¼ �kds

2∅ tð Þ ð43Þ

It can be easily observed that every term on the right-hand side of(35) is bounded, hence _s∅ðtÞ is bounded. This means that _V1ðtÞ is auniformly continuous function of time. Since V1ðtÞ is boundedbelow by 0, and _V1ðtÞr0 for all t, applying Barbalat's lemmaresults in _V1ðtÞ-0 as t-1. This implies that the inequalitysðtÞ�� r∅ is asymptotically satisfied. Also, the asymptotic trackingerrors as written in [25] are asymptotically bounded by:

e ið ÞðtÞ�� r2iλi�nþ1ϕ; i¼ 1;…;n�1

Remark 1. Theorem 2 demonstrates that it is possible to set up aglobally stable adaptive system by tuning the consequence parametersof all blocks of the HFS control systemwhich are appeared nonlinearlyin the system. This can be accomplished according to the proof ofTheorem 2, due to the linear expression of θi f or i¼ 1;…;n�2 inTheorem 1.






Remark 2. Inequality for tracking errors confirms that the track-ing error can be effected by ∅. If ∅-0, then eðtÞ-0. In such a case,sat s tð Þ=∅� �

becomes signðs tð ÞÞ. In this way, the chattering phenom-ena can be overcome, which makes this method more applicable.

Remark 3. As mentioned in Theorem 2, mðtÞ is the output of anordinary fuzzy system. In fact, based on the value of switchingsurface which is the input of modulate fuzzy system, it producesweights for variable structure and HFS control systems. This givesthe ability to the control system to switch softly between the twocontrollers.

Corollary 1. (Generalization). Adaption laws are considered for HFSin the typical structure of Fig. 1. Notice that they can be applied toany HFS with any structure and any number of inputs at each block.In fact, the important thing is the number of layers appeared as theupper bound of summations at adaptation laws. If the number oflayers are less thann�1, then just the upper bound of summationsshould change to the number of layers. This can be more interestingwhen there is just one layer where the HFS convert to the ordinaryfuzzy system. In the latter case, all summations at adaptation lawsmust be considered as zero. Therefore, the proposed method is moregeneral than which is used in the ordinary fuzzy system and it can beapplied to both of HFS (with any structure) and the ordinary fuzzysystem.

4.3. Asymptotically stability of nonlinear systems with consideringb Xð Þa1

In Theorem 2, it is required that the control gain be unit. In thefollowing theorem, the results are extended to nonlinear systemswith non-unity control gains.

Notice. The following general assumptions with respect to thecontrol gain bðXÞ are made.

� The control gain bðXÞ is finite and non-zero.� The functions h Xð Þ ¼ f ðXÞ=bðXÞ and g Xð Þ ¼ 1=bðXÞ are bounded

by known positive values such that hðXÞ�� rM0

and gðXÞ�� rM1.� There exists a known positive function M2ðXÞ, such that

ðd=dtÞgðXÞ�� rM2ðXÞjjXjj.

Let us denote h Xð Þ ¼ θT

hðn�1ÞΓhðn�1ÞðX; θh1; θh2;…; θhðn�2ÞÞ andg Xð Þ ¼ θ

T

gðn�1ÞΓgðn�1ÞðX; θg1; θg2;…; θgðn�2ÞÞ to be the estimates ofthe optimal fuzzy approximator θnT

hðn�1ÞΓn

hðn�1Þðθn

h1;θn

h2;…;θn

hðn�2ÞÞand θnT

gðn�1ÞΓn

gðn�1Þðθn

g1;θn

g2;…;θn

gðn�2ÞÞ, respectively. Theorem 1 canbe applied in order to obtain the following approximation errorproperties:

~h ¼ h� h¼ ~θThðn�1Þ Γhðn�1Þ � ∑

n�2

i ¼ 1

∂Γhðn�1Þ∂θn

hðiÞjθ ¼ θ θhðiÞ

!

þ θT

hðn�1Þ ∑n�2

i ¼ 1


hðiÞjθ ¼ θ

~θhðiÞ

!þRh ð44Þ

~g ¼ g� g¼ ~θTgðn�1Þ Γgðn�1Þ � ∑

n�2

i ¼ 1

∂Γgðn�1Þ∂θn

gðiÞjθ ¼ ΓθgðiÞ

!

þ θT

gðn�1Þ ∑n�2

i ¼ 1

∂Γgðn�1Þ∂θn

gðiÞjθ ¼ θ

~θgðiÞ

!þRg ð45Þ

Furthermore, Rh

�� rθnTh :Yh and Rg

�� rθnTg :Yg . With these error

features, the stability of the closed-loop system when b Xð Þa1 isestablished in the following theorem.

Theorem 3. If adaptive control laws, presented in (46) to (48) byconsidering the parameter adjust rules given in (49) to (55), areapplied to the nonlinear dynamic system introduced in (1) with

b Xð Þa1 and based on the assumptions mentioned above (accordingto Fig. 2), then all states in the adaptive system will remain bounded,and the tracking errors will be asymptotically bounded bye ið ÞðtÞ�� r2iλi�nþ1ϕ; f or i¼ 1;…;n�1.

u tð Þ ¼ �kdsϕ tð Þ�12M2 Xð ÞXsϕ tð Þþ 1�m tð Þð Þufu tð Þþm tð Þusl tð ÞþurðtÞ

ð46Þ

ufu tð Þ ¼ θT

hðn�1Þ:Γhðn�1Þ þ θT

gðn�1Þ:Γgðn�1Þ:ar

� θT

hYhþ θT

gYg : arj j

sats tð Þϕ

ð47Þ

usl tð Þ ¼ �ksl tð Þ:sats tð Þϕ

ð48Þ

_θhðn�1Þ ¼ 1�m tð Þð Þ:sϕ tð Þ:P�1

h n�1ð Þ: ∑n�2

i ¼ 1

∂Γh n�1ð Þ∂θn

h ið Þjθ ¼ θ θi�Γh n�1ð Þ

!

