control design for interval type-2 fuzzy systems under imperfect premise matching · 2014. 10....

956 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 2, FEBRUARY 2014

Control Design for Interval Type-2 Fuzzy SystemsUnder Imperfect Premise MatchingH. K. Lam, Senior Member, IEEE, Hongyi Li, Christian Deters, E. L. Secco,

Helge A. Wurdemann, and Kaspar Althoefer, Member, IEEE

Abstract—This paper focuses on designing interval type-2 (IT2)control for nonlinear systems subject to parameter uncertainties.To facilitate the stability analysis and control synthesis, an IT2Takagi–Sugeno (T–S) fuzzy model is employed to represent thedynamics of nonlinear systems of which the parameter uncertain-ties are captured by IT2 membership functions characterized bythe lower and upper membership functions. A novel IT2 fuzzycontroller is proposed to perform the control process, where themembership functions and number of rules can be freely chosenand different from those of the IT2 T–S fuzzy model. Conse-quently, the IT2 fuzzy-model-based (FMB) control system is withimperfectly matched membership functions, which hinders thestability analysis. To relax the stability analysis for this class of IT2FMB control systems, the information of footprint of uncertaintiesand the lower and upper membership functions are taken intoaccount for the stability analysis. Based on the Lyapunov stabilitytheory, some stability conditions in terms of linear matrix inequal-ities are obtained to determine the system stability and achievethe control design. Finally, simulation and experimental examplesare provided to demonstrate the effectiveness and the merit of theproposed approach.

Index Terms—Fuzzy control, imperfect premise matching,interval type-2 (IT2) fuzzy control, stability analysis.

I. INTRODUCTION

TYPE-1 fuzzy control approach has been successfully ap-plied to a wide range of domestic and industrial controlapplications, which demonstrate that it is a promising con-trol approach for complex nonlinear plants [1]–[4]. Stabilityanalysis and control synthesis are the two main issues to beconsidered in the fuzzy control paradigm. It is well knownthat the Takagi–Sugeno (T–S) fuzzy model [5] (also known asthe Takagi–Sugeno–Kang fuzzy model [6]) plays an importantrole to carry out stability analysis and control design [7]–[13],

Manuscript received April 11, 2012; revised October 7, 2012 andDecember 30, 2012; accepted February 14, 2013. Date of publicationMarch 15, 2013; date of current version August 9, 2013. This work was sup-ported in part by King’s College London, in part by the European Commission’sSeventh Framework Programme (FP7-NMP-2009-SMALL-3, NMP-2009-3.2-2), and project Cost-driven Adaptive Factory based on Modular Self-ContainedFactory Units (COSMOS) (Grant 246371-2.C), and in part by the NationalNatural Science Foundation of China under Grant 61203002.

H. K. Lam, C. Deters, E. L. Secco, H. A. Wurdemann, and K. Althoefer arewith the Department of Informatics, King’s College London, London, WC2R2LC, U.K. (e-mail: [email protected]; [email protected];[email protected]; [email protected]; [email protected]).

H. Li is with the College of Information Science and Technology, BohaiUniversity, Jinzhou 121013, China (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIE.2013.2253064

which provides a general modeling framework for nonlinearsystems. The system dynamics of the nonlinear systems canbe represented as an average weighted sum of some locallinear subsystems, where the weightings are characterized bythe type-1 membership functions.

Lyapunov stability theory is the most popular method toinvestigate the stability of type-1 fuzzy-model-based (FMB)control systems. Basic stability conditions in terms of linearmatrix inequalities (LMIs) [14] were achieved in [15] and[16]. The FMB control system is guaranteed to be asymp-totically stable if there exists a common solution to a set ofLyapunov inequalities in terms of LMIs. With the proposedparallel distributed compensation (PDC) design concept, somestability conditions were relaxed in [16]. More relaxed stabilityconditions under PDC can be found in [17]–[19]. With the con-sideration of the information of type-1 membership functions,stability conditions can be further relaxed [20]–[22]. Also, thefuzzy control concept was extended to other stability/controlproblems such as output feedback control [23], sampled-datacontrol [26], control systems with time delay [8], [24], [25],tracking control [27], large-scale fuzzy systems [28], and evenfuzzy neural networks [29].

Type-1 fuzzy sets are able to effectively capture the systemnonlinearities but not the uncertainties. It has been shown in theliterature that type-2 fuzzy sets [30], which extend the capabil-ity of type-1 fuzzy sets, are good in representing and captur-ing uncertainties, supported by a number of applications suchas adaptive filtering [31], analog module implementation anddesign [32], [33], active suspension systems [34], autonomousmobiles [35], electrohydraulic servo systems [36], extendedKalman filters [37], dc–dc power converters [38], nonlinearcontrol [39], [40], noise reduction [41], video streaming [42],and inverted pendulum control [43]. However, type-2 fuzzyset theory was developed for a general type-2 fuzzy logicsystem but not mainly for FMB control scheme. Consequently,there are few research studies about the type-2 FMB controlsystems in the literature. This motivates the investigation ofthe system stability and control design of type-2 FMB controlsystems.

Recently, some research has been done on system control andstability analysis based on the existing framework of type-2fuzzy systems [39], [44]–[48]. In [31], a basic interval type-2(IT2) T–S fuzzy model was proposed, which was extended toa more general IT2 T–S fuzzy model [39] for a wider classof nonlinear systems suitable for system analysis and controldesign. Preliminary stability analysis work on IT2 FMB systemcan be found in [39] and [48] of which a set of LMI-based

0278-0046 © 2013 IEEE

LAM et al.: CONTROL DESIGN FOR INTERVAL TYPE-2 FUZZY SYSTEMS UNDER IMPERFECT PREMISE MATCHING 957

stability conditions was obtained determining the system sta-bility and facilitating the control synthesis.

In this paper, we investigate the stability of IT2 FMB controlsystems under imperfect premise matching. Unlike the authors’work in [39] under the PDC design concept, it was required thatthe IT2 fuzzy controller shares the same premise membershipfunctions and the same number of rules as those of the IT2 T–Sfuzzy model. These limitations constrain the design flexibilityand increase the implementation complexity of the IT2 fuzzycontroller. The work in this paper eliminates these limitationsby proposing an IT2 fuzzy controller in which the member-ship functions and the number of rules can be freely chosen,enhancing the applicability of the IT2 FMB control scheme. Bychoosing simple membership functions and a smaller numberof rules, it can reduce the implementation complexity of theIT2 fuzzy controller, resulting in a lower implementation cost.However, the IT2 FMB control systems will have imperfectlymatched membership functions, potentially leading to moredifficult stability analysis as the favorable property of the PDCdesign concept vanishes.

To carry out the stability analysis for IT2 FMB controlsystem subject to imperfect premise membership functions, thelower and upper membership functions characterized by thefootprint of uncertainty (FOU) are chosen to be a favorablerepresentation. This favorable representation allows the lowerand upper membership functions to be taken in the stabilityanalysis. Consequently, the stability conditions in terms ofLMIs are membership function dependent, which is appliedto the nonlinear plant under consideration, but not a familyconsidered in some existing work. The preliminary result ofthe authors in [48] provides technical support to the work inthis paper. To further relax the stability conditions, the FOUis divided into a number of sub-FOUs. The information of thesub-FOUs, along with those of lower and upper membershipfunctions, is brought to the stability analysis. Based on theLyapunov stability theory, LMI-based stability conditions areobtained to guarantee the stability of the IT2 FMB controlsystems and synthesize the IT2 fuzzy controller.

