State and output feedback fuzzy variable structure controllers for multivariable nonlinear systems...

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The International Journal ofAdvanced Manufacturing Technology ISSN 0268-3768Volume 71Combined 1-4 Int J Adv Manuf Technol (2014)71:539-556DOI 10.1007/s00170-013-5453-4

State and output feedback fuzzy variablestructure controllers for multivariablenonlinear systems subject to inputnonlinearities

Abdesselem Boulkroune, MohammedMsaad & Mondher Farza

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ORIGINAL ARTICLE

State and output feedback fuzzy variable structure controllersfor multivariable nonlinear systems subjectto input nonlinearities

Abdesselem Boulkroune & Mohammed Msaad &

Mondher Farza

Received: 9 November 2011 /Accepted: 16 October 2013 /Published online: 1 December 2013# Springer-Verlag London 2013

Abstract This paper presents three fuzzy adaptive controllersfor a class of uncertain multivariable nonlinear systems withboth sector nonlinearities and dead zones: two first controllersare state feedbacks and the last controller is an output feed-back. The design of the first controller concerns systems withsymmetric and positive definite control–gainmatrix, while thesecond control design is extended to the case of nonsymmetriccontrol–gain matrix thanks to an appropriate decomposition,namely the product of a symmetric positive definite matrix, adiagonal matrix with diagonal entries +1 or −1, and a unityupper triangular matrix. The third controller is an outputfeedback extension of the second controller. In this controller,a high-gain observer is incorporated to estimate theunmeasurable states. An appropriate adaptive fuzzy logicsystem is used to reasonably approximate the uncertainfunctions. A Lyapunov approach is adopted to derive theparameter adaptation laws and prove the stability of thosecontrol systems as well as the exponential convergence oftheir underlying tracking errors within an adjustable region.The effectiveness of the proposed fuzzy adaptive controllersis illustrated through simulation results.

Keywords Adaptive fuzzy control . Variable structurecontrol . High-gain observer . Multivariable nonlinearsystems . Sector nonlinearity . Dead zone

1 Introduction

The design of robust adaptive controllers for multivariableunknown nonlinear systems remains one of the most chal-lenging tasks in the area of control systems. Based on theuniversal approximation theorem [1], some adaptive fuzzycontrol systems [2–11] have been developed for a class ofmultivariable nonlinear uncertain systems. The stability of theunderlying control systems has been investigated using aLyapunov approach. The robustness issues with respect tothe approximation error and external disturbances have beenenhanced by appropriately modifying the available adaptivefuzzy controllers. The cornerstone of such a modificationconsists in a robust compensator which is conceived using asliding mode control design [3, 5, 6, 8, 10, 11] or anH∞-basedrobust control design [2, 4, 7, 9]. A key assumption in theavailable fuzzy adaptive control systems [2–11] is that theactuator dynamics may be reasonably approximated by alinear model. This is more an exception than a rule in theengineering practice.

The control problem of uncertain multivariable systemswith nonlinear input channels has received a remarkable at-tention because of those ubiquitous actuator nonlinearities,namely saturation, quantization, backlash, dead zone, and soon [12, 13]. It is well known that the existence of inputnonlinearities may lead to poor performance or even instabil-ity of the control system. It is thereby more advisable to takeinto account the actuator nonlinearities in the control design aswell as the stability analysis. Decentralized variable structurecontrollers have been proposed in [14–16] for a class ofsystems with input sector nonlinearities and/or dead zones.

A. Boulkroune (*)Department of Automatic, Faculty of Engineering Sciences,Jijel University, BP. 98, Ouled-Aissa 18000, Jijel, Algeriae-mail: [email protected]

M. Msaad :M. FarzaGREYC, UMR 6072 CNRS, Université de Caen, ENSICAEN,6 Bd Maréchal Juin, 14050 Caen Cedex, France

M. Msaade-mail: [email protected]

M. Farzae-mail: [email protected]

Int J Adv Manuf Technol (2014) 71:539–556DOI 10.1007/s00170-013-5453-4

Author's personal copy

In [13, 17], the authors designed sliding-mode control systemsfor nonlinear multivariable systems subject to both sectornonlinearities and dead zones. The underlying results sufferfrom two fundamental limitations. Firstly, the considered classof systems is relatively reduced. Secondly, the gain reductiontolerances of the nonlinear dead zones and upper bounds ofuncertain nonlinear functions are required to be known.

More recently, adaptive neural or fuzzy control systemshave been respectively proposed for a particular class ofmultivariable nonlinear systems with unknown dead zonesand gain signs in [18] and [19]. These contributions sufferfrom two restrictive modelling assumptions motivated bytechnical purposes regarding the stability analysis and controldesign. The first one consists in assuming a lower triangularstructure for the system under control while the second oneconcerns the boundedness of the high-frequency controlgains. Moreover, an adaptive fuzzy control for a class ofmultivariable nonlinear systems with unknown dead zoneshas been designed in [20]. Note that, in [18–20], simple deadzones having linear and nonlinear functions outside the deadband have been considered.

In this paper, one aims at designing three fuzzy adaptivecontrollers for three different classes of uncertain nonlinearmultivariable systems containing both sector nonlinearitiesand dead zones, namely:

– Multiple input, multiple output (MIMO) systems witha symmetric positive definite control–gain matrix andmeasurable states

– MIMO systems with a nonsymmetric control–gainmatrixhaving non-zero-leading principal minors andmeasurablestates, and

– MIMO systems with a nonsymmetric control–gainmatrixhaving non-zero-leading principal minors but with statesnot being available for measurement.

Though this work borrows from the available results, itpresents a fundamental contribution to the fuzzy adaptivecontrol of uncertain multivariable nonlinear systems from anapplicability point of view. The main contributions of thispaper are emphasized below:

(a) Unlike in contributions [13, 17], the class of the consid-ered systems is relatively larger and the gain reductiontolerances of the nonlinear dead zones and the upperbounds on uncertain nonlinear functions are not assumedto be known. These bounds are indeed estimated usingadaptive fuzzy systems.

(b) And compared with contributions [18–20], there are threefeatures that are of practical interest. Firstly, the consideredclass of systems is larger as the modelling assumptionsmade in [18, 19] are relatively restrictive, namely lowertriangular control structure with bounded high-frequencycontrol gains . Such modelling requirements are mainly

motivated by stability analysis and control design pur-poses. Secondly, the considered model of the input non-linearity includes sector nonlinearities and dead zones andis hence relatively larger than the one considered in[18–20]. Thirdly, the nonsymmetric control–gain matrixis appropriately decomposed into a product of a symmet-ric definite positive matrix, a diagonal matrix with +1 or−1 on the diagonal and a unity upper triangular matrix. Itis worth noticing that the diagonal matrix elements arenothing than the ratios of the signs of the leading principalminors of the control–gain matrix.

(c) Moreover, unlike in the references [13, 17–20], in ourthird controller, the system states are assumed to beunknown. In fact, a high-gain observer is designed toestimate those missing states.

2 Notation and problem statement

Consider the following class of nonlinear multivariable sys-tems described by:

y r1ð Þ1 ¼ f 1 xð Þ þ

Xj¼1

p

g1 j xð ÞΦ j u j

� �;

⋮

yrpð Þ

p ¼ f p xð Þ þXj¼1

p

gpj xð ÞΦ j u j

� �; ð1Þ

where xi ¼ xi1; xi2;…; xiri½ �T ¼ yi; y i;…; y ri−1ð Þi

h iT, for i =1,

…, p , is the state vector of the subsystem i , x =[x1T, x 2

T,…,xp

T]T∈Rr is the overall state vector with r =r1+…+rp,u =[u 1,…,up]

T ∈Rp is the control input vector, y =[y 1,…,yp]

T∈Rp is the output vector, f i(x ) and gij(x ),i ,j =1,…,p are unknown continuous nonlinear functions, andΦ (u )=[Φ1(u1),Φ2(u2),…,Φp(up)]

T is a nonlinear input func-tion vector satisfying some properties which will be given inSection 2.2.

Let us denote

y rð Þ ¼ y r1ð Þ1 …y

rpð Þp

� �T; F xð Þ ¼ f 1 xð Þ… f p xð Þ

h iT;

G xð Þ ¼g11 xð Þ … g1p xð Þ⋮ ⋱ ⋮

gp1 xð Þ … gpp xð Þ

24 35Then, the system (1) can be rewritten in the following

compact form:

y rð Þ ¼ F xð Þ þ G xð ÞΦ uð Þ ð2Þ

where F (.)∈Rp and G(.)∈Rp×p.