ð49Þ

_θhðjÞ ¼ � 1�m tð Þð Þ:sϕ tð Þ:P�1

h jð Þ : θT

h n�1ð Þ:∂Γh n�1ð Þ∂θn

h jð Þjθ ¼ θ

! !T

;

j¼ 1;…;n�2 ð50Þ

_θh ¼ 1�m tð Þð Þ: sϕ tð Þ

�� :P�1h nð Þ:Yh ð51Þ

_θgðn�1Þ ¼ 1�m tð Þð Þ:sϕ tð Þ:P�1

g n�1ð Þ: ∑n�2

i ¼ 1

∂Γg n�1ð Þ∂θn

g ið Þjθ ¼ θ θi�Γg n�1ð Þ

!ar

ð52Þ

_θgðjÞ ¼ � 1�m tð Þð Þ:sϕ tð Þ:P�1

g jð Þ : θT

g n�1ð Þ:∂Γg n�1ð Þ∂θn

g jð Þjθ ¼ θ

! !T

:ar ;

j¼ 1;…;n�2 ð53Þ

_θg ¼ 1�m tð Þð Þ: sϕ tð Þ

�� :P�1g nð Þ:Yg : arj j ð54Þ

_ρ¼ γ: sϕ�� ð55Þ

where θhi f or i¼ 1;…;n�1 and θgi f or i¼ 1;…;n�1 are

the estimates of θn

hi f or i¼ 1;…;n�1 and θn

gi f or i¼ 1;…;n�1,

respectively. Also,ar ¼ x nð Þd tð Þ�ΛT

V~X ðtÞ and ksl tð Þ ¼M0þM1 arj j. As

mentioned in Theorem 2, mðtÞis the output of the zero-orderTakagi-Sugeno fuzzy system with singleton fuzzifier, sum-productinference engine, and center average defuzzifier fuzzy system whoseinput is only sðtÞ and it has two Gaussian membership functions for

the input variable. Also, P�1hðiÞϵR

rh ið Þ�rh ið Þ ; i¼ 1;…;n�1 where rh ið Þ is

the dimension of θh ið Þ, P�1h nð ÞϵR

n�n and P�1gðiÞ ϵR

rg ið Þ�rg ið Þ ; i¼ 1;…;n�1

where rg ið Þ is the dimension of θg ið Þ, P�1g nð ÞϵR

n�n. The satð:Þ is as definedin (34).

Proof 3. With substituting (46) into (23), gðXÞ_s tð Þ can beexpressed as:

g Xð Þ_s tð Þ ¼ �h Xð Þþu tð Þ�g Xð Þarþg Xð ÞdðX; tÞ¼ �kdsϕ tð Þ�1

2M2 Xð ÞXsϕ tð Þ

þ 1�m tð Þð Þ: ufu tð Þ�h Xð Þ�g Xð Þar� �þm tð Þ: usl tð Þ�h Xð Þð

�g Xð ÞarÞþurðtÞþg Xð ÞdðtÞ ð56ÞWith substituting (44) and (45) into (47), it is concluded that

ufu tð Þ ¼ h Xð Þ� θT

hYhsats tð Þϕ

þ g Xð Þar

� θT

gYg arj jsat s tð Þϕ

¼ h Xð Þ� ~h Xð Þ� θ

T

hYhsats tð Þϕ






þ g Xð Þ� ~g Xð Þð Þar� θT


¼ h Xð Þ

þ ~θThðn�1Þ ∑

n�2

i ¼ 1


hðiÞjθ ¼ θ θhðiÞ �Γhðn�1Þ

!

� θT

hðn�1Þ ∑n�2

i ¼ 1


hðiÞjθ ¼ θ

~θhðiÞ

!�Rh� θ

T

hYhsats tð Þϕ

þ g Xð Þþ ~θTg n�1ð Þ ∑

n�2

i ¼ 1


g ið Þjθ ¼ θ θg ið Þ �Γg n�1ð Þ

!� θ

T

g n�1ð Þ

"

� ∑n�2

i ¼ 1


g ið Þjθ ¼ θ

~θg ið Þ

!�Rg

#ar

� θT


ð57Þ

Eq. (56) can then be written as:

g Xð Þ_s tð Þ ¼ �kdsϕ tð Þ�12M2 Xð ÞXsϕ tð Þ

þ 1�m tð Þð Þ: ~θTh n�1ð Þ ∑

n�2

i ¼ 1


h ið Þjθ ¼ θ θh ið Þ �Γh n�1ð Þ

!"

� θT

h n�1ð Þ ∑n�2

i ¼ 1


h ið Þjθ ¼ θ

~θh ið Þ

!#

� 1�m tð Þð Þ: Rhþ θT

hYhsats tð Þϕ

� �

þ 1�m tð Þð Þ: ~θTg n�1ð Þ ∑

n�2

i ¼ 1


g ið Þjθ ¼ θ θg ið Þ �Γg n�1ð Þ

!24

� θT

g n�1ð Þ ∑n�2

i ¼ 1


g ið Þjθ ¼ θ

~θg ið Þ

!#ar

� 1�m tð Þð Þ: dgarþ θT


� �þm tð Þ usu tð Þ�h Xð Þ�g Xð Þarð ÞþurðtÞþg Xð ÞdðtÞ ð58Þ

Consider the Lyapunov function candidate as follow:

V tð Þ ¼ 12

g Xð Þ:s2ϕ tð Þþ ~θThðn�1ÞPhðn�1Þ

~θhðn�1Þ þ ∑n�2

i ¼ 1

~θThðiÞPhðiÞ

~θhðiÞ

þ ~θThPhðnÞ

~θhþ ~θTgðn�1ÞPgðn�1Þ ~θgðn�1Þ þ ∑

n�2

i ¼ 1

~θTgðiÞPgðiÞ ~θgðiÞ

þ ~θTgPgðnÞ ~θg

!þ1γρ�ρn� �2 ð59Þ

As mentioned in Theorem 2, the last term in the Lyapunovfunction is only for designing QUOTE which is added to the controlsignal in order to compensate the disturbance. Taking the deriva-tive of (59) gives

_V tð Þ ¼ 12_g Xð Þs2ϕþsϕ tð Þg Xð Þ_sϕ tð Þ� ~θ

Th n�1ð ÞPh n�1ð Þ

_θh n�1ð Þ

� ∑n�2

i ¼ 1

~θTh ið ÞPh ið Þ

_θh ið Þ � ~θ

ThPh nð Þ

_θh� ~θ

Tg n�1ð ÞPg n�1ð Þ

_θg n�1ð Þ

� ∑n�2

i ¼ 1

~θTg ið ÞPg ið Þ

_θg ið Þ � ~θ

TgPg nð Þ

_θgþ

1γ_ρ ρ�ρn� � ð60Þ

Using (49) to (54) into (60) and using (59) and applying the upperbounds in the triangular inequality leads to