The organization of this paper is as follows. In Section II, theIT2 T–S fuzzy model representing the nonlinear plant subjectto parameter uncertainties, the IT2 fuzzy controller, and theIT2 FMB control systems are presented. In Section III, LMI-based stability conditions are obtained based on the Lyapunovstability theory for the IT2 FMB control systems. In Section IV,simulation and experimental examples are given to illustrate themerits of the proposed IT2 FMB control scheme. In Section V,a conclusion is drawn.

II. PRELIMINARIES

Considering a nonlinear plant subject to parameter uncer-tainties represented by an IT2 T–S fuzzy model [31], [39], anIT2 fuzzy controller is proposed to perform the control process.An IT2 FMB control system is formed by connecting the IT2T–S fuzzy model and the IT2 fuzzy controller in a closed loop.In this paper, it is not required that both the IT2 T–S fuzzymodel and the IT2 fuzzy controller share the same premisemembership functions and the same number of rules.

A. IT2 T–S Fuzzy Model

A p-rule IT2 T–S fuzzy model [31], [39] is employed todescribe the dynamics of the nonlinear plant. The rule is of thefollowing format where the antecedent contains IT2 fuzzy setsand the consequent is a linear dynamical system:

Rule i : IF f1 (x(t)) is M̃i1 AND · · ·AND fΨ (x(t)) is M̃ iΨTHEN ẋ(t) = Aix(t) +Biu(t) (1)

where M̃ iα is an IT2 fuzzy set of rule i corresponding tothe function fα(x(t)), α = 1, 2, . . . ,Ψ and i = 1, 2, . . . , p; Ψis a positive integer; x(t) ∈ �n is the system state vector;Ai ∈ �n×n and Bi ∈ �n×m are the known system and inputmatrices, respectively; and u(t) ∈ �m is the input vector. Thefiring strength of the ith rule is of the following interval sets:

Wi (x(t)) = [wi (x(t)) , wi (x(t))] , i = 1, 2, . . . , p(2)

where

wi (x(t)) =

Ψ∏α=1

μM̃iα

(fα (x(t))) ≥ 0 (3)

wi (x(t)) =

Ψ∏α=1

μM̃iα (fα(x(t))) ≥ 0 (4)

μM̃iα (fα(x(t))) ≥μM̃iα (fα(x(t))) ≥ 0 (5)

wi (x(t)) ≥wi (x(t)) ≥ 0 ∀ i (6)

in which wi(x(t)), wi(x(t)), μM̃iα(fα(x(t))), and

μM̃iα(fα(x(t))) denote the lower grade of membership,upper grade of membership, lower membership function, andupper membership function, respectively. The inferred IT2 T–Sfuzzy model [39] is defined as follows:

ẋ(t) =

p∑i=1

w̃i (x(t)) (Aix(t) +Biu(t)) (7)

where

w̃i (x(t))=αi (x(t))wi (x(t))+αi (x(t))wi (x(t))≥0 ∀ i(8)

p∑i=1

w̃i (x(t)) = 1 (9)

0 ≤ αi (x(t)) ≤ 1 ∀ i (10)0 ≤ αi (x(t)) ≤ 1 ∀ i (11)

αi (x(t)) + αi (x(t)) = 1 ∀ i (12)

in which αi(x(t)) and αi(x(t)) are nonlinear functions notnecessarily be known but exist, w̃i(x(t)) can be regarded as thegrades of membership of the embedded membership functions,and (8) defines the type reduction.

Remark 1: It can be seen from (9) that the actual grades ofmembership w̃i(x(t)) can be reconstructed and expressed as a


linear combination of wi(x(t)) and wi(x(t)), characterized bythe lower and upper membership functions μ

M̃iα(fα(x(t))) and

μM̃iα(fα(x(t))), which are scaled by the nonlinear functionsαi(x(t)) and αi(x(t)), respectively. In other words, any mem-bership functions within the FOU [39] can be reconstructed bythe lower and upper membership functions. As the nonlinearplant is subject to parameter uncertainties, w̃i(x(t)) will dependon the parameter uncertainties, thus leading to the values ofαi(x(t)) and αi(x(t)) to be uncertain. It should be noted thatthe IT2 T–S fuzzy model (7) serves as a mathematical tool tofacilitate the stability analysis and control synthesis and is notnecessarily implemented.

B. IT2 Fuzzy Controller

An IT2 fuzzy controller with c rules of the following formatis proposed to stabilize the nonlinear plant represented by theIT2 T–S fuzzy model (7):

Rule j IF g1 (x(t)) is Ñj1 AND · · · AND gΩ (x(t)) is Ñ

jΩ

THEN u(t) = Gjx(t) (13)

where Ñ jβ is an IT2 fuzzy set of rule j corresponding tothe function gβ(x(t)), β = 1, 2, . . . ,Ω and j = 1, 2, . . . , c; Ωis a positive integer; and Gj ∈ �m×n, j = 1, 2, . . . , c, repre-sents the constant feedback gains to be determined. The firingstrength of the jth rule is the following interval sets:

Mj (x(t)) =[mj (x(t)) , mj (x(t))

], j = 1, 2, . . . , c

(14)

where

mj (x(t)) =

Ω∏β=1

μÑj

β

(gβ(x(t))) ≥ 0 (15)

mj (x(t)) =

Ω∏β=1

μÑjβ(gβ(x(t))) ≥ 0 (16)

μÑjβ(gβ(x(t))) ≥μÑj

β

(gβ(x(t))) ≥ 0 ∀ j (17)

in which mj(x(t)), mj(x(t)), μÑjβ

(gβ(x(t))), and

μÑjβ(gβ(x(t))) stand for the lower grade of membership,

upper grade of membership, lower membership function, andupper membership function, respectively. The inferred IT2fuzzy controller is defined as follows:

u(t) =

c∑j=1

m̃j (x(t))Gjx(t) (18)

where

m̃j(x(t))=βj(x(t))mj (x(t))+βj (x(t))mj (x(t))∑c

k=1

(βk(x(t))mk(x(t))+βk (x(t))mk(x(t))

)≥ 0 ∀ j (19)

c∑j=1

m̃i (x(t)) = 1 (20)

0 ≤ βj(x(t)) ≤ 1 ∀ j (21)

0 ≤ βj (x(t)) ≤ 1 ∀ j (22)βj(x(t)) + βj (x(t)) = 1 ∀ j (23)

in which βj(x(t)) and βj(x(t)) are predefined functions,

m̃j(x(t)) can be regarded as the grades of membership ofthe embedded membership functions, and (19) is the typereduction.

Remark 2: Compared with the IT2 fuzzy controller in [39],the proposed one in (18) has the following two enhancements.