540 Int J Adv Manuf Technol (2014) 71:539–556


Remark 1 The system (1) represents a class of MIMO non-linear systems with input nonlinearities Φ(u ) (i.e., with deadzone and sector nonlinearities). It is worth noticing that theinput nonlinearities exist widely in the practical control sys-tems, such asmechanical connections, hydraulic servo-valves,piezo-positioners, and biomedical systems, but they are gen-erally neglected in the control design for simplicity purposes.Also, the MIMO system (1) without input nonlinearities hasbeen considered by many literatures [2–11] and is of higherpractical significance. Note that many practical systems can berepresented in the form of the MIMO nonlinear systems (1)such as robotic systems, electrical machines, mechanicalsystems, and chaotic systems.

The objective of this paper is to design a stable adaptivecontrol system allowing the system output vector y to followsa specified desired trajectory yd=[yd1,…, ydp]

T∈Rp. We

assume that the vector xd ¼ yd1; y d1;½ …; y r1−1ð Þd1 ; y r1ð Þ

d1 ;…; ydp;

y dp;…; yrp−1ð Þ

dp ; yrpð Þ

dp �T is continuous, bounded, and available

for measurement. Then, xd ∈Ωxd ⊂ R rþp withΩxd is a knowncompact set.

Let us define the tracking error as

e1 ¼ y1−yd1⋮

ep ¼ yp−ydpð3Þ

and the filtered tracking error as

S ¼ S1;…; Sp� �T ð4Þ

with

Si ¼ d

dtþ λi

� �ri−1ei for λi > 0; ∀i ¼ 1;…; p ð5Þ

Then, one can rewrite Eq. (5) as follows

Si ¼ λ ri−1i ei þ ri−1ð Þλ ri−2

i e i þ⋯þ ri−1ð Þλieri−2ð Þi þ e ri−1ð Þ

i

with i ¼ 1;…; p ð6Þ

Notice that if one chooses λ i > 0, with i = 1,…,p , then the

roots of the polynomialHi sð Þ ¼ λ ri−1i þ ri−1ð Þλ ri−2

i sþ⋯þri−1ð Þλisri−2 þ sri−1 related to the characteristic equation ofSi=0 are all in the open left-half plane [21].

The relation (6) can be rewritten in the following compactform

Si ¼ CTi Ei ð7Þ

with

Ei ¼ ei e i…e ri−2ð Þi e ri−1ð Þ

i

h iTð8Þ

CTi ¼ λ ri−1

i ri−1ð Þλ ri−2i … ri−1ð Þλi 1

� � ð9Þ

Consequently, the vector S takes the form:

S ¼ CTE ð10Þ

where

CT ¼ diag CT1C

T2 … CT

p

h iðp� rÞ

ð11Þ

E ¼ ET1E

T2 … ET

p

h iTr�1ð Þ

ð12Þ

And the dynamic of Si is described by:

S⋅i ¼ CT

riEi þ e rið Þi ; and i ¼ 1;…; p ð13Þ

with

CTri ¼ 0 λ ri−1

i ri−1ð Þλ ri−2i … 0:5 ri−1ð Þ ri−2ð Þλ2

i ri−1ð Þλi

� �ð14Þ

The dynamic of S can be written into the followingcompact form

S⋅ ¼ CT

r E þ e rð Þ ð15Þ

where

CTr ¼ diag CT

r1CTr2 … CT

rp

h ip�rð Þ

ð16Þ

e rð Þ ¼ e r1ð Þ1 e r2ð Þ

2 … erpð Þ

p

� �Tð17Þ

with

e rð Þ ¼ y rð Þ−y rð Þd ð18Þ

where y rð Þ ¼ y r1ð Þ1 y r2ð Þ

2 … yrpð Þ

p

� �Tis previously defined and

y rð Þd ¼ y r1ð Þ

d1 y r2ð Þd2 … y

rpð Þdp

� �T; ð19Þ

From (18), one can write (15) as follows

S⋅ ¼ CT

r E þ y rð Þ−y rð Þd ð20Þ

Int J Adv Manuf Technol (2014) 71:539–556 541


Thereafter, (20) will be used in the development of theproposed controllers and the stability analysis.

2.1 Description of the fuzzy logic system

In this paper, a zero-order Takagi–Sugeno fuzzy system will beused to approximate unknown nonlinear functions. The basicconfiguration of a fuzzy logic system consists of a fuzzifier, afuzzy inference engine, and a defuzzifier, as shown in Fig. 1.The fuzzy inference engine uses the fuzzy IF-THEN rules toperform a mapping from an input vector xT ¼ x1; x2;…; xn½ �∈Rn to an output fb∈R . The i th fuzzy rule is written as

R ið Þ : if x1 is Ai1 and…and xn is A

in then bf is f i ð21Þ

where A1i ,A2

i ,…,and Ani are fuzzy sets and f i is the fuzzy

singleton for the output in the i th rule. By using the singletonfuzzifier, the product inference and the center-averagedefuzzifier, the output of the fuzzy system can be expressedas follows:

bf x�

¼Xm

i¼1f i ∏n

j¼1μAijx j� ��

Xm

i¼1∏n


¼ θTy x� ð22Þ

where μAijx j� �

is the degree of membership of x j to Aji, m is

the number of fuzzy rules, θT=[ f1, f2,…, f m] is the adjustableparameter vector (composed of consequent parameters) andyT=[y1 y2 … ym] with

y i x�

¼∏n


Xm

i¼1∏n

j¼1μAijx j� �� ð23Þ

being the fuzzy basis function (FBF ). Throughout thepaper, it is assumed that the FBFs are selected so thatthere is always at least one active rule [1, 22–24], i.e.,

∑mi¼1 ∏n


> 0:

It is worth noting that the fuzzy system (22) is commonlyused in control applications. Following the universal approx-imation results, the fuzzy system (22) is able to approximateany nonlinear smooth function on a compact operating spaceto an arbitrary degree of accuracy [1]. Of particular impor-tance, it is assumed that the structure of the fuzzy system,namely the pertinent inputs, the number of membership func-tions for each input and the number of rules, and the mem-bership function parameters are properly specified before-hand. The consequent parameters θ are then determined byappropriate parameter adaptation algorithms.

2.2 Input nonlinearity

The mathematical model of the input nonlinearity (i.e., thesector nonlinearity and dead zone) under consideration isdescribed by

Φi uið Þ ¼fiþ uið Þ ui − uiþð Þ; ui > uiþ;0; −ui− ≤ ui ≤uiþ;fi− uið Þ ui þ ui−ð Þ; ui < − ui−;

8<: ð24Þ

where f i+(ui)>0 and f i−(ui)>0 are nonlinear functions of ui,and ui+>0 and ui−>0 are known constants. The involvedmodelling assumption is

Assumption 1 The input nonlinearity Φ i(u i) satisfies thefollowing important properties:

ui − uiþð ÞΦi uið Þ≥m�iþ ui − uiþð Þ2 ; for ui > uiþ;

ui þ ui−ð ÞΦi uið Þ≥m�i− ui þ ui−ð Þ2; for ui < −ui−;

ð25Þ

where mi+* and mi−

* are unknown constants which are called“gain reduction tolerances” .

It is worth mentioning that the models (24)–(25) allows toconsider both dead zones and sector nonlinearities with re-duced prior knowledge. Indeed, the gain reduction tolerancesm+

* and m−* are unknown, unlike in [13] to [17], and the input

nonlinearity is only characterized throughout the property (25)together with the knowledge of the constants u+ and u−. Bythe way, notice that the dead zone considered in the contribu-tion [19, 20] is a particular case of the above general form.

3 Design of state feedback fuzzy adaptive controllers

In this section, two state feedback fuzzy adaptive variablestructure control designs are proposed up to an assumptionon the structure of the control–gain matrix. In the first one, thecontrol–gain matrix is assumed to be symmetric and positivedefinite. Such an assumption is relaxed in the second designwhere the control–gain matrix may not be symmetric and

Fuzzy Rule Base

Fuzzifier Defuzzifier

Fuzzy Inference Engine

x )(ˆ xf

Fig. 1 The basic configuration of a fuzzy logic system



positive definite provided that the signs of its leading principalminors are known.

Here, for both controllers, we assume that the system outputsyi, i=1,…,p , and its j th derivatives (for j =1,…, ri−1) areavailable for measurements, i.e., all state information areavailable.

3.1 The first state feedback fuzzy adaptive controller

In the following, we present a fuzzy adaptive variable–struc-ture controller for the class of system (2) under the followingassumption on the control–gain matrix.