_V tð Þr�kds2ϕ tð Þþ1

2_g Xð Þ�M2 Xð ÞXð Þs2ϕ tð Þ� 1�m tð Þð Þ: sϕ tð Þ

�� θnTh Yh

� 1�m tð Þð Þ:sϕ tð Þ:Rh� 1�m tð Þð Þ: sϕ tð Þ�� θnT

g Yg arj j� 1�m tð Þð Þ:sϕ tð Þ:Rg :arþm tð Þ usu tð Þ�h Xð Þ�g Xð Þarð Þ

�ρ: sϕ�� þ s∅

�� :M1:Dþ1γ_ρ ρ�ρn� �

r�kds2ϕ tð Þ

�ρn: sϕ�� þ ρn�ρ

� �: sϕ�� þ s∅

�� :M1:Dþ1γ_ρ ρ�ρn� �

r�kds2ϕ tð Þ�ρn: sϕ

�� þD:M1: sϕ�� ð61Þ

where the facts _gðXÞ�� rM2ðXÞjjXjj, Rh

�� oθnTh Yh, Rg

�� oθnTg Yg and

sϕ tð Þsat s tð Þ=ϕ� �¼ sϕ tð Þ�� have been used. On the other hand, since

ρn is the designing parameter, considering it as D:M1rρn gives_Vr�kds2∅. With the same argument given in the proof of Theorem2, it is proven that the tracking errors will be asymptoticallybounded by e ið ÞðtÞ

�� r2iλi�nþ1ϕ; i¼ 1;…;n�1.□

Corollary 2. (Generalization): The same as corollary 1 at Theorem 2,adaptation laws can be applied to any HFS with any structure and anynumber of inputs at each block especially to ordinary fuzzy systems. So,this theorem can be used in any case if hðXÞor gðXÞ is HFS or theordinary fuzzy system. Furthermore, the proposed theorem is general.

4.4. Robust adaptation laws

In the previous section, the only uncertainty of the dynamicalsystem is considered in unknown controller parameters. However,in practical applications, when real systems describes with fuzzymodels modeling errors are inevitable [26]. Also, there are othersources of modeling errors, such as unmodeled dynamics, mea-surement noise, time variation of parameters and disturbances[27]. Here, it is assumed that the HFS does not exactly describe thenonlinear functions leading to modeling error and a stabilityanalysis is presented under this condition.

Considering Theorem 2, the adaptation laws of (30)–(33) aremodified as follows

_θn�1 ¼ 1�m tð Þð Þ:sϕ tð Þ:P�1

n�1: ∑n�2

i ¼ 1

∂Γn�1

∂θn

i

jθ ¼ θθi�Γn�1

!�σP�1

n�1θn�1

ð62Þ

_θj ¼ m tð Þ�1ð Þ:sϕ tð Þ:P�1

j : θT

n�1∂Γn�1

∂θn

j

jθ ¼ θ

! !T

�σP�1j θj ;

j¼ 1;…;n�2 ð63Þ

_θf ¼ 1�m tð Þð Þ: sϕðtÞ

�� :P�1n :Yf �σP�1

n θf ð64Þ

_ρ¼ γ: sϕ�� σγρ ð65Þ

where σ is a small positive number. Considering the sameLyapunov function as in (8) and following a similar analysis as inthe proof of Theorem 2 one has

_Vr�kds2ϕþσ ~θ

Tn�1θn�1þσ ∑

n�2

i ¼ 1

~θTi θiþσθ

T

f θf �σρðρ�ρnÞ ð66Þ

with some mathematical simplification it is infer that

_Vr�kds2ϕ�σ2

~θTn�1

~θn�1þ θT

n�1θn�1�θn

n�1Tθn

n�1

��

�σ2∑n�2

i ¼ 1

~θTi~θ iþ θ

T

i θi�θnTi θn

i

� ��σ2

~θTf~θ f þ θ

T

f θf �θnTf θn

f

� �

�σ2

ρ�ρn� �� 2þρ2�ρn2

ið67Þ


_Vr�cVþρr ð68Þwhere

ρr ¼σ2θn

n�1Tθn

n�1þσ2

∑n�2

i ¼ 1θnTi θn

i þσ2θnTf θn

f þσ2ρn2 ð69Þ

and






c¼minðσ;2kdÞminðγi; γf ; γÞ2

ð70Þ

in which without loss of generality it is assumed that the matricesof P�1

i f or i¼ 1;…;n�1; f are diagonal matrices with the samediagonal elements as γi f or i¼ 1;…;n�1; f . Therefore, the Lyapu-nov function converges until Vrρr=c.

Let us consider Theorem 3, the modified adaptation laws of(49)-(55) are as follows

_θhðn�1Þ ¼ 1�m tð Þð Þ:sϕ tð Þ:P�1

h n�1ð Þ: ∑n�2

i ¼ 1


h ið Þjθ ¼ θ θi�Γh n�1ð Þ

!

�σP�1hðn�1Þθhðn�1Þ ð71Þ

_θhðjÞ ¼ � 1�m tð Þð Þ:sϕ tð Þ:P�1

h jð Þ : θT

h n�1ð Þ:∂Γh n�1ð Þ∂θn

h jð Þjθ ¼ θ

! !T

�σP�1hðjÞ θhðjÞ;

j¼ 1;…;n�2 ð72Þ

_θh ¼ 1�m tð Þð Þ: sϕ tð Þ

�� :P�1h nð Þ:Yh�σP�1

hðnÞθh ð73Þ

_θgðn�1Þ ¼ 1�m tð Þð Þ:sϕ tð Þ:P�1

g n�1ð Þ: ∑n�2

i ¼ 1


g ið Þjθ ¼ θ θi�Γg n�1ð Þ

!