1) The type reduction for the IT2 fuzzy controller in [39]is characterized by the average normalized membershipgrades of the lower and upper membership functions, e.g.,βj(x(t)) = βj(x(t)) = 0.5 for all j. In this paper, the

type reduction of the proposed IT2 fuzzy controller (18)is characterized by two predefined functions β

j(x(t)) and

βj(x(t)).2) The proposed IT2 fuzzy controller (18) does not need

to share the same lower and upper premise membershipfunctions and the same number of fuzzy rules as thoseof the IT2 T–S fuzzy model (7). These two enhance-ments offer a higher design flexibility to the IT2 fuzzycontroller. Moreover, by employing simple membershipfunctions and a smaller number of fuzzy rules, the imple-mentation complexity of the IT2 fuzzy controller (18) canbe reduced.

C. IT2 FMB Control Systems

From (7) and (18), with the property of∑p

i=1 w̃i(x(t)) =∑cj=1 m̃j(x(t)) =

∑pi=1

∑cj=1 w̃i(x(t))m̃j(x(t)) = 1, we

have the following IT2 FMB control system:

ẋ(t) =

p∑i=1

w̃i (x(t))

⎛⎝Aix(t) +Bi c∑

j=1

m̃j (x(t))Gjx(t)

⎞⎠

=

p∑i=1

c∑j=1

w̃i (x(t)) m̃j (x(t)) (Ai +BiGj)x(t). (24)

The control objective of this paper is to guarantee the systemstability by determining the feedback gains Gj , such that theIT2 fuzzy controller (18) is able to drive the system states tothe origin, i.e., x(t) → 0, as time t → ∞.

Basic LMI-based stability conditions guaranteeing the stabil-ity of the FMB control system in the form of (24) are given inthe following theorem.

Theorem 1 [15]: The FMB control system in the form of(24) is guaranteed to be asymptotically stable if there existmatrices Nj ∈ �m×n, j = 1, 2, . . . , c, and X = XT ∈ �n×nsuch that the following LMIs are satisfied:

X > 0

Qij =AiX+XATi +BiNj +N

Tj B

Ti < 0 ∀ i, j

where the feedback gains are defined as Gj = NjX−1 forall j.


Remark 3: The stability conditions in Theorem 1 are veryconservative as the membership functions of both the fuzzymodel and fuzzy controller are not considered. The stabilityconditions can be reduced to Qij = AiX+XA

Ti +BiN+

NTBTi < 0 for all i by choosing a common feedback gain, i.e.,N = Nj for all j, resulting in a linear controller.

To facilitate the stability analysis of the IT2 FMB controlsystem (24), the state space of interest denoted as Φ is dividedinto q connected substate spaces denoted as Φk, k = 1, 2, . . . , q,such that Φ =

⋃qk=1 Φk. Furthermore, to consider more in-

formation of the IT2 membership functions, local lower andupper membership functions within the FOU are introduced.Considering the FOU being divided into τ + 1 sub-FOUs,in the lth sub-FOU, l = 1, 2, . . . , τ + 1, the lower and uppermembership functions are defined as follows:

hijl (x(t)) =

q∑k=1

2∑i1=1

· · ·2∑

in=1

n∏r=1

vrirkl (xr(t))

× δiji1i2···inkl ∀ i, j, k, l (25)

hijl (x(t)) =

q∑k=1

2∑i1=1

· · ·2∑

in=1

n∏r=1

vrirkl (xr(t))

× δiji1i2···inkl ∀ i, j, k, l (26)

0 ≤hijl (x(t)) ≤ hijl (x(t)) ≤ 1 (27)

0 ≤ δiji1i2···inkl ≤ δiji1i2···inkl ≤ 1 (28)

where δiji1i2···inkl and δiji1i2···inkl are constant scalars tobe determined; 0 ≤ vriskl(xr(t)) ≤ 1 and vr1kl(xr(t)) +vr2kl(xr(t)) = 1 for r, s = 1, 2, . . . , n; l = 1, 2, . . . ,τ + 1; ir = 1, 2; x(t) ∈ Φk; and vrisk(xr(t)) = 0 ifotherwise. As a result, we have

∑qk=1

∑2i1=1

∑2i2=1

· · ·∑2in=1

∏nr=1 vrirkl(xr(t)) = 1 for all l, which is used in the

stability analysis.We then express the IT2 FMB control system (24) in the

following favorable form:

ẋ(t) =

p∑i=1

c∑j=1

h̃ij (x(t)) (Ai +BiGj)x(t) (29)

where

h̃ij (x(t)) ≡ w̃i (x(t)) m̃j (x(t))

=

τ+1∑l=1

ξijl (x(t)) (γijl (x(t))hijl (x(t))

+ γijlhijl (x(t))) ∀ i, j (30)

with

p∑i=1

c∑j=1

h̃ij (x(t)) = 1. (31)

0 ≤ γijl(x(t)) ≤ γijl(x(t)) ≤ 1 has two functions, which are

not necessary to be known, exhibiting the property thatγijl(x(t)) + γijl(x(t)) = 1 for all i, j, and l. ξijl(x(t)) = 1

if the membership function hijl(x(t)) is within the sub-FOU l;otherwise, ξijl(x(t)) = 0.

Remark 4: It should be noted that only one ξijl(x(t)) = 1among the τ + 1 sub-FOUs at any time instant and the restequal zero for the ijth membership function h̃ij(x(t)). It canbe seen from (30) that, the more the sub-FOUs are considered,the more the information about the FOU is contained in thelocal lower and upper membership functions.

Remark 5: The local lower and upper membership functionscan reconstruct h̃ij(x(t)) ≡ w̃i(x(t))m̃j(x(t)) by representingit as a linear combination of hijl(x(t)) and hijl(x(t)) in sub-FOU l as shown in (30).

Remark 6: The IT2 FMB control system in (24) is a subsetof (29). Comparing both the IT2 FMB control systems, the onein (29) demonstrates some favorable properties to facilitate thestability analysis.

1) The partial information of hijl(x(t)) and hijl(x(t))is extracted and represented by the constant scalarsδiji1i2···inkl and δiji1i2···inkl, which are brought to thestability conditions.

2) Referring to (25) and (26), the cross-terms∏nr=1 vrirkl(xr(t)) are independent of i and j and,

thus, can be collected in the stability analysis.3) With the nonlinear functions γ

ijl(x(t)) and γijl(x(t)),

h̃ijl(x(t)) can be reconstructed as shown in (30) as alinear combination of hijl(x(t)) and hijl(x(t)). Fur-thermore, with (25) and (26), the values of hijl(x(t))and hijl(x(t)) are determined by the constant scalarsδiji1i2···inkl and δiji1i2···inkl through

∏nr=1 vrirkl(xr(t)).

As a result, the stability of the IT2 FMB control systemcan be determined by hijl(x(t)) and hijl(x(t)) (the local

lower and upper bounds of h̃ij(x(t))) characterized bythe constant scalars δiji1i2···inkl and δiji1i2···inkl. Theseproperties can be seen in the stability analysis carried outin the next section.