Assumption 2 The control–gain matrix is symmetric, positivedefinite, and of class C1 with the following property

∂gij xð Þ=∂y ri−1ð Þi ¼ 0 ∀i ¼ 1; 2;…; p and j ¼ 1; 2;…; p:

Such an assumption is not too restrictive in nature as thereare many physical mechanical and electrical systems whosedynamics can be described by Eq. (2) with symmetric, positivedefinite, and class C1 control–gain matrix [25]. The positivedefinite property of the control–gain matrix, which is closelyrelated to the system controllability, has been already requiredin recent fundamental contributions on adaptive (fuzzy orneural) control of multivariable systems. Moreover, the requiredproperty on the partial derivatives of the control–gain matrixensures that the time derivative of its inverse, i.e., dG−1(x)/dt ,depends only on the state vector x (i.e., it ensures that dG−1(x)/dt does not depend on the system input u). Notice that thefollowing fundamental results could be adapted, up to someappropriate modifications, to the case where the control–gainmatrix is symmetric and negative definite.

By substituting (2) into (20), the dynamics of S become

S⋅ ¼ CT

r E þ F xð Þ þ G xð ÞΦ uð Þ − y rð Þd ð26Þ

Letting G1(x )=G−1(x ), one has

G1 xð Þ S⋅ ¼ G1 xð Þ CTr E − y rð Þ

d þ F xð Þh i

þ Φ uð Þ ð27Þ

For stability analysis and control design simplicity purposes,it is more advisable to rewrite the dynamics of S as follows

1

2G⋅1 xð ÞS þ G1 xð Þ S⋅ ¼ 1

2G⋅1 xð ÞS þ G1 xð Þ CT

r E−yrð Þd þ F xð Þ

h iþ Φ uð Þ

¼ α x; v; Sð Þ þ Φ uð Þ ð28Þ

with

α x; v; Sð Þ ¼ α1 x; v; Sð Þ;α2 x; v; Sð Þ;…;αp x; v; Sð Þ� �T¼ 1=2ð ÞG⋅1 xð ÞS þ G1 xð Þ vþ F xð Þ½ �

and

v ¼ CTr E − y rð Þ

d

This form makes it possible to introduce a further assumptionthat has to be made to get the fundamental result we areconcerned by, namely

Assumption 3 There exists an unknown continuous positivefunction αi xð Þ such that αi x; v; Sð Þj j≤ηαi xð Þ ∀x ∈Ω x⊂Rr

with η ¼ mini

ηif g where η i=min{mi+* ,mi−

* }

Notice that such an assumption is not restrictive sinceαi xð Þ is assumed to be unknown. Moreover, since v andS are functions of (x , x d), xd∈L∞ and α i(x ,v,S ) is acontinuous function, such a function always exists. Theunknown continuous nonlinear function αi xð Þ can behence approximated, on a compact set Ω x , by the fuzzysystems (22) as follows:

bαi x; θð Þ ¼ θTi y i xð Þ ð29Þ

where y i(x ) is the FBF vector, which is fixed a priori by thedesigner, and θ i is the adjustable parameter vector of the fuzzysystem (composed of consequent parameters). Its adaptivelaw will be designed later.

According to [1, 26, 27], the unknown continuous functionαi xð Þ can be optimally approximated as

α��i xð Þ ¼ bαi x; θ*i

� �þ δi xð Þ

¼ θ�Ti y i xð Þ þ δi xð Þð30Þ

where δ i(x ) is the fuzzy approximation error and θ i* is the

optimal parameter vector defined by

θ�i ¼ arg minθi

supx∈Ωx

αi xð Þ − αbi x; θið Þj i"

ð31Þ

Note that this vector is mainly introduced for analysispurposes as its value is not needed when implementingthe controller [28–31]. As in the open literature [1,25–27], one assumes that the used fuzzy systems donot violate the universal approximator property on thecompact set Ω x , which is assumed large enough so thatthe input vector of the fuzzy system remains in Ω x forthe control system. It is hence reasonable to assume thatthe fuzzy approximation error is bounded for all x ∈Ω x ,i.e.,

δi xð Þj j≤δi; ∀x ∈ Ωx;



where δi is an unknown constant. From the above analysis,one has

bαi x; θið Þ−αi xð Þ ¼ bαi x; θið Þ−bαi x; θ�i

� �þ bαi x; θ�i

� �−αi xð Þ;

¼ bαi x; θið Þ−bαi x; θ�i

� �−δi xð Þ;

¼ θ~ Ti y i xð Þ−δi xð Þ ð32Þ

whereeθi ¼ θi−θ�i is the parameter estimation error.The following fuzzy adaptive variable structure controller

is proposed to achieve the control objective.

ui ¼−ρi tð Þsign Sið Þ−ui−; Si > 0;0; Si ¼ 0;−ρi tð Þsign Sið Þ þ uiþ; Si < 0;

8<: ð33Þ

with

ρi tð Þ ¼ k0i þ k1i Sij j þ θTi y i xð Þ; ∀i ¼ 1;…; p

k0i ¼ −γ0iσ0ik0i þ γ0i Sij j for k0i 0ð Þ > 0ð34Þ

θ i ¼ −γ1iσ1iθi þ γ1i Sij jy i xð Þ with θij 0ð Þ > 0 ð35Þ

where γ0i, γ1i, σ0i, σ1i, k1i>0 are design constants, and k0iand θ i are the online estimates of the uncertain terms k�0i ¼ δiand θ i

*, respectively. It is important to remark that if k0i(0)>0and θ ij(0)>0 for i =1,…,p and j =1,…,m , it follows from theadaptive laws (34–35) that their respective solutions satisfyk0i(t )>0 and θ ij(t )>0, for t >0.

Multiplying (28) by ST and using Assumption 3, one has

1

2ηSTG

⋅1 xð ÞS þ 1

ηSTG1 xð Þ S⋅ ¼ 1

ηSTα x; v; Sð Þ þ 1

ηSTΦ uð Þ

≤Xi¼1

p

Sij jαi xð Þ þ 1

ηSTΦ uð Þ

ð36Þ

And using (32) and (36) yields

1

2ηSTG

⋅1 xð ÞS þ 1

ηSTG1 xð Þ S⋅ ≤

Xi¼1

p

Sij jαi xð Þ þ 1

ηSTΦ uð Þ

≤−Xi¼1

p

Sij jk~0i −Xi¼1

p

Sij jθ~Ti y i xð Þ þ

Xi¼1

p

Sij jk0i

þXi¼1

p

Sij jθTi y i xð Þ þ 1

ηSTΦ uð Þ ð37Þ

where eθi ¼ θi − θ�i and ek0i ¼ k0i − k�0i ¼ k0i − δi:This allows to state the fundamental result concerning the

first fuzzy adaptive control system.

Theorem 1 Consider the system (2) subject to Assumptions1–3. Then, the control law defined by Eqs. (33) to (35)guarantees the following properties:

& All the variables involved in the closed-loop system areuniformly ultimately bounded (UUB).

& The filtered tracking errors S i of the control system expo-nentially converge to an adjustable domain defined by:

ΩSi ¼ Sij Sij j≤ 2ησg1

π1

μ1

� �1=2( )

where π1,μ1 and σg1 will be defined later.

Proof Let us consider the following Lyapunov functioncandidate:

V ¼ 1

2ηSTG1 xð ÞS þ 1

2

Xi¼1

p 1

γ0iek20i þ 1

2

Xi¼1

p 1

γ1ieθTi eθi: ð38Þ

Its time derivative is given by

V⋅ ¼ 1

ηSTG1 xð Þ S⋅ þ 1

2ηSTG

⋅1 xð ÞS þ

Xi¼1

p 1

γ0iek0ik⋅0i þX

i¼1

p 1

γ1ieθTi θ⋅ið39Þ

One can easily show from (25) and (33) that

ui < −ui− for Si > 0 ⇒ ui þ ui−ð ÞΦi ui� �

¼ m�i− ui þ ui−ð Þ2≥η ui þ ui−ð Þ2

and

ui > uiþ for Si < 0 ⇒ ui � uiþð ÞΦi uið Þ

¼ m�iþ ui � uiþð Þ2≥η ui � uiþð Þ2

From this analysis and (33), one can also conclude that forSi>0 one has

ui þ ui−ð ÞΦi ui� � ¼ −ρi tð Þsign Sið ÞΦi uið Þ≥m�

i−ρ2i tð Þ sign Sið Þ½ �2≥ηρ2i tð Þ;

ð40Þ

and for Si<0 one has

ui−uiþð ÞΦi ui� � ¼ −ρi tð Þsign Sið ÞΦi uið Þ≥m�

iþρ2i tð Þ sign Sið Þ½ �2≥ηρ2i tð Þ;