�ar�σP�1gðn�1Þθgðn�1Þ ð74Þ

_θgðjÞ ¼ � 1�m tð Þð Þ:sϕ tð Þ:P�1

g jð Þ : θT

g n�1ð Þ:∂Γg n�1ð Þ∂θn

g jð Þjθ ¼ θ

! !T

:ar

�σP�1gðjÞ θgðjÞ; j¼ 1;…;n�2 ð75Þ

_θg ¼ 1�m tð Þð Þ: sϕ tð Þ

�� :P�1g nð Þ:Yg : arj j�σP�1

g θg ð76Þ

_ρ¼ γ: sϕ�� σγρ ð77Þ

where σ is a small positive number. Assuming the same Lyapunovfunction as in (59) and following a similar analysis as in the proofof Theorem 3, one has

_Vr�kds2ϕþσ ~θ

Thðn�1Þθhðn�1Þ þσ ∑

n�2

i ¼ 1

~θThðiÞθhðiÞ þσθ

T

hθh

þσ ~θTgðn�1Þθgðn�1Þ þσ ∑

n�2

i ¼ 1

~θTgðiÞθgðiÞ þσθ

T

g θg�σρðρ�ρnÞ ð78Þ

some mathematical simplification leads to

_Vr�kds2ϕ�

σ2

~θThðn�1Þ

~θhðn�1Þ þ θT

hðn�1Þθhðn�1Þ �θn

hðn�1ÞTθn

hðn�1Þi�

�σ2

∑n�2

i ¼ 1

~θThðiÞ

~θhðiÞ þ θT

hðiÞθhðiÞ �θn

hðiÞTθn

hðiÞi�σ2

~θTh~θhþ θ

T

hθh�θnTh θn

h

� ��

�σ2

~θTgðn�1Þ

~θgðn�1Þ þ θT

gðn�1Þθgðn�1Þ �θn

gðn�1ÞTθn

gðn�1Þi�

�σ2

∑n�2

i ¼ 1

~θTgðiÞ

~θgðiÞ þ θT

gðiÞθgðiÞ �θn

gðiÞTθn

gðiÞi�

�σ2

~θTg~θgþ θ

T

g θg�θnTg θn

g

� ��σ2

ρ�ρn� �2þρ2�ρn2h i

ð79Þ


_Vr�cVþρr ð80Þwhere

ρr ¼σ2θn

hðn�1ÞTθn

hðn�1Þ þσ2

∑n�2

i ¼ 1θn

hðiÞTθn

hðiÞ þσ2θnTh θn

hþσ2θn

gðn�1ÞTθn

gðn�1Þ

þσ2

∑n�2

i ¼ 1θn

gðiÞTθn

gðiÞ þσ2θnTg θn

gþσ2ρn2 ð81Þ

and

c¼minðσ;2kdÞminðγh ið Þ; γh; γgðiÞ; γg ; γÞ2

ð82Þ

It is assumed that the matrices of P�1hðiÞ ; P�1

gðiÞ ; P�1h ; P�1

g f or i¼1;…;n�1 are diagonal matrices with the same diagonal elementsas γi f or i¼ 1;…;n�1; f ; g. Hence, the Lyapunov function con-verges until Vrðρr=cÞ.

The stable region in which the controlled system is stable afterapplying modifications is as follows

R¼ xρr

coVð:Þ

�� onð83Þ

It is important to note that a high value for σ would make thesystem more robust while a small value of it make the perfor-mance of the system better.

In the next section, simulation results are presented to moreexplain the proposed theorems.

5. Simulation results

In this section, four examples are illustrated; (1) a roboticsystem: the flexible joint robot, notice that in this case b Xð Þ ¼ 1, So,Theorem 2 is applied on it (2) a mechanical system: a quarter caractive suspension system, in this case b Xð Þ ¼ cte and Theorem 2can be applied on it (3) a nonlinear system, notice that in the lastcasesbðXÞa1 So, Theorem 3 is applied on it. (4) a physical system:the hyper chaos system, such that in this case b Xð Þ ¼ 1, So,Theorem 2 is applied on it. The last case is considered to revealthe efficiency of proposed theorems when HFSs are considered forthe estimation of both of hðXÞ and gðXÞ. The results are comparedwith the proposed method in [8].

It is necessary to notice that as mentioned in corollaries,theorems can be applied to the ordinary fuzzy system. Simulationresults of the proposed method with ordinary fuzzy systems withthe same initial condition and control parameters with HFS lead tothe exactly same results. This case is shown in appendix A.3 for thefirst example.

Example 1. (for Theorem 2): Fig. 3 shows the arrangement of theflexible joint robot schematically. Defineq¼ q1; _q1; q2; _q2

�as the

set of generalized coordinates for the system where:q1 is the angleof the link, q2 ¼ � 1=m

� �θ1 is the angular displacement of the

rotor,m is the gear ratio and q1�q2 is the elastic displacement ofthe link, using Euler–Lagrange equations, the analytical model ofthe flexible joint robot is derived as [26].

I €q1þMgLsin q1� �þKðq1�q2Þ ¼ 0

J €q2�K q1�q2� �¼ u

(ð84Þ

The state space representation of the system is derived as:

_x1 ¼ x2

uq2

q1

k

I,ML

Fig. 3. A single flexible joint mechanism.






_x2 ¼ �MgLI

sin x1ð Þ�KIx1�x3ð Þ

_x3 ¼ x4

_x4 ¼KJx1�x3ð Þþ1

JuþdðtÞ

y¼ x1þn tð Þ ð85Þwhere x1 ¼ q1 and x3 ¼ q2. Also, nðtÞ is a white noise signal withmean parameter of zero and standard deviation of 0.1 andd tð Þ ¼ 0:1 sin ðtÞ.

The state space description of (85) is not presented in thenormal form. So, it does not satisfy the conditions stated in (1).With changing the coordinates of the system such that they satisfyconditions of (1), the resulting system will be in the form of

_z1 ¼ z2_z2 ¼ z3_z3 ¼ z4

_z4þMgLI

cos ðz1ÞþKIþK

J

z3�

MgLI

z22�KJ

sin z1ð Þ ¼ K

IJuþdðtÞ

ð86Þin which

z1 ¼ x1z2 ¼ x3

z3 ¼ �MgLI

sin x1ð Þ�KIðx1�x3Þ

z4 ¼ �MgLI

x2 cos x1ð Þ�KIðx2�x4Þ ð87Þ

Eq. (87) indicates that the resulting system is in the normal form.The numerical values of the system's parameters considered in thesimulation depicted in Table 1.

From (86) and Table 1, it is obvious that in this example f ðZÞand bðZÞ are defined as

f Zð Þ ¼ 9:8 cos z1ð Þþ2ð Þz3�9:8 z22�1� �

sin ðz1Þb Zð Þ ¼ 1 ð88ÞThe structure of the hierarchical fuzzy controller applied in thisexample is depicted in Fig. 4. The numerical values of controller'sparameters achieved by trial and error are shown in Table 2.