III. STABILITY ANALYSIS

The stability of the IT2 FMB control system (24) is investi-gated based on the Lyapunov stability theory with the consid-eration of the information of the lower and upper membershipfunctions and sub-FOUs. For brevity, in the following analysis,the time t associated with the variables is dropped for the situa-tion without ambiguity, e.g., x(t) is denoted as x. The variableswi(x(t)), wi(x(t)), w̃i(x(t)), mj(x(t)), mj(x(t)), m̃j(x(t)),

h̃ijl(x(t)), v1i1kl(x1(t)), v2i2kl(x2(t)), . . . , vninkl(xn(t)), andξijl(x(t)) are denoted by wi, wi, w̃i, mj , mj , m̃j , h̃ijl,v1i1kl, v2i2kl, . . . , vninkl, and ξijl, respectively. Furthermore,the property of

∑pi=1 w̃i =

∑cj=1 m̃j =

∑pi=1

∑cj=1 w̃im̃j =∑p

i=1

∑cj=1 h̃ij = 1 is utilized.

The stability analysis result is summarized in the followingtheorem to guarantee the asymptotic stability of the IT2 FMBcontrol system (24) and facilitate the control synthesis.


Theorem 2: Considering the FOU being divided into τ + 1sub-FOUs, the IT2 FMB control system (24) under imperfectpremise matching, formed by a nonlinear plant [represented bythe IT2 T–S fuzzy model (7)] and an IT2 fuzzy controller (18)connected in a closed loop, is guaranteed to be asymptoticallystable if there exist matrices M = M ∈ �n×n, Nj ∈ �m×n,X = XT ∈ �n×n, and Wijl = WTijl ∈ �n×n, i = 1, 2, . . . , p;j = 1, 2, . . . , c; and l = 1, 2, . . . , τ + 1, such that the followingLMIs are satisfied:

X > 0 (32)

Wijl ≥ 0 ∀ i, j, l (33)Qij +Wijl +M > 0 ∀ i, j, l (34)

p∑i=1

c∑j=1

(δiji1i2···inklQij −

(δiji1i2···inkl − δiji1i2···inkl

)

Wijl + δiji1i2···inklM)−M < 0

∀ i1, i2, . . . , in, k, l (35)

where δiji1i2···inkl and δiji1i2···inkl, i = 1, 2, . . . , p; j = 1, 2,. . . , c; i1, i2, . . . , in = 1, 2; k = 1, 2, . . . , q; and l = 1, 2,. . . , τ + 1, are predefined constant scalars satisfying (25) and(26); Qij = AiX+XATi +BiNj +N

Tj B

Ti for all i and j;

and the feedback gains are defined as Gj = NjX−1 for all j.Proof: The proof of Theorem 2 is given in the

Appendix. �Remark 7: The stability conditions in Theorem 1 are a

particular case of Theorem 2. If there exists a solution tothe stability conditions in Theorem 1, X > 0 and Qij <0 for all i and j can be achieved. Choosing M = ε1I >0 and Wijl = −Qij + (−ε1 + ε2)I > 0 for all i, j, and lwith sufficiently small nonzero positive values of ε1 andε2 in Theorem 2, the LMIs (33) and (34) can be satis-fied. As a result, recalling that δiji1i2···inkl ≥ δiji1i2···inkl ≥0, the LMIs in (35) become

∑pi=1

∑cj=1(δiji1i2···inklε2I−

δiji1i2···inklWijl)− ε1I < 0 for all i1, i2, . . . , in, k, and l,which will be satisfied by a sufficiently small value of ε2. Con-sequently, the solution of the stability conditions in Theorem 1is that of Theorem 2 but not the other way round.

IV. SIMULATION AND EXPERIMENTAL EXAMPLES

Simulation and experimental examples are given in thissection to demonstrate the effectiveness and the merit of theproposed IT2 FMB control approach.

Example 1: A three-rule IT2 T–S fuzzy model in the formof (7) is employed to represent a nonlinear plant with A1 =[1.59 −7.290.01 0

], A2=

[0.02 −4.640.35 0.21

], A3=

[−a −4.330 0.05

],

B1=

[10

], B2=

[80

], B3=

[−b+ 6−1

], x=[x1 x2]T, and

a and b being constant system parameters.The IT2 membership functions are chosen to be w̃1(x1) =

μM11 (x1) = 1 − 1/1 + e−(x1+4+σ(t)), w̃2(x1) = μM21 (x1) =

1 − w̃1(x1) − w̃3(x1), and w̃3(x1) = μM31 (x1) = 1/1 +

e−(x14+σ(t)). It should be noted that the IT2 membershipfunctions will lead to uncertain grades of membership becauseof the parameter uncertainty σ(t) ∈ [−0.1, 0.1]. As a result,the existing type-1 stability analysis for FMB control systemunder the PDC design concept cannot be applied.

The lower and upper membership functions for the IT2 T–Sfuzzy model are chosen to be w1(x1)=μM̃11

(x1)=1−1/1+e−(x1+4+d1), w3(x1)=μM̃31

(x1)=1/1+e−(x1−4−d1), w1(x1)=

μM̃11(x1)=1−1/1+e−(x1+4−d1), w3(x1) = μM̃31 (x1)=1/1 +

e−(x14+d1), w2(x1)=μM̃21(x1) = 1− μM̃11 (x1)− μM̃31 (x1),

and w2(x1) = μM̃21 (x1) = 1− μM̃11 (x1)−μM̃31 (x1), where d1is a constant to be determined.

To stabilize the nonlinear plant, a two-rule IT2 fuzzycontroller in the form of (18) is employed. For demonstrationpurposes, the lower and upper membership functions arechosen as m1(x1) = μÑ11

(x1) = 1− 1/e−x1+d2/2, m1(x1) =μÑ11

(x1)=1−1/e−x1−d2/2, m2(x1)=μÑ21 (x1)=1−μÑ11 (x1),and m2(x1) = μÑ21 (x1) = 1− μÑ11 (x1). From (19), we havem̃j(x1) = (βjmj(x1) + βjmj(x1)) / (

∑2k=1(βkmk(x1) +

βkmk(x1))) for j = 1, 2, where βj and βj are chosen to beconstants and d2 is a constant to be determined.

In this example, we consider τ = 0, which means thatno sub-FOUs are considered. For simplicity, the subscriptl is dropped for all variables. To determine the (local)lower and upper membership functions hij(x1) and hij(x1),we consider x1 ∈ [−10, 10] and divide the state space ofx1 into 20 equal-size regions (which is arbitrarily cho-sen for demonstration purposes), i.e., φk : x1,k ≤ x1 ≤ x1,k,k = 1, 2, . . . , 20, where x1,k = (k − 11) and x1,k = (k −10). The lower and upper membership functions hij(x1)and hij(x1) are defined by choosing v11k(x1) = 1− (x1 −x1,k/x1,k − x1,k) and v12k(x1) = 1− v11k(x1), and the con-stant scalars are defined as δij1k = wi(x1,k)mj(x1,k), δij2k =wi(x1,k)mj(x1,k), δij1k = wi(x1,k)mj(x1,k), and δij2k =wi(x1,k)mj(x1,k) for all k.