ð41Þ

Then, for Si<0 and Si>0, one has

−ρi tð Þsign Sið ÞΦi uið Þ≥ηρ2i tð Þ ð42Þ



Using (42) and the fact that Si2>0 and Sisign(Si)=|Si| yields

−ρi tð ÞS2i sign Sið ÞΦi uið Þ≥ηρ2i tð ÞS2i¼ ηρ2i tð Þ Sij j2

ð43Þ

Finally, because ρ i(t)>0, thus for all Si one has

SiΦi uið Þ≤−ηρi tð Þ Sij j ð44Þ

And using the expressions (34), (35), (44), and (37), (39)becomes

V⋅≤Xi¼1

p

Sij jk0i þXi¼1

p

Sij jθTi y i xð Þ þ 1

η

Xi¼1

p

SiΦi uið Þ

−Xi¼1

p

σ0ik~0ik0i −

Xi¼1

p

σ1iθ~iTθi≤

Xi¼1

p

Sij jk0i þXi¼1

p

Sij jθTi y i xð Þ

þXi¼1

p

−ρi tð Þ Sij j−Xi¼1

p

σ0ik~0ik0i−

Xi¼1

p

σ1iθ~iTθi ¼ −

Xi¼1

p

k1iS2i

−Xi¼1

p

σ0ik~0ik0i −

Xi¼1

p

σ1iθ~iTθi ð45Þ

One can henceforth easily check that

−σ0ik~0ik0i≤−

σ0i

2k~0i2 þ σ0i

2k�20i

−σ1iθ~iTθi≤−

σ1i

2eθi 2 þ σ1i

2θ�i

2And using the previous inequalities, (45) becomes

V⋅≤−

Xi¼1

p

k1iS2i −Xi¼1

p σ0i

2k~0i2−Xi¼1

p σ1i

2eθi 2 þX

i¼1

p σ0i

2k�20i

þXi¼1

p σ1i

2θ�i

2 ð46Þ

Taking into account the Assumption 2, there exists a pos-itive scalar σgo such that G (x )≥σg0Ip and henceforth

STG−1 xð ÞS ¼ STG1 xð ÞS≤ 1

σg0Sk k2 ð47Þ

And using (46) and (47) yields

V⋅≤−μ1V þ π1 ð48Þ

with

π1 ¼Xi¼1

p σ0i

2k�20i þ

Xi¼1

p σ1i

2θ�i

2

where μ1 ¼ minfmini

2ησg0k1i� �

;mini

γ0iσ0if g;mini

γ1iσ1if gg .

Multiplying (48) by eμ1t yields

d

dtVeμ1tð Þ≤π1e

μ1t ð49Þ

And integrating (49) over [0,t ], one has

0≤V tð Þ≤ π1

μ1þ V 0ð Þ− π1

μ1

� �e−μ1t ð50Þ

Therefore, all variables of the control system, i.e.,k 0i , θ i , S i , E and x are UUB. And hence the input u i

is bounded.With bearing in mind of (38), one can define V (0) as

follows

V 0ð Þ ¼ 1

2ηS 0ð ÞTG1 x 0ð Þð ÞS 0ð Þ þ 1

2

Xi¼1

p 1

γ0ik0i 0ð Þ−k�0i� �2

þ 1

2

Xi¼1

p 1

γ1iθi 0ð Þ−θ�i� �T

θi 0ð Þ−θ�i� � ð51Þ

Since G1(x ) is symmetric and positive definite, i.e., thereexists an unknown positive constant σ g 1 such that:G1(x )≥σg1Ip, it follows from (50) and (38) that

Sij j≤ 2ησg1

π1

μ1þ V 0ð Þ− π1

μ1

� �e−μ1t

� �� 1=2

Then, the solution of S i exponentially converges toa bounded adjustable domain defined as follows ΩSi ¼Sij Sij j≤ 2η

σg1π1μ1

� 1=2� �

. This ends the proof of the theorem.

Remark 2 Since the fuzzy approximation (30) is only guaran-teed with a compact set, the proposed stability results in thispaper are semiglobal in the sense that for any bounded initialstates, there exists a fuzzy controller with sufficient largenumber of fuzzy rules such that the function approximation(30) holds for possible operating region.

Remark 3 According to the definition of μ1 and π1, it can beseen that the size of μ1 depends on the controller designparameters γ0i, γ1i, σ0i, σ1i, and k1i (which must be chosenstrictly positive) and that of μ1 depends on the controllerdesign parameters σ0i and σ1i. It is very clear that if weincrease γ0i, γ1i and k1i and decrease σ0i and σ1i, it will helpto reduce the term (2ηπ1/σg1μ1)

1/2. This implies that thefiltered tracking errors S i and the tracking error E canbe made arbitrary small by appropriately choosing thosedesign parameters.



3.2 The second state feedback fuzzy adaptive controller

The fuzzy adaptive controller presented above is onlyapplicable for nonlinear systems with a symmetric andpositive definite control–gain matrix, e.g., robotic systemsand some electrical machines. Such a property is notsatisfied for several physical systems, e.g., the visualserving and the automotive thermal management systems[32]. In the following, one presents a fuzzy adaptivecontroller in the case where the control–gain matrix isno longer symmetric and positive definite using a suitablematrix decomposition that has been already introduced in[20, 25, 33–35].

3.2.1 Decomposition of the control–gain matrix

If the control–gain matrix is with non-zero leadingprincipal minors [36], it can be always decomposed intothe product of a symmetric positive definite matrix, adiagonal matrix whose elements are +1 or −1 and aunity upper triangular matrix as pointed out by the followingresult.

Lemma 1 Any real matrix G (x )∈Rp×p with non-zero leadingprincipal minors can be decomposed as follows [34, 35]:

G xð Þ ¼ Gs xð ÞDT xð Þ ð52Þ

where Gs(x )∈Rp×p is a symmetric positive definite matrix,D ∈Rp×p is a diagonal matrix whose elements are +1 or −1and T (x )∈Rp×p is a unity upper triangular matrix.

Proof See [34].It is worth noting that the decomposition (52) of the

matrix G (x ) is very useful. The symmetric positive definitematrix Gs(x ) will be particularly exploited in the stabilityanalysis. The unity upper triangular matrix T (x ) allows foralgebraic loop free sequential determination of the controlcomponents. The diagonal elements are nothing than theratios of the signs of the leading principal minors of thecontrol–gain matrix. This implies the following facts whenthe control–gain matrix has non -zero leading principalminors .

& If G (x ) is positive definite, then D =IP,& If G (x ) is negative definite, then D =−Ip, and& IfG (x ) is indefinite, the diagonal elements of the matrixD

are +1 and −1.

3.2.2 Fuzzy adaptive control design

Consider the system (2) subject to Assumption 1 and thefollowing assumption.

Assumption 4 The control–gain matrix is of class C1, hasnon-zero leading principal minors of known signs and sat-isfies the following property

∂gij xð Þ.∂y ri−1ð Þ

i ¼ 0; ∀i ¼ 1; 2;…; p and j ¼ 1; 2;…; p:

Using the matrix composition (52) and Eq. (20), thedynamics of S can be rewritten as follows.

G −1s xð Þ S⋅ ¼ G−1

s xð Þ vþ F xð Þ½ � þ DT xð ÞΦ uð Þ ð53Þ

with

v ¼ CTr E−y

rð Þd :

Letting Gs1(x )=Gs−1(x ) and F1(x ,u )=Gs

−1(x )[v +F(x )]+[DT(x )−D ]Φ(u ), Eq. (53) becomes

Gs1 xð Þ S⋅ ¼ F1 x; uð Þ þ D Φ uð Þ ð54Þ

Now, the following change of the variables for stabilityanalysis and control design simplicity are introduced

S ¼ D−1S ð55Þ

or Si ¼ diiSi as D =DT=D−1 and dii=+1 or −1.This allows to rewrite (54) under the form

Gs2 xð Þ S⋅ ¼ D−1F1 x; uð Þ þ Φ uð Þ ð56Þ

where Gs2(x )=D−1Gs1(x )D . Due to the special forms of the

matrices D and Gs1(x ), Gs2(x ) preserves the importantproperties of the original matrix Gs1(x ) (or Gs(x )). In-deed, one can easily show that Gs2(x ) is also symmetricand positive definite. This property is of fundamentalinterest when investigating the control system stability.

Similarly to the first fuzzy adaptive control system, (56)can be rearranged as follows

Gs2 xð Þ S⋅ þ 1

2G⋅s2S ¼ α z

� þ Φ uð Þ ð57Þ

where α zð Þ ¼ α1 z1ð Þ;α2 z2ð Þ;…;αp zp� �� T ¼ D−1F1 x; uð Þþ

12G

⋅s2 xð ÞS , with z ¼ zT1 ; z

T2 ;…; zTp

h iT.