Results of simulations are shown in Fig. 5. Two signals areconsidered in order to verify the proposed controller in trackingproblems; sinusoidal and pulse signals. In every simulation, threemembership functions for each input in each block are considered.Therefore, every block has 9 rules for the fuzzy system. Since thereare three blocks, the total number of rules in HFC will be 27. It isobvious that if the ordinary fuzzy control systemwas used, then thetotal number of rules would be 81 rules. On the other hand, thenumber of tunable parameters in HFC with respect to the ordinaryfuzzy system reduces from 81 to 27. Initial values of tunable

Table 1Numerical values of parameters considered in simulation.

Parameters Numerical value

g 9.8 m/s2

M 1 kgK 1 N/mJ 1 K gm2

I 1 K gm2

L 1m

FIS1

FIS2FIS3

z1

z2

z4

z3

Fig. 4. Structure of hierarchical fuzzy controller used in the example.

Table 2Numerical values of parameters considered in simulation of controller.

Parameters Numerical value

∅ 0:05kd 100ksl 100

0 5 10 15 20

-1

-0.5

0

0.5

1

1.5

Time

y

ydesire

yVSHFC

yHFCMAC

0 5 10 15 20-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time

y

yVSHFCydesireyFCMAC

Fig. 5. Time response of the adaptive system; proposed method yVSHFC(‘_’) andmethod in [8] yHFCMAC(‘..’) follows ydesire (‘–’); Tracking of the sinusoidal signal(b) tracking of the pulse signal.

ms

mu

ksBs

U

Controller

kt

x1

x3

Zr

Fig. 6. The mechanism of a quarter car suspension system.






parameters are chosen randomly. Also, the initial values of statesare considered as follows; X 0ð Þ ¼ 1 0 0 0½ �T for the sinusoidal signaland X 0ð Þ ¼ 1 0 0 0½ �T for the pulse signal. Also, in this example thedescription of fuzzy switching system,mðtÞ, and the rule base of HFSare expressed in the appendix for more clarification.

In tracking of the sinusoidal signal, after few seconds theresponse of the controlled signal follows the sinusoidal signalbecause the desired signal has the smooth behavior. Although, intracking of the pulse signal the sudden change in the desired signalcauses it takes time for the adaptation of tunable parameters. As itis depicted in Fig. 4, the control performance of the proposedadaptive control scheme has been demonstrated in simulations andthe proposed control system makes the system states track itsdesired trajectories. Also, the results reveals the performance ofproposed method in comparison with the method in [8].

Example 2. (for Theorem 2): In this example, the regulationproblem of a quarter car active suspension system given by (89)

is considered [28]. Fig. 6 schematically shows the mechanism ofmentioned system._x1 ¼ x2

_x2 ¼1ms

ð�ks x1�x3ð Þ3�Bs x2�x4ð Þ5þuðtÞÞ_x3 ¼ x4

_x4 ¼1mu

ks x1�x3ð Þ3þBs x2�x4ð Þ5�Ktx3þKtZr�uðtÞ� �

þdðtÞy¼ x1þn tð Þ ð89ÞWhere x1; x2 are displacement and vertical velocity of sprungmass and x3; x4 are displacement and vertical velocity of unsprangmass, respectively. ks is the stiffness of the suspension, Bs is thedamping of the suspension, kt is the stiffness of the tire, Zr is theroad displacement input, mu;m_ sare sprung and unsprang massesrespectively and U is the active control. Also, nðtÞ is a white noisesignal with mean parameter of zero and standard deviation of0.1 and d tð Þ is defined in (90). The numerical values of the system'sparameters considered in the simulation depicted in Table 3.

d tð Þ ¼0:08 cos 8πtð Þ; 1oto1:20; otherwise

(ð90Þ

The state space form of (89) is not in the normal form. So, it doesnot satisfy the conditions stated in (1). With changing thecoordinates of the system such that they satisfy the conditions of(1), the resulting system will be in the following form

_z1 ¼ z2_z2 ¼ z3_z3 ¼ z4_z4þ f Zð Þ ¼ bu tð ÞþdðtÞ ð91Þwhere

f Zð Þ ¼ ktmu

½ks x1�x3ð Þ3þBs x2�x4ð Þ5�kt � ð92Þ

b¼ ktmu

ð93Þ

with

x1 ¼1ms

z1þmuz3kt

x2 ¼1ms

z2þmuz4kt

x3 ¼ �z3kt

x4 ¼ �z4kt

ð94Þ

Eq. (91) indicates that the resulting system is in the form of (1).The structure of the HFC control system applied in the example isdepicted in Fig. 7. It is important to note that in this examplecontrol gain is not unity but it is constant and Theorem 2 can beapplied to it. Numerical values of parameters of the control systemare shown in Table 4. These values are achieved by trial and error.

The output of system must approach to zero in this example toshow efficiency of the proposed controller in regulation problems.Also, three membership functions are considered for each input ofeach block in every simulation. Thus, every block has 9 rules andthe total number of rules in HFC will be 27. On the other hand, thetotal number of rules and the number of tunable parameters incomparison with the ordinary fuzzy control system decrease from81 to 27. Initial values of tunable parameters are randomly chosen.Also, the initial values of states are considered as follows;X 0ð Þ ¼ 0:1 0 0 0½ �T . Simulation results for regulation of a quartercar suspension system are shown in Fig. 8.

The regulation problem of the output of quarter car suspensionwhich is displacement of sprung mass is done well. As it is

Table 3Numerical values of parameters considered in simulation of controller.

Parameters Numerical values

mu 40 kgms 400 kgBs 1000 Ns/mks 18000 N/mkt 200000 N/m

FIS1

FIS2FIS3

z1

z2

z4

z3

Fig. 7. Structure of hierarchical fuzzy controller used in the first example.

Table 4Numerical values of parameters considered in the simulation of the HFS controlsystem (Theorem 2).

Parameters Numerical Values

∅ 0:005kd 10ksl 1λ 3

0 5 10 15 20

0

2

4

6

8

10

Time

y (c

m)

yHFCMACyVSHFCydesire

Fig. 8. Time response of the adaptive system; proposed method yVSHFC(‘_’) andmethod in [8] yHFCMAC(‘..’) follows ydesire (‘–’);






depicted in Fig. 8, the suggested controller force the system totrack the desire constant signal with no overshoot or undershoot.Therefore, using proposed control technique regulation problem ofthe typical real world examples is properly possible. Also, compar-ison of the results and the method in [8] shows the performance ofproposed method.