It should be noted that, by employing the same lower andupper membership functions hij(x1) and hij(x1), any βj and

βj in the fuzzy controller will make no difference in thestability analysis result except the implementation of the IT2fuzzy controller. However, by employing different values of β

j

and βj , the IT2 fuzzy controller defined in (18) will affect the

FOU of h̃ij ≡ w̃i(x1)m̃j(x1). As a result, different hij(x1) andhij(x1) fitting better the FOU can be employed for differentcases. In this example, the introduction of d1 and d2 to themembership functions is for the purpose of obtaining fitterhij(x1) and hij(x1) for different values of βj and βj .

The stability of the IT2 FMB control system subject todifferent values of a and b is checked by the LMI-based stabilityconditions in Theorem 2 (l = 1) with the help of the MatlabLMI toolbox. Three cases shown in Table I with differentvalues of β

j, β

j, d1, and d2 are considered to demonstrate

the characteristics of the IT2 fuzzy controller and how theyinfluence the stabilization capability. The values of d1 and d2


TABLE IPARAMETER VALUES FOR β

j, β

j, d1, AND d2 IN EXAMPLE 1

Fig. 1. Stability regions given by the stability conditions in Theorem 2 for(×) Case 1 (5 points), (�) Case 2 (41 points), and (◦) Case 3 (110 points) inExample 1.

are chosen such that h̃ij(x1) in the form of (30) is within thelower and upper membership functions defined in (25) and (26),respectively. We consider 10 ≤ a ≤ 20 at the interval of oneand 3 ≤ b ≤ 8 at the interval of 0.5 for each of the three cases.The stability regions corresponding to Cases 1–3 indicatedby “×,” “�,” and “◦,” respectively, are shown in Fig. 1. Asseen on these figures, different values of β

jand βj leading to

different values of d1 and d2 produce different sizes of stabilityregions.

For comparison purposes, Theorem 1 is employed to checkthe stability of the IT2 FMB control system. However, thereis no feasible solution by using the Matlab LMI toolbox. Itshould be noted that the IT2 FMB control system is underimperfect premise matching and the stability conditions in[39] for perfect premise matching cannot be applied in thisexample. In order to apply the stability conditions in [39], weconsider that the IT2 fuzzy controller shares the same lowerand upper membership functions as those of the IT2 T–S fuzzymodel. However, there is still no feasible solution for thisexample.

Example 2: The simulation results of the system responsesfor the IT2 FMB control system given in the previous ex-ample were performed for the verification of the stabilityanalysis result. The IT2 T–S fuzzy model is given as ẋ =∑3

i=1 w̃i(x1)(Aix+Biu). A two-rule IT2 fuzzy controlleru =

∑2j=1 m̃j(x1)Gjx is proposed to close the feedback

loop. As a result, we have the IT2 FMB control systemẋ =

∑3i=1

∑2j=1 w̃i(x1)m̃j(x1)(Aix+BiGj)x, which can

be represented in the form of (29). The membership func-

TABLE IIFEEDBACK GAINS OF THE IT2 FUZZY CONTROLLER IN EXAMPLE 2

FOR DIFFERENT VALUES OF a AND b CORRESPONDING TO THEPARAMETER VALUES OF β

j, β

j, d1, AND d2 FOR

DIFFERENT CASES AS SHOWN IN TABLE I

Fig. 2. Phase portrait of the system states of IT2 FMB control system subjectto various initial conditions for a = 14 and b = 3, with parameter values ofβj, β

j, d1, and d2 shown in Case 1 in Table I.

tions are defined in the previous example. In this example,we consider that the grades of membership are capped suchthat w̃i(x1) = w̃i(−10), i = 1, 2, 3, and m̃j(x1) = m̃j(−10),j = 1, 2, for x1 ≤ −10 and w̃i(x1) = w̃i(10), i = 1, 2, 3, andm̃j(x1) = m̃j(10), j = 1, 2, for x1 ≥ 10 in order to apply thestability analysis result obtained in the previous example forx1 ∈ [−10, 10].

Referring to Fig. 1, we pick arbitrarily a number of pointscorresponding to the parameter values of β

j, β

j, d1, and d2 as

shown in Table I. We consider the system parameters a = 14and b = 3 for the parameters of Case 1 in Table I, a = 15and b = 5.5 for Case 2, and a = 20 and b = 5.5 for Case 3to perform the simulations. The parameter uncertainty is cho-sen to be σ(t) = 0.1 sin(x1) ∈ [−0.1, 0.1] for demonstrationpurposes. With the Matlab LMI toolbox and the LMI-basedstability conditions in Theorem 2, we obtained the feedbackgains of the IT2 fuzzy controller for different cases as shownin Table II. The phase portraits of x1 and x2 for different caseswith various initial conditions are shown in Figs. 2–4. It canbe seen that the IT2 fuzzy controllers are able to stabilize thenonlinear plant with different values of a and b.

Example 3: In this example, we investigate the effect ofusing the information of sub-FOUs to the size of the stabil-ity region through a computer simulation. Consider the sameIT2 T–S fuzzy model and IT2 fuzzy controller as those inExample 1. The LMI-based stability conditions are employed


Fig. 3. Phase portrait of the system states of IT2 FMB control system subjectto various initial conditions for a = 15 and b = 5.5, with parameter values ofβj, β


Fig. 4. Phase portrait of the system states of IT2 FMB control system subjectto various initial conditions for a = 20 and b = 5.5, with parameter values ofβj, β


to check the stability of the IT2 FMB control system withthe system parameters 10 ≤ a ≤ 20 at the interval of one and14 ≤ b ≤ 50 at the interval of two (a larger parameter rangeis considered compared with Example 1). Three scenarios, withdifferent numbers of sub-FOUs from two to four, are consideredand shown in Tables III–V. For each scenario, we consider theparameter values of β

j, β

j, d1, and d2 as shown in Table I. As

a result, we have nine combinations in total.The lower and upper membership functions hij(x1) and

hij(x1) are defined in Example 1. According to Tables III–V,the local lower and upper membership functions hijl(x1) andhijl(x1) for sub-FOU l, l = 1, 2, . . . , τ + 1, can be defined.