By carefully examining the expression of F1(x ,u) andα zð Þ ,the elements of the vector z can be selected as follows:

z1 ¼ xT ; ST ; u2;…; up� �T

z2 ¼ xT ; ST ; u3;…; up� �T

⋮zp−1 ¼ xT ; ST ; uP

� �Tzp ¼ xT ; ST

� �Tð58Þ



It is clear from the propriety of the matrix of DT (x )−Dthat z1 depends on control inputs u 2,…,up, z2 dependson u 3,…,up, and so on. The nonlinearities α zð Þ has anupper triangular control structure , allowing thereby foralgebraic loop free sequential determination of the controlvariables.

Let us define the following compact sets

Ωzi ¼ xT ; ST ; uiþ1;…; up� �T

x∈Ωx⊂Rr; xd∈Ωxdjn ofor i ¼ 1; 2;…; p−1

and

Ωzp ¼ xT ; ST� �

x∈Ωx⊂Rr; xd∈Ωxdj� �and introduce the following assumption.

Assumption 5 There exists an unknown continuous positivefunction αi zið Þ such that

αi zi� ≤ηαi zi

� ; ∀zi∈Ωzi where η ¼ min

iηif g:

The choice of the vectors zi , i.e., the input argumentsof the unknown functions αi , is not unique. Indeed, sinceS and u are functions of state x and xd, then it can beeasily seen that all vectors zi are functions of x and xd,

e.g., one can choose zi ¼ xT ; xdT½ �T or zi ¼ xT ;ET� �T

fori =1,2,…,p . Moreover, since x d is bounded, one canchoose zi ¼ x .

As in Section 3.1, the unknown continuous function αi zið Þcan be approximated by a fuzzy system as follows

bαi zi; θ�

¼ θTi y i zi�

ð59Þ

where y i zð Þi is the FBF vector, which is fixed a priori by thedesigner, and θ i is the adjustable parameter vector of the fuzzysystem. Furthermore, the functions αi zið Þ can be approximatedoptimally as follows:

αi zi�

¼ bαi zi; θ*i

� þ δi zi

� ¼ θ�Ti y i zi

� þ δi zi

� ð60Þ

and the fuzzy approximation error is assumed to be boundedas usual [1, 26, 27], i.e.,

δi zi� ≤δi; ∀zi∈Ωzi

where δi is an unknown constant.

From the above analysis, one has

bαi zi; θið Þ−αi zið Þ ¼ bαi zi; θið Þ−bαi zi; θ�i

� �þbαi zi; θ�i

� �−αi zið Þ;

¼bαi zi; θið Þ−bαi zi; θ�i

� �−δi zið Þ;

¼ θ~Ti yi zið Þ−δi zið Þ

ð61Þ

The following fuzzy adaptive variable–structure con-troller is proposed to perform the required tracking controlobjective:

ui ¼−ρi tð Þ sign Si

� �−ui−; Si > 0;

0; Si ¼ 0;−ρi tð Þsign Si

� �þ uiþ; Si < 0;

8<: ð62Þ

with

ρi tð Þ ¼ k0i þ k1i Si þ θTi yi zið Þ ∀i ¼ 1;…; p

k⋅0i ¼ −γ0iσ0ik0i þ γ0i Si

; with k0i 0ð Þ > 0ð63Þ

θ⋅i ¼ −γ1iσ1iθi þ γ1i Si

yi zið Þ; with θij 0ð Þ > 0 ð64Þ

where γ0i, γ1i, σ0i, σ1i, k1i>0 are design constants, and k0iand θ i are the online estimates of the uncertain terms k�0i ¼ δiand θ i

*, respectively.

Multiplying (57) by ST

and using Assumption 5 yields

1

2ηSTG⋅s2 xð ÞS þ 1

ηSTGs2 xð Þ S⋅ ¼ 1

ηSTα zð Þ þ 1

ηSTΦ uð Þ

≤Xi¼1

p

Si αi zið Þ þ 1

ηSTΦ uð Þ

ð65Þ

And taking into account (61) and (65), one has

1

2ηSTG⋅s2 xð ÞS þ 1

ηSTGs2 xð Þ S⋅ ≤

Xi¼1

p

Si αi zið Þ þ 1

ηSTΦ uð Þ

≤−Xi¼1

p

Si k

~0i þ θ

~iTyi zið Þ

� �

þXi¼1

p

Si k0i þ θTi yi zið Þ� �þ 1

ηSTΦ uð Þ

ð66Þ

where eθi ¼ θi−θ�i and ek0i ¼ k0i−k�0i ¼ k0i−δi .

Theorem 2 Consider the system (2) subject to Assumptions 1,4 and 5. Then, the control law defined by Eqs. (62) to (64)ensures the following properties:

& All the variables in the closed-loop system are UUB.



& The filtered tracking errors S i of the control system expo-nentially converge to an adjustable domain defined as:


π2

μ2

� �1=2( )

where σg2 satisfies σg2Ip≤Gs2(x ), π2=π1, and

μ2 ¼ min mini

2ησg2k1i� �

;mini

γ0iσ0if g;mini

γ1iσ1if g� �

:

Proof It is carried out using the following Lyapunov functioncandidate

V ¼ 1

2ηSTGs2 xð ÞS þ 1

2

Xi¼1

p 1

γ1ieθTi θi þ 1

2

Xi¼1

p 1

γ0iek 2i : ð67Þ

Since this proof is very similar to that of Theorem 1, it isomitted.

4 Design of the output feedback fuzzy adaptive controller

In previous section, fuzzy adaptive variable structure control-lers have been developed for multivariable systems with ac-tuators nonlinearities using the state feedback , based on theassumption that all the states system are measurable. It may beimpossible in reality to measure all the system states [37, 38].In the case where only the output vector y =[y1,…,yp]

T isavailable, an observer is needed to estimate y i

(j ), for j =1,…,r i−1. Below, a high-gain observer [39–41] is used toreconstruct the missing states of the system.

Lemma 2 Suppose the function yi(t) and its first derivatives riare bounded. Consider the following linear system, for i=1,…,p:

επ i1 ¼ πi2

επ i2 ¼ πi3

⋮επ iri ¼ −bi1πiri−⋯−bi ri−1ð Þπi2−πi1 þ yi

8>><>>: ð68Þ

Where ε is any small positive constant, the param-eters bi1;⋯; bi ri−1ð Þ are chosen so that the polynomial

sri þ bi1s ri−1ð Þ þ⋯þ bi ri−1ð Þsþ 1 is Hurwitz. Then, there

exists positive constants h ik, k =2,…,r i and for i=1,…,p, andt* such that for all t >t* we have:

πi kþ1ð Þεk

− y kð Þi ¼ −εw kþ1ð Þ

i ; k ¼ 1;…; ri−1 ð69Þ

πi kþ1ð Þεk

− y kð Þi

≤εhi kþ1ð Þ; k ¼ 1;…; ri−1 ð70Þ

where wi ¼ πiri þ bi1πi ri−1ð Þ þ⋯þ bi ri−1ð Þπi1 , wi(k) denotes

the k th derivative of wi, and |wi(k)|≤hik.

Proof The proof of this lemma can be found in [41, 42].Now, let us consider the system (2) subject to Assumptions

1, 4, and 5. Having the observers (i =1,…,p ) in (68), we candefine the following variables:

Wi ¼ 0;w 2ð Þi ;…;w rið Þ

i

h iT; i ¼ 1;…; p; W ¼ WT

1 ;WT2 ;…;WT

p

h iT;

πi ¼ πi1;πi2;…;πiri½ �T ; i ¼ 1;…; p; π ¼ πT1 ;π

T2 ;…;πT

p

h iT;

bxi ¼ xi1;πi2

ε;πi3

ε2; ;…;

πiri

εri−1

h iT; i ¼ 1;…; p; bx ¼ bxT1 ;bxT2 ;…;bxTp� �T

;

xdi ¼ ydi; y di;…; y ri−1ð Þdi

h iT; i ¼ 1;…; p; xd ¼ xTd1; x

Td2;…; xTdp

h iT;

bS ¼bS1⋮bSp

264375 ¼

CT1 bx1−xd1�

⋮CT

p bxp−xdp� 264

375;

Now, we can show as in [39, 40] that bS and its dynamicscan be given by:

bS ¼ S−εCT

1W 1

⋮CT

pWp

24 35 ¼ S−εCTW ð71Þ

bS˙ ¼ S˙−εCT W⋅

ð72Þ

The vectors C1, …, Cp have been previously defined insection 2.

From (20), (52), and (72), the dynamics of bS can berewritten as follows.