Example 3. (for Theorem 3): In this example, a nonlinear systemgiven by (95) is considered._x1 ¼ x2_x2 ¼ x3_x3 ¼ x4_x4þ f Xð Þ ¼ b Xð Þu tð ÞþdðtÞy¼ x1þnðtÞ ð95Þwhere

f Xð Þ ¼ �11þx21þx22þx23þx24

ð96Þ

b Xð Þ ¼ 11þ sin x1ð Þ cos x2ð Þþ sin ðx3Þ

ð97Þ

d tð Þ ¼ 0:1 sin ðtÞ ð98ÞWhere nðtÞ is a white noise signal with mean parameter of zeroand standard deviation of 0.1. It is obvious that the nonlinearsystem of (95) is in the form described in (1). Also, it can be easilyshown that the system in (95) satisfies all assumptions which areessential for applying Theorem 3. Structures of HFC systemsapplied in the example for the estimation of functionsh Xð Þ ¼ f ðXÞ=bðXÞ and g Xð Þ ¼ 1=bðXÞ are shown in Fig. 9. Numericalvalues of parameters are shown in Table 5.

Like previous example, three membership functions are con-sidered for each input in each block. Therefore, the number ofrules with respect the ordinary fuzzy system in estimation of hðXÞand gðXÞ reduces from 81 to 27 and from 27 to 18, respectively. Asit is known in the zero-order Takagi-Sugeno fuzzy systems, thenumber of tunable parameters increases with the same relationwith the number of rules. The same as Example 1, initial values ofthe consequent part of different blocks of the HFC are randomlychosen. Initial values of states are considered as follows;X 0ð Þ ¼ 0:8 0 0 0½ �T for the sinusoidal signal, X 0ð Þ ¼ 0:5 0 0 0½ �T forthe pulse signal. Simulation results are shown in Fig. 10.

Like the previous example, results illustrate the efficiency ofthe proposed control system. It is important to notice that like anyadaptive system selection of the matrixes P�1

i which play the roleof the learning rate is very crucial in time which is necessary fortraining of tunable parameters. Especially when the desired signalis pulse, proper selecting of them is very important for trackingproblem due to changing behavior of the signal. Moreover,comparison of the results and the method in [8] shows theperformance of proposed method especially in tracking of pulsesignal.

Example 4. (for Theorem 2): In this example, a 4D integral-orderhyper-chaotic system given by (99) is considered. In hyper-chaotic

systems, the construction of the strange attractor is more complexthan the chaotic attractor [30]._x1 ¼ ax1�x2�u_x2 ¼ x1�x2x23_x3 ¼ �b1x2�b2x3�b3x4_x4 ¼ x3þcx4y¼ x4þnðtÞ ð99Þwhere a¼ 0:56; b1 ¼ 1:0; b2 ¼ 1:0; b3 ¼ 6:0; c¼ 0:8. The statespace form of (99) is not in the normal form and so, does not

FIS1

FIS2FIS3

z1

z2

z4

z3

FIS1

FIS2

z1

z2

z3

Fig. 9. Structure of the HFS control system for estimation of; (a) h Xð Þ (b) g Xð Þ.


Parameters Numerical values

∅ 0:0005kd 10ksl 1λ 3

0 5 10 15 20-1

-0.5

0

0.5

1

1.5

2

Time

y

yVSHFCydesireyHFCMAC

0 5 10 15 20-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time

y


Fig. 10. Time response of the adaptive system; proposed method yVSHFC(‘_’) andmethod in [8] yHFCMAC(‘..’) follows ydesire (‘–’); Tracking of the sinusoidal signal(b) tracking of the pulse signal.






satisfy the conditions stated in (1). With changing the coordinatesof the system such that they satisfy the conditions of (1), theresulting system will be in the following form

_z1 ¼ z2_z2 ¼ z3_z3 ¼ z4

_z4þ f Zð Þ ¼ b1uðtÞþdðtÞ ð100Þwhere

f Zð Þ ¼ �½� b1aþAð Þm1þ b1�b1Bð Þm2� b1þ2b1b2ð Þm2m23

þ A�2b21� �

m22m3þ C�b2Bð Þm3þ C � c�b3 � Bð Þm4

þb1m1m23�2b1b3m2m3m4� ð101Þ

d tð Þ ¼ 0:1 sin ðtÞ ð102ÞWith

A¼ b1nðc�b2Þ;B¼ c2�b3�b2ðc�b2Þ;C ¼ c c2�b3� ��b3ðc�b2Þ;

m1 ¼c�b2ð Þ z3�cz2ð Þþ c2�b3

� �z2�z4

b1;

m2 ¼c�b2ð Þ z2�cz1ð Þþ c2�b3

� �z1�z3

b1m3 ¼ z2�cz1;m4 ¼ z1:

Eq. (100) indicates that the resulting system is in the form of (1). Thestructure of the HFC control system applied in the example isdepicted in Fig. 11. Numerical values of parameters of the controlsystem are shown in Table 6. These values are achieved by trialand error.

The same as previous example, two signals are considered in toshow efficiency of the proposed controller in tracking problems;sinusoidal and pulse signals. Also, three membership functions areconsidered for each input of each block in every simulation. Thus,every block has 9 rules and the total number of rules in HFC will be27. On the other hand, the total number of rules and the number oftunable parameters in comparison with the ordinary fuzzy controlsystem decrease from 81 to 27. Initial values of tunable parametersare randomly chosen. Also, the initial values of states are con-sidered as follows; X 0ð Þ ¼ 1 0 0 0½ �T for the sinusoidal signal andX 0ð Þ ¼ 0 0 0 0½ �T for the pulse signal. Simulation results for trackingof the desire signals are shown in Fig. 12.

Tracking of the both sinusoidal signal and the pulse are donewell. As it is depicted in Fig. 12, although there are sharp changes inthe pulse signal the suggested controller force the system to trackthe desire signal with no overshoot or undershoot. Therefore, usingproposed control technique tracking problem of the typical smoothand sharp signals is properly possible. Also, the results revealsperformance of proposed method in comparison with the methodin [8].

According to results of the examples, it can be demonstratedthat the proposed control approach can guarantee all signals in theclosed-loop system are bounded, and the system output tracks thedesired signal even though the exact information on the nonlinearfunctions in controlled systems is not available.