With the Matlab LMI toolbox and the LMI-based stabilityconditions in Theorem 2, the stability regions for differentscenarios and cases are shown in Figs. 5–7. Referring to thesefigures, it can be seen that different values of β

j, β

j, d1, and

TABLE IIILOWER AND UPPER MEMBERSHIP FUNCTIONS hijl(x1) AND hijl(x1),

l = 1, 2, FOR SCENARIO 1 IN EXAMPLE 3. THE LOWER ANDUPPER MEMBERSHIP FUNCTIONS hij(x1) AND hij(x1)

ARE DEFINED IN EXAMPLE 1

TABLE IVLOWER AND UPPER MEMBERSHIP FUNCTIONS hijl(x1) AND hijl(x1),

l = 1, 2, 3, FOR SCENARIO 2 IN EXAMPLE 3. THE LOWER ANDUPPER MEMBERSHIP FUNCTIONS hij(x1) AND hij(x1)


TABLE VLOWER AND UPPER MEMBERSHIP FUNCTIONS hijl(x1) AND hijl(x1),

l = 1, 2, 3, 4, FOR SCENARIO 3 IN EXAMPLE 3. THE LOWER ANDUPPER MEMBERSHIP FUNCTIONS hij(x1) AND hij(x1)


d2 will produce different sizes of stability regions. It followsthe trend that Case 3 produces a larger stability region thanCase 2 while Case 2 produces a larger stability region thanCase 1. Comparing with Example 1, it can be seen that thestability regions shown in Figs. 5–7 are larger [it should benoted that the scale in Fig. 7 (3 ≤ b ≤ 8) is different from thosein Figs. 5–7(14 ≤ b ≤ 50)]. It is because more information isconsidered by the stability conditions in Theorem 2 throughthe local lower and upper membership functions hijl(x1) andhijl(x1). Comparing the stability regions in Figs. 5–7, it can beobserved that Scenario 3 produces a larger stability region thanScenario 2 while Scenario 2 produces a larger stability regionthan Scenario 1 as more information is utilized when more sub-FOUs are considered.


Fig. 5. Stability regions of Scenario 1 (lower and upper membership functionsdefined in Table III) given by the stability conditions in Theorem 2 for (×)Case 1 (7 points), (�) Case 2 (16 points), and (◦) Case 3 (41 points) inExample 3.

Fig. 6. Stability regions of Scenario 2 (lower and upper membership functionsdefined in Table IV) given by the stability conditions in Theorem 2 for (×)Case 1 (67 points), (�) Case 2 (89 points), and (◦) Case 3 (121 points) inExample 3.

Example 4: In this example, we consider an inverted pendu-lum, as shown in Fig. 8, subject to parameter uncertainties [39]as the nonlinear plant to be controlled. The dynamic equationfor the inverted pendulum is given by

θ̈(t)=gsin(θ(t))−ampLθ̇(t)2sin(2θ(t))/2−a cos(θ(t))u(t)

4L/3−ampL cos2(θ(t))(36)

where θ(t) is the angular displacement of the pendulum,g = 9.8 m/s2 is the acceleration due to gravity, mp ∈[mpmin mpmax ] = [2 3] kg is the mass of the pendulum,Mc ∈ [Mmin Mmax] = [8 12] kg is the mass of the cart,a = 1/(mp +Mc), 2L = 1 m is the length of the pendu-lum, and u(t) is the force (in newtons) applied to the cart.The inverted pendulum is considered working in the operating

Fig. 7. Stability regions of Scenario 3 (lower and upper membership functionsdefined in Table V) given by the stability conditions in Theorem 2 for (×)Case 1 (125 points), (�) Case 2 (144 points), and (◦) Case 3 (168 points) inExample 3.

Fig. 8. Inverted pendulum system.

domain characterized by x1 = θ(t) ∈ [−5π/12, 5π/12] andx2 = θ̇(t) ∈ [−5, 5].

A four-rule IT2 T–S fuzzy model in the form of (7) isemployed to describe the inverted pendulum subject to param-eter uncertainties with x = [x1 x2]T = [θ(t) θ̇(t)]T, A1 =

A2 =

[0 1

f1min 0

], A3 = A4 =

[0 1

f1max 0

], B1 = B3 =[

0f2min

], B2 = B4 =

[0

f2max

], f1min = 10.0078, f1max =

18.4800, f2min = −0.1765, and f2max = −0.0261. The lowerand upper membership functions are defined in Table VI.

A two-rule IT2 fuzzy controller is employed to stabilizethe inverted pendulum with the lower and upper membershipfunctions chosen as m1(x1)=μÑ11

(x1)=m1(x1)=μÑ11(x1)=

e−x21/0.35, m2(x1) = μÑ21

(x1) = m2(x1) = μÑ21(x1) = 1−

μÑ11(x1), and βk = βk = 1/2.

In this example, we consider only one sub-FOU, i.e., τ = 0.For simplicity, the subscript l is dropped for all variables.The number of equal-size regions for x1 is arbitrarily cho-sen to be 500. The lower and upper membership functionshij(x1) and hij(x1) are defined by choosing v11k(x1) =1− (x1 − x1,k/x1,k − x1,k) and v12k(x1) = 1− v11k(x1),


TABLE VILOWER AND UPPER MEMBERSHIP FUNCTIONS OF THE IT2 T–S FUZZY

MODEL OF INVERTED PENDULUM IN EXAMPLE 4

Fig. 9. Phase portrait of the system states of the inverted pendulum subjectto various initial conditions. (Solid lines) mp = 2 kg and Mc = 8 kg. (Dottedlines) mp = 3 kg and Mc = 12 kg.

where x1,k = (10π/12)/500(k − 251) and x1,k = (10π/12)/500(k − 250), k = 1, 2, . . . , 500. The constant scalars are cho-sen as δij1k = wi(x1,k)mj(x1,k), δij2k = wi(x1,k)mj(x1,k),δij1k = wi(x1,k)mj(x1,k), and δij2k = wi(x1,k)mj(x1,k) forall k.

Theorem 2 with l = 1 is employed to determine the sys-tem stability and synthesize the feedback gains. A feasible

solution was found as X =

[0.0983 −0.1870−0.1870 0.4989

], G1 =

[1432.8239 653.0531], and G2 = [1845.9736 849.8562].The IT2 fuzzy controller is employed to stabilize the invertedpendulum with mp = 2 kg and Mc = 8 kg and with mp =3 kg and Mc = 12 kg, respectively. The phase portrait of thesystem states is shown in Fig. 9, which shows that the invertedpendulum can be stabilized subject to different values of mpand Mc and different initial conditions.

For comparison purposes, considering the simulation resultin [39], it can be seen that the IT2 fuzzy controller canalso stabilize the inverted pendulum. However, the number

Fig. 10. Bolt-tightening tool.

Fig. 11. Bolt-tightening tool mounted on a robot arm.

of rules of the IT2 fuzzy controller is required to be fourbecause of the PDC design concept. In this example, the IT2T–S fuzzy model and fuzzy controller do not share the samepremise membership functions and the same number of rules.Consequently, the stability conditions proposed in [39] cannotbe applied in this example. Furthermore, because the numberof rules is two and simpler membership functions are used,the implementation complexity of the IT2 fuzzy controller isreduced.

Example 5: An experiment was done to verify the analy-sis result. A bolt-tightening tool (DSM BL 57/140 MDW),which is shown in Fig. 10, is considered as the plant. In realoperation, the bolt-tightening tool is mounted on a robot arm(Fanuc M6iB) for bolt tightening, as shown in Fig. 11. Anintegrated encoder and a torque sensor are installed to providethe information of angular position (360◦ per revolution) andtorque, respectively. It accepts voltage in the range of −10 to10 V as input.