G−1s xð Þ bS˙ ¼ G−1

s xð Þ vþ F xð Þ½ � þ DT xð ÞΦ uð Þ−εG−1s xð ÞCT W˙

ð73Þ

with v =C rTE − y d

(r ) . Letting G s 1(x ) =G s− 1(x ) and

F 1(x ,u )=Gs−1(x )[v +F (x )]+ [DT (x )−D ]Φ (u ), Eq. (73)

becomes

Gs1 xð Þ bS⋅ ¼ F1 x; uð Þ þ DΦ uð Þ−εGs1 xð ÞCTW⋅ ð74Þ

Now, we use the following change of coordinates

Sb¼ D−1bS ð75Þ

Recall that D =DT=D−1. This allows to rewrite (74) as

Gs2 xð Þ bS⋅ ¼ D−1F1 x; uð Þ þ Φ uð Þ−εGs2 xð ÞD−1CTW⋅ ð76Þ

With Gs2(x )=D−1Gs1(x )D .



Similar to the previous section, (76) can be arranged asfollows

Gs2 xð Þ bS⋅ þ 1

2G⋅s2bS ¼ α z

� þ Φ uð Þ−εGs2 xð ÞD−1CT W

⋅ ð77Þ

where α zð Þ ¼ α1 z1ð Þ;α2 z2ð Þ;…;αp zp� �� T ¼ D−1F1 x; uð Þþ

12G

⋅s2 xð ÞbS , with z ¼ zT1 ; z

T2 ;…; zTp

h iT. Recall that the non-

linearities α zð Þ has an upper triangular control structure ,allowing thereby for algebraic loop free sequential determina-tion of the control variables. Then, the inputs z1 can be selectedas in the previous section. The functions αi zið Þ are assumed tosatisfy Assumption 5.

As in the previous section, the unknown continuous func-tions αi zið Þ (for i =1,…, p ) can be approximated by adaptivefuzzy systems as follows

bαi bzi; θ� ¼ θTi y i

bzi� ð78Þ

where bzi is an estimate of zi . The elements of bzi can bedirectly determined from the designed observers (68).

Similarly to the previous section, we have

αi zi�

¼ θTi y ibzi�

−eθTiy i

bzi� þ δi zi

� þ θ�Ti y i

bzi� −y i zi

� h ið79Þ

This fuzzy approximation error δi zið Þ þ θ�Ti y ibzi� �

−y i zið Þ� �can be assumed to be bounded as usual [1, 26, 27], i.e.,one has

δi zi�

þ θ�Ti y ibzi�

−ψi zi� h i ≤δi; ∀zi∈Ω

zið80Þ

Let us introduce the following assumption

Assumption 6 We assume

εηbSTGs2 xð ÞD−1CT W

⋅≤εκ1

bS 2 þ εκ2 ð81Þ

where κ1 and κ2 are unknown constants.

Remark 4 Note that Assumption 6 is not restrictive for thefollowing reasons:

& W⋅

is generally assumed to be bounded, please see[39–42].

& Gs1(x ) is a symmetric positive-definite matrix, i.e., ∃ σg1

>0 such as Gs1(x )≥σg1I . Then,bSTGs2 xð ÞbS≤ 1

σg1bS 2:

The following fuzzy adaptive variable–structure controlleris proposed to achieve the control objective:

ui ¼− bρi tð Þ sign bSi�

−ui−; bSi > 0;

0; bSi ¼ 0;

−b�ρi tð Þsign bSi� þ uiþ; bSi < 0;

8>><>>: ð82Þ

with

bρi tð Þ ¼ k0i þ k1i bSi þ θTi y i bzi� ; ∀i ¼ 1;…; p

k⋅0i ¼ −γ0iσ0ik0i þ γ0i bSi ; k0i 0ð Þ > 0

ð83Þ

θ⋅i ¼ −γ1iσ1iθi þ γ1i

bSi y ibzi�

; with θij 0ð Þ > 0 ð84Þ

where γ0i, γ1i, σ0i, σ1i, k1i>εκ1 are design constants, andk0i and θ i are the online estimates of the uncertain termsk�0i ¼ δi and θ i

*, respectively.

Multiplying (77) by bST and using Assumptions 5 and 6yields

1

2ηbS TG

⋅s2 xð ÞbS þ 1

ηbS TGs2 xð ÞbS⋅ ¼ 1

ηbS Tα zð Þ þ 1

ηbS TΦ uð Þ− ε

ηbS TGs2 xð Þ D−1CT W

⋅� �≤Xi¼1

p bSi αi zið Þ þ 1

ηbS TΦ uð Þ þ εκ1 bS 2 þ εκ2

ð85Þ

from (79) and (85), one has

1

2ηbS TGs2 xð ÞbS þ 1

ηbS TGs2 xð Þ bS˙ ≤−X

i¼1

p bSi ek0i þ θTi y i zbi� � �

þXi¼1

p bSi k0i þ θTi ψi zbi� � þ 1

ηbS TΦ uð Þ þ εκ1 bS 2

þ εκ2

ð86Þ

where eθi ¼ θi−θ�i and ek0i ¼ k0i−k�0i ¼ k0i−δi .

Theorem 3 Consider the system (2) with observers (68)and subject to Assumptions 1 and 4–6. Then, the controllaw defined by Eqs. (82) to (84) ensures the followingproperties:



& All the variables in the closed-loop system are UUB.& The filtered tracking error Si converges to an adjustable

domain which can be defined by:


π3

μ3

� �1=2

þ εW

( )

where σg2 satisfies σg2Ip ≥ Gs2 (x ), W ≥ CTW ,

π3 ¼ ∑i¼1

pσ0i2 k

�20i þ ∑

i¼1

pσ1i2 θ�i 2 þ εκ2 and μ3 ¼ min

fmini

2ησg2 k1i−εκ1ð Þ� �; min

iγ0iσ0if g;min

iγ1iσ1if gg .

Proof It is carried out using the following Lyapunov functioncandidate

V ¼ 1

2ηSbTGs2 xð ÞSbþ 1

2

Xi¼1

p 1

γ1ieθTieθi þ 1

2

Xi¼1

p 1

γ0iek 2i

ð87Þ

Since this proof is very similar to that of Theorem 1, it isomitted.

Remark 5 Two notes are worth to be made. Firstly, in the casewhere ui+=ui−=ui0, the expressions (33), (62), and (82) arerespectively simplified as

ui ¼ − ρi tð Þ þ ui0ð Þsign Sið Þ ð88Þ

ui ¼ − ρi tð Þ þ ui0�

sign Si�

ð89Þ

ui ¼ − ρbi tð Þ þ ui0�

sign Sbi� ð90Þ

where

ρi tð Þ ¼ k0i þ k1i Sij j þ θTi y i xð Þ;

ρi tð Þ ¼ k0i þ k1i Si þ θTi y i zi

� and

ρbi tð Þ ¼ k0i þ k1i Sbi

þ θTi y i zbi� :

Secondly, the sign function has to be replaced by anyequivalent smooth function to deal with the chattering effects[43–45].

Remark 6 We can show without any technical difficulty thatthe proposed controller remains applicable for MIMO system(2) free of the input nonlinearities (i.e., in the case where ui+=ui−=0, and f i+(ui)=f i−(ui)=1).

Remark 7 In the proposed adaptive laws, the σ −modificationtechnique is used to prevent parameter drift. Also, we can usea projective operator method (1), and e −modification methodin place of σ − modification technique.

Remark 8 Table 1 summarizes the comparison between thethree fuzzy adaptive controllers proposed in the paper:

Remark 9 Although the class of systems considered in our paperand that in [13] are all completely different, a simple theoreticalcomparison is made here and it is summarized in Table 2.

5 Simulation results

Simulation studies are carried out to show the effectiveness ofthe proposed adaptive fuzzy controllers. Two control prob-lems are considered to this end. The first one concerns a two-link rigid robot manipulator (its model has naturally a sym-metric control–gain matrix), while the second one concerns anacademic multivariable nonlinear system (its model has anonsymmetric and indefinite control–gain matrix).

5.1 Test of the fuzzy adaptive state feedback controllers

5.1.1 Example 1 (Test of the first controller)

In the following, one presents simulation results showing theperformances of the first controller applied to a two-link rigid

Table 1 Comparison between the three proposed controllers

Type of thecontroller

Assumptions made

Controller 1 A state feedbackcontroller

The matrix G(x) is symmetric andpositive definite

The state vector is available formeasurement.

Controller 2 A state feedbackcontroller

The matrix G(x) is not necessarysymmetric and but with non-zeroleading principal minors. Theirsigns are assumed to be known.