6. Discussion and conclusion

In this paper, the output tracking control problem has beenaddressed for a class of high-order SISO nonlinear systems with acanonical structure. The nonlinear functions in systems are con-sidered completely unknown. An additional term is added to theoutput control system in order to compensate uncertainties whichare caused because of disturbances. Since consequent part para-meters of fuzzy blocks of the hiβerarchical fuzzy control systemexcept the last layer appeared in the nonlinear form in the output,the Taylor expansion is used for the linearizing them. Stability ofthe closed loop system is guaranteed using the Lyapunov function.

In this study, three theorems are developed. Summary of themis as follows, Theorem 1 proves how the nonlinearly parameterizedtunable parameters of the HFS can be express in the linearlyparameterized form using Taylor series expansion. Also, high orderterms of the expansion can be demonstrate as the linearlyparameterizable structure with respect to the residual term. Inaddition, the residual term is bounded by a linear expression with

FIS1

FIS2FIS3

z1

z2

z4

z3

Fig. 11. Structure of hierarchical fuzzy controller used in the first example.


Parameters Numerical Values

∅ 0:005kd 10ksl 100λ 3

0 5 10 15 20-1.5

-1

-0.5

0

0.5

1

1.5

Time

y


0 5 10 15 20-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time

y


Fig. 12. Time response of the adaptive system; proposed method yVSHFC(‘_’) andmethod in [8] yHFCMAC(‘..’) follows ydesire (‘–’); Tracking of the sinusoidal signal(b) tracking of the pulse signals.






a known vector of functions. Due to the first theorem, Theorems2 and 3 present the globally stable adaptive system by tuning theconsequence parameters of all blocks of the HFS control system.Theorem 2 is applicable when the control gain is unity whileTheorem 3 is used for the non-unity control gain systems. General-ization of the theorems to any structure of hierarchical fuzzysystem and even to ordinary fuzzy systems are expressed as twocorollaries.

It is important to notice that the proposed VSHF control systemdescribed in this paper differs from those described in theliterature from these aspects:

(i) For the first time, the HFC system is combined with thevariable structure control. Also, its nonlinearly appearingadjustable parameters are tuned using adaptation lawsachieved by the Lyapunov approach.

(ii) There is an extra term in the Lyapunov approach to overcomethe bounded disturbances of nonlinear systems.

(iii) In order to gain soft switching system, it is considered as afuzzy system.

Finally, the proposed control system is applied for trackingcontrol and regulation of three nonlinear systems; three withunity control gain, and the other with non-unity control gain.Results of simulations demonstrate the presented control algo-rithm is capable to control nonlinear systems with completelyunknown functions in presence of bounded disturbances. Also,comparison of the results and the method in [8] shows theperformance of proposed method. Moreover, in comparison withthe ordinary fuzzy system, tunable parameters are significantlyreduced.

Appendix A

A.1 Fuzzy switching system

The fuzzy switching system is a single input zero-order Takagi-Sugeno fuzzy system with two Gaussian membership functions,singleton fuzzifier, sum-product inference engine, and centeraverage defuzzifier. Its input considers as sliding surface which isa metric such that determines the role of sliding surface and HFSin the control of the system. Fig. A.1 shows the structure of fuzzyswitching system.

A.2 Rule base of HFS of Example 1

In the Tables A.1–A.3 the rule base of fuzzy systems in thelayers of HFS in the first example are expressed. In this tables the

Ai;Bi;Ci;Di; Ei; Fi f or i¼ 1; ::;3 are the Gaussian membership func-tions of z1; z2; z3; z4; y1; y2 respectively where y1; y2 are intermedi-ate variables due to outputs of fuzzy systems located in the firstand second layer of HFS.

FISs m

-6 -4 -2 0 2 4 60

0.2

0.4

0.6

0.8

1

S

µ(s)

N P

Fig. A.1. Schematic of fuzzy switching system (a) Membership functions of the input (b) Block diagram of fuzzy switching system..

Table A.1Rule base of first layer.

Number of rule Antecedent part Consequence part

1 If z1 is A1 ;&z2 is B1 Then y1 ¼ θ112 If z1 is A1 ;&z2 is B2 Then y1 ¼ θ213 If z1 is A1 ;&z2 is B3 Then y1 ¼ θ314 If z1 is A2 ;&z2 is B1 Then y1 ¼ θ415 If z1 is A2 ;&z2 is B2 Then y1 ¼ θ516 If z1 is A2 ;&z2 is B3 Then y1 ¼ θ617 If z1 is A3 ;&z2 is B1 Then y1 ¼ θ718 If z1 is A3 ;&z2 is B2 Then y1 ¼ θ819 If z1 is A3 ;&z2 is B3 Then y1 ¼ θ91

Table A.2Rule base of second layer.


1 If y1 is E1;&z3 is C1 Then y2 ¼ θ122 If y1 is E1;&z3 is C2 Then y2 ¼ θ223 If y1 isE1 ;&z3 is C3 Then y2 ¼ θ324 If y1 is E2;&z3 is C1 Then y2 ¼ θ425 If y1 is E2;&z3 is C2 Then y2 ¼ θ526 If y1 is E2;&z3 is C3 Then y2 ¼ θ627 If y1 is E3;&z3 is C1 Then y2 ¼ θ728 If y1 is E3;&z3 is C2 Then y2 ¼ θ829 If y1 is E3;&z3 is C3 Then y2 ¼ θ92

Table A.3Rule base of third layer.


1 If y2 is F1 ;&z4 is D1 Then y3 ¼ θ132 If y2 is F1 ;&z4 is D2 Then y3 ¼ θ233 If y2 is F1 ;&z4 is D3 Then y3 ¼ θ334 If y2 is F2 ;&z4 is D1 Then y3 ¼ θ435 If y2 is F2 ;&z4 is D2 Then y3 ¼ θ536 If y2 is F2 ;&z4 is D3 Then y3 ¼ θ637 If y2 is F3 ;&z4 is D1 Then y3 ¼ θ738 If y2 is F3 ;&z4 is D2 Then y3 ¼ θ839 If y2 is F3 ;&z4 is D3 Then y3 ¼ θ93






A.3 Simulation result of applying conventional fuzzy system to thefirst example

In this section, using Corollary 1 the conventional fuzzy systemis replaced with HFS in the proposed control structure in order toshow the comprehensiveness of the suggested method. The resultis shown in Fig. A.2. As it is shown in Fig. B.1 with considering allthe initial values and parameters of controller and the flexiblejoint system as Example 1, the result is exactly the same. The onlydifference is in the number of rules which reduce from 81 rule inconventional fuzzy system to 27 rule in HFS.