An IT2 fuzzy model is constructed to describe the systemdynamics with the Matlab system identification toolbox. Alocal state-space model was obtained using the input–outputdata, which are the input voltage and the output angle positionand angular velocity. Three local state-space models operatingat output angles around −90◦, 0, and 90◦ were obtained underno-load condition. IT2 fuzzy sets are employed to combine thethree local state-space models to form an IT2 fuzzy model tofacilitate the design of the IT2 fuzzy controller. The IT2 fuzzymodel was obtained in the form of (7) with x = [x1 x2]T,where x1 is the angle position in degrees and x2 is the angular

velocity in degrees per second, A1 =

[0.0009 0.00340.0108 −0.0264

],

A2 =

[0.0008 0.00420.098 −0.0161

], A3 =

[0.0008 0.00500.088 −0.0057

],

B1 =

[0.00140.0013

], B2 =

[0.00140.0016

], and B3 =

[0.00140.0018

].

The lower and upper membership functions are chosen as


Fig. 12. State responses and control signals of the IT2-FMB-controlledbolt-tightening tool subject to initial conditions of (dotted line) x(0) =[−180 0]T, (dotted lines) x(0) = [−75 0]T, (solid lines) x(0) =[75 0]T, and (dash-dotted lines) x(0) = [180 0]T.

w1(x1) = μM̃11(x1) = 0.8− (0.8/1 + e−x1+90/15), w3(x1) =

μM̃31

(x1) = 0.8/1 + e−x1−90/15, w1(x1) = μM̃11 (x1) = 1−

1/1 + e−x1+90/15, w3(x1) = μM̃31 (x1) = 1/1 + e−x1−90/15,

w2(x1) = μM̃21(x1) = 1− μM̃11 (x1)− μM̃31 (x1), and

w2(x1) = μM̃21(x1) = 1− μM̃11 (x1)− μM̃31 (x1).

A two-rule IT2 fuzzy controller is employed to stabilizethe angle position, where the lower and upper membershipfunctions are chosen as m1(x1) = μÑ11

(x1) = m1(x1) =

μÑ11(x1) = e

−x21/4000, m2(x1) = μÑ21(x1) = m2(x1) =

μÑ21(x1) = 1− μÑ11 (x1), and βk = βk = 1/2.

Similar to the previous example, we consider only onesub-FOU, i.e., τ = 0, and thus, the subscript l is droppedfor all variables. The number of equal-size regions for x1 isarbitrarily chosen to be 500. The lower and upper membershipfunctions hij(x1) and hij(x1) are defined by choosingv11k(x1) = 1− (x1 − x1,k/x1,k − x1,k) and v12k(x1) =1− v11k(x1), where x1,k = (10π/12)/500(k − 251) andx1,k = (10π/12)/500(k − 250), k = 1, 2, . . . , 500. Theconstant scalars are chosen as δij1k = wi(x1,k)mj(x1,k),δij2k = wi(x1,k)mj(x1,k), δij1k = wi(x1,k)mj(x1,k), andδij2k = wi(x1,k)mj(x1,k) for all k.

A feasible solution to Theorem 2 with l = 1 was found

as X =

[0.3917 −1.3310−1.3310 4.6466

]× 108, G1 = [−15.5289 −

4.6345], and G2 = [−3.9267 − 1.1719]. The IT2 fuzzy con-troller was implemented with a programmable logic controllerwhich integrates Matlab Simulink in real time in a BeckhoffTwinCAT 3 system. The state responses and control signalsof the IT2 FMB control system subject to initial conditionsof x(0) = [−180 0]T, [−75 0]T, [75 0]T, and [180 0]Tare shown in Fig. 12. The system states and control signalswere sampled at 0.05 s and filtered by a tenth-order lower passfilter at the signal collection points. It can be seen from thefigures that the IT2 fuzzy controller is able to stabilize the angle

position, however, with a small steady error, which is due to thefriction of the gearbox.

V. CONCLUSION

The stability of IT2 FMB control systems subject to param-eter uncertainties has been investigated. Under the imperfectpremise matching, the IT2 fuzzy controller can choose freelythe premise membership functions and the number of rulesdifferent from those of the IT2 T–S fuzzy model, enhancing thedesign flexibility and reducing the implementation complexity.To facilitate the stability analysis, a favorable form of lowerand upper membership functions has been proposed, and theinformation of sub-FOUs has been considered. The informa-tion of membership functions has been brought to the LMI-based stability conditions, resulting in a more relaxed stabilityanalysis result. Simulation and experimental results have beengiven to illustrate the merit of the proposed approach. In futurework, we will consider the problems of output-feedback controland sampled-data control for the nonlinear systems subject toparameter uncertainties in the frame of this paper.

APPENDIXPROOF OF THEOREM 2

We consider the following quadratic Lyapunov functioncandidate to investigate the stability of the IT2 FMB controlsystems (24) expressed in the form of (29):

V = xTPx (37)

where 0 < P = PT ∈ �n×n.The main objective is to develop a condition guarantee-

ing that V > 0 and V̇ < 0 for all x �= 0. According to theLyapunov stability theorem, by satisfying V > 0 and V̇ < 0for all x �= 0, the IT2 FMB control system is guaranteed to beasymptotically stable, implying that x → 0 as time t → ∞.

Denote z = X−1x and X = P−1. Define the feedback gainsGj = NjX

−1, where Nj ∈ �m×n, j = 1, 2, . . . , c, representsthe matrices to be determined. From (29) and (37), we have

V̇ = ẋTPx+ xTPẋ

=

p∑i=1

c∑j=1

h̃ijxT((Ai +BiGj)

TP+P(Ai +BiGj))x

=

p∑i=1

c∑j=1

h̃ijxTPP−1

((Ai +BiGj)

TP

+P(Ai +BiGj))P−1Px

=

p∑i=1

c∑j=1

τ+1∑l=1

ξijl

(γijlhijl + γijlhijl

)zTQijz (38)

where Qij = AiX+XATi +BiNj +NTj B

Ti .

Recalling the properties that 0 ≤ hijl ≤ hijl ≤ 1, 0 ≤γijl

≤ 1, 0 ≤ γijl ≤ 1, and γijl + γijl = 1 for all i, j, andl, the information of the sub-FOUs is brought to the stability


analysis with the introduction of some slack matrices throughthe following inequalities using the S procedure [14]:⎛⎝ p∑

i=1

c∑j=1

τ+1∑l=1

ξijl


)− 1

⎞⎠M =0 (39)

−p∑

i=1

c∑j=1

(1− γ

ijl

) (hijl − hijl

)Wijl ≥ 0 (40)

where M = MT ∈ �n×n represents arbitrary matrices and 0 ≤Wijl = W

Tijl ∈ �n×n.