The state vector is available formeasurement.

Controller 3 An output feedbackcontroller

The matrix G(x) is not necessarysymmetric and but with non-zeroleading principal minors. Theirsigns are assumed to be known.

The state vector is not measurable,except the output vector.

Remarks The class considered of the MIMO systems in the designof the second and third controllers is large that the oneconsidered in the first design.

The third controller is an output feedback extension ofthe second controller.



robot manipulator which moves in a horizontal plane. Thedynamic equations of this MIMO system are given by [6, 20,25]:

q⋅⋅1

q⋅⋅2

� �¼ M11 M12

M21 M22

� �−1Φ1 u1ð ÞΦ2 u2ð Þ

� �− −hq 2 −h q 1 þ q 2ð Þ

hq 1 0

� �q 1

q 2

� �� ;

ð91Þ

where

M11 ¼ a1 þ 2a3 cos q2ð Þ þ 2a4 sin q2ð Þ ; M 22 ¼ a2;M21 ¼ M 12 ¼ a2 þ a3 cos q2ð Þ þ a4 sin q2ð Þ; h ¼ a3 cos q2ð Þ−a4 cos q2ð Þ;

with

a1 ¼ I1 þ m1l2c1 þ Ie þ mel

2ce þ mel

21; a2 ¼ Ie þ mel

2ce;

a3 ¼ mel1lcecos δeð Þ; a4 ¼ mel1lcesin δeð Þ:

Note that Φ i(ui), for i =1,2, are the considered input non-linearities. The robot parameters are:

m1 ¼ 1;me ¼ 2; l1 ¼ 1; lc1 ¼ 0:5; lce ¼ 0:6; I1 ¼ 0:12;

Ie ¼ 0:25; δe ¼ 30�:

Let y1; y2½ � ¼ q1; q2½ �; u ¼ u1; u2½ �T ;Φ uð Þ ¼ Φ1 u1ð Þ;½ Φ2

u2ð Þ�T ; x ¼ q1; q 1; q2; q 2½ �T : One can rewrite the robot mod-el with actuator nonlinearities (Eq. 92) as follows

y⋅⋅ ¼ F xð Þ þ G xð ÞΦ uð Þ ð92Þ

where

G xð Þ ¼ g11 xð Þ g12 xð Þg21 xð Þ g22 xð Þ

� �¼ M−1 ¼ M11 M 12

M21 M 22

� �−1

:

F xð Þ ¼ f 1 xð Þf 2 xð Þ

� �¼ −M−1 −hq 2 −h q 1 þ q 2ð Þ

hq1 0

� �q 1

q 2

� �;

The input nonlinearities are described as in [16]:

Φ1 u1ð Þ ¼u1−3ð Þ 1:5−0:3e0:3 sin u1ð Þj j

� ; u1 > 3;

0; u1j j≤3;u1 þ 3ð Þ 1:5−0:3e0:3 sin u1ð Þj j

� ; u1 < −3;

8>><>>:

Φ2 u2ð Þ ¼u2−3ð Þ 1:3−0:2e0:1 cos u2ð Þj j

� ; u2 > 3;

0; u2j j≤3;u2 þ 3ð Þ 1:3−0:2e0:1 cos u2ð Þj j

� ; u2 < −3;

8>><>>:The control objective consists in allowing the system out-

puts q1 and q2 to respectively track the sinusoidal-desiredtrajectories yd1=sin(t ) and yd2=sin(t ). Two square waveshaving an amplitude ±1 with a period of 2π (s ) are added tosystem states as external disturbances. Then, the system (92)becomes

y⋅⋅ ¼ F xð Þ þ G xð ÞΦ uð Þ þ d tð Þ ð93Þ

Table 2 Comparison between our third controller and that proposed in [13]

Comparison Controller 3 proposed in this paper Controller proposed in [13]

The class of MIMOnonlinear systemconsidered

y (r)=F(x)+G(x)Φ(u) x ¼ Axþ B Φ uð Þ þ d x; p; tð Þð Þ with An×n is a state matrix, Bn×p is a constantcontrol gain matrix, d(x ,p ,t)∈Rp is the lumped uncertainties and externaldisturbances, and p(t) is a vector of uncertain parameters.

Type of the controllerdesigned

An output feedback fuzzy adaptivevariable structure controller

A state-feedback sliding mode controller

The control problemconsidered

A tracking problem A stabilization problem

Type of inputnonlinearities

Sector nonlinearities with dead zones Sector nonlinearities with dead zones

Assumptions made The state vector is not measurable The state vector is available for measurement

F(x) and G(x) are unknown The matrices A and B are perfectly known.

The gain reduction tolerances of thenonlinear dead-zones are unknown.

The gain reduction tolerances of the nonlinear dead-zones are known.

The nonlinear term d(x ,p ,t) is bounded by a known function.

Control gain matrix The control gain matrix G(x) is afunction the state vector.

The control gain matrix B is constant. Remark . This assumption can beconsiderably simplifying the controller design .

Singularity of thecontroller

Our controller is free of the singularityproblem.

The proposed controller in [13] can be singular, when ‖S‖=0.

Conclusions Note that the class of the MIMO systems considered in [13] is simple and its model is almost known. Moreover, thecontroller designed is a state-feedback and can be singular when ‖S‖=0.



Recall that the system nonlinearities F(x ) and G (x ) areassumed to be completely unknown except the symmetryproperty and the sign of the matrix G (x ) when designing thefuzzy adaptive controller.

The fuzzy systems θ1Tψ1(x ) and θ2

Tψ2(x ) used to approx-imate the unknown nonlinearities are designed as (22) and

have only the state vector x ¼ q1; q 1; q2; q 2½ �T as input. Foreach variable of the entries of the two fuzzy systems, as in[26], one defines three (one triangular and two trapezoidal)membership functions uniformly distributed on the interval[−2,2]. Thus, the number of the fuzzy rules is 34 (because ourfuzzy rule base is a completed base). The consequent param-eters are estimated online via the adaptive laws (35), they areinitialized as follows: θ1i(0)=θ2i(0)=0.001, for i =1,…,81.

The design parameters used in this simulation are specifiedas follows: γ01=γ02=30, γ11=γ12=700,σ01=σ02=0.001,σ11=σ12=0.001,λ1=λ2=2 and k11=k12=2. The initial con-

ditions are chosen as follows: x 0ð Þ ¼ 0:5 0 0:5 0½ �T ,

and k01(0)=k02(0)=0.0005. It is worth noticing that the signfunction has been replaced in the simulation by a smoothfunction tanh(ksiS i), with ksi=20.

Figure 2 shows proposed fuzzy adaptive controller performswell in spite of the input nonlinearities, uncertainties and statedisturbances. Indeed, the tracking errors for both links arebounded and converge towards small values as pointed outby Fig. 2a, b. Figure 2c, d show that the control variables aswell as the norms of the estimated parameters are bounded.

5.1.2 Example 2 (Test of the second controller)

In the following, one presents simulation results showing theperformance of the second fuzzy adaptive control systeminvolving an academic multivariable nonlinear system havinga nonsymmetric and indefinite control–gain matrix. The dy-namics of this system is given by:

x 11 ¼ x12;

x 12 ¼ x21−0:3sin x11x12ð Þ þ x212 þ 2þ cos x11ð Þð ÞΦ1 u1ð Þ þ 1þ sin x21ð Þð Þ2�

Φ2 u2ð Þ;x 21 ¼ x22;

x 22 ¼ x322 þ ex11 þ x212 − 0:5 Φ1 u1ð Þ − 1þ sin x21ð Þð Þ2�

Φ2 u2ð Þ ;y1 ¼ x11; y2 ¼ x21:

8>>>>><>>>>>:ð94Þ

Let y =[y 1,y 2]T, u =[u 1,u 2]

T,Φ (u )=[Φ 1(u 1),Φ 2(u 2)]T,

and x =[x11,x12,x21,x22]T. Then, the system (94) can be given

in the following form:

y⋅⋅ ¼ F xð Þ þ G xð ÞΦ uð Þ ð95Þ

where

F xð Þ ¼ x21−0:3sin x11x12ð Þ þ x212x322 þ ex11 þ x212

� �;

G xð Þ ¼ 2þ cos x11ð Þð Þ 1þ sin x21ð Þð Þ2−0:5 − 1þ sin x21ð Þð Þð Þ2

� �

The input nonlinearities Φ i(ui) for i =1,2 are described by:

Φi uið Þ ¼ui−3ð Þ 1−0:3sin uið Þð Þ; ui > 3;0; −3≤ui≤3;ui þ 3ð Þ 0:8−0:3cos uið Þð Þ; ui < −3;

8<:Recall that the matrix G (x ) is nonsymmetric and

the system nonlinearities (F (x ) and G (x )) are assumedto be unknown except the signs of its leading

principal minors when designing the fuzzy adaptivecontroller.