References

[1] Raju GVS, Zhou J, Kisner RA. Hierarchical fuzzy control. Int J Control1991;54(5):1201–16.

[2] Wang LX. Universal approximation by hierarchical fuzzy systems. Fuzzy SetSyst 1998;93(2):223–30.

[3] Wang LX. Analysis and design of hierarchical fuzzy systems. IEEE Trans FuzzySyst 1999;7(5):617–24.

[4] Wei C, Wang LX. A note on universal approximation by hierarchical fuzzysystems. Inform Sci 2000;123(3–4):241–8.

[5] Joo MG, Lee JS. Universal approximation by hierarchical fuzzy system withconstraints on the fuzzy rule. Fuzzy Set Syst 2002;130(2):175–88.

[6] Torra V. A. Review of the construction of hierarchical fuzzy systems. Int J IntelSyst 2002;17(5):531–43.

[7] Wang D, Zeng XJ, KeaneJ A. A survey of hierarchical Fuzzy systems. Int J CompCogn 2006;4:18–29.

[8] Armendáriza M, Olveraa C, Rodríguezb F, Rubioa E. Indirect hierarchical FCMACcontrol for the ball and plate system. Neurocomputing 2010;73(13–15):2454–63.

[9] Joo MG, Sudkamp T. A method of converting a fuzzy system to a two-layeredhierarchical fuzzy system and its run-time efficiency. IEEE Trans Fuzzy Syst2009;17(1):93–103.

[10] Hsum IH, Lee LW. A hierarchical structure of observer-based adaptive fuzzy-neural controller for MIMO systems. Fuzzy Set Syst 2011;185(1):52–82.

[11] Emara H, Elshafei A. Robust robot control enhanced by a hierarchical adaptivefuzzy algorithm. Eng Appl Artif Intel 2004;17(2):187–98.

[12] Sun G, Huo W. Adaptive fuzzy predictive control of satellite attitude based onhierarchical fuzzy systems. In: Proceedings of the international conference onintelligent system design and engineering application (ISDEA); 13–14 Oct2010, p. 208–11.

[13] Masmoudi NK, Rekik C, Djemel M, Derbel N. Optimal control for discrete largescale nonlinear systems using hierarchical fuzzy systems. In: Proceedings ofthe 2010s international conference on machine learning and computing(ICMLC), 9–11 Feb 2010. p. 91–5.

[14] Zhou YP, Fang JA. Intrusion detection model based on hierarchical fuzzyinference system. In: Proceedings of the second international conference oninformation and computing science ICIC’09; 21-22 May 2009. p. 144–7.

[15] Hu Y, Chiou A, Qinglong H. Hierarchical fuzzy logic control for multiphasetraffic intersection using evolutionary algorithms. In: Proceedings of the IEEEInternational conference on industrial technology ICIT 2009; 10–13 Feb 2009.p. 1–6.

[16] Naghibi S, Teshnehlab M, Shoorehdeli MA. Breast cancer classification based onadvanced multi dimensional fuzzy neural network. J Med Syst 2012;36(5):13–20.

[17] Ho HF, Wong YK, Rad AB. Adaptive fuzzy approach for a class of uncertainnonlinear systems in strict-feedback form. ISA Trans 2008;47(3):286–99.

[18] Farivar F, Shoorehdeli AM. Fault tolerant synchronization of chaotic heavysymmetric gyroscope systems versus external disturbances via Lyapunov rule-based fuzzy control. ISA Trans 2012;51(1):50–64.

[19] Kahkeshi MS, Sheikholeslam F, Zekri M. Design of adaptive fuzzy waveletneural sliding mode controller for uncertain nonlinear systems. ISA Trans2013;52(3):342–50.

[20] Liu YJ, Zhou N. Observer-based adaptive fuzzy-neural control for a class ofuncertain nonlinear systems with unknown dead-zone input. ISA Trans2010;49(4):462–9.

[21] Boulkroune A, Bounar N, M'Saaad M, Farza M. Indirect adaptive fuzzy controlscheme based on observer for nonlinear systems: a novel SPR-filter approach.Neurocomputing 2014;135:378–87.

[22] Li Y, Li T, Jing X. Indirect adaptive fuzzy control for input and outputconstrained nonlinear systems using a barrier Lyapunov function. Int J AdaptControl 2014;28(2):184–99.

[23] Sanner RM, Slotine E. Gaussian networks for direct adaptive control. IEEETrans Neural Netw 1992;3(6):837–63.

[25] Slotine E, Coetsee JA. Adaptive sliding controller synthesis for nonlinearsystems. Int J Control 1986;43(6):1631–51.

[26] Khanesar MA, Teshnehlab M, Kaynak O. Direct model reference Takagi–Sugeno fuzzy control of SISO nonlinear systems. IEEE Trans Fuzzy Syst2011;19(5):914–24.

[27] Farrell JA, Polycarpou MM. Adaptive approximation based control: Unifyingneural, fuzzy and traditional adaptive approximation approaches, vol. 48. JohnWiley & Sons; 2006.

[28] Zahiripour SA, Jalali AA. Designing an optimal proportional-integral slidingsurface for a quarter car active suspension system with suspension compo-nents possessing uncertain constants and nonlinear characteristics.NASHRIYYAH-I MUHANDISI-I BARQ VA MUHANDISI-I KAMPYUTAR-I2012;10:2.

[29] Hwang CL, Chiang CC, Yeh YW. Adaptive fuzzy hierarchical sliding-modecontrol for the trajectory tracking of uncertain underactuated nonlineardynamic systems. IEEE Trans Fuzzy Syst 2014;22(2):286–99.

[30] Zhang X, Ke S. The control action of the periodic perturbation on ahyperchaotic system. Acta Physica Sinica [overseas edition] 1999;8:651–6.

0 5 10 15 20-1

-0.5

0

0.5

1

1.5

Time

y

yydesire

Fig. A.2. Tracking of sinusoidal signal for ordinary adaptive fuzzy system; outputy (‘…’) follows ydesire (‘_’)..



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