From (30), (38), (39), and (40), we have

V̇ =

p∑i=1

c∑j=1

τ+1∑l=1

ξijl


)zTQijz

≤p∑

i=1

c∑j=1

τ+1∑l=1

ξijl

(γijlhijl +

(1− γ

ijl

)hijl

)zTQijz

−p∑

i=1

c∑j=1

τ+1∑l=1

ξijl

(1− γ

ijl

) (hijl − hijl

)zTWijlz

+

⎛⎝ p∑

i=1

c∑j=1

τ+1∑l=1

ξijl

(γijlhijl

+(1− γ

ijl

)hijl

)− 1

)zTMz

= zT

⎛⎝ p∑

i=1

c∑j=1

τ+1∑l=1

ξijl(hijlQij

−(hijl − hijl

)Wijl + hijlM

)−M

)z

+

p∑i=1

c∑j=1

τ+1∑l=1

ξijlγijl(hijl − hijl

)zT

× (Qij +Wijl +M)z. (41)

Referring to (41), V̇ < 0 for x �= 0 is satisfied by∑pi=1

∑cj=1

∑τ+1l=1 ξijl(x)(hijlQij − (hijl − hijl)Wijl +

hijlM)−M < 0 and Qij +Wijl +M > 0 (because ofhijl − hijl ≤ 0) for all i, j, and l. Recalling that onlyone ξijl = 1 for each fixed value of ij at any time instantsuch that

∑τ+1l=1 ξijl = 1, the first set of inequalities is

satisfied by∑p

i=1

∑cj=1(hijlQij − (hijl − hijl)Wijl +

hijlM)−M < 0 for all i, j, and l. Expressing hijl andhijl with (25) and (26), respectively, and recalling that∑q

k=1

∑2i1=1

∑2i2=1

· · ·∑2

in=1

∏nr=1 vrirkl = 1 for all l and

vrirkl ≥ 0 for all r, ir, k, and l, the first set of inequalities willbe satisfied if the following inequalities hold:

q∑k=1

2∑i1=1

2∑i2=1

· · ·2∑

in=1

n∏r=1

vrirkl

×

⎛⎝ p∑

i=1

c∑j=1

(δiji1i2···inklQij

−(δiji1i2···inkl − δiji1i2···inkl

)Wijl

+δiji1i2···inklM)−M

)

< 0 ∀ i1, i2, · · · , in, k, l. (42)

Consequently,∑p

i=1

∑cj=1(hijlQij − (hijl − hijl)×

Wijl + hijlM)−M < 0 can be guaranteed by∑pi=1

∑cj=1(δiji1i2···inklQij − (δiji1i2···inkl − δiji1i2···inkl)×

Wijl + δiji1i2···inklM)−M < 0.The LMI-based stability conditions previously mentioned are

summarized in Theorem 2. By satisfying those LMIs, the IT2FMB control system (24) is guaranteed to be asymptoticallystable.

Referring to (42), the advantages of representing the IT2FMB control system (24) in the form of (29) can be seen.The membership functions h̃ij are reconstructed by the lin-ear combination of the local lower and upper membershipfunctions hijl and hijl. Consequently, as seen from (41), thestability of the IT2 FMB control system is determined by thelocal lower and upper membership functions hijl and hijl.By expressing hijl and hijl in the form of (25) and (26),respectively, they are characterized by the constant scalarsδiji1i2···inkl and δiji1i2···inkl. Furthermore, as the cross-terms∏n

r=1 vrirkl are independent of i and j, they can be extractedas shown in (42) to facilitate the stability analysis. With thesefavorable properties as previously stated in Remark 6, we onlyneed to check

∑pi=1

∑cj=1(δiji1i2···inklQijl − (δiji1i2···inkl −

δiji1i2···inkl)Wijl + δiji1i2···inklM)−M < 0 at some discretepoints (δiji1i2···inkl and δiji1i2···inkl) instead of every singlepoint of the local lower and upper membership functions hijland hijl to guarantee the holding of the inequality (42).

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H. K. Lam (M’98–SM’10) received the B.Eng.(Hons.) and Ph.D. degrees from the Department ofElectronic and Information Engineering, The HongKong Polytechnic University, Kowloon, Hong Kong,in 1995 and 2000, respectively.

From 2000 to 2005, he was a Postdoctoral Fellowand then a Research Fellow with the Department ofElectronic and Information Engineering, The HongKong Polytechnic University. Since 2005, he hasbeen with King’s College London, London, U.K.,where he was a Lecturer and is currently a Senior

Lecturer. He is the coeditor for two edited volumes: Control of ChaoticNonlinear Circuits (World Scientific, 2009) and Computational Intelligenceand Its Applications (World Scientific, 2012). He is the coauthor of the bookStability Analysis of Fuzzy-Model-Based Control Systems (Springer, 2011). Heis an Associate Editor for the International Journal of Fuzzy Systems. Hiscurrent research interests include intelligent control systems and computationalintelligence.

Dr. Lam is an Associate Editor for the IEEE TRANSACTIONS ON FUZZYSYSTEMS.


Hongyi Li received the B.S. and M.S. degrees inmathematics from Bohai University, Jinzhou, China,in 2006 and 2009, respectively, and the Ph.D. de-gree in intelligent control from the University ofPortsmouth, Portsmouth, U.K., in 2012.

He is currently with the College of InformationScience and Technology, Bohai University. His re-search interests include fuzzy control, robust control,and their applications in suspension systems.

Christian Deters received the Dipl.-Ing. degreein computer science from Hochschule Bremen,Bremen, Germany, in 2008 and the M.Sc. degreefrom King’s College London, London, U.K., in 2009,where he is currently working toward the Ph.D.degree.

His research focus is on control, automation, andmanufacturing.

E. L. Secco was born in 1971. He received the M.Sc.degree in mechanical engineering from the Univer-sity of Padua, Padua, Italy, in 1998 and the Ph.D. de-gree in bioengineering and medical computer sciencefrom the University of Pavia, Pavia, Italy, in 2001.

From 2003 to 2011, he was with diverse institu-tions and research centers (the Rehabilitation Insti-tute of Chicago, Chicago, IL, USA, the University ofBologna, Bologna, Italy, and the European Centre forTraining and Research in Earthquake Engineering,Pavia, Italy). Since 2012, he has been a Research

Associate with the Centre for Robotics Research, Department of Informatics,King’s College London, London, U.K. He has been working on neural networkcontrollers, biomimetic systems, sensor integration and motor learning inhumans, smart garments and wearable sensors, and sensor fusion algorithms.His main interests are in humanlike robotics.

Helge A. Wurdemann received the Dipl.-Ing. de-gree in electrical engineering from Leibniz Univer-sity of Hanover, Hannover, Germany.

In 2006, he was with Auckland University ofTechnology, Auckland, New Zealand. In 2007, hewas with Loughborough University, Loughborough,U.K., where he carried out a research project. Heis currently a Research Associate with the Centrefor Robotics Research, Department of Informatics,King’s College London, London, U.K. His Ph.D.project, which he started in late 2008, at King’s Col-

lege London was funded by the Engineering and Physical Sciences ResearchCouncil. In November 2011, he joined the research team of Prof. KasparAlthoefer working on two European Union Seventh Framework Programmeprojects. His research interests are medical robotics for minimally invasivesurgery and self-adaptive control architectures.

Kaspar Althoefer (M’03) received the Dipl.-Ing.degree in electronic engineering from the Univer-sity of Aachen, Aachen, Germany, and the Ph.D.degree in electronic engineering from King’s CollegeLondon, London, U.K.

He is currently the Head of the Centre for RoboticsResearch, Department of Informatics, King’s Col-lege London, London, U.K., where he is also aProfessor of robotics and intelligent systems. Hehas authored or coauthored more than 180 refereedresearch papers related to mechatronics, robotics,

and intelligent systems.

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