The control objective consists in allowing the system out-

puts y1 and y2 to track the sinusoidal desired trajectories yd1=

sin(t ) and yd2=sin(t ), respectively.The fuzzy system θT1ψ1 z1ð Þ has the vector [xT,u2]

T as

input, while the fuzzy system θT2ψ2 z2ð Þ has the state vectorx as input. For each variable of the entries of these fuzzysystems, as in [26], one defines three (one triangular and twotrapezoidal) membership functions uniformly distributed onthe intervals [−2,2] for x11,x12,x21,and x22, and [−25,25] foru 2. Thus, the number of the fuzzy rules (for both fuzzysystems) is 35 and 34, respectively. Note that our fuzzy rulebases are completed bases. The consequent parameters areestimated online via the adaptive laws (64), they are initializedin a random way: θ1i(0)=0.001, for i =1,…,243 and θ2j(0)=0.002, for j =1,…,81.

The design parameters used in this simulation are chosen asfollows: γ 01=γ 02=20, γ 11=γ 12=1500, σ 01=σ 02=0.01,σ11=0.001,σ12=0.001,λ1=λ2=2 and k11=k12=0.5. The ini-

tial conditions are selected as: x 0ð Þ ¼ 0:5 0 0:5 0½ �Tand k01(0)=k02(0)=0.001,. The sign function has been re-

placed by an equivalent smooth function, i.e., tanh ksiSi� �

;

with ksi=20.



Figure 3 shows that the proposed fuzzy adaptivecontrol system performs well from tracking point ofview. In fact, Fig. 3a, b illustrate the boundedness andconvergence of the tracking errors for both subsystems.The boundedness of the corresponding control signals aswell as of the estimated fuzzy parameters is well illus-trated in Fig. 3c, d, respectively. In spite of the presenceof the unknown input nonlinearities (dead zone andsector nonlinearities) and the uncertainties, those simu-lation results show a good tracking performance.

5.2 Test of the fuzzy adaptive output feedback controller

5.2.1 Example 1

When the states (x12, x22) of the system (92) are not measur-able, a high-gain observer is designed as follows:

επ 11 ¼ π12

επ 12 ¼ −b11π12−π11 þ y1επ 21 ¼ π22

επ 22 ¼ −b21π22−π21 þ y2

8>><>>: ð96Þ

0 5 10 15 20-1.5

-1

-0.5

0

0.5

Tra

ckin

g er

rors

of l

ink

1

0 5 10 15 20-1

-0.5

0

0.5

Tra

ckin

g er

rors

of l

ink

2

0 5 10 15 20-20

-10

0

10

20C

ontr

ol s

igna

ls

0 5 10 15 200

10

20

30

Time(s) Time(s)

Time(s) Time(s)

Nor

ms

of fu

zzy

para

met

ers

a b

c d

Fig. 2 Simulation resultsobtained by the controller 1 of theexample 1. a Tracking errors oflink 1: e1 (dotted line) and e 1

(solid line). b Tracking errors oflink 2: e2 (dotted line) and e 2

(solid line). c Control inputsignals: u1 (dotted line) and u2(solid line; d). Norm of fuzzyparameters: ‖θ1‖ (dotted line),‖θ2‖ (solid line)

0 5 10 15 20-1

-0.5

0

0.5

Tra

ckin

g er

rors

of s

ubsy

1

0 5 10 15 20-1

-0.5

0

0.5

Tra

ckin

g er

rors

of s

ubsy

2

0 5 10 15 20-10

-5

0

5

10

Con

trol

sig

nals

0 5 10 15 200

10

20

30

Nor

ms

of fu

zzy

para

met

ers

time(s)

time(s) time(s)

time(s)

a b

c d

Fig. 3 Simulation resultsobtained by the controller 2 of theexample 2. a Tracking errors: e1(dotted line) and e 1 (solid line).b Tracking errors: e2 (dotted line)and e 2 (solid line). c Controlinput signals: u1 (dotted line) andu2 (solid line; d). Norm of fuzzyparameters: ‖θ1‖ (dotted line),‖θ2‖ (solid line)



with ε =0.01, b 11=b 21=1, and the initial conditionsπ (0)=[0,0,0,0]T. The respective estimates of the vectors z1and z2 are:

zb1 ¼ y1;π12

ε; y2;

π22

ε; u

h iTzb2 ¼ y1;

π12

ε; y2;

π22

ε

h iTFigure 4 illustrates the simulation results of the fuzzy

adaptive output feedback controller. Figure 4a, b show the

boundedness and convergence of the tracking errorse1; e 1; e2; e 2ð Þ towards small values for both subsystems.Figure 4c indicates the boundedness of the applied controlsignals. Figure 4d shows the observation errors (ee12 ¼ bx12−x12 ,ee22 ¼ bx22−x22 ) are very small and bounded.

5.2.2 Example 2

When the states (x12, x22) of the system (94) are not availablefor measurement, the high-gain observer (96) can be designed.

0 5 10 15 20-1.5

-1

-0.5

0

0.5

Tra

ckin

g er

ros

of li

nk1

0 5 10 15 20-1.5

-1

-0.5

0

0.5

Tra

ckin

g er

rors

of l

ink2

0 5 10 15 20-20

0

20

40

Con

trol

sig

nals

0 5 10 15 20-5

0

5x 10-3

time(s)

a

time(s)

b

time(s)

c

time(s)

d

Obs

erva

tion

erro

rs

Fig. 4 Simulation resultsobtained by the controller 3 of theexample 1. a Tracking errors oflink 1: e1 (dotted line) and e 1

(solid line). b Tracking errors oflink 2: e2 (dotted line) and e 2

(solid line). c Control inputsignals: u1 (dotted line) and u2(solid line; d). Observationerrors: ee12 ¼ bx12−x12 (dottedline), ee22 ¼ bx22−x22 (solid line)

0 5 10 15 20-1

-0.5

0

0.5

Tra

ckin

g er

rors

of s

ubsy

1

0 5 10 15 20-1

-0.5

0

0.5

Tra

ckin

g er

rors

of s

ubsy

2

0 5 10 15 20-10

-5

0

5

10

Con

trol

sig

nals

0 5 10 15 20-5

0

5x 10-3

time(s)

a

time(s)

b

time(s)

c

time(s)

d

Obs

erva

tion

erro

rs

Fig. 5 Simulation resultsobtained by the controller 3 of theexample 2. a Tracking errors: e1(dotted line) and e 1 (solid line).b Tracking errors: e2 (dotted line)and e 2 (solid line). c Controlinput signals: u1 (dotted line) andu2 (solid line; d). Observationerrors: ee12 ¼ bx12−x12 (dottedline), ee22 ¼ bx22−x22 (solid line)



The parameters of this observer are selected as ε =0.01 andb11=b21=1, and the initial conditions as π (0)=[0,0,0,0]

T. Therespective estimates of the vectors z1 and z2 can be deter-mined as

zb1 ¼ y1;π12

ε; y2;

π22

ε; u

h iTzb2 ¼ y1;

π12

ε; y2;

π22

ε

h iTThe simulation results of the fuzzy adaptive output feed-

back controller are given in Fig. 5. It can be seen in Fig. 5a, b,and d that tracking errors e1; e 1; e2; e 2ð Þ and observationerrors (ee12 ¼ bx12−x12 , ee22 ¼ bx22−x22 ) are small. Figure 5cindicates the boundedness of applied control signals. Theproposed output feedback controller achieves a satisfied track-ing performance even with little knowledge of the system, inthe presence of input nonlinearities and only the measurementof the output.

6 Conclusion

In this paper, three fuzzy adaptive controllers (the two firstcontrollers are state feedback and but the last one is an outputfeedback) for a class of multivariable unknown nonlinearsystems subject to actuator sector nonlinearities and deadzones have been proposed bearing in mind the usual stabilityand convergence requirements. The first one has been de-signed for systems having a symmetric and positive-definitecontrol–gain matrix, while the second one has been particu-larly designed for systems with a nonsymmetric control gainsmatrix whose leading principal minors are non-zero. A suit-able decomposition of the control–gain matrix has been fullyexploited to carry out the second control design. The third oneis an output feedback extension of the second controller. In thelatter, a high-gain observer has been designed to estimate themissing states. Of fundamental interest, it has been proven thatthe proposed control systems are stable and their underlyingtracking errors converge exponentially to an adjustable do-main. Simulation results have been given to emphasize theeffectiveness of the proposed controllers.

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